Asymptotical Mechanics of Thin-Walled Structures.

I.V. Andrianov, J. Awrejcewicz, L.I. Manevitch

Asymptotical Mechanics of Thin-Walled Structures. A Handbook

Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Tokyo

Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1

ASYMPTOTIC APPROXIMATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fundamental concepts of asymptotics [129] . . . . . . . . . . . . . . . . . . . 1.3 Transformations of asymptotical series [129] . . . . . . . . . . . . . . . . . . 1.4 Nonuniform expansions [129] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Asymptotics of integrals [129] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

REGULAR PERTURBATIONS OF PARAMETERS . . . . . . . . . . . . . . . 47 2.1 Eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2 Stability of oval cylindrical shell uniformly loaded by external pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.3 Stability of the cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4 Adjoint operators method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.5 Transformation of coordinates and variables . . . . . . . . . . . . . . . . . . . 65 2.6 Asymptotic and real error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.7 Numerical verification of asymptotic solution . . . . . . . . . . . . . . . . . 75 2.8 Removal of nonuniformities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.9 Nonlinear vibrations of a stringer shell . . . . . . . . . . . . . . . . . . . . . . . 81 2.10 Non-quasilinear asymptotics of nonlinear system . . . . . . . . . . . . . . 84 2.11 Artificial small parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.12 Method of small δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.13 Method of large δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.14 Choice of zero approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.15 Lyapunov–Schmidt procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.16 Nonlinear periodical vibrations of continuous structures . . . . . . . . . 103

3

SINGULAR PERTURBATION PROBLEMS . . . . . . . . . . . . . . . . . . . . . 117 3.1 The method of Gol’denveizer-Vishik-Lyusternik [313, 645, 672, 673, 674] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.2 Multiscale method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

19 19 21 23 25 27 29

4

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3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

Newton polygon and asymptotic integration parameters . . . . . . . . . 125 Stretched plate bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Simplification of the static equations of a cylindrical shell . . . . . . . 133 Boundary layer: Papkovitch approach . . . . . . . . . . . . . . . . . . . . . . . . 136 Edge boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Incorporating of the singular part of solution . . . . . . . . . . . . . . . . . . 140 Plane theory of elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Asymptotic foundation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Vibrations of reinforced conical shells . . . . . . . . . . . . . . . . . . . . . . . . 149

4

BOUNDARY VALUE PROBLEMS OF ISOTROPIC CYLINDRICAL SHELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.1 Governing relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.2 Operator method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.3 Simplified boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . 160

5

BOUNDARY VALUE PROBLEMS – ORTHOTROPIC SHELLS . . . 169 5.1 Governing relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.2 Statical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.3 Non-linear dynamical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.4 Stability problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5.5 Error estimation using Newton’s method . . . . . . . . . . . . . . . . . . . . . 197

6

COMPOSITE BOUNDARY VALUE PROBLEMS – ISOTROPIC SHELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.1 Statical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.2 Equations of higher order approximations . . . . . . . . . . . . . . . . . . . . . 206 6.3 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 6.4 Dynamical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.5 Non-linear dynamical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

7

COMPOSITE BOUNDARY VALUE PROBLEMS – ORTHOTROPIC SHELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.1 Statical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.2 Dynamical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7.3 Non-linear dynamical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.4 Stability problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

8

AVERAGING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.1 Two-scales approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.2 Visco-elastic problems and ’freezing’ method . . . . . . . . . . . . . . . . . 244 8.3 The successive change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . 246 8.4 Application of the Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 8.5 Whitham method (non-linear WKB approach) . . . . . . . . . . . . . . . . . 256

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5

CONTINUALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

10 HOMOGENIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 10.1 ODEs with rapidly oscillating coefficients . . . . . . . . . . . . . . . . . . . . 267 10.2 Axisymmetric bending of corrugated circle plate . . . . . . . . . . . . . . . 272 10.3 Deformation of reinforced membrane . . . . . . . . . . . . . . . . . . . . . . . . 276 10.4 Ribbed strip – two-scale and Fourier homogenization . . . . . . . . . . . 280 10.5 Ribbed plate – direct homogenization . . . . . . . . . . . . . . . . . . . . . . . . 284 10.6 Perforated membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 10.7 Composite with periodic cubic inclusions . . . . . . . . . . . . . . . . . . . . . 302 10.8 Torsion of bar with periodic parallelepiped inclusions . . . . . . . . . . . 307 10.9 Solution of cell problem: perturbation of boundary form . . . . . . . . 313 10.10 Linear vibartions of a beam with concentrated masses and discrete supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 11 INTERMEDIATE ASYMPTOTICS – DYNAMICAL EDGE EFFECT METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 11.1 Linear preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 11.2 Nonlinear beam vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 11.3 Nonlinear rectangular plate vibrations . . . . . . . . . . . . . . . . . . . . . . . . 340 11.4 Nonlinear shallow shell vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 11.5 Rayleigh-Ritz-Bolotin approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 11.6 Parallelogram plate vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 11.7 Sectorial plate nonlinear vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 360 12 LOCALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 12.1 Localization in linear chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 12.2 Localization in nonlinear chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 12.3 Localization of shell buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 12.4 Localization of vibration in plates and shells . . . . . . . . . . . . . . . . . . 373 13 IMPROVEMENT OF PERTURBATION SERIES . . . . . . . . . . . . . . . . . 377 13.1 Padé approximants (PA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 13.2 The effect of autocorrection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 13.3 Extending of perturbation series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 13.4 Improvement iterational procedures convergence . . . . . . . . . . . . . . . 384 13.5 Nonuniformities elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 13.6 Error estimation of asymptotic approaches . . . . . . . . . . . . . . . . . . . . 388 13.7 Localized solutions and blow-up phenomenon . . . . . . . . . . . . . . . . . 388 13.8 Gibbs phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 13.9 Boundary conditions perturbation method . . . . . . . . . . . . . . . . . . . . 392 13.10 Bifurcation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 13.11 Borel summation and superasymptotics . . . . . . . . . . . . . . . . . . . . . . 402 13.12 Domb–Sykes plot [340, 659] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 13.13 Extraction of singularities from perturbation series [340, 659] . . . . 406

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Table of Contents

13.14 Analytical continuation [407] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 14 MATCHING OF LIMITING ASYMPTOTIC EXPANSIONS . . . . . . . 417 14.1 Two-point Padé approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 14.2 Quasifractional approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 14.3 Post-buckling behaviour of shallow convex shell . . . . . . . . . . . . . . . 429 15 COMPLEX VARIABLES IN NONLINEAR DYNAMICS . . . . . . . . . . 435 15.1 Nonlinear oscillator with cubic anharmonicity . . . . . . . . . . . . . . . . . 436 15.2 System of two weakly coupled nonlinear oscillators . . . . . . . . . . . . 447 15.3 Nonlinear dynamics of an infinite chain of coupled oscillators . . . . 453 15.4 Nonlinear dynamics of an infinite chain of coupled particles . . . . . 460 16 OTHER ASYMPTOTICAL APPROACHES . . . . . . . . . . . . . . . . . . . . . . 463 16.1 Matched asymptotic procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 16.2 Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 16.3 Normal forms in non-linear problems . . . . . . . . . . . . . . . . . . . . . . . . 467 16.4 WKB - approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 16.5 The WKB method and turning points . . . . . . . . . . . . . . . . . . . . . . . . 473 16.6 A distributional approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 AFTERWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Asymptotics and Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Are Asymptotic Methods a Panacea? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

List of Figures

1.1 1.2 1.3 1.4 1.5

A convergent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An asymptotic series: divergent but very useful [660, 663]. . . . . . . . . Surface p(x, y) cross section introduced by p 0 . . . . . . . . . . . . . . . . . . . Deformation of the integration contour. . . . . . . . . . . . . . . . . . . . . . . . . Choice of integration path in the saddle-point method. . . . . . . . . . . . .

2.1 2.2 2.3 2.4 2.5 2.6

2.13

A rod with variable rigidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 The vertical compressed rod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 An estimation of accuracy of the perturbation method. . . . . . . . . . . . 60 The compressed cantilever beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 period of oscillation of a mathematical pendulum. . . . . . . . . . . . . . . . 72 A comparison of results obtained using different analytical methods: a) ω0 = 1, ε = 0.1; b) ω0 = ε = 1. . . . . . . . . . . . . . . . . . . . . 74 Verification of the asymptotic error of the partial sums (2.92). . . . . . 76 Solutions to the Cauchy problem (2.162), (2.163) for A = 1 using the fourth order Runge-Kutta method (——), and the approximations: (2.170) (- - - -); (2.171) ( - - - ); (2.172) (— — —) for n = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Solutions to the Cauchy problem (2.162), (2.163) for A = 1 using the fourth order Runge-Kutta method (——), and the approximations: (2.170) (- - - -); (2.171) ( - - - ); (2.172) (— — —) for n = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Solutions to the Cauchy problem (2.162), (2.163) for A = 1 using the fourth order Runge-Kutta method (——), and the approximations: (2.170) (- - - -); (2.171) ( - - - ); (2.172) (— — —) for n = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Parallelogram membrane and different methods of choosing of zero order approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Trapezoid membrane and different methods of choosing of zero order approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Typical bifurcation diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.1 3.2 3.3

Typical behaviour of a solution with occurrence of a boundary layer. 118 Example of interior boundary layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 The Newton polygon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

2.7 2.8

2.9

2.10

2.11 2.12

19 20 41 42 44

8

List of Figures

3.4 3.5 3.6 3.7 3.8 3.9 3.10

Newton polygon for equation (3.53). . . . . . . . . . . . . . . . . . . . . . . . . . . 128 The partition of the (α, β) plane for problem (3.56). . . . . . . . . . . . . . . 130 The (α, β) plane partition for the problem of the initially stretched plate deflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 The (α, β) plane partition in regard to the x, y co-ordinates for a cylindrical shell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 A scheme of the considered conical shell.. . . . . . . . . . . . . . . . . . . . . . . 151 Non-dimensional minimal frequencies of shells in the case of free support (see text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Non-dimensional minimal frequencies of shells in the case of clamping (see text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.1

A circular cylindrical isotropic shell. . . . . . . . . . . . . . . . . . . . . . . . . . . 158

5.1

A comparison of buckling axial loads obtained using exact and approximate solutions for the stringer cylindrical shell. . . . . . . . . . . . 196

6.1

A comparison of moduli of small roots of the characteristic equation obtained using various approximated theories. . . . . . . . . . . . 210 A comparison of arguments of small roots of the characteristic equation obtained using various approximated theories. . . . . . . . . . . . 210 A comparison of moduli of large roots of the characteristic equation obtained using various approximated theories. . . . . . . . . . . . 211 A comparison of arguments of large roots of the characteristic equation obtained using various approximated theories. . . . . . . . . . . . 211 Small vibration frequencies of a cylindrical shell obtained using various approximated theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Large vibration frequencies of a cylindrical shell obtained using various approximated theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Small vibration frequencies of a cylindrical shell obtained using various approximated theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Large vibration frequencies of a cylindrical shell obtained using various approximated theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

6.2 6.3 6.4 6.5 6.6 6.7 6.8 7.1 7.2 7.3

A comparison of moduli of small roots of the characteristic equation for the waffle shell obtained using various approximating theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 A comparison of arguments of small roots of the characteristic equation for the waffle shell obtained using various approximating theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 A comparison of moduli of large roots of the characteristic equation for the waffle shell obtained using various approximating theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

List of Figures

7.4 7.5 7.6 7.7 7.8

9

A comparison of moduli of large roots of the characteristic equation for the waffle shell obtained using various approximating theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 A comparison of moduli of small roots of the characteristic equation for the stringer shell obtained using various approximating theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 A comparison of arguments of small roots of the characteristic equation for the stringer shell obtained using various approximating theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 A comparison of moduli of large roots of the characteristic equation for the stringer shell obtained using various approximating theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 A comparison of moduli of large roots of the characteristic equation for the stringer shell obtained using various approximating theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

8.1 8.2

Splitting of a solution into a sum of a slow and fast components. . . . 241 Two different ways of transformating y (0) → zt . . . . . . . . . . . . . . . . . . 251

10.1 10.2

10.11 10.12 10.13 10.14

Bending moment in the circled corrugated plate. . . . . . . . . . . . . . . . . 276 Triangulation of the (α 1 , β) plane characterising the different asymptotics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Perforated medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 Cross section of input composite material with chosen periodic cell. 303 Scheme of lubrication approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 The composite material with distinguished unit cell Ω i = Ω+i + Ω−i . . 308 Three-phase model as applied to the composite material under consideration in case of small inclusions (a → 0). . . . . . . . . . . . . . . . 309 Model of the composite material under consideration in the case of large inclusions (a → 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Effective rigidity q as a function of inclusions concentration c = a2 and rigidity λ, according to the analytical solution (10.224). . 312 The analytical solution (10.224) (solid curve) is compared with Bourgat’s numerical results [189] for c = 1/9, (black points). . . . . . . 313 The perforated medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Scheme of beam with concentrated masses and discrete supports. . . 319 Dispersion relation in the low-frequency range of the spectrum. . . . . 324 The sketches of long (a), medium (b), and short wave modes. . . . . . . 330

11.1 11.2 11.3

Inner solution and dynamical edge effects. . . . . . . . . . . . . . . . . . . . . . . 334 period versus amplitude for different support stiffnesses. . . . . . . . . . . 340 Amplitude-frequency relations for a rectangular plate. . . . . . . . . . . . . 345

12.1 12.2

Localised modes generated by two inclusions. . . . . . . . . . . . . . . . . . . 364 Infinite chain with two different masses. . . . . . . . . . . . . . . . . . . . . . . . 365

10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10

10

List of Figures

12.3

Characteristic dimension of the space localization versus wave number for various concentrations of barriers. . . . . . . . . . . . . . . . . . . . 370

13.1 13.2 13.3

Efficiency of the perturbation series and PA for the Duffing equation.383 Illustration of nonuniformity elimination using PA. . . . . . . . . . . . . . . 387 The diagonal PA (dashed curve) for three term of the series (13.34). Solid curve corresponds to the exact solution (13.32). . . . . . 391 First eigenvalue of the eigenproblem (13.37), (13.39) versus the parameter ε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Errors in estimation of first fifteenth eigenvalues to the problem (13.37), (13.38); curves correspond to formulars (13.51) for ε = 1, and (13.57) for ε = 1 respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Eigenvalue λ versus clamping stiffness. Curves 1 and 2 correspond to formulas (13.70), (13.71), respectively. . . . . . . . . . . . . . . . . . . . . . . 400 The Domb-Sykes plot for function f (ε) = ε(1 + ε 3 )(1 + 2ε)−1/2 . . . . 406 The continuous is f (ε) = sin −1 ε; the dotted curves are increasingly higher order expansions of f in terms of ε; the dashed curves are increasingly higher order expansions of the inversion, ε( f ). . . . . . . . . . . . . . . . . . . . . . .√. . . . . . . . . . . . . . . . . . . . . . 407 The continuous curve gives f (ε) = ε −ε/2 1 + 2ε; the dotted curves are increasingly higher order expansions of f in terms of ε; the dashed curves are increasingly higher order expansions of f 2 in terms of ε. . . . . . . . . . . .√. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 The function f (ε) = e −ε/2 1 + 2ε is given by the continuous curve; the dotted curves are for increasingly higher order expansions of f in terms of ε; the dashed curves are for increasingly higher order expansions of f in terms of ε. ˜ . . . . . . . . . . 409 Transformation of the circle |η| < 1 on plane with two cross sections.413 Transformation of circle |η| < 1 into a strip (P). . . . . . . . . . . . . . . . . . 414 Mapping of the circle |η| < 1 into a zone with a cut out ray. . . . . . . . . 414 Plane with cut out strip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Mapping of circle |η| < 1 on the halfplane Reλ ≤ a. . . . . . . . . . . . . . . 415 Mapping of a top halfplane with a cut along imaginary axis into a top halfplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

13.4 13.5 13.6 13.7 13.8

13.9

13.10

13.11 13.12 13.13 13.14 13.15 13.16 14.1 14.2 14.3 14.4 14.5

Frequencies of a chain obtained by various approaches. . . . . . . . . . . . 419 period of the Van der Pol oscillator: comparison of numerical, perturbative and TPPA solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 Laplace transform inverse, the exact and TPPA solutions. . . . . . . . . . 420 The monotone sequences of one-point [M/M] 0 and two-point Padé approximants [M/M] 1 , [M/M]2 uniformly converging to the Stieltjes function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Effective conductivity coefficient k for regular array of spheres. Analytical results (14.20), (14.21) for different values of m, n are compared with experimental data given in [707]. . . . . . . . . . . . . . . . . 428

List of Figures

14.6 14.7

11

Effective conductivity coefficient k for regular array of spheres. Analytical solution is compared with experimental measurements [382]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 Comparing of analytical and numerical results. . . . . . . . . . . . . . . . . . . 433

List of Tables

1.1

Example of asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1 2.2 2.3 2.4 2.5

Slopes of the least-squares fit of log(E M ) as a function of log(x) on different domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Error estimation of asymptotic procedure. . . . . . . . . . . . . . . . . . . . . . . . 96 Comparison of various approximations for parallelogram membrane. 100 Comparison of various approximations for trapezoidal membrane. . . 101 Comparison of asymptotic solution with numerical data. . . . . . . . . . . . 110

3.1 3.2 3.3

Computations of σ for orthotropic materials . . . . . . . . . . . . . . . . . . . . . 143 Computations of σ for isotropic materials . . . . . . . . . . . . . . . . . . . . . . . 143 Computation of non-dimensional frequencies using various methods . 155

4.1 4.2 4.3

Governing boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Canonical form of boundary conditions for isotropic shell. . . . . . . . . . 165 Splitting boundary conditions for isotropic shell. . . . . . . . . . . . . . . . . . 166

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Splitting boundary conditions for waffle shell. . . . . . . . . . . . . . . . . . . . . 176 Canonical forms of the boundary conditions for stringer shell. . . . . . . 179 Splitting boundary conditions for stringer shell. . . . . . . . . . . . . . . . . . . 180 Splitting boundary conditions for ring-stiffened shell. . . . . . . . . . . . . . 181 Splitting boundary conditions of reinforced shell. . . . . . . . . . . . . . . . . . 189 Splitting boundary condition for stringer shell. . . . . . . . . . . . . . . . . . . . 190 Splitting boundary conditions for ring-stiffened shell. . . . . . . . . . . . . . 191 Splitting boundary conditions for waffle shell. . . . . . . . . . . . . . . . . . . . . 192

9.1

Numerical characteristics of a splash effect in discrete mass chain. . . 260

11.1 11.2 11.3 11.4

Comparison of various approximations of frequencies. . . . . . . . . . . . . 356 Frequencies for various angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Comparison of results for circular clamped plate. . . . . . . . . . . . . . . . . . 362 Comparison of results for sector clamped plate. . . . . . . . . . . . . . . . . . . 362

13.1 Comparison of numerical and analytical results for Van der Pol equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 13.2 Results of successive approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . 385

14

List of Tables

13.3 13.4 13.5 13.6

Results of generalized summations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Iterations and results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Results obtained using Padé approximants. . . . . . . . . . . . . . . . . . . . . . . 386 PA for function sign(x) Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

14.1 The constants a i used in formula (14.18). . . . . . . . . . . . . . . . . . . . . . . . . 426 14.2 The constants M 1 , M2 and ϕmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

Preface

In spite of wide development and application of numerical techniques (especially FEM), in Mechanics of Solids at all and in Mechanics of Thin-walled Structures, in particular, the analytical methods retain fully their significance. Because possibility of exact solution is provided by high symmetry of corresponding equations and boundary problems, it is usually rather an exception, if dealing with real technical problems. On the contrary, asymptotic expansions, being by important class of approximations, imply a weak or strong asymmetry of the system under consideration and are distinguished by genuine universality (as we will see, the former case (weak asymmetry) leads to regular asymptotics, and latter one to singular asymptotics). Asymptotic methods allow to penetrate into essence of the problem and to reveal the possibilities of its decomposition that leads to real understanding. From other side, they provide a rational way of numerical simulation if there is a need in making of calculations by more precise. It is worth to mention from very beginning that the thin-walled structures manifest, probably, one of the most suitable fields of Mechanics and Physics at all for using the asymptotic approach. This is a consequence of the presence of natural and small enough parameters in corresponding equations. First of all, any model of thin-walled structure is actually certain asymptotics of continuum model of 3D solid and describes adequately the stress-strain state of thin body far enough from boundaries and concentrated forces where 3D effects may be essential. Engineers reinforce usually these regions by special manners for compensation of such effects. Further, even 2D or 1D equations of thin-walled structures contain themselves the additional small parameters reflecting, e.g., a strong asymmetry of stiffness (tensioncompression rigidity is much more than bend and twist ones) or geometry of the structure. This leads to appearance of supplemented boundary layers which prolongation depends, e.g. from the ratio of different stiffness. New small parameters arise if dealing with reinforced or composite structures that are usually characterized by strong asymmetry with respect to stiffness in different directions. So, there are many different aspects that need in asymptotic consideration to understand specificity of thin-walled structures. Because the main separating like between asymptotic methods is distinction regular or singular perturbation problem, we consider (after introduction of common asymptotic technique in the first Section) main aspects of corresponding procedure with applications to important technical problems (Sections 2 and 3).

Following Sections 4–7 are devoted to systematic consideration of boundary value problems (statical and dynamical, linear and nonlinear) for homogeneous shells. The role of mentioned small parameter (ratio of bend-twist and tensioncompression stiffeners, as well as parameters, characterizing strong anisotropy of structurally-orthotropic shells), is fully clarified. In Sections 8–10 we consider averaging, continualization and homogenization technique for various problems of discrete and continuous media. Final Sections of handbook are devoted to more special and refined problems that are not usually (or rarely) discussed in Textbooks and even in Monographs. They are: intermediate asymptotics (Section 11), localization (Section 12), Padé approximants (Section 13) and matching the limiting asymptotic approach (Section 14), complex variables in Nonlinear Dynamics (Section 15), and other asymptotic ideas that may be useful in different problems for thin-walled structures (Section 16). Some general questions of asymptotics methods are discussed in Section 17 and Afterwords. Finally, the book contains numerous results obtained earlier by the authors [2195, 103-110, 427, 453-460, 531, 532, 637-639, 669, 670].

Acknowledgments

Our first debt are to R.G. Baranstev, V.N. Engelgart, A.Yu. Evkin, A.I. Potapov, V.N. Pilipchuk and R.A. Ibrahim, who agreed to include their research into some chapters of the book. We express our sincere thanks to Professor V.I. Babitsky, who strongly supported the idea of writing this monograph. We are indebted to a number of other colleagues for helpful comments and criticism; in particular to M.G. Dmitriev, I. Elishakoff, W.T. van Horssen, A.L. Kalakanov, G.L. Litvinov, B.V. Nemebaylo, I.V. Novozhilov, Ya.G. Panovko , Kh. Semerdjiev, S. Tokarzewski, P.Ye. Tovslik, A.F. Vakakis. Many of the ideas presented also show the influence of our year of collaboration with N.S. Bulanova, V.V Danishevs’kyy, A.A. Diskovsky, A.O. Ivankov, M.V. Matyash, E.G. Kholod, V.I. Olevsky, G.A. Krizhersky, V.A. Lesnichaya, V.V. Loboda, A.N. Pasechnik, V.V. Shevchenko , G.A. Starushenko, V.M. Verbonol’, V.G. Oshmyan, V.N. Pilipchuk, Yu.V. Mikhlin, A.V. Pavlenko, S.G. Koblik, A.D. Shampovskii, A.Yu. Evkin.

1 ASYMPTOTIC APPROXIMATIONS

1.1 Asymptotic series There is always a certain charm in tracing the evolution of theories in the original papers; often such study offers deeper insights into the subject matter than the systematic presentation of the final result, polished by the words of many contemporaries. A. Einstein [632,p.244]

The series generated by a perturbation approach does not necessarily converge. Asymptotic methods use a mathematical apparatus of a somewhat peculiar nature - asymptotic series. They diverge but still approximate the functions in hand in a certain sense. Briefly, we can say that a convergent series represents a function at x = x0 , n → ∞ (Fig. 1.1), while an asymptotic series is valid for n = n 0 , x → x0 (Fig. 1.2).

Fig. 1.1. A convergent series.

To understand difficult things, there is often nothing better than to read the classics. Let us give a long quotation from “The new methods of celestial mechanics” by Poincaré, 1892 [560]:

20


“Geometers and astronomers (today we would say ‘pure and applied mathematicians’ - authors) understand the word ‘convergence’ in different ways. Geometers are concerned entirely about attaining impeccable rigor and often are absolutely indifferent to the length of difficult calculations (they assume their realizability and do not think about its practical realization). They say that a series is convergent, when the sum of its terms tends to a certain limit, even if the first terms decrease very slowly. In contrast, astronomers usually say that certain series is convergent, if, e.g., twenty of its first terms diminish very fast, in spite of unlimited growth of further terms.

Fig. 1.2. An asymptotic series: divergent but very useful [660, 663]. n

As a simple example let us consider two series with common terms 1000 n! and n! , respectively. Geometers would say that the first series is rapidly convergent, n 1000 since its milionth term is much less than 1/999999. They will consider the second series as divergent, because its general term grows infinitely. Astronomers, on the contrary, would consider the first series as divergent, because the first 1000 terms grow, while the second one would be treated as convergent, since its first 1000 terms diminish, their decrease being fastest at the beginning. Both points of view are valid, the former in theoretical investigations, the latter in numerical applications. They both dominate completely, but in different areas, and the boundaries of these areas of applicability must be clearly distinguished.

1.2 Fundamental concepts of asymptotics [129]

21

Astronomers do not always know clearly the limits of applicability of their methods, but they are rarely mistaken. The approximations they are satisfied with usually lie within the limits where their methods are applicable. Moreover, intuition allows them to foresee the right result. If they made a mistake, the comparison with observations would allow them to correct it. Nevertheless, I think that it will be appropriate to introduce some more preciseness in this question. This is just what I am going to do, though the question involved is not very convenient for this purpose by its nature. So, we have to consider a relationship of a new nature, which can exist between a function of arguments x and ε which will be denoted by φ(x, ε), and a divergent power series in ε (1.1) f0 + ε f1 + ε2 f2 + ... + ε p f p + ... . The coefficients f0 , f1 , . . . can be functions depending upon only ε and not depending upon x or they can depend upon both x and ε. Let φ p = f0 + ε f1 + ε2 f2 + ... + ε p f p . If

lim φ − φ p ε−p = 0,

ε→0

(1.2)

then I will say that the series (1.1) is an asymptotic representation of the function φ and use the notation: φ (x, ε) = f0 + ε f1 + ε2 f2 + ... . I will call relationships like (1.2) asymptotic equalities. It is clear that, if the parameter ε is very small, then the difference φ − φ p also will be very small, and, though the series (1.1) will diverge, the sum of its first (p+1) terms will give a very good approximation of the function φ.”

1.2 Fundamental concepts of asymptotics [129] In this chapter the fundamental symbols and definitions of the asymptotical analysis will be introduced. For this purpose we consider function f (x), x ∈ S , for x → x 0 . Its limit f0 can be bounded, unbounded, zero or even non-existing. In the asymptotical approach an attention is focused rather on a behaviour in a limit neighbourhood of f (x). Our aim is to find a different and more simple function ϕ(x), which represents f (x) for x → x0 with increasing accuracy. The qualitative estimations are mainly supported by a definition of an order of a variable quantity. In what follows the fundamental estimating relations and the associated symbols will be introduced. A function f (x) is said to be of ϕ(x) order for x → x 0 , i.e. f (x) = (Oϕ(x)), x → x0 , if there exists a such number A, that in a certain neighbourhood ∆ of the point x0 the following estimation holds: | f (x)| ≤ A|ϕ(x)|. A function f (x) is said to be of an order smaller then ϕ(x) for x → x 0 , i.e. f (x) = o(ϕ(x)), x → x 0 , if for any ε > 0 there is a neighbourhood ∆ ε of the point x 0 , where | f (x)| ≤ ε|ϕ(x)|.

22


Observe that in the first case the ratio | f (x)|/|ϕ(x)| is bounded in the space ∆, whereas in the second case it approaches zero for x → x 0 . For example, sin x = O(1), x → ∞; ln x = o(x α ), α > 0, x → ∞. The first relation clearly allows for extension into the bounded space ∆. (Note that first characters O and o come from the word ‘order’.) In addition, two more estimating relations follow. A function f (x) is said to be of order in exactness of ϕ(x) for x → x 0 , i.e. f (x) = Oe(ϕ(x)), x → x 0 , if there exists such numbers a > 0 and A, that in a certain neighbourhood ∆ of the point x 0 the following estimation holds: a|ϕ(x)| ≤ | f (x)| ≤ A|ϕ(x)|. For example, 1 − cos x = Oe(x 2 ), x → 0. Furthermore, a function f (x) is said to be asymptotically equal ϕ(x) for x → x 0 , i.e. f (x) ∼ ϕ(x), x → x 0 , if f (x)/ϕ(x) → 1. For example, sin x ∼ x, x → 0. One may prove, that the relation f (x) = Oe(ϕ(x)) is equivalent to two relations f (x) = O(ϕ(x)), ϕ(x) = O( f (x)), whereas f (x) ∼ ϕ(x) is equivalent to f (x) = ϕ(x)[1 + o(1)]. Consequently, the symbols Oe and ∼ may be reduced to O and o and they are used for a short notation purpose. Note that the equality sign in order estimation relations is used only in a specific sense, since the symbols O, o, Oe can appear only to the right of the mentioned sign. One may distinguish the following asymptotical steps of approximation. First, the above (below) estimations of the type f (x) = O(ϕ(x)) are constructed. Usually the obtained estimations are to high, i.e. f (x) = O(ϕ(x)). In order to improve the obtained estimation the exact estimation is found, i.e. f (x) = Oe(ϕ 0 (x)). Then the following asymptotical equality is achieved: f (x) ∼ a 0 ϕ0 (x). For instance, for sin ax for x → 0 the mentioned steps are as follows: sin ax = O(1), sin ax = o(1), sin ax = Oe(x), sin ax ∼ ax. Observe that more and more information is delivered by achieving successive steps. Having carried out the described cycle, one may also obtain the asymptotical equality f (x) = a 0 ϕ0 (x) ∼ a1 ϕ1 (x), and then one may also achieve more advanced estimations. However, for this purpose new concepts are required. A sequence {ϕn (x)}, n = 0, 1, . . ., x → x 0 is called the asymptotical one, if ϕn+1 (x) = O(ϕn (x)). For example, {x n } for x → 0. A series an ϕn (x), where a m are arbitrary numbers, is called asymptotical one, if {ϕn (x)} is an asymptotical sequence. A function f (x) has an asymptotical expansions with respect to {ϕ n (x)}, i.e. f (x) ∼

N

an ϕn (x),

N = 0, 1, 2, . . . ,

n=0

if f (x) =

m

an ϕn (x) + o(ϕm (x)),

m = 0, 1, . . . , N.

n=0

For instance, consider the series ex ∼

∞ n=0

xn /n!,

x → 0.

1.3 Transformations of asymptotical series [129]

23

This series is convergent for any fixed x. More interesting example is represented by the exponential integral y Ei(y) = eξ ξ−1 dξ, y < 0. −∞

Integration by parts yields a divergent asymptotical series ∞ (n − 1)!y−n , y → −∞ . n=1 −y

Denoting f (x) = e Ei(y), y = −x−1 , one may get the following estimation ∞ f (x) ∼ (n − 1)!(−x) n.

(1.3)

n=1

In order to outline a better meaning of this divergent series, we denote by S N (x) its sum and introduce the following quantity ∆N (x) = | f (x) − S N (x)| , characterizing an accuracy of estimation. In order to solve a problem on the series convergence in the point x, one may consider ∆ N (x) for N → ∞, x = const. In an asymptotical approach ∆ N (x) is analysed for x → 0, N = const. Therefore, one may achieve a suitable approximation even when a series is divergent. In the Table 1.1 ∆N (x) for the case (1.3) is reported. Table 1.1. Example of asymptotic series ∆2 ∆3 ∆4 ∆5 ∆6 x f (x) ∆1 1/3 0.2619 0.071 0.040 0.034 0.040 0.059 0.106 1/5 0.1764 0.030 0.010 0.006 0.040 0.037 0.040 1/7 0.1267 0.016 0.004 0.016 0.08 0.06 0.057

∆7

∆8

0.040

0.047

It is clear that for a fixed x (with increase of N) the accuracy first increases, and then decreases, which is in agreement of our expectations owing to the mentioned divergency. However, the mentioned convergency at the beginning observed in the beginning is more strongly outlined for a small x. The a priori given accuracy for a fixed x can be only achieved on a certain finite interval N. An increase of accuracy requirements yields a decrease of the internal of x, where it can be achieved. To sum up, three quantities: ∆, x and N, characterizing accuracy, locality and asymptotical simplicity, respectively, are coupled in pairs via supplemented relations.

1.3 Transformations of asymptotical series [129] Theorem of uniqueness. Let f (x), x ∈ S , for x → x 0 be developed into the asymp totical series {ϕn (x)}, i.e. f (x) ∼ ∞ n=0 an ϕn (x). Then, the coefficients a n are defined uniquely.

24


Proof. This theorem will be proved by contradiction. Assume that there exists one more series f (x) ∼ ∞ n=0 bn ϕn (x), bn an . Therefore we have f (x) ∼ a 0 ϕ0 (x) + o(ϕ0 (x)), f (x) ∼ b 0 ϕ0 (x) + o(ϕ0 (x)). The assumed difference of both series yields a0 − b0 = o(1), which means that a 0 = b0 . Applying the mathematical induction one gets the proof. In words, for a given asymptotical series {ϕ n (x)}, the coefficients a n of the asymptotical series of f (x) for x → x 0 are defined uniquely via the relation   n−1   an = lim  f (x) − ak ϕk (x) ϕ−1 n (x). x→x0 k=0

However, one may expect that the analysed function f (x) can be expanded using the different asymptotical sequence {x n (x)}. It is clear that there are different coefficients associated with the second sequence. For instance: 1 ∼ xn , 1 − x n=0 ∞

x → 0,

1 ∼ (1 + x)x2n , 1 − x n=0 ∞

x → 0.

On the other hand, one asymptotical series can be represented by a few functions. For example: ∞ 1 + ce−1/x n ∼ x , x → 0. 1− x n=0 In words, any asymptotical series represents not only one but rather a class of asymptotically equal functions. Summations. The asymptotical series of the function f (x) and g(x), x ∈ S ⊂ R 1 , x → x0 , with respect to the sequence {ϕ n (x)}, f (x) ∼ an ϕn (x), g(x) ∼ bn ϕn (x), can be summed and multiplied by the constants, i.e. α f (x) + βg(x) ∼

∞

(αan + βbn ) ϕn (x).

n=0

Multiplication. Note, that generally, a product of asymptotical series is not defined, since {ϕn (x) · ϕm (x)}, m, n = 0, 1, . . ., not always yields an asymptotical series. However, if the latter is achieved (for instance for ϕ n (x) = xn ), then the product is defined. The same holds for division, assuming b 0 0. Logarithms and exponents. Taking the logarithm of asymptotics does not lead to special difficulties, whereas during exponentiating a rather√careful considerations are required. The following example is considered: f (x) = ( x ln x + 2x)e x = [2x + o(x)]e x , x → ∞. If g(x) ≡ ln f (x), hence g(x) = x + ln[2x + o(x)] = x + ln x + ln 2 + o(1) ∼ x + o(x), x → ∞. Now looking for the corresponding exponent for g(x), one gets f (x) ∼ e x , x → ∞. Observe that in the main term 2x multiplier is lost. The

1.4 Nonuniform expansions [129]

25

latter occurred owing to not accounted terms ln x and ln 2, which have an influence on the main asymptotical term of f (x) (only the quantities o(1) have no influence, since exp{o(1)} ∼ 1). Integration and differentiation. If in a polynomial asymptotical series f (x) ∼ an x−n , x → ∞, a0 = a1 = 0, then it can be integrated term-by-term, i.e. ∞ ∞ an 1−n x . f (x)dx ∼ 1−n x n=2 In general, the asymptotical series can not be differentiable. For example, f (x) = e−1/x sin(e1/x ) has the singular polynomial representation f (x) ∼ 0·1+0·x+0·x e+. . .. Although a derivative exists, but it does not possess a polynomial series representation. If a continuous derivative f (x) for x ≥ d > 0 has (similarly to f (x)) a polynomial series for x → ∞, than it can be obtained via term-by-term differentiation process of the function f (x).

1.4 Nonuniform expansions [129] Earlier, the function f (x) depended on one argument, has been considered. Consider now a function of two variables f (x, ε), x ∈ D ⊂ R n , which can be represented by the series N an (x)ϕn (ε), ε → 0. (1.4) f (x, ε) ∼ n=0

The coefficients a n in (1.4) depend on x in R n . The introduced series differs from the asymptotical series, since in the latter ones the sequence {ϕ n } depends not only on the parameter ε, but also on the variable x. If the relation (1.4) holds for all x ∈ D, then the sequence is called uniform with respect to x in D. In addition f (x, ε) −

N

an (x)ϕn (ε) = o[ϕN (ε)],

∀x ∈ D.

(1.5)

n=0

Instead of ‘uniform asymptotical’ series often the equivalent terms are used either as ‘uniformly suitable’ or ‘uniformly correct’ series. If in the space D ⊂ Rn there exists the manifold L of a smaller dimension such that the relation (1.5) is not satisfied for an arbitrary subspace D ⊂ D of dimension n, which includes L, than the series (1.4) is called nonuniform one. One may say also that f (x, ε) has a singularity with respect to ε for x ∈ L, or that f (x, ε) is a singular function, and L is called a boundary layer. Consider, for example, the function f (x, ε) = e−x/ε − 1, It is clear, that for all x ∈ [α, 1], α > 0,

x ∈ [0, 1], ε → 0.

(1.6)

26


f (x, ε) = −1 + o(εn ).

(1.7)

An asymptotical representation (1.7) for the function (1.6) is nonuniform in the vicinity of x0 = 0. In this example L corresponds to the point x 0 = 0. Two fundamental questions appear. What kind of sources generate nonuniformities? Is it possible a priori to predict if a considered problem as regular or singular? There are the following main sources of nonuniformities: (i) an occurrence of ε standing by the highest derivative of a studied differential equation; (ii) a type change of partial differential equations for ε → 0; (iii) a qualitative change of boundary and initial conditions for ε → 0, as well as a change of a structure of the domain D for ε → 0; (iv) unboundness of domain D. For example, in the plane theory of elasticity the characteristic asymptotical problems are the boundary value problems for 4-th order elliptic equations, where a solution should be defined in the multiply domain D. The latter is defined by an occurrence of thin cracks, holes, etc. Now, if a perturbation parameter ε represents a characteristic thickness of a crack or a hole diameter, then for ε → one gets simply connected domain D 0 , in which the mentioned cracks and holes do not exist. A solution of the problem in D 0 will be suitable everywhere in D except of the manifold L, which is exactly constituted by the mentioned cracks and holes. In other words, if a structure of the limiting domian D 0 essentially differs from the structure of D, then one may expect singularities. Consider now the following nonlinear problem governed by the Duffing equation ˙ = 0, ε → 0. (1.8) U¨ + U + εU 3 = 0, U(0) = a, U(0) The following approximate solution is sought U=

∞

εn Un (x).

(1.9)

n=0

Substituting (1.9) into (1.8) and equating the terms standing by the same powers of ε one obtains

a3 3 (cos 3t − cos t) + O(ε2 ). U = a cos t + εa3 − t sin t + (1.10) 8 32 For x = O(ε−1 ), the second term in (1.10) is of the same order as the first one. It means that for an unbounded increase of t, the series (1.9) has no physical interpretation due to occurrence the so called secular terms of the type t sin t in asymptotes. This observation is typical for all problems of non-linear oscillation. A physical criterion of singular perturbation is motivated by an occurrence of two or more characteristic scales of any physical or geometrical quantities.

1.5 Non-dimensionalization

27

Observe that an asymptotical series (suitable in D everywhere except of the boundary layer L) is called regular or external asymptotics, whereas that governing behaviour in the vicinity of the boundary layer is called either local or internal asymptotics. However, not all of singularly perturbed problems are characterized via a boundary layer L concept. For instance, the function f (x, ε) = e−x/ε sin(1/(xε)), x ∈ [0, 1] for x ∈ [α, 1], α > 0 can be estimated via f (x, ε) = O(e−1/ε ), ε → 0, and for x = o(ε) via f (x, ε) ∼ (1/(xε)). It is clear that an asymptotical estimation for f (x, ε), x ∈ [0, 1] is non-unique one, but in the neighbourhood of x = 0 the function f (x, ε) can not be developed into an asymptotical series, because it has no limit for ε → 0. In contrary, the boundary layers are characterized by the following behaviour: limit for ε → 0 exists in both local and external spaces.

1.5 Non-dimensionalization It is obvious that before starting with application of any asymptotical approach a non-dimensialization process is highly required. In this process very often some nondimensionalized quantities appear, which can be treated as parameters of asymptotic integrations. A brief review of dimensional analysis will be addressed now [131, 601]. Let one deals with dimensional quantity a, which is the function of mutually independent dimensional quantities a = f (a1 , a2 , ..., ak , ak+1 , ..., an).

(1.11)

We are going to exhibit a structure of the function f (a 1 , ..., an) assuming that this function expresses a certain physical rule independently on a choice of the system measurement units. Let among dimensional quantities a 1 , a2 , ..., an first k quantities posses independent dimensions. Assume k independent quantities a 1 , a2 , ..., ak as the fundamental ones, and let us introduce their dimension via notations [a1 ] = A1 , [a2 ] = A2 , ..., [ak ] = Ak . Dimension of the rest quantities have the form mk 1 m2 [a] = Am 1 A2 ...Ak ,

[ak+1 ] = A1p1 A2p2 ...Akpk , . . . . . . . . . . . . . . . . . . [an ] = Aq11 Aq22 ...Aqk k . Let us change units of quantities a 1 , a2 , ..., ak measurement into α 1 , α2 , ..., αk times. Numerical values of these quantities and the quantities a, a k+1, ..., an in the new units system have the form:

28


a 1 = α1 a1 ,

mk 1 m2 a = αm 1 α2 ...αk a,

a 2 = α2 a2 ,

a k+1 = α1p1 α2p2 ...αkpk ak+1 ,

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

a k

= αk ak ,

a n = αq11 αq22 ...αqk k an .

In the new system of measurement the relation (1.11) has the form mk m1 m2 mk 1 m2 a = αm 1 α2 ...αk a = α1 α2 ...αk f (a1 , a2 , ..., an ) =

f (α1 a1 , ..., αk ak , α1p1 α2p2 ...αkpk ak+1 , ..., αq11 αq22 ...αqk k an ).

(1.12)

This equality expresses a property of homogenity with respect to scales α 1 , α2 , ..., αk . The scales α1 , α2 , ..., αk are arbitrary. We are going to find such scales choice that the number of the function f argument is decreased. Let us assume α1 =

1 1 1 , α2 = , ..., αk = . a1 a2 ak

In words, system of units measurement is taken in such way, that the values of first k arguments in right hand side of relation (1.12) should be equal to one. In other words, since the relation (1.11) does not depend on the units system measurement, we establish units system measurement in a way that k arguments of function f have fixed constant values equal to one. In relative units system measurement, numerical values of parameters a, a k+1 , ..., an are governed by formulas a Π = m1 m2 , k a1 a2 · · · am k ak+1 Π1 = p1 p2 p , a1 a2 · · · ak k . . . . . . . . . . . . . . . . . . an Πn−k = q1 q2 q , a1 a2 · · · ak k where a, a1 , a2 , ..., an are numerical values of the considered units system measurement. Notice that the values of Π, Π 1 , ..., Πn−k do not depend on the choice of initial units system measurement, since they have zero order dimension with respect to measurement units A 1 , A2 , ..., Ak . Values of Π, Π1 , ..., Πn−k in general do not depend on the choice of the system units. They are used to define k units measurement for quantities a1 , a2 , ..., ak . Therefore, they can be treated as non-dimensional ones. Applying a relative units system measurement, the relation (1.11) can be recast into the form (1.13) Π = f (1, 1, ..., Π1 , ..., Πn−k ). Therefore, a relation between n + 1 dimensional quantities a, a 1, ..., an , independent on a units system measurement choice, exhibits a relation between n + 1 − k dimensional quantities Π, Π 1 , ..., Πn−k . This general conclusion of theory of dimensionality is known as Π-theorem.

1.6 Asymptotics of integrals [129]

29

1.6 Asymptotics of integrals [129] Many mechanical problems are governed by integrals depended on parameters. In particular, it is important to trace their behaviour in singular cases. Asymptotical methods allow for integrals estimation, especially those having either complex analytical structure or being unsuitable for numerical computations due to occurrence of various singularities. Obviously, the asymptotic integrations creates a hide branch of mathematical analysis [141, 196, 206, 236, 268, 286, 287, 340, 343, 506, 507, 511, 535]. Our attention is focused on a brief review of this field beginning from simple examples. Direct asymptotic series application. Consider the first order integral A(ε) =

/2

1 − ε sin2 x

−1/2

dx,

ε → 0.

(1.14)

0

The under-integral function can be developed into the series 1 3 (1 − ε sin2 x)−1/2 ∼ 1 + ε sin2 x + ε2 sin4 x + . . . , 2 8 which is uniformly defined for x ∈ [0, π/2]. Its integration yields

9 2 ε 3 A(ε) = 1 + + ε + 0(ε ) . 2 4 64 Similarly, applying the expansion e−x = 1 −

x2 x3 x + + + ..., 1! 2! 3!

x ∈ [0, ε],

the following asymptotic approximation is obtained ε 0

4 x−3/4 e−x dx = 4ε1/4 − ε5/4 + 0(ε9/4 ), 5

ε → 0.

(1.15)

An integration of the integral ∞ B(ε) =

2

e−x dx,

ε → 0,

ε

can be reduced to the so far described method applying the formula B(ε) = B1 (ε) − B2 (ε),

(1.16)

30


where: B1 (ε) =

∞ 0

2

e−x dx; B2 (ε) =

ε

2

e−x dx.

0

The integral B 1 (ε) is equal to π/2, whereas computing B 2 (ε), a series of exponential function for x → 0 is used. In result, the following approximation is applied √ π 1 − ε + ε3 + 0(ε5 ). B(ε) = 2 3 Recall that a direct series representation with respect to a parameter is valid only when it is uniformly suitable on the whole integration interval (see examples (1.14) and (1.16)). Otherwise, a false result can be obtained. For example, 1 ε

√ √ √ dx ε = 1 + ε − 2ε = 1 − 2ε + + 0(ε2 ), √ 2 2 x+ε

ε → 0.

However, substituting the under integral function as the product [x(1+ε/x)] −1/2, and then substituting second multiplier through its series representation with respect to ε/x, the integration of two first series terms yields 1 ε

3√ 1 −1/2 1 1ε x + . . . dx = 1 − ε + ε + ..., 1− 2 2x 2 2

which is not in agreement with the exact asymptotics. A reason is that a direct series representation of the under integral function is suitable only in the domain ε = 0(x), and hence it is not uniformly suitable for all x = 0(ε). Series development in the vicinity of a peak kernel. Let an integral depended on the parameter has the form b K(x, ε)ϕ(x)dx.

J(ε) =

(1.17)

a

The function K(x, ε) is called a kernel, whereas ϕ(x) is a loading function. It is required that K(x, ε) is relatively simple but representing the fundamental properties of the input functions. Let K(x, ε) possesses a peak for x → x 0 . Then, it is recommended to develop ϕ(x) into an asymptotical series for x → x 0 . In the case of analytical function ϕ(x) the series overlaps with its Taylor series. The latter can be used during integration (1.17), assuming that a fundamental role in the asymptotics A(ε) plays the neighbourhood of the point x = x 0 . As an example, the following integral is analysed ∞ A(ε) = 0

e−x/ε ϕ(x)dx.


31

The kernel K(x, ε) = e −x/ε has a peak for x 0 = 0, and hence A(ε) = ϕ(0)ε + ϕ (0)ε2 +

ϕ (0) 3 ε +... . 2!

Integration by parts. The following integral is going to be investigated x E(x) = −∞

eξ dξ, ξ

x → ∞.

(1.18)

ξ

Let K = e , ϕ = 1/ξ. Integrating (1.18) infinitely times by parts, differentiating on each step the loading function, and integrating the kernel, one gets ∞ (n − 1)!

E(x) = e x

n=1

xn

.

Therefore, the following asymptotics is found ∞ −x e dx 1 1 2 = − 2 + 3 + 0(t−4 ), t+x t t t 0

t → ∞.

Observe that integration by parts to construct asymptotic series of an integral can be only applied, when the integrals occurred on each step are convergent and when the obtained expression is asymptotic in a limit with respect to a parameter. Assume now, that the under integral function f (x, ε), x ∈ [0, 1], ε → 0 can be simplified for xε = o(x), f ∼ f (ε) , and for x = o(x ε ), f ∼ f (i) , where xε −−−→ 0. ε→0

In other words, here we deal with a boundary layer; f (ε) , f (i) are external and local asymptotics, correspondingly. The point x = x ε is said to be the transitional point. Since the function x ε (x) is defined up to a constant, in general the whole transitional subspace exists, where x = 0e(x ε ). Assume, that the integral 1 f (x, ε)dx

J(ε) = 0

represents the main contribution generated by the transitional domain. The following question arises: how to find the asymptotics of A without a direct integration of f (x, ε) along the transitional domain. Note, that in this domain a simplification of f (x, ε) is not known, and the asymptotics f (e) can not be here applied. It occurs, that in some cases this is possible owing to integration by parts, decreasing the weight of under-integral function in the transition zone. As an example the following integral is considered l A(ε) =

f (x) ε

√

x2 − ε2 − x dx,

32


where: xε (ε) = ε; f (e) = −0.5 f (x)ε2 /x. Obviously, the transition zone influences essentially the main term of asymptoties A(ε). Assuming f (x) ∈ C 1 [0, 1], the following integration by parts is carried out 1 1 f (x)A1 (x, ε)dx, A(ε) = A1 (x, ε) f (x) − 0.5 ε

where: A1 (x, ε) =

ε

x √

x2 − ε2 − x dx =

ε

√ √ ε (ln ε + 1) − ε ln x + x2 − ε2 + x x2 − ε2 − x . 2

2

The integral term is divided into three following terms 1

2

ε (ln ε + 1)

f (x)dx − ε

2

ε

1

√ f (x) ln x + x2 − ε2 dx+

ε

1

f (x)x

√

x2 − ε2 − x dx.

(1.19)

ε

The first integral in (1.19) is trivial, 1 f (x)dx ∼ f (1) − f (0) . ε

The second is asymptotically equivalent to integral ε

1

f (x) ln(2x)dx,

whereas the third integral can be investigated dividing the interval of integration into two parts by the point x ∗ (ε) such that ε = o(x ∗ ):

x∗

ε

f (x)x

√

x2

−

ε2

1

f (x)x

− x dx +

√

x2 − ε2 − x dx.

x∗

In what follows x∗

f (x)x ε

and

√

x2

−

ε2

x∗ √ − x dx ∼ max( f ) x x2 − ε2 − x dx = o(ε2 ),

x

ε


1

f (x)x

√

x2

−

ε2

1

− x dx ∼

x∗

x∗

ε2 =− 2

1 x∗

33

4 ε 1 ε2 + 0 3 dx = f (x)x − 2 x x

ε4 ε2 f (x)dx + 0 =− f (1) − f (0) + O(ε2 ). x∗ 2

and finally  1    f (0) 2 ε2  f (0)  2  f (x) ln(2x)dx − f (1) − A(ε) = ε ln ε +  + O(ε ). 2 2  2 

(1.20)

ε

To conclude, in this example owing to an integration by parts, a direct integration through a transitional zone is omitted. Observe that the integral occurred in (1.20) can be easily calculated, since the underintegral function does not depend on ε. Even if the presented so far approach is not effective, one may apply the following approach. In order to construct an integral series, the asymptotical extensions of external f (e) and internal f (i) asymptoties up to the point x ∗ = Cxε (ε) are applied 1

Cx ε (ε)

f (x, ε)dx = 0

1

(i)

f dx + 0

f (e) dx.

Cxε (ε)

The constant C is defined via the condition ∆ 1 = ∆2 , where: ∆1 = 1 ∆2 = x f − f (ε) dx. 0 The following example supports our considerations. Let

x 0

f − f (i) dx,

f (x, ε) = ε2 + ε exp(−x/ε), x ∈ [0, 1] . In this case x∗ = Cε, f (i) = ε, f (e) = ε2 , and from the relation ∆ 1 = ∆2 one finds C = 1. It means that 1 ε 1 f (x, ε)dx ∼ εdx + ε2 dx = 2ε2 . 0

ε

0

Note that exact value of this integral is equal to 2ε 2 . Singularities isolation. Assume that an essential influence into asymptotics of the integral f (x, ε)dx A(ε) = S

is expected in the vicinity of the point x = x 0 . If fA (x, ε) is the asymptotics of f (x, ε) for ε → 0, x → x0 , than the integral can be presented in the following form

34


x0 +δ

f (x, ε)dx =

f1 (x, ε)dx + x0 −δ

S

x0 +δ

f − f1 dx +

x0 −δ

f!(x, ε)dx,

(1.21)

{S −[x0 −δ,x0 +δ]}

where the quantity δ(ε) is defined owing to convergence condition of the first integral in the right hand side of (1.21). It can be proved that the second and third integrals in (1.21) are negligible in comparison to the first one. In result, S

f (x, ε)dx ∼

x0 +δ

fA (x, ε)dx,

x0 −δ

and this estimation should be independent on δ. For instance, consider the following integral 1 A(ε) =

ex

2

−ε2

x2

0

cos(xε) dx, + ε2

ε → 0.

In the above, the main contribution is expected in the neighbourhood of the point x0 = 0, and hence δ A(ε) = 0

1 δ

dx + 2 x + ε2

1 dx ∼ 2 2 x +ε ε 1 0

δ/ε 0

 1  x2 −ε2  e cos(xε) 1   dx; − 2  x 2 + ε2 x + ε2 0

dξ + 2 ξ +1

δ 0

x2 (1 + ε2 /2) − ε2 dx+ x 2 + ε2

 x2  1/ε  e dξ 1  1 ,  2 − 2  dx + 2 ε x x ξ +1 δ/ε

where: ε = o(δ). After estimation of two last integrals one gets 1 J(ε) = ε

∞ 0

ξ2 + 1

−1

dξ =

π + 0(1). 2ε

In this case, when f (x, ε) possesses a few points of transition {x εk }, k = 1, 2, . . ., it is expected to find one or a few transitional points, which represent the main contribution to the considered integral. For example, let us derive A(ε) =

0

1

−1 ε2 + εα xβ + x2 dx, α + β < 2, 0 < β < 1 .


35

Comparing first and second terms in the under-integral function with x 2 , transition points are found, i.e. xε1 = ε and xε2 = εα/(2−β) . Since xε1 = o(xε2 ), one may conclude that a fundamental contribution is yielded by integral in the neighbourhood of x ε2 , and hence A(ε) = ε

−α/(2−β)

∞

τβ + τ2

−1

dτ.

0

Laplace method. Let us consider the integral of the form b f (t) =

ϕ(x) exp [th(x)] dx,

t → ∞,

(1.22)

a

within the following assumptions: (i) an integration interval is finite; (ii) ϕ(x) and h(x) are continuous on [a, b]; (iii) h(x) possesses only one maximum x 0 ∈ (a, b); (iv) ϕ(x0 ) 0; (v) h(x) = h 0 − h2 (x − x0 )2 + 0[(x − x0 )3 ], x → x0 , h2 > 0. Note that an estimation of f (t) = O[exp(th 0 )] is obviously to tough, since the under-integral function in (1.22) has a peak-like character. Indeed, isolating the constant exponent exp(th 0 ), the function exp{−t[h 0 − h(x)]} is obtained. The latter one approaches zero except of the point x 0 , where it is equal 1 for all t. In what follows, increasing t a localization in the neighbourhood of x 0 appears, and one may transform f (t) to more simpler form. In order to find this asymptotical representation, the integration interval is divided into three parts: [a, b] = [a, x0 − ε] + [x0 − ε, x0 + ε] + [x+ ε, b], assuming ε = ε(t) −−−→ 0. In order to obt→∞ tain the main term of asymptotics in the vicinity of x 0 , the integral is estimated in average on the level of asymptotical equality. Owing to the introduced assumptions x0 +ε x0 +ε 2 th(x) th2 ϕ(x)e dx ∼ ϕ(x0 )e e−th2 (x−x0 ) dx, x0 −ε

x0 −ε

Introducing

√ th2 (x − x0 ) = ξ, one gets

if tε3 → 0.

√

ε th2 x0 +ε 1 π 2 2 exp −th2 (x − x0 ) dx = √ exp −ξ dξ ∼ , th th2 √ 2

x0 −ε

if tε2 → ∞.

−e th2

Observe that both conditions tε 3 → 0 and tε2 → ∞ are satisfied simultaneously, only if 1 1 0;

h(x) ∼ −cxα , c > 0, α > 0.

Therefore, the following estimations take place ε f (t) ∼

x 0

β−1 −tcxα

e

dx ∼ (ct)

−β/α

∞ 0

α

ξ β−1 e−ξ dξ,

for tεα (t) → ∞.

(1.25)


37

Expressing the last integral via Γ function, one gets 1 f (t) ∼ (ct)−β/α Γ(β/α). α

(1.26)

Observe that increase of α yields the peak extension, and f (t) is decreased slowly. Parameter β works oppositely to α, since by increasing β the peak effect disappears. One obtains (1.24) from (1.26) for β = 1, α = 2, C = h 2 with the coefficient 0.5, since now only one peak side is considered. More interesting case occurs, when the load possesses also exponential character, for instance of the form ϕ(x) ∼ exp −x−β , β > 0. Owing to analysis of the function Ψ (t, x) = x −β + ctxα , the main contribution is expected in the moving point x 0 (t) = [β/(αct)]1/(α+β) , where Ψ has a minimum. In order to stabilize the peak, the transformation x = ξ(ct) −1/(α+β) is applied. Therefore one gets ε(ct) 1/(α+β)

f (t) ∼ (ct)

−1/β β/(α+β) −β α exp −(ct) et∗ h∗ (ξ) dξ, ξ + ξ dξ ∼ t∗

−1/(α+β)

∞

0

0

ε(t)t1/(α+β) → ∞. where: t∗ = (ct)β/(α+β) , h∗ = −ξ−β − ξα , and the conditions, for which the formula (1.24) holds, are satisfied. Consequently π t∗ h3 −1/β f (t) ∼ t∗ e , (1.27) t∗ h4 where: h3 = −(α + β)α−α/(α+β) β−β/(α+β) ,

h4 =

0.5(α + β) (β+2)/(α+β) (α−2)/(α+β) α β . 2

Two following examples support out considerations. Stirling formula. In the integral ∞ Γ(t + 1) = e−x xt dx, t → +∞, 0

the function Ψ (t, x) = x − t ln x possesses a minimum in the point x 0 = t, tending to infinity. Introducing x = tξ, one obtains Γ(t + 1) = t

t+1

∞ 0

exp t(ln ξ − ξ) dξ,

38


where: h(ξ) = ln ξ − ξ, ξ 0 = 1, h(ξ0 ) = −1, h (ξ0 ) = −1, and finally √ Γ(t + 1) ∼ 2πtt+1/2 e−t .

(1.28)

Integral of relaxation. In the field of kinetics often the following integral appears ∞ f (t) =

exp −x2 − t/x dx,

t → +∞.

(1.29)

0

Function Ψ (t, x) = x 2 + t/x minimum is achieved in the point x 0 = (t/2)1/3. Variable change x = ξt 1/2 yields the function f (t) = t1/3

∞

exp − t2/3 ξ2 + ξ−1 dξ = t∗1/2 exp t∗ h∗ (ξ) dξ, ∞

0

0

where: t∗ = t2/3 , h∗ (ξ) = −ξ2 − ξ−1 . Using (1.24) and computing ξ 0 = (1/2)1/3, h∗ (ξ0 ) = −3(1/2) 2/3, h ∗ (ξ0 ) = −6, one finally gets π exp −3(t/2)2/3 . (1.30) f (t) ∼ 3 Stationary phase method. This method is applicable to integrals of the form b f (t) =

ϕ(x)eith(x) dx,

t → +∞.

(1.31)

a

Observe that this class of integrals differs from (1.22) of the imaginary one in the exponent. It means that the previously peak-shaped function is now rapidly oscillating. Approximation f (t) = O(1) is overestimated, since for rapid oscillations the integrals (owing to positive and negative parts) contributions mutually cancel each other. This observation is exhibited, for instance, by the known Riemann-Lebesque lemma ∞ ϕ(x)eitx dx → 0 for t → ∞, ϕ ∈ L1 (R). −∞

If h (x) on [a, b] is not equal to zero, then integration by part yields the following estimation

b 1 ϕ(x) ith(x) e . f (t) ∼ it h (x) a Roots of equation h (x) = 0 are called stationary points. In their vicinity a frequency of oscillations t(∆x) 2 is smaller in comparison to regular points. Therefore,


39

the main contribution of asymptotics into integral (1.31) is expected from stationary points. Let us assume in the beginning, that there is one stationary point x 0 ∈ (a, b), and an integration interval is finite, ϕ(x) and h(x) are continuous, ϕ(x 0 ) 0, h(x) = h0 + 0.5h (x0 )(x − x0 )2 + O[(x − x0 )3 ], x → x0 . Applying the Laplace method and using Poisson’s integral ∞ π 2 , exp(±iξ )dξ = ∓i −∞

in stationary point neighbourhood the following estimation holds π 1 f (t) ∼ ϕ(x0 )eith(x0 ) , h2 = − h (x0 ). ith2 2

(1.32)

In the rest of integration by parts intervals the following estimation holds: (iε) −1 = o(t−1/2 ) for tε2 → ∞. Formula (1.32) differs from (1.24) by imaginary one for t. In the case of (1.25), it allows for generalization analogous to (1.26): 1 β f (t) ∼ (ict)−β/α Γ , 0 < β < α, (1.33) α α where a sign of c can be arbitrary. Two examples follow. Fresnel’s integral. In this case ∞

2

∞

exp(ix )dx = t

f (t) = t

1

exp(it2 ξ2 )dx ∼

i it2 e for t → ∞, 2t

since there is a lack of stationary points in the interval [1, ∞). Bessel function. In this case 1 cos(t sin x − nx)dx. Jn (t) = π 0

Therefore, h(x) = ± sin x, x 0 = π/2, ϕ(x0 ) = exp(∓0.5 sin π), h 2 = ±0.5, and applying the formula (1.32) one gets π π R Jn (t) ∼ cos t − n − , t → ∞. (1.34) πt 2 4 Saddle-point method. A behaviour of integrals (1.22) and (1.31) studied via both Laplace and stationary phase methods is qualitatively different. However, their analysis and corresponding

40


results (1.24), (1.32) are formally close to each other. A reason is mainly because both of the mentioned methods represent particular cases of one general method using theory of complex variables functions. Advantages and possible extensions of the method occur when localizations on the real axis for large t do not appear. In order to give a background of the saddle-point method, the following integral is considered exp [th(z)] dz, t → +∞, (1.35) f (t) = c

where: h(z) is holomorphic function, c is contour is a complex plane z with ends approaching infinity. Let h(z) = p(x, y) + iq(x, y), z = x + iy, p → −∞ on the contour ends. Rough estimation gives f (t) = 0[exp(t max c p)]. However, in accordance with the Cauchy theorem, the contour c can be deformable, which is equivalent to max p variation. Central idea of the method is focused on achieving an integrating route to get min max p via contour deformation. Searching of this route, a qualitative investigation of the surface p(x, y) is carried out. An essential role play the points for which h (z) = 0. In their neighbourhoods the surface p(x, y) creates a saddle, and one may expect that an optimal integration route has a saddle characteristic. Let us investigate a structure of the saddle point z 0 , introducing the polar coordinates z − z0 = ρ exp(iϕ). Let h (z0 ) 0 and 0.5h (z0 ) = ρ0 exp(iϕ0 ). Therefore, h(z) − h(z0 ) ∼ ρ0 ρ2 [cos(2ϕ + ϕ0 ) + i sin(2ϕ + ϕ0 )], and p − p 0 changes its sign on the radius ϕ = −ϕ0 /2 + π/4(2K + 1), K = 0, 1, 2, 3. Neighbourhood of the saddle point is divided into four sectors: in two of them the level of p is higher than saddle level, whereas in two other the level p is lower than initial one. For z 0 = 0, ϕ0 = 0 the cross section of the surface p(x, y) by the plane p 0 is shown in Figure 1.3, where ‘positive’ sectors (p − p 0 > 0) are shaded. Minimal value of max c p is achieved, when the integration contour path goes through negative sectors via the saddle point. Although direction of transition does not influence a result, but an estimation procedure depends on it. If a passage is realized via bisectrix of the negative sector, i.e. through the fastest rise and descent, then ϕ = 3π/2 and π/2, p = p0 − ρ0 ρ2 + O(ρ3 ), q = q0 + O(ρ3 ), and finally Laplace method can be applied. However, using the boundary sector radius ϕ = π/4(2K + 1), it follows that of the stationary phase p = p0 + 0(ρ3 ), q = q0 ± iρ0 ρ2 + 0(ρ3 ), and an application √ method holds. Observe that in the latter case 1/ i in (1.32) is compensated by a corresponding multiplier in dz ∼ e iϕ dρ, since a difference in directions is equal to π/4. To conclude, a common aspects of two methods have been outlined. Moving in other directions within the negative sector, the following integral is applicable ∞ i −1/4 a exp −bx2 ± iax2 dx = π1/2 a2 + b2 exp ± arctan , 2 b

a > 0, b > 0,

−∞

and it leads to the same, as earlier discussed, result, but via more complicated way. In practice a passage should be crossed using estimation (1.24).


41

y

x

Fig. 1.3. Surface p(x, y) cross section introduced by p0 .

In the case of finite integration interval in (1.35) the contour deformation should not include the ends A and B. The following series of actions is recommended. First, max p on the initial contour c is found. If it is achieved on ends, then the problem of deformations does not appear. If it is achieved inside, then the passage points should be found and the contour should be suitably deformated. Then again maximum of p should be found, but already on the new contour c, since owing to deformation the internal maximum can be dropped down below p values on the ends. Passage point z0 contributes to the main term of asymptotics only if p(z 0 ) ≥ p(A), p(z 0 ) ≥ p(B). The following example is studied (1.36) f (t) = exp t ix − 0.5x2 dx on the interval [−l, l] of the real axis. Since a localization in the point x = 0 does not appears, the peak effect is cancelled via rapidly oscillated exponent. During search of the passage points, the function h(z) = iz − 0.5z 2 is differentiated to yield h (z) = l − z, z0 = i, h(z0 ) = 0.5. The contour is deformable in the way shown in Figure 1.4. On the new contour, p(z 0 ) is maximal if l > 1. In addition, owing to (1.24) one gets 2π −t/2 e , t → +∞. (1.37) f (t) ∼ t Note that for l → ∞ the asymptotic relation becomes exact one. For l ≤ 1 one may expect a contribution from both ends A and B. Estimation (1.37) is essentially better than rough estimation f (t) = O(1). Observe that exp(−0.5t) appeared owing to decrease of max p by the contour deformation. After introduction of the method using the earlier example, possible generalizations are further discussed. First of all if z 0 is a multiple root of equation h (z) = 0,

42

1 ASYMPTOTIC APPROXIMATIONS y

-l

-1

1

l

x

Fig. 1.4. Deformation of the integration contour.

the surface p(x, y) becomes more complex. In the passage point of k-th order the k + 1 curves of passage level intersect, and the neighbourhood is partitioned into k + 1 positive and k + 1 negative sectors, which interleave during a transition around z0 . Occurrence of the function ϕ(z) under integral (1.35) yields the problem of its analytical continuation in the domain of contour deformation and a question of overcoming of possible singularities occur. Parameter t can be complex, and the asymptotic is constructed for |t| → ∞ in a certain sector arg t. Finally, contour of integration may depend on t. In general case, the main difficulty of the saddlepoint method application is focused on a proof of observation that in the point z 0 the function p possesses not only local, but also global maximum for whole chosen integration contour. In each of the mentioned cases the separate analysis is required. Finally, the following illustrative example is considered. Airy integral. Consider the following integral 1 3 1 (1.38) exp tw − w dw, for t → ∞, f (t) = 2πi 3 c

where the infinite contour c comes into the point w = 0 along the radius arg w = 43 π and goes into direction arg w = 23 π. Differentiating the integral (1.38) with respect to t two times, one may conclude that the function f (t) satisfies the equation f (t) − t f = 0. The following cases are considered: t < 0 and t > 0. (i) For t > 0 the following transformation is applied: t = ∂ 2 , w = ∂t,

(1.39)


f (t) =

ν 2πi

c

43

x3 exp ν3 z − dz, 3

where: h(z) = z − z3 /3. Denoting z = zeiν , on the input contour p = 0.5z−z 3 /3, since max p = p(0) = 0, a rough estimation yields f (t) = 0(ν). Let us find the passage points from the equation h (z) = 1 − z2 = 0, which read z 1,2 = ±1, h(z1,2 ) = ±2/3. In order to decrease max p, only second passage point is applied, where h (z2 ) = 2. As a new integration contour the straight line x = −1 serves (−∞ < y < +∞). On this line h(z) = − 23 − y2 + 3i y3 , since max p is achieved in the saddle point. Limiting our considerations to the main asymptotics terms, the following estimation is obtained ν f (t) ∼ 2πi

∞ −∞

2 3 1 3 2 2 + y idy = √ e− 3 ν , exp −ν 3 2 πν

t = ν2 → +∞. (1.40)

(ii) For t < 0, let t = −λ2 , w = λz. Hence λ z3 f (t) = exp λ3 −z − dz, 2πi 3 c

3 3 /3 and on the input contour where: h(z) = −z − z√ √ p = 0.5z − z /3, since max p is achieved for z = 1/ 2 and it is equal to 1/(3 2). Looking for the saddle point we get: h (z) = −1 − z2 , z1,2 = ±i, h(z1,2 ) = ∓ 23 i, h (z1,2 ) = ∓2i. Level of the surface p in both points is lower of the input maximum. Accounting of positions of positive and negative sectors, the integration path is taken in the way shown in Figure 1.5. Maximum of p on the new contour is achieved in the passage points, and each of these points introduces a contribution to the main asymptotics term. Calculating its sum, and applying the relation dz = ∓ exp(∓ π4 i)dρ one gets

λ f (t) ∼ 2i

 ∞

 ∞       2   3 2 3 2 2 − = exp λ exp λ i − ρ i dρ + i − ρ i dρ − − +       3 4 3 4   −∞

∞

1 2 √ sin λ2 + 3 4 λ

for t = −λ2 → ∞.

(1.41)

Note that integral (1.38) with the displayed contour c gives only one of solutions to equation (1.39) (second one is found if one of contour radius is taken in direction arg w = 0). In this case t > 0 the saddle point z = 1 is accounted, and it yields exponentially increasing asymptotics. In the case t < 0 the deformed contour intersects only one the saddle points z 1,2 = ±i. Changing argument t into s = −t, for U(s) = f (t) from (1.39) the following Airy equation is obtained U + sU = 0.

44

1 ASYMPTOTIC APPROXIMATIONS y

1

x

Fig. 1.5. Choice of integration path in the saddle-point method.

For s < 0 the solution U 0 (s) is defined by the following conditions U0 (0) = 1,

U 0 (−∞) = 0.

Hence, owing to (1.40) and (1.38) one gets

2 32/3 2 (−s)−1/4 exp − (−s)3/2 1 + O(−s)−3/2 for s → −∞. U0 (s) = √ Γ 3 2 π 3 For s > 0, U 1 (s) and U 2 (s) are introduced via the relations U1 (0) = 1,

U 1 (0) = 0;

U2 (0) = 0,

U 2 (0) = 1.

Combining (1.41) with the analogous asymptotics of the second relation, one finds

31/6 2 −1/4 2 3/2 π −3/2 ) , cos s − U1 (s) = √ Γ s + 0(s 3 3 12


45

3−1/6 1 −1/4 π 2 U2 (s) = √ Γ sin s3/2 + s + O(s−3/2 ) for s → +∞. 3 3 12 π Finally, the following coupling between both Airy and Bessel functions is detected 2 √ 2 sJ−1/3 s3/2 , U1 (s) = 3−1/3 Γ 3 3 1 √ 2 3/2 −2/3 Γ sJ1/3 s . U2 (s) = 3 3 3

2 REGULAR PERTURBATIONS OF PARAMETERS

A difference between real and idealized systems is very often reduced to perturbation of the input parameters. For instance, a thickness of a plate (or shell) is described via formula h = h 0 + εh(x, y) (h0 = const, ε 1); contour of the circle plate slightly differs from a circle via relation r(θ) = r 0 + ε cos nθ, etc. Although often the considered system does not follow Hook’s principle, but a difference is small. Non-linearity of many systems only slightly differs from linearity, and this system is said to be a quasi-linear one. The material of an object is weakly anisotropic, and so on. In all cited examples an influence of deviations (or perturbations) is small, and it can be estimated applying the method of regular perturbations. A being sought solution can be presented in the form of the following series f (x, ε) ∼

∞

δn (ε) fn (x) ,

n=0

where δn (ε) is the asymptotic sequence depends upon the small parameter ε.

2.1 Eigenvalue problems The following algebraic system of linear algebraic equations is considered ¯ (A0 + εA1 ) ¯x = b,

(2.1)

where: A0 and A1 are matrices of dimension n × n, and ¯x and b¯ are n-dimensional vectors. Assume that the problem (2.1) is solvable, and its solution has the form ¯x = ¯x0 + ε ¯x1 + +ε2 ¯x2 + ... .

(2.2)

Substitution (2.2) into (2.1) and comparison of terms standing by the same powers of ε give the following recurrence system ε0 :

¯ A0 ¯x0 = b;

ε1 :

A0 ¯x1 = −A1 ¯x0 ;

ε2 :

A0 ¯x2 = −A1 ¯x1 ;

48


.

.

. .

.

.

. .

.

.

The equation (2.1) is called the generative (unperturbed or zero order approximation) equation. If the problem has a solution, then there exists A −1 0 and the iterational process follows ¯ x1 = −A−1 A1 ¯x0 ; ¯x2 = −A−1 A1 ¯x1 ; ... . ¯x0 = A−1 0 b; ¯ 0 0 However, if det A 0 = 0, then the unperturbed equation (2.1) is solved but not for all ¯x [672]. In this case the point x0 = 0 belongs to a spectrum of the operator A0 (a linear problem is situated on the spectrum) and a construction of asymptotics becomes more difficult. A detailed description of this problem is addressed in references [672]. We consider now a longitudinal deformation of a rod with variable rigidity EF (Figure 2.1) governed by the equation d du EF =q (2.3) dx dx and with the attached boundary condition u = 0 for

x = 0, L.

(2.4)

Let EF = E 0 F0 + εE1 F1 , where E 0 F0 = const, ε ≤ 1. A solution to the boundary value problem (2.3), (2.4) is sought in the form u = u0 + εu1 + ε2 u2 + ... .

Fig. 2.1. A rod with variable rigidity.

Comparing terms standing by the same powers of ε the following recurrent system of equations and boundary condition is obtained: ε0 :

E0 F0

d 2 u0 = q; dx2

2.1 Eigenvalue problems

ε1 : .

d 2 u1 du0 d =− E 1 F1 ; dx dx dx2 . . . . . . . .

E0 F0 .

For x = 0, L : .

49

.

. .

.

.

ε0 :

u0 = 0;

ε1 :

u1 = 0;

. .

.

.

From these boundary value problems one easily obtains: u0 = q u1 = (E0 F0 )2

qx (x − L) ; E0 F0

  x L   x   E1 F1 (2x − L) dx ; ... .  E1 F1 (2x1 − L) dx −  L 0

0

A similar approach can be applied for partial differential equations. It is worth noticing, that the described procedure becomes more tedious when a corresponding linear problem lies on the spectrum (see [672]). Two main approximate approaches, i.e. asymptotical and successive approximation methods, are applied. Each method has its advantages and disadvantages, which can sometimes be exploited by working with a combination of the two. The most important of these differences can be summarized as follows: (a) Iteration can be started only if an appropriate initial approximation is knows. Series expansion is more automatic, because it can generate the basic approximation if one substitutes a series with the asymptotic sequence left unspecified. (b) Iteration eliminates the need to guess the asymptotic sequence. It is therefore safer than assuming an expansion, unless one leaves the asymptotic sequence unspecified. For example, it will often – though not always, in singular perturbation problems – produce logarithms in higher-order terms that are missed is a power series is assumed. (c) Beyond the second term the series expansion is more systematic because it produces only significant results, whereas iteration will, in nonlinear problems, generate some higher-order terms, to which no significance can be attached because others of equal order are missing. (d) Iteration will yield in a single step groups of terms of nearly the same order that require several steps in a series expansion [660,p.34,35]. It is worth noticing, that in an unperturbated boundary value problem yielded either by linear or non-linear equation, all of the higher order approximations are defined by linear equations with linear boundary conditions. In order to discuss the eigenvalue problems, we recall first briefly some necessary background from linear algebra and differential equations.

50


The matrix A is called self-adjoint if A = A ∗ , where A∗ is adjoint to A. Note that an adjoint matrix is obtained from the initial one via transposition and application of complex conjugancy. The self-conjugated matrices play a role similar to that of real numbers among all complex one since any arbitrary matrix may be presented in the form A = A 1 + iA2 , where A1 , A2 are the self-adjoint matrices. If A is self-adjoint matrix, then for any n-th order vectors ¯x, ¯y, the following relation holds (A ¯x, ¯y) = ( ¯x, A¯y) , (2.5) where (. . . , . . .) denotes the scalar product. The self-adjoint matrices have only real eigenvalues. The equation (2.5) is often used to verify if any operator is a self-adjoint one. Let us consider the following ordinary differential equation (a similar approach also holds for a partial differential equations): L (y) − λM (y) = 0,

(2.6)

where:

dn dn−1 + a + ... + a0 , n−1 dxn dxn−1 dm dm−1 M = bm m + bm−1 m−1 + ... + b0 , dx dx and the following homogeneous boundary conditions are attached L = an

Q1 (x)| x=a = 0;

Q2 (x)| x=b = 0.

(2.7)

Let us introduce a definition of testing function u(x) satisfying the boundary conditions (2.7). The equation (2.6) is said to be self-adjoint if b [uL (υ) − υL (u)] dx = 0, a

b [uM (υ) − υM (u)] dx = 0,

(2.8)

a

where u(x) and v(x) satisfy the given boundary conditions (2.7). The self-adjoint property of a linear differential operator may be verified via integrations by parts. As an example, a self-adjoincy of the boundary value problem governing a stability loss of a compressed rod will be further investigated. The following equation is analysed EIw xxxx + T wxx = 0 with the boundary conditions x = 0, l w = w xx = 0,

(2.9)


51

where: EI is the bending rod’s rigidity, whereas T denotes the compressing load. For the given case the following transformation holds l 0

d4 u d4υ u 4 − υ 4 dx = dx dx

l l 3 du d 3 υ dυ d3 u d 3 u d υ − dx = u 3 − υ 3 − dx dx3 dx dx3 dx dx 0 0

−

dυ d 2 u du d 2 υ − 2 dx dx dx dx2

l l 2 2 d u d υ d2 υ d2 u + − dx = 0, dx2 dx2 dx2 dx2 0 0

and hence, the relations (2.8) are satisfied. In a similar way, one may show that also the operator d 2 /dx2 is self-adjoint. Now the following eigenvalue problem for a system of algebraic equations possessing small parameter is considered (A0 + εA1 − λE) ¯x = 0,

(2.10)

where E is the unity matrix. It is assumed that A 0 is self-adjoint matrix (the selfadjointy property of the matrix A 0 + εA1 is not required). To solve the eigenvalue problem the following series are introduced ¯x = ¯x0 + ε ¯x1 + ε2 ¯x2 + ...; λ = λ0 + ελ1 + ε2 λ2 + ... .

(2.11)

Assuming that the eigenvalue problem does not lie on the spectrum, the series (2.11) is substituted into the system (2.10) to get ε0 : ε1 : .

.

(A0 − λ0 E) ¯x0 = 0,

(2.12)

(A0 − λ0 E) ¯x1 = λ1 ¯x0 − A1 ¯x0 ,

(2.13)

. .

.

.

. .

.

.

Let the homogeneous problem has n eigenvalues λ (i) 0 and corresponding eigen(i) vectors ¯x0 . First, assume that all eigenvalues are different, and we are going to find first correcting term for the eigenvalue λ (i) 0 . Since det(A0 − λ0 E) = 0, the nonhomogeneous equation (2.13) will have a solution when its right-hand side is orthogonal to the eigenvectors of its left hand-side. Multiplying all terms of equation (2.13) scalarly by ¯x(i) 0 we get

(i) 2 (i) x1 = λ1 ¯x(i) − A1 ¯x(i) x(i) ¯x(i) 0 , A0 − λ0 E ¯ 0 0 , ¯ 0 .

52


Since the matrix (A 0 − λ0 ¯x(i) 0 E) is self-adjoint, hence

(i) (i) (i) (i) x1 = ¯x1 , A1 − λ(i) x0 = 0. ¯x(i) 0 , A0 − λ0 E ¯ 0 E ¯

The being sought correcting term reads (i) 2 (i) (i) λ(i) = A ¯ x , ¯ x x0 . 1 1 0 0 / ¯ Consider now the differential equations M0 (y) + εM1 (y) = λ N0 (y) + εN1 (y)

(2.14)

with the attached boundary conditions Q¯1 (y) = 0 for y = a, Q¯2 (y) = 0 for y = b.

(2.15)

Here M0 , M1 , N0 and N1 denote the differential operators, and the orders of operators M1 , N1 is not higher than orders of the operators M 0 , N0 . Besides, we assume that the homogeneous boundary value problem M 0 (y) = λN0 (y) is selfadjoint for the boundary conditions (2.15) (the non-homogeneous problem does not require the self-adjoint property). The following series are applied (n) (n) λ(n) = λ(n) 0 + ελ1 + ελ2 + ... , (n) (n) y(n) = y(n) 0 + εy1 + y2 + ... .

Assume that λ(n) 0 is the simple eigenvalue. Equating terms standing by the same powers of ε we obtain (n) ε0 : M0 y0(n) = λ(n) N y0 , 0 0 = 0 for y = a, Q¯1 y(n) 0 (n) Q¯2 y0 = 0 for y = b,

(2.16)

(n) ε1 : M0 y(n) − λ(n) = 1 0 N0 y1 (n) (n) (n) + λ(n) −M1 y0 + λ(n) 0 N1 y0 1 N0 y0 , = 0 for y = a, Q¯1 y(n) 1 (n) Q¯2 y1 = 0 for y = b,

(2.17)

.

.

. .

.

.

. .

.

.


53

The self-adjoint property of homogeneous problem is used in a way similar to the linear algebraic equations. Multiplying (2.17) by y (n) 0 and integrating on the whole space (i.e., introducing the scalar product by y (n) ), one obtains 0 b

(n) (n) M0 y(n) − λ(n) y0 dx = 1 0 N0 y1

b

b

M0 y0(n) − λ0(n) N0 y0(n) y(n) 1 dx = 0,

a

and the following correcting term to λ (n) 0 is found (n) (n) (n) (n) M1 y0 , y0 − λ(n) N1 y0 , y0 0 (n) (n) . λ1 = N y0 , y0 Further correction terms may be found similarly. Additionally, one more example will be considered, where an influence of the density of a vertical rod on its buckling load during the axial compression are investigated (Figure 2.2). The problem is reduced to the following ordinary differential equation d d2 w d4 w dw + T 2 = 0, EI 4 + ρgF x dx dx dx dx and the boundary conditions (2.9) are applied. Introducing the non-dimensional parameters ξ= one obtains

T l2 πx ¯ ρgFl3 , T= , ε= , 2 l EIπ EIπ3

2 d4 w d dw ¯ d w = 0, + ε ξ + T dξ dξ dξ4 dξ2 w=

d2 w =0 dξ2

f or ξ = 0, π.

(2.18)

For practically used rods ε 1, hence the solution of equation (2.18) has the form w = w0 + εw1 + ε2 w2 + ..., T¯ = T¯0 + εT¯1 + ε2 T¯2 + ... . In the first approximation the Euler’s stability problem of longitudinally compressed rod is obtained d4 w0 ¯ d2 w0 + T 0 2 = 0; dξ4 dξ

54


EI, rg

Fig. 2.2. The vertical compressed rod.

w0 =

d2 w0 = 0 for ξ = 0, π. dξ2

The first eigenvalue (buckling load) T 0 = 1, and the corresponding buckling form w0 = A sin ξ0 . From the first approximation equation d2 w0 d dw0 d4 w1 ¯ d2 w1 + T = − (2.19) ξ + T¯1 2 0 4 2 dξ dξ dx dξ dξ one obtains T¯1 = −0.5, and then T¯ ≈ 1 − 0.5ε. Observe that this problem is also addressed in [231], where in the analysed equation before the underlined term (representing influence of the rod’s mass) instead of ‘+’ the sign ‘-’ is taken. In the result T ≈ 1 + 0.5ε, i.e. an influence of a specific mass yields an increase of the critical stresses, which is in contradiction to elementary physical imagination. Consider now more complicated problems of multiple eigenvalues. First, the system of algebraic equations is considered. Assume that (A0 − λ0 E) ¯x = 0 has the eigenvalue λ 0 of multiplicity k, with the associated mutually orthogonal eigenvectors ¯e1 , ¯e2 , . . . , ¯ek . For a non-homogeneous problem the so called splitting of multiple eigenvalue occurs. Indeed, let for instance A 0 = E and let A1 be a self-adjoint matrix with the different eigenvalues λ (1) , λ(2) , . . . , λ(n) . Then the unperturbed problem possesses the eigenvalue equal to 1 with n multiplicity, whereas the perturbed problem has n different values 1 + λ (1) ε, 1 + λ(2) ε, . . . , 1 + λ(n) ε. The perturbations of the eigenvalue and the associated eigenvector are: 2 (i) λ(i) = λ0 + ελ(i) 1 + ε λ2 + ...,


55

¯x(i) = ¯x0(i) + ε ¯x1(i) + ε2 ¯x2(i) + ..., i = 1, 2, ..., k. Note that a difference between simple and multiple eigenvalues is evident, since x(i) e1 , . . . , ¯ek , it is not clear how to find ¯x(1) 0 . One may expect that ¯ 0 is a combination of ¯ but this combination is not a priori known. In words, contrary to the case of simple eigenvalue now ¯x(1) 0 should be also defined. Comparing the terms standing by ε the following equation is obtained A1 ¯x(i) x1(i) = λ1 ¯x1(i) − λ(i) x0(i) , 0 + A0 ¯ 1 ¯

(2.20)

ek , i.e. where ¯x(i) 1 is the linear combination of the eigenvectors ¯ ¯x0(i) = a1 ¯e1 + a2 ¯e2 + ... + ak ¯ek .

(2.21)

x(i) We are going to find λ (1) 1 and the vector ¯ 0 . Multiplying scalar both sides of (2.20) by ¯er , one obtains er + A0 ¯x1(i) , ¯er = λ0 ¯x1(i) , ¯er + λ1(1) ¯x(i) er . A1 ¯x(i) (2.22) 0 ,¯ 0 ,¯ Since

A0 ¯x1(i) , ¯er = ¯x1(i) , A0 ¯er = λ0 ¯x0(i) , ¯er ,

the equation (2.22) can be rewritten in the form er . A1 ¯x0(i) , ¯er = λ1(1) ¯x(i) 0 ,¯

(2.23)

Now, instead of ¯xi0 we take (2.21) and noting that ( ¯x(i) er ) = ar the following 0 ,¯ equation is obtained k brp a p = λ1(i) ar , (2.24) p=1

where:

brp = A1 ¯ep , ¯er .

&& && & & It is clear now that λ(i) 1 are the eigenvalues of the matrix B = brp , r, p = 1, 2, . . . , k, defined by the equation det(B − λ1 E) = 0. The vector ¯x(i) 0 is defined by the formula (2.21), where a n are found from the equations (2.24). (i) The corrected terms of the eigenvectors ¯x(i) 1 and the eigenvalues λ 2 are found in a similarly way. Degenerate multiple roots can lead to an expansion in non-integral powers of ε [340,p.18]. Consider the n-degenerate eigensolution in the Jordan Normal Form A0 ¯x(1) = λ0 ¯x(1) ;

56


A0 ¯x(2) = λ0 ¯x(2) + C2 ¯x(1) ; .

.

. .

.

.

. .

.

.

A0 ¯x(n) = λ0 ¯x(n) + Cn ¯x(n−1) . Then if the perturbation has a component in the direction ¯xn , say εbn , an expression is needed in power of ε 1/n , i.e. ¯x = ¯x(1) + ε1/n α2 ¯x(2) + ε2/n α3 ¯x(3) + . . . + ε(n−1)/n αn ¯x(n) + . . . , λ = λ0 + ε1/n λ1 + . . . with solution α2 = λ1 /c2 , α3 = λ21 /c2 c3 , . . . , αn = λn−1 1 /c2 c3 . . . cn , λ1 = (c2 c3 . . . cn bn )1/n . If the components of εA 1 ( ¯x) vanish in the directions of ¯x(k+1) , ¯x(k+2) , . . ., ¯x(n) , then an expression in powers of ε 1/k is needed. Analogously, the case of multiple eigenvalues while considering the differential equations may be analyzed. Consider the following unperturbed self-adjoint boundary value problem M0 (y0 ) + λ0 N0 (y0 ) = 0, Q1 (y0 )| x=a = 0,

Q2 (y0 )| x=b = 0,

where: λ0 is k-order multiplicity eigenvalue with k associated eigen-functions ϕ1 , ϕ2 , . . . , ϕk . In addition, we assume that the eigenfunction are orthonormalized, i.e. b ϕr N0 ϕ p dx = 0, p r, a

b ϕr N (ϕr ) dx = 1.

(2.25)

a

The small perturbation yields splitting of eigenvalue λ 0 . The associated with zero order approximation eigenvectors are sought in the form y(0) =

k

cr ϕr

p = 1, 2, ..., k.

(2.26)

i=1

The first order relations read approximation (r) (r) (r) M0 y(r) = −M1 y(r) 1 − λ0 N0 y0 0 + λ1 N0 y0 + λ0 N1 y0 , Q1 y(r) 1 x=a = 0,

Q2 y(r) 1 x=b = 0.

(2.27)


57

Multiplying (2.27) successively by ϕ p p = 1, 2, . . . , k, and using self-adjoint property of the unperturbed operator as well as the property of the generalized orthonormality (2.25), the following system of linear algebraic equations is obtained     k  b  k        M1  c p ϕ p  ϕr − λ0 N1  c p ϕ p  ϕr −     p=1 p=1 a

 k         λ1 N0  c p ϕ p  ϕr  dx = 0, r = 1, 2, ..., k,    p=1 which yields λ(r) 1 and cr . Now the problem of a rod on the elastic foundation buckling is analysed, which is governed by the equation wIV + (4 + ε) w + λw = 0, w (0) = w (0) = w (π) = w (π) = 0. The smallest eigenvalue λ 0 = 5 of the unperturbed problem is twicely degenerated, and two functions sin x and sin 2x are associated with λ 0 . First order approximation equation yields w1IV + 4w1 + λ0 w1 + w0 + λ1 w0 = 0. Substituting w0 = a1 sin x + a2 sin 2x, we get a1 − a1 λ1 = 0;

a2 − 4a2 λ1 = 0,

and hence λ 1 = 1 and λ1 = 0.25. In words, in the presented approach first the eigenvalues are found and then the eigenvectors on each of the computational steps. However, there exists another approach [302]. The latter one is illustrated using an example of equation governing longitudinal vibrations of the rod with variable rigidity: du d E0 + εE1 (x) + ρλu = 0, dx dx with the attached boundary conditions (2.4), where: ρ ≡ const, E 0 ≡ const, ε 1. Zero order approximation relations yields E0

d 2 u0 + ρλ0 u0 = 0, dx2

for x = 0, L u0 = 0. The input equation is rewritten by adding and substracting of the term ρλ 0 u

58


E0

du d2 u d (x) − ρλ u + ε E = −ρ (λ − λ0 ) u. 0 1 dx dx dx2

(2.28)

Both sides of (2.28) are multiplied by u 0 and then integrated along x from 0 to L. Taking into account the self-adjoint property of the operator E 0 (d 2 /dx2 ) − λ0 ρ the following formula is obtained L −ρ (λ − λ0 ) = ε 0

 L −1

  du d   E1 (x) u0 dx  uu0 dx .   dx dx

(2.29)

0

The eigenfunction u is normalized in the following way L uu0 dx = 1. 0

get

Next, the eigenvalue λ is eliminated by the equation (2.29), and from (2.28) we

L du du d2 u d d E0 2 − ρλ0 u + ε E1 (x) = εu E1 (x) udx. dx dx dx dx dx

(2.30)

0

The obtained nonlinear equation with boundary conditions (2.4) can be solved using either perturbation series or successive approximations. Knowing the eigenvector the corresponding eigenvalue can be found from the formula (2.29).

2.2 Stability of oval cylindrical shell uniformly loaded by external pressure The mentioned problem can be reduced to integration of differential equation of inextensional theory (see Chapter 4.3) ∂8 w q ∂6 w Eh ∂4 w + + = 0, (2.31) ∂y8 R (y) ∂y6 R2 (y) ∂x4 where: R (y) = R0 1 + (µ cos(4πy/L1), D = Eh 12 1 − v2 ; q is the external normal pressure; L1 is the parameter of transversal cross section; µ is the eccentricity parameter (0 ≤ µ ≤ 1). The following boundary conditions are applied D

w=

∂2 w = 0 for x = 0, L. ∂x2

Taking w = f (y) sin(πx/L), the following eight order differential equation is obtained with the variable coefficients with respect to y:

2.2 Stability of oval cylindrical shell uniformly loaded by external pressure

59

d8 f 1 h2 q R0 d6 f 6+ ' ( 8 − 2 4πy E h 1 + µ cos 12 1 − v dy dy L1

2 1 4πy π 4 f = 0. 1 + µ cos L1 L R20

(2.32)

For µ 1 the coefficients of the equation (2.32) only slightly differ from the coefficients of the differential equations governing a buckling of a circular cylinder. Since a solution to the latter differential equation is known, the perturbation technique may be used successfully to solve the input problem. The unperturbed problem (µ = 0) is governed by the following equation d8 f0 q0 R0 d6 f0 1 π 4 h2 − + f0 = 0. ' ( E h dy6 12 1 − ν2 dy8 R20 L For v = 0.3 we obtain 5 R0 R0 ER0 h 2 y 2 q0 = 0.92 . , f0 = cos n , n = 2π L R0 R0 h L

(2.33)

(2.34)

One may check that the unperturbed problem is self-adjoint. The being sought solution of the input problem is developed into the series of small parameter µ f = f0 + µ f1 + µ2 f2 + . . . , (2.35) q = q0 + rq1 + r2 q2 + . . . .

(2.36)

The following recurrent system of equations is obtained M ∗ ( f0 ) = q0 N ∗ ( f0 ) , M ∗ ( f1 ) + M¯ ( f0 ) = q1 N ∗ ( f0 ) + q0 N ∗ ( f1 ) + N¯ ( f0 ) , .

.

where: M∗ ( f ) =

. .

.

.

. .

.

.

d8 f h2 1 π 4 + f, ' ( 12 1 − v2 dy8 R20 L

1 R0 d 6 f , E h dy6 4 ¯ f ) = 2 cos 4πy π f, M( L1 L R20 N∗ ( f ) =

¯ f ) = −N ∗ ( f ) cos 4πy . N( L1 Applying the self-adjoint property of the unperturbed boundary problem the eigenvalues q 1 are found from the formula:

60


2π q1 =

f0 M¯ ( f0 ) − q0 N¯ ( f0 ) dy

0

2π

. f0

N∗

( f0 ) dy

0

Now the intervals of application of the formula (2.36) are estimated. The comparison of the results obtained using (2.36) and numerically with the help of double trigonometric series taken from the reference [699] is carried out for the following values of the parameters: h = 0.00124 m; L = 0.254 m; v = 0.3; L 1 = 0.494 m.

Fig. 2.3. An estimation of accuracy of the perturbation method.

The results of comparison are reported in Figure 2.3, where Q = q ∗ /q; q∗ corresponds to the values given in reference [699], and q are computed using formula (2.36). The reasonably good results agreement is observed in the region 0 ≤ µ ≤ 0.25. For example, for µ = 0.25 the critical load obtained via the formula (2.36) is of 12.5% higher, than that found in reference [699], but already for µ = 0.15 this error achieves 2.5%.

2.3 Stability of the cantilever beam In the stability theory the Ishlinskii-Leibenzon method is widely used to analyse constructions with free edges [353, 424]. The idea is focused on that the in the equilibrium equations the parametric terms are neglected keeping them only in the boundary conditions. The problem of the physical motivation of those simplifications is still under considerations in the literature. One may also construct the corresponding simplifications formally, introducing into the input equations the parameter ε standing by the parametric terms, which firstly is assumed to be small and finally is taken as ε = 1. In other words, we are going to apply formally a small parameter ε.

2.3 Stability of the cantilever beam

61

Fig. 2.4. The compressed cantilever beam.

A stability of the cantilever beam is analysed (Figure 2.4). The governing buckling equation and the boundary conditions are EIw xxxx + εT wxx = 0,

(2.37)

w (0) = w x (0) = 0,

(2.38)

w xx (L) = 0,

EIw xxx (L) + T wx (L) = 0.

(2.39)

The functions w and T are sought in the form of the series w = w0 + w1 ε + w2 ε2 + ..., T = T 0 + T 1 ε + T 2 ε2 + ... .

(2.40)

The splitting with respect to ε after substituting (2.40) into (2.37)–(2.39) gives ε0 :

EIw0xxxx = 0, w0 = w0x = 0 for x = 0, w0xx = 0, EIw0xxx + T 0 w0x = 0 for x = L;

ε1 :

EIw1xxxx + T 0 w0xx = 0, w1 = w1x = 0 for x = 0, EIw1xxx + T 0 w1x + T 1 w0x = 0 for x = L;

ε2 :

EIw2xxxx + T 0 w1xx + T 1 w0xx = 0, w2 = w2x = 0 for x = 0, w2xx = 0,

EIw2xxx + T 0 w2x + T 1 w1x + T 2 w0x = 0 for x = L.

62


For the zero order approximation we get EIw0xxxx = 0, w0 (L) = 0, and hence

w0 (0) = w0xx (0) = 0,

EIw0xxx (L) + T 0 w0x (L) = 0

T 0 = 2EI/L2 ,

w0 = Ax2 (x − 3L) .

For the first order approximation EIw1xxxx + T 0 w0xx = 0, w1 (0) = w1x (0) = 0, w1xx (L) = 0, EIw1xxx (L) + T 0 w1x (L) + T 1 w0x (L) = 0, and hence

T 1 = EI/ 3L2 ,

x5 x4 2 3 − w1 = A −5x L + x + . 2L 10L2 From the equations and boundary conditions of the second order EIw2xxxx + T 0 w1xx + T 1 w0xx = 0, w2 (0) = w2x (0) = 0, w2xx (L) = 0, EIw2xxx (L) + T 0 w2x (L) + T 1 w1x (L) + T 2 w0x (L) = 0 one obtains

T 2 = 4EI/ 45L2 .

Finally we get T=

4 2 1 EI ε + ε + ... . 2 + 3 45 L2

For ε = 1 we obtain T = 2.4020EI/L 2. Note that exact solution is T = 2.4674EI/L 2 (error of the approximate solution is only 1.83%).

2.4 Adjoint operators method Let us consider a self-adjoined eigenvalue problem of the form: (A + εB)ϕ = λϕ, where: λ, ϕ denotes a being sought eigenvalue and an eigenfunction of the problem, correspondingly.

2.4 Adjoint operators method

63

Usually eigenvalues and eigenfunctions are obtained step by step. On the other hand, knowledge of the n-th eigenfunction allows to define 2n + 1 eigenvalues [462, 463, 464, 465, 466]. Let us suppose ϕ = ϕ0 + εϕ1 + ε2 ϕ2 + . . . , λ = λ0 + ελ1 + ε2 λ2 + . . . . After splitting in respect to ε we have the following set of perturbation equations Aϕ0 = λ0 ϕ0 ,

(2.41)

Aϕ1 − λ0 ϕ1 = −Bϕ0 + λ1 ϕ0 ,

(2.42)

Aϕ2 − λ0 ϕ2 = −Bϕ1 + λ1 ϕ1 + λ2 ϕ0 ,

(2.43)

Aϕ3 − λ0 ϕ3 = −Bϕ2 + λ1 ϕ2 + λ2 ϕ1 + λ3 ϕ0 ,

(2.44)

.

.

. .

.

.

. .

.

.

From equations (2.41), (2.42) we define λ 0 , λ1 and ϕ0 , ϕ1 . Then λ2 is defined by equation (2.43): ((B − λ1 )ϕ1 , ϕ0 ) . λ2 = (ϕ0 , ϕ0 ) Now, multiplying scalarly (2.44) by ϕ 0 we obtain λ3 =

((B − λ1 )ϕ2 , ϕ0 ) − λ2 (ϕ1 , ϕ0 ) . (ϕ0 , ϕ0 )

(2.45)

Eigenfunction ϕ 2 is not known. Let us modify the first term of the numerator of the fraction (2.45) in the following way: ((B − λ1 )ϕ2 , ϕ0 ) = ((B − λ1 )ϕ0 , ϕ2 ). From the equation (2.42) we get (B − λ1 )ϕ0 = −(A − λ0 )ϕ1 , and therefore ((B − λ1 )ϕ0 , ϕ2 ) = −((A − λ0 )ϕ1 , ϕ2 ) = −((A − λ0 )ϕ2 , ϕ1 ) = −((−Bϕ1 + λ1 ϕ1 + λ2 ϕ0 ), ϕ1 ) = ((B − λ1 )ϕ1 , ϕ1 ) − λ2 (ϕ0 , ϕ1 ). Finally one obtains λ3 =

((B − λ1 )ϕ1 , ϕ1 ) − 2λ2 (ϕ1 , ϕ1 ) . (ϕ0 , ϕ0 )

Further the method can be applied in similar way. Having obtained the values of λ0 , λ1 , . . . , λn and ϕ0 , ϕ1 , . . . , ϕn one can also find λ n+1 , . . . , λ2n+1 .

64


Assuming that the input eigenproblem has the form Aϕ = λ(N + εM)ϕ. After substituting (2.42) and splitting due to ε one gets Aϕ0 = λ0 Nϕ0 ,

(2.46)

Aϕ1 = λ0 Nϕ1 + λ1 Nϕ0 + λ0 Mϕ0 ,

(2.47)

Aϕ2 = λ0 Nϕ2 + λ1 Nϕ1 + λ2 Nϕ0 + λ0 Mϕ1 + λ1 Mϕ0 ,

(2.48)

Aϕ3 = λ0 Nϕ3 + λ1 Nϕ2 + λ2 Nϕ1 + λ3 Nϕ0 + λ0 Mϕ2 + λ1 Mϕ1 + λ2 Mϕ2 , .

.

. .

.

.

. .

.

(2.49)

.

In an usual way one can find λ 0 , λ1 , λ2 as well as ϕ0 , ϕ1 . The expression for λ 3 includes a being unknown function ϕ 2 λ3 =

((λ1 N + λ0 M)ϕ2 , ϕ0 ) + ((λ2 N + λ1 M)ϕ1 , ϕ0 ) + λ2 (Mϕ0 , ϕ0 ) . (Nϕ0 , ϕ0 )

(2.50)

We note that ((λ1 N + λ0 M)ϕ2 , ϕ0 ) = ((λ1 N + λ0 M)ϕ0 , ϕ2 ),

(2.51)

whereas from equation (2.47) we have (λ1 N + λ0 M)ϕ0 = (A − λ0 N)ϕ1 .

(2.52)

Taking into account (2.52), the relation (2.51) can be presented in the form ((λ1 N + λ0 M)ϕ0 , ϕ2 ) = ((A − λ0 N)ϕ1 , ϕ2 ) = ((A − λ0 N)ϕ2 , ϕ1 ). Now from equation (2.48) one gets ((A − λ0 N)ϕ2 , ϕ1 ) = ((λ1 Nϕ1 + λ2 Nϕ0 + λ0 Mϕ1 + λ1 Mϕ0 ), ϕ1 ) = ((λ2 N + λ1 M)ϕ0 , ϕ1 ) + ((λ1 N + λ0 M)ϕ1 , ϕ1 ) = ((λ2 N + λ1 M)ϕ1 , ϕ0 ) + ((λ1 N + λ0 M)ϕ1 , ϕ1 ), and finally from relations (2.50) and (2.53) we obtain λ3 =

2((λ2 N + λ1 M)ϕ1 , ϕ0 ) + ((λ1 N + λ0 M)ϕ1 , ϕ1 ) + λ2 (Mϕ0 , ϕ0 ) . (Nϕ0 , ϕ0 )

Then, the procedure can be extended.

(2.53)

2.5 Transformation of coordinates and variables

65

2.5 Transformation of coordinates and variables Now examples of non-dimensionalising procedures, introduction of perturbation parameters as well as effective transformation of coordinates and variables will be given. The equations of axisymmetric vibrations of an orthotropic cylindrical shell in projections on the axes of an undeformed coordinate system have the form ∂2 u ∂M ∂ ¯ ¯ T 1 cos θ − sin θ − ρ 2 = 0, ∂x ∂x ∂t T2 ∂2 w ∂M ∂ cos θ¯ + + ρ 2 = 0. (2.54) T 1 sin θ¯ + ∂x ∂x R ∂t Here ρ is the mass per unit area. We will write the geometric and elasticity relations in the form ε¯1 =

2 ∂u 1 ∂w + , ∂x 2 ∂x

ε¯2 =

T 1 = B1 ε¯1 ,

w , R

∂w θ¯ = arcsin , ∂x

T 2 = B2 ε¯2 ,

κ¯ =

∂θ¯ , ∂x

M = D¯ κ.

Here, for the sake of simplicity, it is assumed that the Poisson ratio is equal to zero. We will introduce the notation α 1 = D/(B1R2 ), α2 = B2 /B1, α3 = H0 /(B1R2 ) (H0 is the initial energy level). In the above the parameter α 1 characterizes a ratio of reduced bending membrane rigidity to a membrane rigidity; α 2 is a ratio of a rigidities in various directions; α 3 describes a degree of nonlinearity. Now we examine affine transformations leads to various limiting systems: 1.) ρ 2 R τ. u = α3 RU, w = α3 RW, t = D The transformed system has the form ∂κ ∂2 U ∂ sin θ − α1 2 = 0, α ε cos θ − α−1 1 3 ∂ξ ∂ξ ∂τ ∂ ∂κ ∂2 W cos θ + α2 W + α1 2 = 0. α ε sin θ + α−1 1 3 ∂ξ ∂ξ ∂τ Here

2 ∂W ∂U 1 + α3 ε= , ∂ξ 2 ∂ξ

∂W θ = arcsin α3 , ∂ξ

κ=

∂θ . ∂ξ

(2.55)

In the case α2 → 0, the limiting system describes nonlinear vibrations of a rod. When α3 → 0, it describes linear axisymmetric vibrations of a cylindrical shell. With α1 → 0, α2 → 0, α3 ∼ 1, we arrive at the system

66


∂ε0 ∂θ0 cos θ0 − ε0 sin θ0 = 0, ∂ξ ∂ξ ∂ε0 ∂θ0 sin θ0 + ε0 cos θ0 = 0. ∂ξ ∂ξ Hence ε0

(2.56)

∂θ0 = 0, ∂ξ

from which it follows that ε0 =

2 ∂W0 ∂U0 1 + α3 = 0. ∂ξ 2 ∂ξ

(2.57)

We will represent the sought functions with respect to the small parameter α 1 ε = ε0 + α1 ε1 + α1 ε2 + . . . , etc. and suppose α2 ∼ α21 . Let us derive the equation of the first approximation (remember ε0 = 0): ∂ ∂κ0 ∂2 U 0 sin θ = 0, ε1 cos θ0 − α−1 0 − 3 ∂ξ ∂ξ ∂τ2 ∂ ∂κ0 ∂2 W0 cos θ = 0. (2.58) ε1 sin θ0 − α−1 0 − 3 ∂ξ ∂ξ ∂τ2 Excluding the function ε 1 from (2.58), we obtain α−1 3

∂κ0 ∂θ0 ∂U0 ∂2 W0 − 2 cos θ0 − sin θ0 = ∂ξ ∂ξ ∂τ ∂τ2

 −1  ∂θ0  ∂2 U 0 ∂  −1 ∂2 κ0 ∂2 W0 + cos θ0 + sin θ0 −  α3  . ∂ξ ∂ξ ∂ξ2 ∂τ2 ∂τ2

(2.59)

Equations (2.57) and (2.59) constitute the limiting system corresponding to nonlinear vibrations of a flexible rod with its edges free in the axial direction. This system was obtained by using equations of the first approximation, which is an interesting feature of this case. 2.) ρ 2 1/2 1/2 R τ. u = α1 α3 RU, w = α3 RW, x = α1 Rξ, t = D The transformed system has the form ∂2 U ∂ 1/2 −1 ∂κ sin θ − α1/2 = 0, ε cos θ − α1 α3 1 ∂ξ ∂ξ ∂τ2 ∂ ∂2 W 1/2 −1 ∂κ sin θ + α2 W + = 0. ε sin θ + α1 α3 ∂ξ ∂ξ ∂τ2

(2.60)


67

Here 2 ∂U α−1 1 α3 ∂W + ε= , ∂ξ 2 ∂ξ

∂θ κ= , ∂ξ

∂W 1/2 θ = arcsin α1 α3 . ∂ξ

With α1 → 0, α3 ∼ α1 , α2 ∼ 1, we obtain ∂ε0 = 0, ∂ξ

∂4 W0 ∂2 W0 ∂2 W0 + ε + α W + = 0. 0 2 0 ∂ξ4 ∂ξ2 ∂τ2

(2.61)

The problem is linearized if constant axial forces are applied to the ends, and nonlinear effects can be revealed in the equations of the first approximation. Let the edge of a shell be fixed so as to prevent displacements in the axial direction. Then, the nonlinear effects in system (2.60) are preserved, since it follows from the first equation that ε0 =

α−1 1 α3 2l

l 0

∂W0 ∂ξ

2 dξ,

where l = L/R, L is the shell length. At the same time, with simply supported ends the variables can be separated: w = wm sin

mπx βm (τ), l

m = 1, 2, . . . .

We obtain for the time function an ordinary differential equation which has the following form 1 B2 Dm4 π4 B 1 m4 π 4 2 3 β¨ m + βm + + wm βm = 0. (2.62) 2 4 ρ R l 4ρl4 The equation (2.62) is solved in elliptic functions. For example, with β m (0) = 1, βm (0) = 0, we have βm = sn(Km t, S m ), (2.63) where −1 B1 w2m B1 w2m M 2 π2 D3/2 1/2 B1 wm 2 A , Sm = , Km = 1+ 1+ A 2A A 4l2 ρ1/2 B2 l4 A = 4D 1 + , DR2 m2 π4 sn(. . . , . . .) is the elliptic sine [5]. Thus, the limiting system (2.62) has exact solutions corresponding to normal vibration modes [656, 681]. 2 1/2 3.) 5 ρR u = α3 RU, w = α3 RW, x = Rξ, t = τ. B1

68


The transformed equations of motion have the form ∂2 U ∂ −1 ∂κ sin θ − 2 = 0, ε cos θ − α1 α3 ∂ξ ∂ξ ∂τ ∂ ∂κ ∂2 W cos θ + α2 W + = 0, ε sin θ + α1 α−1 3 ∂ξ ∂ξ ∂τ2 where ε, x, θ are defined by (2.55). With α1 → 0, α2 ∼ 1 we arrive at the limiting system ∂2 U ∂ (ε cos θ) − 2 = 0, ∂ξ ∂τ

∂ ∂2 W (ε sin θ) + α2 W + = 0. ∂ξ ∂τ2

(2.64)

Now let us examine plane nonlinear vibrations of a circular ring (a cylindrical shell of infinite length). We will write the equations of motion in a local coordinate system, the axes of which correspond to the tangential and radial directions at a point on the undeformed axis of the ring: ¯ 1 ∂M ∂2 v ∂T − − ρ1 2 = 0, ∂y E ∂y ∂t where

∂M ¯ sin θ, T¯ = T cos θ¯ − ∂y

∂2 w ∂2 M¯ 1 ¯ + T + ρ1 2 = 0, 2 R ∂y ∂t

(2.65)

¯ ∂M ∂M ¯ = T sin θ¯ + cos θ, ∂y ∂y

and ρ1 is the density. We will write the physical and geometric relations in the form T = EF ¯ε,

2 dv w 1 ∂w v ∂θ¯ + + − M = EI¯ κ, κ¯ = , ¯ε = , ∂y dy R 2 ∂y R ∂w v − θ = arcsin . ∂y R

Let α1 = I/(FR2 ), α3 = H0 /(EFR). Subjecting the variables v, w, y, and t to affine transformations, we obtain the following classes of nonequivalent systems: v = α3 RV,

w = α3 RW,

y = Rη,

t=

ρ 1/2 R2 τ. EI

The transformed equations have the form ∂κ ∂2 V ∂ −1 ∂κ sin θ cos θ − α − ε sin θ − α α = 0, ε cos θ − α1 α−1 1 1 3 3 ∂η ∂η ∂η ∂τ2 ∂ ∂κ ∂2 W −1 ∂κ cos θ + ε cos θ − α1 α−1 sin θ + α1 2 = 0. ε sin θ + α1 α3 3 ∂η ∂η ∂η ∂τ Here


ε=

2

∂W 1 ∂V + W + α3 −V , ∂τ 2 ∂η ∂W θ = arcsin −V . ∂η

κ=

69

∂θ , ∂η (2.66)

With α3 → 0, we obtain the limiting system corresponding to the linear theory. The second limiting system is obtained as α 1 → 0: ∂ (ε0 cos θ) − ε0 sin θ = 0, ∂η

∂ (ε0 sin θ) + ε0 cos θ = 0. ∂η

(2.67)

The condition of inextensibility follows from (2.67): 2 ∂V0 ∂W0 1 + W0 + α3 − V0 = 0. ∂η 2 ∂η

(2.68)

We will write out the equations of the first approximation: ∂ −1 ∂κ0 sin θ0 − ε1 sin θ0 − ε1 cos θ0 − α3 ∂η ∂η ∂ 2 V0 ∂κ0 cos θ0 − = 0, ∂η ∂τ2 ∂κ0 ∂ cos θ ε1 sin θ0 + α−1 0 + ε1 cos θ0 − 3 ∂η ∂η α−1 3

α−1 3

∂κ0 ∂2 W0 sin θ0 + = 0. ∂η ∂τ2

(2.69)

After making the obvious transformations, we may exclude the functions ε 1 from equations (2.69). Considering condition (2.68), we obtain the following relations connecting the functions V 0 and W0 : 2 ∂V0 ∂W0 1 + W0 + α3 − V0 = 0, ∂η 2 ∂η ∂κ0 ∂2 W0 ∂θ0 ∂ 2 V0 cos θ − sin θ0 = α−1 1 + + 0 3 ∂η ∂η ∂τ2 ∂τ2  −1  ∂  −1 ∂2 κ0 ∂2 V0 ∂2 W0 ∂θ0  −  α3 + sin θ0 + cos θ0 1 +  . ∂η ∂η ∂η2 ∂η2 ∂η2

(2.70)

The nonlinear system (2.70) is complex, so it is natural to take the following for the limiting linearized system (α 1 → 0, α3 → 0) ∂V0 + W0 = 0, ∂η

∂ 6 V0 ∂ 4 V0 ∂ 2 V0 ∂ 2 V0 ∂ 4 V0 + 2 + − + = 0, ∂η6 ∂η4 ∂η2 ∂τ2 ∂τ2 ∂η2

(2.71)

70


and to evaluate the nonlinear effects by using the equations of the first approximation W = W0 + α3 W1 + α23 W2 + . . . , V = V0 + α3 V1 + α23 V2 + . . . , 2 1 ∂W0 ∂V1 + W1 = − − V0 , ∂η 2 ∂η ∂ 4 V1 ∂ 2 V1 ∂ 2 V1 ∂ 4 V1 ∂ 6 V1 + 2 + − + = ∂η6 ∂η4 ∂η2 ∂τ2 ∂τ2 ∂η2 ∂ ∂2 κ0 ∂θ0 ∂2 W0 ∂θ0 ∂2 V0 ∂κ0 ∂θ0 ∂2 W0 − + − θ0 . − 2 θ0 + ∂η ∂η ∂η ∂η2 ∂η ∂τ ∂τ2 ∂η ∂τ2

(2.72)

The n-th mode of free vibrations is a particular solution of system (2.72): W0,n = wn cos(nη) exp(iω n τ), Vo,n =

1 wn sin(nη) exp(iωn τ), ωn = n2 (n2 −1)(n−1). n

Then, the solution of system (2.72) satisfying the periodicity conditions has the form B 2 V1,n = w sin(2nη) exp(2iw n τ), 2n n   2  1  1  W1,n =  n − − B w2n cos(2nη) exp(2iω n τ)− 4 n 1 1 2 w n− exp(2iwn τ), (2.73) 4 n 4 where: B=

−1 1 (n2 + 1)(4n2 − 1) 4n2 + 1 − . 2 (n2 − 1)2 4n2 − 1

Thus, in the case of nonlinear vibrations, the n-th harmonic with respect to the coordinate η is accompanied by a zero harmonic. w = α3 RW,

v = α1/2 1 α3 V,

y = α1/2 1 Rη,

t=

ρ 1/2 R2 τ. EI

The limiting (α1 → 0, α3 ∼ α1 ) system has the form α−1 α3 ∂W 2 ∂V +W+ 1 = 0, ∂η 2 ∂η

(2.74)

∂2 W ∂2 W + = 0. ∂η2 ∂τ2

(2.75)

The equation (2.75) is linear, and the nonlinearity of the problem is determined by equation (2.74). With certain additive terms, this system was used earlier to investigate nonlinear vibrations of a ring [254, 255, 633]. It turns out that these additive

2.6 Asymptotic and real error

71

terms play a minor role in investigating the given type of vibrations and are of the same order as the terms. The particular solutions of system (2.74), which satisfy the periodicity conditions, have the form ) * ) * n2 −1 w1n cos(nη) w1n exp(iωn τ) − α1 α3 exp(2iωn τ), Wn = w2n sin(nη) w2n 4 Vn =

* *1/2 ) 2 ) 1 −w1n sin(nη) n w1n sin(2nη) α exp(2iωn τ). exp(iωn τ) + α−1 3 −w22n sin(2nη) n w2n cos(nη) 8 1

The axisymmetric component of the radial displacement coincides with the nonlinear correction for W 0 determined from (2.74). w = α3 RW,

v = α3 RV,

y = Rη,

t=

ρ 1/2 Rτ. EF

With α1 → 0, this transformation makes it possible to obtain the following limiting system: ∂2 V ∂(ε cos θ) − ε sin θ − 2 = 0, ∂η ∂τ

∂2 W ∂(ε sin θ) + ε cos θ + = 0, ∂η ∂τ2

where ε and θ are defined by (2.66).

2.6 Asymptotic and real error In practice one expects a real error and not asymptotical one. We give some examples illustrating that an asymptotical series better describe a sought solution if their analytical structure is better approximated by a particular structure of the problem. For instance, from a point of view of asymptotical approach for the Duffing equation two amplitude-frequency characteristics ω = ω0 + εω1 + ε2 ω2 + ..., ω2 = ω20 + εω21 + ε2 ω22 + ... are essentially equivalent. However, from a point of view of numerical values the second result possesses better accuracy. We consider an example of pendulum oscillations governed by the equations θ¨ + sin θ = 0, θ(0) = θ0 ;

˙ = 0. θ(0)

(2.76) (2.77)

One of the typical approaches is to develop sin θ into series of θ and then to apply the perturbation technique. However, it is more efficient to apply the transformation

72


sin θ = ϕ.

(2.78)

Then the input Cauchy problem defined by (2.76) and (2.77) is substituted by the following one + ϕϕ˙ 2 ˙ ϕ¨ + +ϕ 1 − ϕ2 = 0, (2.79) 1 − ϕ2 ϕ(0) ≡ ϕ0 = sin θ0 ;

ϕ(0) ˙ = 0.

(2.80)

The solutions to the problems (2.76), (2.77) and (2.79), (2.80) can be found using perturbation technique. In the first case the following value of the period is obtained T = 2π(1 + θ02 /16 + ...). (2.81) In the second case, we have T = 2π[1 + 0.25 sin2 (0.5θ0 ) + (9/64) sin4 (0.5θ0 ) + ...].

(2.82)

Fig. 2.5. period of oscillation of a mathematical pendulum.

The numerical results are shown in Figure 2.5. Curve 1 corresponds to the linear solution, whereas the curves 2 and 3 correspond to solution (2.81) and (2.82), respectively. The curve 4 corresponds to exact solution which in this case can be expressed via the elliptic functions [451]. It is obvious, that when an analytical structure of the problem is taken via transformation (2.78) then the obtained accuracy of the results increases. In the reference [480,pp.2900-2902] the following interesting problem has been stated. Consider the Duffing equation

2.6 Asymptotic and real error

x¨ + ω20 x + εx3 = 0

73

(2.83)

with the following initial conditions x(0) = A,

x(0) ˙ = 0.

(2.84)

An application of the perturbation method in the form introduced by LindstedtPoincaré results in the series approximation to the solution defined by the initial problem (2.83), (2.84) in regard to the powers of small parameter ε (Chapter 2.8). However, there are different ways to fulfil the initial conditions (2.84). If in zero approximation the following conditions are taken: x0 (0) = A,

x˙ 0 (0) = 0,

then the series of a being sought amplitude-frequency characteristics has the form [480] 3A2 21A4 81A6 ε3 ω = ω0 + ε− 2 3 + + ... . (2.85) 8ω0 16 ω0 8 · 16ω50 A.H. Nayfeh [507, 511] proposed another approach, where in the first approximation the amplitude α is unknown. Then one gets ω = ω0 +

3α2 ε α4 ε4 − 15 2 3 + ... , 8ω0 16 ω0

(2.86)

α3 ε 5α5 ε2 − 2 4 + ... . 2 32ω0 16 ω0

(2.87)

and α is defined by the equation A= α+

Both of the described approaches are equivalent from a point of view of asymptotic series. In fact, making an inverse of expression (2.87) and substituting α = A + εϕ1 (A) + ε2 ϕ2 (A) + ... into equation (2.86), one obtains equation (2.85). However, asymptotical and real errors belong to qualitatively different aspects of the problem. Among others, we are going to show that the exactness of the solution of equation (2.83) depends on the form on the expansion of its solution, and that the exactness of the solution of equation (2.83) depends on the chosen variables. We begin with a numerical experiment. The results are shown in Figure 2.6, where the curves 1-3 correspond to the following solutions: governed by (2.85); exact solution which can be expressed via the elliptic functions [507, 511]; the solution defined by (2.86) and (2.87). In the last case first α has been found numerically from the equation (2.87), and then it has been substituted to (2.86). It is seen that the method proposed by Nayfeh leads to better results. In accordance with a subject of this section let us consider the following nonlinear equation

74


Fig. 2.6. A comparison of results obtained using different analytical methods: a) ω0 = 1, ε = 0.1; b) ω0 = ε = 1.

x¨ + γ x˙ + ω2 x + εxn = 0, n = 3, 5, 7, ... .

(2.88)

For |ε| 1 and relatively small values of n (n = 3, 5) a solution to equation (2.88) can be obtained using either the standard Lindstedt-Poincaré or averaging methods. For the large values of n the standard approach seems not lead to correct results. Let us suppose that we are going to find a solution to equation (2.88) in the form (2.89) x = x0 + εx1 + ε2 x2 + ... . Then the nonlinear term is approximated by the formulae n x0 + εx1 + ε2 x2 + ... = nn0 + εnxn−1 + ... . 0 Thus, a role of the real “small” parameter plays εn instead of ε. It seems that this changing is not so important for n = 3 and n = 5, however it is expected to play a crucial role for large values of n (especially for n → ∞). That is why the following Ansatz may be useful x1 (2.90) x = x0 n 1 + εn + ... . x0

2.7 Numerical verification of asymptotic solution

75

For small n values the expressions (2.89) and (2.86) are equivalent. For n → ∞, a high order of singularity caused by the n-power root seems to play an important role. A typical for quasi-linear approach term x 1 can be obtained from the equation x¨1 + γ x˙1 + ω2 x1 = −xn0 . In other words, known qausi-linear solutions can be transformed to the formulae (2.90).

2.7 Numerical verification of asymptotic solution Let an asymptotic expansion be constructed in terms of a small parameter ε > 0. Typically, the verification of this expansion consists of comparing a known exact or a numerically calculated solution with the asymptotic solution for a specific value of ε and showing that the difference is relatively small. This typical procedure, however, gives no evidence that the asymptotic solution is accurate to a specific order. To demonstrate that the error decreases asymptotically as expected, several values of ε must be used and the expected rate of decrease in the error verified [185]. The expansion is constructed so that in the limit as ε → 0, it will approach the exact solution u e at a specified rate. Specifically, we say that an asymptotic solution ua is accurate to O(δ(ε)) if the limiting condition

ue (x; ε) − ua (x; ε) lim =0 ε→0 δ(ε) is satisfied for all x in the domain of interest. This indicates that the error is given by E = ue (x; ε) − ua (x; ε) = O(γ(ε)), (2.91) where γ(ε) δ(ε). Some single-valued measure of the error that decreases monotonically in ε is chosen. In an initial value problem we might choose to compare the numerical solution to the asymptotic solution at the final integration time, or perhaps the most uniform measure would be the maximum absolute difference between the numerical and asymptotic solutions on the domain of interest. If we assume that the error (2.91) is represented to leading order by E = Kε α , then we can find a good estimate of the order α by graphing log(E) versus log(ε) for several different values of ε. The function will be nearly linear and the slope α is easily determined using a linear least-squares fit of the data. The factor K may depend on a number of different parameters, such as the value of x or initial conditions. This dependence may be removed in a manner that is appropriate to the problem under consideration. The range of ε over which the computations are performed is somewhat problem dependent, with two limiting constraints. If the values of ε are too large, then the asymptotic behavior of the constructed solution will not be observed. If the values of

76


ε are too small, numerical routines will have difficulty obtaining the required accuracy and the numerical error will exceed the asymptotic error. Fortunately, in most problems, there does exist a range of ε for which both numerical and asymptotic methods are efficient and accurate.

10

-2

10

-4

10-6 10

-8

10

-10

10

-12

10

-14

|Inum - I1| |Inum - I2| |Inum - I3| |Inum - I4| |Inum - I5| 5

10

50

100

x

500

Fig. 2.7. Verification of the asymptotic error of the partial sums (2.92).

As example, consider the asymptotic expansion of the integral ∞ −t e x dt I(x) = xe t x for large x. The example is interesting because the infinite series I(x) ∼

∞ (−1)n n! n=0

xn

for the asymptotic behavior of I(x) is divergent for all values of x. This does not negate the usefulness of the expansion, however, since the partial sums I M (x) =

N (−1)n n! n=0

xn

(2.92)

are asymptotically accurate to O(x −M ), with an error of O(x −M−1 ) as x → ∞. Figure 2.7 plots E M (x) = |Inum (x) − I M (x)| versus x on a log-log scale [185].

2.8 Removal of nonuniformities

77

Table 2.1. Slopes of the least-squares fit of log(EM ) as a function of log(x) on different domains. E M (x) |Inum − I1 | |Inum − I2 | |Inum − I3 | |Inum − I4 | |Inum − I5 |

Slope for x ∈ [5, 50] -1.861 -2.823 -3.789 -4.758 -5.729

Slope for x ∈ [50, 200] -1.972 -2.963 -3.954 -4.945 -5.937

Slope for x ∈ [200, 500] -1.991 -2.988 -3.985 -4.981 -5.999

Asymptotic slope -2.0 -3.0 -4.0 -5.0 -6.0

Note that the slopes are not quite constant over the full range of x, but continue to decrease for larger values of x. The least-squares slopes for different portions of the domain of x to show that the slopes approach the asymptotically expected results (Table 2.1).

2.8 Removal of nonuniformities Non-uniformity of the obtained results belongs to one of the main drawbacks of the asymptotical series. Mainly non-uniformities are either generated by infinity (or long scale of the considered space) or by occurrence of singularities. Consider one-degree-of-freedom oscillator with the Duffing stiffness governed by the equation (2.93) u¨ + u + εu3 = 0, u (0) = a, u˙ (0) = 0. Assuming small non-linearity (ε 1), the approximating solution is sought in the form of the series ∞ u= (2.94) εm um (t). m=0

Substituting (2.94) into (2.93) and comparing the terms standing by the same orders of ε the following equations are obtained u¨ 0 + u0 = 0, u0 (0) = a, u˙ 0 (0) = 0,

(2.95)

u¨ 1 + u1 = −u30 , u1 (0) = 0, u˙ 1 (0) = 0,

(2.96)

.

.

. .

.

.

. .

.

.

A solution to the problem (2.95) has the form u0 = a cos t.

(2.97)

Substituting u0 into (2.96) and using the formula cos 3t = 4 cos 3 t − 3 cos t, one obtains cos 3t + 3 cos t u¨ 1 + u1 = −a3 . (2.98) 4

78


A solution to the equation (2.98) with the initial conditions (2.96) is: u1 =

a3 3a3 (cos 3t − cos t) . t sin t + 8 32

Finally one obtains the solution

εa3 1 u = a cos t + −3t sin t + (cos 3t − cos t) + O ε2 . 8 4

(2.99)

(2.100)

Since the term t sin t yields u → ∞ for t → ∞, the used series are not appropriate for all values of t. The term t sin t is called secular and it tends to infinity for t → ∞. Note that the solution (2.100) regularity is violated not only for infinite t, but also for t = O(ε−1 ) (the second term becomes of the order of the first term, which is in contradiction to our assumption that u 1 is small in comparison to u 0 ). Further computation leads to detection of the secular terms t n cos t, tn sin t. Although the obtained series is convergent, but this convergence is very slow and it is impossible to find solution representation using only few terms, which is valid for all t values. The occurrence of secular terms is typical for the problems of the non-linear vibrations, and the direct series application is not uniformly suitable. To avoid the described drawback, one may apply the Lindstedt-Poincaré method. We illustrate this approach using the solution (2.100). Observe that for ε → ∞ the solution is periodic with the period 2π. However, the occurrence of term εµ 3 in the equation (2.93) can not guarantee a periodicity with the same period for ε 0. Since the period of the unperturbed solution (2.97) is constant and equal to 2π, the successive correcting terms change the period, and in result the secular terms appear. For example, the following series development holds sin (ω + ε) t = sin ωt + εt cos ωt −

ε2 t 2 sin ωt − ... . 2!

In order to avoid the secular terms, the new variable is introduced t = τ 1 + εω1 + ε2 ω2 + ... ,

(2.101)

and ω1 , ω2 , . . . are appropriately chosen (this is the main idea of the LindstedtPoincaré method). Using this change of variables the following transformed Duffing equation is obtained 2 d2 u + 1 + εω1 + ε2 ω2 + ... u + εu3 = 0. 2 dτ

(2.102)

Comparing the terms standing by the same powers of ε we get d 2 u0 + u0 = 0, dτ2

(2.103)

d 2 u1 + u1 = −u30 − 2ω1 u0 , dτ2

(2.104)

2.8 Removal of nonuniformities

d 2 u2 + u2 = −3u20 u1 − 2ω1 u1 + u20 − ω21 + 2ω2 u0 , 2 dτ . . . . . . . . . .

79

(2.105)

The general solution of equation (2.103) has the form u0 = a cos (τ + ϕ) , where ϕ is the integration constant. From equation (2.104) one gets d 2 u1 1 3 3 2 (τ a a + u = − cos 3 + ϕ) − + 2ω 1 1 a cos (τ + ϕ) . 4 4 dτ2

(2.106)

(2.107)

In order to avoid the secular terms, the term standing by cos(τ + ϕ) should be equal to zero in the right hand side of the equation (2.107), and one gets 3 ω1 = − a2 . 8 Then solution of the equation (2.107) has the form u1 =

1 3 a cos 3 (τ + ϕ) . 32

Substituting u0 , u1 and ω1 into equation (2.105), we get 51 4 d2 u a − 2ω2 a cos (τ + ϕ) + ψ, + u2 = 128 dτ2 where by ψ the non-resonance terms are denoted. Suppose ω2 = (51/256)a 4, one obtains u = a cos (ωt + θ) +

ε 3 a cos 3 (ωt + ε) + O ε2 , 32

where θ is the integration constant. So, −1 3 2 51 4 2 3 51 4 2 a ε + ... = 1 + a2 ε − a ε + O ε3 . ω= 1− a ε+ 8 256 8 256

(2.108)

(2.109)

Both relations for ω are asymptotically equivalent, however first one yields a better real accuracy. Note that other forms of the illustrated method may be applied. The main idea is to find a parameter changing with a change of perturbation (for example, frequency, wave number, eigenvalue) and to develop it with respect to an order of perturbation intensity. The coordinates which lead to uniform developments into series, are refereed as the optimal coordinates.

80


Note that in some cases the singular terms can be canceled using the so-called ‘Tisserand method’ [383]. Assume that after perturbation procedure the following solution is obtained x (t) = cos ω0 t + εat sin ω0 t, then the following appropriate solution is recommended x (t) ≈ cos (ω0 + εa) t. Consider now a case, when an asymptotic series exhibits some singularities which do not occur in the initial solution. One may expect in this case, that in the higher order terms the singularities order increases. The exact solution is x 1+x x 2 +4− . (2.110) +2 y= ε ε ε On the other hand, the formal asymptotical solution reads y=

(1 − x)(1 + 3x) (1 + x)(1 − x)(1 + 3x) 1+x −ε + ε2 + ··· . 3 x 2x 2x5

(2.111)

Note that the formal solution, contrary to the exact solution (2.110), possesses a singularity in the point x = 0. Following the Lighthil’s method, both y value y = y1 (s) + εy2 (s) + ε2 y3 (s) + · · · ,

(2.112)

as well as x intendent value x = s + εx1 (s) + e2 x2 (s) + · · · ,

(2.113)

are sought in the form of ε power series. The initial equation yields d (sy1 )3 = 1; ds d dy1 dx1 (sy2 ) = − x1 − y1 ; s ds ds ds . . . . . . . . . .

(2.114) (2.115)

The solution satisfying the boundary condition of equation (2.114) reads y1 =

1+s . s

In what follows equation (2.115) takes the form

d d (1 + s)2 x1 d 1 + 2s x1 1 (sy2 ) = − + + + = − . ds ds s ds 2s2 s 2 2s2

(2.116)

(2.117)

2.9 Nonlinear vibrations of a stringer shell

81

An arbitrary way of x 1 choice is introduced in order to avoid singularities occurred during integration of the right hand sides of equation (2.117). Note that the terms yielding singularities are of higher orders than s −1 . The simplest way of the choice is defined by the relation x1 = −

1 + 2s . 2s

(2.118)

Hence, one gets

1+s +··· , (2.119) s 1 + 2s x= s−ε +··· . (2.120) 2s If the parameter s is reduced from relations (2.119), (2.120) then the following y value is obtained x 1+x x 2 y= +1− . (2.121) +2 ε ε ε Exact solution (2.110) gives (for x = 0): 2 2 +4= +ε +··· . ε ε y=

On the other hand, approximated solution (2.121) yields for x = 0: 2 2 ε +1= + + ··· . ε ε 4 To conclude, owing to the Lighthill method the solution in the point x = 0 is highly improved. Consider the so-called Lighthill’s problem, defined by the equation [437, 660, 663] dy (x + εy) + y = 1, dx y(1) = 2 . (2.122) This equation possesses the singularities located on the straight x = −εy.

2.9 Nonlinear vibrations of a stringer shell Dynamics of a structurally orthotropic stringer shell for large displacements (achieving its thickness order) is analysed. Applying the semi-inextensional theory the following governing equations are used (see chapter 5) L1 (w) = ∇41 w − R

∂2 Φ ∂2 w + ρ 2 − L (w, Φ) = 0, 2 ∂x ∂t

(2.123)

82


1 ∂4 Φ 1 ∂2 w 1 + = L (w, w) , B1 ∂y4 R ∂x2 2R2 where: ∇41

(2.124)

1 ∂4 ∂4 ∂4 = 2 D1 4 + 2D3 2 2 + D2 4 , R ∂x ∂x ∂y ∂y

∂2 w ∂2 Φ ∂2 w ∂2 Φ ∂2 Φ ∂2 w + −2 , 2 2 ∂x∂y ∂x∂y ∂x2 ∂y2 ∂x ∂y w is normal Φ is Airy function; D 1 = D + NE1 I/ (2πR), D1 = displacement; Eh3 / 12 1 − ν2 ; E, E1 are Young moduluses of shell and rib respectively; ν is Poisson coefficient; h is shell thickness; R is shell radius; ρ = ρ 0 h + Nρ1 F/(2πR); ρ0 , ρ1 are densities of shell and rib respectively; n is number of stringers; F is square stringer cross-section; I is statical moment of stringer cross-section; B 1 = Eh/ 1 − ν2 + NE1 F/ (2πR), N is number of stringers. We suppose ribs symmetric with respect to shell middle surface. Equations govern dynamics for low frequencies, the most important from practical standpoint. For simply supported shell L (w, Φ) =

w=

∂2 w = 0 for ∂x2

x = 0, L,

normal displacement w is approximated by w (x, y, t) = f1 (t) sin (m1 x) cos (n1 y) + f2 (t) sin2 (m1 x) ,

(2.125)

where: m1 = πm/L, n1 = n/(Rm), m and n are the parameters characterizing waves into x and y directions. Note that f 1 and f2 are not independent, and its relation is defined from a continuity condition of the displacement υ along a ring: 2π 0

∂υ dy = 0. ∂y

∂v ∂y

In our case = w (see Chapter 5), which gives f 2 = f12 n2 /(4R). The Airy function is found from the equation (2.124): m 2 f 5 m1 2 1 1 (m (n B−1 Φ = sin x) cos y) − × 1 1 1 h Rn2 16 h 2 3 f1 f1 1 cos (2n1 y) + m21 sin (m1 x) cos (2m1 x) cos (n1 y) . R 2 R Knowing Φ one may apply the Bubnov–Galerkin method to the equation (2.123) to get 2πRL L1 (w) sin (m1 x) cos (n1 y) dxdy = 0, 0

0

2.9 Nonlinear vibrations of a stringer shell 2

83

2

where: L1 (w) = ∇41 w − R ∂∂xΦ2 − L (w, Φ) + ρR2 ∂∂tw2 . After some computations the following second order differential equation with constant coefficient is obtained:  2   dξ d2 ξ d 2 ξ   + aξ  + ξ 2  + A1 ξ + A2 ξ3 + A3 ξ5 = 0, (2.126) dt1 dt12 dt1 where

, t1 =

B1 t , A1 = ε1 + 2ε2 p−2 + ε2 p−4 + n−4 , ρ R

A2 = 0.0625 + 0.5n 4ε1 − 0.75, A3 = 0.25n4, a = 0.09375n 4, p = m1 /n, ξ = f1 /h, ε1 = D11 / B1 R2 , ε2 = D2 / B1 R2 , ε3 = D3 /(B1R2 ). The perturbation method is used to analyse periodic vibrations of the equation (2.126). The independent variable is substituted by τ = ωt 1 , where ω is the unknown frequency of the periodic solution. From (2.126) one obtains ω2 ξ¨ + aω2 ξ ξ˙2 + ξξ¨ + A1 ξ + A2 ξ3 + A3 ξ5 = 0, (. ˙. .) = d(. . .)/dτ.

(2.127)

Since the periodic solutions are considered the following initial conditions can be applied: for τ = 0 ξ = f, ξ˙ = 0. (2.128) The solution ξ(τ) is developed into the series of formal small parameter ε of the form (2.129) ξ (τ) = εξ1 (τ) + ε2 ξ2 (τ) + ε3 ξ3 (τ) + ..., and ω is also sought in the form of ε series ω = ω0 + εω1 + ε2 ω2 + ... .

(2.130)

Substituting (2.129)–(2.130) into (2.127) the following differential equations are obtained ε1 : ω20 ξ¨1 + A1 ξ1 = 0, (2.131) ε2 : ω20 ξ¨2 + A1 ξ2 = −2ω0 ω1 ξ¨1 , ε3 : ω20 ξ¨3 + A1 ξ3 = − ω23 + 2ω0 ω2 ξ¨1 − 2ω0 ω1 ξ¨2 − aω20 ξ1 ξ˙12 + ξ¨1 ξ12 − A2 ξ13 , .

.

. .

.

.

. .

.

(2.132)

(2.133)

.

The initial conditions (2.128) give ξ1 (0) = f, ξ˙1 (0) = 0;

(2.134)

84


ξi (0) = 0, ξ˙i (0) = 0, i = 2, 3, ....

(2.135)

A solution of (2.131) for the initial conditions (2.134) has the form ξ1 = f cos τ, ω0 = A1/2 1 .

(2.136)

The equation (2.132), using (2.136), can be rewritten in the form ξ¨2 + ξ2 = 2A1/2 1 f ω1 cos τ.

(2.137)

Since ω1 = 0, the equation (2.137) (for the initial conditions (2.135)) has the solution ξ2 = 0. The equation (2.138) taking into account the initial conditions (2.135) and the periodicity condition, yields ξ¨3 + ξ3 = 2 f A1/2 ω2 − 0.125A1/2 3c1 − 2a f 2 cos τ− 1 1 0.25A1 (c1 − 2a) f 3 cos 3τ , c1 = A2 A−1 1 .

(2.138)

Solution of equation (2.138) satisfying an initial condition and a periodicity condition has the following form 2 ω2 = 0.125A1/2 1 (3c1 − 2a) f ,

ξ3 = −0.03125c 1 (cos τ − cos 3τ) f 3 + 0.0625a (cos τ + cos 3τ) f 3 . In analogous way obtains ω 3 = 0, ω4 = 0.03906A 1/2 γ1 (γ2 − 6c1 ) − 2γ2 (γ2 + 8a) + 80c2 f 4 , 1 c2 = A3 A−1 1 , γ1 = c1 − 2a, γ2 = 3c1 − 2a. Finally, the following amplitude-frequency dependence is obtained ω = A1/2 1 + 0.125 γ2 + 0.03125 γ2 (γ1 − 1 . 2γ2 − 16a) − 6c1 γ1 + 80c2 f 2 ε2 f 2 ε2 .

2.10 Non-quasilinear asymptotics of nonlinear system1 In many references the asymptotics of the following equation is discussed: x¨ + f (x) = 0,

(2.139)

where: f (x) = c1 x + c2 xn , n = 3. In this case one may construct a quasilinear asymptotics in a relatively simple manner. However, for n → ∞, the situation is qualitatively changed and one obtains, roughly speaking, vibro-impact 1

By courtesy of V.N. Pilipchuk

2.10 Non-quasilinear asymptotics of nonlinear system

85

regime. In words, in order to describe the vibro-impact state, the infinitely many quasi-harmonic approximations are required. To avoid this problem in references [546, 547, 548, 549, 550, 551, 552, 553, 556, 557] the following method was proposed. Let us suppose the following initial conditions are attached to equation (2.139) x = 0, x˙ = υ for t = 0.

(2.140)

Let P(ϕ) be a saw-tooth periodic piecewise smooth function wit the unit amplitude πϕ 2 P (ϕ) = arcsin sin , P (ϕ + 4) = P (ϕ) . (2.141) π 2 Since the diagram of this function is composed of straight lines, the calculus is reduced to relatively simple operations. The following observation holds in the framework of distributions: P 2 (ϕ) = 1, P (ϕ) = 2

∞

δ (ϕ + 1 − 4k) − δ (ϕ − 1 − 4k) ,

k=−∞

−∞ < ϕ < +∞.

(2.142)

A solution to the Cauchy problem (2.139), (2.140) is sought in the form x = ψ + X (ψ) , ψ = AP (ϕ) , ϕ = υt/A.

(2.143)

The period of solution with respect to time T = 4A/υ, where the parameter A and the function X are to be defined. Differentiating (2.143) twice with respect to time, and taking into account (2.142) the following equation is obtained 2 υ ' ( 2 x¨ = υ X + 1 + X P . (2.144) A Since the acceleration in this system should be bounded, hence X ψ=A = −1.

(2.145)

Because ψ = AP(ϕ) is periodic and X (ϕ) is even, the relation (2.145) takes place for all points ϕ = ±1 + 4k, k = 0, ±1, ±2, . . .. The equation (2.145) serves for determination of the parameter A. Substituting (2.143) into the initial equation (2.139), taking into account (2.144), (2.145) the following differential equation is obtained υ2 X = − f (ψ + X) = − f (ψ) − f (ψ) X −

1 f (ψ) X 2 − ... . 2

The initial relations (2.140) yield the following initial conditions

(2.146)

86


ψ = 0 for X = 0, X = 0.

(2.147)

A solution to the problem (2.146), (2.147) is sought in the form of the series of successive approximations X = X (1) + X (2) + X (3) + ..., A = A(1) + A(2) + A(3) + ... .

(2.148)

Substituting first series of (2.148) into (2.146) the following sequence of equations is obtained

X (1) = −υ−2 f (ψ) , X (2) = −υ−2 f (ψ) X (1) ,

2 1 X (3) = −υ−2 f (ψ) X (2) + f (ψ) X (1) , ... . 2 These equations, with the initial conditions (2.147) yield the terms of the first series of (2.148). Integrating by parts one gets X

(1)

= −υ

−2

ψ ψ f (ψ) dψdψ, 0

X (2)

0

 ψ  ψ ψ     = −υ−2  f (ψ) X (1) dψ − f (ψ) X (1) dψdψ ,   0

X (3)

0

0

  ψ    (1)2 (2) (1) (1) −2  1 (ψ) (ψ) f X = −υ  + f − 2X X dψ+ X    2 0

ψ

ψ

+ 0

0

f (ψ) X (1) X (1)

 

    (2) − X dψdψ , ... .    

(2.149)

Substituting (2.149) into (2.145), and developing the derivatives X i (i = 1, 2, . . .) into the power series in the vicinity of the point ψ = A ( 1), the following chain of equations is obtained X (2) (1) (2) X ψ=A(1) = −1, A = − (1) , X ψ=A(1) A

(3)

=−

1

X (1)

1 (1) (2)2 X A + X (2) A(2) + X (3) 2

..., ψ=A(1)

which serves to get A j ( j = 1, 2, . . .). Taking into account (2.150) from the first equation of (2.150) one gets

(2.150)

2.10 Non-quasilinear asymptotics of nonlinear system

A(1)

f (ψ) dψ = υ2 .

87

(2.151)

0

Therefore, first term of the second series of (2.150) is equal to amplitude of vibrations, which occurs for the twicely enlarged initial energy. Note that the terms of (2.149) do not include derivatives of the function f (ψ). In addition, the introduced iterational procedure does not require any differentiation of this function, i.e. the analyticity requirement may be omitted. Assumed, for instance X

(0)

i

≡ 0, X = −υ

−2

ψ ψ 0

f ψ + X i−1 dψdψ, i = 1, 2, ... .

0

A function X (N) , for large values of N, is the approximated solution to the equation (2.139). Although for X (1) we have relative simple relation (2.149), but for higher approximations the calculations using quadratures do not belong to simple and clear procedure. In order to illustrate the peculiarities of the described approach the following example is considered. Let f (x) = x n , where n is add value. The main problem during computations related to the limiting case n → ∞. On the other hand, this case belongs to relatively simple one, because between impacts the uniform inertial motion is observed, and the corresponding solution has the form x = P (ϕ) , ϕ = υt. (2.152) Observe that considering the relations (2.152) by the variable change in (2.139), than in the equation with respect to new variable ϕ the singular functions which correspond to impact interactions vanish, and the equation is ϕ¨ = 0. The appropriate solution for large and finite n may be now obtained. After integration of (2.149) the following solution is found nψ2n+3 ψn+2 + − (n + 1) (n + 2) 2υ4 (n + 1)2 (n + 2) (n + 3) n n n−1 + ψ3n+4 + ..., 6υ6 (n + 1)3 (n + 2) (3n + 4) n + 2 2n + 3 x = ψ−

υ2

(2.153)

ψ = AP (υt/A) . In addition, (2.145) yields A = A(1) + 1−

2υ2 (n

n+2 n A(1) + 2 + 1) (n + 2)

2n+3 n+2 n n−1 n2 A(1) − − + ..., 3n 6n + 9 4n + 4 2υ4 (n + 1)3 (n + 2)2

(2.154)

88


1/(n+1) A(1) = (n + 1) υ2 . Expansions (2.153) and (2.154) possess the following asymptotics as n → ∞ A → 1, X = x − ψ → 0, x → P (υt) . To estimate convergence the most unsuitable case is further analysed for n = 1 (harmonic oscillator). The expressions (2.153) and (2.154) for n = 1 have the following form ψ 1 ψ3 1 ψ5 1 ψ7 ψ − x=υ + − + ... = υ sin , υ 3! υ3 5! υ5 7! υ7 υ √ 3 π 1 + + ... = υ A = 2υ 1 + 12 360 2 and they may be treated as a general solution of the linear equation of the form x = υ sin τ, τ =

1 πP (2t/π) . 2

Contrary to the classical form of solution (x = υ sin t) the approximation of sin t by a part of the perturbation series does not loose the property of periodicity which is now represented by ’oscillating time’ τ |τ| ≤ π2 , but the smoothness of the approximating function is lost. The occurred discontinuities are smoothened by introduction of the successive terms of the series, and they practically vanish only for a large number of the series terms. In the points of discontinuity the velocity function possesses first order discontinuities, which correspond to certain fictive impacts. However, for large n the approximation of the fast velocity jumps by discontinuities, and the force impulses by the sudden impulses seems to be appropriate.

2.11 Artificial small parameters Typically, in a being considered system one may introduce small parameters either in result of non-dimensionalization or scaling procedures. A serious question appears, however, if such parameters do not exist or their corresponding applicability domains are so small that they are in practice no useful. In this case one may introduce the so called artificial perturbation (small) parameters [147, 148, 149, 150, 151, 152, 252, 335, 336, 337, 433]. In order to illustrate an idea of artificial small parameters let us consider the following algebraic equation (2.155) x5 + x = 1. Our aim is to find a real root of this equation. Note that its numerical value is equal to 0.75487767.... A small parameter does not appear explicitly in equation (2.155), and we are going to introduce a small artificial parameter ε. Below different cases are studied.

2.11 Artificial small parameters

89

1. Approximation yielded by weak coupling [151] (terminology comes from physics). εx5 + x = 1.

(2.156)

The variable x is represented by its series x = a0 + a1 ε + a2 ε + ...,

(2.157)

Substituting (2.157) into (2.156) and splitting with respect to ε (i.e. comparing terms standing by the same powers of ε), one gets a0 = 1, a1 = −1, a2 = 5, a3 = −35, a4 = 285, a5 = −2530, a6 = 23751. Note that a closed expression for a n can be obtained, namely an =

(−1)n5n! n!(4n + 1)!

and the radius of series (2.157) convergence R = 4 4 /55 = 0.08192. Consequently, for ε = 1 the series (2.157) is fastly diverged, and already the sum of first six terms it equal to 21476. A situation can be improved via help of Padé approximation (see Chapter 12). Constructing the approximation [3/3] (i.e. this including three term in both numerator and denominator) and computing it for ε = 1, one gets the value of x = 0.76369. The latter differs from exact solution on amount of 1.2%. 2. Approximation of strong coupling [152]. Let us introduce now the small parameter by the linear term in equation (2.155) x5 + εx = 1. A solution is sought in the series form x = b0 + b1 ε + b2 ε2 + ...,

(2.158)

and applying a standard perturbation technique one gets b0 = 1, b1 = −1, b2 = −1/25, b3 = −1/125, b4 = 0, b5 = 21/15625, b 6 = 78/78125. Also in this case one may get a general expression for coefficients bn = −

Γ[4n − 1]/5 2Γ[(4 − n)/5]n!

and hence define the radius of convergence R = 5/4 4/5 = 1.64938.... Observe that the value of x(1), accounting of first six terms of series (2.158), is equal to x = 0.75434, and it differs from the exact value of amount of 0.07%.

90


The described methods are simple and illustrative, and in many branches of computational science many problems are solved in satisfactory manner. However, sometimes the series are diverged so fast, that it is either very difficult or even impossible to find the appropriate method of summation. One may try to find other parameters, but they not always possess a clear physical meaning. 3. Perturbed exponents. Quasi-linear asymptotics [152]. Let us introduce a “small parameter” δ into an exponent in the following way x1+δ + x = 1 and let us use the following series x = c0 + c1 δ + c2 δ2 + ... .

(2.159)

The series (2.159) coefficients are easily found: c 0 = 0.5, c1 = 0.25 ln 2, c2 = −0.125 ln 2,... . Radius of series convergence is equal 1, and it can be computed for δ = 4. Applying again Padé approximation [3/3] one gets x = 0.75448. The latter value differs on 0.05% from the exact result. Furthermore, calculating ci up to i = 12 and constructing Padé approximation [6/6] one finds x = 0.75487654 (the error is of amount of 0.00015%). The described approach is called “small δ method” and it is widely used in physics. 4. Perturbed exponent. The case of strongly nonlinearity [34, 108]. Since in all earlier described examples in order to get a realiable result one has to construct large series terms, it is hardly to believe on their practical application. Let us try to construct asymptotics of the considered example assuming now exponent to large parameter by consideration of the equation xn + x = 1. Assuming n → ∞ (“large δ method”method!large n

variable y = x is introduced accounting that y2/n = 1 + In result one gets

δ [34, 108]), the new

1 ln y + . . . . n

ln n y≈ n

n

ln n − ln ln n y≈ n

,

(2.160)

1/n , ... .

(2.161)

A characteristic feature of this asymptoties relies on occurrence of its complexity exhibited via logarithms, logarithms of logarithms,

2.12 Method of small δ

91

and so on. Besides, whole expression is in power of inversed large parameter. For n = 2 the formula (2.160) gives x = 0.58871; the error related √ to exact solution (0.5 5 − 1 ≈ 0.618034) is of amount of 4.7%. For n = 5 one gets x = 0.79745 (error is of amount of 4.4%). Formula (2.161) for n = 5 gives x = 0.74318 (error ∼ 2.7%). Therefore, already first terms of asymptotics give very good results. To cunclude, in the considered case the “large δ method” is challenging one, since it secures high accuracy even for low orders of the applied perturbation method. 2.12 Method of small δ We exhibit main features of the small δ method using example of equation x¨ + x(1/2n+1) = 0,

n = 1, 2, . . .

(2.162)

with the initial conditions x(0) = 0,

x(0) ˙ =A.

(2.163)

We suppose n → ∞. A similar like model has been very often used in the theory of vibro-impact system [117]. A solution to equation (2.162) can be found exactly [347, 578, 579]. However, here a construction of an asymptotical solution using small δ method is proposed. In the limit n → ∞ we get the following equation x¨0 + sign(x0 ) = 0,

(2.164)

which has been analysed using various methods in references [316, 440, 550]. An analytical solution to equation (2.164) can be presented in the form of the Fourier series [316], piecewise continuous functions [440], or by the “saw-tooth” functions [550]. By taking δ = (2n + 1) −1 and supposing δ 1 we use the following approximation (considering only the values x > 0, since a solution for x < 0 can be obtained using a symmetric mapping) xδ = 1 + δ ln |x| + ... .

(2.165)

For equation (2.162) with the initial conditions (2.163) the following solution is sought x = x0 + δx1 + ... . (2.166) Substitution (2.165), (2.166) into equations (12.24) and (2.163) yields (in the first approximation) the following result

92


Fig. 2.8. Solutions to the Cauchy problem (2.162), (2.163) for A = 1 using the fourth order Runge-Kutta method (——), and the approximations: (2.170) (- - - -); (2.171) ( - - - ); (2.172) (— — —) for n = 2.

x¨1 = − ln |x0 | .

(2.167)

In order to avoid singularities we consider a solution to (2.167) on the quarter part of its period. A solution to the boundary value problem (2.164) can be presented in the form [316]  t   − (t − 2A) , 0 ≤ t ≤ 2A,     2 x(t) =    t2    − 3At + 4A2 , 2A ≤ t ≤ 4A, 2 x(t + nT ) = x(t), T = 4A. Therefore, the first approximation on the interval 0 ≤ t ≤ A yields t2 x¨1 = − ln tA − 2 x1 (0) = 0,

x(0) ˙ = 0.

(2.168) (2.169)

Integrating equation (2.168) and taking into account (2.169) gives (in this case Mathematica computations have been used): x1 (t) = 2At − 3t 2 − t2 ln 2 − 4A(A − t) ln |−2A| + t 2 ln |2A − t| +

2.12 Method of small δ

93


t2 ln t + 4A2 ln |−2A + t| − 4At ln |−2A + t| . Although the solution can be extended up to the terms of δ 2 , δ3 , . . ., only zero and first order approximations are used: x ≈ x0 + δx1 .

(2.170)

The Padé approximants (see Chapter 13) can be used to extend an application area of the solution (2.166). For our case one gets x≈

x20 . x0 − δx1

(2.171)

In addition also the exponential approximation can be applied x ≈ x0 exp(δx1 /x0 ).

(2.172)

Some of the numerical results for A = 1 are presented in Figures 2.8–2.10. The analysis leads to the following conclusions. Only the small δ method allows for estimation of all of the periods in the considered interval of changes of n values. Examination of the figures shows that only approximation (2.170) gives the correct value for small n. In addition, it gives the best approximation to the numerical values of the being sought periods. The relative errors decrease quickly with an increase of n. Although the latter behaviour can be also observed in a case of the

94



Padé approximants and exponential approximations, but it seems that the exponential approximations is more suitable than the Padé approximants one. However, with an increase of n all of them are suitable to approximate the periods.

2.13 Method of large δ Consider the following nonlinear equation x¨ + xn = 0,

n = 3, 5, 7, ...

(2.173)

which attracted an attention of many researchers [546, 547, 548, 549, 550, 551, 552, 553, 556, 557]. Among others, for large values of n it can model the vibro-impact processes [117]. In particular, this equation can be integrated using the special function (cam, sam or the Ateb-functions) introduced by Rosenberg [578, 579]. From a mathematical point of view the mentioned functions belong to a class of inversion of the uncompleted Beta-functions [605]. The parameter n can be used into two ways. For small n the approximation the small δ (see Section 2.12) may be applied. For n → ∞, a corresponding asymptotics can be formulated using the parameter n −1 . In section 2.10 in order to integrate equation (2.173) saw-tooth functions have been applied. Here a construction of the successive asymptotics will be outlined.

2.13 Method of large δ

95

Let us consider the equation (2.173) with the initial conditions x(0) = 0,

x(0) ˙ = 1,

(2.174)

for n → ∞. We are going to define a periodic solution to the Cauchy problem (2.173), (2.174). The first integral related to the problem defined by (2.173), (2.174) is as follows 2xn+1 p2 = 1 − . (2.175) n+1 √ A change of variables x = n+1 0.5(n + 1)ξ and an integration allows for transformation of (2.175) to get the form n+1

2 t= n+1

0≤ξ≤1

0

dξ . / 1 − ξn+1

(2.176)

On the other hand, the implicit solution (2.176), after a change of the variables ξ = sin2/(n+1) θ, yields the expression including explicitly the small parameter ε = 2/(n + 1) of the form π 0≤θ≤ 2 εε/2 t = ε sin−1+ε θdθ. (2.177) 0

First we consider the following expression (see (2.177)) sin−1+ε θ = θ−1+ε

1 θ 1−ε 0 θ θ − ε ln + ... . = θ−1+ε sin θ sin θ sin θ

(2.178)

Using a series approximation θ θ2 =1+ +... sin θ 3! and leaving the terms with ε order in the right hand side of equation (2.178), one obtains θ2 + . . . + 0(ε). (2.179) sin−1+ε θ = θ−1+ε + 3! It is clear, that an essential part during an integration procedure is introduced by the first term of the series (2.179). Therefore, in the first approximation we take −1 εε/2 t ≈ θε . It is easy to find that θ ≈ ε1/2 tε , which reads in the input variables   2  n+1  2   n+1 n + 1  x≈ sin n+1  t 2  . (2.180) 2 n+1 The found solution (2.180) can be calculated only on a 1/4 part of the period T , and then it can be periodically extended. The period of solution (2.180) reads

96


 2   π n + 1  n+1  . T = 4  2 2 

(2.181)

It should be emphasized that for ε = 1 the exact solutions are obtained: x = sin t, T = 2π. When n → ∞, one gets T → 4, which should be expected. When the solution (2.181) is expanded into the series in relation to ‘t’ and taking into account only the first term the non-smooth solution obtained by Pilipchuk occurs [546, 547]. Now we are going to estimate an accuracy of the solution (2.181). To this aim the following expression is used: 2 n+1

π/2 2 sin−1+ n+1 θdθ = 0

1 1 1 B , = A1 , n+1 n+1 2

(2.182)

where: B(. . . , . . .) is the uncompleted Beta-function. An approximate value of the integral in the left hand side of expression (2.182) has the form 2 π n+1 . A2 = 2 The numerical comparison between the values of A 1 and A2 and accuracy estimation is reported in Table 2.2. Table 2.2. Error estimation of asymptotic procedure. n 1 3 5 ... ∞

A1 π/2 1.30 1.20 ... 1

A2 π/2 1.25 1.16 ... 1

Error % 0% 5% 3% ... ∼ 0%

The obtained results lead to the following conclusion. Already the first approximation of the asymptotics for n → ∞ gives reasonable accuracy for practical applications (even for small values of n). It should be pointed out that the expressions (2.180), (2.181) are essentially generalizing an inversion of the uncompleted Beta-function for n = 1 (sine) and for n = ∞ (linear function). In reference [619] the following approximations to the uncompleted B-functions are given xν for small x, B(ν, 0.5, x) ≈ ν and √ πΓ(ν) for x close to 1. B(ν, 0.5, x) ≈ Γ(0.5 + ν)

2.13 Method of large δ

97

It is not difficult to observe (using asymptotic Γ(ν) → ν −1 [619]) that for small ν from the above formula the following uniform approximation of the Beta-function occurs: xν B(ν, 0.5, x) ∼ for ν → 0, ν which proves a validity of our results. Now let us examine the equation x¨ + x + xn = 0.

(2.183)

Taking into account the initial conditions (2.174), a solution to (2.183) has the following implicit form x t= 0

dx + 1 − x2 −

2 n+1 n+1 x

.

(2.184)

The following change of variables x2 +

2 n+1 x = sin2 θ n+1

(2.185)

is introduced. A solution to equation (2.185) with regard to x we will search in the form / n−1 (2.186) x = sin θ 1 + x1 , and it yields:

− sinn−1 θ . 1 + sinn−1 θ Finally, the expression (2.184) can be transformed to the form x1 ≈

 1 n−1 n−1  sinn−1 θ   1 − sin θ  dθ, −  1 + sinn−1 θ 1 + sinn−1 θ 

0≤θ≤π/2 

t= 0

(2.187)

with the 1/n accuracy. The expression standing by the integral (2.187) can be changed by the following one 1/n 1 sinn θ sinn θ = + O(1/n). − 1+ n 1 + sin θ 1 + sinn θ 1 + sinn θ Consider the following approximation ) n θ , 0≤θ< n sin θ = 1, θ = π/2

1 √ n n

.

Using two-point Padé approximants (see Chapter 13) one obtains

98


sinn θ = Therefore

θ 0

θn . 1 + θn

1 + θn 1 ln(1 + 2θn ), dθ = θ − 1 + 2θn 2n

and the implicit solution has the form t=θ−

1 ln(1 + 2θn ). 2n

Furthermore, the being sought function θ can be presented in the form /n θ = t 1 + 0.5θ1 /t. The θ1 are given by the transcendental equation θ1 − ln 1 + 2tn + tn−1 θ1 = 0, solutions of which can be obtained numerically for each t. The obtained equations (2.186) and (2.187) define a being sought asymptotics.

2.14 Choice of zero approximation A construction of an asymptotical solution to a physical problem can be separated into three parts. First, one has to choose small (large) parameters. Then, one of the well developed asymptotical approaches can be applied. Finally, an estimation of validity of the obtained results should be added. Observe that although the first step becomes the most important it is formalized at the very least. Let us begin to find an asymptotic of nontrivial solution. One has to guess initially a kind of sought asymptotics. Observe that this initial guess can not be formalized in practice. Many researchers mention the following properties helping for the initial guess: analogy, experience, physical feeling, intuition [288]. Very often the only method to estimate a validity of a choice of the zero order approximation can be done in a posteriori way by comparing the approximate results with those obtained using either numerical and experimental results. In addition, it can happen that the successive approximations do not lead to a solution improvement. In principle, it can even lead to conclusion that the applied asymptotic approaches are not appropriate. From this point of view seems to be interesting to consider oscillations of parallelogram and trapezoidal membranes. The mentioned problems have been solved using various approaches including also perturbation methods. A promising hope to solve the problem in full give numerical results obtained in reference [320]. In the mentioned monograph the recurrent perturbation formulas are given with respect to

2.14 Choice of zero approximation

99

‘small’ parameter λ = tanα (see Figs. 2.11, 2.12). It has been shown, among others, the one needs 16 approximations to get a convergence of the process for λ ≤ 0.21. Increase λ leads to to greater of number of approximations. In fact, the observed property does not allow to get a solution using perturbation technique for large values of λ [320]. y

y

a)

y

b)

b

c) b

b

a a 0

0

x

a

x

a cos a

0

x

Fig. 2.11. Parallelogram membrane and different methods of choosing of zero order approximation.

y

a

b

a)

x

0

-b

a

y

y

b

b

b)

x

0

-b

a

c)

x

0

-b

a cos a

Fig. 2.12. Trapezoid membrane and different methods of choosing of zero order approximation.

Observe that higher approximations which not only does not improve a solution but sometimes even lead to uncorrect results is typical for the asymptotic series. However, in the discussed case the problem is caused by other reasons which we are going to illustrate and discuss. The governing equation is taken in the form u xx + uyy − ω2 u = 0. Boundary condition are u = 0 on the contour. The membrane shown in Figure 2.11b is taken in monograph [320] as the input one to proceed with an asymptotical analysis. The fundamental frequency of the membrane can be estimated via the following formula:

100


ω2 =

π2 a2 b2 . a2 + b2

(2.188)

In Table 2.3 the computational results due to the formulae (2.188) are given. It is seen that the error is large and it increases with increase of λ. Table 2.3. Comparison of various approximations for parallelogram membrane. λ 0.5 0.7 1.5 2.0

Formula (2.188) 19.74 19.74 19.74 19.73

Error % 7 14 46 62

Formula (2.189) 22.21 24.57 41.94 59.22

Error % 4 7 13 12

Numerical solution 21.32 22.93 36.89 52.63

Let us take another membrane shown in Figure 2.11c which serves for zero order approximation in the asymptotical procedure. In this case we have: ω2 =

π2 a2 b2 cos2 α . a2 cos2 α + b2

(2.189)

The errors related to the formula (2.189) are essentially smaller in comparison to those generated by formula (2.188) (see Table 2.3). A similar observation holds also with a trapezoidal membrane (see Figure 2.12a). Analogously, let us take the membranes shown in Figure 2.12b and Figure 2.12c as the initial ones. One obtains the following frequency estimation: ω2 = and ω2 =

b2

π2 a2 b2 b2 + 0.25a2

(2.190)

π2 a2 b2 + 0.25a2 cos2 α

(2.191)

respectively. The computational results derived from the formula (2.190) and (2.191) are included in Table 2.4. The obtained results clearly show an advantage of zero order approximation governed by the membrane shown in Figure 2.12c. Note that in the obtained result there is nothing surprised. A frequency dependence on the integral system property, i.e. area. A choice of the zero order approximation in the cases reported in Fig. 2.11c and Fig. 2.12c is based on the invariance of area. The included results and discussion lead to the following conclusion. It can happen (in fact, very often) that an improvement of an asymptotical solution mainly depends on a proper choice of initial approximation rather than on tedious procedure related to calculation of higher order approximations.

2.15 Lyapunov–Schmidt procedure

101

Table 2.4. Comparison of various approximations for trapezoidal membrane. λ 0.5 0.7 1.5 2.0

Formula (2.188) 19.74 19.74 19.74 19.73

Error % 7 14 46 62

Formula (2.189) 22.21 24.57 41.94 59.22

Error % 4 7 13 12

Numerical solution 21.32 22.93 36.89 52.63

2.15 Lyapunov–Schmidt procedure Now we propose one of possible approaches to solve a bifurcation problem using an example of bending of an axially compressed homogeneous rod [138, 646, 647]. The angle of rotation θ(x) of a tangent to the rod satisfies the following equation and boundary conditions θ xx + T sin θ = 0, 0 ≤ x ≤ 1, θ x (0) = θ x (1) = 0.

(2.192)

Ql2 EI ,

where: T = x = ¯x/l, ¯x is axial coordinate, l is length of road, Q is axial load. Note that the function θ 0 (x) = 0 is the solution to the equation (2.192) for all T values (trivial solution). The following problems are addressed: (i) find the bifurcation values of T 0 , where a change of solutions number occurs; (ii) define number of solutions in the vicinity of T 0 ; (iii) investigate solutions behaviour in the neighbourhood of T 0 . The following associated linearized (in the θ 0 (x) vicinity) is analysed: θ xx + T θ = 0,

(2.193)

θ x (0) = θ x (1) = 0.

(2.194) 2

This problem yields the eigenvalues T i = (πi) , i = 1, 2, 3, . . ., and the corresponding eigenfunctions θ i (x) = cos(πix), which are mutually orthogonal in the interval [0, 1], i.e. 1 θi (x) · θ j (x) dx = 0 for i j, 0

1

θi2 (x) dx = 0.5 for i = j.

0

If the function θ(x) is continuously differentiable in the interval [0, 1] and satisfies the boundary conditions (2.194), then it can be uniformly projected into eigenfunctions of the linearized problem (2.193), (2.194):

102


θ (x) =

∞

1 cn θn (x), cn = 2

n=1

θ (t) θn (t) dt. 0

Consider a solution to the considered problem in the neighbourhood of the first eigenvalue T 1 = π2 . Assuming T = π2 + λ, the equation (2.192) can be presented in the following form ∞ (−1)i+1 2 . θ2i−1 Bθ ≡ θ xx + π θ = −θλ − π + λ (2i − 1)! i=2 2

(2.195)

A solution to the non-homogeneous equation BVθ = h(x) exists if and only if the the following condition holds 1 h (x) cos (πx) dx = 0.

(2.196)

0

The solution θ of the equation (2.195) is sought in the form of the series with respect to small parameter ξ: θ = ξ cos (πx) +

∞

ξ2i+1 θ2i+1 , ξ 1.

(2.197)

i=1

Note that add values of ξ powers applied, since the operator B is also the add function. Substituting (2.197) into (2.195) and remaining only main terms (i.e. neglecting high order small terms with respect to ξ and λ), the following boundary value problems are found: ξ3 Bθ3 =

π2 3 ξ cos3 (πx) − λξ cos (πx) , 6

(2.198)

θ3x (0) = θ3x (1) = 0.

(2.199)

The condition (2.196) of existence of a solution of the problem (2.198), (2.199) yields the following bifurcation condition π2 ξ3 /8 − λξ = 0.

(2.200) √ For λ > 0 the equation (2.200) possesses two solutions ξ = ± λ and the trivial one ξ ≡ 0. The particular solution of (2.198), (2.199) has the form θ3 = −

cos (3πx) . 192

(2.201)

2.16 Nonlinear periodical vibrations of continuous structures

103

The general solution of (2.198), (2.199) constitutes of the sum of the particular solution (2.201) and the solution of the corresponding homogeneous problem, i.e. θ3 = −

cos (3πx) + C cos (πx) . 192

The constant C is defined by a solvability condition of the following problem 5

ξ5 Bθ5 = −

λξ3 π2 5 π2ζ ξ cos5 (πx) + cos3 (πx) − λξ3 θ3 + cos2 (πx) , 120 6 2 θ5x (0) = θ5x (1) = 0.

Therefore, the rotational angle of the rod for a load close to buckling one is defined approximately by the equation θ ξ cos (πx) −

ξ3 5ξ3 cos (3πx) − cos (πx) + . . . . 192 128

The dependence of ξ on the A parameter in the vicinity of the critical value Λ = π2 is reported in Figure 2.13.

Fig. 2.13. Typical bifurcation diagram.

One may proceed analogously to investigate bifurcation behaviour in the neighbourhood of all other eigenvalues.

2.16 Nonlinear periodical vibrations of continuous structures Non-linear vibrations of continuous structures (such as rods, beams, plates, shells, etc.) have been attracting many researchers. A construction of asymptotics for infinite systems with respect to spatial coordinates is well known [506, 509, 513]. However, this case differs significantly from that of finite systems analysis. A significant peculiarity of these problems is the phenomena of internal resonances,

104


arising in non-linear multi-degree-of-freedom structures when natural frequencies become commensurable with each other. In general, it causes the coupling of normal modes and results in multi-mode and multi-frequency response. Simple models that describe these vibrations can involve non-linear second and fourth order partial differential equations. The rigorous proof of the existence of periodic solutions was given in [569]. This work inspired additional investigations, some of them are reviewed in [195]. Construction of periodic solutions for this case leads to the wellknown problem of small denominators. In order to omit the occured difficulties in the series of works a modification of the KAM (Kolmogorov-Arnold-Moser) theory is applied was originally developed to overcome small denominator problems in celestial mechanics [98, 99]. Since then it has been extended for a wide range of multi-degree-of-freedom structures (see, for example, references [685, 686]). Another approach, based on the Newton’s method, was proposed in [188]. This procedure may be used for obtaining periodic solutions for equations on spatial domains of arbitrary dimension as well as quasiperiodic solutions for equations on one- and two-dimensional domains. However, an application of both KAM-theory and the Newton method yield only a qualitative estimation of solutions. They are rather used as a tool for proofs of solutions existence theorems rather, than to constructions of trajectories. Therefore, such methods as Ritz, Bubnov–Galerkin and harmonic balance are of wider use [429, 430, 431, 439]. Unfortunatelly, while applying these approachies a serious problem related to truncation of the series appears, which need a careful considerations. One of the most popular analytical approaches for studying non-linear structural vibrations are perturbation methods [216, 217, 380, 381, 494, 513, 648]. The methods have returned to the Lindstedt-Poincaré procedure, which was employed by earlier astronomers and laid the foundations of the modern perturbation theory. It allows the periodic vibrations to be determined directly. In contrast to the classical Lindstedt-Poincaré technique, the method of multiple time scales can provide more general solutions (periodic as well as quasiperiodic), which are able to treat internal resonance phenomena and to investigate stability of motions [173, 174, 175, 218, 415, 421, 512]. Detailed investigations on comparing the multiple time scales procedure with low order Bubnov–Galerkin’s method are presented in references [414, 510]. Continuous structures are of infinite degrees of freedom. In this case, use of known methods leads to infinite systems of non-linear algebraic or ordinary differential equations. For its solution truncation procedures are applied. Many studies have been restricted by considering only a few (usually two) mode representations. However, in some instances neglecting the subsequent modes without justification is not suitable and may even cause significant errors. For example, it has been showed in [173, 174, 175] that for bridges under non-linear drag and lift loads caused by wind flow at least four modes have to be taken into account for some values of restoring force. Here an asymptotic approach for deriving periodical solutions of non-linear vibration problems of continuous structures is proposed. An advantage of the pro-


105

cedure is the possibility of taking a number of vibration modes into account. As examples, free longitudinal vibrations of a rod (the second order PDE) and lateral vibrations of a beam (the fourth order PDE) under a cubically non-linear restoring force αu + εu3 are considered. Here u is a displacement, α ≥ 0, ε 0. Studying free vibrations provides one with the basic knowledge of the proper characteristics of a structure, and this is the starting point for investigations of more complicated dynamical problems, such as forced vibrations, dissipation effects, etc. Initially, the well-known perturbation technique is used and the independent variable u and frequency expanded in power series of the natural small parameter ε. It leads to infinite systems of the interconnected non-linear algebraic equations governing relationships between mode amplitudes and frequencies. For its solution, a non-trivial asymptotic approach is used, based on the introduction of an artificial small parameter. As the result, resonance interactions between different modes are investigated and the corresponding backbone curves are evaluated. Free longitudinal vibrations of a clamped rod in non-linear elastic medium are considered. The governing equation can be written in the form ES ¯u¯x ¯x = ρS ¯ut¯t¯ + F ¯u,

(2.202)

where: S is the area of cross-section, and F(u) is the restoring force per unit length acting on the rod from the surrounding medium. The length of the rod equals l. Suppose that the non-linear force has a symmetric characteristic and can be expanded in a Taylor series with F (¯u) = g1 ¯u + g3 ¯u3 + g5 ¯u5 + ... . This expansion is restricted by two leading terms.

1/2 Let the dimensionless variables x = πl¯x, u = Sl¯u, t = πE/ρl2 t¯be introduced equation (2.202) becomes (2.203) u xx = utt + αu + εu3 , where: α = g1 l2 / π2 ES , ε = g3 S/ π2 E . Restoring force is supposed to be weakly non-linear, so that ε 0. The case of one potential well is studied, α ≥ 0. As an illustrative example consider the case of zero initial displacement, although the proposed method can be extended for different initial conditions. Now the input boundary value problem can be formulated as follows: equation (2.203) with conditions u (0, t) = u (π, t) = 0, u (x, 0) = 0, ut (x, 0) = u∗ (x) .

(2.204)

In the linear case, displacement u can be found as superposition ul =

∞ i=1

ai sin (ix) sin ωli t ,

(2.205)

106


√ where: ωli = i2 + α. Next, stationary solutions of the non-linear problem (2.203), (2.204) are sought. Let a change of the time scale be introduced τ = ωt

(2.206)

and present the solution in the form of asymptotic expansions u = u0 + εu1 + ε2 u2 + . . . ,

(2.207)

ω2 = ω20 + εξ1 + ε2 ξ2 + . . . ,

(2.208) √ where: ω0 = ωl1 = 1 + α. Substituting series (2.207), (2.208) into problem (2.203), (2.204) and splitting it with respect to gives a recurrent system of linear PDEs: u0xx = ω20 u1ττ + αu0 ,

(2.209)

u1xx − ω20 u1ττ − αu1 = ξ1 u0ττ + u30 ,

(2.210)

.

.

. .

.

.

. .

.

.

The boundary conditions can be rewritten as follows: ui (0, τ) = ui (π, τ) = 0, i = 0, 1, 2, . . . .

(2.211)

Solution of the boundary value problem (2.209), (2.211) yields the O(1) approximation:  lin  ∞ ∞  ω  u0 = ai sin (ix) sin  i τ = ai sin (ix) sin (Ωi t). (2.212) ω0 i=1 i=1 In the linear case expression (2.212) coincides with representation (2.205). For non-linear vibrations, unknown frequencies of the modes with respect to time t are + ωlin i2 + α i (1 + α) + εξ1 + ε2 ξ2 + . . ., i = 1, 2, 3, . . . . Ωi = ω= (2.213) ω0 1+α The next approximation is derived from the boundary value problems (2.210) and (2.211). The terms containing sin(ix) sin(ω lin τ/ω0 ) in the right-hand side of equation (2.210) will produce secular terms, which should not be parts of the uniformly valid expansion (2.207). In order to eliminate the secular terms, coefficients of sin(ix) sin(ωlin τ/ω0 ) have to be zero. This condition leads to an infinite system of non-linear algebraic equations ∞ ∞ ∞ i2 + α Ci(k,l,m) ak al am , i = 1, 2, 3, . . . . (2.214) = ai ξ1 (1 + α) k=1 l=1 m=1


107

Here coefficients C i(k,l,m) are evaluated after substituting expression (2.214) into the right-hand side of equation (2.210) and expanding it using the goniometric relation sin β sin γ sin θ = 0.25 (sin (β + γ − θ) − sin (β − γ − θ) − sin (β + γ + θ) + sin (β − γ + θ)) . (2.215) Solution of system (2.214) provides the next term ξ 1 of the frequency expansion (2.208) and allows relationships between the amplitudes a i to be obtained. Having known ξ1 and ai the term u1 can be determined from the boundary value problems (2.210) and (2.211). Later, the asymptotic procedure can be continued routinely for evaluating higher order approximations. Infinite systems such as (2.214) may be obtained in various ways (e.g., by means of Bubnov–Galerkin method [214, 414, 430, 451, 510], by multiple time scales technique [173, 174, 175, 218, 415, 421, 512] or by the averaging procedure [494]. Existence of non-trivial solutions in the case of internal resonance is shown in papers [188, 295, 569, 685, 686]). In practical problems, these systems are usually treated by truncation procedures, and many studies consider only a few mode representations. Meanwhile, in some instances, neglecting the subsequent modes leads to losing information of higher order internal resonances, which can produce significant errors in the solution. Due to this, solving system (2.214) should be started with a detailed investigation of probable mode interactions, and truncation can be allowed only for those modes which are not involved in the resonance coupling. Here, the asymptotic approach for analytical solution of the non-linear system is proposed. This procedure gives the possibility of taking an arbitrary number of modes into account. The proposed technique is based on the introduction of an artificial small parameter. The artificial small parameter µ is introduced into the right-hand side of system near the terms satisfying condition (k > i) ∪ (l > i) ∪ (m > i). For example, consider three leading equations of system (2.214) a1 ξ1 − C1(1,1,1) a31 = C1(1,1,2) a21 a2 + C1(1,2,2) a1 a22 + C1(2,2,2) a32 + C1(1,1,3) a21 a3 + C1(1,3,3) a1 a23 + C1(1,2,3) a1 a2 a3 + C1(2,2,3) a22 a3 + C1(2,3,3) a2 a23 + C1(3,3,3) a33 + . . . , a2 ξ1

(2.216)

(4 + α) − C2(111) a32 − C2(1,1,2) a21 a2 − C2(1,2,2) a1 a22 − C2(2,2,2) a32 = (1 + α) C2(1,1,3) a21 a3 + C2(1,3,3) a1 a23 + C2(1,2,3) a1 a2 a3 + C2(2,2,3) a22 a3 + C2(2,3,3) a2 a23 + C2(3,3,3) a33 + . . . , a3 ξ1

(9 + α) − C3(111) a31 − C3(1,1,2) a21 a2 − C3(1,2,2) a1 a22 − (1 + α)

(2.217)

108


C3(2,2,2) a32 − C3(1,1,3) a21 a3 − C3(1,3,3) a1 a23 − C3(1,2,3) a1 a2 a3 − C3(2,2,3) a22 a3 − C3(2,3,3) a2 a23 − C3(3,3,3) a33 = . . . ,

(2.218)

‘Small parameter’ µ is introduced near the right hand terms of the equations (2.216)–(2.218). This perturbation is such that at µ = 0 system (2.216)–(2.218) turns into a triangular form and is reduced to a recurrent equations sequence, and at µ = 1 it restores to the original form (2.216)–(2.218). Here, ξ1 can be considered as a kind of eigenvalue, which is associated with the specific set of eigensolutions a i . Perturbed eigensolutions of system (2.214) are sought near the given unperturbed eigensolutions. Unperturbed eigenvalue ξ 1(0) can be obtained from the equation (2.216) when µ = 0: ξ1(0) = C1(1,1,1) a21 .

(2.219)

The physical sense of solution (2.219) is that interconnections between different modes are neglected and vibrations in only one fundamental mode with amplitude a 1 are considered. The following formal asymptotic expansions in powers of µ starting from the unperturbed eigensolution are proposed ξ1 = ξ1(0) + µξ1(1) + µ2 ξ1(2) + ..., (1) 2 (2) a j = a(0) j + µa j + µ a j + ...,

j = 2, 3, 4, . . . .

(2.220)

Starting values of amplitudes a (0) j are calculated by substituting expression (2.219) into the subsequent (second, third and so on) equations of system at µ = 0. In the subsequent approximations by µ, representations (2.220) allow modes resonance interactions to be taken into account and the eigenvalue ξ 1 to be defined. Calculating coefficients in expansions (2.220) it is supposed finally that µ = 1. Now an investigation of system (2.214) is considered. On the basis of expression (2.212) and relation (2.215) one could determine that only some terms a k al am with specific combinations of k, l, m contribute to the right-hand side of system (). So,     ±k ± l ± m = ±i, (k,l,m) √ √ √ √ 0 if  Ci (2.221)   ± k2 + α ± l2 + α ± m2 + α = ± i2 + α. In general case α ∈ (0, ∞) system (2.214) can be written as follows:  i−1  ∞ 9 3 3  2 2  i2 + α ai ξ1 = a + ai  a + a  , i = 1, 2, 3, . . . . 1+α 16 i 4  k=1 k k=i+1 k 

(2.222)

Infinite algebraic systems with cubic non-linearity such as (2.214) may have three non-trivial solutions, which describe different resonance interactions between modes. However, in the case under consideration α ∈ (0, ∞) simple numerical analysis can show that system (2.222) does not have real solutions describing resonance interactions: all three non-trivial solutions are imaginary. If real roots are sought


109

by means of the approach of artificial small parameter, then it would be found that expansion (2.220) diverges rapidly. The only possible solutions are: ξ1 =

9 (1 + α) 2 ( a , i = 1, 2, 3, . . . , ' 16 i2 + α i

a j = 0,

j = 1, 2, 3, . . . ,

(2.223)

j i.

Relations (2.223) correspond to the case when in the O(1) approximation the rod is able to vibrate in only one ith mode (i = 1, 2, 3, . . .). These vibrations are periodical with frequency   √   9 a2i 2 Ωi = i + α 1 + ε + O ε2 . (2.224) 2 32 i + α The amplitude of the excited mode a i can be evaluated from initial conditions: 2 3 3 3 π 4 2 ai = u∗ (x)2 dx. (2.225) πΩ2i 0

Amplitudes of all other modes equal zero: a j = 0, j = 1, 2, 3, . . ., j i. In the O(1) approximation more than one mode cannot be excited simultaneously. The structure could be reduced to a one-degree-of-freedom oscillator. In this case, mode interactions may take place only between modes with non-zero initial energy (up to O(ε)). So, if one starts with zero initial energy in the jth mode there will be no energy present up to O(ε). This allows truncation to the modes with non-zero initial energy. When the linear part of restoring force is absent (α = 0), extra contributions to the right-hand side of system (2.214) occur. System (2.214) has the form a1 ξ1 −

9 3 3 2 a1 = a1 a2 + a23 + a24 + a25 + 16 4

3 + (a1 a2 a4 + a2 a3 a4 + a1 a3 a5 + a2 a4 a5 ) + 8 3 2 + a1 a3 + a22 a3 + a22 a5 + a23 a5 + ..., 16 4a2 ξ1 −

(2.226)

9 3 3 2 3 a2 − a1 a2 = a2 a23 + a24 + a25 + 16 4 4

3 + (a1 a2 a3 + a1 a3 a4 + a1 a2 a5 + a1 a4 a5 + a3 a4 a5 ) + 8 3 2 a1 a4 + a23 a4 + ..., 16

(2.227)

110


9a3 ξ1 −

3 1 3 9 3 3 2 3 a1 − a3 − a3 a1 + a22 − a1 a22 = a3 a24 + a25 + 16 16 4 16 4

3 2 3 (a1 a2 a4 + a2 a3 a4 + a1 a3 a5 + a2 a4 a5 ) + a1 a5 + a24 a5 + ..., 8 16 16a4 ξ1 −

(2.228)

3 9 3 3 2 3 2 a4 − a4 a1 + a22 + a23 − a1 a2 a3 − a1 a2 + a2 a23 = 16 4 8 16

3 2 3 3 a4 a25 + (a2 a3 a5 + a3 a4 a5 ) + a2 a1 + a2 a23 + ..., 4 8 16 25a5 ξ1 −

(2.229)

3 9 3 3 2 a5 − a5 a1 + a22 + a23 + a24 − (a1 a2 a4 + a2 a3 a4 ) − 16 4 8 −

3 2 a1 a2 + a21 a3 + a23 a1 + a3 a24 = ... 16 . . . . . . . . . .

(2.230)

Note that expressions (2.223) satisfy system (2.226)–(2.230). Besides this, system (2.226)–(2.230) may have three non-trivial solutions describing resonance interactions. In the case under consideration (α = 0) two of them are imaginary, and only one solution is real and has got physical sense. For its evaluation an artificial small parameter µ is introduced near the right hand sides of equations (2.226)–(2.230). The solution is sought as asymptotic series (2.220). Being restricted by two leading terms in expansion (2.220), one obtains a j = 0, j = 2, 4, 6, . . . , a3 = 1.449 · 10 −2 a1 , a5 = 2.071 · 10 −4 a1 , . . . , ξ1 =

3 9 2 3 2 3 3 a23 a5 a1 + a3 + a25 + a3 a5 + a1 a3 + + ... ≈ 0.565376a 21. (2.231) 16 4 8 16 16 a1

Here, the approach of an artificial small parameter yields very accurate results. In Table 2.5, expressions (2.231) are compared with numerical solutions of the non-linear system (2.227)–(2.229). The numerical data were calculated in the Mathematica program package by truncating system (2.231) to five leading equations. Table 2.5. Comparison of asymptotic solution with numerical data. Variable ξ1 a2 a3 a4 a5

Asymptotic solution (2.231) 0.565376a21 0 a3 = 1.449 · 10−2 a1 0 a5 = 2.071 · 10−4 a1

Numerical results 0.565360a21 0 a3 = 1.442 · 10−2 a1 0 a5 = 2.049 · 10−4 a1


111

According to expressions (2.231), in the O(1) approximation an infinite number of all odd modes are involved in resonance interactions. This provides energy transfers between odd modes, and the truncation may not be valid. The physical meaning of this phenomenon is that if the rod initially vibrates in a high mode, then low modes can be excited. This can lead to large-amplitude oscillations of the structure. Taking into account relations (2.231), mode frequencies can be expressed as functions of the fundamental amplitude (2.232) Ωi = i 1 + 0.282688a 21ε + O ε2 , i = 1, 3, 5, . . . , where: a1 is determined from the initial conditions 2 2 3 3 3 3 3 3 π π 4 4 2 2 2 ∗ (x) dx − 9a2 − 25a2 − ... ≈ a1 = u u∗ (x)2 dx. 3 5 πΩ21 1.001891πΩ 21 0

0

It should be pointed out that for α = 0 the ratio of frequencies of interacting modes equals the ratio of their wave numbers: Ωm m = , m, n = 1, 3, 5, . . . . Ωn n

(2.233)

Therefore frequencies of all excited modes are commensurable with each other. In this case the O(1) approximation formula (2.212) describes periodical vibrations with the general period T = 2π/Ω 1 . Now we consider the behaviour of the rod near the resonance. In order to introduce detuning it is supposed that α → 0, but α 0. Here, the parameter α shows how far the structure is from the pure resonance state. Changing the scale of time (2.206), the solution of the input boundary value problem (2.203), (2.204) as asymptotic expansions by powers of a is as follows: u = u(0) + αu(1) + α2 u(2) + ...,

(2.234)

ω = ξ(0) + αξ(1) + α2 ξ(2) + ... .

(2.235)

Here, each term is represented by a series u(n) = u(n,0) + εu(n,1) + ε2 u(n,2) + ...,

(2.236)

ξ(n) = ξ(n,0) + εξ(n,1) + ε2 ξ(n,2) + ..., n = 0, 1, 2, . . . ,

(2.237)

(0,0)

where: ξ = 1. Splitting problem (2.203), (2.204) with respect to α and ε gives the recurrent sequence of equations (0,0) (2.238) u(0,0) xx − uττ = 0, 3 (0,1) (0,1) (0,0) uττ + u(0,0) , (2.239) u(0,1) xx − uττ = 2ξ (1,0) (1,0) (0,0) u(1,0) uττ + u(0,0) , xx − uττ = 2ξ

(2.240)

112


(1,1) (1,0) (0,1) (0,0) u(1,1) uττ + u(0,1) + 2 ξ(0,1) ξ(1,0) + ξ(1,1) uττ + xx − uττ = 2ξ 2 (1,0) +2ξ(0,1) uττ + 3 u(0,0) u(1,0) .

.

. .

.

.

. .

.

(2.241)

.

with boundary conditions u(n,m) (0, τ) = u(n,m) (π, τ) , m, n = 0, 1, 2, . . . .

(2.242)

Solution of the boundary value problems (2.238) and (2.242) provides u(0,0) =

∞

Ai sin (ix) sin (iτ).

(2.243)

i=1

The next approximation u (0,1) is evaluated from the boundary value problems (2.239) and (2.242). In order to prevent secular terms in expansion (2.236), coefficients of sin ix sin iτ in the right-hand side of equation (2.239) have to be equated to zero. This condition leads to an infinite system of non-linear algebraic equations 2Ai ξ(0,1) i2 =

∞ ∞ ∞

Ci(k,l,m) Ak Al Am , i = 1, 2, 3, . . . .

(2.244)

k=1 l=1 m=1

System (2.244) gives the solution A j = 0, j = 2, 4, 6, . . . , A3 = 1.449 · 10 −2 A1 , A5 = 2.071 · 10 −4 A1 , . . . , ξ(0,1) =

3 9 2 3 2 A1 + A3 + A25 + A3 A5 + 32 8 16

3 3 A23 A5 A1 A3 + + ... ≈ 0.282688A 21. 32 32 A1

(2.245)

Function u(0,1) can be represented as the harmonic expansion by τ: u(0,1) =

∞

fi (x) di(1) sin iτ + di(2) cos iτ ,

(2.246)

i=1

The term u(0,1) is determined from the boundary value problems (2.240) and (2.242). Eliminating secular terms in expansion (2.234), the right-hand side of equation (2.240) yields 1 ξ(1,0) = 2 , (2.247) 2i where i is the mode number. Then u(1,0) =

∞ i=1

Bi sin (ix) sin (iτ).

(2.248)


113

The boundary value problems (2.241) and (2.242) allow u (1,1) to be evaluated. Satisfying the condition of absence of secular terms in expansion (2.236), the righthand side of equation (2.241) gives an infinite linear system for coefficients B i 9 2 1 2 L (A1 , B1 ) = A + L1 (A1 ) + A (A1 A3 + A2 A4 + A3 A5 ) B1 + 3 16 1 8 1 1 A1 A2 + (A2 A3 + A1 A4 + A3 A4 + A2 A5 + A4 A5 ) B2 + 2 8 1 1 M1 + A1 A3 + (A2 A4 + A1 A5 + A3 A5 ) B3 + 2 8 1 1 A1 A4 + (A1 A2 + A2 A3 + A2 A5 ) B4 + 2 8 1 1 M1 + A1 A5 + (A1 A3 + A2 A4 ) B5 + ..., 2 8 1 1 8 L (A2 , B2) = A1 A2 + (A2 A3 + A1 A4 + A3 A4 + A2 A5 + A4 A5 ) B1 + 3 2 8 1 9 2 A2 + L1 (A2 ) + (A1 A3 + A1 A5 ) B2 + 16 8 1 1 A2 A3 + (A1 A2 + A1 A4 + A3 A4 + A4 A5 ) B3 + 2 8 1 1 M2 + A2 A4 + (A1 A3 + A1 A5 + A3 A5 ) B4 + 2 8 1 1 A2 A5 + (A1 A2 + A1 A4 + A3 A4 ) B5 + ..., 2 8 1 1 6L (A3 , B3 ) = M1 + A1 A3 + (A2 A4 + A1 A5 + A3 A5 ) B1 + 2 8 1 1 A2 A3 + (A1 A2 + A1 A4 + A3 A4 + A4 A5 ) B2 + 2 8 1 9 2 1 2 A3 + A1 + A22 + A24 + A25 + (A2 A4 + A1 A5 ) B3 + 16 4 8 1 1 A3 A4 + (A1 A2 + A2 A3 + A2 A5 + A4 A5 ) B4 + 2 8 1 1 M3 + A3 A5 + (A1 A3 + A2 A4 ) B5 + ..., 2 8

114


1 32 1 L (A4 , B4 ) = A1 A4 + (A1 A2 + A2 A3 + A2 A5 ) B1 + 3 2 8 1 1 M2 + A2 A4 + (A1 A3 + A1 A5 + A3 A5 ) B2 + 2 8 1 1 A3 A4 + (A1 A2 + A2 A3 + A2 A5 + A4 A5 ) B3 + 2 8 1 9 2 1 2 A4 + A1 + A22 + A23 + A25 + A3 A5 B4 + 16 4 8 1 1 A4 A5 + (A1 A2 + A2 A3 + A3 A4 ) B5 + ..., 2 8 50 1 1 L (A5 , B5) == M4 + A1 A5 + (A1 A3 + A2 A4 ) B1 + 3 2 8 1 1 A2 A5 + (A1 A2 + A1 A4 + A3 A4 ) B2 + 2 8 1 1 M3 + A3 A5 + (A1 A3 + A2 A4 ) B3 + 2 8 1 1 A4 A5 + (A1 A2 + A2 A3 + A3 A4 ) B4 + 2 8 9 2 A + L1 (A5 ) B5 + ... , 16 5 . . . . . . . . . . where:

(2.249)

L (Ai , Bi ) = ξ(0,1) ξ(1,0) Ai + ξ(1,1) Ai + ξ(0,1) Bi , L1 (Ai ) = 0.25

5

A2k ,

ki

M3 =

M1 =

1 2 A1 + A22 , 16

1 2 A1 + A24 , 16

M5 =

M2 =

1 2 A2 + A23 , 16

1 2 A2 + A23 , 16

It provides B i and ξ (1,1) : B j = 0, j = 2, 4, 6, . . . , B3 = 1.443 · 10 −2 B1 , B5 = 2.004 · 10 −4 B1 , . . . , A21 . (2.250) i2 Even in the pure resonance case α = 0 all odd modes take part in resonance interactions in the O(1) approximation. From expressions (2.245), (2.247) and (2.250), the asymptotic formula is given for unknown frequencies as: ξ(1,1) ≈ 0.565352A 1 B1 − 0.141344


115

  √ A21 i2   2  Ωi = i + α 1 + 0.282688 2 ε + 0.565352iA 1 B1 εα− i +α A21 εα + O (ε) + O (α) + O (εα) , i = 1, 3, 5... . (2.251) i2 The mode amplitudes being equal to a i = Ai + αBi , a1 is evaluated by formula (2.232). For simplicity it can be assumed that B i = 0, ai = Ai . The solution correctly represents the behaviour of the structure in ‘limiting’ cases. So, for ε = 0 formula (2.251) is in agreement with relations (2.224), and for α = 0 formula (2.251) coincides with expression (2.231). It should be noted that in the detuning case the ratio of frequencies of interacting modes differs from the ratio of their wave numbers: −0.141344

Ωm m = + O (α) , Ωn n

m, n = 1, 3, 5, . . . .

(2.252)

Actually, in this case the ratios of the frequencies (2.252) may be irrational numbers. The general solution (2.212) can then describe quasiperiodical motions. The asymptotic approach can be extended to study the solutions of the fourth order PDE. For example free vibrations of a simply supported beam on a non-linear elastic foundation are considered. The governing equation can be written as follows: EIw xxxx + ρS wtt + F (w) = 0,

(2.253)

where: w is the lateral displacement; I is the moment of inertia of the cross section. As in the previous example, the non-linear restoring force per unit length is ¯ + g3 w ¯3 . assumed to be in the form F( w ¯ ) = g1 w √ 2 EI t¯ , the input Introducing the dimensionless variables x = πl¯x, w = Sw¯ l, t = πl2 √ρS boundary value problem may be formulated as follows: w xxxx + wtt + αw + εw3 = 0, w (0, t) = w (π, t) = 0, w(x, 0) = 0,

w (O, t) xx = w xx (π, t) = 0,

(2.254) (2.255)

∗

wt (x, 0) = w (x) .

where: α = g1 l4 /π4 EI, ε = g3 l2 S 2 /π4 EI if ε 1 and α ≥ 0. For solving problem (2.254), (2.255) the asymptotic technique is used. All intermediate evaluations remain the same, and only final results are displayed below. In the O(ε0 ) approximation w=

∞

Di sin (Ωi t) sin (ix) + O (ε) .

(2.256)

i=1

In the general case for α = [0, ∞), α m 2 n2 , m, n = 1, 3, 5, . . ., there are no mode interactions up to O(ε). Frequencies Ω i are determined as follows:   √  9 D2i   + O ε2 , i = 1, 2, 3, . . . , Ωi = i4 + α 1 + ε (2.257) 4 32 i + α

116


where amplitudes are ai =

2 3 3 3 4

2 πΩ2i

π

w∗ (x)2 dx.

(2.258)

0

The structure can be truncated to the modes with non-zero initial energy. Resonance coupling in the O(1) approximation can occur only for two odd modes m and n if m4 + α Ωm m = , m, n = 1, 3, 5, . . . . (2.259) = n Ωn n4 + α Equation (2.259) determines the specific values of α which allow interactions and energy transfers between the mth and nth modes: α = m 2 n2 . In this case, independent of the distribution of the initial energy, both the mth and nth modes can be excited. For example, when α = 9, there is a coupling of the first and third modes, and D3 = 1.449 · 10 −2 D1 , √ ε 9 2 3 3 2 Ωi = i 1 + α 1 + D1 + D1 D3 + D3 + O ε2 , i = 1, 3. (2.260) 32 32 8 1+α The asymptotic relations (2.260) may be rewritten in the form √ Ωi = i 10 1 + 0.0282679A 21ε + O ε2 , i = 1, 3.

(2.261)

where the fundamental amplitude D 1 is evaluated from the initial conditions: 2 2 3 3 3 3 3 3 π π 4 4 2 2 2 2 ∗ D1 = w (x) dx − 9a3 = w∗ (x)2 dx. (2.262) πΩ21 1.001869πΩ 21 0

0

In the detuning case α → 9, α 9 condition (2.259) is not satisfied strictly. The frequencies Ω1 , Ω3 are determined as follows: √  A2  √ 1+α Ωi = i  1 + α + ' 4 ( (α − 9) + 0.282679 √ 1 ε+ 2 i +α 1+α  2 A1  A1 B1 (α − 9) ε − 0.141339 √ 0.565357 √ '4 ( (α − 9) ε + 1+α 1+α i +α O (ε) + O (α − 9) + O (ε (α − 9)) , i = 1, 3. (2.263) The mode amplitudes are a i = Ai + (α − 9)Bi; here A3 = 1.441 · 10 −2 A1 , B3 = 1.441 · 10 −2 B1 . One can suppose that B i = 0, Di = Ai . ai is given by expression (2.262). Equation (2.263) correctly represents ‘limiting’ cases: for ε = 0 it corresponds to relation (2.256) and for α = 9 it takes the form of equation (2.261). Observe that the described method of a use of an artificial parameter can be treated as the supplementary one to truncated approach. A simultaneous application of these two approaches gives hope to obtain the real solution.

3 SINGULAR PERTURBATION PROBLEMS

In this chapter the problems when the small parameter stands by a highest order derivatives are considered. Note that for ε = 0 a qualitative change of the system occurs since the system order of the analysed differential equation is decreased. The similar like asymptotics is called the singular one.

3.1 The method of Gol’denveizer-Vishik-Lyusternik [313, 645, 672, 673, 674] Consider the first simple problem εz + z = 1, z (0) = 0,

(3.1)

with the following exact solution z = 1 − exp −ε−1 x . Observe that for ε = 0 one gets z 0 = 1, which does not satisfy the boundary condition; both exact and zero order approximation solutions are shown in Figure 3.1. The error is located essentially in the ε neighbourhood of the point x = 0. The domain called boundary layer shrinks with decrease of ε. Note that there is no way to obtain a solution to singularly perturbed system using the direct series application z = z0 + εz1 + ε2 z2 + ... .

(3.2)

One should, in addition, use a boundary layer part of solution exp(−ε −1 x), which should change rapidly with respect to the corresponding coordinate. The particularities of the singular cases are analysed using the following equation (3.3) εz + z + z = 0 with the boundary conditions z (0) = 1, z (0) = 0.

(3.4)

A solution far from the point t = 0 is sought in the form of the series (3.2). Substituting (3.4) into (3.3) and carrying out the splitting with respect to ε one gets

118


~e

Fig. 3.1. Typical behaviour of a solution with occurrence of a boundary layer.

z 0 + z0 = 0,

(3.5)

z 1 + z1 = −z 0 , ... .

(3.6)

Using equations (3.5) only one initial condition can be satisfied. In order to satisfy the boundary conditions (3.4) a boundary layer solution z b should be additionally constructed. It is known that the solution rapidly changes with respect to x, but a speed of the change is not yet known. Let us introduce the parameter of asymptotic integration α using the following relation: z b ∼ ε−α z. a > 1, limiting equation z b = 0, a = 1, limiting equation z b + z b = 0, a < 1, limiting equation z b = 0. Since the second case includes two other, one may take α = 1. A solution of the boundary layer type reads zb (x, ε) = ε p [zb0 (x) + εzb1 (x) + ...] . Note that a parameter p is not yet defined, since the order of smallness of the boundary layer is not yet known. It depends on the smallness order of unperturbed solution, and a relation between z 0 and zb is realized via boundary conditions. Substituting zb into (3.3) and carrying out the splitting with respect to ε one obtains ε p : ¯z 0 + ¯z 0 = 0, (3.7) .

ε p+1 :

zb1 + z b1 = −zb0 ,

.

.

. .

.

. .

.

(3.8)

.

Formally, equations (3.7), (3.8) has the second order. However, since we are looking for rapidly changed solution, the equations (3.7) may be replaced by the following ones: εz b0 + zb0 = 0,

3.1 The method of Gol’denveizer-Vishik-Lyusternik [313, 645, 672, 673, 674]

εz b1 + zb1 = − .

.

. .

.

.

119

x zb0 dx, 0

. .

.

(3.9) .

because slow changed solution components belongs to z 0 , z1 , etc. The equations (3.9) are refereed as the boundary layer equations. Now the boundary conditions for equations (3.5), and (3.9) are formulated: (z0 + εz1 + ...) + ε p (zb0 + εzb1 + ...) = 1, z 0 + εz 1 + ... + ε p z b0 + εz b1 + ... = 0 for x = 0.

(3.10)

Since zb ∼ ε−1 zb , the first term in the second bracket in (3.10) has the order ε . We are going to find p, for which the boundary conditions for equations (3.5) and (3.9) have a solution. It means, that for each of the equations from the given recurrent series only one boundary condition should be attached. If p > 1 (p < 1) then both of boundary conditions go to equations (3.5) ((3.9)). Therefore, p = 1 should be chosen. Taking p = 1, comparing the terms standing by the same powers of ε equating them zero and taking into account that from equation (3.5) we have z 0 = −z0 , one gets (3.11) z0 = 1; z b0 = z0 = 1; p−1

z 1 = z0 = 1; z b1 = zb0 ; for x = 0, .

.

. .

.

.

. .

.

(3.12)

.

An experience in application of asymptotical methods indicates that a fundamental part of an error is introduced not by splitting of the initial equation, but owing to splitting of boundary conditions. With this respect, Yu.M. Vachromyev and V.M Kornev [653] proposed a procedure of boundary layer removal. In many cases a solution describing boundary layer can be construct in an analytical way. For instance, in our case z b0 = C1 exp(−ε−1 x). Owing to the boundary condition (3.12) one gets C 1 = 1. Now a condition for z 0 can be transformed into the following one: z0 = 1 − ε.

(3.13)

Observe that owing to asymptotics, conditions (3.11) and (3.13) are equivalent. However, a real error introduced by condition (3.13) is smaller. Note that even for relatively complicated problems the boundary layer often is governed by ordinary differential equations with constant coefficients. A reason is mainly caused by a possibility of ‘freezing’ of the variable coefficients in the boundary layer equations. The method of ‘freezing’ is further explained using an example of a pendulum with small variable mass: εϕ (t) x¨ + x˙ + x = 0, x (0) = x0 , x˙ (0) = 0, where: 0 < ϕ(t) < 1, ϕ(t) ∼ ϕ(t). ˙ For ε = 0 one obtains:

120


x˙ + x = 0.

(3.14)

The boundary layer equation has the form εϕ (t) x¨b + x˙b = 0.

(3.15)

The equation (3.15) includes time depended coefficient. However, it can be treated as the constant, i.e. one may assume in the first approximation ϕ(t) = ϕ(0). Really, function ϕ(t) changes slowly, and one can assume it constant on the sport distance, where boundary layer solution is essential. Therefore, the following equation with constant coefficients after neglecting of slow solution of equation (3.15) εϕ (0) x˙b + xb = 0

(3.16)

is used to define the boundary layer. Note that components of boundary layer type have the ε order in comparison with the fundamental solution. However, although the boundary layer effect may be neglected during displacement estimation, since x = x 0 + O(ε), it should be included while estimating the velocity x, i.e. x˙ = x˙0 + x˙b + O (ε) . Since in the theory of plates and shells the derivative define stresses, it is in practice impossible to get a full of stress-strain state without taking into account a boundary layer (edge effect). Note that also if a boundary layer is described by ordinary differential equation with constant coefficients, then the negative real values of the roots of the characteristic equation guarantees a solution regularity [672]. Now an example of application of boundary layer method to solution of nonlinear problem with small parameter standing by highest derivatives is illustrated. Then Van der Pol pendulum with small masses is governed by the equation ε x¨ + 1 − x2 x˙ + x = 0, (3.17) x (0) = a, x˙ (0) = 0, a = const.

(3.18)

It is assumed, that a < 1, but a ∼ 1, which means that the coefficient standing by the second term of equation (3.17) is not small one. For ε = 0 1 − x20 x˙0 = −x0 , and after integrating of this equation one obtains ln |x0 | = 0.5x20 + t + C. In this case, a construction of a boundary layer becomes difficult. Note also, that contrary to the linear cases, in spite of exponential boundary layers one may construct other boundary layers, like, for example, power-type boundary layer. In

3.1 The method of Gol’denveizer-Vishik-Lyusternik [313, 645, 672, 673, 674]

121

the considered case, taking into account a solution of the linearized problem, the following boundary layer change is taken ∂xb ∼ ε−1 xb . ∂t

(3.19)

Therefore, x b ∼ εx0 , and the splitted initial conditions have the form x0 (0) = a,

(3.20)

a . (3.21) 1 − a2 The assumed solution form x = x 0 + xb is substituted in the equation (3.17), taking into account the relation (3.19), as well as the smallness of x b in comparison with x0 (xb ∼ ε x¨0 ), the following linear equation is obtained (3.22) ε x˙b + 1 − x20 xb = 0. x˙b (0) = −

The slow-variable coefficient x 20 should be ‘freezen’ on the boundary, and one finally obtains ε x˙b + 1 − a2 xb = 0. The general solution of the latter equation has the form xb = C1 exp − 1 − a2 ε−1 . The constants C and C 1 are found from the splitted boundary conditions (3.20), (3.21): εa a2 C = ln |a| − , C1 = − ' ( . 2 1 − a2 2 However, an order decrease of the input equation may cause some problems related to satisfaction of the boundary (initial) conditions as well as the occurrences of discontinuities of the solution or their derivatives. In the latter case the solutions localized in the neighborhoods of the points or curves (in the case of partial differential equations) appear, which are called internal boundary layers. Note that the separation of boundary layers essentially simplify the numerical solution. The combined analytical-numerical algorithms, where boundary layer effects are constructed in the analytical way, whereas slow solution part numerically, are often useful [691]. An interior boundary layer belongs to one of most interesting cases. This problem is described via the following example [138], governed by the differential equations dz dy = z; ε = 1 − z2 ; (3.23) dx dx y(0) = y(1) = 0. (3.24) The exact solution of boundary value problem (3.23), (3.24) is easily written as

122


Fig. 3.2. Example of interior boundary layer.

5 1 2x − 1 cosh , y = ln cosh 2ε 2ε 1 − z2 = 0.

z = tanh

2x − 1 , 2ε

(3.25) (3.26)

The asymptotic approach considered here shows that the equation (3.26) has two solutions: z = −1, y = −x + C and z = 1, y = x + C 1 . From the boundary conditions (3.24) one obtains C = 0;

C1 = −1.

The straight lines y = −1 and y = x − 1 intersect at the interior point (0.5, −0.5). In a neighbourhood of this point, there is an internal boundary layer which smoothens the jumps at t = 0.5 (Fig. 3.2). In order to construct an internal boundary layer a matching procedure will be applied 138. Namely, having known a solution to left A l and to right A r from a discontinuity, and also knowing a solution B representing the boundary layer peak compatibility conditions of smooth matching of A r and B, and A l and B in some not knows so far points, are formulated. Note that both coordinates of these points and unknown constants are defined via conditions of smooth matching, i.e. in the mentioned points the corresponding function and their derivatives should coincide.

3.2 Multiscale method The multiple scale method can be also applied to analyse singularity perturbed boundary problems. This method found wide application in averaging and homogenization theories. We illustrate the method using two scale approach and the mathematical pendulum with small mass. The following differential equation is considered ε x¨ + x˙ + x = 0;

(3.27)

3.2 Multiscale method

x (0) = a, x˙ (0) = 0, a ≡ const.

123

(3.28)

Two variables, slow (t 1 = t), and fast (τ = ε−1 t) are taken instead of original one t. The variable t 1 is used during a description of the limiting (ε → 0) state, whereas the variable τ governs the state of a boundary layer. Therefore, one obtains d ∂ ∂ = + ε−1 . dt ∂t1 ∂τ

(3.29)

Note that x depends now on two arguments, t 1 and τ. Substituting (3.29) into (3.27) and (3.28), we get 2 ∂ ∂2 ∂2 x ∂x 2∂ x + ε x + ε + + 1 + 2 = 0, (3.30) ∂τ2 ∂τ ∂t1 ∂τ∂t1 ∂t12 ∂x ∂x +ε = 0. (3.31) x (0, 0) = a; ∂τ ∂t1 τ=t1 =0 The partial differential equation is obtained instead of singularly perturbed ordinary differential equation. Since now the small parameter does not stay by the highest derivative, the solution is sought in the form of the asymptotic series x = x0 (t1 , τ) + εx1 (t1 , τ) + ε2 x2 (t1 , τ) + ... .

(3.32)

Substituting (3.32) into (3.30) and into initial conditions (3.31), the following system of equations is obtained ∂2 x0 ∂x0 = 0, + ∂τ2 ∂τ ∂2 x1 ∂x1 ∂x0 ∂ 2 x0 =− + − x0 − 2 , 2 ∂τ ∂t1 ∂τ∂t1 ∂τ . . . . . . . . . . x0 (0, 0) = a, ∂x0 (0, 0) = 0, ∂τ x1 (0, 0) = 0,

.

∂x1 (0, 0) ∂x0 (0, 0) = , ∂τ ∂t1 . . . . . . . . .

(3.33) (3.34) (3.35) (3.36) (3.37) (3.38)

The general solution (3.33) has the form x0 = C (t1 ) + C1 (t1 ) e−τ . The initial conditions (3.35) and (3.36) yield

(3.39)

124


C1 = 0, C (0) = a.

(3.40)

In order to find the function C(t) the equation (3.34) is rewritten: ∂C ∂2 x1 ∂x1 ∂ 2 x0 +2 + = ϕ (t1 ) = − − C. 2 ∂τ ∂τ∂t1 ∂t1 ∂τ

(3.41)

If ϕ(t1 ) differs form zero, then x 1 includes the secular term ϕ(t 1 )τ. A lack of secularity yields the equation ∂C + C = 0. ∂t1 The last equation together with the condition (3.40) yields C = ae −t1 . The general solution of the equation (3.34) has the form x1 = A (t1 ) + A1 (t1 ) e−τ . The initial conditions (3.37) yield A1 (0) = −A (0) . The function A 1 (t) is defined by a condition of the lack of singularities in the equations of the second order approximation. Function A 1 (t1 ) can be ‘frozen’ for t1 = 0. A particular advantage of two scales method is exhibited while analysing the non-regular singularities. To illustrate the mentioned advantage the following equation is considered (3.42) ε2 y + y + y = 0 with the attached initial conditions y (0) = a, y (0) = 0, y (0) = 0, a = const.

(3.43)

It is easy to check that in this case a solution of (3.42) can not be separated into the fundamental and boundary layer states. Indeed, choosing slow and fast variables in the form x 1 = x and ξ = ε−1 x and expressing y = y0 (x1 , ξ) + εy1 (x1 , ξ) + ε2 y2 (x1 , ξ) + . . . , the following recurrent equations and initial conditions are obtained ∂3 y0 ∂y0 = 0, + ∂ξ ∂ξ3 ∂3 y1 ∂y1 ∂y0 ∂ 3 y0 = − + − y − 3 , 0 ∂ξ ∂x1 ∂ξ3 ∂ξ2 ∂x1 . . . . . . . . . . ∂y0 (0, 0) ∂2 y0 (0, 0) = 0, y0 (0, 0) = a, = 0, ∂ξ ∂ξ2

(3.44) (3.45)

(3.46)

3.3 Newton polygon and asymptotic integration parameters

∂y1 (0, 0) ∂y0 (0, 0) ∂2 y1 (0, 0) =− , = 0, ∂ξ ∂x1 ∂ξ2 . . . . . . . . . .

y1 (0, 0) = 0,

125

(3.47)

From the first equation of (3.44) and taking into account the initial conditions (3.46) one obtains y = C (x1 ) , C (0) = a. The conditions of avoiding the secular terms yield C (x1 ) = ae−x1 . Finally, the fast part of solution to equation (3.45) takes the form y1 = −a sin ξ. In other words, the method of boundary layer is suitable to solve problems where a part of solution is localized in a vicinity of points or curves. The two-scale method is recommended in the case when a solution with significantly different variations is sought and when there is no localization. The multiple scales method is a generalization of two-scales method. One may often take ‘fast’ variable in a more generalized form, i.e. as the function depended on old variable and small parameter (for instance, τ = ϕ(t, ε)).

3.3 Newton polygon and asymptotic integration parameters Here we are going to remind a reader the so-called Newton’s polygon [138, 646]. Let n f s (ε)y s, (3.48) f (y, ε) = s=0

where the coefficients f s (ε) are represented in the vicinity of the point ε = 0 in the form of the convergent series f s (ε) = ερs

∞

frs εr/p ,

(3.49)

r=0

where ρ s are the rational numbers, and p denotes a general natural number for each possible value of f s . Assuming that f00 0, i.e. f (0, ε) 0, we are going to detect the solution y = y(ε) of the equation f (y, ε) = 0, (3.50) represented by the form y = yk εk + O(εk ),

yk 0.

(3.51)

126


In order to find possible values of the exponent k and the coefficient y k , it is necessary to introduce (3.51) into (3.50) and equating a coefficient, standing by the lowest power of ε to zero. However, because the exponent k is not known, therefore it is difficult to judges, which terms are of the lowest order. It is only known that the sought terms belong to the following family f00 ερ0 , ..., f0k yk ερk +k , ..., f0n yn εn ,

(3.52)

where k has the values of 1, 2, . . ., for which f 0l (ε) 0. Because f00 0 and f0n 0, then at least two terms in series (3.52) are different from zero. In order to exclude the lowest terms, we need to choose the exponent k in a such way that at least two of the exponents ρ 0 , ρ1 + k , ρn + nk overlap, and the other ones should be not smaller than the two mentioned above. This condition allows one to find all possible k values and the corresponding values of the coefficients y k . For this purpose Newton polygon is used (Figure 3.3).

Fig. 3.3. The Newton polygon.

The points (0, ρ 0 ), (ell, ρ ), (n, ρn ) are depicted in the Cartesian co-ordinate systems ( corresponds to the same values as in series (3.52)). Let us draw a vertical line including (0, ρ 0 ) and overlapping the co-ordinate axis. Let us now start to rotate it around the point (0, ρ 0 ) against a clock arrow rotation until the second of the depicted points occurs on the line (for instance, (m, ρ m )). The tangents of an angle between that L line position and the negative direction of the horizontal co-ordinate is equal to one of the possible k values defined by the equation tgα = (ρ0 − ρm )/m = k. Now, using α we can define new lines including points (s, ρ s ) different from those lying on L. The new lines are lying above L, and therefore ρ s + k s > ρm + mk .


127

On the line L linking the points (0, ρ 0 ) and (m, ρm ), some additional points (, ρ ) may occur. Now we start to rotate the straight line L around the point (m, ρ m ) (lying on L), for which the abscissa is the largest one until an additional point appears, say (p, ρ p ), on L. The tangents of an angle between a new straight line direction and a negative direction of the abscissa axis defines the second possible value of k: tgα = (ρm − ρl )/( − m) = ε. The straight lines intersecting the other points (s, ρ s ) are parallel to the new L direction and they are situated above, which means that ρ s + k s > ρm + km = ρ p + k p . Proceeding in a similar way all possible values of k may be obtained. The broken line in Figure 3.3 is called the Newton polygon. The diagram corresponding to the first k exponents possesses a general number of links equal to n. In a general case, it can be divided into three parts: a decreasing part, a constant part and an increasing part. The increasing part defines positive values of k. It defines the solutions to equation (3.50) of the form y(0) = 0. The constant part corresponds to the case k = 0 and, according to (3.50), it defines the solution y = y(a) in the form y(ε) = y 0 + 0(ε) for ε = 0, where y 0 = 0. Finally, the increasing part of the Newton polygon leads to detection of the ‘large solution’ to equation (3.50), which approaches infinity for ε → 0 (the values of k are negative). In order to determine stretching and compressing transformations to the linear ordinary differential equation in regard to the abscissa also the orders of derivative n are depicted. The constant parts of Newton polygon correspond to the fundamental state, the increasing parts - to the boundary layers, whereas the decreasing parts - to small roots (their occurrence implies that the initial problems are on the spectrum). For instance, for the equation (3.53) εz + z + z = 0 an application of the Newton polygon gives ad hoc the results such that in the boundary layer one should apply stretching of ε −1 (Fig. 3.4). Recently a very active development of two new and theoretically coupled branches of asymptotical analysis is observed: power geometry [198, 199, 200] and idempotent analysis [392]. Power geometry is natural generalization of Newton polygon. A fundamental idea of the power geometry is focused on investigations of non-linear problems using logarithmic co-ordinates instead of classical ones. Therefore many of the input non-linear properties become linear in new logarithmic co-ordinates. Algorithms of power geometry using those linear relations lead to simplifications of equations and to isolations of their singularities as well as to establish their first approximations and to find either the solution or their asymptotics. In particular, an application of the power geometry methods gives rights to find the asymptotics using different approaches.

128


Fig. 3.4. Newton polygon for equation (3.53).

Effective numerical algorithms related to this field are proposed in [608, 609]. Consider heuristic aspects of the idempotent calculus following reference [443]. There is a correspondence between important, useful and interesting constructions and results over the field of real (or complex) numbers and similar constructions and results over idempotent semirings in the spirit of N. Bohr’s correspondence principle in Quantum Mechanics. It is very important that some problems nonlinear in the traditional sense turn out to be linear over a suitable semiring; this linearity considerably simplifies the explicit construction of solutions. In this case we have a natural analog of the socalled superposition principle in Quantum Mechanics. The theory is well advanced and includes, in particular, new integration theory, new linear algebra, spectral theory and functional analysis. Applications include various optimization problems such as multicriteria decision making, optimization on graphs, discrete optimization with a large parameter (asymptotic problems), optimal design of computer systems and computer media, optimal organization of parallel data processing, dynamic programming, discrete event systems, computer science, discrete mathematics, mathematical logic and so on. Let R be the field of real numbers, R + the subset of all non-negative numbers. Consider the following change of varibales: u → w = h ln u,

(3.54)

where: u ∈ R+ , h > 0; thus u = ew/h , w ∈ R. We have got a natural map Dh : R+ → A = R ∪ {−∞}

(3.55)

defined by the formula (3.54). Denote by the “additional” element −∞ and by the zero element of A (that is = 0); of course = D h (0) and = Dh (1). Denote by Ah the set A equipped with the two operations ⊕ (generalized addition) and (generalized multiplication) borrowed from the usual addition and multiplication in R+ by the map D h ; thus w1 w2 = w1 + w2 and w1 ⊕ w2 = h ln ew1 /h + ew2 /h . Of


129

course, Dh (u1 + u2 ) = Dh (u1) ⊕ Dh (u2 ) and Dh (u1 u2 ) = Dh (u1 ) Dh (u2 ). It is easy to prove that w 1 ⊕ w2 = h ln ew1 /h + ew2 /h → max{w1 , w2 } as h → 0. Let us denote by R max the set A = R ∪ {−∞} equipped with operations ⊕ = max and = +; set = −∞, = 0. Algebraic structures in R + and Ah are isomorphic, so Rmax is a result of a deformation of the structure in R + . There is an analogy to the quantization procedure, and h is an analog for the Planck constant. Thus R + (or R) can be treated as a “quantum object” with respect to R max and Rmax can be treated as a “classical” or “semiclassical” object and as a result of a “dequantization” of this quantum object. Similarly denote by R min the set R ∪ {+∞} equipped with operations ⊕ = min and = +; in this case = +∞ and = 0. Of course, the change of variables u → w = −h ln u generates the corresponding dequantization procedure for this case. The set R ∪ +∞ ∪ −∞ equipped with the operations ⊕ = min and = max can be obtained as a result of a “second dequantization” with respect to R (or R + ). In this case = ∞, = −∞ and the dequantization procedure can be applied to the subset of negative elements of R max and the corresponding change of variables is w → v = h ln(−w). Let us now consider the partial differential equations. The variety of possible occurrence of limiting equations will be discussed using the following equation as an example: (3.56) ε(w xx + wyy ) + w = 0. Equation (3.56) governs the behaviour of the membrane on the elastic support with high stiffness. For an asymptotic integration of input boundary value problem and a qualitative study of its solution as ε → 0, it is necessary to have a method to describe the rate of change of w(x, y, ε) with respect to x and y, i.e. ∂w x and ∂wy , as ε → 0. A.L. Gol’denveizer [313, 314] introduced the index of variation of a function - very convenient tool of asymptotic analysis. Parameters α, β describe the rate of change of w with respect to x, i.e. w x as ε → 0 (index of variation of function w). Observe that parameters governing an asymptotical process can represent also a link between different small parameters describing the relative relations between various physical parameters and functions occurring in the differential equations, the order of non-linearities, and so on. The sought solution w can exhibit a different behaviour along the co-ordinates x and y. In order to take into account this observation, the parameters α and β of the asymptotic integration are introduced in the following way: w x ∼ εα w, wy ∼ εβ w, −∞ < α, β < ∞.

(3.57)

Let us consider possible values of α and β. It allows us to exhibit different simplified equations, which by supplementing each other can explain the behaviour governed by input equations (3.56).

130


0.5

0.5

Fig. 3.5. The partition of the (α, β) plane for problem (3.56).

Below are given all exponents of the ε order to equation (3.56): 1 − 2α, 1 − 2β, 0. Let us consider the plane αβ (Fig. 3.5) and let us mark the areas corresponding to the smallest values of exponents. The exponent 1 − 2α is the smallest one for any α and β in zone 4. The exponent 1 − 2β is the smallest one in zone 1, whereas the exponent 1 − 2α is the smallest one in zone 6 (zones 1, 4, 6 are treated as the open ones - the boundary lines are not included). During the choice of the parameters α and β in zones 1, 4, 6, the limiting equations have the following form wyy = 0,

w xx = 0,

w = 0.

(3.58)

The above equations are extremely simple. The values of α and β which correspond to two terms of equation (3.58) belong to the boundary lines. For instance, for zones 3, 5, 7, they have the following form w xx + wyy = 0,

εw xx + w = 0,

εwyy + w = 0.

(3.59)

Finally, the point of intersection between the boundary lines (zone 2) gives the values of α and β, for which equation (3.56) includes all terms (α = β = 0.5). To conclude, a whole possible set of limiting equations is represented by (3.56), (3.58) and (3.59). The described graphic procedure allows for a relatively simple choice of two parameters of the asymptotic integration also with a use of computer [608, 609]. Let us consider now two important examples.

3.4 Stretched plate bending

131

3.4 Stretched plate bending Let us consider a differential equation governing the stressed state of a stretched plate of the form (3.60) ε∇4 w + ∇2 w = q(x, y), with the following boundary conditions: for x = 0, a w = ∂w/∂x = 0, for y = 0, b w = ∂w/∂y = 0. In order to detect various limiting systems, the parameters α and β, characterising a variations of the being sought function w in regard to x and y correspondingly, are introduced. Consider first equation (3.60) for q = 0 and let us estimate orders of all its terms: ε1+4α , ε1+2α+2β , ε1+4β , ε2α , ε2β .

(3.61)

An algorithm for searching the limiting systems and the corresponding exponents has been described earlier. It can be additionally simplified. Receiving in equation (3.60) firstly one term, then two terms, and so on, we can make a choice of the exponents α and β. If we cannot find the exponents, it means that the limiting systems being sought do not exist. Some of possible cases can be ad hoc omitted. For instance, in sequence (3.61) the second term cannot be the largest one; it is impossible to remain the first and third terms without the second one; and so on. To conclude, the whole plane (α, β) should be divided into the equivalent classes. Note that the pairs of the parameters (α 1 , β1 ) and (α2 , β2 ) are equivalent if they lead to the same limiting systems. The above algorithm can be described using inequalities. The conditions necessary for existence of a limiting system including only the first term εw xxxx are represented by the inequalities 1 + 4α > 1 + 2α + 2β, 1 + 4α > 1 + 4β, 1 + 4α > 2α, 1 + 4α > 2β. The conditions for existence of limiting systems of the equation εw xxxx + wyy = 0 have the form 1 + 4β = 2α, 2α > 2β, 2α > 1 + 2α + 2β, and so on. It should be emphasised that the considerations can be simplified using a symmetry property (we can interchange x and y). Below the possible limiting systems are given: 1) εw xxxx = 0, 2) εwyyyy = 0,

132


3) w xx = 0,

4) wyy = 0,

6) ∇2 w = 0,

5) ε∇4 w = 0,

7) εw xxxx + w xx = 0,

8) εwyyyy + wyy = 0,

9) ε∇4 w + ∇2 w = 0.

(3.62)

The (α, β) plane is divided into the equivalent parts (classes), which include the point (−0.5, −0.5), four radii and four plane quarters (Fig. 3.6). Other limiting systems do not exist.

0.5

0.5

Fig. 3.6. The (α, β) plane partition for the problem of the initially stretched plate deflection.

The found limiting systems can be ordered in a sequence corresponding to the simplification order: ε∇4 w + ∇2 w = 0, (3.63) ∇2 w = 0, 4

ε∇4 w = 0,

ε

∂ w ∂ w + 2 = 0, ∂x4 ∂x

ε

∂4 w ∂2 w + 2 = 0, ∂y4 ∂y

∂2 w = 0, ∂x2

(3.64)

2

ε

∂4 w = 0, ∂x4

(3.65)

3.5 Simplification of the static equations of a cylindrical shell

∂2 w = 0, ∂y2

ε

133

∂4 w = 0. ∂y4

Let us comment on possible applications of the found approximate formulas. Suppose that an index of variation of the load q is small: ∂q ∼ ε0 q, ∂x

∂q ∼ ε0 q. ∂y

Thus, in the first approximation the following limiting system should be applied ∇2 w0 = q.

(3.66)

Now, similarly to earlier considerations, the boundary conditions should be splitted. It is not difficult to show that the following boundary conditions are attached to equation (3.66): for x = 0, a w0 = 0, for y = 0, b w0 = 0. The other boundary conditions are related to the boundary layer, w b1 . On the boundaries x = 0, a, the boundary layer is governed by the equation ε

∂2 wb1 + wb1 = 0 ∂x2

(3.67)

with the attached boundary condition wb1x = −w0x

for x = 0, a.

On the boundaries y = 0, b, the boundary layer w b2 is governed by the equation ε

∂2 wb2 + wb2 = 0. ∂y2

(3.68)

and the attached boundary condition for y = 0, b has the form wb2y = −w0y . The constructed boundary layers extend applicability of the approach except of thin zones in the area angles, where the so-called corner boundary layers occur (see Chapter 3.7).

3.5 Simplification of the static equations of a cylindrical shell Consider the stress-strain state of a circular cylindrical shell. The governing equation has the following form

134


∇81 Φ + ε−2 d−4 8d −2

∂4 Φ ∂6 Φ + (8 − 2ν2 )d −4 4 2 + 4 ∂ξ ∂ξ ∂η

4 ∂6 Φ ∂6 Φ ∂4 Φ −2 ∂ Φ + 2 + 4d + 2 = 0, ∂ξ2 ∂η4 ∂ξ2 ∂η2 ∂η4 ∂η6

where: ∇81 = ∇41 ∇41 , ∇41 = d −4

(3.69)

∂4 ∂4 ∂4 + 2d −2 2 2 + 4 , 4 ∂ξ ∂ξ ∂η ∂η

ξ = x/L, η = y/R, d = L/R, ε2 = h2 /(12(1 − ∂ 2 )R2 ). Two parameters, ε 2 and d, are attached to equation (3.69). For an asymptotic analysis we need to find a relation between them. As an example we consider a case d ∼ 1. Let us introduce the parameters α and β characterising a change of the sought function Φ in regard to ξ and η: ∂Φ ∼ εα Φ, ∂ξ

∂Φ ∼ εβ Φ. ∂η

Let us describe all possible limiting systems for homogeneous equation (3.69): a) the limiting systems, including only one term of equation (3.69) 1) ∂8 Φ = 0, ∂η8 2)

3)

4)

∂8 Φ = 0, ∂ξ8

(3.71)

∂4 Φ = 0, ∂ξ4

(3.72)

∂4 Φ = 0; ∂η4

(3.73)

b) the limiting systems, including two terms of equation (3.69) 5) ∂8 Φ ∂4 Φ ∂4 Φ + ε−2 d−2 4 = 0, or Φ + ε−2 d−2 4 = 0, 8 ∂ξ ∂ξ ∂ξ 6)

(3.70)

∂8 Φ ∂4 Φ + ε−2 d−4 4 = 0; 8 ∂η ∂η

c) the limiting systems with higher number of terms 7) ∇81 Φ = 0,

(3.74)

(3.75)

(3.76)

3.5 Simplification of the static equations of a cylindrical shell

8) ∇81 Φ + ε−2 d−4 9)

∂4 Φ = 0, ∂ξ4

2 ∂4 ∂2 ∂4 Φ + 1 Φ + ε−2 d−4 4 = 0, 4 2 ∂η ∂η ∂ξ

135

(3.77)

(3.78)

10)

2 ∂4 ∂2 + 1 Φ = 0. (3.79) ∂η4 ∂η2 All of the above-obtained limiting equations have a physical meaning: equation (3.73) governs the membrane theory; equations (3.75) and (3.78) – the semimomentous theory (with large and small variability in the η direction, respectively). According to the Gol’denveizer terminology, those equations govern the fundamental states. Equation (3.74) corresponds to the edge effect, whereas equation (3.77) to the theory of shallow shells (sometimes it is called the state with a large variability; the technical shells equation or Vlasov-Donnel theory). Equation (3.76) governs a bending plate deformation and an in-plane stress state, whereas equation (3.79) a bending deformation of a ring. Equations (3.70) and (3.71) govern a bending and longitudinal deformation of a beam in the η and ξ direction, respectively. Equation (3.72) governs longitudinal deformations of a beam in the η direction. Each of the above mentioned limiting systems corresponds to the following intervals of the parameters α, β: 1) α < −0.5, β > α, 2) β < −0.5, α > β and − 0.5 < β < 0, α < 0.5 + 2β, α α 1 3) β > − , α > −0.5, 4) 0 < β < − 0.25, α > 0.5, 2 4 2 5) β > α, α = −0.5, 6) β > −0.5, α = 0.5 + 2β, 7) β > −0.5, α = β, 8) β = α = −0.5; 9) β = 0, α = 0.5, 10) β = 0, α > 0.5. To conclude, the whole (α, β) plane is divided into the relatively simple parts for further considerations (Fig. 3.7). Among the limiting sets there are two points, three rays, four parts of a plane, and one area including a finite interval without the boundary points and a ray. It is interesting to note that the parameters (α, β), for which we need to use input equations (3.69), do not exist. Equation (3.77) is the most general one because it includes almost all - limiting states, without only the states (3.78) and (3.79). The equation of the membrane theory is represented by a singular equation on the spectrum [672]. The equations governing the semi-momentous theory (3.75) and (3.78) play a role of the supplementary ones. The edge effect governed by (3.74) gives a possibility to satisfy the boundary conditions.

136


0.5

0.5 -0.75 0.5

Fig. 3.7. The (α, β) plane partition in regard to the x, y co-ordinates for a cylindrical shell.

3.6 Boundary layer: Papkovitch approach It often happens that one may reduce dimension of the analysed equation in order to obtain an asymptotic solution. We consider a bending of the clamped narrow plate, i.e. the plate which one dimension (l) is sufficiently smaller than the second one (L). The initial equation and the boundary conditions are as follows: D∇4 w = q,

(3.80)

for x = 0, L w = w x = 0, for y = 0, l w = wy = 0.

(3.81)

Introducing the change of variables ξ = x/L, η = y/l the equation (3.80) is transformed to the form ε4 wξξξξ + 2e2 wηηηη + wηηηη = ¯q, where

¯q = ql4 /D, ε = l/L 1.

The boundary conditions (3.81) have the form

(3.82)

3.6 Boundary layer: Papkovitch approach

137

for ξ = 0, 1 w = wξ = 0,

(3.83)

for η = 0, 1 w = wη = 0.

(3.84)

A solution is sought in the form w = w0 + ε2 w1 + ε4 w2 + ...,

(3.85)

and the following recurrent sequence of the boundary value problems is obtained: w0ηηηη = q,

(3.86)

w1ηηηη = −2w0ηηξξ ,

(3.87)

wiηηηη = −2wi−1ηηξξ − wi−2ηηηη , i = 2, 3, ...,

(3.88)

for η = 0, 1, w j = w jn = 0, j = 0, 1, . . .. The found solution does not satisfy the boundary conditions for ξ = 0, 1. Since the plate is narrow, the solution can be treated as a boundary layer. In order to construct the boundary layer the variation of the variables ϕ = x/L, η = y/l is required in the homogeneous equation obtained from the input one, and ω n is sought in the following series form wb = εα1 wb0 + εα2 wb1 + εα3 wb2 + ... . The following equations serve for construction of a boundary layer 4 ∂ ∂4 ∂4 + 2 2 2 + 4 wbi = 0. ∂ϕ4 ∂ϕ ∂η ∂η

(3.89)

The parameters α i are yielded by a procedure of the asymptotic splitting. It is easy shown, that α1 = 1, α2 = 2, αi = i. For ϕ = 0, ε−1 we have: wb0 = 0, wb0ϕ = −w0ξ ξ=0 , wb1 = −w0 , wb1ϕ = −w1ξ ξ=0 ,

.

.

. .

.

.

. .

wbi = −wi−1 , wbiϕ

.

. = −wiξ ξ=0 ,

(3.90)

for η = 0, 1 wn j = wn jη , j = 0, 1, 2, . . .. Since the distance between the sides ϕ = 0 and ϕ = ε −1 is large, the corresponding solutions can be constructed separately in the vicinity of each edge. The boundary value problems (3.89), (3.90) are significantly two-dimensional. In order to solve them the Kantorovich [369] or Papkovitch [540] methods are recommended. We turn our attention to Papkovitch method. Following the Papkovitch method a solution of the equations (3.89) and the boundary conditions (3.90) are sought in the form (we consider only the edge ϕ = 0, since for ϕ = ε−1 we proceed in a similar way)

138


wb j =

∞ ¯ Yk (η1 ) αk e−λk ϕ + Y¯k (η1 ) ¯αk e−λk ϕ ,

(3.91)

k=1

where: η1 = η − 0.5. The bar in (3.91) denotes a complex conjugate. For even components, with respect to the line η 1 = 0, we get Yk =

ch (sk η1 ) 2η1 sh (sk η1 ) − , ch (0.5sk ) sh (0.5sk )

(3.92)

sh (sk η1 ) 2η1 ch (sk η1 ) − . sh (0.5sk ) ch (0.5sk )

(3.93)

and for odd ones, we obtain Yk =

The constant sk are the roots of the transcendental equation: in the case (3.91) sh (sk ) = −sk , in the case (3.93) sh(sk ) = sk . The exponents in the expression (3.91) have the form: √ λk = isk , i = −1. The roots of transcendental equations are found numerically. In order to define the constants a k , ¯ak in the solution (3.91), the boundary conditions (3.83) should be projected into the functions Y k , Y¯k . In other words, first the solutions are presented in the form (3.91), and then the conditions (3.83) are successively multiplied by Y k , Y¯k (k = 1, 2, . . .), and then the integration is carried out with respect to η from 0 to 1.

3.7 Edge boundary layer Let us come back to problem (3.60). In order to compensate errors in the neighbourhood of the angle point the homogeneous equation must be solved ε∇4 w˜ + ∇2 w˜ = 0

(3.94)

for the following boundary conditions: w˜ = −wb1 − wb2 ;

w˜ x = −wb2x

w˜ = −wb1 − wb2 ; w˜ y = −wb1y

for x = 0, a; for y = 0, b.

(3.95)

The boundary problem (3.94), (3.95) is essentially two-dimensional. Sice its solutions does not belong to trivial ones, we treat this problem in more detailed

3.7 Edge boundary layer

139

manner. First we use the polar coordinates, therefore the Laplace operator in the education (3.94) has the form ∇2 =

1 ∂2 ∂2 1 ∂ + + . ∂r2 r ∂r r2 ∂θ2

The boundary conditions can be rewritten in the form w˜ = wi (r);

wθ = w1i (r) for θ = 0, i = 1, θ = π/2, i = 2,

where: wi , w1i are the known values. Instead of the boundary problem (3.94), (3.95) with non-homogeneous equation with homogeneous boundary conditions applying the formula 1 2 π π 0 w˜ = w∗ (r, θ) − θ− w1 (r0 + θ w11 (r) − w1 (r) + π 2 2 2 π 2 θ w2 (r) + θ − w2 (r) − w12 (r) . π 2 π A solution of the non-homogeneous equation ε∇4 w∗ + ∇2 w∗ = f (r, θ)

(3.96)

with the homogeneous boundary conditions w˜ = w˜ θ = 0 for θ = 0, π/2 can be found using various methods. One may apply the Mellin transformation [651] and then the transformation yields the sought solution. Although the obtained solution exhibits all solution singularities, but this is not enough suitable for a computation. One may also apply the Papkovitch method. Finally, the Kantorovich method can be used, and the solution of the equation (3.96) is presented in the form w∗ =

∞ i=1

π 2 π 2i−2 ϕi (r)θ2 θ − . ψ− 2 4

Applying the standard procedure of the Kantorovich method one obtains a nonhomogeneous ordinary equation with respect to ϕ i (r). It is composed of a particular and general part, and a general solution has the form 4j=0 Ci j rλi j . Since the solution is sought as a boundary layer, the decaying conditions hold for r → ∞. Already this analysis yields that a decaying of the boundary layer solution is not exponentional [667].

140


3.8 Incorporating of the singular part of solution A solution in the angle point may have singularities [395, 515, 519, 520]. In general, they should be taken into account. A solution in the vicinity of the angle point can be presented in the form [542] wa = Crα ψ(θ) + ... .

(3.97)

For the considered case a = 1.5. The solution (3.97) should be matched with the solution of the angle boundary layer. Consider, for example, the first approximation of the Kantorovich method π 2 w∗ = ϕ(r) ˜ + C11 rλ11 + C12 rλ12 θ2 θ − , (3.98) 2 where: ϕ(r) is the particular solution. 2 Let us project the function ψ(θ) into θ 2 θ − π2 : π 2 ψ(θ) = C1 θ2 θ − + ..., 2 where:

C1 =

π/2

0

π 2 ψ(θ)θ2 θ − dθ. 2

Now we require a condition of smooth matching of the solution (3.98) and the solution π 2 , w(y) = Cr1.5 θ2 θ − 2 up to the third derivative in the ceratin, not known yet point r 0 : (k) (k) = ϕ(r) + C11 rλ11 + C12 rλ12 for r = r0 , k = 0, ..., 3. Cr1.5

(3.99)

The system (3.99) yields the constants C, C 11 , C12 and the matching points.

3.9 Plane theory of elasticity In this section our attention is focused on interesting and important example of singular asymptotics. According to the linear theory of elasticity, the equilibrium equation have the following form [500] eu xx + Guyy + e(µ + G/e)v xy = 0; Gv xx + evyy + e(µ + G/e)u xy = 0.

(3.100)

Here e = E/(1 − µ2 ), E is the Young’s modulus, G is the shear modulus, µ is the Poison’s coefficient.

3.9 Plane theory of elasticity

141

In Cartesian coordinates the components of the stress tensor have the following form [500] σy = e(vy + µu x ); σ x = e(u x + µvy ); τ xy = G(v x + uy ).

(3.101)

Here σ x (σy ) is the stress in Ox (Oy) direction, τ xy is the shear stress. N. Muskhelishvili [500] successfully solved the problem of plane elasticity for the isotropic body by methods of complex variables. Meanwhile, the transition to the anisotropic case generally involves much higher complexity [425, 426]. For a slight deviation from the isotropic case it is possible to introduce a small parameter γ = (Ba − Bi )/Bi 1, where Ba and Bi are corresponding properties of the anisotropic and isotropic media. Then the asymptotic solution with respect to γ can be developed [426]. On the contrary, for the strong anisotropy one can take into account the smallness of the parameter 1/γ. It is obvious that loading in Ox (Oy) direction causes mainly u (v) displacement. These reasonable approximation have been used for a long time by many engineers, especially in the fields of aircraft [409] and rocket [126] design. Furthermore the were successfully applied to the theory of composite and nonhomogeneous materials [220, 278, 279, 620]. Most of the authors referred above used only the zero order approximation of the procedure. However, further development of this partially-empirical engineering approach was restricted by the evident drawbacks: the boundary conditions were not satisfied, the choice of the appropriate approximation was not unique and clear, there was a lack of error’s estimations, etc. Beginning from the paper [459], a special asymptotic technique using expansions with respect to γ was developed. A singular character of the asymptotic solution was detected, and the input biharmonic equation of the plane problem was reduced to two Laplace equations. The link between them was introduced via split boundary conditions, which allowed the application of a powerful procedure of the theory of potential. Then, the decomposition of the governing boundary value problem was carried out (including those of mixed boundary value problems), and finally the higher order approximations were derived [457, 458] (see also [177, 178, 363, 364, 394]). Moreover, it has been also shown that in the isotropic case (which is the most unsuitable for this procedure) an error introduced by the firs approximation is rather low. Now let us briefly describe the asymptotic procedure used as follows. We introduce the dimensionless parameters ε = G/e, ¯µ = ε−1 µ. Then the input equations (3.100) can be rewritten as follows u xx + εuyy + ε( ¯µ + 1)vxy = 0; εv xx + vyy + ε( ¯µ + 1)uxy = 0.

(3.102)

As soon as ε < 1, ¯µ ∼ 1, we can use the parameter ε for asymptotic splitting of system (3.102). We pose expansions for the functions u, v u = u0 + εu1 + ε2 u2 + ...;

142


v = v0 + εv1 + ε2 v2 + ... .

(3.103)

Derivatives of displacements are estimated using indices of variation α 1 , α2 ∂(u; v) ∼ εα1 (u; v); ∂x

∂(u; v) ∼ εα (u; v). ∂y

(3.104)

The order of the function u with respect to function v can be estimated using indice of intensity α 3 u ∼ εα3 v. (3.105) Substituting anzatz (3.103) and estimations (3.104), (3.105) into the equation (3.102) and comparing the coefficients of ε, we conclude that asymptotics are strongly depend on the values of the parameters α k . There upon, we look over all possible values of αk and search all possible cases, when asymptotics (ε → 0) have a mathematical (well - posedness) and a physical meaning. It is remarkable that by this simple (but laborious) procedure we obtain only two systems which are analysed below: 1. α1 = 0.5; α2 = 0; α3 = 1.5. (1) u(1) 0xx + εu0yy = 0;

(3.106)

v(1) µ + 1)u(1) 0y = −ε( ¯ 0x .

(3.107)

(2) εv(2) 0xx + v0yy = 0;

(3.108)

µ + 1)v(2) u(2) 0x = −ε( ¯ 0y .

(3.109)

2. α1 = −0.5; α2 = 0; α3 = 1.5.

Analysis of the stress relations (3.101) leads to the following results. For the first and second type of stress-strain state (3.106), (3.107) and (3.108), (3.109), the main stresses are, respectively (1) σ(1) 0x = eu0x ;

(1) τ(1) 0xy = Gu0y ;

(3.110)

(2) σ(2) 0y = ev0y ;

(2) τ(2) 0xy = Gv0x .

(3.111)

Simplified equations (3.106), (3.107), (3.110) and (3.108), (3.109), (3.111) describe all possible stress-strain states. Links between these states can be established after splitting the input boundary conditions. Let, for instance, on a boundary of the half-plane (|y| < ∞; 0 < x < ∞) the following conditions are given for

x = 0 σ x = ϕ(y) τ xy = 0.

Hence, for equation (3.106) we have the following boundary condition: for

x = 0 eu(1) 0x = ϕ(x),

3.9 Plane theory of elasticity

143

whereas for equation (3.108) for

(1) x = 0 v(2) 0x = −u0y .

Equation (3.106) and (3.108) can be reduced to Laplace equations due to a simple affine transformation of the independent variables. So, we can use the highly developed theory of complex variables. The described method is particularly suitable for analysis of strongly anisotropic media, where as small parameter the following quantity appears: ε = G(1−ν 1ν2 )/E 1 , where G, E 1 - are shear and extension compression moduli in different directions; ν1 , ν2 are Poisson’s coefficients. Note that this method is also suitable for analysis of isotropic media. Although its first approximation give satisfying results, but a perturbation series is divergent. Padé transformation (see Chapter 12) removes this drawback. Consider the model-type problem, i.e. an action of concentrated force P on an elastic anisotropic plane. Exact relation of longitudinal stress has the form   √ P qx   1 1   , σ11 = − − (3.112)  π(a2 − a1 ) a21 x2 + y2 a22 x2 + y2 / where: a2i = 0.5qε−1 1 − q−1 ε2 (b2 − 1) − (−1)i (1 − q−1 ε2 (b2 − 1))2 − 4q−1 ε2 ; b = 1 + ν1 qε−1 = 1 + ν2 ε−1 ; q = E 2 /E1 , i = 1, 2. In Tables 3.1, 3.2 results of computation of the quantity σ = (−πx/P)σ 11 on the curve y = 0 for orthotropic (Table 3.1, where ε = 0.135, ν 1 = ν2 = 0.14) and isotropic (Table 3.2, where ν = 0.3) materials are reported. The exact solution σ in the first case is 2.9688, in the second case it is equal to 2. By index 1 in Tables 3.1, 3.2 series related to formula (3.112) with respect to ε (n - approximation number); by index 2 in Tables 3.1, 3.2 Padé approximation is reported. Table 3.1. Computations of σ for orthotropic materials n σ1 σ2

0 2.69 2.69

1 3.01 3.01

2 2.94 3.01

3 2.97 2.96

Table 3.2. Computations of σ for isotropic materials n σ1 σ2

0 1.69 1.69

1 2.07 2.07

2 1.78 2.06

3 1.82 2.022

4 1.72 2.00

5 1.66 2.00

Therefore, already the first approximation yields good results, and additionally the Padé approximation improves them.

144


3.10 Asymptotic foundation model Various models, taking into account of the specifies of real soils, are based on a some a priori assumptions on the stress distribution and can be considered as an approximation of a three-dimensional elastic solid. The equation describing the work of the base have been obtained in a number of investigations, directly from the relationships of three-dimensional elasticity theory, hence the hypotheses about the absence of tangential stresses were introduced [499]. Such a foundation model turns out to be more exact than the Winkler and twoparameter models, but still does not permit a correct description of the stress–strain state in the whole domain (particularly near the boundary of the elastic half-space), nor satisfaction of all the boundary conditions. A new approach is proposed herein for the construction of the approximate equations for three-dimensional transversally isotropic bases, which is based on an asymptotic analysis of the equilibrium equations. It is shown that equations which contain both the particular cases of the elastic base models examined up to now can be obtained in a zero approximation by just means, but also permit satisfaction, in contrast, of not only the normal but also the tangential boundary conditions on the surface of the base. The equilibrium equations of a transversally isotropic solid subjected to axisymmetric stress are ∂2 u kε ∂2 u ∂2 w 1 ∂u u − + + + = 0; 2 ∂z2 ∂r∂z r ∂r r ∂r2 ∂2 w k ∂u ∂w ∂2 w k ∂2 u + 2 + 2 + + = 0, (3.113) 2 ∂r∂z ∂r 2r ∂z ∂r ∂z where: r, z are cylindrical coordinates; u, w are displacement vector components; ε = E /E; E, E and G, G are the elastic and shear moduli in the plane of isotropy and in a perpendicular direction; G = E/2, G = kE /2; and the Poisson ratios are assumed zero. Let us introduce affine transformations of variables a) r = r1 ; z = z1 ; u = εU1 ; w = W1 ;

(3.114)

b) r = r2 ; z = ε1/2 z2 ; u = ε1/2 U2 ; w = εW2 .

(3.115)

Substituting (3.114) and (3.115) into equations (3.113), we, respectively, obtain   ∂2 W1  1 ∂U1 U1 ∂2 U1 k  ∂2 U1 +  2 + − = 0;  + 2 ∂z1 ∂r1 ∂z1 r1 ∂r1 r1 ∂r12   k  ∂2 U1 ∂2 W1  ∂2 W1 k ∂U1 ∂W1 + + + ε = 0; (3.116)  + ε 2 ∂r1 ∂z1 2r1 ∂z1 ∂r1 ∂r12 ∂z21   ∂2 W2  1 ∂U2 U2 ∂2 U2 k  ∂2 U2 +  2 + ε − = 0;  + 2 ∂z2 ∂r2 ∂z2 r2 ∂r2 r2 ∂r22

3.10 Asymptotic foundation model

  ∂2 W2  ∂2 W2 k ∂U2 ∂W2 k  ∂2 U2  = 0. + ε 2  + + + ε 2 ∂r2 ∂z2 2r2 ∂z2 ∂r2 ∂r2 ∂z22

145

(3.117)

Let us assume E > E , then the quantity ε can be considered as a small parameter in an asymptotic analysis (3.116), (3.117). In the case of a half-space, the solutions obtained by asymptotic integration of the system (3.116) will correspond to a stress state varying relatively slowly along the z axis, which is realized far from the boundary. Analogous solution of (3.117) correspond to the stress state varying rapidly along the z axis which is localized near the half-space boundary. Let us represent the displacement vector components as the sum of solutions of both kinds (3.118) U = U1 + U2 ; W = W1 + W2 . Let us substitute (3.118) into (3.113) and the boundary conditions by applying the appropriate transformations (3.114), (3.115) to the coordinates and the desired functions. Let us seek the functions U i , Wi as asymptotic series in the parameter ε 1/2 : U1 = U10 +ε1/2 Ui1 +εUi2 +. . . ; Wi = Wi0 +ε1/2 Wi1 +εWi2 +. . . , i = 1, 2 . (3.119) We introduce the coordinate transformation ζ 1 = αz1 , ζ2 = βz2 , and to represent α and β by the following expansions in ε: α = α0 + α1 ε + α2 ε2 + . . . ;

β = β0 + β1 ε + β2 ε2 + . . . .

Collecting terms of identical powers in ε, we arrive at a system of equations in the functions U im , Wim (m = 0, 1, 2, . . .) and appropriate boundary conditions. The coefficients α0 and β0 should be set equal to one since the zero approximation equations should agree with the limit systems obtained from (3.116) and (3.117) as ε → 0. The coefficients α k , βk (k = 1, 2, . . .) can be selected in such a manner that all the independent equations in the successive approximations (i.e., the equations in W1m , U1m and in U 2m , W2m ) would agree with the appropriate equations for the limit systems. In this case we have; for the stress state for W 1m , U1m k ∂2 W1m k ∂W1m ∂2 W1m + + = 0; 2 2 ∂r1 2r1 ∂r1 ∂ζ12 k ∂2 W1m ∂2 U1m 1 ∂U1m U1m + + − = f1m ; r1 ∂r1 r1 2 ∂r1 ∂ζ1 ∂r12

(3.120)

and for the stress state for U

∂2 U2m 1 ∂U2m U2m k ∂2 U2m + − = 0; + r2 ∂r2 r2 2 ∂ζ22 ∂r22 k ∂2 U2m k ∂U2m ∂2 W2m + + = f2m . 2 ∂r2 ∂ζ2 2r2 ∂ζ2 ∂ζ22

(3.121)

146


The right sides f1m and f2m depend on the functions U in , Win and the numbers α n , βn determined in the previous approximations (n = 0, 1, 2, . . . , m − 1), i.e., are known functions of the coordinates. The boundary conditions for equations of each of the approximations can be formulated after extraction of appropriate terms in the expansions of the boundary conditions. Hence, besides the expansions (3.119), it is necessary to take account of the appropriate series for the stresses

∂W10 ∂W20 ∂W12 ∂W21 ∂W10 1/2 ∂W11 +ε +... ; +ε + α1 + + σ2 = E ∂ζ1 ∂ζ1 ∂ζ2 ∂ζ1 ∂ζ1 ∂ζ2 E ∂U20 ∂W10 ∂W11 1/2 ∂U 21 τrz = + +ε + + 2 ∂ζ2 ∂r1 ∂∂ζ2 ∂r1

∂U2 2 ∂U10 ∂U20 ∂W12 ∂W20 + + β1 + + +... . ε ∂ζ2 ∂ζ1 ∂ζ2 ∂r1 ∂r2 Therefore, the solution of the system (3.113) has been reduced to successive integration of the equations for the functions W im , Uim with appropriate boundary conditions, where the equations (3.121) for each of the approximations are integrated after the appropriate equations (3.120) have been solved. The connection between this solutions is determined by the boundary conditions. Since the principal components of the normal and radial displacements are the functions W10 and U 20 , respectively, an approximation of the stress-strain state in the whole domain, acceptable for practical purposes, is already achieved upon integrating the equations k ∂2 W10 k ∂W10 ∂2 W10 + + = 0; 2 2 ∂r1 2r1 ∂r1 ∂ζ12

(3.122)

∂2 U20 1 ∂U20 U20 k ∂2 U20 + − = 0. + 2 r2 ∂r2 r2 2 ∂ζ22 ∂r2

(3.123)

These equations coincide with foundation model, described in reference [499]. Putting U 20 = 0 in these equations, we obtain the model of a foundation examined in [675]. Moreover, if the function W 10 is represented as W10 (r1 , ζ1 ) = ϕ(r1 )ψ(ζ1 );    ψ(ζ1 ) =  

1, for ζ1 = 0, 0, for ζ1 = H, H is the thickness of the lager.

and the procedure corresponding to the Galerkin method is applied to (3.122), then we arrive at the two-parameter model of the foundation [675]. Equations (3.122) and (3.123) which describe the proposed model of an elastic foundation, are considerably simpler than the original system (3.113) since each contains only one unknown. At the same time, they permit satisfaction of both the normal and tangential boundary conditions on the boundary surfaces. There follows

3.10 Asymptotic foundation model

147

from an asymptotical analysis of the boundary conditions that a boundary value problem is first formulated from (3.122) and after it has been solved, the tangential boundary conditions for (3.123) are determined. Let us note, finally, the possibility of refining the solution because of taking account of subsequent approximations. As an illustration of the proposed method, let us examine the problem of the vibrations of an infinite plate on a transversally isotropic elastic half-space, and subjected to a concentrated impact. This problem has been examined on the basis of a two-parameter model of an elastic foundation, without permitting the tangential boundary conditions on the contact plane to be taken into account. The equation of plate motion in cylindrical coordinates is D∇2r ∇2r υ(r, t) = −q(r, t) − ρ

∂2 υ(r, t) , ∂t2

(3.124)

where: ∇2r is the Laplace operator in cylindrical coordinates; ρ is the mass per unit square surface of the plate; and q(r, t) is the reactive pressure transmitted to the plate by the elastic foundation (it is assumed that the plate does not separated from foundation). Neglecting the density of foundation, let us describe its deformation by the approximate equations obtained above. Let us hence consider the case when there is no friction between plate and foundation. Under the assumption that the plate stiffness is negligible, let us write the initial conditions as ∂υ = V0 , υ|t=0 = 0, ∂t t=0,r=0 where: V0 is the velocity of the impacting mass at the time of collision. To seek W10 let us consider (3.122) with the boundary conditions ∂W10 q(r1 , t) =− , W10 |ζ1 →∞ = 0, W10 |r1 →∞ = 0. ∂ζ1 ζ1 =0 E Let us represent the solutions (3.122) as W10 = ϕ(r1 , t)ψ(ζ1 )

(3.125)

and let us apply the Galerkin method. The function ψ(ζ 1 ) characterizing the change in displacement over the thickness of the foundation can be selected analogously to [675]. Then we obtain the relationship  2   ∂ ϕ(r1 , r) 1 ∂ϕ(r1 , t)    − n1 ϕ(r1 , t) = −q(r1 , t), + (3.126) kl1  r1 ∂r1 ∂r12 where: E l1 = 2

∞

2

ψ (ζ1 )dζ1 , 0

n1 = E

∞ 0

dψ(ζ1 ) dζ1

2 dζ1 .

148


Taking account of the condition that the normal displacements of the plate and the foundation are equal on the boundary and eliminating the reaction q(r 1 , t) from (3.124) and (3.126), we arrive at the equation ∇2r1 ∇2r1 υ(r1 , t) − 2m2 ∇2r1 υ(r1 , t) + s4 υ(r1 , t) = −m∗

∂2 υ(r1 , t) , ∂t2

(3.127)

where: m2 = kl1 /2D, s4 = n1 /D, m∗ = ρ/D. Equation (3.127) agrees with the equations of plate vibrations on a two-parameter foundation [675]. Its solution is υ(r1 , t) =

V0 J0 (br1 ) sin ωt, ω

+ √ m4 + λ4 − m2 , λ4 is found in conformity with []. where: ω2 = (s4 + λ4 )/m∗ , b = The the normal displacements in the half-space are determined in the initial variables by the relationship ω(r, z, t) =

V0 J0 (br)ψ(z) sin ωt. ω

Without examining the possible refinement of the solution W 10 because of error of the approximation (3.125), which is particularly important for estimating the stresses and displacements in the half-space, let us spend our main attention in taking account of the influence of the tangential boundary conditions. To determine the tangential displacements, let us consider (3.123). Since τ rz |z=0 = 0, we have the following boundary condition ∂U20 ∂W10 V0 J1 (br2 ) sin ωt. =− = ∂ζ2 ζ2 =0 ∂r1 ζ1 =0 ω Moreover U20 |ζ2 →∞ = 0,

U 20 |r2 →∞ = 0.

Applying a Hankel transform to (3.123) and the boundary conditions, we determine the transform of the function U 20 . Inverse transform gives us   √   2 V0 k  ζ2 sin ωt , U20 = − √ J1 (br2 ) exp −b (3.128) k 2ω which characterizes the tangential displacement of points of the half-space not taken into account in the two-parameter model of the base. It is essential here that the tangential stress resultants associated with the mentioned displacement and the appropriate stress state which varies rapidly with distance from the boundary, be of the same order near the boundary as are the tangential stresses dependent on the normal displacement. The normal stress components σ r and σθ are determined in terms of the function U 20 .

3.11 Vibrations of reinforced conical shells

149

Taking account of the next approximation, the setting of the foundation the contact pressure can be fined, i.e., the influence of the tangential boundary conditions on the fundamental system characteristics can be exposed. To this end, let us consider the second equation in (3.121) for m = 0. Taking account of the equality (3.128), we find     V0 k 2 b 2 J0 (br2 ) exp −b ζ2 sin ωt . W20 = − 4ω k It is now possible to write the boundary conditions for the first equation of the system (3.120) for m = 1: ∂W20 V0 k3/2 b ∂W11 =−− = √ J0 (br2 ) sin ωt; ∂ζ1 ζ1 =0 ∂ζ2 ζ2 =0 2 2ω W11 |ζ1 →∞ = 0,

(3.129)

W11 |r1 →∞ = 0.

Applying a Hankel transform to the mentioned equation and the boundary conditions (3.129), we obtain the following solution     2 V0 k  J − 0(br1 ) exp −b ζ1 sin ωt . W11 = wω k Taking into account of the two approximations the settling of the foundation under the plate is written as V0 1/2 1/2 k 1+ε J0 (br) sin ωt. w = w1 + w2 = W10 + e W11 = ω 2 The second term determines the correction to the solution obtained with the twoparameter model. The contact pressure is found from the formula   E V0 b  k  1/2 q(r, t) − 1 − ε  J0 (br) sin ωt. ω 2

3.11 Vibrations of reinforced conical shells Vibrations of constructively-isotropic conical shells are investigated by many researchers. In general, the applied methods can be reduced either to application of energy based methods, or to direct numerical integration of the governing equations. A comparison of the results obtained using energy approach, perturbation techniques, successive approximations and integral equations with numerical solutions [201] (which are further referred as exact ones) yields a conclusion, that only minimal frequencies are well approximated. Besides, a domain of application of the obtained solutions is limited to shallow shells close to cylindrical ones. On the other

150


hand, for a wide class of essentially shallow shells the discussed approximating solutions may principally differ from exact ones. A reason is mainly motivated by observation that variational methods require a priori definition of the being sought functions, which usually are chosen in a way analogous to cylindrical shells analysis and do not describe in a satisfactory manner shallow shells behaviour with a large conical angle. Therefore, this section in devoted to formulation of analytical solutions in the low frequency spectrum part of an arbitrary shallow shell. Theory of perturbations is applied in order to obtain frequencies and corresponding modes. Depending on the geometrical parameters of a cone (or a plate on a stiff support), a semi-momentous cylindrical shell theory with averaged parameters is used to begin a perturbational procedure. Matching of these models has yielded satisfactory results in whole considered spectrum part. It is known, that in the case of general non-axially symmetric deformation of a shallow shell a number of half-waves in the radial direction is essentially larger than in meridial one. Owing to this observation, one may apply semi-momentous theory. The system of differential equations governing vibrations of a shallow shell consists of three equations of motion and three compatibility equations ∂S ∂(BS 1) +A = 0; ∂s ∂β A

∂S 2 ∂(BS ) ∂B γh ∂2 V ∂G2 + + S − k2 A − AB = 0; ∂β ∂s ∂s ∂s g ∂t2

k2 S 2 + A

1 ∂2G2 γh ∂2 W − = 0; g ∂t2 B2 ∂β2

∂κ1 ∂(Bτ) ∂Bτ ∂ε1 − − − k2 a = 0; ∂β ∂s ∂s ∂β

∂τ ∂(Bκ2) −A = 0; ∂s ∂β k2 κ1 +

1 ∂ 2 ε1 = 0, B2 ∂β2

(3.130)

where: s, β are coordinates in meridial and circle directions, respectively; k 2 is the second main shell curvature; s 1 , s2 , s are tangential stresses; G 2 is the banding moment; ε1 is the relative extension of mean shell surface in the direction s; κ 1 , κ2 are variations of main curvatures of mean shell surface; τ denotes a torsion of mean shell surface; V, W are components of displacement in β direction and shell’s normal; γ is the specific gravity of the shell material; g is the gravitational acceleration; h is the shell thickness and t denotes time. Observe that a neglection of inertial force component in meridial direction yields an error of the same order as an error introduced by application of the semimomentous theory. The system (3.130) yields the following equation governing behaviour of vibration modes ˜ d2 3 d2 W ˜ = 0, (3.131) ξ + Ψ (ξ, ω∗ )W dξ3 dξ2 where:


151

˜ ˜ = (sW)/R2 ; W(ξ, β, t) = W(ξ) cos nβeiωt ; (ξ, W) Ψ (ε, ω∗ ) = ω2 =

ε2 tg6 αn41 (n21 − 1)2 sin2 αn21 (n21 + 1)ξω2∗ − ; ξ3 (1 − λ)2

hπp Ehπp B gω2∗ n R1 ; ε2 = ; λ= ; B = ; ; n1 = 2 2 2 sin α R2 hγl (1 − ν ) 1 − ν2 12R2

In the above E is the Young modulus, ν is Poisson’s coefficient, and h r is reduced thickness of a constructively-orthotropic shell (in the case when the ribs stiffness changes linearly it is constant value). Other fundamental characteristics are shown in Figure 3.8.

g

S

H

l

R1

a

_ R2

a 2R Z

Fig. 3.8. A scheme of the considered conical shell..

A non-perturbed (initial) equation is obtained from equation (3.131) via a substitution of non-constant coefficients by their averaged values on (ξ 1 , ξ2 ): 2ω20∗ sin2 αn21 n21 (n21 + 1) 2ε2 tg6 αn41 (n21 − 1) ˜0 d4 W 4 4 − . (3.132) − k w = 0; k = 0 dξ4 (1 − λ)2 (ξ12 + ξ22 ) ξ12 ξ12 (ξ12 + ξ22 ) Observe that equation (3.132) governs vibrations within semi-momentous shell theory with the following averaged parameters: −1/2 1/2 1 + λ2 h sin α h 1 + λ2 l = (1 − λ) tgα; = . R cil 2 R cil λ R2 2

(3.133)

152


Applying the operator notation to equations (3.131), (3.132) and boundary conditions, one gets ω∗1

T1 = ; T1 = T2

ξ

2

6 . ˜ 0 ] − M ∗ [W ˜ 0 ] − ω2∗0 N[W ˜ 0 ] dξ; ˜ 0 ] − N ∗ [W ˜ 0 (ξ) M[W W

ξ1

ξ2 T2 =

(3.134) ˜ 0 ]dξ; ˜ 0 (ξ)N ∗ [W W

ξ1

˜ 1] − M [W ∗

where:

˜ 1] ω2∗0 N ∗ [W

˜ 0 ] + M ∗ [W ˜ 0 ] + ω2∗0 N[W ˜ 0 ]+ = −M[W ˜ 0 ]; ω∗1 − ω2∗0 N ∗ [W

˜ ≈W ˜0 +W ˜ 1; ω2∗ ≈ ω2∗0 + ω∗1 ; W ω2∗0 =

2 2k4 (1 − λ)2 (1 + λ2 ) ε2 tg6 α sin α(1 − λ)2 n21 (n21 − 1)2 + ; λ2 (n21 + 1) n21 (n21 + 1) sin2 2α

ε2 tg6 αn41 (n21 − 1)2 d2 ( ) d ( ); M[( )] = 2 ξ3 2 + dξ dξ ξ3

N ∗ [( )] =

(3.135) (3.136) (3.137)

sin2 αn21 (n21 + 1)(ξ12 + ξ22 ) sin2 αn21 (n21 + 1)ξ ( ); N[( )] = ( ); 2(1 − λ)2 (1 − λ)2

M ∗ [( )] =

(ξ1 + ξ2 )(ξ12 + ξ22 ) d 4 ( ) ε2 tg6 αn41 (n21 − 1)2 (ξ1 + ξ2 ) + ( ); 4 dξ4 2ξ12 ξ22

where: W0 (ξ) denotes beam function for the corresponding boundary conditions; k are the corresponding numbers of boundary value problems with averaged coefficients. It is worth noticing that the results obtained using the perturbation theory can be improved by decreasing a difference of the coefficients of “perturbed” (3.131) and “non-perturbed” (3.132) equations. Increasing a conical angle the error introduced by formulas (3.134)–(3.137) increases, and it may achieve an essential values for small α. Therefore, in order to include into considerations the whole class of conical shells, one has to analyse separately vibrations of conical shallow shells. Fundamental equations of a shallow conical shell read ˜ n2 ˜ d2 Φ ˜ 1 dΦ ˜ d 2 (˜zc − δ) 1 d(˜zc − δ) 2 4 ˜ ˜ ˜ − 2 Φ + ω∗2 W; + ε∇W= ξ dξ ξ dξ dξ2 d 2 ξ2 ξ 2 2 ˜ ˜ 2˜ ˜ ˜ = − 1 d(˜zc − δ) d W − d (˜zc − δ) 1 d W − n W ˜ . ∇4 Φ (3.138) ξ dξ ξ dξ dξ2 dξ2 ξ2 where:


153

˜ = (zc , δ)/R2; δ(s) = (R2 − s2 )1/2 − ˜ = Φ/ER22 ; (˜zc , δ) Φ (R2 − s)tgα − (R2 − R22 )1/2 ; R2 =

R21 + R22 + 2R1 R2 cos 2α sin2 2α

; ω∗2 =

γR22 (1 − ν2 )ω2 ; Eg

zc = (R2 − s2 )1/2 − (R22 − R2 )1/2 ; ∇4 = ∇2 ∇2 ; ∇2 = d 2 /dξ2 − (2n/ xi)(d/dξ) − n 2 /ξ2 . In the above Φ is the stress function; δ denotes the distance between averaged surfaces of conical and related spherical shells; z c is the distance between the mean surface of the spherical shell and the plane of bottom cone support. System of equations (3.138) governs vibrations of conical shells whose parameters satisfy one of the following conditions: (1 − λ)tgα < 0.2; (1 − λ)tgα > 0.2 and n ≥ (R/h) 1/2.

(3.139)

Observe that δmax for shallow shells is small, since

2 2 H (1 + λ)ctg2 α 1 − 4λ sin2 α ˜δmax = H −1 = F ∗ (α, λ). R2 2(1 − λ)2 sin α (1 + λ) cos α R2 One may check that F ∗ (α, λ) = O(1) and hence δ˜ max = O(H1 /R2 )2 ( O(a) denotes a quantity of “order”). Hence, δ max can serve as the perturbation parameter ˜ and Φ ˜ 0 and taking δ˜ = 0) yields equations and the system (3.138) (changing W of non-perturbed problems governing vibrations of shallow spherical shell with the radius R. The latter one, following static problems of spherical shells theory, can be considered as a ring-type plate located on a stiff support with the stiffness coefficient c = Eh/R2 . The governing equation for modes reads 2 ˜ 0 − k4 W ˜ 0 = 0; k4 = γhR42 ω∗2 ∇4 W 0 /gD − Eh/DR .

(3.140)

In what follows the eigenfrequencies, deflections, moments and transversal forces of the spherical shell for given radial boundary conditions can be yielded by the methods devoted to computation of a ring-type plate lying on a stiff support. Tangential displacements and stresses are expressed via the scalar function F, which is found from the differential equation ∇2 ∇2 F = W and the attached boundary conditions. The mentioned conditions correspond to inplane stress plate state. Correction to the first approximation can be computed form (3.134) and (3.135), where the following values should be taken: M[( )] = ε2 ∇8 ( ) + LL[( )]; M ∗ [( )] = ε2 ∇8 ( ) + ∇4 ( );

154


N[( )] = N ∗ [( )] = ∇4 ( ); L[( )] =

1 dzk d 2 ( ) ; ξ dξ dξ2

4 2 2 4 2 2 2 ω∗2 0 = k h l /12R2 + (1 − ν )l /R .

(3.141)

In the above z k (ξ) = xc (ξ) − δ(ξ) denotes the distance from the averaged surface of the conical shell to the below support; W 0 (ξ) and k are eigenfunctions and eigenvalues of ring-type plates for the corresponding boundary conditions. Using the perturbation method averaged analytical solutions yielding frequencies and mode of vibrations of thin conical shells are formulated. Owing to application of two non-perturbed objects (a cylindrical shell with averaged parameters and a shell on the stiff support), the whole interval of geometrical parameters variation is accounted. Boundaries between spaces of application of solutions (3.134), (3.137) and (3.134), (3.141) define relations (3.139). This conclusion will be further illustrated on examples of freely supported and clamped shells. w* .10

4

III 0.20

II I

0.15

0.10

0.05

0

15

30

45

60

75

a

Fig. 3.9. Non-dimensional minimal frequencies of shells in the case of free support (see text).

In Figures 3.9, 3.10 dependencies of non-dimensional minimal frequencies ω∗ · 104 = ωR2 [ρ(1 − γ2 )/Eg]1/2 · 104 of shells with parameters λ = 1/3, R 2 /h = 400 versus conical angle γ = π/2 − α in the case of free support (Fig. 3.9) and clamping (Fig. 3.10) are reported. Curve 1 corresponds to exact solution, curves 2, 3 correspond to solutions obtained using the formulas (3.134), (3.137) and (3.141), respectively. For the considered shells, variating γ from 3 to 75 ◦ the minimal frequencies are 1 − 7% (0.5 − 6%) higher for clamped (free) boundary conditions. Curves 2, 3 corresponding to two different approximated solutions intersects for γ = 73◦ (Fig. 3.9) and γ = 76 ◦ (Fig. 3.10) (relations (3.139) give γ = 74 ◦ ) and cover in full the variational interval of γ. Observe that the numbers n in circle direction may not coincide. However, already an account of the first approximation correction removes this drawback. The occurred improvement achieves 2 − 18% in the case of nondimensional frequencies ω ∗ . Consider now more higher frequencies of the lower part of spectrum. Owing to comparison of the parameters ω ∗ · 103 for free supported shells with h/R 2 =

3.11 Vibrations of reinforced conical shells w* .10

155

4

III 0.20

II I

0.15

0.10

0.05

0

15

30

45

75

60

g

Fig. 3.10. Non-dimensional minimal frequencies of shells in the case of clamping (see text).

0.003 and h/R 2 = 0.03 (Table 3.3) obtained using the perturbation technique via the formulas (3.134), (3.137) with exact ones [201], the non-dimensional frequencies coincide well for all shells besides of two (h/R 2 = 0.03, α = 45 ◦ and α = 15◦ ) for n = 2. Table 3.3. Computation of non-dimensional frequencies using various methods Method

α

Exact Formulas (3.134), (3.137) Exact Formulas (3.134), (3.137) Formula (3.141) Exact Formulas (3.134), (3.137) Formula (3.141)

75

45

14

2 223 276 336 319 574 136 162 241

h/R2 = 0.003 n 3 4 161 168 177 166 223 164 210 159 582 589 103 86.3 109 89.6 270 300

5 195 204 141 138 593 84.2 86.2 352

2 494 562 551 1113 595 252 643 265

h/R2 = 0.03 n 3 4 539 727 510 781 430 451 513 446 600 606 281 352 315 289 290 358

5 974 1011 552 537 615 455 387 462

In the latter case, however, one may use relations (3.139) and (3.141). The parameters ω∗0 and ω∗1 are computed from the formulas ω2∗0

2 (1 + λ2 )ctg2 αm4 π4 (1 − λ)2 tg2 α sin2 α h n21 (n21 − 1)2 = + ; R2 12λ2 2(1 − λ)2 n21 (n21 + 1) n21 + 1

ω∗1 = −

2 3ctg2 αm2 π2 (1 + λ2 )tg2 α sin2 α h n21 (n21 − 1)2 − × R2 12λ2 2n21 (n21 + 1) n21 + 1

(1 − λ)2 4m2 π2 λ2 J ; − 1 + λ2 1 − λ4

156

where:


˜ 0 (ξ) = sin k(ξ − ξ1 ); k = mπ cos α/(ξ2 − ξ1 ); W J = cos 2kξ1 [Ci(2kξ2 ) − Ci(2kξ1 )] + sin 2kξ1 [S i(2kξ2 ) − S i(2kξ1 )];

m denotes numer of half-waves in meridial direction (m = 2 has been taken): ξ S i(ξ) = 0

sin ξ dξ; Ci(ξ) = − ξ

∞ ξ

cos ξ dξ. ξ

4 BOUNDARY VALUE PROBLEMS OF ISOTROPIC CYLINDRICAL SHELLS

4.1 Governing relations We use as the governing equations of the general shells’ theory presented in reference [313]. Denote by R, h, L the radius, the thickness and the length of a shell, correspondingly. The dimensionless co-ordinates are defined by ξ = x/R and η = y/R. The surface load components are defined by P 1 (x, h), P2 (x, h) and P 3 (x, h). The displacements u(ξ, η), υ(ξ, η), w(ξ, η) are measured along the axes x, y, z, correspondingly. The geometric quantities, stress and load components are defined in Fig. 4.1. The governing equations for the circle cylindrical shell have the form T 1ξ + S η + RP1 = 0, S ξ + T 2η + N2 + RP2 = 0, N1ξ + N2η − T 2 − RP3 = 0, Hξ + M2η − RN2 = 0, M1ξ + Hη − RN1 = 0.

(4.1)

Above T 1 (T 2 ) and S denote the membrane forces in ξ(η) directions and shear force; M1 , M2 , H are the bending and the twisting moments; N 1 , N2 are the transverse shear forces. The indexes ξ, η denote the derivations in relation to ξ and η, respectively. We take as strain-displacement relations the set ε11 =

1 uξ , R

ε22 =

1 υη − w , R

ε12 =

1 υξ + uη , R

1 1 1 w , χ = − + υ = − + υ w , χ w . (4.2) ξξ 22 ηη η 12 ξη ξ R2 R2 R2 The stress-strain relations for the circle cylindrical shell have the following form [313] 1−ν Bε12 , T 1 = B (ε11 + νε22 ) , T 2 = B (ε22 + νε11 ) , S = 2 M1 = D (χ12 + νχ22 ) , M2 = D (χ22 + νχ11 ) , H = (1 − ν)Dχ12 , (4.3) χ11 = −

158


Fig. 4.1. A circular cylindrical isotropic shell.

where following notations are used: D is the cylindrical stiffness, D = Eh 3 / the 2 12 1 − ν ; B is the membrane stiffness, B = Eh/(1 − ν 2 ); ν is the Poisson’s coefficient; E is the Young’s modulus. From the relations (4.2), (4.3) the following equations are obtained T1 =

B uξ + ν υη − w , R

T2 =

B υη − w + νuξ , R

D (1 − ν) B υξ + uη , M1 = − 2 wξξ + ν wηη + υη , 2 R R D D M2 = − 2 wηη + υη + νwξξ , H = (1 − ν) 2 wξη + υξ . (4.4) R R They link the tangential stresses and the moments with the middle surface displacements. Eliminating the transverse shear forces from the first three equations of the system (4.1) and substituting the stresses and moments by the displacements of the middle surface, the following equilibrium equations related to the displacements are obtained 1−ν 1+ν uξξ + uηη + vξη − νwξ = −R2 p1 , 2 2 1−ν 1+ν uξη + vξξ + vηη + ε2 2(1 − ν)vξξ + vηη + 2 2 S =

4.2 Operator method

* ∂2 ∂2 −1 + ε2 (2 − ν) 2 + 2 wη = −R2 p2 , ∂ξ ∂η )

* ∂2 ∂2 νuξ − −1 + ε2 (2 − ν) 2 + 2 vη − 1 + ε2 ∇4 w = Rp3 , ∂ξ ∂η

159

)

where: ∇4 =

(4.5)

∂4 ∂4 ∂4 Pi + 2 + , pi = , i = 1, 2, 3, 4 2 2 4 B ∂ξ ∂ξ ∂η ∂η + 7 7 √ ' ( ε = D BR2 = h 2 3R .

Suppose that a particular solution to the differential equations (4.5), corresponding to the right-hand sides, is already found. We are going to analyse a homogeneous system of equations corresponding to (4.5).

4.2 Operator method The partial differential equations (4.5) have constant coefficients, and one can use the operator method of solution of differential equations with constant coefficients [313]. The main idea of this method is to operate with derivatives as with constants and to use the methods of linear algebra. We introduced a so-called potential function φ. According to that function a solution has the following form L11 L12 L13 Zφ ≡ L21 L22 L23 φ = 0. L31 L32 L33 Above Li j denote differential operators acting on u, υ, w in the 1st, 2nd and 3rd equations of the system (4.5). They can be treated as constant during formal further transformations. It is not difficult to prove a validity of this formal transformation using Fourier (or Laplace) transforms in regard with x and y. Therefore equations (4.5) may be reduced to a single equation for the potential function φ ∂8 φ ∂8 φ 2 + 4 1 + ε + Zφ ≡ 1 + 4ε2 ∂ξ8 ∂ξ6 ∂η2 ∂8 φ ∂8 φ ∂8 φ + 4 + + 6 + ε 2 1 − ν2 ∂ξ4 ∂η4 ∂ξ2 ∂y6 ∂η6 (8 − 2ν2 ) (1 + ν2 )(4 +

∂6 φ ∂6 φ ∂6 φ + 8 + 2 + ∂ξ4 ∂y2 ∂ξ2 ∂y4 ∂y6 1 ∂4 φ ∂4 φ ∂4 φ ) 4 + 4 2 2 + 4 = 0. 2 ε ∂ξ ∂ξ ∂y ∂y

(4.6)

160


The displacements are defined by the potential function φ ) ∂5 ∂3 ∂3 2 (1 + ν) (2 − ν) u= − + ν + ε + 1−ν ∂ξ∂η2 ∂ξ3 ∂ξ3 ∂η2

* ∂3 2ν2 ∂3 1 + ν ∂5 + 4ν 3 + φ, 1 − ν ∂ξ∂η4 ∂ξ 1 − ν ∂ξ∂η2 ) ∂3 ∂3 2 (2 − ν) ∂5 + v = (2 + ν) 2 + 3 − ε2 1 − ν ∂ξ4 ∂η ∂ξ ∂η ∂η

* 4 − 3ν + ν2 ∂5 ∂5 + φ, 1 − ν ∂ξ2 ∂η3 ∂η5      4 4    ∂4 2 2 − 2ν − ν2    ∂ ∂  4  2   + ε + + w= ∇ 4 φ.       4 2 2 4   ∂ξ 1−ν ∂ξ ∂η ∂η   

(4.7)

The stresses and moments are defined by the potential function φ substituting the equations (4.7) into (4.4). For a closed circle cylindrical shell we need to satisfy four boundary conditions on each edge for ξ = 0, d (d = L/R, where L is the shell’s length) (see Table 4.1). The following notation is used ∂H ∂w , wξ = , N˜ 1 ≡ N1 + ∂η ∂ξ 6 . ˜ W; ˜ T˜ ; W ˜ ξ ; M; ˜ N˜ = {Un ; Wn ; T n ; Wn ; Mn ; Nn } cos nη, U; 6 . ˜ S˜ = {Vn ; S n } sin nη, V; and the quantities with the ‘n’ index are the constants. Therefore, in a general case an investigation of a circle cylindrical shell is reduced to the boundary value problem of the eighth order for the partial differential equation (4.6) and one of the variants of the boundary conditions, presented in Table 4.1.

4.3 Simplified boundary value problems The asymptotic analysis of the governing equation (4.6) in regard to ε parameter leads to various reduced limiting cases. Develop the potential function φ into the asymptotic series (4.8) φ = φ0 + φ1 ε2 + φ2 ε3 + ... . Let us introduce the parameters of the asymptotic integration α and β which characterise the variation of the stressed shell’s state in relation to the special coordinates of the form −α −β φξ ∼ ε2 φ, φη ∼ ε2 φ. (4.9)

4.3 Simplified boundary value problems

161

Table 4.1. Governing boundary conditions. Type of boundary conditions A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

Boundary conditions for ξ = 0, d v = T˜ , v = T˜ , v = T˜ , v = T˜ , S = S˜ , S = S˜ , S = S˜ , S = S˜ , v = T˜ , v = T˜ , v = T˜ , v = T˜ , S = S˜ , S = S˜ , S = S˜ , S = S˜ ,

˜ u = U, T 1 = T˜1 , ˜ u = U, T 1 = T˜1 , ˜ u = U, ˜ w = W, ˜ u = U, ˜ w = W, ˜ u = U, T 1 = T˜1 , ˜ u = U, T 1 = T˜1 , ˜ u = U, wξ = W˜ ξ , ˜ u = U, T 1 = T˜1 ,

˜ w = W, ˜ w = W, ˜ w = W, ˜ w = W, ˜ w = W, T 1 = T˜1 , ˜ w = W, T 1 = T˜1 , ˜ N˜1 = N, ˜ N˜1 = N, ˜ N˜1 = N, ˜ N˜1 = N, ˜ N˜1 = N, ˜ N˜1 = N, ˜ N˜1 = N, ˜ N˜1 = N,

wξ = W˜ ξ wξ = W˜ ξ ˜ M1 = M ˜ M1 = M wξ = W˜ ξ wξ = W˜ ξ ˜ M1 = M ˜ M1 = M wξ = W˜ ξ wξ = W˜ ξ ˜ M1 = M ˜ M1 = M wξ = W˜ ξ T 1 = T˜1 ˜ M1 = M ˜ M1 = M

The input problem allows for the variable separation. It means that the potential function can be presented in the form φ (ξ, η) = ϕ (ξ) exp (im η) .

(4.10)

Now we can summarise the asymptotic procedure used as follows. Substituting ansatz (4.8) into the governing boundary value problems and comparing coefficients of ε k , we conclude that the limiting (ε → 0) systems are strongly dependent upon the values of the parameters α, β. Now we are going to find all possible α, β and search for all sensible value of these parameters, for which limiting systems have mathematical (well posedness) and physical sense. It is remarkable that as the result of this (but very laborious) procedure we obtain only the few limiting systems analysed below. α = −1/2, β = 0. This case corresponds to a general variant of the semiinexstensional theory. The corresponding equation has the form φξξξξ +

ε2 ∇2 φηηηη = 0, 1 − ν2 0η

7 where: ∇0η = 1 + ∂2 ∂η2 . The displacement and stress-strain relations have the following form u = −φξηη , T1 = −

B1 φξξηη , R

v = φηηη , w = φηηηη , B1 ε2 T2 = − ∇ φ φξξξξ + 0η ηηηη , R 1 − ν2

(4.11)

162


B1 D φξξξη , M1 = −ν 2 ∇0η φηηηη , R R D D M2 = − ∇0η φηηηη , H = (1 − ν) 2 ∇0η φξηηη , R R D D N1 = − 3 ∇0η φξηηη , N2 = − 3 ∇0η φηηηη , R R S =

(4.12)

where: B1 = Eh. The given expressions are called the interior-zone equations. The physicalgeometrical hypotheses, corresponding to the relations of the semi-inextensional theory, lead to the following simplified equations for displacements uη + vξ = 0,

vη − w = 0,

ε2 ∇2 wηηηη − Rp3 = 0. (4.13) 1 − ν2 0η α = 1/2, β < α. This case corresponds to an edge effect. The potential function φ is defined by the equation wξξξξ +

1 − ν2 φ = 0. ε2

φξξξξ +

(4.14)

The expressions for displacements, forces and moments have the form u = −νφξξξ , T1 = − S = M2 = −ν

υ = (2 + ν)φξξη , B1 φξξηη , R

B1 φξξξη , R D ∂6 φ , R2 ∂ξ6

T2 = − M1 = − H=

w = φξξξξ , B1 φξξξξ , R

D ∂6 φ , R2 ∂ξ6

D ∂6 φ . (1 − ν) R2 ∂ξ5 ∂η

(4.15)

The physical-geometrical hypothesis corresponding to the equation (4.14) allows to get the following relations 1+ν 1−ν uξη + vξξ − wη = 0, 2 2 4 2 ∂ νuξ − 1 + ε w = 0. ∂ξ4

uξξ − νwξ = 0,

(4.16)

α = β = 1/2. This case corresponds to the theory of shallow shells (DonnelVlasov theory [251, 694, 695]). The governing equation has the form ∇4 ∇4 φ +

1 − ν2 φξξξξ = 0, ε2

(4.17)


163

where: ∇2 = (∂2 /∂ξ2 ) + (∂2 /∂η2 ). The displacements, forces and the moments are expressed by the potential function φ in the form u = −φξηη + νφξξξ , v = φηηη + (2 + ν)φξξη , w = ∇4 φ, 2 2 1 ∂2 1 ∂2 ∂ ∂ ε11 = − − φξξ , ε22 = φξξ , ν ν R ∂ξ2 ∂η2 R ∂η2 ∂ξ2

(4.18)

2(1 + ν) B1 B1 φξξξη , T 1 = − φξξηη , T 2 = − φξξξξ , R R R B1 1 4 1 4 φξξξη , χ11 = − 2 ∇ φξξ , χ22 = − 2 ∇ φηη , S = R R R 1 4 D 4 χ12 = − 2 ∇ φξη , M1 = − 2 ∇ φξξ + νφηη , R R D 4 D M2 = − 2 ∇ νφξξ + φηη , H = (1 − ν) 2 ∇4 φξη , R R D D (4.19) N1 = − 3 ∇6 φξ , N2 = − 3 ∇6 φη , R R 7 7 where: ∇6 = ∇4 ∂2 ∂ξ2 + ∂2 ∂η2 . α ∼ β, 1/2 < β. This case corresponds to the state with the fast circumferential and longitudinal variations. The governing equation splits into two separate groups of the generalized plane stress state and bending of the plate. The first group consists of only the membrane displacements, strain and stresses (u, υ, ε 11 , ε22 , ε12 , T 1 , T 2 , S ), defined by the expression (4.19). The potential function is defined by the following equation (4.20) φξξξξ + 2φξξηη + φηηηη = 0. ε12 =

The equations for displacements have the form uξξ +

1−ν 1+ν νηη + νξη = 0, 2 2

1+ν 1−ν uξη + νξξ + νηη = 0. (4.21) 2 2 The second group consists of moments, transverse shear forces, components of bending deformation and a normal displacement defined by expressions (4.19). The potential function may be obtained from the following equation φξξξξ + 2φξξηη + φηηηη = 0.

(4.22)

The equilibrium equations for displacements lead to one equation in relation to the normal displacement wξξξξ + 2wξξηη + wηηηη = 0.

(4.23)

164


The asymptotic analysis of the governing relations of the theory of isotropic cylindrical shells leads to an order reduction of the fundamental equation. Therefore, in order to formulate appropriate boundary problems for limiting equations we need to split the boundary conditions. This part belongs to the most complicated ones. A full classification of the simplified boundary problems including the different variations of boundary conditions, is presented here. Consider the splitted boundary conditions for the given limiting equations using the canonical form of the boundary conditions [674]. We assume the slow variation in the interior zone. It allows for the separation of the input stressed state for the semi-inextensional theory (4.11) and edge eefect (4.14). We assume, that a shell is long enough for the interactions of the edge effects to be neglected. As an example the A 6 variant of Table 4.1 the boundary conditions splitting will be considered. Using the expression (4.12) and (4.15) for displacements, we estimate the order of the terms in boundary conditions: S (1) ∼ ε1.5−7β φ(1) ,

S (2) ∼ ε−1.5−β φ(2) ,

T 1(1) ∼ ε1−6β φ(1) ,

T 1(2) ∼ ε−1−2β φ(2) ,

w(1) ∼ ε−4β φ(1) , 0.5−6β (1) w(1) φ , ξ ∼ε

w(2) ∼ ε−2 φ(2) , −2.5 (2) w(2) φ . ξ ∼ε

The superscript (1) and (2) define the states for which the splitting is carried out. The characteristic variation of a stress or a displacement Q is defined by µ(Q) using the following relation 7 εµ(Q) ∼ Q(2) φ(1) Q(1) φ(2) . Then we get µ(S ) = −3 + 6β, µ(w) = −2 + 4β,

µ (T 1 ) = −2 + 4β, µ wξ = −3 + 6β.

The boundary condition A 6 will be splitted in a way corresponding to the decrease of the characteristic variation (it is related to their canonical form [674]) T 1 , w, S, wξ . ˜ 0 (or T˜ 0) a being sought splitting has the It implies that for S˜ = 0, W˜ ξ , W form: ˜ for ξ = 0, d T 1(1) = T˜ , w(1) = W, (1) S (2) = −S (1) , w(2) ξ = −wξ .

˜ = T˜ = 0, S˜ 0 (or W˜ ξ = 0) we get: In a case for W for ξ = 0, d

˜ S (2) = S˜ , w(2) ξ = Wξ ,

(4.24)


165

(1) T 1(1) = −T (2) = −w(2) . 1 , w

It leads to the conclusion that, depending on the given load, an calculation of a shell can be initiated using semi-inextensional theory splitting (4.24) or the edge effect. For the splitting boundary conditions A 5 ÷ A12 , taking into account the canonical forms given in Table 4.1, the stress states can be sequentially considered with the help of equations (4.11) and (4.14). Table 4.2. Canonical form of boundary conditions for isotropic shell. Variants of boundary conditions A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

Boundary conditions in the canonical form υ υ υ υ u T1 u T1 υ υ υ υ u T1 u T1

∼ ∼ ∼ ∼

u T1 u T1 w w w w u T1 u T1 S S S S

∼ ∼ ∼ ∼

∼ ∼ ∼ ∼

w w w w S S S S wξ wξ M1 M1 wξ wξ M1 M1

∼ ∼ ∼ ∼

wξ wξ M1 M1 wξ wξ M1 M1 N˜1 N˜1 N˜1 N˜1 N˜1 N˜1 N˜1 N˜1

For the rest of the variants, additional analysis is needed. Consider the A 1 ˜ W, ˜ W˜ ξ differs from zero, then from the variant. If at least one of the variables V, canonical form we get the following ‘unsuitable’ splitting for ξ = 0, d

˜ w(1) + w(2) = W,

˜ w(2) ξ = −Wξ , ˜ v(1) = V,

(4.25) (4.26)

u(1) = −u(2) .

However, taking into account the relations (4.13), the condition (4.25) takes the form: for ξ = 0, d w(2) = (Wn − nVn ) cos nη. (4.27) The splitting defined by (4.26) and (4.27) allows for separate considerations of the semi-inextensional theory and the edge effect. Suppose that only U˜ 0. The canonical form leads to the following splitting

166


for ξ = 0, d

˜ w(1) = w(2) . v(1) = 0, w(1) = 0, u(1) = U, ξ ξ

(4.28)

Table 4.3. Splitting boundary conditions for isotropic shell. Variants of boundary conditions A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

Boundary conditions for ξ = 0, d ˜ − nV), ˜ w(2) = W˜ ξ , v(1) = V, ˜ u(1) = −u(2) w(2) = (W ξ (1) (1) (1) (2) ˜ v = 0, u = U, w = −ν12 φξξηη , w(2) ξ = −wξ (2) (1) (2) (1) ˜ ˜ ˜ ˜ w = (W − nV), wξ = Wξ , v = V, T 1 = −T (2) 1 (1) v(1) = 0, T 1(1) = T˜ , w(2) = −ν12 φξξηη , w(2) ξ = −wξ ˜ − nV), ˜ M (2) = −M(1) (or M (2) = M) ˜ , v(1) = V, ˜ u(1) = −u(2) w(2) = (W 1 1 1 (2) (1) (1) (2) ˜ w = −ν12 φξξηη , M = −M (1) v = 0, u = U, 1 1 (2) (2) ˜ − nV), ˜ M = −M(1) (or M (2) = M) ˜ , v(1) = V, ˜ T (1) = −T (2) w = (W 1 1 1 1 1 v(1) = 0, T 1(1) = T˜ , w(2) = −ν12 φξξηη , M1(2) = −M(1) 1 (1) ˜ S (2) = S˜ , w(2) = −u(2) , w(1) = −w(2) ξ = Wξ , u (1) (1) (1) (2) ˜ ˜ u = U, w = W, S = −S (1) , w(2) ξ = −wξ (1) (2) (1) (1) (2) ˜ S = −S , w = −w(1) T 1 = T˜ , w = W, ξ ξ (2) (2) (1) S = S˜ , wξ = W˜ ξ , T 1(1) = −T (2) = −w(2) 1 , w ˜ S (2) = −S (1) , M (2) = −M(1) w(1) = W, 1 1 (2) ˜ u(1) = −u(2) , w(1) = −w(2) S = S˜ , M1(2) = M, ˜ S (2) = −S (1) , M (2) = −M(1) T 1(1) = T˜ , w(1) = W, 1 1 (2) (2) ˜ T (1) = −T (2) , w(1) = −w(2) S = S˜ , M1 = M, 1 1 ˜ u(1) = U, ˜ w(2) = −w(1) , N˜ (2) = 0 v(1) = V, ξ ξ 1 (2) (2) ˜ u(1) = −u(2) , v(1) = 0 wξ = W˜ ξ , N1 = N, ˜ T (1) = T˜ , w(2) = −w(1) , N˜ (2) = 0 v(1) = V, ξ ξ 1 1 (2) ˜ T (1) = −T (2) , v(1) = 0 wξ = W˜ ξ , N1(2) = N, 1 1 ˜ u(1) = U, ˜ M (2) = −M (1) , N˜ (2) = 0 v(1) = V, 1 1 1 (2) (2) ˜ N = N, ˜ u(1) = −u(2) , v(1) = 0 M1 = M, 1 ˜ T (1) = T˜ , M (2) = −M (1) , N˜ (2) = 0 v(1) = V, 1 1 1 1 (2) ˜ N (2) = N, ˜ T (1) = −T (2) , v(1) = 0 M1 = M, 1 1 1 ˜ S (1) = (S˜ + nN), ˜ w(2) = −w(1) , N (2) = N˜ u(1) = U, ξ ξ 1 (2) (2) ˜ u(1) = −u(2) , S (1) = 0 wξ = W˜ ξ , N1 = N, ˜ w(2) = −w(1) , N (2) = N˜ T 1(1) = T˜ , S (1) = (S˜ + nN), ξ ξ 1 (2) (2) ˜ N = N, ˜ T (1) = −T (2) , S (1) = 0 M1 = M, 1 1 1 ˜ S (1) = (S˜ + nN), ˜ M (2) = −M(1) , N (2) = N˜ u(1) = U, 1 1 1 (2) (2) ˜ N˜ = 0, u(1) = −u(2) , S (1) = 0 M1 = M, 1 ˜ M (2) = −M(1) , N (2) = N˜ T 1(1) = T˜ , S (1) = (S˜ + nN), 1 1 1 (2) (2) ˜ N˜ = 0, T (1) = −T (2) , S (1) = 0 M1 = M, 1 1 1

The first two conditions are linearly dependent (see the relations (4.13)). The additional boundary condition is achieved in the following way. The first order terms of the displacements v and w are expressed by the potential functions φ (1) and φ(2) :


167

(1) v = φ(1) ηηη + (2 + ν)φηηη , (1) (2) w = φ(1) ηηη + 2φηηη + φηηη .

Using the above expressions, the splitting (4.28) can be substituted by the following one: for ξ = 0, d

˜ υ(1) = 0, w(2) = −νφ(2) , w(2) = −w(1) . u (1) = U, ξξηη ξ ξ

An analogous procedure is applied for splitting of the boundary condition A 2 – A4 , A13 –A16 . For the latter one the combination of S and N 1 is needed. Final results are shown in Table 4.3. The splitting boundary conditions for the relations of the plane stress state (4.20) and the plate bending (4.22) result indirectly from the limiting cases. The boundary conditions, including the in-plane factors u, υ, T 1 , T 2 , S should be given for a plane stress state (4.20). The boundary conditions including a normal displacement, its derivative and the moments should be referred to as the bending state (4.22). Therefore, for small and large variations of a strain state a solution to the governing boundary value problem for the eight-order differential equation is reduced to that of finding solutions of the fourth order along the longitudinal co-ordinate for the interior and edge zones. However, in order to investigate the stress state with the α = β = 0.5 we need to solve a boundary value problem for the eighth order equation of the shallow shell theory, which does not lead to the reduction of the input problem.

5 BOUNDARY VALUE PROBLEMS – ORTHOTROPIC SHELLS

5.1 Governing relations In this Chapter we consider a closed circle cylindrical shell supported in two principal directions. Supporting ribs are the one-dimensional elastic elements, situated uniformly with the same constant distance between them. The boundary value problems of the theory of closed circular cylindrical shells, eccentrically reinforced in the two principal directions, are investigated within the framework of the structurally orthotropic scheme. The supporting ribs are placed dense enough and we can homogenize their stiffness and mass characteristics. For the whole shell, the hypothesis about undeformable normal is valid. We assume that the ribs’ height is small in comparison with the curvature radius. There is no interaction between the two ribs lying in two directions. In the calculation scheme we take the shell as the structural orthotropic. Therefore, we can define with high accuracy displacements, membrane forces and vibration frequencies [11, 12, 13, 14, 454]. The values of moments are considered as the first approximation of homogenization procedure (see Chapter 10). The equilibrium equations have the form (4.1). According to the reference [8, 9, 10, 452], the relations between the forces (moments) and deformations (curvatures and torsion) can be written as follows: T 2 = B21 ε11 + B22 ε22 + K21 χ11 + K22 χ22 , S = B33 ε12 + K33 χ12 , M1 = D11 χ11 + D12 χ12 + K11 ε11 + K12 ε22 , M2 = D12 χ11 + D22 χ22 + K21 ε11 + K22 ε22 , H = D33 χ12 + K33 χ12 .

(5.1)

The middle surface of the shell serves as a reference surface defining the stiffness Di j and Ki j . Because the transversal interaction between the ribs is small, the quantities K12 and K21 can be neglected. It is assumed that the interaction between torsion stiffness and the in-plane displacement is small, and the homogenized torsion is equal to the sum of the torsion stiffness of the shell and the ribs, so B33 =

Eh , 2(1 + ν)

K33 = 0,

D33 =

D E s It E r It + + . 2 4d1 4d2

(5.2)

170


The other stiffness coefficients are given in the reference [452] and they have the following form B11 = B +

EsFs , d1

B22 = B +

Er Fr , d2

B12 = B21 = ν21 B11 = ν12 B22 ≈ νB, K11 =

E sS s , d1

K22 =

Er S r , d2

D11 = D +

E s Is , d1

D22 = D +

E r Ir , d2

D12 = D21 = ν41 D11 = ν14 D22 ≈ νD, B=

Eh , 1 − ν2

D=

Eh3 ' (. 12 1 − ν2

(5.3)

Above F s (F r ), I s (Ir ), It (It ), S s (S r ) denote the square of the transverse section, moment of inertia, torsion moments, and the static and of inertia of the stringer (ring), E s , (E r ) denotes the Young’s modulus of the shell stringer (ring), and B 12 , D12 denote the “Poisson’s coefficients”, d 1 (d2 ) denotes distances between stringers (rings). The ribs location in relation to the middle shell surface (the ribs eccentricity) is characterised by the values K 11 and K12 . For the internal support, we have K 11 , K22 > 0, whereas for the external support we have K 11 , K22 < 0. In the case of a central support, we have K 11 = K22 = 0. Note that the eccentricity of the supporting ribs essentially complicates the problem. The expression (5.1) shows that for this case the additional relation between the tangential and the bending factors occurs. The geometrical relations have the form 1 1 υη − w , ε12 = υξ + uη , R R 1 1 1 (5.4) χ11 = − 2 wξξ , χ22 = − 2 wηη + vη , χ12 = − 2 wηη . R R R Substituting the expressions (5.1), (5.4) by (4.1) we obtain the following equilibrium equations for the displacements ε11 =

1 uξ , R

ε22 =

K11 wξξξ = −P1 R2 , R K22 D22 D33 (B21 + B33 ) uξη + B22 − 2 + 2 υηη + B33 + 2 υξξ + R R R D + 2D K D K 22 22 22 33 22 − wξξη + B22 − wηηη + wη = −P2 R2 , R R R2 R2 D K11 K22 D12 + 2D33 22 uξξξ + − υξξη + −B21 uξ − υηηη + 2 R R R R2 K22 D11 2 B22 − υη + 2 wξξξξ + 2 (D12 + 2D33 ) wξξηη + R R R K22 D22 wηη + B22 w = P3 R. wηηηη + 2 (5.5) R2 R The equations (5.1)–(5.5) should be supplemented by one of the boundary conditions given in Table 4.1. B11 uξξ + B33 uηη + (B12 + B33 ) υξη − B12 wξ −

5.1 Governing relations

171

We set out that a particular solution to the equation (5.5), corresponding to their right hand sides, is found. Using the operator method, the homogeneous equations (5.5) are reduced to solving the partial differential equation of the eighth order of the form ) 8 ∂8 2 ∂ −1 −2 Zn φ ≡ 1 + ν12 ε−2 ε + 2 ε + ε ε − ν ε ε ε + 3 4 12 6 e 1 6 1 5 ∂ξ8 ∂ξ6 ∂η2 −2 2 ε2 + 4ε3 ε4 ε−1 5 + ε1 εe + 2ν12 ε6 ε7

∂8 + ∂ξ4 ∂η4

8 ∂8 2 ∂ + ε2 ε4 + ν221 ε−2 − 1 ε7 2 6 ∂η8 ∂ξ ∂η ∂6 ∂6 + 2 2ε3 ε4 (ε5 + ν12 ε3 ε4 ) + ε−2 + 2ν12 ε−2 1 ε6 1 εe 6 ∂ξ4 ∂η2 ∂ξ ∂6 −2 2 2ε3 ε4 ε−1 + ν ε ε + ε ε − ν ε ε + 12 2 4 3 4 21 7 1 5 ∂ξ2 ∂η4 4 6 2 ∂ −2 ∂ + ε + 2 ε2 ε4 ε−1 2 ε2 ε4 + ν221 ε−2 1 ε7 1 5 + ν12 ε2 ε4 − ∂ξ4 ∂η6 ∂4 4 * −2 2 ∂ ν21 ε−2 ε + ε ε + ν ε ε φ = 0, 2 4 21 1 7 1 7 ∂ξ2 ∂η2 ∂η4

−2 2 ε2 ε4 ε−1 5 + ε3 ε4 − ν12 ε1 ε7 εe

where:

7 1/2 ε1 = D1 B 2 R 2 , ε4 = B2 /B1 , 7 Di = Dii − Kii2 Bi ,

ε2 = D2 /D1 ,

ε5 = B3 /B1 ,

ε6 = e1 /R,

εe = ε6 + ε7 , ε8 =

ε21 ε4

ε3 = D3 /D1 , ε7 = e2 /R,

D3 = D12 + 2D33 + ν0 e1 e2 B12 ,

B3 = 2B1 B22 B33 /(B1 B22 − 2B12 ), + ε6 ,

ei = Kii /Bi ,

ε70 = ε21 ε2 + ε27 , ν10 = ν12 +

ε−1 5 ,

(5.6)

Bi = ν0 Bii , i = 1, 2,

ν0 = 1 − ν12 ν21 , ν11 = 1 + ν12 ν21 .

D1 , D2 , D3 denote the bending rigidities and the torsion rigidity; B 1 , B2, B3 denote the membrane rigidities and shear rigidity. The ε1 parameter characterizes the relative thickness of the shell; ε 2 characterizes the ratio of the bending rigidities; ε 3 characterizes the ratio of the torsion rigidity and bending rigidity in longitudinal direction; ε 4 characterizes the ratio of the membrane rigidities; ε 5 characterizes the ration of shear rigidity and in-plan membrane rigidity; ε 6 and ε7 are related to the eccentricities of stringers and rings. The displacements are defined by the potential function of the form ) ∂4 −1 2 u = ν0 ε−1 + 1 + 2ν ε ε + ν ν ε − ν ε ε ε + 12 4 12 21 3 12 6 7 4 ε6 1 5 ∂ξ4

172


∂4 + 1 + ν12 ν21 + 2ε4 ε−1 ε70 − 5 2 2 ∂ξ ∂η ∂4 ∂2 −1 + ν + 2ν ε − ν ε7 ε ε70 + 1 + ν21 ν10 + ε−1 12 12 4 12 5 5 ∂η4 ∂ξ2 ∂2 * − 3ν ν − 1 φξ , ε 1 − 2ν12 ε−1 12 21 7 5 ∂η2 ) ∂4 −1 + ν ε − υ = − ν10 ε5 ε21 − ν12 ε6 ε7 + 2ν12 ε−1 ε6 11 4 5 ∂ξ4 ∂4 − ν10 ε70 − 2ν10 ε7 + ν0 ε21 ε2 − ν12 ε6 ε7 ∂ξ2 ∂η2 * ∂4 ∂2 ∂2 ν0 (ε70 − ε7 ) 4 − (ν12 − 2ν10 ) 2 + (1 − ν0 ε7 ) 2 φη , ∂η ∂ξ ∂η 4 4 ∂ ∂ w = ε−1 + 2ε−1 + 4 5 − 4ν10 + 2ν0 ν10 ε70 ∂ξ4 ∂ξ2 ∂η2

∂4 (1 − 2ν0 ε7 + ν0 ε70 ) 4 φ. ∂η

ε−1 5 + ν10 ε6

(5.7)

The reinforced shell can be classified on the basis of various supports: A. Waffle shells. It this case the stiffness in two main directions are approximately equal. The following relations are true ε1 1,

ε2 ∼ 1,

ε4 ∼ 1,

ε3 ∼ 1

ε5 < 1,

(or ε3 ∼ ε1 ) ,

ε 6 ∼ ε7 ∼ ε1 .

(5.8)

B. Stringer shells. The following relations are valid for this case ε1 1,

ε 2 ∼ ε1 ,

ε3 ∼ ε1

ε4 ∼ ε5 < 1,

(or ε3 ∼ 1) ,

ε 6 ∼ ε1 .

(5.9)

C. Ring-stiffened shells. The following estimations are true in this case ε1 0, ε21 ε4 ∇41 φηη + 0.5ε5 ∇1η φξξ = 0, (5.18) where:

∂2 , ∂η2 υ = 2ε−1 5 ε7 φηη + 2φ

∇1η = 1 + (ε6 + ε7 ) u = 2ε−1 5 ε6 φξη − φ

ξξη

,

T 1 = −B1 R−1 ∇1η φξξηη ,

ξξη

,

S = B1 R−1 ∇1η φξξξη .

(5.19)

The two other systems define the splitting plane stress plate state: γ = δ + 0.5. It corresponds to the prevalent deformation in the ξ direction φξξ + 2ε4 ε−1 5 φηη = 0, υ = 2ε−1 5 φξξη ,

S = B1 R−1 φξξξη .

(5.20) (5.21)

δ − 0.5. It corresponds to the prevalent deformation in the η direction φξξ + 0.5ε5 φηη = 0, u = −φξηη ,

T 1 = −B1 R−1 φξξηη ,

S = −0.5B1R−1 ε5 φξξηη .

(5.22) (5.23)

While the variations increase, the equations (5.20), (5.22) do not change, and in the equation (5.18) the underlined term should be neglected. Then, the relation’s (5.18), (5.20) and (5.22) govern both bending and plane deformation of the plate with a waffle - like support. In order to formulate the boundary conditions for the simplified equations (5.11), (5.12), it is necessary to split them. In accordance with the splitting algorithm described in the Chapter 4.3, we get the canonical system of the boundary conditions

5.2 Statical problems

175

of the structurally orthotropic waffle cylindrical shells. It is the same as the one presented in Table 4.2, and then we may use splitting boundary conditions given in Table 4.3. The analysis of the boundary conditions for the simplified equations (5.18), (5.20), (5.22) possesses some differences in comparison with the earlier considered cases. It is realised in two steps. In order to illustrate them, we consider the variant A2 . We introduce the characteristic exponents µ 1 (P) and µ2 (P) using the following relations 7 7 ε1µ1 (P) ∼ P(1) φ(2) P(2) φ(1) , ε1µ2 (P) ∼ P(3) φ(1) P(1) φ(3) . Estimating orders of the quantities in the boundary conditions A 2 we obtain µ1 (υ) = −µ1 (w) = −3, µ1 (T 1 ) = −1, µ1 wξ = −3.5, µ2 (υ) = −1, µ2 (w) = 1, µ2 (T 1 ) = −1, µ2 wξ = 0.5 . The two series are introduced in the form T1,

υ ∼ w,

wξ ,

(5.24)

w1 ,

wξ ,

υ ∼ T 1.

(5.25)

The splitting (5.24) and (5.25) is characterized by a decrease of the µ 1 and µ2 parameter. The series (5.24) allows for splitting of the state (5.20), whereas the series (5.25) allows for a final splitting. ˜ =W ˜ ξ = V˜ = 0, then, taking into accounts the first If, for instance, T˜ 0, W canonical expression, we obtain: for ξ = 0, d

T 1(2) = T˜ ,

w(1) + w(2) + w(3) = 0,

υ(1) + υ(2) + υ(3) = 0, (3) w(1) ξ = −wξ .

Using the expressions (5.25) we get the final splitting for ξ = 0, d

T 1(2) = T˜ ,

w(1) = −w(2) ,

υ(3) = −υ(1) − υ(2) , w(1) ξ = 0.

A detailed analysis shows that two additional splitting variants can be realised: ˜ ˜ for ξ = 0, d w(1) = W, w(1) ξ = Wξ , v(3) = −v(1) , for ξ = 0, d

˜ v(3) = V,

T 1(2) = −T 1(1) − T 1(3) ; or

T 1(2) = −T 1(3) , w(1) = 0,

(3) w(1) ξ = −wξ .

˜ 0 (or W ˜ ξ 0), V˜ = T˜ = 0, whereas The first one should be used in the case W ˜ ˜ ˜ ˜ the second one in the case V 0, W = Wξ = T = 0. The splitting for all the variants boundary conditions are given in Table 5.1.

176

5 BOUNDARY VALUE PROBLEMS – ORTHOTROPIC SHELLS Table 5.1. Splitting boundary conditions for waffle shell. Variants of boundary conditions A1

A2

A3

A4

A5

A6

A7

A8

A9

A10

A11

A12

Boundary conditions for ξ = 0, d. ˜ w(1) = −w(2) , w(1) = 0, v(3) = −(v(1) + v(2) ) u(2) = U, ξ (1) ˜ w(1) = W ˜ ξ , u(2) = −u(1) , v(3) = −v(1) w = W, ξ ˜ u(2) = −u(3) , w(1) = −w(2) , w(1) = 0 v(3) = V, ξ (2) (3) (1) ˜ T 1 = T , w(1) = −w(2) , w(1) = 0, v = −(v + v(2) ) ξ (1) (2) (1) (1) (3) (1) ˜ ˜ w = W, wξ = Wξ , v = −v , T 1 = −(T 1 + T 1(3) ) ˜ T (2) = −T (3) , w(1) = 0, w(1) = −w(3) v(3) = V, ξ ξ 1 1 (2) ˜ w(1) = −w(2) , M (1) = 0, v(3) = −(v(1) + v(2) ) u = U, 1 ˜ M (1) = M, ˜ u(2) = −u(1) , v(3) = −v(1) v(1) = V, 1 (3) (2) ˜ v = V, u = −u(3) , w(1) = −w(2) , M1(1) = 0 (2) T 1 = T˜ , w(1) = −w(2) , M1(1) = 0, v(3) = −(v(1) + v(2) ) ˜ M (1) = M, ˜ v(3) = −v(1) , T (2) = −(T (1) + T (3) ) w(1) = W, 1 1 1 1 (2) (3) ˜ v = V, T 1 = −T 1(3) , w(1) = 0, M1(1) = −M1(3) ˜ S (3) = −S (2) , w(1) = −w(3) , w(1) = 0 u(2) = U, ξ ξ (1) (2) ˜ ˜ w = W, w(1) = −u(1) , S (3) = −S (1) ξ = Wξ , u (3) (2) S (3) = S˜ , w(1) = 0, w(1) = −u(3) ξ = −wξ , u (2) (1) (3) (3) (2) ˜ T 1 = T , S = −S , wξ = −wξ , w(1) = 0 ˜ w(1) = W ˜ ξ , T (2) = −T (1) , S (3) = −S (1) w(1) = W, ξ 1 1 (3) (2) (3) (3) (1) ˜ S = S , w = 0, w(1) ξ = −wξ , T 1 = −T 1 (1) (3) (2) (3) (2) (1) ˜ S = −S , M = −M , w = 0 u = U, 1 1 ˜ M (1) = M, ˜ u(2) = −u(1) , S (3) = −S (1) w(1) = W, 1 (3) S (3) = S˜ , w(1) = 0, M1(1) = −M (3) = −u(1) 1 , u (2) (1) (3) (2) (3) ˜ T 1 = T , S = −S , M1 = −M 1 , w(1) = 0 ˜ M (1) = M, ˜ T (2) = −T (1) , S (3) = −S (1) w(1) = W, 1 1 1 (2) (3) (3) (1) ˜ S = S , w = 0, M1(1) = −M (3) 1 , T 1 = −T 1 (1) (1) (3) ˜ v(3) = −v(2) , w = 0, N = −N u(2) = U, ξ 1 1 (1) ˜ ξ , N (1) = N, ˜ v(3) = v(1) , u(2) = −u(1) wξ = W 1 ˜ N (1) = −N (3) , w(1) = 0, u(2) = −u(1) v(3) = V, ξ 1 1 (2) (1) (3) T 1 = T˜ , v(3) = −v(2) , w(1) ξ = 0, N1 = −N 1 (1) (1) (2) (1) (3) (1) ˜ ˜ wξ = Wξ , N1 = N, v = −v , T 1 = −(T 1 + T 1(3) ) ˜ w(1) = 0, N (1) = −N (3) , T (2) = −T (3) v(3) = V, ξ 1 1 1 1 (2) ˜ v(3) = −v(2) , N (1) = −N (3) , M (1) = 0 u = U, 1 1 1 ˜ M (1) = M, ˜ v(3) = −v(1) , u(2) = −u(1) N1(1) = N, 1 (1) (3) (3) ˜ v = V, N1 = −N 1 , M1(1) = 0, u(2) = −(u(1) + u(3) ) T 1(2) = T˜ , v(3) = −v(2) , M1(1) = 0, N1(1) = −N (3) 1 (1) ˜ N (1) = N, ˜ v(3) = −v(1) , T (2) = −(T (1) + T (3) ) M1 = M, 1 1 1 1 ˜ M (1) = 0, N (1) = −N (3) , T (2) = −T (3) v(3) = V, 1 1 1 1 1


177

Table 5.1. Cont.

A13

A14

A15

A16

˜ S (3) = −S (2) , w(1) = 0, N (1) = N (2) u(2) = U, ξ 1 1 (1) ˜ ξ , N (1) = N, ˜ S (3) = −S (1) , u(2) = −u(1) wξ = W 1 (1) (2) S (3) = S˜ , N1(1) = −N (3) = −(u(1) + u(3) ) 1 , wξ = 0, u (2) (1) T 1 = T˜ , S (3) = −S (2) , wξ = 0, N1(1) = −N (3) 1 (1) ˜ ˜ (3) = −S (1) , T (2) = −T (1) w(1) ξ = Wξ , N1 = N, S 1 1 (1) (2) (1) (3) S (3) = S˜ , N1(1) = −N (3) 1 , wξ = 0, T 1 = −(T 1 + T 1 ) ˜ S (3) = −S (2) , M (1) = 0, N (1) = −N (3) u(2) = U, 1 1 1 (1) ˜ N (1) = N, ˜ S (3) = −S (1) , u(2) = −u(1) M1 = M, 1 (1) (2) S (3) = S˜ , N1(1) = −N (3) = −(u(1) + u(3) ) 1 , M1 = 0, u (2) (2) T 1 = T˜ , S (3) = −S , M1(1) = 0, N1(1) = −N (3) 1 ˜ N (1) = N, ˜ S (3) = −S (1) , T (2) = −T (1) M1(1) = M, 1 1 1 (1) (2) (1) (3) S (3) = S˜ , N1(1) = −N (3) 1 , M1 = 0, T 1 = −(T 1 + T 1 )

Stringer shells In this case, the obtained limiting relations of the semi-inexstensional theory and the edge effect overlap with the corresponding expressions (5.11) and (5.12), (5.14), when ε7 = 0. The equations governing the state with fast variation (the shallow shells’ theory) lead to further splitting. For the stringer shells for ε 3 ∼ ε1 we obtain the new approximate equations which do not have their analogies in the isotropic case. α = 0, β = 0.5. This case corresponds to the state with fast variation in the longitudinal direction. The potential function is defined by the equation ∇22η φξξξξ + ε21 ε4 ∇41 φηηηη = 0, where: ∇2η = 1 + ε6

(5.26)

∂2 . ∂η2

The displacements are defined by the potential functions in the following way u = −φξηη ,

υ = φηηη ,

w = φηηηη .

(5.27)

α = β = 0.5. It corresponds to the edge effects: 2 ∂2 ∂2 ∂2 6 ∇42 φ + ε−2 ε φ + ε − 2ν 1 − ν 2 − ε φηη = 0. 12 6 6 12 1 ∂ξ2 ∂η2 ∂ξ2 The displacements are

∂4 ∂4 ∂2 ∂2 u = ν0 ε−1 ε + 2ν ε + ν − φξ , 6 10 6 12 4 ∂ξ4 ∂ξ2 ∂η2 ∂ξ2 ∂η2

(5.28)

(5.29)

178


∂2 ∂4 −1 ∂2 −1 υ = − 2ν12 ε−1 + ν ε + ε + ν − φη , ε 11 4 6 10 5 5 ∂ξ4 ∂ξ2 ∂η2 4 w = ε−1 4 ∇2 φ.

0.5 < β. It corresponds to the state with a fast variation. The governing equation is splitting into the equation with a prevalent bending deformation of a plate: α = β − 0.5. ε21 ε4 ∇41 φ + ε26 φηηηη = 0 (5.30) and prevalent plane deformation of a plate: α = β. 2 ∂2 ∂2 ε21 ∇42 φ + ε26 ∇2η 2 − 2 φ = 0. ∂ξ ∂η

(5.31)

For ε6 = 0 we obtain the classic equations governing the bending and the plane stress state of the stringer type plate. The expressions for displacements can be obtained from the relation (5.16). In order to formulate the boundary conditions for the simplified equations (5.26), (5.28) we need to split the boundary conditions. The canonical forms of the boundary conditions are given in Table 5.2. ˜ 0), A10 − A16 the splitting can be obtained For the variants A 6 − A8 (for W using the canonical form. For the other variants the obtained splitting relations are wrong, because using the expression (5.27) we obtain w (1) = υ(1) η . ˜ 0) is found by using a linear Proper splitting of the variants A 2 , A3 , A5 , A8 (W ˜ combination of the variants of the canonical forms. In the case of A 2 , A3 , A5 (W 0), A3 (V˜ 0) we need to apply the expressions for u, v, w, using the first order approximation terms. The combinations of the A 1 , A2 , A13 variants are found without the use of the canonical forms. Final results are given in Table 5.3, where the following notation (2) (2) is applied: L1 = υ(2) L2 = u(2) η −w , ηη + wξ . Ring-stiffened shells The given equations of the semi-inexstensional theory and the simple edge effect (5.11) are valid in the considered case. We only must take ε 6 = 0. The shallow shells’ equations are splitting into the following cases. α = β = 0.25. It corresponds to the state with fast in both directions variations. The governing equation has the form (5.32) ε21 ε2 ∇42 + ε27 ∇23η φηηηη + ε7 ∇3η φξξηη + φξξξξ = 0, 2

2

∂ ∂ where: ∇3η = ∂ξ 2 − ν21 ∂η2 . The relations define the displacements

∂2 ∂4 ∂2 −1 u = − 2ν21 ε5 + ν11 ε7 4 + ν12 2 − 2 φξ , ∂η ∂ξ ∂η


179

Table 5.2. Canonical forms of the boundary conditions for stringer shell. Variants of boundary conditions A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

υ = 2ν0 ε7

Boundary conditions in the canonical form υ υ υ υ w w w w υ υ υ υ u wξ M1 M1

∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼

w w w w u wξ M1 M1 u wξ M1 M1 wξ T1 u T1

∼ ∼ ∼

∼

u wξ M1 M1 wξ T1 u T1 wξ T1 u T1 S S S S

∼

∼ ∼ ∼ ∼

wξ T1 u T1 S S S S N1 N1 N1 N1 N1 N1 N1 N1

∂2 ∂4 ∂4 −1 ∂2 + ν ε + ε + ν + φη , 11 7 10 5 ∂ξ2 ∂η2 ∂η4 ∂ξ2 ∂η2 4 w = ε−1 4 ∇2 φ.

(5.33)

α = 0.25, β < α. It corresponds to the edge effect. The potential function is defined by the equation 2 ∂2 2 4 ε1 ∇1 φ + 1 + ε7 2 φ = 0. (5.34) ∂η The following relations describe the displacements 2 ∂2 u = ν12 φξξξ , υ = 2ν10 1 + ε7 2 φξξη , ∂η

w = ε−1 4 φξξξξ .

(5.35)

α > 0.25, β ∼ α. It corresponds to the state with large fast variations. The governing equation is splitting into the equations with a prevalent in-plane and a prevalent bending deformation of a plate ε21 ε2 ∇42 φ + ε27 ∇23η = 0,

(5.36)

ε21 φξξξξ + ε70 φηηηη = 0.

(5.37)

The above equations for ε 7 = 0 are transformed into the equations governing the plane stress state and the bevelling of a plate. The splitting procedure for the boundary conditions of the ring-stiffened shells (equations (5.32), (5.34)) give results from the canonical forms and are given in Table 5.4.

180

5 BOUNDARY VALUE PROBLEMS – ORTHOTROPIC SHELLS Table 5.3. Splitting boundary conditions for stringer shell. Variants of boundary conditions A1 A2

A3

A4 A5 A6

A7

A8 A9 A10 A11 A12

A13 A14 A15 A16

Boundary conditions for ξ = 0, d ˜ ˜ L2 (u(2) , w(2) ) = (W ˜ ξ − n2 U) L1 (v(2) , w(2) ) = (nV˜ − W), ξ (1) (2) (1) (2) ˜ ˜ v = V − v , wξ = Wξ − wξ ˜ v(1) = V, ˜ w(1) = −w(2) T 1(2) = T˜ , L1 = nV˜ − W, ξ ξ (1) (2) (1) ˜ v = 0, wξ = Wξ , T 1 = −T (1) , L = 0 1 1 ˜ L1 = nV, ˜ u(1) = −u(2) , ν(1) = V˜ M1(2) = M, v(1) = 0, u(1) = u˜ , L1 = 0, M1(2) = −M (1) 1 ˜ M (1) = 0, L1 = −W, ˜ u(2) = −u(1) w(1) = W, 1 ˜ R−1 ∇2η M (2) + (ε6 + ε7 )T (2) + ε6 T (2) − L1 = nV˜ − W, 1 1ηη 1 (2) −1 ˜ − ν21 B1 (ε6 + ε7 )V˜ ηηη + B1 R (ε6 + ε7 )ν12 ∇2η Vηηη = R−1 ∇2η M ε6 T˜ + (ε6 + ε7 )T˜ ηη , v(1) = V˜ − v(2) , T 1(1) = T˜ − T 1(2) ˜ w(1) = W ˜ ξ − n2 U, ˜ ξ − w(2) , w(1) = 0, S (2) = S˜ , L2 = W ξ ξ (1) (1) ˜ w = W, U = 0, L2 = 0, S (2) = −S (1) ˜ w(1) = W ˜ ξ , T (2) = −T (1) , S (2) = 0 w(1) = W, ξ 1 1 (2) (1) T 1(2) = T˜ , S (2) = S˜ , w(1) =0 ξ = −wξ , w (2) (1) (1) (1) (2) ˜ u = U, M1 = −M 1 , u = −u , S (2) = 0 ˜ S (2) = S˜ , u(1) = −u(2) , w(1) = 0 M1(2) = M, ˜ M (1) = 0, u(2) = −u(1) , S (2) = 0 w(1) = W, 1 (2) S (2) = S˜ , R−1 ∇0η M1(2) + (ε6 + ε7 )T 1ηη + ε6 T 1(2) − (2) ˜ + ε6 T˜ + (ε6 + ε7 )T˜ ηη , B1R−1 ∇0η (ε6 + ε7 )ν21 wηη = R−1 ∇0η M T 1(1) = T˜ − T 1(2) , w(1) = 0 ˜ N (2) = N, ˜ ξ − n2 U, ˜ v(1) = 0, w(1) = W ˜ ξ − w(2) L2 = W 1 ξ ξ (2) ˜ u(1) = 0, L2 = 0, N = −N (1) v(1) = V, 1 1 ˜ w(1) = W ˜ ξ , T (2) = −T (1) , N (2) = 0 v(1) = V, ξ 1 1 1 (2) (2) ˜ w(1) = −w(2) , v(1) = 0 T 1 = T˜ , N1 = N, ξ ξ ˜ M (2) = M, ˜ u(2) = −u(1) , N (2) = 0 v(1) = V, 1 1 ˜ N (2) = N, ˜ v(1) = −v(2) , M (1) = −M (2) u(2) = U, 1 1 1 ˜ M (2) = M, ˜ T (2) = −T (1) , N (2) = 0 v(1) = V, 1 1 1 1 ˜ v(1) = −v(2) , M (1) = −M (2) T 1(2) = T˜ , N1(2) = N, 1 1 (2) L = W − nU, ∇0η Q(2) − (ε6 + ε7 )S ηηη + ε6 S η(2) + −1 2 ˜ ˜ B1R−1 ε21 ε3 ε4 ∇0η u(2) ηηηη = ∇0η N + B1 R ε1 ε3 ε4 U ηηηη − (1) (2) (ε6 + ε7 )S˜ ηηη + ε6 S˜ η , u = U˜ − u , S (1) = S˜ − S (2) (1) ˜ ˜ (2) = −S (1) , N (2) = −N (1) w(1) ξ = Wξ , T 1 = T , S 1 1 (2) (2) ˜ T (1) = T (2) , w(1) = 0 S = S˜ , N1 = N, ξ 1 1 ˜ u(1) = U, ˜ S (2) = −S (1) , N (2) = −N (1) M1(2) = M, 1 1 (2) (2) ˜ u(1) = −u(2) , M (1) = 0 S = S˜ , N1 = N, 1 ˜ T (1) = T˜ , S (2) = −S (1) , N (2) = −N (1) M1(2) = M, 1 1 1 (2) ˜ T (1) = T (2) , M (1) = 0 S = S˜ , N1(2) = N, 1 1 1

5.2 Statical problems Table 5.4. Splitting boundary conditions for ring-stiffened shell. Variants of boundary conditions A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

Boundary conditions for ξ = 0, d ˜ u(1) = U, ˜ w(2) = −w(1) , w(2) = 0 v(1) = V, ξ (2) (2) ˜ ˜ ξ , u(1) = −u(2) , v(1) = 0 w = W, wξ = W ˜ T (1) = T˜ , w(2) = −w(1) , w(2) = 0 v(1) = V, ξ 1 (2) ˜ ˜ ξ , T (1) = −T (2) , v(1) = 0 w = W, w(2) = W ξ 1 1 ˜ u(1) = U, ˜ w(2) = −w(1) , M (2) = 0 v(1) = V, 1 ˜ M (2) = M, ˜ u(1) = −u(2) , v(1) = 0 w(2) = W, 1 ˜ T (1) = T˜ , w(2) = −w(1) , M (2) = 0 v(1) = V, 1 1 (2) ˜ T (1) = −T (2) , v(1) = 0 w = T˜ , M1(2) = M, 1 1 ˜ w(2) = −w(1) , w(2) = 0 S (1) = S˜ , u(1) = U, ξ (2) (2) ˜ ˜ w = W, wξ = Wξ , S (1) = −S (2) , u(1) = −u(2) S (1) = S˜ , T 1(1) = T˜ , w(2) = −w(1) , w(2) ξ = 0 (2) ˜ w(2) = W ˜ ξ , S (1) = −S (2) , T (1) = −T (2) w = W, ξ 1 1 ˜ w(2) = −w(1) , M (2) = 0 S (1) = S˜ , u(1) = U, 1 ˜ M (2) = M, ˜ S (1) = −S (2) , u(1) = −u(2) w(2) = W, 1 (1) (1) ˜ S = S , T 1 = T˜ , w(2) = −w(1) , M1(2) = 0 ˜ M (2) = M, ˜ S (1) = −S (2) w(2) = W, 1 (2) (2) (1) (1) ˜ ˜ v = V, u = U, wξ = −w(1) ξ , N1 = 0 (2) (2) (1) (2) ˜ ˜ wξ = Wξ , N1 = N, u = −u , v(1) = 0 ˜ T (1) = T˜ , w(2) = −w(1) , N (2) = 0 v(1) = V, ξ ξ 1 1 (2) ˜ ξ , N (2) = N, ˜ T (1) = −T (2) , v(1) = 0 wξ = W 1 1 1 ˜ u(1) = U, ˜ M (2) = −M (1) , N (2) = 0 v(1) = V, 1 1 1 (2) (2) ˜ N = N, ˜ u(1) = −u(2) , v(1) = 0 M1 = M, 1 ˜ T (1) = T˜ , M (2) = −M (1) , N (2) = 0 v(1) = V, 1 1 1 1 (2) ˜ N (2) = N, ˜ T (1) = −T (2) , v(1) = 0 M1 = M, 1 1 1 ˜ w(2) = −w(1) , N (2) = 0 S (1) = S˜ , u(1) = U, ξ ξ 1 (2) (2) ˜ ξ , N = N, ˜ S (1) = −S (2) , u(1) = −u(2) wξ = W 1 (1) (2) S (1) = S˜ , T 1(1) = T˜ , w(2) ξ = −wξ , N1 = 0 (2) (2) (1) (2) (1) ˜ ξ , N = N, ˜ S = −S , T = −T (2) wξ = W 1 1 1 ˜ M (2) = −M (1) , N (2) = 0 S (1) = S˜ , u(1) = U, 1 1 1 ˜ N (2) = N, ˜ S (1) = −S (2) , u(1) = −u(2) M1(2) = M, 1 (1) (2) (1) S = S˜ , T 1 = T˜ , M1(2) = −M (1) 1 , N1 = 0 (2) (2) (1) (2) (1) ˜ N = N, ˜ S = −S , T = −T (2) M1 = M, 1 1 1

181

182


5.3 Non-linear dynamical problems It is needless to say that from engineering point of view we need more accurate results related to thin shells. It means that we need to consider the non-linear effects [677]. The governing equations of motion are written in the form proposed by Sanders [593] (in addition to the terms present in the original equations from reference [593] dynamic terms are added): T 1ξ + S η + Hη /(2R) − 0.5[θ(T 1 + T 2 )] − ρRutt = 0, S ξ + T 2η − N2 − Hξ /(2R) + (θ1 S + θ2 T 2 )η − 0.5[θ(T 1 + T 2 )]ξ − ρRυtt = 0, N1ξ + N2η + T 2 − (θ1 T 1 + θ2 S )ξ − (θ1 S + θ2 T 2 )η − ρRwtt = 0, θ1 = −wξ /R,

θ2 = −(wη + v)/R,

θ = (υξ − uη )/(2R).

(5.38)

The stress-strain relations and the expressions for the transverse shear forces have the same character as in the linear case, see (5.1). The strain-displacement relations are defined as follow [593]: 7 7 ε11 = uξ R + 0.5 θ12 + θ2 , ε22 = υη − w R + 0.5 θ22 + θ2 , 7 7 ε12 = υξ + uη (2R) + 0.5θ1 θ2 , χ1 = θ1ξ R, 7 7 χ2 = θ2η R, χ12 = θ2ξ + θ1η − θ (2R).

(5.39)

The following boundary conditions are assumed for ξ = 0, d v = 0 or S − 1.5R −1 H + 0.5 (T 1 + T 2 ) θ = 0, u = 0 or T 1 = 0, w = 0 or

N1 + R−1 Hη − θ1 T 1 − θ2 S = 0, θ1 = 0 or

M1 = 0.

(5.40)

¯ v, w) have the form The initial conditions for the vector of displacements U(u, U¯ = U¯0 ,

˙¯U = U¯00 ,

for t = 0.

(5.41)

Below a full classification of the simplified non-linear boundary value problems for structural - orthotropic cylindrical shell is given. The asymptotic integration parameters α and β characterise longitudinal and circumferential variations. In addition, we introduce the γ, δ 1 , δ2 and δ3 parameters characterising the variation in time, the non-linearity order and the magnitude of the in-plane displacements in relation to normal displacement, of the following form wt ∼ εγ1 w,

w ∼ εδ11 R,

u ∼ εδ12 w,

v ∼ εδ13 w.

5.3 Non-linear dynamical problems

183

The limiting systems obtained while applying the asymptotic procedure are given below. All of them posses their analogy in the linear case. Thus, the stressstrain relations are defined by the corresponding expressions for analogous linear equations. β < 0.5, α = β, γ = 0, δ1 = 0, δ1 = 2β, δ2 = δ3 = β. This case corresponds to the non-linear membrane vibrations of a shell. The governing equations have the following form T 1ξ + S η − 0.5 [θ (T 1 + T 2 )]η = 0, S ξ + T 2η + 0.5 [θ (T 1 + T 2 )]ξ + θ1 S + θ2 T 2 = 0, RT 2 − R (θ1 T 1 + θ2 S )ξ − R (θ1 S + θ2 T 2 )η − ρR2 wtt = 0, 7 7 ε11 = uξ R + 0.5θ12 , ε22 = vη − w R + 0.5θ22 , 7 ε12 = 0.5R−1 uη + vξ + 0.5θ1 θ2 , θ1 = −wξ R, θ2 = −R−1 wη − [v] /R, θ = 0.5 vξ − uη /R.

(5.42)

The term in square brackets is included only if β = 0. β < 0.5, α = 0.5 + 2β, γ1 = −1 + 2β, δ1 = 2β, δ2 = 2β, δ3 = β. This case corresponds to the non-linear semi-inextensional theory. The governing equations’ system has the following form T 1ξ + S η − 0.5 [θT 1 ]η = 0, 7 S ξ + T 2η − M2η R − 0.5 (θT 1 )ξ − θ1 S − θ2 T 2 = 0, M2ηη + RT 2 − R (θ1 T 1 + θ2 S )ξ − R (θ1 S + θ2 T 2 )η − ρR2 wtt = 0, 7 7 ε11 = uξ R + 0.5θ12 , 0 = υη − w R + 0.5θ22 , 0 = 0.5R−1 uη + υξ + 0.5θ1 θ2 , θ1 = −R−1 wξ , θ2 = −R−1 wη − [v/R] , θ = 0.5R−1 υξ − uη .

(5.43)

In this case, the non-linear dependencies concerning shear lack and no stretching of the middle surface of the shell are valid. The terms in the square brackets are used only if β = 0. For δ 1 > 2β the equations (5.43) can be linearised. Now we are going to define an edge effect in order to compensate a discrepancy in the boundary conditions. We begin with an additive state to the semi-inextensional theory equations. The edge effect is characterised by a fast longitudinal variation. Its circumferential variation and variation time should be similar to those in the interior-zone state. Further, an arbitrary component P of the input realtions can be presented in the form (5.44) P = P(0) + P(e) . The superscript (0) and (e) correspond to the interior-zone state and the edge effects, respectively.

184


The relations between the magnitude orders of the semi-inextensional thory and edge effect are defined by the boundary conditions. The splitting procedure of the boundary conditions for the nonlinear case is realised in a way similar to that of the linear one. For all variants of the boundary conditions the following estimation is valid w(0) . (5.45) w(e) ∼ ε(1−2β) 1 From the equation (5.42), (5.43) we automatically estimate of the non-linearity order of the edge effect. Substituting relation (5.44) in the governing equations and carrying out the asymptotic splitting (taking into account estimation (5.45)), the following limiting system governing the edge effect is obtained: α = 0.5, β < 0.5, δ2 = 0.5, δ3 = 1 − β. (e) T 1ξ + S η(e) = 0;

(e) ε12

(e) S ξ(e) + T 2η = 0,

(e) (e) M1ξξ + RT 2(e) + T 1(0) wξξ = 0,

7 2 8 7 (e) (e) 2 ε11 = uξ(e) R + wξ(e) (2R2 ) + w(0) ξ wξ R , 7 7 (e) (e) 2 = −w(e) R + w(0) ε22 η wη R , 7 7 (0) (e) (0) (e) = vξ(e) + uη(e) 2R + wξ(e) w(e) R2 . η + wξ wη + wη wξ

(5.46)

In the relation (5.46) the varying coefficient should be ‘frozen’. For instance, (0) when the edge ξ = 0 is analysed then instead of w (0) ξ (or wη ) we have to take

(0) (0) is w(0) ξ |ξ=0 (or wη |ξ=0 ). Such a frozen procedure is true, because the function w (e) changed along ξ slower than w . Therefore, the coefficient of equation (5.46) is changing only along the co-ordinate η. The edge effect relations for the non-momentous equations differ from the equation (5.46) only by the dynamical term ρR 2 w(e) tt . α = β = 0.5, γ1 = 0, δ1 = 1, δ2 = δ3 = 0.5. It corresponds to the theory of shallow shells. The equations have the form

T 1ξ + S η = 0,

ε11

S ξ + T 2η = 0, M1ξξ + 2Hξη + M2ηη + RT 2 + wξ T 1 + wη S ξ + wξ S + wη T 2 − ρR2 wtt = 0, η 7 2 8 7 2 8 = uξ R + wξ (2R2 ), ε22 = υη − w R + wη (2R2 ), 7 7 ε12 = uη + υξ (2R) + wξ wη (2R2 ), 7 7 7 χ1 = θ1ξ R, χ2 = θ2η R, χ12 = θ2ξ + θ1η (2R).

(5.47)

(5.48)

(5.49)

The equations (5.47), (5.49) can be linearized for δ 1 > 2β. The relations, describing dynamics of the supported plate, coincide with the equations (5.47), (5.48), if the underlined term is not used.


185

The obtained limiting systems have the second order in relation to time t, whereas the input system has had the order equal to six. Therefore, only a lower part of the free vibration spectrum is represented. In order to analyse high frequency vibrations or the nonstationary process the following limiting system should be used β < 1, α = β, γ1 = β, δ1 = 2β, δ2 = δ3 = −β. T 1ξ + S η − ρRutt = 0, S ξ + T 2η − ρRvtt = 0, RT 2 + wξ T 1 + wη S + wξ S + wη T 2 − ρR2 wtt = 0. ξ

η

(5.50) (5.51)

The geometrical relations are linear. The equations (5.50) describe a motion of the structurally-orthotropic plate in its plane (u, υ w). The complementary to (5.47)–(5.49) equation, which give possibility to satisfy the boundary conditions, has the form M1ξξ + RT 2 − ρR2 wtt = 0.

(5.52)

The further asymptotic analysis is carried out for each class of the reinforced shells. Stringer shells For the stringer shells the following system describes the dynamical state with the fast variations in circumferential direction α = 0, β = 0.5, γ = −1, δ 1 = 1, δ2 = 1, δ3 = 0.5. T 1ξ + S η = 0,

S ξ + T 2η = 0, (1) (1) (1) M1ξξ + 2Hξη + M2ηη + RT 2(1) − R θ1(1) T 1(1) + θ2(1) S (1) − ξ (1) (1) (1) (1) − ρR2 wtt = 0, R θ2 S (1) + θ2 T 2 η 7 (1) 2 8 (1) = u ε(1) 2R2 , R + wξ ξ 11 7 2 8 (1) 0 = υ(1) 2R2 , R + w(1) η −w η 7 7 (1) (1) (2R) + w(1) 2R2 , 0 = u(1) η + υξ ξ wη 7 7 7 (1) (1) (1) (1) 2 (1) 2 2 = −w , χ = −w , χ = −w χ(1) R R ηη ξξ ξη R . 11 22 12

(5.53)

(5.54)

The additive state to that governed by equations (5.53), (5.54) is localized in the neighbourhood of the shell’s edges α = β = 0.5, γ = −1, δ1 = 1(or δ1 = 1.5), δ2 = δ3 = 0.5. T 1ξ + S η = 0,

S ξ + T 2η = 0,

(2) (1) (2) + RT 2(2) − Rθ2η T 2 = 0, M1ξξ

186


7 7 (2) (2) (2) ε(2) ε(2) R + θ2(1) θ2(2) , 11 = uξ R, 22 = υη − w 7 (2) (2) (2R) + 0.5θ2(1) θ1(2) . ε(2) 12 = υξ + uη

(5.55)

β = 0.5 + k, α = k, γ = −1 + 2k, δ 1 = 1 + 2k, δ2 = 1 + k, δ3 = 0.5 + k; k > 0. This is the flexural vibrations of the stringer plate. The limiting equations have the form: T 1ξ + S η = 0,

S ξ + T 2η = 0,

M1ξξ + 2Hξη + M2ηη − ρR2 wtt = 0.

(5.56)

β > 0.5, α = β, γ = −1 + 2β, δ1 = 1, δ2 = δ3 = 1 − β. This is the stringer type plate vibrations with higher frequencies than in the previous case T 1ξ + S η = 0,

S ξ + T 2η = 0,

M1ξξ − R (θ1 T 1 + θ2 S )ξ − R (θ1 S + θ2 T 2 )η − ρR2 wtt = 0, 7 7 ε11 = uξ R + 0.5θ12 , ε22 = υη R + 0.5θ22 , 7 ε12 = υξ + uη (2R) + 0.5θ1 θ. Ring - stiffened shells For the shells with the dominating ring support there are states with fast variations in both longitudinal and circumferential directions. Both of the states are dynamical ones, and a solution to the problem may begin with a calculation of one of them. It depends on the needs which part of frequency spectrum is analysed. If we begin the calculation from the interior-zone state, then we obtain the following limiting systems 1. α = β = 0.25, γ = 0, δ 1 = 0.5, δ2 = δ3 = 0.25. T 1ξ + S η = 0, S ξ + T 2η = 0, (1) M2ηη + RT 2(1) − R θ1(1) T 1(1) + θ2(1) S (1) − ξ (1) (1) (1) (1) 2 (1) R θ2 S + θ2 T 2 − ρR wtt = 0. η

For the additive state we have β = 0.25, α = 0.5, γ = 0, δ 1 = 1, δ2 = 0.5, δ3 = 0.75. T 1ξ + S η = 0,

S ξ + T 2η = 0,

(2) (2) (1) (2) M1ξξ + M2ηη + RT 2(2) − Rθ2η T 2 − ρR2 w(2) tt = 0, 7 (2) 2 (2) + θ1(1) θ1(2) , ε(2) 11 = uξ R+0.5 θ1 7 (2) ε(2) R + θ2(1) θ2(2) , 22 = −w

(5.57)


ε(2) 12

7 (2) (2R) + 0.5θ1(2) θ2(2) + 0.5 θ1(1) θ2(2) + θ2(1) θ1(2) . = υ(2) ,ξ + u,η

187

(5.58)

2. α = β = 0.25, γ = 0, δ 1 = 0.75, δ2 = δ3 = 0.25. T 1ξ + S η = 0,

S ξ + T 2η = 0,

(1) + RT 2(1) − ρR2 w(1) M1ξξ tt = 0.

(5.59)

The geometrical relations are here linear. The complementary state is described by the following relations β = 0.25, α = 0.5, γ = 0, δ 1 = 1, δ2 = 0.5, δ3 = 0.75. T 1ξ + S η = 0,

ε(2) 12

S ξ + T 2η = 0, (2) (2) (2) (2) (2) M1ξξ + M2ηη + RT 2(2) − RT 1(1) θ1ξ − RS (1) θ2ξ + θ1η − (1) (1) (1) (1) − RS (1) θ2ξ + θ1η − ρR2 w(2) RT 1(1) θ1ξ − RT 2(1) θ2η tt = 0, 7 2 (2) (2) 2 + θ1(1) θ1(2) + 0.5 θ1(1) , ε(2) 11 = uξ R+0.5 θ1 7 2 (2) ε(2) R + 0.5 θ2(1) , 22 = −w (1) (2) 7 (2) (2) (1) (2) (2) = υ(2) + u θ + 0.5 θ θ + θ θ (2R) + 0.5θ + 0.5θ1(1) θ2(1) . η ξ 1 2 1 2 2 1

α = β = 0.25, γ = 0, δ1 = 1, δ2 = δ3 = 0.25. For this case the equations for the interior-zone state overlap with the equations (5.59). To analyse the additive state the following relations are used β = 0.25, α = 0.5, γ = 0, δ 1 = 1, δ2 = 0.5, δ3 = 0.75. T 1ξ + S η = 0,

S ξ + T 2η = 0,

(2) (2) (2) + M2ηη + RT 2(2) − RT 1(1) θ1ξ − ρR2 w(2) M1ξξ tt = 0, 7 7 (2) (2) 2 (2) (2) ε(2) = u , ε = −w R+0.5 θ R, ξ 11 1 22 7 (2) (2) (2R) + 0.5θ1(2) θ2(2) + 0.5θ2(1) θ1(2) . ε(2) 12 = υξ + uη

If we consider in the beginning a state with a prevalating longitudinal variation, then the following limiting systems are obtained β = 0.25, α = 0.5, γ = 0, δ 1 = 1, δ2 = 0.5, δ3 = 0.75. T 1ξ + S η = 0,

S ξ + T 2η = 0,

(2) (2) + M2ηη + RT 2(2) − ρR2 w(2) M1ξξ tt = 0, 8 7 7 (2) (2)2 (2) ε(2) R, ε(2) R, 11 = uξ R+θ1 22 = −w 7 (2) (2) (2R) + 0.5θ1(2) θ2(2) . ε(2) 12 = υξ + uη

Additional state: β = α = 0.25, γ = 0, δ1 = 0.75, δ2 = 0.25, δ3 = 0.25.

188


T 1ξ + S η = 0,

S ξ + T 2η = 0,

(1) (2) (1) + RT 2(1) − Rθ1ξ T 1 − ρR2 w(1) M2ηη tt = 0.

The geometrical relations for this case are linearized. β = 0.25 + k, α = 0.5 + k, γ = 2k, δ 1 = 1 + 2k. δ2 = 0.5 + k, δ3 = 0.75 − k, k > 0. T 1ξ + S η = 0, ε11

S ξ + T 2η = 0,

M1ξξ + M2ηη + ρR2 wtt = 0, 7 7 = uξ R1 + 0.5θ12 , ε22 = −w/R, ε12 = υξ (2R).

(5.60)

The equations (5.60) govern the prevelating bending vibrations of a plate supported in the y direction. Additional state: β > 0.5, α = β, γ = −0.5 + 2β, δ 1 = 0.5, δ2 = δ3 = 0.5 − β. T 1ξ + S η = 0,

S ξ + T 2η = 0,

M2ηη − R (θ1 T 1 + θ2 S )ξ − R (θ1 S + θ2 T 2 )η + ρR2 wtt = 0.

(5.61)

The limiting equation (5.61) governs the vibrations of the structural - orthotropic plate with higher frequency values than in the previous case. Waffle shells For the waffle shells we carry out the splitting in relation to the ε 5 parameter. We introduce the new parameters of asymptotic integration γ 1 , δ, and also γ0 , δ4 , δ5 and δ6 according to the formulas: ∂ 1 ∼ ε−γ 5 , ∂ξ

∂ ∼ ε−δ 5 , ∂η

∂ 0 ∼ ε−γ 5 , ∂t

R−1 w ∼ εδ54 ,

u ∼ εδ55 w,

v ∼ εδ56 w.

As a result of the asymptotic splitting in relation to the equations (5.47)–(5.49) we obtain the following limiting system γ1 = δ, γ0 = δ4 = 2δ, δ5 = δ6 = δ. M1ξξ + 2Hξη + M2ηη + RT 2 − ρR2 wtt = 0, 7 2 8 ε11 = uξ R + wξ 2R2 , 7 7 ε22 = υη − w R + w2η 2R2 , 7 7 ε12 = uη + υξ (2R) + wξ wη 2R2 .

(5.62)

The equations (5.62) can be obtained from the governing system using the hypotheses

5.4 Stability problems

2 8 uξ + wξ 2R2 − ε6 wξξ = 0,

υη − w + wη

2 8

2R2 − ε7 wηη = 0.

189

(5.63)

The expressions (5.20) and (5.22) describe the complementary state. Methodology of the boundary conditions splitting does not differ from that described earlier. The splitting boundary conditions for the equations of non-linear semi-inextensional theory (5.43) and the edge effect (5.46) are given in the Table 5.5. If the interior-zone state is described by the membrane relations’ (5.42), then the splitting conditions given in Table 5.5 can be used in cases where the membrane dynamical theory is applied. During the investigations of the non-linear dynamical boundary value problems for the shallow stringer type shells (the equations (5.53)–(5.55)) the Table 5.6 can be used. For the ring stiffened shells (the equations (5.57)–(5.58)) and the waffle shells (the equations (5.62), (5.20), (5.22)) one needs to use the boundary conditions given correspondingly in the Tables 5.7, 5.8. Table 5.5. Splitting boundary conditions of reinforced shell. Variants of conditions A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

Boundary conditions for ξ = 0, d (2) w(1) = 0, w(1) ξ = 0, wξ = 0, 2 L(2) = v(2) − w(2) + 0.5R−1 (w(2) η ) = 0 (2) (2) w(1) = 0, w(1) = 0, L = 0, w = −w(1) ξ ξ ξ w(1) = 0, Wξ(1) = 0, L(2) = 0, M1(2) = −M(1) 1 w(1) = 0, T 1(1) = 0, L(2) = 0, M1(2) = −M(1) 1 (2) (2) w(1) = 0, w(1) = −S (1) ξ = 0, wξ = 0, S (1) (2) (1) (1) (2) w = 0, T 1 = 0, wξ = −wξ , S = −S (1) (2) w(1) = 0, w(1) = −S (1) , M1(2) = −M (1) ξ = 0, S 1 (1) (1) (2) w = 0, T 1 = 0, S = −S (1) , M1(2) = −M (1) 1 (1) w(1) = 0, u(1) = 0, N1(2) = 0, w(2) ξ = −wξ (1) (2) (2) (1) w = 0, T 1 = 0, N1 = 0, wξ = −w(1) ξ (2) (2) (1) w(1) = 0, w(1) ξ = 0, N1 = 0, M1 = −M 1 w(1) = 0, T 1(1) = 0, N1(2) = 0, M1(2) = −M (1) 1 (1) u(1) = 0, S (1) = 0, N1(2) = 0, w(2) ξ = −wξ (1) T 1(1) = 0, S (1) = 0, N1(2) = 0, w(2) ξ = −wξ (2) (2) (1) (1) u = 0, S = 0, N1 = 0, M1 = −M (1) 1 T 1(1) = 0, S (1) = 0, N1(2) = 0, M1(2) = −M(1) 1

5.4 Stability problems A stability investigation of the eccentrically reinforced shells belongs to very important problems, what is expressed in references [317, 362, 452, 644]. However, the obtained results often become too complex for design engineers.

190

5 BOUNDARY VALUE PROBLEMS – ORTHOTROPIC SHELLS Table 5.6. Splitting boundary condition for stringer shell. Variants of conditions A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

Boundary conditions for ξ = 0, d (2) (2) (2) w(1) = 0, w(1) = 0, u(2) =0 ηη + w ξ = 0, vη − w (1) (2) (2) (1) (2) w = 0, wξ = 0, vη − w = 0, T 1 = −T (1) 1 (2) (2) w(1) = 0, w(1) = 0, M1(2) = −M(1) ξ = 0, vη − w 1 (2) w(1) = 0, T 1(1) = 0, v(2) = 0, M1(2) = −M (1) η −w 1 (2) (2) w(1) = 0, w(1) = 0, S (2) = −S (1) ξ = 0, uηη + w (2) w(1) = 0, w(1) = 0, T 1(2) = −T (1) ξ = 0, S 1 (1) w(1) = 0, wξ = 0, S (2) = 0, M1(2) = −M (1) 1 w(1) = 0, M1(1) = 0, S (2) = 0, T 1(2) = −T (1) 1 (2) (2) w(1) = 0, w(1) = 0, N1(2) = −N (1) ξ = 0, uηη + w 1 (1) (2) (1) w = 0, wξ = 0, N1 = 0, T 1(2) = −T (1) 1 (2) (2) (1) w(1) = 0, w(1) ξ = 0, N1 = 0, M1 = −M 1 (1) (2) (2) (1) (1) w = 0, M1 = 0, N1 = 0, T 1 = −T 1 (2) (1) (2) U (1) = 0, w(1) = −S (1) ξ = 0, N1 = −N 1 , S (1) (1) (2) (1) (2) wξ = 0, T 1 = 0, N1 = −N 1 , S = −S (1) (1) (2) (1) (2) w(1) = −S (1) ξ = 0, M1 = 0, N1 = −N 1 , S (1) (1) (2) (1) T 1 = 0, M1 = 0, N1 = −N 1 , S (2) = −S (1)

In this Chapter the approximate equations for different stability problems will be given. The heart of these investigations is focused on the following approach. The variations of the stress state are not a priori known, and it is defined by rigidity parameters and by a load. The occurring problems do not belong to the principal ones, because design engineer’s experience and the experimental data give enough qualitative information about possible shell behaviour. The pre-buckling state of the shells is assumed to be homogeneous and membrane. This hypothesis is true because the influence of the moment - type state on the buckling is much smaller than in the isotropic case (when the external load possesses no eccentricity) [317, 362, 452]. The stability equations, stress-strain and stress-displacement relations and the boundary conditions are obtained after the linearization of the nonlinear equations (5.38), (5.39). Using the operator method the system of stability equations is reduced to the one of the eight orders in relation to the potential function φ (it is not presented here because of its complexity). We use, as previously, the α and β parameters characterising the changes along the circle and longitudinal directions, respectively. In addition, we introduce the parameters λ 1 , t1 , t2 , t3 , using the following relations ε2 ∼ ε4λ 1 ,

T¯10 ∼ εt11 ,

T¯20 ∼ εt12 ,

S¯0 ∼ εt13 .

(5.64)

Here {T¯10 , T¯20 , S¯0 } = {T 10 , T 20 , S 0 }/B2 ; T 10 , T 20 , S 0 denote the forces in the prebuckling state. We take λ = −0.25; 0.5; 0.25 for a waffle a stringer shell and a ring-stiffened shell.


191

Table 5.7. Splitting boundary conditions for ring-stiffened shell. Variants of conditions A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

Boundary conditions for ξ = 0, d (2) u(1) = 0, v(1) = 0, w(2) = −2(1) ξ = 0, w (2) (2) (1) (1) w = 0, wξ = 0, v = 0, u = −u(2) (2) v(1) = 0, T 1(1) = 0, w(2) = −w(1) ξ = 0, w (2) (1) (2) (1) (2) w = 0, wξ = 0, v = −v , T 1 = −T (2) 1 u(1) = 0, v(1) = 0, M1(2) = 0, w(2) = −w(1) w(2) = 0, M1(2) = 0, v(1) = 0, u(1) = −u(2) v(1) = 0, T 1(1) = 0, M1(2) = 0, w(2) = −w(1) (2) w = 0, M1(2) = 0, v(1) = −v(2) , T 1(1) = −T (2) 1 (2) u(1) = 0, S (1) = 0, w(2) = −w(1) ξ = 0, w (1) w(2) = 0, w(2) = −u(2) , S (1) = −S (2) ξ = 0, u (1) (2) (1) T 1 = 0, S = 0, wξ = 0, w(2) = −w(1) (1) (1) w(2) = 0, w(2) = −S (2) ξ = 0, T 1 = 0, S (2) (1) (1) (2) u = 0, S = 0, M1 = 0, w = −w(1) (2) w = 0, M1(2) = 0, u(1) = −u(2) , S (1) = −S (2) T = 0, S = 0, M = 0, w(2) = −w(1) (2) w = 0, M1(2) = 0, T 1(1) = 0, S (1) = −S (2) (1) u(1) = 0, v(1) = 0, N1(2) = 0, w(2) ξ = −wξ (2) (2) (1) (1) (2) wξ = 0, N1 = 0, v = 0, u = −u (1) v(1) = 0, T 1(1) = 0, N1(2) = 0, w(2) ξ = −wξ (2) (2) (1) (1) (2) wξ = 0, N1 = 0, v = −v , T 1 = −T (2) 1 u(1) = 0, v(1) = 0, N1(2) = 0, M1(2) = −M (1) 1 M1(2) = 0, N1(2) = 0, v(1) = 0, u(1) = −u(2) v(1) = 0, T 1(1) = 0, N1(2) = 0, M1(2) = −M(1) 1 M1(2) = 0, N1(2) = 0, v(1) = −v(2) , T 1(1) = −T (2) 1 (1) u(1) = 0, S (1) = 0, N1(2) = 0, w(2) ξ = −wξ (2) (1) w(2) = −u(2) , S (1) = −S (2) ξ = 0, N1 = 0, u (1) (2) (1) (1) T 1 = 0, S = 0, N1 = 0, w(2) ξ = −wξ (2) (2) (1) (1) wξ = 0, N1 = 0, T 1 = 0, S = −S (2) u(1) = 0, S (1) = 0, N1(2) = 0, M1(2) = −M (1) 1 M1(2) = 0, N1(2) = 0, u(1) = −u(2) , S (1) = −S (2) T 1(1) = 0, S (1) = 0, N1(2) = 0, M1(2) = −M(1) 1 M1(2) = 0, N1(2) = 0, T 1(1) = 0, S (1) = −S (2)

According to the earlier described procedure of the asymptotic integration, the following limiting systems are obtained: α = 2β − 0.5 − λ, β < min{0.5, 0.5 + λ}, t 1 = 1 + 2λ, t2 = 1 + 2λ + 2β, t3 = 1 + 2λ + β. This is the semi-inextensional theory. The approximate stability equation has the form 2 ∂2 ∂2 ∂2 2 (0) −1 (0) ε + 1 φ + ε φ + + 1 × ε21 ε2 + ν221 ε−1 ηηηη 4 7 4 ξξξξ ∂η2 ∂η2 ∂η2

192

5 BOUNDARY VALUE PROBLEMS – ORTHOTROPIC SHELLS Table 5.8. Splitting boundary conditions for waffle shell. Variants of conditions A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

Boundary conditions for ξ = 0, d (2) w(1) = 0, w(1) = −u(1) , v(3) = −v(1) ξ = 0, u (1) (3) w = 0, wξ = 0, v = −v(1) , T 1(2) = −(T 1(1) + T 1(3) ) w(1) = 0, M1(1) = 0, u(2) = −u(1) , v(3) = −v(1) (1) w = 0, M1(1) = 0, v(3) = −v(1) , T 1(2) = −(T 1(1) + T 1(3) ) (2) w(1) = 0, w(1) = −u(1) , S (3) = −(S (1) + S (2) ) ξ = 0, u (1) (2) (1) (3) w = 0, wξ = 0, T 1 = −T (1) = −(S (1) + S (2) ) 1 , S (1) (1) (2) (1) (3) w = 0, M1 = 0, u = −u , S = −(S (1) + S (2) ) (3) w(1) = 0, M1(1) = 0, T 1(2) = −T (1) = −S (1) 1 , S (1) (1) (2) (1) (3) wξ = 0, N1 = 0, u = −u , v = −v(1) (1) wξ = 0, N1(1) = 0, T 1(2) = −(T 1(1) + T 1(3) ), v(3) = −v(1) M1(1) = 0, N1(1) = 0, u(2) = −u(1) , v(3) = −v(1) (1) M1 = 0, N1(1) = 0, v(3) = −v(1) , T 1(2) = −(T 1(1) + T 1(3) ) (1) (2) w(1) = −u(1) , S (3) = −(S (1) + S (2) ) ξ = 0, N1 = 0, u (1) (1) (3) wξ = 0, N1 = 0, T 1(2) = −T (1) = −S (1) 1 , S (1) (1) (2) (1) (3) M1 = 0, N1 = 0, u = −u , S = −(S (1) + S (2) ) (3) M1(1) = 0, N1(1) = 0, T 1(2) = −T (1) = −S (1) 1 , S (1)

∂2 ∂2 ∂2 ¯ ¯ ¯ + T ε + T + 2 S −2ν12 ε−1 φ(0) = 0. 10 0 20 4 7 ∂ξ2 ∂ξ∂η ∂η2

(5.65)

α = 0.5, β < min{0.5, 0.5 + λ}, t 1 = 1 + 2λ, t2 = 1 + 2λ + 2β, t3 = 1 + 2λ + 2β. In this case the edge effect type occurs: (1) + −2ν12 ε6 + T¯10 φ(1) (5.66) ε21 + ν221 ε26 φ(1) ξξξξ + φ ξξ = 0. The analysis of the obtained results shows that the ribs eccentricity sign influences only on the interior-zone state (in the case of stringers on the edge effect). The equation (5.65) corresponds to relations of the semi-inextensional theory and it can be applied for slightly expressed waveforms in the longitudinal direction. This kind of behaviour is typical for action of the external pressure, and of the buckling of long shells. The equation (5.66) can be used to describe axially symmetric buckling. The use of the splitting equations and the boundary conditions allows, in many practically important cases, to obtain the eigenvalues using the 4th order equation because of the ξ co-ordinate. In order to formulate the modes of buckling we need to take both equations (5.65) and (5.66), because the states governed by them complement each other. α = 0, 5, β < min{0.5, 0.5 + λ}, t 1 = 1 + 2λ, t2 = 1 + 2λ + 2β, t3 = 1 − 2λ + β. This is the theory of shallow shells. The resolving equation has the form ∇42

2 2 ∂2 ∂2 ∂2 4 2 ¯ ∂ −2 4 ¯ ¯ + T 20 2 φ + ε1 ∇3 − 2 φ = 0. ∇1 + ε1 T 10 2 + 2S 0 ∂ξ∂η ∂ξ ∂η ∂ξ

(5.67)


193

The equation (5.67) is analogous to the Mushtary-Donnel-Vlasov one [251, 501, 694, 695] in the isotropic case. It gives good results for real reinforced shells. However, it possesses the eighth order. Further simplifications are possible when some characteristic relations for the fundamental types of the supported shells are taken into account. Stringer shells α = 0, β = 0.5, t1 = 2, t2 = 2 + 2β, t3 = 2 + β. The following equation corresponds to the modes of the buckling with a relatively fast variations is circumferential direction 2 4

∂2 ∂4 ∂4 2 ∂ ε−1 φ + ε + 2ε + ε 1 + 2ε φηηηη + 6 ξξξξ 3 2 4 1 ∂η2 ∂ξ4 ∂ξ2 ∂η2 ∂η4

∂2 ∂2 ∂2 ¯ ¯ ¯ + T 20 2 φηηηη = 0. T 10 2 + 2S 0 ∂ξ∂η ∂ξ ∂η

(5.68)

The equation (5.68) can be obtained using the variational method taking into account the hypotheses of the shear lack and no stretching of the middle surface of the shell in the ring direction. α = 0.5, β = 0.5. This case corresponds to both space co-ordinate fast variations. The corresponding resolving equation has the form 2 ∂ ∂2 2 2 2 φ + 2ε6 − ν + ε + ν ε φ φξξξξ + 12 1 12 6 ∂η2 ∂ξ2 2 2 2 2 ε21 ε4 ε−1 5 − ν12 ε6 φξξηη + ε1 ε4 + ε6 φηηηη = 0.

(5.69)

The equation (5.69) governs a state of the edge effect type. It can be found using the variational method when one takes ε 11 , ε12 , ε22 , χ11 as the fundamental deformations. The parametric terms appear only in the equation (5.68) and, therefore, the eigenvalues can be found using only this equation. In order to determine a mode of buckling solutions of two equations (5.68) and (5.69) are used. Their matching is realised through boundary conditions. It is seen from the equations (5.68) and (5.69), that the critical stresses and the modes of buckling depend essentially on the magnitude and on the sign of the stringers eccentricity. For β > 0.5 the equation (5.68) is transformed into the plate stability equation. The equations (5.69) then are governed by the state with a prevalence in-plane plate deformation eccentrically supported by ribs in one direction. For this case we need to neglect underlined terms in the equations (5.68) and (5.69).

194


Ring-stiffened shells α = β = 0.25, t1 = 1, t2 = 0.5, t3 = 0.75. The waveforms with a fast variation of SSS are governed by the following fourth order equation in relation to χ 2 ∂4 ∂4 −2 ∂ + 2ε7 − ν21 4 + ε1 ∂ξ2 ∂ξ2 ∂η2 ∂η ξξ ε27

2 ∂4 ∂2 ∂2 ∂2 4 −2 ¯ − ν φ + ∇ + ε ε T 21 2 20 φηη = 0. 2 1 ∂η4 ∂ξ2 ∂η2 ∂η2

(5.70)

From the energetic point of view, the fundamental deformations are: ε 11 , ε22 , ε12 , χ22 . α = 0.5, β = 0.25, t1 = 1, t2 = 0.5, t3 = 0.75. This case corresponds to the buckling due to short longitudinal waves. The corresponding approximate equation has the following form φ + 2ε7 φηη + ε21 φξξξξ + ε21 ε2 + ε27 φηηηη+

∂2 ∂2 ∂2 + T¯20 2 φ = 0. T¯10 2 + 2S¯0 ∂ξ∂η ∂ξ ∂η

(5.71)

The fundamental deformations are consisted of ε 12 , χ11 , χ22 and χ22 . The equation (5.70) must be used in the case of a normal pressure, whereas the equation (5.71) - in the case of the axial compression or twisting. A mode of buckling (similarly to a stringer shell) is presented as a sum of solutions to the equations (5.70) and (5.71). The coefficients of the equations (5.70), (5.71) strongly depend on the magnitude and the sign of eccentricity. For β > 0.25 the relation (5.70) is transformed into the stability equation of the eccentrically-supported plate (if the underlined terms are neglected) and the equation (5.71) has the form ∇42

4 2 ∂2 ∂2 −2 ¯ 2 ∂ − ν21 2 φηηηη = 0. ε2 2 + ε1 T 20 φηη + ε7 ∂η ∂ξ4 ∂η

(5.72)

For T¯20 = 0 the equation (5.72) governs an in-plane deformation of the plate. For β7 = 0 it is splitting into the equations of the in-plane deformation of the plate and the bar stability with the compressing load T¯20 . In a general case that splitting is not possible. Waffle shells α = β = 0.5. In this case we obtain equation (5.67). Further simplifications can be achieved using the parameter ε 5 . As a result we obtain γ = δ = 0, t 1 = 1, t2 = 2, t3 = 1.5.


2 ∂2 2 4 −1 ε1 ∇1 φηη + 0.5ε4 ε5 1 + (ε6 + ε7 ) 2 φξξ + ∂η

∂2 ∂2 ∂2 ¯ ¯ ¯ + T 20 2 φηη = 0. T 10 2 + 2S 0 ∂ξ∂η ∂ξ ∂η

195

(5.73)

The equation (5.73) can be obtained using the variation method and taking the hypotheses of an unstretched neutral layer of the shell. The equation (5.73) is valid for a large twisting ribs’ stiffness (ε 3 ∼ 1). If this stiffness is small (ε3 ∼ ε1 ), then in the operator ∇ 41 the second term should be neglected. The complementary states are governed by equations (5.20), (5.22). The parametric terms appear only in fourth-order equation in relation to ξ (5.73). The coefficients of (5.73) essentially depend on ε 6 and ε7 . The influence of the rib eccentricity on the buckling loads and the buckling modes are significant. The complementary states governed by equations (5.20) and (5.22) do not dependent on the support eccentricity. α ∼ β, β > 0.5. ∂2 ∂2 ∂2 ∇42 ∇41 + T¯10 2 + 2S¯0 + T¯20 2 φ + ∇43 φ = 0. (5.74) ∂ξ∂η ∂ξ ∂η For a centrally supported shell the equation (5.74) is splitting into the stability equation and the in-plane deformation of a plate. In a general case this splitting is not possible. In order to solve the corresponding problems for the eigenvalues the splitting boundary conditions should be used (given in the Tables 5.1, 5.2, 5.4). Now we consider some of the examples of the given approximate systems’ application. Suppose that a cylindrical structurally orthotropic shell is under a uniform axial pressure. First, consider the case of a classic simple support for ξ = 0, d

w = υ = T 1 = M1∗ = 0,

(5.75)

where: M1∗ = M1 = e1 T 1 . Let us the function φ in the form, satisfied the boundary conditions (5.75) φ = A sin( m ¯ ξ) cos(nη),

m ¯ = mπ/d.

We obtain the following results for different types of reinforced shells. Stringer shells In this case m = 1. A minimum in relation to ‘n’ can not be found analytically, and for T¯10 we obtain  2 2  π 2  π 2 n d  −1 2 2 2 2 ¯  + 2ε3 n + ε2 1 − ε6 n + ε1  T 10 = ε4 (5.76) . 2 d π  n d

196


To illustrate an application of the approximate equation, the results of the critical stresses on the basis of the eighth order equation (5.67) are presented in Fig. 5.1. The results are taken from the reference [452] (curve 1) and due to the formula (5.76) (curve 2). The following geometrical and stiffness parameters are taken: ε 2 = 0.008, ε3 = 0.024, ε4 = ε5 = 0.7, |ε6ε−2 1 | = 0.4. The solid line corresponds to the external support whereas the dashed line corresponds to the internal support. 7 √ We √ also denote Z = d 2 R B2 /(12D1 ), Z is Batdorf’s parameter, T ∗ = 0.5T 10 R B1 D2 .

Fig. 5.1. A comparison of buckling axial loads obtained using exact and approximate solutions for the stringer cylindrical shell.

The obtained curves correspond to each other very well in the whole-considered range of the parameter Z change. It should be noted that the taken geometrical - stiffness parameters correspond to the ribs with a relatively small stiffness. Therefore, this case appear to be rather worth one for the presented asymptotic analysis. For centrally supported stringer shells and for a small twisting stiffness of the ribs (ε6 = ε3 = 0) after minimisation in relation to ’n’ the following expression for the critical stress of the axial compression is obtained −1/2 + ε21 π2 d−2 . T¯10 = 2ε1 ε1/2 2 ε4

Ring-stiffened shells In this case we have

0 2 1 T¯10 = ε21 m ¯2 + ε21 ε2 n4 + 1 − ε7 n2 m ¯−2 .

(5.77)

−1 A minimum of the expression (5.78) is obtained for n 2 = ε7 ε21 ε2 + ε27 . Therefore, an axially unsymmetrical form of the buckling is only possible for a relatively large absolute magnitude of the internal ring eccentricity. In other cases the axially symmetric form of the buckling (n = 0) occurs. Then we have / −1 2 −1/2 . T¯10 = (2/R) B2 D1 1 + ε−2 1 ε2 ε7

5.5 Error estimation using Newton’s method

197

Waffle shells Consider the case of a small twisting ribs’ stiffness (ε 3 ∼ ε1 ). The critical stress of the axial compression is given by

m 2 1 ¯ −2 2 2 −2 ¯ T 10 = ε1 m (5.78) ¯ π + ε2 2 + ε−1 1 − (ε6 + ε7 ) n2 . 4 ε5 n 2 n Minimising the expression (5.78) in relation to ‘n’ and ‘m’ we find m ¯ = ε1/4 2 n,

−1 2 −1/2 −1/2 ) (ε n2 = 4ε21 ε1/2 ε ε + + ε ε . 4 6 7 5 2 2

(5.79)

For a shell, regularly supported by the ribs without eccentricity (ε 2 = ε4 = 1, ε6 = ε7 = 0), the expression (5.79) has the form T¯10 = 2ε1 ε1/2 5 .

(5.80)

The exact solution is given in [452], and it reads (1 + ε5 )1/2 . T¯10 = 2ε1 ε1/2 5

(5.81)

As it has been expected, the approximate solution (5.80) corresponds to the first term of the exact solution (5.81) series development in relation to the perturbation parameter ε5 . The equations’ and boundary conditions splitting leads to an effective search for the eigenvalues using the variational methods. It is related to the reduction of the equation order. Another advantage is that the variational method is used to obtain a “smooth” part of the solution.

5.5 Error estimation using Newton’s method In order to quantify errors introduced by zero order approximation of asymptotical process it is not necessary to construct next successive approximations. Instead, one may treat zero order approximations as zero approximation for Newton method and then construct first approximation of this method [232]. As an example, illustrating fundamental peculiarities of this method, a computation of excentrically supported cylindrical shell subjected to an action of small ) is considered. oscillating boundary load (m < ε −1/4 1 In this case, a shell equations can be splitted into semi-inextensial theory (5.11) and edge effect (5.12). The values of roots of the characteristic equation (the semiinextensioanl theory is considered, since for edge effect the results can be obtained in an analogical way) have the form √ m m2 − 1 (0) (b11 ± ib12 ) , j = 1, 2, 3, 4. (5.82) Kj = ± √ 2

198


where: i=

√ −1,

a=

(a21

b1p = a − (−1) p a1 1/2 ,

+ a2 )

1/2

,

a1 = ν21 ε7 ,

p = 1, 2;

a2 = ε2 ε4 ε21 .

In order to improve the found roots values and estimation of error occurred during their determination, the simplified Newton’s method is used []. It is simple and supported by very well developed mathematical background for convergence investigation. The improved (up to an order of ε 1/2 1 ) roots (5.82) have the form a2 (0) K (1) (b13 ± ib14 ) , j = Kj ± √ 2 2 b13 + b214

j = 1, 2, 3, 4.

If one considers an isotropic shell (in this case ε 1 = a2 /(1 − ν2 ), ε2 = ε3 = ε4 = ε5 = 1, ε6 = ε7 = 0), the formula (5.82) coincides in full with analogical expression obtained applying another process in [482]. Analogously are found improved values of the roots of characteristic equations for all obtained earlier variants of approximating equations. Owing to complexity of the corresponding expressions, they are reported only for the case of shallow waffle shell (5.18). For zero order approximation one gets N1(0)j = ±(b21 ± ib22 ), N2(0)j = ±m(0.5ε5)1/2 ,

1/2 N3(0)j = ±(wε4 ε−1 , 5 )

and in the first order approximation N1(1)j = N1(0)j = ±ε5 (A11 ± iA12 ),

j = 1, 2, 3, 4,

   ε−1 ε3 ε3/2  N2(1)j = ±m (0.5ε4 )1/2 + 2 √ 5  , d 2 (1) −1 1/2 , N3 j = ±m (2ε4 ε5 ) + 0.5ε3 ε4 ε1/2 5 where:

1/2 1 p ; b2p = √ m2 ε1/2 2 − (−1) c11 2

p = 1, 2;

−1 −2 −1 −2 2 c11 = ε3 m2 − 0.25ε−2 − 0.5ε−2 1 ε5 ε4 m 1 (ε6 + ε7 )ε5 ε4 + 0.5ε1 (ε6 + ε7 ) ; A11 = c3 4b23 c12 + 8b21 b22 b24 − 2ε3 m2 (c21 + c12 + 2b21 b22 c23 )+

d12 c1 m4 (ε2 + ε4 )d11 − ε3 ε4 m6 + ε2 ε4 m8 ; 2 2 2 2 d12 + d21 4 c1 + c22 A12 = c3 4b24 c12 − 8b21 b22 b23 − 2ε3 m2 (c23 c12 − 2b21 b22 c21 ) +

5.5 Error estimation using Newton’s method

b23

199

d21 c1 m4 (ε2 + ε4 )d22 − ε3 ε4 m6 + ε2 ε4 m8 ; 2 2 2 2 d12 + d21 4 c1 + c22 −1 c3 = 16 c212 + b221 b222 ; c1 = c22 c12 − 2b21 b22 c23 ; c2 = 2b21 b22 c22 + c23 c12 ; d1p = b21 c12 − (−1) p 2b222 ; d2p = b22 c12 − (−1) p 2b221 ; c12 = −m2 b221 + b222 + c11 ; c2p = b21 b221 + (−1) p 3b222 ; c23 = b22 3b221 − b222 ; = c21 b221 − b222 − 2b21 b22 c23 ; b24 = c23 b221 − b222 + 2b21 b22 c21 ; p = 1, 2.

In all of the considered cases a convergence of the simplified Newton’s method is proved. Numerical analysis of the obtained solutions shows, that for real parameters values an error of asymptotical splitting of equations governing behaviour of stringer, ring and waffle shells (in the last case, small oscillations of a boundary load are ) does not achieve 6 ÷ 7%. applied, m < ε−1/2 1

6 COMPOSITE BOUNDARY VALUE PROBLEMS – ISOTROPIC SHELLS

The method of asymptotic analysis of the fundamental equation of the shells’ theory allows to reduce the problem to investigation of the limiting equations. These equations solve many practical problems analytically, but unfortunately they also posses some drawbacks. For different variation of the stress-strain state, we have to apply different approximate fromulas. In order to avoid difficulties, one can use method of composite equations [660]. In hydromechanics a method of composite equations was developed by C.W. Oseen [538], M.J. Lighthill [436], M. Van Dyke [660]. A fundamental idea of the composite equations’ method may be formulated in the following manner ([660], p. 195): a) Identify the terms in the differential equations whose neglect in the straightforward approximation is responsible for the nonuniformity. b) Approximate those terms insofar as possible while retaining their essential character in the region of nonuniformity. The method of composite equations has been succesfully applied in order to avoid nonhomogeneities due to the space co-ordinates of the hydromechanics equations. Here we use it in the theory of shells. We construct a composite equation in such a way, that it includes the limiting equations, which makes the composite equation valid for small and large values of perturbation parameters. According to the proposed algorithm, for the problems of shells theory a composite equation of the interior-zone is obtained as a result of matching of the semi-inextensional theory and in-plane deformation of plate. The composite equation governing the edge-zone includes the relations of a simple edge effect and the plate’s bending. The expressions for displacements, deformations, stresses and moments are obtained as a result of the described procedure. The formulated composite equations have the fourth order on the longitudinal co-ordinate, and they are applied in a whole range of the stress-strain state variations. It is worth noticing, that in some sense, the method of etalon functions is similar to the composite equations’ method. In the neighbourhood of the turning points, the governing equation has been changed by the etalon one. The etalon equations are taken from the following requirements: 1. Their solution should possess the same asymptotic properties as the governing equation.

202


2. It should be simpler than the governing equation. The algorithm of those equations formulation is proposed in [116]. In the neighbourhood of the turning point, the coefficient of the governing equation are substituted by the polynomials of the unknown variable with a small number of terms.

6.1 Statical problems Limiting boundary value problems mentioned in the Chapter 4 essentially reduce the solution of the governed problem for small and large variations of solutions. However, some problems also occur. For different changes of the solution, different limit relations are needed and in the case of the equation governing by a shallow shell theory a solution is not simplified. Therefore, the traditional equations governing the interior and edge zones are changed by the composite equations, obtained by matching the limiting expressions of the asymptotic splitting. The interior-zone equation, which is further referred to as the equation of the fundamental state, includes relations of semi-inexstensional theory and a generalized plane stress state of the form  2   ∂4 ∂4 ∂2  1 − ν2  φξξξξ = 0. (6.1) Z0 φ ≡  4 + 2 2 2 + 1 + 2  φηηηη + ∂ξ ∂ξ ∂η ∂η ε2 Equation (6.1) can not be obtained using an asymptotic procedure. However, this equation serves for arbitrary changes of the stress-strain state. For small variations the terms related to the semi-inextensional theory play a fundamental role, whereas for large variations the generalized plane stresses state is of most importance. Equation (6.1) differs from the equations of the semi-inextensional theory (4.11) by the underlined terms. In order to get the relations for displacements, deformations, stresses and the moments we need to include all the terms of the limiting cases. Proceeding in this way, we get the following expressions for deformations stresses, and moments u = −φξηη + νφξξξ ,

υ = φηηη + (2 + ν)φξξη ,

w = ∇4 φ,

1 1 νφξξξξ − φξξηη , ε22 = νφξξηη − φξξξξ , R R 1+ν 1 1 φξξξη , χ11 = 2 ∇4 φξξ , χ22 = 2 ∇4 φηη , ε12 = 2 R R R 1 B B ε2 = 2 ∇4 φξη , T 1 = − φξξηη , T 2 = − φξξξξ + ∇ φ 0η ηηηη , R R R 1 − ν2 ε11 =

χ12

S =

B φξξξη , R

M1 = −ν

D ∇0η φηηηη , R2

M2 = −

D ∇0η φηηηη, R2

H = (1 − ν)

D ∇0η φξηηη , R2

N1 = −

D ∂5 ∇0η φ, R3 ∂ξ∂η4


203

D ∂5 ∇0η φ. R3 ∂η5

(6.2)

N2 = −

The expressions for displacements and deformations for equation (6.1) cover corresponding results of the shallow shell’s theory, and the expressions for stresses and moments have only slight differences. Therefore, the obtained component relations can be considered as a simplification of the shallow shells’ theory. They extend a field of the semi-inextensional theory and the generalized plane stress state to the whole range of the stress-strain state variations. The resolving equation is of fourth order in relation to the longitudinal coordinate. During its integration, only two boundary conditions are satisfied on each of the shell’s edges. We call it further the composite equation of the edge effect. It has the following form Ze φ ≡ φξξξξ + 2φξξηη + φηηηη +

1 − ν2 φ = 0, ε2

(6.3)

and it includes limiting relations of a simple edge effect and a plate bending. Equation (6.3) differs from the equation (4.14) by the underlined term. Matching the corresponding relations for limiting cases, we get the following equations for displacements, deformations, stresses and moments u = νφξξξ ,

υ = (2 + ν)φξξη ,

w = ∇4 φ,

ε11 =

ν φξξξξ , R

1 2(1 + ν) φξξξη , νφξξηη − φξξξξ , ε12 = R R 1 1 1 χ11 = − 2 ∇4 φξξ , χ22 = − 2 ∇4 φηη , χ12 = − 2 ∇4 φξη . (6.4) R R R Expressions for stresses and moments for the composite equation of an edge effect (6.3) cover the expressions of the shallow shell’s theory. Efficiency of the approximated equations depends on the taken hypotheses and the simplification related to a real state of a shell. Let us to discuss the physical and geometrical hypotheses related to the composite equations. The composite equation of the interior-zone (6.1) can be obtained on the basis of the following hypotheses: ε22 =

1. In the geometrical relations a curvature in the circle direction is larger than the curvatures in the longitudinal and circumferential directions, χ11 χ22 ,

χ12 χ22 .

(6.5)

2. In the third equation the term M 1 can be neglected. The above hypotheses are equivalent to the following conclusion. Instead of using the relations (4.2)–(4.3) one can use the following system of equations. Geometrical relations

204


ε11 =

1 uξ , R

1 1 υη − w , ε12 = υξ + uη , R R 1 = − 2 wηη + υη . R

ε22 = −

(6.6)

χ22

(6.7)

Physical relations T 1 = B (ε11 + νε22 ) , M1 = νDχ22 ,

T 2 = B (ε22 + νε11 ) , M2 = Dχ22 ,

S =

1−ν Bε12 , 2

H = (1 − ν)Dχ12 .

(6.8)

Equilibrium equations T 1ξ + S η + RP1 = 0, N2η − T 2 − RP3 = 0,

S ξ + T 2η + N2 + RP2 = 0,

M1ξ + Hη − RN1 = 0,

M2η − RN2 = 0.

(6.9)

Expression (6.6)-(6.9) are the composite equations of the interior-zone state. These relations lead to the following composite equations for displacements 1−ν 1+ν uηη + υξη − νwξ = 0, 2 2 2 1+ν 1−ν 2 2 ∂ uξη + υξη + 1 + ε υηη − 1 − ε wη = 0, 2 2 ∂η2 ∂2 ∂4 −νuξ − 1 − ε2 2 υη + 1 + ε2 4 w = 0. ∂η ∂η uξξ +

(6.10)

The composite equation governing a fundamental state can be obtained on the basis of the mentioned hypotheses. Using the operator approach for the homogeneous equations of system (6.10) we get the following equation governing the interior zone  2   ∂4 ∂4 ∂2   + 2 2 2 + 1 + 2  φηηηη + ∂ξ4 ∂ξ ∂η ∂η 1 − ν2 φξξξξ + 2(1 + ν)φξξηη + 2(2 + ν)φξξηηηη = 0. (6.11) ε2 Equation (6.11) consists of two last terms more than equation (6.1). However, as it is shown by numerical experiences, their influence is not important. It should be mentioned that the equation governing the interior zone could be obtained using the following hypotheses υη − w = 0,

uη + υξ = 0.

(6.12)

Contrary to the classical semi-inextensional theory the quantities χ 11 and χ12 should be taken into account. Hypotheses (6.12) lead to the following simplified equations. Geometrical relations


ε11 =

1 uξ , R

χ22 = −

0 = υη − w,

0 = υξ + uη ,

1 + υ w , ηη η R2

χ12 = −

χ11 = −

205

1 wξξ , R2

1 + υ w . ξη ξ R2

(6.13)

Physical relations T 1 = B 1 − ν2 ε11 ,

0 = ε22 − νε11 ,

M1 = D (χ11 + νχ22 ) ,

S =

1−ν ε12 , 2

M2 = D (νχ11 + χ22 ) .

(6.14)

Equilibrium equations T 1ξ + S η + RP1 = 0,

S ξ + T 2η + N2 + RP2 = 0,

N1ξ + N2η − T 2 − RP3 = 0,

M1ξ + Uη − RN1 = 0,

Hξ + M2η − RN2 = 0.

(6.15)

Composite equations for displacements, related to the hypothesis (6.12), have the following form uη + υξ = 0, υη − w = 0,  2   ε2  ∂4 ∂2   ∂4 wξξξξ +   ∂ξ4 + 2 ∂ξ2 ∂η2 + 1 + ∂η2  wηηηη + 1 − ν2  . (1 − ν)wξξηη + (3 − 2ν)wξξηηηη = 0. (6.16) Applying the operator method for the homogeneous system (6.13)–(6.15) (for pi = 0) we get the following equation  2   ∂4 ∂4 ∂2   + 2 2 2 + 1 + 2  φηηηη + ∂ξ4 ∂ξ ∂η ∂η 1 − ν2 φξξξξ + (1 − ν)φξξηη + (3 − 2ν)φξξηηηη = 0. (6.17) ε2 The above equation (6.17) differs from (6.1) of two last terms, but their influence can be negligible. The composite equation of an edge effect can be obtained using the following physical and geometrical hypotheses: 1. A normal deflection is of more importance, w u,

w υ.

(6.18)

2. The following assumption is valid ε11 + νε22 = 0.

(6.19)

206


3. In the second equation we can take N2 = 0.

(6.20)

The assumed hypotheses allow modelling a composite additional state using the following system of simplified equations. Geometrical relations 1 w 1 ε11 = uξ , ε22 = − , ε12 = υξ + uη , R R R χ11 = −

1 wξξ , R2

χ22 = −

1 wηη , R2

χ12 = −

1 wξη . R2

(6.21)

Physical relations T 1 = B (ε11 + νε22 ) , M1 = D (χ11 + νχ22 ) ,

T 2 = B (ε22 + νε11 ) ,

M2 = D (χ22 + νχ11 ) ,

S =

1−ν Bε12 , 2

H = (1 − ν)Dχ12 .

(6.22)

Equilibrium equations T 1ξ + S η = 0,

S ξ + T 2η = 0,

M1ξ + Hη − RN1 = 0,

N1ξ + N2η − T 2 − RP3 = 0, Hξ + M2η − RN2 = 0.

(6.23)

Using the relations (6.21)-(6.23) we get the following composite equation for an edge effect for displacements uξξ − νwξ = 0, 1+ν 1−ν uξη + υξη − wη = 0, 2 2 −νuξ + 1 + ε2 ∇4 w − R2 p3 = 0.

(6.24)

Using the operator method for the system of equations (6.21)–(6.24) we get the composite equation of the edge effect (6.3). It can be also derived using a variational method, the hypotheses (6.18)-(6.20) and including χ 22 and χ12 for energy expressions. The boundary value problems for the obtained constitutive equations are defined by the splitting boundary conditions in Table 4.3.

6.2 Equations of higher order approximations The obtained above limiting equations of a zero order can be improved introducing higher order terms. Consider a construction of higher approximations for the interior zone and the edge effect. Substituting the potential function (4.8) in the resolving equation (4.6) and taking into account the terms up to O(ε 2 ) order we get the following equations:

6.2 Equations of higher order approximations

207

1. This case corresponds to the interior zone state   4 2 2 4  ∂ ∂ ∂ φ 2 2 2 + 1 + 2  4 + ∂ξ ∂η ∂η ∂η 4 ∂φ ∂4 1 − ν2 ∂ 4 φ ∂2 + = 0. 2 2+4 2 + 4 2 2 ∂η ∂η ∂ξ ∂η ε2 ∂ξ4 The displacements are defined as follows 3 5 ∂φ 2ν2 ∂3 φ 21+ ν ∂ φ −1 + ε , u = ν 3 + ε2 1−ν 1 − ν ∂ξ∂η4 ∂ξ ∂ξ∂η2 3 2 ∂ φ ∂3 φ 2 ∂ υ = (2 + ν) 2 + 1 − ε , ∂ξ ∂η ∂η2 ∂η3 ∂4 φ ∂4 φ . w = 2 2 2 + 1 + ε2 ∂ξ ∂η ∂η4 2. It corresponds to the additive state of the edge effect. 4 4 ∂ φ 1 − ν2 ∂ 4 φ ∂ ∂4 + 4 + = 0, ∂ξ4 ∂ξ2 ∂η2 ∂ξ4 ε2 ∂ξ4 u=−

(6.25)

(6.26)

(6.27)

5 3 ∂3 φ 2 ∂ φ 2 (1 + ν)(2 − ν) ∂ φ + ν 1 + 4ε + ε , 1−ν ∂ξ∂η2 ∂ξ3 ∂ξ3 ∂η2

∂3 φ (2 − ν) ∂5 φ ∂3 φ + 3 − 2ε2 , 2 1 − ν ∂ξ4 ∂η ∂ξ ∂η ∂η ∂4 φ ∂4 φ w = 2 2 2 + 1 + 4ε2 . ∂ξ ∂η ∂ξ4

υ = (2 + ν)

(6.28)

The second order approximations lead to the following relations. 3. It corresponds to the interior zone state  2   ∂4 ∂4 ∂2  ∂4 φ  + 4 2 2 + 1 + 2  4 + ∂ξ4 ∂ξ ∂η ∂η ∂η 4 ∂φ ∂2 ∂4 2 2+4 2 + 4 + ∂η ∂η ∂ξ2 ∂η2 6 ∂ φ ∂2 1 − ν2 ∂ 4 φ 2 8 − 2ν + 5 2 + = 0, ∂η ∂ξ4 ∂η2 ε2 ∂ξ4 ∂3 φ ∂3 φ ∂5 2 (1 + ν)(2 − ν) u=− + ν + ε + ∂ξ∂η2 ∂ξ3 1−ν ∂ξ3 ∂η2

1 + ν ∂5 ∂3 2ν2 ∂3 + 4ν + φ, 1 − ν ∂ξ∂η4 ∂ξ3 1 − ν ∂ξ∂η2

(6.29)

(6.30)

208


5

∂3 φ 4 − 3ν + ν2 ∂5 ∂3 φ 2 ∂ − ε + + φ, 1 − ν ∂ξ2 ∂η3 ∂ξ2 ∂η ∂η3 ∂η5 4

∂ ∂4 φ ∂4 φ 2 − 2ν − ν2 ∂4 ∂4 φ + 2 φ. w = 4 + 2 2 2 + 4 + ε2 1 − ν ∂ξ2 ∂η2 ∂ξ ∂ξ ∂η ∂η ∂η4 υ = (2 + ν)

4. It corresponds to the additive state of the edge effect type 4 ∂ ∂4 ∂ 4 ∂ 4 φ 1 − ν2 ∂ 4 φ +4 2 2 +6 4 + = 0, (6.31) ∂ξ4 ∂ξ ∂η ∂η ∂ξ4 ε2 ∂ξ4 ∂5 ∂3 φ ∂3 φ 2 (1 + ν)(2 − ν) u=− + ν + ε + 2 3 1−ν ∂ξ∂η ∂ξ ∂ξ2 ∂η3

1 + ν ∂5 ∂3 2ν2 ∂3 + 4ν + φ, 1 − ν ∂ξ∂η4 ∂ξ3 1 − ν ∂ξ∂η2

5 ∂3 φ 4 − 3ν + ν2 ∂5 ∂3 φ 2 2(2 − ν) ∂ + υ = (2 + ν) 2 + 3 − ε φ, (6.32) 1 − ν ∂ξ4 ∂η 1 − ν ∂ξ2 ∂η3 ∂ξ ∂η ∂η 4

∂4 φ ∂4 φ 2 − 2ν − ν2 ∂4 ∂ ∂4 φ 2 φ. w = 4 +2 2 2 + 4 +ε 4 4 +2 1 − ν ∂ξ2 ∂η2 ∂ξ ∂ξ ∂η ∂η ∂ξ The constructed relations lead to more accurate solutions in comparison with those obtained by means of the first order asymptotic approximations and they are relatively simple.

6.3 Error estimation For estimation of an error and the area of applicability of the composite equation we compare the second powers of the characteristic equations roots corresponding to the governing equation (4.6) and the composite equations (6.1) and (6.2). A potential function has the form ϕ = Ceλξ cos nη.

(6.33)

Substituting the expression (6.32) in the equation (4.6) we obtain the following characteristic equation 6 1 + 4ε2 λ8 − 4 1 + ε2 n2 λ6 + 6 + ε2 1 − ν2 n4 − 8 − 2ν2 n2 + * 2 1 2 4 2 2 2 4 2 1−ν + 4 λ − 4n − 1 λ + n − 1 = 0. n n ε2

(6.34)

For the composite equation of the interior zone state (6.1) we obtain the following characteristic equation

2 n4 + a λ4 − 2n6 λ2 + n2 − 1 n4 = 0,

6.3 Error estimation

where: a = (1 − ν2 )ε−2 . It has the following solutions 9 1 : −1 λ21,2 = n6 ± n2 2n2 − 1 n4 + a − n2 a 2 n4 + a .

209

(6.35)

The characteristic equation relating to the composite equation of the edge effect (6.3) and its solution have the forms λ4 − 2n2 λ2 + n4 + a = 0, √ λ23,4 = n2 ± i a.

(6.36)

The characteristic equations for the first order approximation equations (6.25) and (6.27) have the form 2 2 aλ4 − 4n2 n2 − 1 λ2 + n2 − 1 n4 = 0,

λ21,2

λ4 − 4n2 λ2 + a = 0, ) 0 1 12 * 2 2 = 2 ± 4n2 n2 − 1 − a n2 n2 − 1 a−1 , λ23,4 = 2n2 ±

√

4n4 − a.

(6.37) (6.38)

The characteristic equations for the second order approximations equations (6.29) and (6.31) have the form 2 2 6n4 − 8 − 2ν2 n2 + a λ4 − 4n2 n2 − 1 λ2 + n2 − 1 n4 = 0,

(6.39)

λ4 − 4n2 λ2 + 6n4 + a = 0, 0 1 12 * 2 2 2 2 4 2 2 = 2 ± 4n n − 1 − 6n + 8 − 2ν n − a n2 n2 − 1 [6n4− )

λ21,2

8 − 2ν2 n2 + a]−1 , √ λ23,4 = 2n2 ± i 2n4 + a.

(6.40)

A solution to the equation (6.34) has been found numerically for the following parameters: ε2 = 10−5 , ν = 0.3. A comparison of the obtained results with the solution of the composite equations (6.35), (6.36) and the high order approximations equations is given in the Figures 6.1–6.4. In the Figures 6.1, 6.3 the modulus values are given, whereas in the Figures 6.2, 6.4 the values of the arguments are given. The curves 1 denote the solutions of equation (6.34); the curves 2 denote the solutions obtained using the composite equations (6.35), (6.36); the curves 3, 4 denote the solutions of a simple edge effect and the semi-inextensional theory; the curves 5, 6 denote the solutions obtained using the shallow shell’s theory (4.17); the curves 7, 8 denote the solutions of limiting equations of the first approximation of the edge

210


Fig. 6.1. A comparison of moduli of small roots of the characteristic equation obtained using various approximated theories.

Fig. 6.2. A comparison of arguments of small roots of the characteristic equation obtained using various approximated theories.

effect type and the interior-zone state (6.37), (6.38); curves 9, 10 denote the solutions of the limiting equation of the second order approximation of the edge effect type and interior-zone state (6.39), (6.40). As it has been seen from the Figures 6.1–6.4, the first order approximation limiting equations secure a reasonable accuracy only for slow variation of the stress-strain state in a circumferential direction (n < 15). The limiting second order approximation equations secure a reasonable accuracy for the stress-strain state slow variations up to n < 25. The composite equations are useful for all n and allow

6.4 Dynamical problems

211

Fig. 6.3. A comparison of moduli of large roots of the characteristic equation obtained using various approximated theories.

Fig. 6.4. A comparison of arguments of large roots of the characteristic equation obtained using various approximated theories.

for smooth matching of solutions, which are valid for slow and fast variations of the stress-strain state.

6.4 Dynamical problems A classification of the limiting cases obtained using the asymptotic techniques is given in [69]. Here we generalize of the composite equation for the analysis of the linear and non-linear shells’ vibrations.

212


At the beginning we consider the linear vibrations of isotropic cylindrical shells. The dynamical components have the form P1 = −

1 uττ , R2

P2 = −

1 vττ , R2

P3 =

1 wττ , R

(6.41)

where: τ = (E/ρ0 )1/2 t; ρ0 denotes the materials density, mass per unit volume. Substituting the relations (6.41) into the equations (4.5) we obtain the following equations governing the dynamics of the elastic cylindrical shell 2 ∂ ∂w 1 − ν ∂2 ∂2 1 + ν ∂2 υ −ν = 0, + − u + 2 ∂η2 ∂τ2 2 ∂ξ∂η ∂ξ ∂ξ2 )

* 1 + ν ∂2 u 1 − ν ∂2 ∂2 ∂2 ∂2 2 + + + ε + 2(1 − ν) υ− 2 ∂ξ∂η 2 ∂ξ2 ∂η2 ∂ξ2 ∂η2

* ) ∂2 ∂w ∂2 ∂2 υ 2 = 0, + −1 + ε (2 − ν) 2 + 2 2 ∂τ ∂ξ ∂η ∂η )

* ∂υ ∂u ∂2 ∂2 2 ν + −1 + ε (2 − ν) 2 + 2 + ∂ξ ∂η ∂ξ ∂η   2 2 2 2   ∂ ∂ ∂  2 + 2 + 2  w = 0. (6.42) 1 + ε 2 ∂ξ ∂η ∂τ A boundary value problem of the equations (6.42) is defined using one of the variants given in Table 4.1 and the following initial conditions u = U 0 (ξ, η) , uτ = U˙ 0 (ξ, η) ,

υ = V0 (ξ, η) ,

υτ = V˙ 0 (ξ, η) ,

w = W0 (ξ, η) ,

˙ 0 (ξ, η) , wτ = W

for τ = 0.

(6.43)

Using the operator method and the equations (6.42) we get the following equation governing dynamics of isotropic cylindrical shells  2 2  2 ∂6 φ  ∂2 ∂2  ∂ 2 2 ∂ ε (1 − ν)Zφ + 2 6 + 2 − (3 − ν) + + + 2ε × ∂ξ2 ∂η2 ∂ξ2 ∂η2  ∂τ  2 2 2 ∂ ∂2 ∂2 ∂4 φ  2 ∂ − (1 − ν) 2 + 3 − ν − 2ν − (1 − ν) + + ∂τ4  ∂η ∂ξ2 ∂ξ2 ∂η2 2 3  ∂ ∂2  ∂2 φ 2 ε (3 − ν) + = 0,  ∂ξ2 ∂η2  ∂τ2

(6.44)

where Z denotes the differential operator of static of the thin elastic cylindrical shell (see equation (4.6)). We introduce the parameter γ, which characterises variation in time, of the form −γ φτ ε2 φ. (6.45)


213

Using a procedure of the asymptotic splitting of the equation (6.44) we obtain the following asymptotic parameters values α, β, γ defining the limiting dynamical equations. α = −0.5, β = 0, γ = −1. It corresponds to the semi-inextensional shell’s vibrations. The fundamental equation has the form

1 − ν2

2 4 ∂2 ∂4 φ ∂ ∂2 2 1 + φττ = 0. + ε + − ∂ξ4 ∂η2 ∂η4 ∂η4 ∂η2

∂4 φ

(6.46)

α = 0.5, β < α, γ = 0. It corresponds to the dynamic edge effect. The potential function is defined by the following equation ∂4 φ + 1 − ν2 φ + φττ = 0. 4 ∂ξ

ε2

(6.47)

α = β = 0.5, γ = 0. It corresponds to the vibration theory of shallow shells. The resolving equation has the form ε2

∂2 ∂2 + 2 2 ∂ξ ∂η

4

2 ∂4 φ ∂2 ∂2 φ + 1 − ν2 + + φττ = 0. ∂ξ4 ∂ξ2 ∂η2

(6.48)

α = β > 0.5, γ = 2α − 1. In this case the potential function is defined by the following equation ε2

∂2 ∂2 + 2 2 ∂ξ ∂η

4 φ+

∂2 ∂2 + 2 2 ∂ξ ∂η

2 φττ = 0.

(6.49)

For rapid circumferential and longitudinal variations of the stress-strain state the equation (6.49) is splitting into the following two equations, which correspond to the bending vibrations and the in-plane plate’s deformation ε

2

∂2 ∂2 + ∂ξ2 ∂η2

2

∂2 ∂2 + ∂ξ2 ∂η2

φ + φττ = 0,

(6.50)

2 φ = 0.

(6.51)

α = β = 0.5, γ < 0. It corresponds to the static theory of shallow shells. The fundamental equation has the form

∂2 ∂2 + 2 2 ∂ξ ∂η

4 φ+

1 − ν2 ∂ 4 φ = 0. ε2 ∂ξ4

(6.52)

α = β < 0.5, γ = 0. The potential function is defined by 2 ∂4 φ ∂2 ∂2 1 − ν2 + + φττ = 0. ∂ξ4 ∂ξ2 ∂η2

(6.53)

214


α ∼ β ∼ γ, 1 < β. It corresponds to the dynamical SSS of shells with fast circumferential and longitudinal variations. The potential function is defined by the following resolving equation (1 − ν)

∂2 ∂2 + ∂ξ2 ∂η2

4

∂2 ∂2 φ+2 + ∂ξ2 ∂η2

∂2 ∂2 (3 − ν) + ∂ξ2 ∂η2

2 φττττ−

3 φττ = 0.

(6.54)

The equation (6.55) is splitting into two equations. The first one governs the in-plane vibrations (1 − ν)

∂2 ∂2 + 2 2 ∂ξ ∂η

2

φ + 2φττττ − (3 − ν)

∂2 ∂2 + φττ = 0. ∂ξ2 ∂η2

(6.55)

The second one governs the plate’s bending

∂2 ∂2 + ∂ξ2 ∂η2

2 φ = 0.

(6.56)

α = 2β − 0.5, 0 < β < 0.5, γ = 0. The composite equation has the form ε2

∂4 φ ∂8 ∂6 φ + 1 − ν2 + 4 2 = 0. 8 4 ∂η ∂ξ ∂ξ ∂τ

(6.57)

α = β = γ = 0. The potential function is defined by the following equation 2 ∂4 φ ∂ ∂6 φ ∂2 ∂4 φ (1 − ν) 1 − ν2 + 2 + 2 − (3 − ν) + − ∂ξ4 ∂ξ2 ∂η2 ∂τ4 ∂τ6  2  2 2 2 2 2   ∂ φ (1 − ν) ∂ + 3 − ν − 2ν2 ∂ − (1 − ν) ∂ + ∂ = 0. (6.58)  ∂η2 ∂ξ2 ∂ξ2 ∂η2  ∂τ2 α = β − 0.5, β < 0, γ = −1. The potential function is defined by equation

1 − ν2

∂4 φ ∂ξ4

+ ε2

∂4 φ ∂4 φ − = 0. ∂η4 ∂η2 ∂τ2

(6.59)

α < −0.5, β = 0, γ = −1. This case corresponds to the ring vibrations. The motion equation has the form 2 4 2 ∂ ∂ φ ∂2 ∂4 φ ε 1+ 2 + −1 = 0. 4 2 ∂η ∂η ∂η ∂η2 ∂τ2 2

(6.60)

α ≤ −0.5, β = 0, γ < −1. It corresponds to the deformation of a ring. The potential function has the form


1+

∂2 ∂η2

2

∂4 φ = 0. ∂η4

215

(6.61)

The limiting equations given above fully define the asymptotic splitting of the governing equation describing dynamics of thin elastic cylindrical shells. The obtained limiting equations allow for the formulation of the composite equations, applied for the arbitrary space and time variations of the state. Linking the limiting relations (6.56) and (6.61), valid for fast and slow variations of the dynamical state, the following composite equation of the interior-zone is obtained   4 2 2 4  ∂4 ∂ ∂ ∂ φ (1 − ν)  4 + 2 2 2 + 1 + 2  4 − ∂ξ ∂ξ ∂η ∂η ∂η 4 4 ∂ ∂ φ ∂4 ∂4 ∂4 φ + 2 +2 4 4 =0 (6.62) (3 − ν) 4 2 2 2 2 ∂η ∂ξ ∂η ∂η ∂τ ∂η ∂τ Equation (6.62) has the fourth order in relation to time and allows for definition of the middle and high parts of the spectrum. In order to define the low frequencies it is necessary to use the semi-inextensional theory. It can be obtained on the basis of equations (6.46) and the in-plane deformation of the plate (6.51) 2 2 ∂2 2 ∂ ε + 2 φ + 1 − ν2 φ + φττ = 0. (6.63) 2 ∂ξ ∂η The composite equation (6.63) allows for getting the low part of the frequency spectrum. In order to get the highest frequencies we need to include a composite equation of the edge zone. Matching the relations of the dynamic edge effect (6.47) and of the plate’s bending (6.56), the following dynamical composite equation of the additive state of the edge effect type is obtained  2   ∂4 ∂4 ∂2  ∂4 φ 2  ε  4 + 2 2 2 + 1 + 2  4 + ∂ξ ∂ξ ∂η ∂η ∂η 2 2 4 ∂ ∂2 ∂2 φ 2 ∂ φ + + = 0. 1−ν ∂ξ4 ∂ξ2 ∂η2 ∂τ2

(6.64)

The obtained composite equations fully agree with the observation which holds, that the characteristic equation corresponding to the resolving equation governing dynamics of thin shells always has three real frequencies. It corresponds to different relations between the amplitudes u, υ, w and to the different variation in circumferential and longitudinal variations. In addition, two of them are larger than the third one. The lowest frequency (ω 21 ) is obtained using the composite equations (6.63), (6.64). It corresponds to the mode when the normal displacement w is more than two others, u and υ. The frequency values ω 22 , ω23 are defined by the composite equation of the interior-zone state (6.62).

216


Here it is necessary to give the following comment. The very high frequencies (when the half waves of vibrations along the co-ordinates have an order of a shell thickness) can not be obtained in frame of the classical theory of shells. In this book we address the vibration problems governed by equations used in the theory of shells. Therefore, by ’high frequency’ we understand a frequency higher than that obtained within the semi-inextensional theory. In order to formulate correct boundary value problems for the composite dynamical equations (6.63), (6.64), the boundary conditions should be added from the Table and the initial conditions are defined by (6.43). For error estimation of the obtained equation free vibrations of a cylindrical shell with simply supported edges (the variant A 1 , Table 4.3) are considered. A solution is being sought in the form φ = A sin( m ¯ ξ) sin(nη)eiωτ ,

m ¯ =

mπ . d

(6.65)

Substituting (6.65) in the equation (6.44), we obtain the following third order algebraic equation in relation to ω 2 3 3 − ν + 4ε2 (1 − ν) 2 3 − ν + 2ε2 2 m ¯ + n + ω2 − 1 + 2 2  2 2  2 1 2  (1 − ν) 1 + 4ε  1 + ν + 2ε  n2 + m ¯4 + ω2 +  ¯ + n2 ε2 m   2 2  1 − ν − ε2 (3 + ν) 4 n + 1 − ν − ε 2 2 − ν2 m ¯ 2 n2 + 2 3 * 3−ν 2 1 − ν 6 2 ε2 ω2 − ¯8 + m ¯ + n2 ε 1 + 4ε2 m 2 2 4 1 + ε2 m ¯ 2 n6 + ¯ 6 n 2 + 6 + ε 2 1 − ν2 m ¯ 4 n4 + 4 m n 8 − 2 4 − ν2 m ¯ 2 n4 − 2n6 + ¯ 4 n2 − 8 m . 1 − ν2 1 + 4ε2 m ¯ 4 + ε2 n 2 4 m ¯ 2 + n2 = 0.

(6.66)

This equation serves for the natural vibration frequency determination. In order to estimate the accuracy of the composite equations we substitute the solution (6.65) to the equation (6.55) governing the in-plane vibration of a plate. We obtain 3−ν 2 1−ν 2 [m ¯ + n2 ]ω2 + [m ¯ + n2 ]2 = 0. (ω2 )2 − (6.67) 2 2 Proceeding in the analogous way, using the dynamic equations of the edge effect (6.47) and the semi-inextensional vibrations equations (6.46) we obtain the following expressions ¯4 , (6.68) ω2 = 1 − ν2 + ε2 m


ω2 =

¯4 1 − ν2 m n2 [n2 + (1 + ν)/(1 − ν)]

+ ε2

2 n2 1 − n 2 n2 + (1 + ν) (1 − ν)

.

217

(6.69)

Substituting the solution (6.65) to the composite equation of the interior-zone state (6.62), the following equation is obtained 3−ν 2 ¯ 2 ω2 + n + 2m 2 0 2 1 1−ν 4 ¯2 + 1 − n2 = 0, (6.70) m ¯ + 2n2 m 2 which serves for the frequency determination. The composite dynamical equations of the semi-inexstensional theory (6.63) and the edge effect (6.64) give the following relations for the second powers of frequencies 2 ¯ 2 + n2 , ω2 = 1 − ν2 + ε2 m (6.71) : 9 2 m ¯2 + n2 − 2n2 + 1 n4 1 − ν2 m ¯4 2 . (6.72) ω2 = +ε 2 m ¯ 2 + n2 2 m ¯ + n2 2

ω2

2

−

In the Figures 6.5–6.6, the frequency versus n dependencies are given. The results are obtained for cylindrical shells with different length and with relative thickness of ε2 = 10−5 . The curve 1 denotes solutions of the equation (6.66); the curves 2 represent the solutions obtained using the composite equation of the interior-zone state (6.69); the curves 3, 4 are related to the results obtained using the composite equations of the edge effect and semi-inextensional theory (6.67), (6.68); curve 5 describes the edge effect (6.70); curve 6 corresponds to the in-plane plate vibrations (6.67); curve 7 is related to the semi-inextensional theory of vibrations (6.71). Analysis of the obtained results displays that the composite equation of the interior-zone state (6.62) gives approximated solution for different variations in both longitudinal and circumferential directions. The composite dynamical equation governing the semi-inextensional theory of vibrations serves for low frequencies estimation for a longitudinal direction for m < 4, when ω 2 1. When the waves number is increased in the longitudinal direction (m ≥ 4), the low frequency range is obtained using the composite equation of the of the edge effect type.

6.5 Non-linear dynamical problems Now we are going to analyse the non-liner vibrations of shells. We consider the non linear limiting equations governing cylindrical shells’ dynamics. They are obtained using the asymptotic splitting of the governing system. The non-linear semi-inextensional theory of vibrations are defined by the following equations

218


T 1ξ + S η − 0.5 (θ1 T 1 )η = 0, S ξ + T 2η −

1 M2η − 0.5 (θT 1 )ξ + θ1 S + θ2 T 2 = 0, R

M2ηη − RT 2 − R (θ1 T 1 − θ2 S )ξ − R (θ1 S + θ2 T 2 )η − ρR2 wττ = 0, (6.73) 1 υξ − uη . where: θ1 = − R1 wξ , θ2 = − R1 wη − υ , θ = 2R The underlined terms are taken into account only if the circumferential variations are equal to zero. For the latter case, the following hypotheses are using 1 wξ wη = 0, R 1 2 (6.74) υη − w + wη = 0. 2R The non-linear dynamical equations of the edge effect have the following form υξ + uη +

T 1ξ + S η = 0,

S ξ + T 2η = 0,

M1ξξ − RT 2 − RT˜ 1(1) θ1ξ = 0.

(6.75) (6.76)

Fig. 6.5. Small vibration frequencies of a cylindrical shell obtained using various approximated theories.

In equation (6.76) the variable coefficients should be “frozen” in respect of ξ on the shell edges  (1)  ξ = 0,   T 1 (0), (1) T˜ 1 =    T (1) (d), ξ = d. 1


219

Fig. 6.6. Large vibration frequencies of a cylindrical shell obtained using various approximated theories.

Fig. 6.7. Small vibration frequencies of a cylindrical shell obtained using various approximated theories.

This is valid because the outer solution variability in the axial direction is much less then the edge effect one, and, in the zone localised near the shell edges, the inner solution may be assumed constant with respect to the ξ variables. The non-linear dynamical equations of the shallow shells’ theory have the following form T 1ξ + S η = 0, S ξ + T 2η = 0, M1ξξ + 2Hξη + M2ηη − RT 2 − R (θ1 T 1 + θ2 S )ξ −

220


Fig. 6.8. Large vibration frequencies of a cylindrical shell obtained using various approximated theories.

R (θ1 S + θ2 T 2 )η + ρR2 wττ = 0.

(6.77)

For the small non-linearity the equations (6.77) can be linearized and the limiting system is splitting into the in-plane stress state and the bending vibration of a plate. The non-linear equations governing the vibrations of a plate have the from T 1ξ + S η = 0,

S ξ + T 2η = 0,

M1ξξ + 2Hξη + M2ηη − RT 2 − R (θ1 T 1 + θ2 S )ξ − R (θ1 S + θ2 T 2 )η − ρR2 wττ = 0.

(6.78)

The given limiting relations (6.73)–(6.78) allow finding the terms having important meaning for fast and slowing variations of the shell state. They can be included to the composite equations. According to the proposed algorithm, matching the relations of semi-inextensional theory and in-plane vibrations of a plate, the following non-linear composite equations of the interior-zone state are obtained T 1ξ + S η − 0.5 (θ1 T 1 )η − ρRuττ = 0, S ξ + T 2η −

1 M2η − 0.5 (θT 1 )ξ + θ1 S + θ2 T 2 − ρRυττ = 0, R

M2ηη − RT 2 − R (θ1 T 1 − θ2 S )ξ − R (θ1 S + θ2 T 2 )η + ρR2 wττ = 0.

(6.79)

The underlined terms are taken only for the case of zero circumferential variation. For the state described by equations (6.79), the physical and geometrical hypotheses (6.5) are satisfied. The expressions for deformation, forces and moments


221

needed for equations (6.79) are obtained via matching the following expressions for the generating equations ε11 =

1 1 1 1 uξ + θ12 , ε22 = υη , ε12 = υξ + uη , R 2 R 2R

1 1 1 θ1ξ , χ22 = − θ2η , 0 = θ2ξ + θ1η , R R 2R 1 1 1 θ1 = − wξ , θ2 = − wη , θ = υξ − uη , R R 2R T1 = B (ε11 + υε22 ) , T2 = B (ε22 + υε11 ) ,

0=

1−ν Bε12 , M1 = νDχ22 , M2 = Dχ22 . (6.80) 2 Matching the relations of the edge effect and the bending vibration of a plate, the following non-linear additive state is obtained S=

T 1ξ + S η = 0,

S ξ + T 2η = 0,

M1ξξ + M2ηη − RT 2 − RT˜ 1(1) θ1ξ − ρRwττ = 0, where: T˜ 1(1)

(6.81)

 (1)    T 1 (0), ξ = d, =   T (1) (l), ξ = l. 1

In the above, the force T 1 is a reaction, and it can be obtained from equations (6.81). The expressions for deformation, forces and moments are defined as follows ε11 =

1 1 uξ + θ12 + θ˜1(1) θ1 , R 2

ε22 = −

w ˜(1) + θ2 θ2 , R

1 1 1 υξ + uη + θ1 θ2 + θ˜1(1) θ2 + θ1 θ˜2(1) , 2R 2 2 1 1 θ1 = − wξ , θ2 = − wη , R R T 1 = B (ε11 + νε22 ) , T 2 = B (ε22 + νε11 ) ,

ε12 =

1−ν Bε12 , M1 = D (χ11 + νχ22 ) , 2 M2 = D (νχ11 + χ22 ) , H = (1 − ν)Dχ12 . S =

where:

(6.82)

 (1)   θi (0), ξ = 0, ˜θ(1) =   i  (1)  θi (l), ξ = d,

The non-linear composite dynamical equations (6.79), (6.81) can not be obtained as a result of the asymptotic splitting of the initial system. However, they

222


are valid for all possible variations of a shell dynamical state. The equation (6.79) has the sixth order with respect to time. A full spectrum of free vibrations can be obtained using the equation (6.79). The composite boundary conditions for the equation (6.79) and (6.81) are defined by splitting boundary conditions given in Table 4.3 and by the initial conditions (6.43).

7 COMPOSITE BOUNDARY VALUE PROBLEMS – ORTHOTROPIC SHELLS

7.1 Statical problems The given in Chapter 5 simplified equations allow for a simple solution of a wide class of practically important problems. However, when a high number of the limiting simplified relations is needed, then the some problems with a practical application occur. Therefore, we propose a procedure to formulate the approximate equations guaranting the simplicity of the limiting equations of the asymptotic analysis. For every type of the shell’s support we include the terms playing the most important role for low and fast variations of the being searched solution. Waffle shells In this case, in the composite equation of the fundamental state, the relations of the semi-inextensional theory (5.11) as well as in-plane deformation of the plate (5.17) are included:   4 2 2  ∂4 ∂ ∂  2 −1 + ε4 1 + 2  φηηηη + Z1 φ ≡ ε1 ε2  4 + 2ε4 ε5 2 2 ∂ξ ∂ξ ∂η ∂η

2 ∂2 ∂2 ∂2 ∂2 ∂4 ν21 ε7 1 + 2 − φ + εe 2 + ε e 2 φηη = 0. ∂η ∂η2 ∂ξ2 ∂η ∂ξ4

(7.1)

Equation (7.1) differs from the equations of the semi-inextensional theory (5.11) by underlined terms. We obtain the following expressions for displacements ∂4 ∂4 ∂4 −1 + 2ν ε − 2ν ε + ν + ε u = ν0 ε−1 10 6 21 11 7 4 ε6 5 ∂ξ4 ∂ξ2 ∂η2 ∂η4

∂2 ∂2 ν21 2 − 2 φξ , ∂ξ ∂η ∂4 ∂4 −1 υ = − 2ν12 ε−1 ε6 4 + 2ν10 ε7 2 2 + 5 + ν11 ε4 ∂ξ ∂ξ ∂η

224


2ε−1 5 + ν12

∂2 ∂4 ∂2 + + ν ε φη , 0 7 ∂ξ2 ∂η2 ∂η4

(7.2)

4 w = ε−1 4 ∇2 φ.

The following expressions define the forces and moments ∂2 B1 ∂2 T1 = ν21 ε4 ∇0η 2 − 2 φηη , R ∂η ∂ξ B1 ∂2 ∂2 T2 = ν21 ε7 ∇0η 2 − 2 φξξ − R ∂η ∂ξ B2 2 ε1 ε2 + ν221 ε4 ε7 ∇0η φηηηη − ν21 ε−1 4 ε7 φξξηη , R ∂3 B1 ∂ 3 − ν21 ε7 ∇0η 3 φη , M1 = ν21 B1 ε7 ∇0η φηηηη , S = R ∂ξ3 ∂η

4 ∂2 2 ∂ −1 M2 = B2 ∇0η + ε21 ε2 + ν221 ε−1 ε − ν ε ε φηη , 21 4 7 4 7 ∂η4 ∂ξ2 H = B1 ε21 ε3 ε4 − ν21 ε8 ∇0η φξηηη .

(7.3)

The equation (7.1) is of the fourth order in respect to the longitudinal coordinate. In order to satisfy the boundary condition, an equation of supplementary state is needed. Therefore, we obtain the following equation as a result of matching the equations governing an edge effect (5.12) and the deflection of a plate (5.17):    ∂4 ∂4 ∂4  2  Z2 φ ≡ ε1  4 + 2ε3 2 2 + ε2 4  φ+ ∂ξ ∂ξ ∂η ∂η 2  2 2   ∂ ∂ ν12 ε6 − εe 2 − 1 φ = 0. 2 ∂ξ ∂η

(7.4)

Equation (7.4) differs from the equations of the edge effect (5.12) by underlined terms. The expressions for the displacements, forces and moments defined by the equation (7.4) by matching the corresponding expressions for a state of the edge effect and bending deformation of a plate. As a result, we obtain u = ν0 ε−1 4 ε6 φξξξξξ + ν21 φξξξ , −1 4 w = ε−1 υ = − 2ν12 ε−1 φξξξξη + 2ε−1 5 + ν11 ε4 5 + ν10 φξξη , 4 ∇2 φ, B1 B1 ∂2 T1 = ν12 ε6 2 − 1 φξ , T 2 = − φξξξξ , R R ∂ξ


225

∂2 B1 1 − ν12 ε6 2 φξξξη , R ∂ξ 2 4 M1 = −B1 ε−1 4 ε1 + ε6 ∇2 φξξ + ν41 φηη −

∂2 φ ε + ν ε6 ν0 ε−1 12 ξξξξ , 4 6 ∂ξ2 6 2 4 M2 = −B1 ε21 ε2 ε−1 4 + ε7 ∇2 ν14 φξξ + φηη + S =

−1 ε6 ε4 ε7 2ν12 ε−1 5 + ν11 ε4

* 2 ∂4 ∂2 −1 ∂ φξξ , − ν + ε 12 4 ∂ξ2 ∂η2 ∂η2 ∂ξ2 B1 2 H=− (7.5) ε1 ε3 − ν41 ε21 + ε26 − ν21 ε6 ε7 ∇42 φξη . 2ε4 It is clear that efficiency and advantages of the approximated equations depend on how close the applied simplification describes a real shell’s state under the load. In order to analyse this problem we consider the physical-geometrical hypotheses corresponding to the composite equations (7.1), (7.4). The following hypotheses correspond to the composite equations of the interiorzone state. 1. The curvature in the longitudinal direction and the torsion are small χ22 χ11 ,

χ22 χ12 .

(7.6)

2. In the third equilibrium equation the moment M 1 must be neglected. The assumed hypotheses are equivalent to the following simplified system of equations. Geometrical relations: 1 1 ε11 = uξ , ε22 = υη − w , R R 1 1 ε12 = χ22 = 2 wηη + υη . (7.7) υξ + uη , R R Stress-strain relations: T 1 = B11 ε11 + B12 ε22 , S = B33 ε12 ,

T 2 = B21 ε11 + B22 ε22 + K22 χ22 , M1 = D12 χ22 + K11 ε11 ,

M2 = D22 χ22 + K22 ε22 ,

H = D33 χ12 .

Equilibrium equations: T 1ξ + S η + RP1 = 0, N2η − T 2 − RP3 = 0,

S ξ + T 2η + N2 + RP2 = 0, M1ξ + Hη − RN1 = 0,

(7.8)

226


M2η − RN2 = 0.

(7.9)

Substituting expression (7.7), (7.8) for forces and moments to equation (7.9), the following composite equations of the interior-zone state for displacements is got B11 uξξ + B33 uηη + (B12 + B33 ) υξη − B12 wξ = −R2 P1 , 2K22 D33 (B21 + B33 ) uξη + B22 − + 2 υηη + B33 υξξ + R R D K22 K22 22 − + B − w wη = −R2 P2 , ηηη 22 R R R2 D K22 K22 22 − + B − −B21 uξ + υ υη + ηηη 22 R R R2 K22 D22 wηη + B22 w = R2 P3 . wηηηη + 2 (7.10) 2 R R Note that the equation (7.10) can be obtained using the variation method and the hypothesis (7.6). In the analogous way to that of the isotropic case, the composite equation of the fundamental state can be obtained using the hypotheses of the lack of extension of the middle surface in the circumferential direction. In this case the forces S 1 , T 2 are defined by the equation (7.9). The composite equation of the edge effect can be obtained using the following hypotheses. 1. A normal displacement plays a dominating role in the stress-strain relations, w u,

w υ.

(7.11)

2. In the first three equilibrium equations we can take H = N2 = 0.

(7.12)

The above hypotheses lead to the following simplification of the governing equations. Geometrical relations ε11 =

1 uξ , R

w ε22 = − , R

ε12 =

1 1 wξξ , χ22 = − 2 wηη , R2 R Stress-strain relations are the same as (5.4). Equilibrium equations χ11 = −

T 1ξ + S η = 0,

S ξ + T 2η = 0,

Hη + M1ξ − RN1 = 0,

1 υξ + uη , R χ12 = −

1 wξη . R2

(7.13)

N1ξ + N2η − T 2 = 0,

Hξ + M2η − RN2 = 0.

(7.14)


227

Equation (7.14) are the composite equations of the edge effect. Taking into account (7.13), (7.14), the composite equations of the edge effect for displacements have the form K11 wξξξ = 0, B11 uξξ − B12 wξ − R D33 K22 (B21 + B33 ) uξη + B33 + 2 υξξ + B22 − wη = 0, R R K11 D11 2 uξξξ + 2 wξξξξ + 2 (D12 + 2D33 ) wξξηη + −B21 uξ − R R R K22 D22 wηηηη + 2 (7.15) wηη + B22 w = 0. R R2 The composite equation of the edge effect can be obtained using the variational method and the same hypotheses as in the case of an edge effect (see Chapter 5). In the expression for energy the terms χ 22 and χ12 are additionally taken. The composite equations of the interior-zone state and the edge effect for cylindrical waffle shells have their analogy in the isotropic case. Therefore splitting of boundary conditions for the equations (7.1), (7.4) overlaps with the results given in Table 4.3. Stringer shells For the shells with a stringer support, the shallow shells’ equations, obtained as a result of the asymptotic analysis, have the fourth order in respect to the longitudinal co-ordinate. It is valid in the whole interval of variation except for low solution changes (2-3 waves) in the circumferential direction. Therefore, in order 7 to formulate the composite equation the interior-zone state the operator 1 + ∂2 ∂η2 instead of operator ∂ 2 /∂η2 is included in the equation governing the shallow shell’s behaviour  2   ∂4 ∂4 ∂2  2  Z3 φ ≡ ε1 ε4  4 + 2ε3 2 2 + ε2 1 + 2  φηηηη + ∂ξ ∂ξ ∂η ∂η

∂2 1 + ε6 2 ∂η

2 φξξξξ = 0.

(7.16)

The additional state of the edge effect type is defined by the equation Z4 φ ≡

ε21 ∇42 φ

2 ∂2 ∂2 + ε6 ν12 2 − 2 − 1 φ = 0. ∂ξ ∂η

(7.17)

The equation (7.16) is distinguished from the equation (5.11) by the underlined term. The equation (7.17) overlaps with the equation (5.12). The first of the given relations includes the equations of the interior-zone state with fast circumferential variation and of the plates’ bending. The equation (7.17) includes the relations of an edge effect, the strain state with a fast in both direction variations and the in-plane

228


deformation of a plate. The equation (7.17) is applied in the whole interval of the stress-strane state variation. It results from the asymptotic analysis of the governing equation of the structurally - orthotropic cylindrical shells. The displacements of that state (7.16) are governed by relation’s (5.11), whereas the forces and moments are defined as follows ∂2 ∂2 B1 T1 = ν12 ε6 2 − ε6 2 − 1 φξξηη , R ∂ξ ∂η ∂2 ∂2 B1 T2 = ν12 ε6 2 − ε6 2 − 1 φξξξξ , R ∂ξ ∂η 2 ∂ ∂2 B1 S =− ν12 ε6 2 − ε6 2 − 1 φξξξη , R ∂ξ ∂η )

∂2 ∂2 2 M1 = B1 ε21 + ν12 ε26 + ν ε ε ∇4 + 12 1 4 ∂ξ2 ∂η2 2

2 * 4 ∂ ∂4 ∂2 ∂2 2 ∂ 1 + 2ν12 ε4 ε−1 φ, + ν ε ε + ε − ν ε 6 12 4 6 6 12 5 ∂ξ2 ∂η2 ∂η4 ∂ξ2 ∂η2 ∂ξ2 2 ∂2 2 −1 ∂ M2 = B2 ε1 ε2 ε4 + ν12 2 ∇42 φ, ∂η2 ∂ξ 4 H = −ν12 B1 ε−1 4 ε8 ∇2 φξη .

(7.18)

The displacements, forces and moments for the equation (7.17) are defined by the following expressions ∂2 ∂2 B1 T1 = ν12 ε6 2 − ε6 2 − 1 φξξηη , R ∂ξ ∂η ∂2 ∂2 B1 T2 = ν12 ε6 2 − ε6 2 − 1 φξξξξ , R ∂ξ ∂η 2 ∂ ∂2 B1 S =− ν12 ε6 2 − ε6 2 − 1 φξξξη , R ∂ξ ∂η )

∂2 ∂2 2 M1 = B1 ε21 + ν12 ε26 + ν ε ε ∇4 + 12 1 4 ∂ξ2 ∂η2 2

2 * 4 ∂ ∂4 ∂2 ∂2 2 ∂ 1 + 2ν12 ε4 ε−1 φ, + ν ε ε + ε − ν ε 6 12 4 6 6 12 5 ∂ξ2 ∂η2 ∂η4 ∂ξ2 ∂η2 ∂ξ2 2 ∂2 2 −1 ∂ M2 = B2 ε1 ε2 ε4 + ν12 2 ∇42 φ, ∂η2 ∂ξ 4 H = −ν12 B1 ε−1 4 ε8 ∇2 φξη .

(7.19)

The state of the considered types essentially depends on the magnitude and the sign of the stringer eccentricity. The composite equation (4.16) can be obtained


229

using the variational method. The corresponding forces S , T 2 are the static reactions and are defined by equation (7.14). From the energetic point of view, for the equation (7.17) the deformations ε 22 , χ11 , χ22 , χ12 play a fundamental role. In order to obtain the boundary value problems for equations (7.16), (7.17) we can use the splitting limiting boundary conditions of Table 5.3.First, according to this method, the equations of edge effect are solved. Then from the boundary conditions the corresponding constants are excluded. The essential error (when the asymptotic matching is applied) is introduced due to splitting of the boundary conditions and, therefore, the described procedure leads to an increase in the calculation accuracy. Ring-stiffened shells The composite equation of the interior-zone state for the structural - orthotropic 7 ring-stiffened cylindrical shells is obtained by introducing the operator 1 + ∂2 ∂η2 to the equation (5.32). We obtain ∂4 ∂4 2 −1 2 + 2 ε ε ε ε − ν ε + Z5 φ ≡ ε21 ε2 + ε27 2 4 12 1 7 5 ∂ξ4 ∂ξ2 ∂η2 2  ∂2  2 2 2 ε1 ε2 ε4 + ν21 ε7 1 + 2  φηηηη + ∂η 2 2 ∂ ∂ 2ε7 −ν21 + 1 φξξηη + φξξξξ = 0. (7.20) ∂ξ2 ∂η2 The additional state of the edge effect type is governed by the equation

Z6 φ ≡

ε21 ∇41 φ

∂2 + 1 + ε7 2 ∂η

2 φ = 0.

(7.21)

The equation (7.20) differs from the equation (5.11) by the underlined terms, and the equation (7.21) overlaps with the equation (5.12). The equation (7.20) governs the interior-zone state behaviour, the state with the fast variations, and the in-plane deformation of a plate. Equation (7.21) governs the behaviour of an edge effect, the state with the fast longitudinal variations, and the bending deformation of a plate. The expressions (5.33) and (5.35) correspond to equations (7.20), (7.21). The splitting of the boundary conditions for the composite equations (7.20), (7.21) is the same as the results given in Table 5.4. The composite equations (7.20), (7.21) be obtained using the variational method by taking the ε 22 , ε12 , χ11 and ε22 , ε11 , χ22 as the fundamental deformations, correspondingly. Therefore, it has been shown that the investigation of the state of cylindrical shells with the waffle, stringer and ring types of support is possible. Formulated composite boundary value problems define the differential equations of the fourth order in respect to the longitudinal co-ordinate.

230


Fig. 7.1. A comparison of moduli of small roots of the characteristic equation for the waffle shell obtained using various approximating theories.

Fig. 7.2. A comparison of arguments of small roots of the characteristic equation for the waffle shell obtained using various approximating theories.

The next step includes investigation of the accuracy and the area of application of the obtained composite equations. For this purpose, a comparison of the second power of the characteristic equations’ roots has been carried out. The characteristic equations correspond to the input resolved equation (5.6) and to the given composite equations for waffle ((7.1), (7.4)) and stringer ((7.16), (7.17)) shells.


231

Fig. 7.3. A comparison of moduli of large roots of the characteristic equation for the waffle shell obtained using various approximating theories.

Fig. 7.4. A comparison of moduli of large roots of the characteristic equation for the waffle shell obtained using various approximating theories.

The analysis has been carried out for the following geometrical and rigidity parameters:

232


Fig. 7.5. A comparison of moduli of small roots of the characteristic equation for the stringer shell obtained using various approximating theories.

Fig. 7.6. A comparison of arguments of small roots of the characteristic equation for the stringer shell obtained using various approximating theories.

for a waffle-type shell: ε 21 = 2.2 · 10−6 , ε2 = 10−2 , ε3 = 10−2 , ε4 = 1, ε5 = 0.3, ε6 = ε7 = 0, ν12 = ν21 = 0.2; for a stringer-type shell: ε 21 = 2.2 · 10−6 , ε2 = 10−4 , ε3 = 10−2 , ε4 = 0.6, ε6 = ε7 = 0, ν12 = ν21 = 0.2.


233

Fig. 7.7. A comparison of moduli of large roots of the characteristic equation for the stringer shell obtained using various approximating theories.

Fig. 7.8. A comparison of moduli of large roots of the characteristic equation for the stringer shell obtained using various approximating theories.

A comparison of the numerical solutions with the solutions of the composite equations for the waffle shells is given in Figures 7.1–7.4 and for the stringer shells in Figures 7.5–7.8. In the Figures 7.1, 7.2, 7.5 and 7.6, a comparison of the modulus of the second power of the characteristic equations’ roots is given. In the Figures 7.3, 7.4, 7.7 and 7.8 the corresponding arguments are shown. The curves 1 denote

234


the exact solutions; curves 2 denotes the solutions obtained using the composite equations; curves 3 correspond to the edge effect; curves 4 correspond to semiinextensional theory; curves 5, 6 correspond to the theory of plates. The results display validity and a high accuracy of the composite equations for all value of n. They guarantee sufficient approximation accuracy, smoothly matching the solutions with fast and slow variations.

7.2 Dynamical problems Now we are going to formulate the composite dynamical equations of the structuralorthotropic cylindrical shell theory. We consider first the linear vibrations. Waffle shells We have the following composite equation of the interior-zone state, valid for arbitrary frequencies and variations of the SSS

) 4 ∂2 ∂2 ∂ ∂4 −1 + + 2ε5 + ε4 1 + 2 Z1 φ − Z11 φττ ≡ Z1 φ − ∂ξ4 ∂ξ2 ∂η2 ∂η ∂η2 ∂2 ∂2 ∂2 −1 −1 −1 ε21 ε4 1 + 2 ε−1 + ν + ε + 2 ε + ν + ν 12 0 4 12 ν0 5 5 ∂ξ2 ∂η2 ∂τ2 * 4 −1 ∂ ε4 2 ε−1 + ν (7.22) φττ = 0, ν 12 0 5 ∂τ4 where: τ = (B1 /ρ)1/2 . Here Z1 is defined by equation (7.1). The displacements corresponding to the equation (7.22) have the form −1 u = ν0 ε−1 4 ε6 φξξξξξ + 2ν10 ε6 φξξξηη − 2ν21 ε5 + ν11 ε7 φξηηηη +

∂3 ∂ ν12 φξξξ − φξηη + ν0 ε6 3 + ν12 φττ , ∂ξ ∂ξ −1 v = − 2ν12 ε−1 ε6 φξξξξη + 2ν10 ε7 φξηηηη + 2ε−1 5 + ν11 ε4 5 + ν12 φξξη + ∂ ∂3 −1 φηηη + ν0 ε7 φηηηηη + 2ε5 ν0 ε7 3 + φττ, ∂η ∂η 2 2 −1 4 −1 −1 ∂ −1 ∂ w = ε4 ∇2 φ − ε4 ν0 + 2ε5 + ν0 + 2ε4 ε5 φττ + ∂ξ2 ∂η2 −1 2ε−1 4 ε5

−1 2ν0 ε−1 4 ε5 φττττ .

Dynamical composite equation of the edge effect is

(7.23)


Z2 φ − φττ = 0.

235

(7.24)

Here Z2 is defined by equation (7.4). The displacements, forces and moments for the equation (7.24) are defined by the equations (7.5). Equation (7.22), (7.24) should be supplemented by the splitting boundary conditions from the Table 4.3 and the initial conditions (5.42). Stringer shells In this case the composite dynamical equations have the form Z3 φ − Z11 φττ = 0,

(7.25)

Z4 φ − φττ = 0.

(7.26)

Here Z3 , Z4 and Z11 are defined by equations (7.16), (7.17) and (7.22), respectively. The displacements for the equation (7.25) are defined by relations (7.23) for ε7 = 0, whereas for the equation (7.26) the are defined via relations (7.5). The splitting boundary conditions for the equation (7.25), (7.26) are given in Table 5.4. The initial conditions have the form (5.42). Ring-stiffened shells In this case we have the following composite dynamical equations Z5 φ − Z11 φττ = 0,

(7.27)

Z6 φ − φττ = 0.

(7.28)

Here equations (7.20), (7.21) and (7.22) define Z 5 , Z6 and Z11 , respectively. The displacements related to the equation (7.27) are defined by the expressions (7.28), where ε 6 = 0 in case of the equation (7.28) they are defined via the relations (7.5). For the equations (7.27), (7.28) the corresponding boundary conditions are given in Table 5.5, and the initial conditions have the form (5.42).

7.3 Non-linear dynamical problems A synthesis of the obtained in Chapter 5.3 limiting equations leads to the following non linear composite equations governing the dynamics of the structurally orthotropic cylindrical shells.

236


Waffle shells As a result of matching the asymptotic equations for the waffle shells the following composite equations are obtained 1 ∂ (θ1 T 1 ) − ρRutt = 0, 2 ∂η

1 1 ∂ (θT 1 ) + θ1 S + θ2 T 2 − ρRυtt = 0, S ξ + T 2η − M2η − R 2 ∂ξ T 1ξ + S η −

M2ηη + RT 2 − R

∂ ∂ (θ1 T 1 + θ2 S ) − R (θ1 S + θ2 T 2 ) − ρR2 wtt = 0. ∂ξ ∂η

(7.29)

Geometrical and physical relations for this state have the from 1 1 1 1 uξ + θ12 , ε22 = υη − w + θ22 , R 2 R 2 1 1 1 ε12 = υξ + uη + θ1 θ2 , χ22 = θ2η , 2R 2 R 1 1 θ1 = − wξ , θ2 = − wη , R R T 1 = B11 ε11 + B12 ε12 , T 2 = B21 ε11 + B22 ε22 + K22 χ22 , ε11 =

S = B33 ε12 ,

M1 = D12 χ22 + K11 ε11 ,

M2 = D22 χ22 + K22 ε22 .

(7.30)

The obtained non-linear interior-zone equations for the cylindrical waffle shells can be used either for low and fast variations of the solution in the whole range of the vibration frequency spectrum. As a result of matching the non-linear relations of the edge effect and bending vibrations of the plate we obtain the following composite dynamical equations of edge effect T 1ξ + S η = 0, S ξ + T 2η = 0, M1ξξ + M2ηη + RT 2 − RT˜ 1(1) θ1ξ − ρR2 wtt = 0.

(7.31)

The deformations, forces and moments for equations (7.31) are defined as follows 1 1 1 ε11 = uξ + θ12 + θ¯1(1) θ2 , ε22 = w + θ¯2(1) θ2 , R 2 R 1 1 1 ε12 = υξ + uη + θ1 θ2 + θ¯1(1) θ2 + θ1 θ¯2(1) , 2R 2 2 1 1 1 χ11 = θ1ξ , χ22 = θ2η , χ12 = θ2ξ + θ1η , R R 2R 1 1 θ1 = − wξ , θ2 = − wη , R R T 1 = B11 ε11 + B12 ε22 + K11 χ11 , T 2 = B21 ε11 + B22 ε22 ,


S = B33 ε12 ,

M1 = D11 χ11 + D12 χ22 + K11 ε11 ,

M2 = D21 χ11 + D22 χ22 + K22 ε22 , where: θ¯i(1)

237

(7.32)

   θi(1) (0, η, τ), ξ = 0,  = ,   θ(1) (d, η, τ), ξ = d, i = 1, 2, i

θi(1)

is defined by equations (7.29), (7.30). The obtained non-linear composite dynamical equations have the fourth order in relation to the longitudinal co-ordinate. In order to formulate the boundary value problems for equations (7.29), (7.30), the splitting boundary conditions (Table 4.3) and the initial conditions (5.42) should be applied. Stringer shells Matching the non-linear limiting relations for the stringer shells we obtain the following non-linear composite equations of the interior-zone state: 1 (θ1 T 1 )η − ρRutt = 0, 2

1 1 S η + T 2η − M2η − (θT 1 )ξ + θS + θ2 T 2 − ρRvtt = 0, R 2 T 1ξ + S η −

M1ξξ + 2Hξη + M2ηη + RT 2 − R (θ1 T 1 + θ2 S )ξ − R (θ1 S + θ2 T 2 )η − ρR2 wtt = 0.

(7.33)

The stress-strain and strain-displacement relations are follows 1 1 1 1 uξ + θ12 , ε22 = υη − w + θ22 , R 2 R 2 1 1 1 ε12 = υξ + uη + θ1 θ2 , χ22 = θ2η , 2R 2 R 1 1 θ1 = − wξ , θ2 = − wη , R R T 1 = B11 ε11 + B12 ε22 , T 2 = B21 ε11 + B22 ε22 + K22 χ22 , ε11 =

S = B33 ε12 ,

M1 = D12 χ22 + K11 ε11 ,

M2 = D22 χ22 + K22 ε22 .

(7.34)

The complementary state has fast longitudinal and circumferential variation. It is localised in the neighbourhood of the edges of the shell and it is governed by the following composite equations of the edge effect T 1ξ + S η = 0,

S ξ + T 2η = 0,

M1ξξ + RT 2 − RT¯2(1) θ2η − ρR2 wtt = 0, where:

(7.35)

238


T 2(1)

 (1)    T 2 (0, η, τ), =   T (1) (d, η, τ), 2

ξ = 0, ξ = d,

T 2(1) is defined by equations (7.29), (7.30). The deformations, forces and moments are as follows 1 1 uξ , ε22 = υη − w + θ¯2(1) θ2 , R R 1 1 1 ε12 = υξ + uη + θ¯2(1) θ1 , χ11 = θ1ξ , 2R 2 R 1 1 1 χ22 = θ2η θ1 = − wξ , θ2 = − wη , R R R T 1 = B11 ε11 + B12 ε22 + K11 χ11 , T 2 = B21 ε11 + B22 ε22 , ε11 =

S = B33 ε12 , where: θ¯i(1)

M1 = D11 χ11 + D12 χ22 + K11 ε11 ,

(7.36)

   θi(1) (0, η, τ), ξ = 0,  , =   θ(1) (d, η, τ), ξ = d, i = 1, 2, i

θi(1)

is defined by equations (7.29), (7.30). The boundary problems for the composite equations (7.33)–(7.36) are defined by one of the variants of the boundary conditions, given in the Table 5.4, and the initial conditions (5.42). Ring-stiffened shells The non-linear composite dynamical equations of the interior-zone state have the from 1 T 1ξ + S η − (θ1 T 1 )η − ρRutt = 0, 2

1 1 S ξ + T 2η − M2η − (θT 1 )ξ + θS + θ2 T 2 − ρRvtt = 0, R 2 M2ηη + RT 2 − R (θ1 T 1 + θ2 S )ξ − R (θ1 S + θ2 T 2 )η − ρR2 wtt = 0.

(7.37)

The deformations, forces and moments for this state are 1 1 1 1 uξ + θ12 , ε22 = υη − w + θ22 , R 2 R 2 1 1 1 1 ε12 = υξ + uη + θ1 θ2 , χ11 = θ1ξ , χ22 = θ2η , 2R 2 R R T 1 = B11 ε11 + B12 ε22 , T 2 = B21 ε11 + B22 ε22 + K22 χ22 , ε11 =

S = B33 ε12 ,

M1 = D11 χ11 + D12 χ22 + K11 ε11 ,

M2 = D21 χ11 + D22 χ22 + K22 ε22 .

(7.38)


239

The additive state to the equations (7.37), (7.38) is described by the following composite equations of the edge effect T 1ξ + S η = 0,

S η + T 2η = 0,

M1ξξ + M2ηη + RT 2 − RT¯2(1) θ2η − ρR2 wtt = 0, where: T 2(1)

 (1)    T 2 (0, η, τ), =   T (1) (d, η, τ), 2

(7.39)

ξ = 0, ξ = d,

T 2(1) is defined by equations (7.29), (7.30). The geometrical and the physical relations are governed by the following expressions 1 1 1 ε11 = uξ + θ12 + θ¯1(1) θ2 , ε22 = w + θ¯2(1) θ2 , R 2 R 1 1 1 (υξ + uη ) + θ1 θ2 + (θ¯1(1) θ2 + θ¯2(1) θ1 ), ε12 = 2R 2 2 1 1 1 1 χ11 = θ1ξ , χ22 = θ2η , θ1 = − wξ , θ2 = − wη , R R R R T 1 = B11 ε11 + B12 ε22 , T 2 = B21 ε11 + B22 ε22 + K22 χ22 , S = B33 ε12 , where:

M1 = D11 χ11 ,

M2 = D22 χ22 + K22 ε22 ,

(7.40)

   θi(1) (0, η, τ), ξ = 0,  (1) ¯ , θi =   (1)  θ (d, η, τ), ξ = d, i = 1, 2, i

θi(1)

is defined by equations (7.29), (7.30). For equations (7.37)–(7.38) we need the splitted boundary conditions (Table 5.5) and the initial conditions (5.42).

7.4 Stability problems Here we formulate the composite equations for stability problem. The stability estimation for the waffle shells is governed by equation  2   ∂4 ∂4 ∂2  ¯  Z1 φ + T 20  4 + 2ε3 2 2 + 1 + 2  φηη + ∂ξ ∂ξ ∂η ∂η ∂2 ∂2 ∂2 ¯ ¯ + T 10 2 φ = 0, 1 + 2 2S 0 ∂ξ∂η ∂η ∂ξ

(7.41)

Z2 φ + T¯10 φ = 0.

(7.42)

Here Z1 , Z2 are defined by equations (7.1) and (7.4), respectively.

240


The displacements, forces and moments for the equations (7.41) and (7.42) are defined by the relations (7.23), (7.5), respectively. The boundary value problems for the equations (7.41), (7.42) are defined by one of the variants of the splitting boundary conditions given in Table 4.3. The composite stability equations for the stringer shells have the following form ∂2 ∂2 ∂2 ∂2 ¯ ¯ ¯ + T 20 2 φηη = 0, Z3 φ + ε4 1 + 2 T 10 2 + 2S 0 (7.43) ∂ξ∂η ∂η ∂ξ ∂η Z4 φ + T¯10 φ = 0.

(7.44)

Here Z3 , Z4 are defined by equations (7.16), (7.17) respectively. The displacements, forces and moments corresponding to the equations (7.43), (7.44) are defined by the expressions (7.23) and (7.5), respectively. The boundary value problems for equations (7.43), (7.44) are defined by one of the variants of the splitting boundary conditions given in Table 5.4. For the ring-stiffened shells we obtain the following composite stability equations   4 2 2  ∂4 ∂ ∂  Z5 φ + T¯20  4 + 2ε3 2 2 + 1 + 2  φηη + ∂ξ ∂ξ ∂η ∂η ∂2 ∂2 ∂2 + T¯10 2 φ = 0, (7.45) 1 + 2 2S¯0 ∂ξ∂η ∂η ∂ξ ∂2 ∂2 ∂2 + T¯20 2 φ = 0. Z6 φ + T¯10 2 + 2S¯0 (7.46) ∂ξ ∂ξ∂η ∂η Here Z5 , Z6 are defined by equations (7.20) and (7.21), respectively. The displacement for the equations (7.45), (7.46) are defined by the expressions (7.23) and (7.5), respectively. The boundary value problems for equations (7.45), (7.46) are defined by one of the variants of the boundary conditions given in Table 5.5.

8 AVERAGING

A wide spectrum of references has been devoted to the averaging method [180, 592, 706]. Although the roots of the method come from pioneering works of H. Poincaré and B. Van der Pol, the kernel has been developed by the works of N. Krylov and N.N. Bogolyubov. In general, the averaging method uses splitting of fast and slow solution components. Assume that a solution to a certain problem has the form shown in Figure 8.1. x

t

Fig. 8.1. Splitting of a solution into a sum of a slow and fast components.

One can separate a slow solution component x 0 (t) as well as fast oscillating components. Both of the components can be found separately using the corresponding simplification.

8.1 Two-scales approach First of all observe that a realization of the averaging can be different as well in different problems the different averaging procedures can be used. Recently an actuation has been focused on a multiple scale approach. We also apply here two scales approach, since it can be easily applied to many different problems including those with rapidly oscillating coefficients. Let us begin with the Duffing equation with small non-linearities

242

8 AVERAGING

x¨ + ω2 x + εx3 = 0, where:

ω2 ≡ const,

ω2 ∼ 1,

(8.1)

ε 1.

The generating linear equation (ε = 0) has the following solution x0 = A cos ωt + B sin ωt,

(8.2)

where: A, B = const. We assume that for 0 < ε 1 a solution to input equation (8.1) can be described approximately by (8.2), where A and B are functions, slowly changing in time. According to two scale method, we introduce a ‘slow time’, where t denotes the ‘fast’ time. We get ∂ ∂ d = + ε. dt ∂t ∂τ Looking for a solution in the form x = x0 (t, τ) + εx1 (t, τ) + ..., we get:

∂x0 ∂x1 dx ∂x0 = +ε + + ..., dt ∂t ∂τ ∂t 2 ∂ x0 ∂ 2 x1 d 2 x ∂ 2 x0 + 2 + ... . = +ε 2 ∂τ∂t dt2 ∂t2 ∂t

After splitting due to ε, the following recurrent equations are obtained ∂ 2 x0 + ω2 x0 = 0, ∂t2

ε0 : ε1 : .

∂ 2 x1 ∂ 2 x0 2 − x30 . + ω x = 2 1 ∂t∂τ ∂r2 . . . . . . . . .

(8.3) (8.4)

A solution (8.3) has the form x0 = A(τ) cos ωt + B(τ) sin ωt. Then, the equation (8.4) reads ∂ 2 x1 + ω2 x1 = P(t, τ), ∂τ2

(8.5)

dB 3 where: P(t, τ) = 2 dA dτ ω sin ωt − 2 dτ ω cos t − (A cos ωt + B sin ωt) . The solution regularity is obtained requiring the vanishment of secular terms. It means, that

8.1 Two-scales approach 2π

243

2π

ω

ω P(t, τ) cos ωtdt = 0,

0

P(t, τ) sin ωtdt = 0.

(8.6)

0

The conditions (8.6) give the following two ordinary differential equations with the unknowns A(τ) and B(τ): −

B dA 3 2 = (A + B2 ) ε, dτ 8 ω

3 dB A = − (A2 + B2 ) ε. dτ 8 ω Observe that an analogous system of averaged equations can be obtained if we look for the following solution to equation (8.1): x = A(t) cos(ωt + θ(t)),

(8.7)

with the condition

dx = −A(t)ω sin(ωt + θ(t)). dt The functions A(t) and θ(t) play a role of slowly changing amplitude and phase. Differentiating (8.7) due to t we get dA dθ dx = Aω sin(ωt + θ) + cos(ωt + θ) − A sin(ωt + θ). dt dt dt

(8.8)

From (8.1) we get ω

dθ dA sin(ωt + θ) + Aω cos(ωt + θ) = εA3 cos3 (ωt + θ). dt dt

(8.9)

Solving (8.8) and (8.9), we get dA εA3 = cos3 (ωt + θ) sin(ωt + θ), dt ω

(8.10)

dθ εA2 = cos4 (ωt + θ). (8.11) dt ω Because A and θ are slowly changed with time (because ε is small), therefore a change of period within the time T = 2π/ω it is also small. Averaging (8.10) and (8.11) in the interval [t, t + T ], where A and θ can be assumed as the constant values, yields T 1 cos3 (ωt + θ) sin(ωt + θ)dt = T 0

1 2π

2π 0

sin(ωt + θ) cos3 (ωt + θ)d(ωt + θ) = 0,

244

8 AVERAGING

1 2π

2π

cos4 (ωt + θ)d(ωt + θ) =

0

and then we finally obtain

3 , 8

dA = 0, dt

(8.12)

dθ 3 εA2 = . dt 8 ω The equation (8.12) yields A = const, whereas from (8.13) we get θ=

(8.13)

3 A20 ε t + θ0 . 8 ω

Therefore, in first step of averaging we have    3 A20  u = A0 cos ω 1 + ε 2  t + O(ε). 8 ω0 The obtained correction to the frequency coincides with those obtained using other methods (for instance, the Poincare-Linstedt method). An application of an averaging approach to the dynamical non-linear problems exhibited by plates and shells is motivated by an exact or approximate (Bubnov–Galerkin method) variables separation. The obtained system of ordinary differential equations can be than averaged in time. An interesting approach to increase accuracy of the averaging method, when non-linear system is taken as input and a solution is sought in the form of elliptic functions ir reported in reference [211, 700].

8.2 Visco-elastic problems and ’freezing’ method The calculation of visco-elastic plates and shells is reduced to consideration of integro-differential equations. For instance, dynamics of visco-elastic rectangular (0 ≤ x ≤ a, 0 ≤ y ≤ b) plate with a geometrical non-linearity is governed by the equation (see Chapter 8.5) ¯ 2 w + ρhwtt = 0, DΓ∇4 w − BΓ I∇ where:

t Γϕ = ϕ +

R(t − τ)ϕ(τ)dτ, 0

1 I= 2ab

a b 0

0

B=

Eh , 1 − ν2

(w2x + w2y )dxdy;

8.2 Visco-elastic problems and ’freezing’ method

245

and R denotes a relaxation kernel. For simply supported edges, the variables can be separated via the following form nπy mπx sin , w(x, y, t) = A(t) sin a b which yields the following non-linear integral-differential equation Dπ

4

2 m a

+

n 2 2 b

Γ(A + BA3 ) + ρhAtt = 0.

(8.14)

Since it is difficult to find an exact solution to equation (8.14), we simplify the problem considering only low frequency oscillations. Indeed, let us consider the integral t A(τ)R(t − τ)dτ. (8.15) J= 0

Assuming a slow change of A(τ) in comparison with a relaxation kernel change R(t − τ), one can freeze it for t = τ. Thus, instead of (8.15) we get t J = A(t)I1 ,

I1 =

R(t − τ)dτ. 0

The described approach is called the method of freezing [294]. Applying it to equation (8.14), we get Dπ

4

2 m a

+

n 2 2 b

I1 (A + BA3 ) + ρhAtt = 0,

(8.16)

which is already ordinary differential equation with constant coefficients. One can also proceed in a different way by applying averaging in expression for J:   T t   l   (t I2 = lim R − τ) dτ J = A(t)I2 , (8.17)   dt. T →∞ T 0

0

The equation (8.14) becomes now the ordinary differential equation with constant coefficients of the from 2 2 2 n m 4 + I2 (A + BA3 ) + ρhAtt = 0, Dπ a a which is well known Duffing equation.

246

8 AVERAGING

8.3 The successive change of variables In this section our considerations follow reference [706]. Consider the following system of differential equations d¯z ¯ = f (t, ¯z) + εF¯ (t, ¯z, ε) , dt

(8.18)

where: ¯z = (x1 , x2 , . . . , xn ); f¯ = ( f1 , f2 , . . . , fn ); F = (F1 , F2 , . . . , Fn ); 0 ≤ ε 1. In the classical averaging method the system (8.18) is reduced to the form called the standard one. This reduction is realized with the change of variables ¯z → ¯x by the formula ¯z = G¯ (t, ¯x) . (8.19) Differentiating the relation (8.19), we obtain d¯z ∂G¯ ∂G¯ = + ¯x, dt ∂t ∂x where:

 ∂G ∂G1  1   ∂x · · · ∂x  1 n   ∂G¯  =  · · · · · · · · · ·  . ∂ ¯x    ∂Gn ∂Gn  · · · ∂x1 ∂xn

Substituting the obtained relations into the system (8.18), and solving it with ¯ ¯x is invertible) respect to d ¯x/dt, one obtains (if ∂G/∂

−1 ) * ¯ ∂G¯ (t, ¯x) d ¯x ¯ (t, ¯x) , ε . ¯ (t, ¯x) − ∂G (t, ¯x) + εF¯ t, G, = f¯ t, G, dt ∂ ¯x ∂t The obtained equations can be presented in the form d ¯x = εX¯ (t, ¯x, ε) , dt

(8.20)

which is said to be the standard form. Observe that the multiplier ε stands in the right hand side of (8.20). The arbitrary systems governed by (8.18) can be reduced to the form (8.20) assuming that a general solution of a zero-order approximation is known. In addition, the variable change G¯ (t, ¯x) = ¯g(t, ¯x) + ε ¯q(t, ¯x) , also transforms a system to standard form. Function q(t, x) denotes the arbitrary differentiable function, which must be chosen to transform the right hand side function ¯ ¯x, ε) to a form more suitable for further investigations. X(t,

8.3 The successive change of variables

247

Further, a solution is presented in the form of a slowly changed part ¯x0 (t) (the system evolution) and rapidly oscillating improvement part ¯y(t). In order to average ¯x0 (t) we have the following averaged equation d ¯x0 = ε1 X¯0 ( ¯x0 , ε) , dt where: ¯x0 =

1 T

and X¯0 = lim

T →∞

T 0 T

X (t, ¯x, ε) dt, if ¯x is the periodic function, ¯x(t + T, ¯x, ε) = X¯ (t, ¯x, ε) ; X¯ (t, ¯x, ε)dt, if X¯ is any unperiodic function.

0

In order to construct higher approximations the method of successive change of variables is applied. The input system (8.20) can be transformed to the form d ¯x = εX¯0 ( ¯x, ε) + εX¯1 (t, ¯x, ε) . dt Note that ¯ X¯0 + X¯1 = X,

T

(8.21)

X¯1 (t, ¯x, ε) dt = 0.

0

We would like to change the variables in order to get the following system d ¯x = εX¯0 ( ¯x, ε) + ε2 X¯2 (t, ¯x, ε) . dt The following close to identity, change of variables is introduced ¯x = ¯y + εS¯ (t, ¯x, ε) .

(8.22)

Differentiating relation (8.22) and taking into account (8.21), one gets d¯y ∂S¯ ∂S¯ d¯y +ε +ε = εX¯0 ¯y + εS¯, ε + εX2 t, ¯y + εS¯, ε , dt ∂t ∂y dt and hence

(8.23)

∂S¯ d¯y = εX¯0 (¯y, ε) + ε X¯1 (t, ¯y, ε) − + ε2 Q¯ (t, ¯y, ε) . dt ∂t

All terms of second and higher order are included in ε 2 Q(t, ¯y, ε). In order to satisfy the required condition S¯(t, ¯y, ε) is chosen as follows X2 (t, ¯y, ε) −

∂S¯ ≡ 0, ∂t

and hence one gets S¯ (t, ¯y, ε) =

t X2 (t, ¯y, ε) dt + C (¯y, ε). 0

248

8 AVERAGING

Therefore, the input system (8.21) via transformations (8.22), (8.23) is reduced to the system d¯y = εX¯0 (¯y, ε) + ε2 Q¯ (t, ¯y, ε) . (8.24) dt Applying averaging to (8.24), the following averaged system is obtained d¯y = εX¯0 (¯y, ε) + ε2 X¯2 (¯y, ε) , dt where: 1 X¯2 (¯y, ε) = T

(8.25)

T Q (t, ¯y, ε) dt. 0

Note that instead of the second order approximation solution (8.25), the first order approximation solution may be substituted into transformation (8.22), i.e. the approximated solution reads ¯x(t) = ¯y(t) + εS¯ t, ¯y(t) , ε , where the additional term εS [t, ¯y(t), ε] is added. The improved first approximation satisfies the system (8.21) in more accurated way, since only second order term ε 2 Q is not compensated. The computational time required to built first order approximation is essentially smaller in comparison to second order approximation. In order to construct an arbitrary approximation the method of induction can be applied. Let the system is reduced to the form where the neglected terms during the averaging procedure are of ε n order: d ¯x = εX¯0 ( ¯x, ε) + ε2 X¯2 ( ¯x, ε) + . . . + εn X¯n ( ¯x, ε) + εn ¯y(t, ¯x, ε) , dt

(8.26)

where ¯y(t, ¯x, ε) the function with mean value equal to zero is denoted. The change of variables ¯x = ¯y + εn S¯ (t, ¯y, ε) is to required to transform the system (8.26) to the form d¯y = εX¯0 (¯y, ε) + ε2 X¯1 (¯y, ε) + . . . + εn+1 X¯n+1 (¯y, ε) + εn+1 Q (t, ¯y, ε) . dt

(8.27)

Repeating the earlier given recipes we find that the being sought function S¯(t, ¯y, ε) can be expressed by the formula (8.23), where only correspondingly ˜ ¯y, ε) appear. The averaged equation for (8.27) has the form X(t, d¯y = εX¯0 (¯y, ε) + ε2 X¯2 (¯y, ε) + . . . + εn+1 X¯n+1 (¯y, ε) , dt

8.4 Application of the Lie groups

1 X¯n+1 (¯y, ε) = T

T

249

Q¯ (t, ¯y, ε) dt.

0

The discussed procedures lead to the following construction. For the given system

d ¯x = εX¯ (t, ¯x, ε) dt the following change of variables is introduced ¯x = ¯y + εS¯1 (t, ¯y, ε) + ε2 S¯2 (t, ¯y, ε) + ...,

(8.28)

and the system is transformed to the following autonomous one d¯y = εX¯0 (¯y, ε) + ε2 X¯2 (¯y, ε) + . . . . dt

(8.29)

The obtained series do not belong to the Taylor series with respect to the powers of ε because the standing terms by ε depend on ε. Note that the transformation (8.28), yielding the autonomous form (8.29) instead of the non-autonomous one, can be also achieved without application of the averaging procedure. However, the averaging procedure guarantees that the coefficients of the series (8.28) are bounded functions of time. In words, an arbitrary order averaging transforms a non-autonomous system to autonomous one with a given accuracy using bounded transformations for arbitrary t. This idea is a generalization of the Poincaré-Lindstedt method to avoid the secular terms.

8.4 Application of the Lie groups1 Like Moliére’s Monsieur Jouradin, who had been speaking prose all his life and did not know it, we have just been, perhaps unknowingly, speaking the language of Lie groups. A. Iserles [351]

The theory of Poincaré normal forms [509], which is similar to averaging techniques, retains resonance terms, since all nonresonance terms are removed from the equations of motion by means of a special coordinate transformation. In this case the Poincaré normal forms is qualified as the simplest possible form of the equations of motion. Alternatively, one can use Lie group operators, which can lead to the simplest form of the system equations of motion. The Lie group theory has become a powerful tool for studying differential equations among mathematicians and specialists, and needs to be adapted for dynamicists. In [139] presented an overview of 1

By courtasy V.N. Pilipchuk and R.A. Ibrahim [554]

250

8 AVERAGING

the mathematical structure of Lie groups and Lie algebras with applications to nonlinear differential equations. Hori [507] used Lie series to construct an additional first integral in an autonomous Hamiltonian system. Zhuravlev [705, 706] developed an algorithm for the asymptotic integration of nonlinear differential equations as monomial Lie group transformation of the phase space into itself. An essential ingredient of the Lie group operators is the Hausdorff formula. This formula relates the Lie group operators of the original system and the new one, and the operator of coordinate transformation. Most of averaging techniques reproduce this formula, each time implicity, during the transformation process. There is no need to do this, since it is reasonable to start the transformation using Hausdorff’s relationship. The theory of Lie groups deals with a set of transformations. In other words, an original dynamical system, y˙ = f (y, ε) is transformed into its simplest form, z˙ = g (z, ε) by means of a coordinate transformation, y → z, which is given as the solution z = z(y, ε) of the third dynamical system, dz/dε = T (z, ε), z| ε=0 = y, in the same phase space. The selection of the vector-function T (z, ε) depends on the desired properties of the transformed system. One of the advantages of the group formulation is that it specifies a general class of near identical transformations. Specifically, one should select the expression z = z(y, ε) among solutions of a dynamical system, but not among all classes of the near identical transformations. Another basic advantage is that all manipulations of the scheme can be done in linear terms, i.e., the monomial Lie group operators. Moreover, the result of transformation in general terms of operators is well known and is given by the Hausdorff formula. First we briefly describe the Hausdorff formula. Consider the Cauchy problem for the vector-function y(t) : y˙ = f (y), y| t=0 = y(0) . Using the monomial Lie group operator, A = f (y)(∂/∂y), the equation is rewritten as y˙ = Ay, and its formal solution can be represented in the exponential form y = etA y(0) .

(8.30)

Here the operator A must act on y (0) components, i.e. A = f (y 0 )(∂/∂y0). Similarly, one has the exponential form of transformation depending on parameter ε: (8.31) z = eεU y. For the Cauchy problem in the new coordinates z˙ = Bz, z| t=0 = z0 on can write a formal solution as (8.32) z = etB z(0) . Now consider two different ways of transformating y (0) → z(t) as indicated in Figure 8.2. Taking into account (8.30)–(8.32), these transformations can be written in the form


z = eεU y = eεU etA y(0)

251

and z = etB z(0) = etB eεU y(0) .

Equating the right-hand sides, and recalling that y (0) is an arbitrary initial point, gives (8.33) eεU etA = etB eεU when ε = 0 the transformation becomes identical and the operator of transformed system coincides with the original one: A = B. For small ε, one obtains A = B+

dB 2 ε=0 ε + O ε . dε

(8.34)

Fig. 8.2. Two different ways of transformating y(0) → zt .

Now consider expression (8.33) with the time parameter t equal to ε. Expanding the exponents in power series, in leading-order terms with respect to ε, one obtains A + εUA = B + εBU. Substituting (8.34) into the last expression gives dB = [B, U] , dε

(8.35)

where: [B, U] = BU − U B is the commutator. This is the Hausdorff equation for the operator of the transformed system. The equation is supposed to be solved under the initial condition B| ε=0 = A. A solution of the Cauchy problem (8.35) can be found in the power series form B = A + ε [A, U] +

ε2 [[A, U] , U] + ... . 2!

252

8 AVERAGING

This expression is called the Hausdorff formula. Consider now the normal form coordinates. In terms of principal coordinates q k , a nonlinear dynamical system of n-degrees-of-freedom may be described by a set of n + 1 autonomous differential equations written in the standard form q¨ k + ω2k qk = εFk (q1 , . . . , qn+1 , q˙ 1 , . . . , q˙ n+1 ) , k = 1, . . . , n + 1,

(8.36)

where a dot denotes differentiation with respect to time t, ε is a small parameter, and an external excitation has been replaced by the coordinate q n+1 . The functions Fk include all nonlinear terms and possibly parametric excitation terms, and ω k are the principal mode frequencies. The Poincaré normal form theory deals with sets of first-order differential equations written in terms of normal form coordinates. In this case it is convenient to transform the (n + 1) second-order differential equation (8.36) into (n + 1) first-order differential equations plus their conjugate set. This can be done by introducing the complex coordinates y k (t) yk = q˙ k + iωk qk , qk =

1 (yk − ¯yk ) , 2iωk

q˙ k =

1 (yk − ¯yk ) . 2

(8.37)

The physical meaning of these coordinates can be understood by considering the linear case (obtained after setting ε = 0). The linear solution of (8.36) and its first derivative (the velocity) are, respectively, qk = Ak eiωk t + A¯k e−iωk t , i2 = −1, q˙ k = Ak eiωk t + A¯k e−iωk t iωk where Ak and A¯k are complex and an overbar denotes conjugate. The complex coordinates y k and ¯yk can be viewed as vectors rotating in the complex plane with angular velocities ω k and −ωk , respectively, yk = q˙ k + iωk qk = 2iωk Ak eiωk t , ¯yk = −2iωk Ak e−iωk t . Introducing the transformation (8.37) into the equations of motion (8.36) gives q¨ k + ω2k qk =

or

d (q˙ k + iωk qk ) − iωk (q˙ k + iωk qk ) = dt

dyk − iωk yk = εFk (y1 , . . . , yn+1 ; ¯y1 , . . . , ¯yn+1 ) dt y˙ k = iωk yk + εFk (y1 , . . . , yn+1 ; ¯y1 , . . . , ¯yn+1 ) , k = 1, . . . , n + 1,

and the corresponding complex conjugate (cc) set of equations where Fk (y1 , . . . , yn+1 ; ¯y1 , . . . , ¯yn+1 ) =

(8.38)


Fk (q1 , . . . , qn+1 , q˙ 1 , . . . , q˙ n+1 ) q j =(1/2iω j )(y j −¯yj ),q˙k =(1/2)(yk−¯yk ) . These terms can be represented in the polynominal form mn+1 l1 1 n+1 Fk = Fkσ ym y1 . . . ¯yln+1 , 1 . . . yn+1 ¯

253

(8.39)

(8.40)

|σ|=2,3,...

where the Taylor coefficients are defined by the partial differentiation of (8.39) as Fkσ =

∂|σ| Fk 1 y=0 . m l m l σ! ∂y 1 . . . ∂y n+1 ∂¯y 1 . . . ∂¯y n+1 1

n+1

1

n+1

Here the multi-index notations σ = {m 1 , . . . , mn+1 , l1 , . . . , ln+1 }, |σ| = m1 , + . . . + mn+1 + l1 + . . . + ln+1 , σ! = m1 ! . . . mn+1 !l1 . . . ln+1 !, have been used. Equations (8.38) correspond to the standard form, which is ready for analysis in terms of Lie group operators. Applying the Lie group operators, first of all we rewrite equation (8.38) in the form (8.41) y˙ = Ay, A = A0 + εA1 where

y = (y1 , . . . , yn+1 ; ¯y1 , . . . , ¯yn+1 )T

and A0 =

n+1 k+1

iωk yk

∂ + c.c. ∂yk

and

A0 =

n+1

Fk

k+1

∂ + c.c. ∂yk

(8.42)

are operators of linear and nonlinear components of the system, respectively. In order to bring the equations of motion to their simplest form, we introduce the coordinate transformation in the Lie series form y = e−εU z = z − εUz +

ε2 2 U z −... 2!

y = (y1 , . . . , yn+1 ; ¯y1 , . . . , ¯yn+1 )T → z = (z1 , . . . , zn+1 ; ¯z1 , . . . , ¯zn+1 )T

(8.43)

where the operator of transformation U is represented in the power series form with respect to the small parameter ε U = U0 + εU1 + . . .

(8.44)

The coefficient of this series are Uj =

n+1 k=1

where

T j,k

∂ + c.c. ∂zk

(8.45)

254

8 AVERAGING

T j,k = T j,k (z1 , . . . , zn+1 ; ¯z1 , . . . , ¯zn+1 ) , j = 0, 1, . . . ,

k = 1, . . . , n + 1,

(8.46)

are unknown functions to be determined. One of the advantages of this process is that the inverted coordinate transformation to the form (8.43) can be easily written as eεU y = z

(8.47)

where one simply replaces z with y in the operator of transformation, U. If ε = 0 transformation (8.47) becomes identical, y = exp(0)z = z. In this case of equation (8.41) has the simplest linear form and there is no need to transform the system. For ε 0 transformation (8.43) converts the system (8.41) into the following one: ¯x = Bz, (8.48) where the new operator B is given by the Hausdorff formula, B = A + ε [A, U] +

ε2 [[A, U] , U] + . . . , 2!

(8.49)

[A, U] = AU − UA is the commutator of operators A and U. Substituting the power series expansions for A and U given by relations (8.41) and (8.44) into (8.49) gives 1 2 B = A0 + ε (A1 + [A0 , U0 ]) + ε [A0 , U1 ] + [A1 , U0 ] + [[A0 , U0 ] , U0 ] + . . . 2! (8.50) A simple calculation gives B=

n+1 k=1

; ∂ + c.c. + O ε2 iωk zk + ε Fk + (A0 − iωk ) T 0,k ∂zk

(8.51)

where the terms of order ε 2 have been ignored, F k = Fk |y→z , A0 = A0 |y→z . The above relationships shows that a transformation of the system y˙ = Ay → z˙ = Bz can be considered in terms of operators A → B. To bring the system into its normal (the simplest) form, one must eliminate as many nonlinear terms as possible from the transformed system such that the system dynamic characteristics are preserved. It follows from (8.51) that all nonlinear terms of order ε would be removed if Fk + (A0 − iωk ) T 0,k = 0 and c.c. Representing the unknown functions in the polynomial form σ m1 n+1 l1 n+1 T 0,k = T 0,k z1 . . . zm z1 . . . ¯zln+1 n+1 ¯

(8.52)

|σ|=2,3,...

and taking into account (8.40) one obtains the left-hand sides of the above written conditions as


Fk + (A0 − iωk ) T 0,k =

|σ|=2,3,...

where

255

mn+1 l1 σ 1 n+1 z1 . . . ¯zln+1 and c.c. Fkσ + i∆σk T 0,k zm 1 . . . zn+1 ¯

' ( ∆σk = (m1 − n1 − δ1k ) ω1 + . . . + mn+1 − ln+1 − δn+1,k ωn+1 .

mn+1 l1 1 n+1 To reach zeroth coefficient of nonlinear form z m z1 . . . ¯zln+1 , one must put 1 . . . zn+1 ¯ σ =i T 0,k

Fkσ , ∆σk

if only ∆σk 0. If ∆σk = 0 for definite k and σ then the corresponding nonlinear term cannot be eliminated from the transformed equation since it is qualified as a resonance term. Finally, the result of transformation is summarized as follows. The Original System. mn+1 l1 1 n+1 Fkσ ym y1 . . . ¯yln+1 , k = 1, 2, 3. (8.53) y˙ k = iωk yk + ε 1 . . . yn+1 ¯ |σ|=2,3,...

The Transformation of Coordinates. yk = zk − ε

i

|σ|=2,3,... ∆σk 0

Fkσ m1 mn+1 l1 n+1 z1 . . . ¯zln+1 + O ε2 . σ z1 . . . zn+1 ¯ ∆k

The Transformed Systems. mn+1 l1 1 n+1 Fkσ zm z1 . . . ¯zln+1 + O ε2 . z˙k = iωk yk + ε 1 . . . zn+1 ¯

(8.54)

(8.55)

|σ|=2,3,... ∆σk =0

This is the normal form of the system, and the summation in this form is much simpler then the one in the original set (8.53). The summation in (8.55) contains only those terms that give rise to resonance while the first term on the right-hand side stands for the fast component of the motion. The fast component of the motion can be extracted by introducing the complex amplitudes a k (τ) as zk = ak (τ) exp (iωk τ) .

(8.56)

Substituting (8.56) into (8.55) and taking into account the resonance condition, ∆ σk = 0, gives mn+1 l1 1 n+1 Fkσ am a1 . . . ¯aln+1 + O ε2 . (8.57) a˙ k = ε 1 . . . an+1 ¯ |σ|=2,3,... ∆σk =0

Note that relationship (8.54)–(8.57) have been obtained under non concrete assumptions regarding the type of resonance.

256

8 AVERAGING

8.5 Whitham method (non-linear WKB approach) It should be strongly emphasized that the averaging method can be applied in various modifications. In particular, Whitham [691, 692] proposed the interesting method for a wide class of equations. This method represents the rapidly oscillating solution of a partial differential equation in the form of the asymptotic expansions (8.58) w = w0 ε−1 S ( ¯x, t) , ¯x, t + εw1 (S ( ¯x, t) , ¯x, t) + . . . , where the phase S ( ¯x, t) and the functions w j (τ, ¯x, t) are periodic in τ, are to be determined. Observe that Witham method can be treated as two scale method, since one may introduce ’fast’ τ = ε −1 S ( ¯x, t) and ’slow’ ( ¯x, t) variables. Substituting (8.58) into input equation and equating the coefficients at powers of ε to zero, one obtains a chain of ordinary (in τ) differential equations, the equation for F 0 (τ, ¯x, t) being nonlinear, and those for F1 (τ, ¯x, t), i ≥ 1, being nonhomogenous linear. The conditions for existence of ε-periodic solutions of these equations lead to additional relations (which form a system of nonlinear partial differential equations) that enable one to determine the phase S ( ¯x, t) and the dependence of the functions F j (τ, ¯x, t) on the ‘slow’ variables ¯x and t. The described modification of the Witham method belongs to Luke [446]. Besides, the representation governed by (8.58) is natural extension to the nonlinear case of the classical WKB expansions for a solution to a linear equations (see Chapter 16). For nonlinear ordinary differential equations, a representation of the solution in a form similar to (8.58) was apparently first proposed by Ku˙zmak [413]. Now an application of the Witham method in the non-linear theory of plates and shells is illustrated. The Berger equation [157, 634] is analysed and its generalization to shell investigation is discussed. The formulation for the Berger equations and the domain of their applicability have repeatedly been discussed in the literature [202, 318, 354, 356, 357, 505, 566, 596, 597]. Berger [157] simply omitted the second invariant of the strain tensor in the expression for the potential energy of the basis that the numerical computations display its slight influence on the state of bending stress. Other authors performed the same, taking little care about the foundation for similar simplifications, which sometimes resulted in false results. Hence, it is important to arrive at equations of the Berger type without utilizing the hypothesis about the smallness of the second invariant of the strain tensor. We write the nonlinear equations of motion of a rectangular plate . 6 I1x + (1 − ν) 0.5ε12y − ε2x − ρ 1 − ν2 ε−1 utt = 0, . 6 I1y + (1 − ν) 0.5ε12x − ε1y − ρ 1 − ν2 ε−1 vtt = 0, ' ( ' ( D 1 − ν2 ∇4 w − Eh I1 w x x + I1 w x x + 9' ( (1 − ν) ε2 w x x + ε1 wy − 0.5 ε12 wy − y

x

(8.59)

8.5 Whitham method (non-linear WKB approach)

257

' ( . 0.5 ε12 w x y + ρ 1 − ν2 hwtt = 0, where:

2 ' ( I1 = ε1 + ε2 , ε1 = u x + 0.5 w x 2 , ε2 = vy + 0.5 wy , ε12 = uy + v x + w x wy .

Here I1 is the first invariant of the strain tensor and the terms in the braces in (8.59) are obtained because of varying the second invariant of strain tensor. The appropriate boundary conditions should supplement (8.59). For instance, let U = 0, w x = 0 for x = 0, a, U = 0, wy = 0 for y = 0, b,

(8.60)

where: U = (u, v, w). We first turn to the spatially-one-dimensional case [376] and examine the nonlinear vibrations of a rod. The Berger equation then agrees with the known equation obtained on the basis of the Kirchhoff hypothesis (neglecting longitudinal inertia) in [376]. As is shown in Chapter 2.5, such an equation is obtained as one of the possible limit cases (for sufficiently large variability in the space variable) because of an asymptotic analysis of the initial system. √ Let us attempt to construct such an asymptotic even in this case using ε = h/2 3a as a small parameter. We first perform the following change of variable ' ( y U ρ 1 − ν2 x at, U¯ = . ξ= , η= , τ= a a E a Equations (8.59) are then rewritten in the following form (the dot now denotes differentiation with respect to τ) I 1ξ + (1 − ν) 0.5ε12η − ε2ξ − uττ = 0, I 1η + (1 − ν) 0.5ε1ξ − ε1η − vττ = 0, 9

(8.61)

0 + I 1 wη + (1 − ν) ε2 wξ + ε1 wη − ξ η ξ η 1: 0.5 ε12 wη ξ − 0.5 ε12 wξ η + ε−2 wττ = 0;

∇4 w − ε−2

I 1 wξ

2 2 I 1 = ε1 + ε2 , ε1 = uξ + 0.5 wξ , ε2 = vη + 0.5 wη , ε12 = uη + vξ + wξ wη , ∇2 =

∂2 ∂2 + 2. 2 ∂ξ ∂η

Following the Witham method, we represent the displacement vector in the form U = U (εα θ (ξ, η) , ξ, η, ε) .

(8.62)

258

8 AVERAGING

As usual, the exponent α is now selected from the condition of non-contradiction of the appropriate limit systems. In this case one such value is α = −0.5. We substitute (8.62) into (8.61) and write the limit system (ε → 0). Here we consider the function θ(ξ, η) as a new independent variable and take into account that ∂ ∂ ∂ = + ε−1/2 θξ , ∂ξ ∂ξ ∂θ

∂ ∂ ∂ = + ε−1/2 θη . ∂η ∂η ∂θ

We finally have

1 0 2 I 1θ θξ + 0.5 (1 − ν) uθθ θη − υθθ θξ θη = 0, (8.63) 1 0 2 I 1θ θη + 0.5 (1 − ν) uθθ θξ − υθθ θξ θη = 0, (8.64) 0 2 2 12 6 0 1 2 2 2 . wθθθθ θξ + θη − I 1θ wθ θξ + θη − I1 wθθ θξ + θη + wττ = 0. (8.65) There follows from the equations (8.63), (8.64) I 1θ = 0.

(8.66)

Then the term in braces drops out of the equation in (8.65). Now, if we return to the variables x, y, t, we then obtain from (8.66) I1x = I1y = 0. Taking into account the boundary conditions (8.60), we hence determine b a 0 2 1 (w x )2 + wy dxdy. I1 = 0.5 0

(8.67)

0

Taking (8.67) into account, we obtain the Berger equation for the vibrations of a rectangular plate h2 ∇4 w − 12∇2 w + 12ρ 1 − ν2 wtt = 0. (8.68) It is possible to arrive analogously at (8.68) by set off from the Kármán system of nonlinear equations. The simplified equations governing dynamics of a shallow shell (0 ≤ x ≤ a, 0 ≤ y ≤ b) with the curvature radiuses R 1 , R2 have the form a b D 4 1 ∂2 1 ∂2 E 2 ∇ w+( + )F − (w2x + w2y )dxdy+ [0.5∇ w[ h R2 ∂x2 R1 ∂y2 ab(1 − ν2 ) 0

a

b (

w xx 0

0

ν 1 + )wdxdy+wyy R1 R2

a

b (

0

0

0

ν 1 + )wdxdy] + ρwtt = 0, R2 R1

1 ∂2 1 ∂2 1 4 + )w = 0, ∇ F +( E R2 ∂x2 R1 ∂y2 where F is Airy function.

(8.69) (8.70)

9 CONTINUALIZATION

Although a change of a discrete medium by the continuous one can be considered as the particular case of the averaging method, it has the series of its own particularities. Therefore, the so called continualization process is further presented within the frame of the separated section. The fundamental peculiarities of the considered system will be illustrated on example of a one dimensional system. So, the one-dimensional case of the chain composed of n material points with the same masses m distributed along the x axis, and having the coordinates jh ( j = 0, 1, . . . , n − 1), is considered. The masses are coupled by the massless stiffness with the coefficient c. This system is also called n-mass oscillator. Let for t = 0 the force f (t) is applied to a zero number point. Then the dynamics of the described chain is governed by the following system of ordinary differential equations (9.1) mÿ j (t) = σ j+1 (t) − σ j (t), σ0 (t) = − f (t), σn (t) = 0,

j = 0, 1, ..., n − 1,

(9.2)

where: y j (t) denotes the longitudinal displacement of the j − th point; σ j (t) = c(y j (t) − y j−1 (t)); σ j (t) is the force of interaction between ( j − 1)th and j − th points. The considered system can be reduced to the form mσ j (t) = c(σ j+1 − 2σ j + σ j−1 ),

j = 0, 1, ..., n − 1,

(9.3)

which is used in our further considerations. The following initial conditions are applied σ j (t) = σ jt (t) = 0 f or t = 0.

(9.4)

Usually, for large values of n the following continuous approximation to the above discrete problem is applied: mσtt (x, t) = ch2 σ xx (x, t), σ(0, t) = − f (t),

σ(l, t) = 0,

σ(x, 0) = σt (x, 0) = 0, where: l = (n + 1)h.

(9.5) (9.6) (9.7)

260

9 CONTINUALIZATION

Getting a solution to the above formulated boundary value problem, the following relation between both systems holds σ j (t) = σ( jh, t),

j = 0, 1, ..., n.

Assuming (without a loss of generality) f (t) = −1 one may find exact solution to the boundary value problem using the d’Alembert method and operational calculus [449] π c t −x , σ(x, t) = H nh arcsin sin 2n m where H(. . .) is the Heaviside function. It yields a direct estimation of |σ(x, t)| ≤ 1 for all time values. A similar like estimation is obtained for instance by N. E. Joukowski [360], and for a long time it has been accepted. However, further numerical and analytical investigations showed that one has to pay attention due to a way of approximation of global and local characteristics of a discrete system. The transition to the continuous system holds during investigation of only lower part of the spectrum of eigenfrequencies of a discrete system. However, already V. P. Maslov [467] warned that the solution derivatives of the discrete problem may not correspond to the solution derivatives of a wave equation for h → 0. In the references [295, 296, 297, 298, 299, 412] is shown that for some masses in the chain the magnitude of σ may essentially differ from 1. This phenomenon is referred as the splash-effect. In particular, for some n the following splashes of P n are found [295]: Table 9.1. Numerical characteristics of a splash effect in discrete mass chain. n Pn

8 1.7561

16 2.0645

32 2.3468

64 2.6271

128 2.9078

256 3.1887

n→∞ Pn → ∞

The increase of splashes amplitudes is accompanied by an increase of n (number of material points); however, for each of fixed n the energy conservation holds. Explanation of the mentioned phenomenon follows. For free oscillations low and high harmonics take part. The latter ones are either estimated by the continuous relations with high errors and they can not be described by the equations (9.5). Therefore, an important problem appears: How to construct theory of continuous medium catching the observed splashes? The theory is expected to give the solution harmonics with a relatively high accuracy. From the mathematical point of view we are going to approximate unlocal (difference) operator by the local (differential) one. In order to solve this problem the relation between eigenfrequencies of both discrete (9.2)–(9.4) and continual (9.5)–(9.7) systems is analysed for f (t) = 0. The discrete system possesses n + 1 eigenfrequencies governed by the formulas

9 CONTINUALIZATION

ωk = 2

kπ c sin , n = 1, 2, . . . , n + 1, m 2 (n + 1)

whereas a continuous system has the following discrete infinite spectrum c k αk = π , k = 1, 2, . . . . mn+1

261

(9.8)

(9.9)

The expressions (9.9) approximate frequencies of the discrete system (9.8) reasonably good for the first frequencies, whereas worse for a large values of k parameter. For instance, ωn+1 is defined with the error higher than 50% (instead of 2 we have π). Note that frequencies ω n+2 , ωn+3 , . . . do not have any relation to the input discrete system. They are parazite frequencies, and they should be omitted while analysing the discrete system (9.2)–(9.4). Consider now the boundary value problem (9.2)–(9.4) for f (t) = −1. A solution sought in the form x (9.10) σ = 1 − + u (x, t) , l where the function u(x, t) is defined by relations: m

2 ∂2 u 2∂ u = ch , ∂t2 ∂x2

(9.11)

u (0, t) = u (l, t) = 0, (9.12) x (9.13) u (x, 0) = −1 + ; ut (x, 0) = 0. l A solution to the boundary value problem (9.11)-(9.13) is found using the Fourier method: ∞ kπx −2l sin l u= cos (αk t), (9.14) π k=1 k and finally we get 2l sin −x +1− l π k=1 k ∞

σ=

kπx l

cos (αk t).

(9.15)

In order to get physically reliable results for discrete system one has to include the frequency series up to (n + 1) harmonics. The rest of infinite frequency series should be neglected, since they have no any relation to the motion of the mass chain. In words, the motion of the discrete system (9.2)–(9.4) can be approximated by the formula n+1 kπx 2l sin l −x +1− cos (αk t). (9.16) σ= l π k=1 k The numerical experiments confirm the expectation.

262

9 CONTINUALIZATION

Although the solution (9.16) qualitatively approximates the chain motion, but its quantitative accuracy is rather low. In words, the vibration modes in the neighbourhood of (n + 1) mass are badly approximated by (9.16). The system (9.5) for f (t) = 0 can be represented by the following pseudodifferential equation [467] ∂2 σ ih ∂ 2 m 2 + 4c sin − σ = 0. (9.17) 2 ∂x ∂t The following McLaurin series is applied h4 ∂4 h6 ∂6 ih ∂ h2 ∂2 + − + ... . sin2 − =− 2 ∂x 4 ∂x2 48 ∂x4 1440 ∂x6

(9.18)

Remaining only the first term in the series (9.18), the classical continuous approximation (9.5) is obtained. Taking into account three first terms the following higher order approximation is found 2 ∂ σ h4 ∂4 σ h6 ∂6 σ ∂2 σ − + (9.19) m 2 = ch2 360 ∂x6 ∂y ∂x2 12 ∂x4 with the following attached boundary conditions σ = σ xx = σ xxxx = 0

f or

x = 0, l.

(9.20)

A comparison of (n + 1)th frequency of the continuous system (9.19), (9.20) with the corresponding frequency of the discrete system shows that the accuracy is essentially increased (the numerical coefficient 2.11 instead of 2 in the exact solution appears, which gives the error of 5.5%). Therefore, the solution (9.16) can be used to describe the masses chain, free oscillations, where , c k π2 k 2 π4 k 4 αk = π 1− + . (9.21) Mn+1 12 (n + 1)2 360 (n + 1)4 The used error estimation of continuous approximation is obtained with respect to maximal frequency of the discrete system. Although it is rather artificially one, but seems to be the most simple. In a general case, a similar like model called in reference [296] the intermediate continuous model, can be found m

N (−1)k−1 h2k ∂2k σ ∂2 σ = 2 , (2k)! ∂x2k ∂t2 k=1

(9.22)

∂2k σ = 0 f or x = 0, l; k = 0, 1, ..., N − 1 (9.23) ∂x2k In order to get the boundary conditions (9.23) either the variational principle [296] or the following motivation can be applied. The equation (9.22) should be

9 CONTINUALIZATION

263

satisfied for any power h 2k , beginning with k = 0. Assuming k = 0, 1, . . . , N − 1, the boundary conditions from equation (9.22) are obtained. Intermediate continuous boundary value problem will be well-posed only if N is odd. Note that intermediate continuous models are used while applying the differential approximations to estimate the difference schemes accuracy [612] as well as in the momentous elasticity theories [410, 411]. Note that an application of intermediate continuous models yields the splash effect. A construction of intermediate continuous models relies on the development of a difference operator into the Taylor series. However, more effective results may be obtained using Padé approximation [257, 683, 684]. Let us construct continual models using one point Padé approximation (see Chapter 13) (the so called quasicontinuous approximation [683, 684]). The Padé approximation of the differential operator (9.18) can be associated with either Fourier or Laplace transformations. The approximation [2/2] and [4/2] have the forms 5 5 h2 ∂2 h2 ∂2 ∂2 h2 ∂2 ∂2 1 − and 1 + 1 − 12 ∂x2 20 ∂x2 ∂x2 30 ∂x2 ∂x2 which correspond to the following quasi-continuous models h2 ∂2 σtt − ch2 σ xx = 0; m 1− 12 ∂x2 h2 ∂2 h2 ∂2 2 − ch m 1− σ 1 + σ xx = 0. tt 30 ∂x2 20 ∂x2

(9.24) (9.25)

The boundary conditions for the equation (9.24) have the form (9.6), whereas for the equation (9.25) they read σ = σ xx = 0 for x = 0, l . The error in estimation of (n + 1)th frequency (with respect to discrete chain) achieves 16.5% (3%) for the equation (9.24) ((9.25)). The equation (9.25) has the lower order in comparison to the equation (9.19). Both of equations (9.24) and (9.25) exhibit the splash effects. Since for the high chain frequencies continualization procedure gives high errors, the so called ’envelope continualization’ is used [393]. In the latter case the following variables change is applied σk = (−1)k Ωk ,

(9.26)

and the following differential equations are found mΩ¨ k + c (4Ωk + Ωk−1 − 2Ωk + Ωk+1 ) = 0, Ω0 = 1,

Ωn+1 = 0,

(9.27) (9.28)

264

9 CONTINUALIZATION

Ωk = Ωkt = 0 for t = 0, k = 0, 1, ..., n + 1.

(9.29)

Next, a series development of the difference operator (9.18) is used and the following continuous equations for the envelope continualizations are yielded:

.

.

mΩ + c4Ω = 0,

(9.30)

mΩ¨ + c4Ω + cΩ xx = 0,

(9.31)

. .

.

.

. .

.

.

The obtained equations required attachment of both initial Ω = Ωt = 0 for t = 0 and boundary conditions. For example, for the equation (9.30) we have Ω = 1 for x = 0, Ω = 0 for x = l.

(9.32)

Observe that the whole interval of the discrete model frequencies is covered by two mutually complimentary approximations (9.5) and (9.30). The mentioned circumstances give serious rights to application od two-point Padé approximants. Two-points Padé approximation of the difference operator, taking into account the first term of the series (9.18) is applied. In the second interval it is required that the frequency of vibrations of continuous approximation should coincide with the frequency ω n+1 of the damped system. √ Requiring αn+1 = 2 c/m, the being sought operator reads 5 2 2 2 ∂ 2 2 ∂ ch 1−α h , ∂x2 ∂x2 where: α2 = 0.25 − π−2 . Hence a continuous approximation (called further as ‘pseudo-continuous’) is governed by the equation 2 2 2 ∂ (9.33) σtt − ch2 σ xx = 0 m 1−α h ∂x2 with the boundary conditions (9.6). The largest error in eigenfrequencies estimation is achieved for k = [0.5(n + 1)], and is of 3% amount. Observe that the equation (9.33) is of second order with respect to spatial coordinate, i.e. the essential increase of accuracy is not achieved in a way of increasing the order of differential operator. Note that the equation (9.33) governs the effect of splashes. Let us turn now to analysis of free oscillations. The system (9.2)–(9.4) for f (t) = −1 has the following exact solution

9 CONTINUALIZATION

σ j (t) =

n 1 πk πk j ctg sin cos ωk t, n + 1 k=1 n + 1 2 (n + 1)

j = 0, 1, . . . , n.

265

(9.34)

Summing up the first part of the series (9.34), we obtain [412] l πk jl πk j − ctg sin cos ωk t, n + 1 n + 1 k=1 n + 1 2 (n + 1) n

σj = 1 −

(9.35)

j = 0, 1, . . . , n + 1. We are going to compare (9.35) and (9.15). For small k (k ≤ n + 1) one has ctg

πk πk 2 (n + 1) − = − ..., 2 (n + 1) πk 6 (n + 1)

and taking into account the first term of this series, from (9.35) one obtains 2l jl πk j cos ωk t − , sin n + 1 π k=1 n+1 k n

σj ≈ 1 −

j = 0, 1, . . . , n + 1.

(9.36)

Comparing the expressions (9.36) and (9.15) in the points x = kl (k = 0, 1, . . . , n + 1) two sources of errors of the continuous approximations are detected. The first one is responsible for the error of frequencies estimations. The second one is related to the error of estimation of the series coefficients. In fact, although a solution is sought in the infinite Fourier series (9.14), only the initial series part (9.16) is used. However, this error can be omitted using the following procedure. A solution of the boundary value problem (9.11)–(9.13) is sought in the form u (x, t) =

n+1 k=1

Ak sin

kπx cos (ωkt ) . l

(9.37)

The collocation method is used to estimate the coefficients A k via satisfying the first of the initial conditions (9.13) in the points x = kl (k = 0, 1, . . . , n + 1). To sum up, a suitable approximation to the vibration frequencies of the chain of discrete masses is obtained in the whole considered interval. It has been achieved by the simultaneous continuous approximation for low- and high-vibration frequencies and two-points Padé approximants. Only one, governing whole spectrum equation is constructed. However, another approach should be applied while analysing forced oscillations. As it has been already mentioned, the error is introduced by uncorrection of frequencies estimation and by approximation of the coefficients series. To avoid the mentioned drawback one may apply the truncated Fourier series in combination with the collocation method, as it has been already described. In spite of this there are other potential possibilities to avoid the mentioned drawback. The Padé approximation may be used for the truncated Fourier series. The low frequency harmonics can be described using classical relations (continuous or quasi-continuous

266

9 CONTINUALIZATION

models), whereas the high frequency parts can be approximated using asymptotics (9.30), (9.31). A matching point of solution can be found from a condition of the same errors obtained using both mentioned methods. The problem has been fully described by Ulam [652] (p. 89, 90): “The simplest problems involving an actual infinity of particles in distributions of matter appear already in classical mechanics. A discussion of these will permit us to introduce more general schemes which may possibly be useful in future physical theories. Strictly speaking, one has to consider a true infinity in the distribution of matter in all problems of the physics of continua. In the classical treatment, as usually given in textbooks of hydro-dynamics and field theory, this is, however, not really essential, and in most theories serves merely as a convenient limiting model of finite systems enabling one to use the algorithms of the calculus. The usual introduction of the continuum leaves much to be discussed and examined critically. The derivation of the equations of motion for fluids, for example, runs somewhat as follows. One images a very large number N of particles, say with equal masses, constituting a net approximating the continuum which is to be studied. The forces between these particles are assumed to be given, and one writes Lagrange equations for the motion of N particles. The finite system of ordinary differential equations “becomes” in the limit N = ∞ one or several partial differential equations. The Newtonian laws of conservation of energy and momentum are seemingly correctly formulated for the limiting case of the continuum. There appears at once, however, at least one possible objection to the unrestricted validity of this formulation. For the very fact that the limiting equations imply tacitly the continuity and differentiability of the functions describing the motion of the continuum seems to impose various constraints on the possible motions of the approximating finite systems. Indeed, at any stage of the limiting process, it is quite conceivable for two neighbouring particles to be moving in opposite directions with a relative velocity which not need tend to zero as N becomes infinite, whereas the continuity imposed on the solution of the limiting continuum excludes such a situation. There are, therefore, constraints on the class of possible motions which are not explicitly recognized. This means that a viscosity or other type of constraints must be introduced initially, singling out the “smooth” motions from the totality of all possible ones. In some cases, therefore, the usual diefferntial equations of hydrodynamics may constitute a misleading description of the physical process.” The occurred situation may be improved within the frame of intermediate continuous model, application of quasi- or pseudo-continuous models.

10 HOMOGENIZATION

10.1 ODEs with rapidly oscillating coefficients In order to introduce to the problem we follow one dimensional example given in reference [154]: d x du a = q(x), (10.1) dx ε dx u = 0 for x = 0, l. (10.2) Observe that a(x/ξ) is a periodic function due to x with the period ε. The introduced equation can govern, for instance, a longitudinal deformations of a rod with rapidly changing thickness. Observe also that a change of the right hand side of equation (10.1) (external load) is small, whereas a change of the coefficient a(x/l) is large. Therefore, instead of introduction of one input variable x we call apply two variables: the “fast” one η = x/ε and the “slow” one y = x. The differential operator reads ∂ ∂ d = + ε−1 , (10.3) dx ∂y ∂η and instead of ordinary differential equation we get a partial differential equation. Its solution is going to be found as the following series u = u0 (η, y) + εu1 (η, y) + ...,

(10.4)

where: u0 , u1 , ... found are periodic functions due to η, u i (η + 1, y) = ui (η, y). On a basis of two (or multiple) scales method two-scales homogenization is developed [3, 4, 154, 590, 591]. Substituting (10.3), (10.4) in the input equation (10.1) and into the boundary conditions (10.2), equating the coefficients at powers of ε to zero, one obtains the following recurrent system of equations

∂ ∂u0 a(η) = 0, ∂η ∂η

∂u20 ∂ ∂ ∂u0 ∂u1 + a(η) + a(η) a(η) = 0, ∂η ∂y ∂y∂η ∂η ∂η

268

10 HOMOGENIZATION

∂u20 ∂ ∂ ∂u2 ∂u1 ∂2 u1 = q(y), a(η) + a(η) 2 + a(η) + a(η) ∂η ∂η ∂η ∂y ∂y∂η ∂y . . . . . . . . . .

(10.5)

for y = 0, 1; η = 0, l 1 , u j = 0 j = i, 1, 2, ..., where: l1 = l/ε. Because u0 is periodic with regard to η, the first equation of (10.4) yields u0 = u0 (y). It means, that u 0 represents a certain homogenized function which does not depend on the fast variable. In many physical cases an occurrence of an homogenized term is clear due their statements and therefore in the beginning one can assume that the first term of (10.5) does not depend on the fast variable. The second equation of the recurrent system (10.5) can be presented in the following way

∂ ∂u1 da du0 . (10.6) a(η) =− ∂η ∂η dη dy Since the equation (10.6) is considered on the period 0 ≤ η ≤ 1, it is referred as the local (or cell) problem. A construction of solution to a local problem belongs to the most tedious steps of an homogenization procedure. It is very often find numerically. For our case, y appears in (10.6) as a parameter and therefore we obtain du0 C(y) ∂u1 =− + . ∂η dy a

(10.7)

The constant C(y) can be found from a periodicity condition for u 1 , u1 |10 = 0: 1 −1 −1 o C = a du dy , a = [ a dη] . 0

Eliminating ∂u1 /∂η from the third equation of the system (10.5) we get

2 ∂ ∂u2 ∂ ∂u1 d u0 a + a + a 2 = q(y). ∂η ∂η ∂η ∂y dy Now, in the order to extract from this equation the slow components, we apply an homogenization operator 1 (. . .)dη (10.8) 0

to each of the equation term. The first two terms are equal to zero because the functions au 2y , au1y are periodic, and we finally obtain

a

d 2 u0 = q(y). ∂y2

(10.9)

The following boundary condition must be attached to equation (10.9) u0 = 0 for y = 0, l.

(10.10)

10.1 ODEs with rapidly oscillating coefficients

269

The function u 1 can be found from the condition (10.7)   η   du0   −1 u1 = a a dη − η , 0 ≤ η ≤ 1. dy 0

Then the function u 1 can be periodically extended along η with the period one. In general, the found value of u 1 does not satisfy the boundary conditions (10.2), but the errors are of ε order. In order to eliminate them we need to solve the following problem d x du a = 0, dx ε dx , u| = B ≡ u | . u| = A ≡ u 1 y=η=0

x=0

1 y=l,η=l1

x=l

We apply again here the homogenization and we get

a

d 2 u10 (y) = 0, dy2

u10 |y=0 = A,

u10 |y=l = B.

The obtained result gives an idea to look for the following solution u = u0 (y) + ε[u10 (y) + εu20 (y) . . .] + ε[u1 (η, y) + εu2 (η, y) . . .].

(10.11)

Now let us consider the following model example governing by the non-linear equation x d x du (10.12) a +b u3 = q, dx ε dx ε u = 0 for

x = 0, l.

(10.13)

Introducing again fast (η) and slow (y) variables and looking for a function u in the form (10.4), we obtain the following recurrent equations

∂ ∂u1 da du0 = 0, (10.14) a(η) + ∂η ∂η dη dy

∂ d 2 u0 ∂u2 ∂ ∂u1 ∂2 u1 + a(η) 2 + b(η)u30 = q(y), (10.15) a(η) + a(η) + a(η) ∂η ∂η ∂η ∂y ∂y∂η dy . . . . . . . . . . u0 = 0 for y = 0, l; u1 = 0 for y = 0, l, .

.

. .

.

η = 0, l 1 , i = 1, 2, . . . . .

. .

.

(10.16)

.

The equation (10.14) overlaps with the corresponding equation (10.5). It means that the “local problem” is not changed due to an introduction of new terms if there

270

10 HOMOGENIZATION

is no change of higher derivatives. Taking into account (10.7) the following homogenized equation is obtained d 2 u0 a 2 + bu3 = q(y), dy

1

b=

b(η)dη.

(10.17)

0

The boundary conditions for equation (10.17) have the form (10.16). Observe that u = u0 + O(ε), but

du du0 ∂u1 = + + O(ε). dx dy ∂η

In the other words, although a solution u 0 of the homogenized equation approximates the function u with an accuracy of ε order, one needs to take into account in the expression of differentiation the terms with u 1 , because their influence strongly increases due to the differentiation. Obviously, their occurrence leads to complications during the computations. Let us comment on physical meaning of the coefficients of the homogenized equation (10.17). It is seen that both the stiffness b and compliance 1/a are homogenized. It corresponds to the known physical rule, that the additive functions must be homogenized. We use one more analogy. The electrical resistance of paralelly joined resisn tances is calculated using the formula R −1 = R−1 i , whereas “in series” joined are calculated by R =

n i=1

i=1

Ri . Therefore, a “direct” homogenization corresponds to the

series joints, whereas the invert quantities are computed using the parallel joined elements. The direct homogenization is called also Voight homogenization [220], whereas the homogenization using the inverse quantities is referred as the Reuss homogenization [220]. It is known, that for a rather wide class of problems the true values of the coefficients of the homogenized equations (ã i j ) lie between those ho

mogenized by Voight (¯ai j ) and by Reuss (ai j ) :

ai j ≤ a˜ i j ≤ ¯ai j .

(10.18)

The estimation (10.18) is called the “Hill’s pitchfork” [220]. Unfortunately, its width is very often too large to be easy applied in practice. In what follows some more general questions of homogenization theory will be addressed (see [123, 124, 159]). It happens often that a space of periodicity exhibits a certain symmetry, which usually yields to more easier determination of the effective characteristics owing to the following circumstances: (i) Some of the characteristics become automatically equal to zero. For instance, in some conditions an isotropic matrix with isotropic cells can be substituted by an isotropic material with reduced (or artificial) characteristics.

10.1 ODEs with rapidly oscillating coefficients

271

(ii) A solution of cell problems can be reduced to solution of problems with a smaller dimension. Furthermore, recall that the periodical problems are investigated. However, for example, in theory of composites a key problem occurs while analysing stochastically distributed cells. It occurs rather that a strictly periodic cells distribution belongs rather to be rarely met in nature. Assume that a matrix characteristics is represented by 1, whereas a cell is characterized by δ. Observe that a sign of increase of averaged characteristics associated with small stochastic excitations is the same as that of δ − 1 [160, 399]. Let us consider now the following eigenvalue problem d x du a + λu = 0. (10.19) dx ε dx Assume that the being sought eigenfunctions has the form (10.11) and the eigenvalue λ is sought in the form λ = λ0 + ελ1 + . . . .

(10.20)

After introduction of (10.11) and (10.20) into the input equation and the boundary conditions (10.13) and taking into account (10.3) the following recurrent equations are obtained ∂ ∂a du0 ∂u1 + a = 0, (10.21) ∂η dy ∂η ∂η ∂ ∂u2 ∂ ∂u1 a + a + ∂η ∂η ∂η ∂y a

d 2 u0 ∂2 u1 da du10 + + a 2 + λ0 u0 = 0, ∂η∂y dη dy dy

∂ ∂2 u2 ∂u3 ∂ ∂u2 da du20 +a + a + a + ∂η ∂η ∂η ∂y dη dy ∂η∂y λ1 u0 + λ0 (u10 + u1 ) = 0, .

.

. .

.

.

. .

.

(10.22)

(10.23)

.

u0 = 0 for y = 0, l,

(10.24)

u1 + u10 = 0 for y = 0, l; η = 0, l 1 ,

(10.25)

.

.

. .

.

.

. .

.

.

Substituting to (10.22) and (10.24) the being found u 1 from (10.21) and applying to the following boundary value problem homogenization operator (10.8), one obtains

272

10 HOMOGENIZATION

d 2 u0 + λ0 u0 = 0, dy2 u0 = 0 for y = 0, l, aˆ

which yields u0 and λ0 . From (10.22) one finds: ∂u1 du10 C1 (y) ∂u2 =− − + . ∂η ∂y dy a The periodicity condition for the function u 2 with regard to η yields

C1 = a

du10 +a dy

1 0

∂u1 dy. ∂y

Substituting the found values of u 1 and u2 into equation (10.23) and applying homogenization operator (10.8), we get

a

2 d2 u10 d u1 + λ u + a + λ0 u1 + λ1 u0 = 0. 0 10 dy2 dy2

(10.26)

From the above equation λ 1 can be found. Let the boundary condition for the equation (10.26) have the form (10.24). A slow corrector of the homogenized solution u 10 can be found from a solution to the boundary value problem (10.26) and (10.25). The described approach can be used for arbitrary terms of ε powers. In addition, it has a very important general property. Indeed, if a solution of the local problem can be found, than without any problems one can find a solution to the eigenvalue problem. If we add the non-linear terms in a way not to change the higher derivatives, than a homogenization procedure again becomes relatively simple. The local problem remains exact the same as for the linear case. Also the higher order approximations will be linear. The non-linearity will be taking into account in the homogenized boundary value problems with smooth coefficients. Especially, in a case of theory of shells it means that solving one of the local problems, we in fact solve a class of problems linear and non-linear, as well as statical or dynamical. Observe that the local problems of the shell theory usually can be analyzed in a frame of plates theory.

10.2 Axisymmetric bending of corrugated circle plate The corrugated plates and shells are typical examples representing a construction with periodically repeated geometry. As an example we consider a relatively simple and well studied problem on axially symmetrical deformation of a circle corrugated plate under the normal load q. One of the peculiarities related to the homogenization

10.2 Axisymmetric bending of corrugated circle plate

273

procedure applied to corrugated plates requires reformulation of the all quantities in relation to the middle plane (and not middle surface). The equilibrium equations being the projection into the middle plane have the form d (rN1 ) − N2 = rq1 , dr d (rQ1 ) = rq2 , dr d (10.27) (rM1 ) − M2 − rQ1 = 0, dr where: r is the polar coordinate, M i are the bending moments, N i , Q1 , qi are the membrane projections of membrane and transverse shear force forces into coordinates at the middle surface. The projected physical and geometrical relation have the forms Eh N1 + βQ1 = (ε1 + νε2 ), A 1 − ν2 N2 =

Eh A(ε2 + νε1 ), 1 − ν2

M1 − zN1 = D(χ1 + νχ2 ), dw 1 du u +β M2 − zN2 = D(χ2 + νχ1 ), ε1 = 2 , ε2 = , dr r A dr du du 1 d 1 dw 1 dw χ1 = −β −β , χ2 = 2 , A dr A2 dr dr dr rA dr

(10.28)

2 1/2 ; z(r) characterizes a corrugation geometry; u and w where: β = dz dr ; A = (1 + β ) are the radial and normal displacements related to middle surface, ε 1 , ε2 , χ1 , χ2 are the deformations and curvatures changes of the middle surfaces, correspondingly. For the clamped plate contour we have

u=w=

dw = 0 for r = r0 . dr

(10.29)

Let n be the number of a corrugation, which is sufficiently large. Then the homogenization procedure is applicable, and one can take as the perturbation parameter ε = n−1 . Let us introduce the ‘fast’ variable ξ = ε −1 r, whereas a ‘slow’ one will be denoted by r. Then ∂ ∂ d = + ε−1 . (10.30) dr ∂r ∂ξ A solution to the boundary value problem (10.27)–(10.29) has the form Ni = Ni(0) (r, ξ) + εNi(1) (r, ξ) + . . . ,

i = 1, 2;

274

10 HOMOGENIZATION

Mi = Mi(0) (r, ξ) + εMi(1) (r, ξ) + . . . ,

i = 1, 2;

(1) Q1 = Q(0) 1 (r, ξ) + εQ1 (r, ξ) + . . . ,

u = u(0) (r, ξ) + εu(1) (r, ξ) + . . . , w = w(0) (r, ξ) + εw(1) (r, ξ) + . . . .

(10.31)

Observe that all the series coefficients are periodic because of ξ with the period r0 . Substituting (10.30) and (10.31) into the input equations and the boundary conditions (10.27)–(10.29), the following recurrent systems of equations are obtained (we neglect the terms ν 2 /A 1) ∂N1(0) = 0, ∂ξ

∂Q(0) 1 = 0, ∂ξ

∂w(0) = 0, ∂ξ r

∂M1(0) = 0, ∂ξ

∂u(0) = 0, ∂ξ

(10.32)

∂N1(0) ∂ + (rN1(0) ) − N2(0) = rq1 , ∂ξ ∂r ∂Q(1) ∂ r 1 + (rQ(0) 1 ) = rq2 , ∂ξ ∂r

r

∂M1(1) ∂ + (rM1(0) ) − M2(0) − Q(0) 1 = 0, ∂ξ ∂r

(10.33)

(0)

N ∂u(1) ∂u(0) + = k(ξ) 1 , ∂ξ ∂r Eh M1(0) ∂2 w(1) ∂2 w(1) + = A(ξ) , D1 ∂ξ2 ∂r2 N2(0) = Eh

A(ξ) (0) u , r

M2(0) = D1 k(ξ)

∂w(0) , ∂r

A ∂w(1) N2(1) = Eh u(1) , M2(1) = D1 k , r ∂r . . . . . . . . . . u(0) = w(0) =

∂w(1) = 0 for ξ = ε−1 r0 , ∂ξ . . . . . . . .

u(1) = w(1) = . 2

.

, D1 = where: k(ξ) = A−1 + 12z h2 The relations (10.32) yield

∂w(0) = 0 for r = r0 , ∂r

Eh3 12 .

(10.34) (10.35)

(10.36) (10.37)

10.2 Axisymmetric bending of corrugated circle plate

N1(0) ≡ N1(0) (r),

(0) Q(0) 1 ≡ Q1 (r),

u(0) ≡ u(0) (r),

275

M1(0) ≡ M1(0) (r),

w(0) ≡ w(0) (r).

A solvability conditions of the equation (10.33), (10.34) in relations to period(1) (1) (1) ically unknown functions N 1(1) , Q(1) give together with (10.35) the 1 , M1 , u , w following homogenized equations d (rN1(0) ) − m(N2(0) ) = rm(q1 ), dr

d (rQ(0) 1 ) = rm(q2 ), dr

d (rM1(0) ) − m(M2(0) ) − Q(0) 1 = 0, dr N1(0) =

Eh du(0) , m(k) dr

N2(0) = Eh

m(A) (0) u , r

u(0) = w(0) = where: m(. . .) =

1 r0

r0

M1(0) =

D1 d2 w(0) , m(A) dr2

M2(0) = D1 m(k)

dw(0) dr

(10.38)

dw(0) , dr

for r = r0 ,

(. . .)dξ.

0

The obtained equations are the same as the earlier obtained equations derived on a basis of physical assumptions [19]. The equations (10.33)–(10.35) yield also the rapidly oscillating correctors. From (10.33) and (10.34) and taking into account (10.35) we get ∂N (1) r 1 = r q1 − m(q1 ) + N2(0) − m N2(0) , ∂ξ ∂M1(1) 1 (0) = M2 − m(M2(0) ) , ∂ξ r N1(0) ∂u(1) = [k − m(k)] , ∂ξ Eh

∂Q(1) 1 = q2 − m(q2 ), ∂ξ

M1(0) ∂2 w(1) = [A − m(A)] , D1 ∂ξ2

M2(1) = D1 [k − m(k)]

∂w(1) , ∂ξ

u N2(1) = EhA . r

(10.39)

However, rapidly oscillating correctors do not satisfy (in general) the boundary conditions (10.29). The obtained small discrepancies (of the ε order) are compensated by a solution to the homogenized equations of the form (10.38). In result one will get slow and small corrections to the homogenized solution. When the projections are found then the displacement u r , wn , forces Nr , Nθ and the moments Mr , Mθ can be found using the formals ur =

u + βw , A

wr =

w + βu , A

Nr =

N1 + βQ1 , A

276

10 HOMOGENIZATION

N2 , Mr = M1 − zN1 , Mθ = M2 − zN2 . A Observe that the stress-strain state can not be fully described using only the homogenized relations because the rapidly oscillating correctors introduce the important part while calculating the circled tangential and bending stresses. An influence of the rapidly oscillating correctors leads to very reliable estimation of the stress-strain state for small values of n. For example, in Figure 10.1 a comparison among different approaches to compute the bending moment are compared for n = 4. The circled corrugated plate clamped along the external contour with a rigid cylindrical body with the radius r 00 in its has been analysed. We suppose z = H sin[n(r − r00 )], M ∗ = Mθ /(πr02 q), r1 = r/r0 . The following parameters have been fixed: r 0 = 28.3 mm, r00 = 1.9 mm, h = 0.22 mm, H = 0.75 mm, E = 105 N/mm2 , q = 0.01 N/mm2 , ν = 0.33. Nθ =

Fig. 10.1. Bending moment in the circled corrugated plate.

The numerical solution obtained in [168] is denoted by curve 1; the curve 2 corresponds to the solution obtained using the homogenized and the curve 3 corresponds to a solution with first rapidly oscillating correctors, which already for the first approximation order well coincide with the numerical values.

10.3 Deformation of reinforced membrane Let us consider homogenization method applied to find a solution to the differential equations with periodically discontinuous coefficients. As the representive example we consider a deflection of a membrane strengthed by the fibres. In the intervals kl < y1 < (k + 1)l, k = 0, ±1, ±2, . . . the following equilibrium equations hold

10.3 Deformation of reinforced membrane

∂2 u1 ∂2 u1 + = q1 (x1 , y1 ). ∂x21 ∂y21

277

(10.40)

The conjugation conditions have the form lim ≡ u+ ≡ lim ≡ u− ,

y1 →kl+0

∂u ∂y1

y1 →kl−0

+

−

∂u ∂y1

− = g1

∂2 u , ∂x21

(10.41)

where: g1 is the parameter characterizing the relative fiber stiffness. The boundary conditions have form u = 0 for

x1 = 0, H.

(10.42)

Assume that the external load is periodic due to y 1 , and its period L is essentially larger then a distance between the fibres. Then the following small parameter can be taken: ε = l/L. Instead of y 1 we introduce the fast variable, η = y 1 /l, and the slow one, y = y1 /L. Therefore, the following relation holds ∂ 1 ∂ −1 ∂ +ε = . (10.43) ∂y1 L ∂y ∂η We also introduce dimensionless variable x = x 1 /L. The function u can be expressed by the series u = u0 (x, y) + εα1 u10 (x, y) + u1 (x, y, η) + εα2 u20 (x, y) + u2 (x, y, η) + . . . , 0 < α1 < α2 < . . . .

(10.44)

Introducing (10.44) into (10.40), (10.41) and taking into account (10.43), we get 2 2 ∂2 u0 ∂2 u0 α1 −2 ∂ u1 α1 −1 ∂ u1 + + + ε + 2ε ∂y∂η ∂x2 ∂y2 ∂η2

∂2 u2 ∂2 u2 + O(εα1 ) = q(x, y), + 2εα2 −1 2 ∂y∂η ∂η u0 + εα1 (u10 + u1 ) + . . . + = u0 + εα1 (u10 + u1 ) + . . . − , 2

+ − ∂u1 ∂ u0 ∂u1 α1 − + O(ε ) , εα1 −1 + O(εα1 ) = g ∂η ∂η ∂x2 εα2 −2

(10.45) (10.46)

where: q = L2 q1 , g = g1 /L. A character of the being constructed asymptotics essentially depends on the relative fibre stiffness g in comparison to ε. Let us introduce the parameter β characterizing this relation (g ∼ ε β ) and let us discuss the possible limiting cases with regard to α1 and β.

278

10 HOMOGENIZATION

The equation (10.45) yields the following limiting cases: 0 < α 1 < 2, α1 = 2 and α1 > 2: for 0 < α1 < 2 for α1 = 2 for α1 > 2

∂2 u1 = 0, ∂η2 ∂2 u0 ∂2 u0 ∂2 u1 2 2 + + = q, ∂x ∂y2 ∂η2 ∂2 u0 ∂2 u0 2 2 + = q. ∂x ∂y2

(10.47) (10.48) (10.49)

The limiting relations obtained from (10.42), (10.46) for ε → 0 have the forms: for β < α1 − 1 for β = α1 − 1 for β > α1 − 1

∂2 u0 = 0, ∂x2 + − ∂u1 ∂u1 ∂2 u0 − = gε1−α1 2 , ∂η ∂η ∂x + − ∂u1 ∂u1 = . ∂η ∂η

(10.50) (10.51) (10.52)

Fig. 10.2. Triangulation of the (α1 , β) plane characterising the different asymptotics.

The plane quadrant β > 0, α 1 > 0 is divided into nine subspaces (Figure 10.2). We are going to analyse them in more details. Let us consider the case β < α 1 < 1 (zone 1-3). It means physically that one deals with the stiff fibres. In this case

10.3 Deformation of reinforced membrane

279

u0 = 0, and the averaged description can not be applied. In this case, in the first approximation, each of the parts where threads create a zone should be analysed separately; on the lines y = kl, u = 0, ±1, ±2, . . ., the clamping conditions should be constructed, u = 0. The non-homogeneity introduced by the fibers is small and the limiting equation has the form (10.50). For zones 7 and 9 one obtains contradictory systems. A special attention is paid to zone 8 (α 1 = 2, β = 1), where the fibres with “mean” stiffness are represented. The limiting system includes the equations (10.48) and (10.51), and the conjugation conditions have the form u+ = u− , + − ∂u1 ∂u1 ∂2 u0 − = gε−1 2 . ∂η ∂η ∂x

(10.53) (10.54)

The equation (10.48) yields u1 = 0.5 q − ∇2 u0 η2 + C(x, y)η + C1 (x, y), 2

2

∂ ∂ where: ∇2 = ∂x 2 + ∂y2 . The constant C 1 (x, y) is related to the component u 10 which is found from the successive homogenized equations. From the conditions (10.54) one obtains

C(x, y) = −0.5(q − ∇2 u0 )L.

(10.55)

It seems to be strange that we need in addition to satisfy the condition (10.54), and we do not have any arbitrary constants more. However, observe that the condition (10.54) already yields the homogenized equation. Indeed, substituting the found u1 into (10.54), we get g1 ∂2 u0 ∇2 u0 + = q. (10.56) l ∂x2 The equations must be integrated for the following boundary conditions u0 = 0 for

x = 0, l1 ,

(10.57)

where: l1 = H/L. A transition to the equation (10.56) corresponds physically that rib rigidics are uniformly distributed over the strip surface (a transition to the structurallyorthotropic theory). The being sought function has u 1 the form u1 =

g1 ∂2 u0 η(η − l). 2l ∂x2

However, the boundary conditions for x = 0, H in general are not satisfied. The boundary discrepancies is rapidly oscillating and leads to an occurrence of the boundary layer u b . To construct the boundary layer we introduce the fast variable ξ = x1 /l, then

280

10 HOMOGENIZATION

∂ ∂ 1 ∂ + ε−1 = , ∂x1 L ∂x ∂ξ

(10.58)

and the series ub = εγ1 ub1 (x, y, ξ, η) + εγ2 ub2 (x, y, ξ, η) + . . . , 0 < γ1 < γ2 < . . . .

(10.59)

Substituting (10.58), (10.59) into input equations (10.40), (10.41) the following equations define u b1 : ∂2 ub1 ∂2 ub1 + = 0, ∂ξ2 ∂η2 ub1 |η=kl = 0, k = 0, ±1, ... .

(10.60) (10.61)

We consider the boundary conditions only for edge x = ξ = 0 (examination for the second edge is analogous): ub1 = −u1

for

x = ξ = 0.

(10.62)

For H 2l we also suppose ub1 → 0 for ξ → ∞.

(10.63)

For a practical construction of the boundary layer we may apply the Kantorovich method, where (10.64) ub1 = Φ(ξ)η(η − l), and it satisfies the boundary conditions (10.61). Let us apply now the standard Kantorovich procedure [369]. In words, the expression (10.64) satisfying the boundary conditions on the edges η = 0.1 (10.61) is substituted into equation (10.60). Then the result is multiplied by the function η(η − 1) and integrated with respect to η from 0 to 1. In result, the following ordinary differential equation is obtained: d2 Φ − 10Φ = 0. dξ2

(10.65)

A solution of the equation (10.65), satisfying the boundary condition (10.63), has the form √ Φ (ξ) = C1 exp − 10ξ . When on a zone boundary slowly variated corrections occur, they can be compensated by homogenized equalizations of the higher orders.

10.4 Ribbed strip – two-scale and Fourier homogenization We consider deformation of a ribbed strip (0 ≤ X ≤ H, −∞ < y < ∞). A rib is modelled as one-dimensional element having concentrated bending stiffness E 1 I

10.4 Ribbed strip – two-scale and Fourier homogenization

281

and located symmetrically in relation to middle plate surface. We consider the ribs with equal geometrical and stiffness characteristics and located in equidistances between each other. Between two neighbourhood ribs the equilibrium equation has the form (10.66) D∇4 w = q. The following conjugation conditions will be used + − 2 + 2 − ∂w ∂w ∂w ∂ w + − = , = , w =w , ∂y ∂y ∂y2 ∂y2 3 + 3 − ∂ w ∂w E1 I ∂4 w − = . D ∂x4 y=kl ∂y3 ∂y3

(10.67) (10.68)

Here (. . .)− = lim (. . .); (. . .)+ = lim (. . .). y→kl−0

y→kl+0

The conditions (10.67) govern the continuity of the displacements, rotation angles and moments. The condition (10.68) describes a jump in the transverse shear plate forces. Assume that plate is clamped along the edges w=

∂w = 0 for ∂x

x = 0, H.

(10.69)

Let the characteristic period L of the external load is more larger that the distance between ribs (L l). Then we can introduce the small parameter ε = l/L and the fast variable η = y/ε (the slow variable is denoted by y), and we get ∂ ∂ ∂ = + ε−1 . ∂y ∂y ∂η

(10.70)

The displacement w has the form w = w0 (x, y) + εα1 w10 (x, y) + εα2 w20 (x, y) + . . . + εβ1 w1 (x, y, η) + εβ2 w2 (x, y, η) + . . . ,

(10.71)

where: wio denote slow correctors to the homogenized solution, w I are the periodic functions because of η with the period L; α 1 < α2 < ...; β1 < β2 < ...; αi , βi are the parameters found during the asymptotic process (see Chapter 3.3). In our case α1 = β1 = 4. Substituting (10.71) and (10.70) into the governing equation (10.66) and into the boundary conditions (10.69), we get ∂4 w1 + ∇4 w0 = q1 , ∂η4 . . . . . . . . . . ∂w0 w0 = 0, = 0 for x = 0, H, ∂x

(10.72)

(10.73)

282

10 HOMOGENIZATION

w1 + w10 = 0, .

.

∂w1 ∂w10 + = 0 for ∂x ∂x . . . . . . . .

x = 0, H,

(10.74)

where: q1 = q/D. The equation (10.72) yields w 1 : w1 = (q1 − ∇4 w0 )

η4 + C1 η3 + C2 η2 + C3 η + C4 . 24

The constants C i (x, y) are obtained from the conditions (10.67), (10.68) which may be rewritten as follows ∂w1 ∂2 w1 ∂w1 ∂2 w1 ; ; (10.75) w1 ; = w1 ; , ∂η ∂η2 η=0 ∂η ∂η2 η=L ∂3 w1 ∂3 w1 ∂4 w0 − = Lp , ∂η3 η=L ∂η3 η=0 ∂x4

(10.76)

where: p = E 1 I/Dl. A further construction of the asymptotics depends on the relative ribs stiffness p. We consider only ribs with mean stiffness (p ∼ 1). From (10.76) one obtains ∇4 w0 + p

∂4 w0 = q. ∂x4

(10.77)

This is equation of the structurally-orthotropic theory. The attached boundary conditions have the form (10.73). The first correctors has the form w1 = −Lp

∂4 w0 2 η (η − L)2 . ∂x4

(10.78)

The found displacement w 1 does not satisfy (generally) the boundary conditions on the edges x = 0, H, and one needs to construct a boundary layer. Let us introduce fast variable ξ = x/ε (x denotes the slow variable). Then ∂ ∂ ∂ = + ε−1 . ∂x ∂x ∂ξ

(10.79)

The function w b (x, y, ξ, η) is sought in the form wb = εγ1 wb1 + εγ2 wb2 + . . . ,

(10.80)

γ1 < γ2 < ..., γ1 are chosen in a way to satisfy the given boundary conditions. In our case we have γ 1 = 4. Substituting (10.79), (10.80) in the input equation (10.66) and into the conditions (10.67), (10.68), we get

10.4 Ribbed strip – two-scale and Fourier homogenization

283

∂4 wb1 ∂4 wb1 ∂4 wb1 +2 2 2 + = 0, 4 ∂ξ ∂ξ ∂η ∂η4

(10.81)

∂wb1 = 0 for y = 0, L. ∂η

(10.82)

wb1 =

The boundary conditions in relation to ξ for H l can be presented in the form (for ξ = 0, for ξ = ε −1 H boundary conditions may by obtained analogously): w1 + wb1 = 0, wb1 → 0,

∂wb1 = 0 for ξ = 0, ∂ξ ∂wb1 → 0 for ξ → ∞. ∂ξ

(10.83) (10.84)

In the order to solve the boundary value problem (10.81)–(10.84), the Kantorovich method can be used. The function w b1 can be assumed in the form 2 wb1 = w(ξ)η ˜ (η − L)2 ,

(10.85)

and it satisfies the boundary conditions for η = 0, L. Further, a standard Kantorovich approach is applied. The slow components w i0 appearing in (10.71) should compensate the slowly discrepancies in the boundary conditions as well as in the equations of successive approximations. The obtained solution (10.78), (10.85) give possibility to estimate of an accuracy of the structurally-orthotropic theory. The following estimations hold w1 ∼ m4 ε4 w0 , (1) (0) M12 ∼ m4 ε3 M12 ,

M1(1) ∼ m4 ε4 M1(0) , M2(1) ∼ m4 ε2 M2(0) ,

(10.86)

where: M1 , M2 , M12 are the bending moments along the axes x, y and twisting moment, correspondingly; m is the parameter, characterizing a change of the structurally-orthotropic solution along x axis. To conclude, only M 2 can be corrected whereas other mentioned quantities are satisfactory approximated using the structurally-orthotropic theory. A comparison of the obtained results with the known numerical solutions [11, 12, 13, 14] indicates satisfactory estimation given by the first order approximation governed by (10.78), (10.85). Let us describe another way of verification obtained results. Let us consider a simple supported strip under constant normal pressure q. In this case exact solution may be constructed [11]:

jπx S 0 j (y) 2qL4 1 4 sin I j , (10.87) w= 1 − E 1 L Db + E 1 I jS 0 j (0) Dπ5 j=1,3,5,... j5 where:

284

10 HOMOGENIZATION

∞ cos πky 1 b S 0 j (y) = 4 + 2 ' 2 ( . −2 k2 2 j + ε j k=1 The expansion of the function S 0 j into a series using ε leads to the expression w=

2qbL41 π5 (E1 I + Db)

qE 1 I 12πD (E1 I + Db)

j=1,3,5,...

πx 1 sin j − L j5

πx 1 sin j y2 (y − b)2 + ... . j L j=1,3,5,...

The first term of this expression corresponds to the structurally-orthotropic theory solution w 0 , while the second to the component w 1 . Because the function (w0 + w1 ) satisfies the given boundary conditions, the boundary-layer-type component in solution is absent. Thus, the obtained asymptotic solution corresponds to the first terms of the expansion of the exact solution (when it can be contracted) into a series using ε. Would emphasize the main advantage of the described approach. The constructed improved solutions are of the same order complexity as those obtained from the structurally-orthotropic theory. The later ones can be found either analytically or numerically and the influence of ribs discreteness can be described via the simple analytical relations. The last example can be also treated as Fourier homogenization [234, 235]. It is worth noticing that in this example the exact solution is obtained. Owing to typical application of the Fourier method, the infinite system of linear algebraic equations is obtained. A separation from this system equations, governing slow champing solution, characterizes main feature of the Fourier homogenization approach.

10.5 Ribbed plate – direct homogenization We are going to present the new approach using the example of flexural oscillations of the rectangular plate (0 ≤ x ≤ L 1 , −L2 ≤ y ≤ L2 ), supported by N = 2k + 1 ribs, regularly and symmetrically in relation to the middle plate surfaces distributed. The governing equation can be formulated in the following way: D∇4 w + E1 Iφ(y)w xxxx + c1 w + [ρ0 h + ρ1 Fφ(y)]wtt = 0, where:

0.5(N−1)

δ(y − ib),

(10.88)

b = 2L2 /(N + 1),

i=−0.5(N−1)

where: E 1 is rib material Young modulus; ρ 1 is rib density; F is area of rib cross function; I 1 is statical moment of rib cross-section. Without loss of generality we take

10.5 Ribbed plate – direct homogenization

for y = ±L2 or w = wyy = 0; for x = 0, L1

285

w = wy = 0 ,

(10.89)

w = wxx = 0 orw = w xx = 0 .

(10.90) (10.91)

The conjugation conditions of the plate and rib have the form w+ = w− = w,

w+y = w−y ,

w+yy = w−yy .

−D(w+yy − w−yy ) = E 1 Iw xxxx + ρ1 Fwtt , −

(10.92)

+

where: (. . .) = lim(. . .), (. . .) = lim(. . .). y→ib−0

y→ib+0

Let us consider natural oscillations

w = w(x, y) exp(iωt).

(10.93)

Substituting (10.93) into (10.88), we transform the later one into the following non-dimensional form ∇4 w + cw + aφ(ϕ)wξξξξ − λ[1 − ρφ(ϕ)]w = 0, where: ∇4 = η1 = λ=

(10.94)

∂4 ∂4 ∂4 x + 2 + , ξ= , 4 2 4 2 ∂ξ 2L2 ∂ξ ∂η1 ∂η1

16c1 L42 E1 I y y , a= , ϕ= , , c= 2L2 D Db b

0.5(N−1) 16ω2 ρ0 hL42 ρ1 F , ρ= , φ(ϕ) = δ(ϕ − i). D ρ0 hb i=−0.5(N−1)

The conjugation conditions (10.92) in new variables read 6 . 6 . w+ ; w+η1 ; w+η1 η1 = w− ; w−η1 ; w−η1 η1 , −w+η1 η1 η1 + w−η1 η1 η1 = ε(awξξξ − λρw). where:

(10.95)

(...)± = lim (...), ε = b/(2L2 ), k = 0, ±1, ±2... . η→k+0

We assume ε 1 and begin the asymptotic analysis of the equation (10.94). Let us estimate the relations between the system parameters. The parameter ρ does not have any essential influence on the asymptotic and we take ρ ∼ 1. The estimation α = ε−1 is used for the near ribs stiffness. Physically it means that the assummed stiffness of the each rib has the same order as the stiffness of the a skin. Applying the two scales method we introduce two variables: the slow one y = y 1 and the fast one y/b = η 1 /ε = ϕ instead of the η1 . Then the differentiation for η 1 has the form

286

10 HOMOGENIZATION

∂ ∂ ∂ + ε−1 . = ∂η1 ∂η ∂ϕ

(10.96)

The normal displacement w and λ can be presented in the form: w = w0 (ξ, η, ϕ) + εw1 (ξ, η, ϕ) + ...,

(10.97)

λ = ε−1 λ0 + λ1 + ελ2 + ... .

(10.98)

Substituting expressions (10.97), (10.91) into (10.94) and taking into account the formula (10.96), after splitting due to ε we get the recurrent system of equations. It is seen that w0 = w00 (ξ, η) and the series (10.97) can be presented in the more suitable form w = w00 (ξ, η) + εw01 (ξ, η) + ... + ε3 w1 (ξ, η, ϕ) + εw2 (ξ, η, ϕ) + ... ,

(10.99)

where: wi are the periodic functions with regard to φ with period 1. Finally we obtain the following successive recurrent equations: w1ϕϕϕϕ + Π0 w00 = 0, w2ϕϕϕϕ + Π0 w01 − λ1 (1 + ρφ)w00 = Π1 w1 − ∇4 w00 , w3ϕϕϕϕ + Π0 w02 − (1 + ρφ)(λ1 w01 + λ2 w00 ) = Π1 w2 + Π2 w1 − ∇4 w01 , w4ϕϕϕϕ + Π0 w03 − (1 + ρφ)(λ1 w02 + λ2 w01 + λ3 w00 + λ0 w1 ) = Π1 w3 + Π2 w2 + Π3 w1 − ∇4 w02 ,   4    w5ϕϕϕϕ + Π0 w04 − (1 + ρφ)  λi w04−i + λ0 w2 + λ1 w1  = i=1

3

Πi w5−i − ∇4 (w03 + w1 ),

i=1

  k−1 k−4   wkϕϕϕϕ + Π0 wok−1 − (1 + ρφ)  λi w0k−1−i + λi wk−3−i  = i=1

3

i=0

Πi wk−i − ∇4 (w0k−2 + wk−4 ),

i=1

where: Π0 = c + aφ

∂4 ∂4 − λ (1 + ρφ), Π = −4 , 0 1 ∂ξ4 ∂ϕ3 ∂η

Π2 = −6

∂4 ∂4 − 2 2 2 , Π3 = Π31 + Π30 , 2 2 ∂ϕ ∂η ∂ϕ ∂ξ

Π31 = −4

∂4 4∂4 ∂4 − , Π = −c − aφ . 30 ∂ϕ∂η∂ξ2 ∂ξ∂η3 ∂ξ4

(10.100)


287

The boundary conditions (10.89)-(10.91) have now the following form for ξ = 0, l w0i = −wi−2 , w0iξξ = −w(i−2)ξξ , or w0i = −wi−2 , w0i2 = −w(i−2)ξ , for η = ±0.5w0i = −wi−2 , w0iη = −w(i−1)ϕ − wi−2 ,

(10.101) (10.102)

where: i = 0, 1, 2, 3, . . ., l = L/(2L 2 ), wi = 0 for i ≤ 0. Let us now homogenized the relations (10.100)–(10.102) because of φ applying the operator 0.5(N+1) 1 (. . .)dϕ. (.< . .) = N +1 −0.5(N+1)

Observe that w =0i = w0i , φ˜ = 1, and because w i are periodic functions, we get Π1 wi = Π2 wi = Π31 wi = 0. Note that in this case the averaging process is carried out directly in input equations, what is a key feature of this averaging modification [454]. This method is particulary effective in application to the problems with periodically discontinuous coefficients, for instance, in the theory of reinforced plates and shells [70, 454]. Taking into account the latter result the following homogenized boundary value problems are obtained: 4

∂ (10.103) Π00 w00 = a 4 + c − (1 + ρ)λ0 w00 = 0, ∂ξ Π00 w01 − λ1 (1 + ρ)w00 = −∇4 w00 ,

(10.104)

Π00 w02 − (1 + ρ)(λ1 w1 + λ2 w01 ) = −∇4 w01 ,

(10.105)

Π00 w03 − (1 + ρ)(λ1 w02 + λ2 w01 + λ3 w00 ) − λ0 (1 + ρφ)w1 = Π30 w1 − ∇4 w02 , .

.

. .

.

.

. .

(10.106) .

.

˜ , woiξξ = −w(i−2)ξξ ˜ , for ξ = 0, l w0i = wi−2

(10.107)

or w0i = −wi−2 ˜ , woiξ = −w(i−2)ξ ˜ ,

(10.108)

˜ , woiη = −w(i−2)η ˜ , for η = ±0.5 w0i = −wi−2

(10.109)

where: w >i = 0 for i ≤ 0. The following boundary value problems serve to define the fast periodic functions wi : 4 ∂ w1ϕϕϕϕ = Π01 w00 ≡ a 4 − λ0 ρ w00 , (10.110) ∂ξ w2ϕϕϕϕ = Π1 w1 + Π01 w01 − λ1 ρw00 ,

(10.111)

288

10 HOMOGENIZATION

wkϕϕϕϕ =

3

Πi wk−i − ∇4 wk−4 + Π01 wo(k−1) −

i=1 k−1 i=1

λi wo(k−1−i) +

k−4

λi wk−3−i (1 + ρφ),

(10.112)

i=0

for ϕ = ±k, k = 0, 1, . . . , 0.5(N + 1) w i = wiϕ = 0,

(10.113)

where: (. . .) = (. . .) − (. ˜. .). Observe that the conditions (10.95) are automatically satisfied in each homogenization step, for the appropriate choice of the functions w i , w0i . Let us clarify the conditions (10.113). Formally, they should be written in the form: for ϕ = ±k wi = Ci0 (ξ, η), wiϕ = Ci1 (ξη). However the functions C i0 , Ci1 can be associated with the slow components of the solution w0i . If for y = ±L2 the given conditions are different of the clamping (10.89), then the boundary conditions (10.113) for ϕ = ±0.5(N + 1) must be changed. For example, if we have conditions of simple support, (10.90), then for ϕ = ±0.5(N + 1) w = wiφφ = 0. Generally the functions w i do not satisfy the boundary conditions for ξ = 0, l. The solution w0i can compensate a slow part of the discrepancy. Therefore we must construct a rapidly decaying boundary layer. Let us introduce the new fast variable ψ = ξ/ε. Then ∂ ∂ ∂ = + ε−1 . (10.114) ∂ξ ∂ξ ∂ψ Since the discrepancy is periodic, therefore a solution w b of the boundary layer can be sought only in one period 0 ≤ φ ≤ 1. Let us assume its following form (10.115) wb = ε3 wb1 (ξ, η, ψ, ϕ) + εwb2 (ξ, η, ψ, ϕ) + . . . . Substituting (10.114), (10.115) into the input relations and after splitting due to ε, the following recurrent system of the boundary value problems is obtained 4 ∂4 ∂4 ∂ ∇1 wb1 ≡ + 2 2 2 + 4 wb1 = 0, (10.116) ∂ψ4 ∂ψ ∂ϕ ∂ϕ ∇1 wbk =

3 i=1

Ai wb(k−i) − cwb(k−3) − ∇4 wb(k−4) +

k−4

λi wb(k−3−i) ,

(10.117)

i=0

for ϕ = ±k wb1 = 0, wbiϕ = 0, k = 2, 3, . . . , 4 k−5 ∂4 wbp = −A w + A w + ρ λi wb(p−3−i) , 0 b(p−1) 1i b(p−i) ∂ψ4 i=1 i=0

(10.118)


wbpϕ = 0, p = 2, 3, . . . ,

289

(10.119)

for ξ = 0, l, ψ = 0, l1 wb j = −w j , wb jψψ = −w jξξ − 2w(b j−1)ψξ − w(b j−2)ξξ , (10.120) or wb j = −w j , wb jψ = −wb jξ − w(b j−1)ξ , j = 1, 2, . . . , (10.121) where: wbp = 0 for p ≤ 1, l 1 = ε−1 l, 4 ∂ ∂4 ∂4 ∂4 + + + , A1 = −4 ∂ψ3 ∂ξ ∂ψ∂ξ∂ϕ2 ∂ψ2 ∂ϕ∂η ∂ψ3 ∂η ∂4 ∂4 ∂4 ∂4 ∂4 + 3 + 3 + 4 + , A2 = −2 ∂ξ∂ψ∂ϕ∂η ∂ψ2 ∂ξ2 ∂ξ2 ∂ϕ2 ∂ϕ2 ∂η2 ∂ψ2 ∂η2 4 ∂4 ∂ ∂4 ∂4 + , + + A4 = −4 ∂ψ∂ξ3 ∂ξ2 ∂ϕ∂η ∂ψ∂ξ∂η2 ∂ϕ∂η3   3  ∂3 (...)  (...) ∂ ∂4 −1  − A0 (...) = (aε)   , A11 = −4 3 , 3 3 ∂ϕ ∂ϕ ∂ψ ∂ξ ϕ=1

A12 = −6

4

∂ , ∂ψ2 ∂ξ2

ϕ=0

A13 = −4

∂4 , ∂ψ∂ξ3

A14 = −

∂4 . ∂ξ4

The proposed method yields the being sought frequencies and the corresponding modes with an arbitrary accuracy because of ε. However, it is well known that in practice only a few terms of the series are taken. We are going to analyse this procedure in more detail. Let us suppose that λ 0 is the simple eigenvalue. Therefore, in order to find λ 1 one must multiply both hand sides of the equation (10.104) by w 00 and than integrate it l 0.5 (...)dξdy, taking into account the boundary conditions (10.107) (or (10.108)) 0 −0.5

and (10.109). In result we obtain the equation to find λ 1 and the function w 00 will satisfy the following equation ∇4 w00 − λ1 (1 + ρ)w00 = 0.

(10.122)

Assuming λ1 simple eigenvalue we obtain λ2 = 0, w01 = w02 = 0, λ03 − λ3 =

l 0.5 o −0.5

λ0 (1 + ρφw1 ) + Π30 w1 w00 dξdη

(1 + ρ)

l 0.5 0 −0.5

l 0.5 λk = {λ0k −

[(1 + ρ) 0 −0.5

k−3 i=0

, w200 dξdη

λi w0(k−i) +

k−3 i=0

λi wk−2−i (1 + ρΦ)+

(10.123)

290

10 HOMOGENIZATION

Π30 wk−2 + ∇4 wo(k−1) + w˜ k−3 ]w00 dξdη}× l 0.5 {(1 + ρ)

w200 dξdη}−1 , k = 4, 5, ....

(10.124)

0 −0.5

When λ0 or λ1 is multiple eigenvalue, one may use known algorithm (see Chapter 2). Eigenvalue λ o j depends on the boundary conditions for ξ = 0, l. For the conditions (10.107) one obtains ξ=l 0.5 λ0 j = a (w˜ ( j−2)ξξ w00ξ +w˜ j−2 w00ξξ ) dη. −0.5 ξ=0 Whereas for the case (10.108) we get 0.5 λ0 j = a −0.5

ξ=l (w˜ ( j−2)ξ w00ξξ +w˜ j−2 w00ξξξξ ) dη, j = 3, 4, ... . ξ=0

The slow components w 0k (k ≥ 3, 4, . . .) are defined by the equations (10.106) after substituting to them earlier found λ i values. Observe that the equations (10.103), (10.104) can be matched into one governing equation of the form D1 w0xxxx + 2Dw0xxyy + Dw0yyyy + cw0 − ω ˜ 2 ρ1 w0 = 0,

(10.125)

where:

E1 I ρ1 F , ρ1 = ρ0 h + , ω ˜ 2 = ε−1 ω20 + ω21 . b b This is equation of the structurally-orthotropic theory. The boundary conditions to the equation (10.125) have the form D1 = D +

for x = 0, L1 w0 = w0xx = 0, for y = ±L2 w0 = w0y = 0,

or w0 = w0x = 0,

(10.126)

or w0 = w0yy = 0.

(10.127)

Let us construct now fast components w i of the solution. Using the equation (10.110) and the boundary conditions (10.113) one obtains w1 =

1 Π01 w0 F4 (ϕ), 24

(10.128)

where: F 4 ϕ is the periodic function of the form F 4 = ϕ2 (ϕ − 1)2 defined in the interval 0 ≤ ϕ ≤ 1. In the governing variables we get


w1 =

1 ∂4 (E 1 I 4 − ω20 ρ1 F)w0 y2 (y − b)2 . 24bD ∂x

291

(10.129)

The wi for i > 1 have the form 1 (Π01 w0(i−1) − λ j wo(k−1− j) )F 4 (ϕ)+ 24 j=1 i−1

wi =

3 i=4 3 j λ j wi−3− j (1 + ρΦ)]dϕdϕdϕdϕ + C (i) [ Π j wi− j − ∇4 wi−4 + j ϕ , j−1

j=0

j=0

(10.130) where the ‘constants’ C ij (ξ, η) are chosen in a way to satisfy the boundary conditions (10.113). Observe that the relations (10.110)–(10.113) physically mean, that an action of the rapidly varying along y load the plate bending is reduced practically to a cylindrical bending between ribs. Finally let us consider the boundary layer problem. The Kantorovich method can be applied to solve the equations (10.116) with the attached boundary conditions (10.118). Assuming wb1 =

∞

P j (ψ)ϕ2 (ϕ − 0.5)2 j−2.

(10.131)

j=1

Taking into account only first term of the (10.131) series the following ordinary differential equation is obtained P1ψψψψ − 8P1ψψ +

63 P1 = 0. 8

(10.132)

A general solution to equation (10.132) has the form P1 = exp(σ1 ψ)[C10 cos(σ2 ψ) + C20 sin(σ2 ψ)]+ exp[σ1 (ψ − l1 )]{C30 cos[σ2 (ψ − l1 )] + C40 sin[σ2 (ψ − l1 )]}, where: ρ1 = 4.150, ρ2 = 2.286, and C 10 − C40 are the constants. In the governing variable the boundary layer solution has the form wb1 = [exp(−4.150x/b)[C 10 cos(2.286x/b)+ C20 sin(2.286x/b)] + exp[4.150(x − L 1 )/b]× [C30 cos[(x − L1 )/b] + C 40 sin(2.286(x − L 1 )/b]}}y2 (y − b)2 . Observe that if L1 > 2, b, then the interaction between boundary layers can be neglected while finding the constants C i0 wb1 → 0, wb1ψ → 0 for |ψ| → ∞.

292

10 HOMOGENIZATION

In order to solve the problems related to forced oscillations one can apply the series related to oscillation eigenforms. The ribs discretness must be taken into account when the bending moments occur. A projection of the external load into the eigenforms can be realized without principal difficulties within the structurallyorthotropic theory. Observe that the problem of free oscillations can be reduced to that of forced oscillations, since the initial conditions w = w(0) (x, y), wt = w(1) (x, y) for t = 0, are equivalent to the following load q = w(0)

dδ(t) + w(1) δ(t). dt

10.6 Perforated membrane We consider the Poisson equation ∇2 u = f (x, y)

(10.133)

in domain Ω which consists of a perforated medium with a large number of circular holes arranged in a periodic pattern (Fig. 10.3).

Fig. 10.3. Perforated medium.

Let us introduce small parameter ε, characterizing the ratio of structure period to the typical size Ω of the region.

10.6 Perforated membrane

293

Let the boundaries of holes ∂Ω i be free (Neumann problem), so that ∂u = 0 on ∂Ωi , ∂ni

(10.134)

here ni is outer normal to holes’s contour. Boundary conditions (without loss of generality) along the domain boundary ∂Ω may be assumed to be u = 0 on ∂Ω. (10.135) Let us introduce new ‘fast’ variables ξ = x/ε, η = y/ε. The solution is written in the form of a formal expansion u = u0 (x, y) + εu1 (x, y, ξ, η) + ε2 u2 (x, y, ξ, η) + ... .

(10.136)

period 1 with respect to variables ξ, η for u j ( j = 1, 2, ...) is assumed. The operators ∂/∂x and ∂/∂y applied to a function have the form ∂/∂x = ∂/∂x + ε−1 ∂/∂ξ, ∂/∂y = ∂/∂y + ε−1 ∂/∂η.

(10.137)

Substituting series into (10.136) into the boundary value problem (10.133)(10.135), taking into account relations (10.137) and splitting it according to the powers of ε, one obtain the recurrent sequence of boundary value problems ∂2 u1 ∂2 u1 + = 0 in Ωi , ∂ξ2 ∂η2 2 ∂ u1 ∂2 u1 ∂2 u0 ∂2 u0 ∂2 u2 ∂2 u2 + + + 2 + = f, + ∂x∂ξ ∂y∂η ∂x2 ∂y2 ∂ξ2 ∂η2 ∂u1 ∂u0 + =0 ∂n ∂n u0 = 0 on ∂Ω.

(10.138)

(10.139)

Here n - outer normal expressed in ‘fast’ variables. We consider the averaging operator defined upon the variables (ξ, η) of a periodic function Φ(x, y, ξ, η) ˜ (x, y) = 1 Φ (x, y, ξ, η) dξdη. (10.140) Φ Ω∗i Ωi

Here |Ω∗1 | is area of the region occupied by periodically repeating cell of the structure; Ωi is cell without a hole (Fig. 10.3). This is easily obtained from (10.138) by applying the averaging operator defined by (10.140) 2 2 ∂2 u1 ∂ u1 ∂ u0 ∂2 u0 + dξdη = 0. (10.141) + − f |Ω | + i ∂x2 ∂y2 ∂x∂ξ ∂y∂η Ωi

294

10 HOMOGENIZATION

We now consider the cell problem (here x, y are parameters) ∂2 u1 ∂2 u1 + = 0 in Ωi , ∂ξ2 ∂η2 ∂u1 ∂u0 + = 0 on ∂Ωi , ∂n ∂n and conditions of periodic continuation {u1 , ξ} = 0, ξ ⇔ η * ) ∂u1 , ξ = 0, ξ ⇔ η. ∂ξ

(10.142) (10.143)

(10.144)

Here the following notations have been introduced: {Φ, ξ} = Φ ξ=0.5 − Φ ξ=−0.5 . The asymptotic method of domain perturbation (Section 10.9) is to be used for solving the boundary value problem (10.145)–(10.147). Let us consider the case, when hole diameter 2a is small in comparison with cell’ size. Then, in the first approximation for the function u (1) 1 one obtains the boundary value problem for the hole in the infinite plane ∂2 u(1) ∂2 u(1) 1 1 + = 0 in Ωi , ∂ξ2 ∂η2

(10.145)

∂u(1) ∂u0 1 =− on ∂Ωi , ∂¯n ∂n

(10.146)

2 2 u(1) 1 → 0 when ξ + η → ∞.

(10.147)

Condition (10.147) means that we are not taking into account, in the first approximation, the boundaries ξ, η = ±0.5. In the polar coordinates, the boundary value problem (10.145)–(10.147) may be written in the form (1) (1) ∂2 u(1) 1 ∂2 u1 1 ∂u1 1 + + = 0, r ∂r ∂r2 r2 ∂θ2 ∂u(1) ∂u ∂u 1 = − 0 cos θ − 0 sin θ, ∂r ∂x ∂y r=a

u(1) 1 → 0 when r → ∞.

(10.148) (10.149) (10.150)

Solution of the boundary value problem (10.148)–(10.150) may be written in polar coordinates in the following form: ∂u0 a2 ∂u0 cos θ + sin θ . u(1) = 1 r ∂x ∂y


295

Function u(1) 1 does not satisfy the conditions of periodicity, and that is why we obtain, in the second approximation, the following problem for the function u (2) 1 (we take into account only the principal terms of the series) ∗ ∆u(2) 1 = 0 in Ωi , . 6 (1) u1 + u(2) 1 , ξ = 0, ξ ⇔ η,   (1)     ∂u(2)   ∂u1 1 + , ξ = 0, ξ ⇔ η.     ∂ξ ∂ξ 

(10.151) (10.152) (10.153)

We consider problem (10.151)–(10.153) in the simply connected domain Ω ∗i ((|ξ| ≤ 0.5; |η| ≤ 0.5). Let us represent u (2) 1 as ¯ (2) u(2) (10.154) u(2) 1 = ¯ 1 + u1 , where ¯u(2) 1 is the solution of the following boundary value problem: ∗ ∆¯u(2) 1 = 0 in Ωi ,  (2)    6 (2) .     ∂ ¯u1 , ξ ¯u1 , ξ = 0,  = 0,    ∂ξ  . 6 (2) ¯u1 + u(1) 1 , η = 0,   (2)     ∂u(1)   ∂ ¯u1 1 + , η = 0.     ∂η ∂η 

(10.155) (10.156)

(10.157)

One easily obtains the solution of (10.155), ¯u(2) 1 = A0 + B0 η +

∞

(An ch (2πnη) + Bn sh (2πnη)) cos (2πnξ) +

n=1

(Cn ch (2πnη) + Dn sh (2πnη)) sin (2πnξ)

(10.158)

where: arbitrary constants A n , Bn, Cn , Dn , n = 0, 1, ... are obtained from the boundary conditions. Solution in the form (10.158) satisfies the boundary conditions (10.156). To satisfy the conditions (10.157), rewrite them in the form . −1 ∂u0 2 2 , ¯u(2) 1 , η = − ∂y a ξ + 0.25  (2)    −2   ∂u0 2 2   ∂ ¯u1 , η a ξ ξ + 0.25 . =2     ∂η  ∂x 6

(10.159)

Expanding right-hand sides of the equation (10.159) into the Fourier series and equating the corresponding coefficients, one obtains

296

10 HOMOGENIZATION

An = Dn = 0 n = 0, 1, ... , ∂u0 ∗ ∂u0 2 B =− πa , ∂y 0 ∂y

B0 =

Bn = −

∂u0 ∗ B = ∂y n

∂u0 2a2 ' −πn e Im E 1 (πn (i − 1)) −eπn Im E 1 (πn (i + 1)), ∂y shπn

∂u0 ∂u0 ⇒ , n = 0, 1, 2, ... . (10.160) ∂y ∂x √ Here E 1 is exponential integral; i = −1; Im(. . .) denotes imaginary part of (. . .). So, for ¯u(2) 1 we have ∂u0 ∗ ¯u(2) 1 = ∂y B0 η+ ∞ ∂u0 ∂u0 sh (2πnη) cos (2πnξ) + ch (2πnη) sin (2πnξ) . B∗n (10.161) ∂y ∂x n=1 Cn ⇒ Bn when

(2) Substituting solutions of the cell boundary value problems u 1 = u(1) 1 + u1 into equation (10.141), one obtains the homogenized equation 2 ∂ u0 ∂2 u0 = ¯q f. (10.162) + q ∂x2 ∂y2

Here

q = 1 − 2πa2 + π2 a4 + 8π2 a4

∞ n=1

( n ' −πn e Im E 1 (πn (i − 1)) − eπn Im E 1 (πn (i + 1)) , shπn

(10.163)

¯q = 1 − πa2 . Series in expression (10.163) is absolutely convergent with rapidly decreasing ' ( terms |an−1 /an | −˜ exp (−πn) . The homogenized boundary conditions are: u0 = 0 on ∂Ω. For Ωi /Ω∗i = 8/9, the homogenized coefficient, obtained by the method mentioned above, is q = 0.79. We note that the results obtained with the use of the above-mentioned asymptotic procedure are in good agreement with the results of numerical computations [188]. For estimation of the reliability of solution (10.163) we will solve the boundary value problem (10.142)–(10.144) in another way. Actually, we will use the Galerkin method. Expansion for u 1 is written in the following form:


u1 =

∞ ∞

297

(A1mn sin (2mπξ) cos (2nπη) + A2mn cos (2mπξ) sin (2nπη) +

m=0 n=0

A3mn cos (2mπξ) cos (2nπη) + A4mn sin (2mπξ) sin (2nπη)) .

(10.164)

Here, constants Akmn (k = 1 − 4) are obtained from the condition of vanishing variation of Galerkin’s functional. Assumption of function u 1 in the form (10.164) enables us to fulfill the boundary conditions (10.144) on the opposite sides of the cell. Boundary conditions on the contour of the holes will be fulfilled only in the average. Variation of Galerkin’s functional is written in the following form: ∂u1 ∂u0 + ∆u1 δu1 ds + δu1 dl = 0. − ∂n ∂n Ωi

∂Ωi

From the conditions of symmetry we have A3mn = A4mn = 0. Coefficients A1mn and A2mn are obtained by using Galerkin’s procedure A1mn = a

∂u0 ∗ ∂u0 ∗ A , A2mn = a A , A∗mn ≡ const. ∂x mn ∂y mn

After substitution of the expression (10.164) into (10.141), and after some transformations, one obtains homogenized equation in the form (10.162), where the homogenized coefficient is q = 1 − πa2 − 2πa2

∞ ∞

A∗mn

√ −1 √ m2 + n2 I1 2πa m2 + n2 .

m=0 n=0

function. Here I1 is Bessel √ For a = 1/ 3 π , the homogenized coefficient is equal to 0.826, and this result is slightly different from the previously obtained value. Let us now, consider the following eigenvalue problem: ∇2 u + ω2 u = 0,

(10.165)

where ω is the natural frequency. For equation (10.165) we may formulate the boundary conditions (10.134), (10.135). We represent eigenvalue λ and eigenfunction u in the following form ω2 = ω20 + εω21 + ε2 ω22 + ... . u = u0 (x, y) + ε u10 (x, y) + u1 (x, y, ξ, η) + ε2 u20 (x, y) + u2 (x, y, ξ, η) + ... .

(10.166)

(10.167)

298

10 HOMOGENIZATION

Substituting expansions (10.166), (10.167) into equation (10.165) and boundary conditions (10.134), (10.135) and splitting it into the powers of ε, one obtains the recurrent systems of boundary value problems. The first step of solving is the same that above - solving the local problem (10.142)–(10.144). It means that eigenvalue problem for perforated membrane is quasi-static. Homogenized eigenvalue problem may be obtained by applying the averaging operator defined by (10.140) 2 ∂ u0 ∂2 u0 + ¯qω20 u0 = 0. + q ∂x2 ∂y2 This equation must be supplied by homogenized boundary conditions (10.139). For rectangular membrane (0 ≤ x ≤ l 1 ), (0 ≤ y ≤ l2 ), solution of the eigenvalue problem is mπx nπy sin , u0 = sin l1 l2  2 2   m n  q 2 2 (10.168) + ω0 = π   . l1 l2  ¯q Calculation of the first correctness term to the frequency square needs obtaining function u 2 due to the following boundary value problem ∂2 u2 ∂2 u2 ∂2 u0 ∂2 u0 + = − − − ∂ξ2 ∂η2 ∂x2 ∂y2 2

(10.169)

∂2 u1 ∂2 u1 −2 − ω20 u0 in Ωi , ∂x∂ξ ∂y∂η

∂u2 ∂u1 ∂u10 =− − in Ωi , (10.170) ∂n ∂n ∂n with conditions of periodic continuation, analogous to (10.144). Using for solution of the boundary value problem (10.169), (10.170), any of the methods mentioned above (for example, Galerkin procedure), one can obtain function u 2 . It may be written (taking into account only the principal terms) as u1 ⇒ u2 when

∂u10 ∂u0 ∂u10 ∂u0 ⇒ ; ⇒ . ∂x ∂x ∂y ∂y

After homogenization of the equation 2 ∂2 u1 ∂2 u1 ∂2 u10 ∂2 u10 ∂ u2 ∂2 u2 + + + + + 2 + ∂x∂ξ ∂y∂η ∂x2 ∂y2 ∂x2 ∂y2 ∂2 u3 ∂2 u3 + + ω21 u0 + ω20 (u1 + u10 ) = 0, ∂ξ2 ∂η2 we obtain the homogenized equation


q

299

∂2 u10 ∂2 u10 + + ¯q ω20 u10 + ω21 u0 = 0 2 2 ∂x ∂y

(10.171)

with the boundary condition u10 = −˜u1 on ∂Ω.

(10.172)

Here u˜ 1 is obtained by applying the averaging operator defined by (10.140) to the function u 1 . Routine procedure of the perturbation method (multiplying relation (10.171) by u0 and integrating by parts over the region Ω ∗ , and taking into account the boundary conditions (10.172)) leads to the following expressions for ω 21 l2 ω21

ϕdy +

q0 = ¯q l1 l2 0 0

Here ϕ =

x=l1 ∂u0 ∂x u10 x=0 ,

ψ=

One obtains:

l1

ψdx

0

.

(10.173)

u20 dxdy

y=l2 ∂u0 ∂y u10 y=0 .

if u˜ 1 = 0, then ω21 = 0 and ω2 = ω20 + O(ε2 ), if u˜ 1 0, then ω2 = ω20 + O(ε). Now we consider the boundary value problem for a composite membrane with periodic circular elastic inclusions. In the domain Ω +i (matrix) we have equation (10.133), in the domain Ω −i (inclusion) λ

∂2 u ∂2 u = f. + ∂x2 ∂y2

The conditions of continuity may be written in the form u+ = u− ,

∂u− ∂u+ =λ on ∂Ωi . ∂n ∂n

The cell boundary value problem may be written in the following form:

u+ = u− ,

∂2 u+1 ∂2 u+1 + = 0 in Ω+i , ∂ξ2 ∂η2

(10.174)

∂2 u−1 ∂2 u−1 + = 0 in Ω−i , ∂ξ2 ∂η2

(10.175)

∂u− ∂u0 ∂u+1 (λ − 1) on ∂Ωi . −λ 1 = ∂n ∂n ∂n

(10.176)

300

10 HOMOGENIZATION

After solution of equation (10.174) and (10.175), and assuming that the inclusion is small, we satisfy the continuity conditions (10.176) and obtain u +1 , u−1 ∂u0 λ − 1 a2 ∂u0 cos θ + sin θ , λ + 1 r ∂x ∂y ∂u0 λ − 1 ∂u0 u−1 = − r cos θ + sin θ , λ + 1 ∂x ∂y

u+1 = −

(10.177) (10.178)

and the homogenized equation

∂2 u0 ∂2 u0 q = f. + ∂x2 ∂y2 Here

2λ πa2 . (10.179) λ+1 Constructing for function u +1 the second approximation (10.154), (10.161), one obtains the homogenized coefficient  ∞ (λ − 1)πa2  n ' −πn q=1+ e Im E 1 (πn (i − 1)) − 2 − πa2 − 8πa2 λ+1 shπn n=1 q = 1 − πa2 +

eπn Im E 1 (πn (i + 1))) .

(10.180)

Various asymptotic for the limiting values λ may be obtained. If λ ∼ ε, relation for the homogenized coefficient coincides with that obtained earlier for a perforated membrane (10.163). For λ ∼ ε−1 (absolutely rigid inclusions) we have q = 1 + 2πa2 − π2 a4 − 8π2 a4

∞ n=1

n ' −πn e Im E 1 (πn (i − 1)) − shπn

−eπn Im E 1 (πn (i + 1))) . Numerical results obtained on the basic of formula (10.180) are in good agreement with the known numerical results [189]. The eigenvalue problem for a membrane with small circular elastic inclusions may be reduced to the following problem ∇2 u+ + ω2 u+ = 0 in Ω+ , ∇2 u− + λω2 u− = 0 in Ω− , ∂u− ∂u+ =λ on ∂Ωi . ∂n ∂n Boundary condition on the outer contour are written as follows: u+ = u− ,


301

u = 0 on ∂Ω. Expansion of every function u + , u− and eigenvalue ω 2 are analogous to (10.166) and (10.167), respectively; then the solution of the local problem, on using the above obtained relations (10.177), (10.178), (10.154), (10.160), (10.161), and the homogenized equation, may be written as 2 ∂ u0 ∂2 u0 q + + ¯qω20 u0 = 0, ∂x2 ∂y2 where

¯q = 1 − πa2 + λπa2 .

(10.181)

Homogenized coefficient q is defined by formula (10.180). Now we deal with square perforations. Boundary conditions on the square hole boundaries 2a may be reduced to the form ∂u0 ∂u1 , ξ ⇔ η, x ⇔ y. =− ∂ξ ξ=±a ∂x We present the solution of the cell problem by trigonometric series (10.164). Using the Galerkin procedure, one obtains an infinite system of algebraic equations ∗ 0 for A1mn = a ∂u ∂x Amn : ∞ ∞ k2 + l2 sin (2πa (k − m)) − A1kl β1kl + l+n k−m k=1 l=0 sin (2πa (k + m)) sin (2πa (n + l)) + k+m 2k cos (2πak) sin (2πam) sin (2πa (n + l)) − l+n sin (2πa (k − m)) sin (2πa (k + m)) 2l − sin (2πal) cos (2πan)] = k−m k+m −

2a ∂u0 sin (2πam) sin (2πan) , m = 1, 2, ..., n = 0, 1, 2, ... . πn ∂x

Here β1kl

  −2π2 , m = k, n = l = 0,      2 = −π , m = k, n = l 0,      0, m k.

∗ 0 Analogous system of equations may be written for A 2mn = a ∂u ∂y Amn . The homogenized equation may be represented as 2 ∂ u0 ∂2 u0 q + = ¯q f. ∂x2 ∂y2

302

10 HOMOGENIZATION

Here q = 1 − 4a2 −

1 ∗ A sin (2mπa) sin (2nπa), π m=0 n=0 mn ∞

∞

(10.182)

¯q = 1 − 4a2 .

(10.183)

For a = 1/6, the homogenized coefficient, obtained by the method mentioned above is 0.823; the numerical result obtained by finite element method was 0.81 [187]. Eigenvalue problem for a rectangular membrane with periodic square perforations may be written as ∇2 u + ω2 u = 0, ∂u = 0 on ∂Ωi , ∂n u = 0 for x = 0, l 1 ; y = 0, l2 . For solution of local problem, the Galerkin variational procedure with function u1 , u2 written in the form (10.164) was used. Homogenized equation and relations for u0 , ω20 , ω21 are of forms analogous to equations (10.167), (10.173) written above. Here one must take into account relations (10.182), (10.183) for the homogenized parameters q, ¯q.

10.7 Composite with periodic cubic inclusions We consider effective characteristics of a matrix with periodic (with a period 2) cubic (with the side 2a) inclusions (Fig. 10.4). The following input equations are taken into considerations: ∆1 u+ = f in Ω+ , u+ = u− ,

+

∆1 u− = f in Ω− ,

(10.184)

−

∂u ∂u =λ on ∂Ωi , ∂n ∂n u = 0 on ∂Ω.

(10.185) (10.186) +

In the above indexes ‘+’ and ‘-’ correspond to the matrix (Ω ) and to the elastic ∂2 ∂2 ∂2 + − + ∂z inclusions (Ω− ); ∆1 = ∂x 2 + 2 , λ = G /G ; n is the external normal to the ∂y21 1 1 inclusion contour ∂Ω 1 ; ∂Ω is the border of the space Ω (Ω = Ω+ ∪ Ω− ); 2L is the characteristic dimension. In addition, we take ε = 1/L (ε 1), we introduce the fast variables ξ = ε −1 x, η = ε−1 y, ζ = ε−1 z, slow variables x = x1 , y = y1 , z = z1 , and then a multiple scale method is used in a spirit of the standard homogenization approach. Note that the differential operator ∂x∂1 , ∂y∂1 , ∂z∂1 acting on the function u possesses now the following form

10.7 Composite with periodic cubic inclusions

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ε−1 , + ε−1 , + ε−1 . = = = ∂x1 ∂x ∂ξ ∂y1 ∂y ∂η ∂z1 ∂z ∂ζ

303

(10.187)

A solution to the stated boundary value problem is sought in the form of the series u± = u0 (x, y, z) + εu±1 (x, y, z, ξ, η, ζ) + ε2 u±2 (x, y, z, ξ, η, ζ) + ... .

(10.188)

Substituting (10.188) into (10.184)–(10.186), and taking into account (10.187), the following recurrent set of boundary value problem is obtained after splitting procedure with respect to the parameter ε. The associated local problem is defined in the following form ∆u+1 = f in Ω+i , u+1 = u−1 , u+1 ξ=1 = u+1 ξ=−1 where: ∆ =

∂2 ∂ξ2

+

∂2 ∂η2

+

∂2 , ∂ζ 2

∆u−1 = f in Ω−i ,

∂u+1 ∂u− ∂u0 on ∂Ωi , − λ 1 = (λ − 1) ∂n1 ∂n1 ∂n ∂u+1 ∂u+1 , (ξ → η → ζ), = ∂ξ ξ=1 ∂ξ ξ=−1

(10.189) (10.190) (10.191)

n1 is the external normal expressed via fast variables.

Fig. 10.4. Cross section of input composite material with chosen periodic cell.

In order to solve the local problem the so called three-phase model is applied [220]. In words, the whole considered space, in spite of one cell, is substituted by a homogeneous medium with unknown characteristics (Figure 10.4). This substitution is equivalent to the change of conditions of periodic continuation on the cell boundaries by the conditions of smooth junction of the cell material with the homogeneous medium. Therefore, the problem is reduced to two phase inclusion in the infinite space. In the next step, one may use the method of perturbation of the

304

10 HOMOGENIZATION

boundary form [329], substituting (in the first approximation) the cube contour a sphere. The following boundary value problem is obtained: u +1 = u−1 ∂u− ∂u+1 ∂u0 ∂u0 ∂u0 − λ 1 = (λ − 1) cos θ sin ϕ + sin θ sin ϕ + cos ϕ , (10.192) ∂r ∂r ∂x ∂y ∂z √3 for r = a 6/π, /3 for r = a 6/π, (10.193) u−1 → u˜ 1 ; λ → λ˜ u˜ 1 → 0; ∂˜∂ru1 → 0 for r → ∞.

(10.194)

A solution to the boundary value problem (10.192)–(10.194) has the following form u−1 = A1 r cos θ sin ϕ + A2 r sin θ sin ϕ + A3 cos ϕ, u˜ 1 = D1 /r2 cos θ cos ϕ + D2 /r2 sin θ cos ϕ + D3 /r2 cos ϕ, u+1 = B1 r + C1 /r2 cos θ sin ϕ+ B2 r + C2 /r2 sin θ sin ϕ + B3 r + C3 /r2 cos ϕ, where

˜ ∂u0 ; A1 = − 1 + 9λΛ ∂x

˜ ∂u0 ; B1 = − 1 + 3 (λ + 2) λΛ ∂x ∂u 0 ; C1 = (18/π) a3 (λ − 1) λ˜ Λ ∂x ∂u0 D1 = (3/π) 1 + 3 2a3 (λ − 1) + λ + 2 Λ ; ∂x −1 Λ = 2 λ˜ − 1 (λ − 1) a3 − (λ + 2) 2λ˜ + 1 ;

A1 → A2 → A3 ;

B1 → B 2 → B3 ;

D1 → D2 → D3

when

C1 → C2 → C3 ;

∂u0 ∂u0 ∂u0 → → . ∂x ∂y ∂z

The following equation governs the effective parameters of the homogeneous medium ' ( ' ( ˜ ' ( Φ u+1 + λΦ u−1 + λΦ u˜ 1 = f. ∼

where: Φ (u) = ∆0 u0 + 2 ∆u1 +∆2 u2 ; ∆0 = ∆2 =

∂2 ∂ξ2

+

∂2 ∂η2

+

∂2 ∂ζ 2

∂2 ∂x2

+

∂2 ∂y2

+

∂2 ; ∂z2

∆˜ =

∂2 ∂x∂ξ

+

∂2 ∂y∂η

     1  (. . .) = ∗  Φ (. . .) dV + λ Φ (. . .) dV + λ˜ Φ (. . .) dV  = f ;  |Ω |  ∼

Ω+1

Ω−1

˜ dV = dξdηdζ. Ω∗ = Ω+i ∪ Ω−i ∪ Ω; Finally, the following averaged equation is obtained

Ω˜

+

∂2 ∂z∂ζ ;

10.7 Composite with periodic cubic inclusions

305

    ∼ ∼ ∼  1  + − ∆u0 + ∗  ∆u1 dV + λ ∆u1 dV + λ˜ ∆u1 dV  = f,  |Ω |  Ω+1

Ω−1

Ω˜

and the effective medium characteristic λ 1 + 2a3 + 2 1 − a3 . λ˜ = ( ' λ 1 − a3 + 2 + a3

(10.195)

A comparison of the results obtained by the formula (10.195) and reported in reference [123] indicates its high accuracy. Consider now the various asymptotical expressions obtained from the relation (10.195): 1. a → 0, λ˜ = 1 is homogenized characteristic is defined by the matrix; 2. a → 1, λ˜ = λ is homogenized characteristic is defined by the inclusions; 3. λ → 0 – perforated medium; for small holes λ = 1 − 1, 5a 3 ; for large holes λ˜ = 2(1 − a3 )/3, which is in agreement with the results reported in reference [123]; 4. λ → 1, λ˜ = 1 – both matrix and inclusions parameters are equal; 5. λ → ∞ – for a 1 we have q = 1 + 3a 3 , whereas for a → 1 we obtain q = 1/(1 − a).

Fig. 10.5. Scheme of lubrication approach.

The case of large inclusions a → 1 (Fig. 10.5) is considered separately. The previous approximation to solve the problem on a cell can not be applied. Nevertheless, a new perturbation parameter may be used, i.e. the wall width between two cubes (Fig. 10.5). Hence, in order to construct an asymptotical solution the singular perturbation technique, can be applied. Since the symmetry holds, each of crosssections can be analysed separately. Consider one of zones Ω +i1 . It is not difficult to

306

10 HOMOGENIZATION

check that the terms u +11ξξ , u+11ηη may be neglected in comparison with u +11ζζ , and the following equations hold: ∂2 u+11 = 0 in Ω+i1 , ∂ζ 2 u+11 = u−1 ,

λ∆u−1 = 0 in Ω−i ,

∂u− ∂u+11 ∂u0 − λ 1 = (λ − 1) for ζ = a. ∂ζ ∂ζ ∂z

(10.196)

Note that in theory of composites a similar approach is referred as lubrication theory [218]. The periodicity conditions are substituted by the following ones: u+11 = 0 form ζ = 1.

(10.197)

The boundary value problems (10.196), (10.197) has the following solution u+11 = P(1 − ζ),

u−1 = P1 ζ,

(10.198)

where: P = (1 − λP2 )

∂u0 ∂u0 ; P1 = − (1 − P2 ) ; P2 = (a + λ (1 − a))−1 . ∂z ∂z

The effective medium characteristics λ˜ is defined via the following averaged equation λ 1 − a 2 + a4 + a2 1 − a 2 . (10.199) λ˜ = λ (1 − a) + a The following limiting cases are further analysed: 1. a → 1, λ˜ = λ for arbitrary λ; 2. λ → 0, perforated medium; for a → 0 this formula (10.199) overlaps with that obtained using the three phase model and with the results reported in monograph [123]; 3. λ → 1 ⇒ λ˜ = 1; 4. λ → ∞, λ˜ = 1/(1 − a) − a 2 (1 + a); for a → 1 the asymptotics is the same as that obtained using the three phase model. To conclude, the following important and somehow surprising result is obtained; three phase model for cubic inclusions has the appropriate asymptotics for arbitrary inclusions dimensions and their characteristics. Finally, the multi-points Padé approximants (see Chapter 14) for construction of an effective parameters estimation and for an estimation of accuracy of three-phase model for limiting values of parameters are applied. For a → 1 the formula (10.199) is applied, whereas for a → 1 the known Voight-Reuss estimators are used [218]: λ˜ = 1 − a3 for λ → 0;

−1 λ˜ = 1 − a3 for λ → ∞.

(10.200)

10.8 Torsion of bar with periodic parallelepiped inclusions

307

Note that while constructing three points PA, the formula (10.200) and the relation λ˜ = 1 for λ = 1 is taken into account. In the results, the following expression is obtained 1 − a 3 + λ + λ2 (10.201) λ˜ = ' (. 1 + λ + λ2 1 − a 3 Both formulas (10.199) and (10.201) are matched via the variable ‘a’ applying three-point Padé approximants: 1 + λ + λ2 − 2λ2 a + a2 2λ2 − 1 (10.202) λ˜ = ' ( (. ( ' ' 1 + λ + λ2 − 2λ2 a + a2 2λ2 − 1 − a3 λ2 − 1 Observe that the yielded by formula (10.202) values of the coefficient λ˜ are in a good agreement with the results obtained using the three-phase model. For an arbitrary values of a the formula (10.202) gives upper bound for the three-phase model for λ > 1, and lower bound for λ < 1. To conclude, the multi-points Padé approximations yield effective top and low boundaries for estimation of the effective composite characteristics with cubic inclusions. On the other hand, the three-phase model yields estimations of those characteristics for arbitrary types of inclusions. Observe that in above asymptotic - Padé procedure of cell problem solving procedure is considered . Namely, a solution to cell-problem is constructed for limiting value of a certain parameter (in this case - volume fracture). Then, being found asymptotics are formulated with a help of a multi-point Padé approximants.

10.8 Torsion of bar with periodic parallelepiped inclusions We study the effective shear rigidity of an infinite simple square array of cylinders with square cross section embedded in a matrix material of unit rigidity (Figure 10.6). The governing relations may be written as follows ∂2 U + ∂2 U + + = f + in Ω+i , ∂x21 ∂y21   2 −  ∂ U ∂2 U −   = f − in Ω−i , + λ  ∂x21 ∂y21 +

∂U ∂U − =λ [U + = U − ]∂Ω , , U|∂Ω = F (x1 , y1 ) . i ∂n1 ∂n1 ∂Ωi

(10.203)

(10.204)

Here U is the function of displacements in the axial direction of the rod (U + , U − respectively, in the matrix (Ω +i ) and in the inclusion (Ω −i )); f is the density of the mass forces, ( f + , f − respectively in the matrix and in the inclusion); λ is the inclusions rigidity; ∂/∂n 1 is the derivative normal to the boundary ∂Ω i between the phases; ∂Ω is the outer boundary of the composite material.

308

10 HOMOGENIZATION

Fig. 10.6. The composite material with distinguished unit cell Ωi = Ω+i + Ω−i .

Let us consider a cell of the studied periodical structure with typical size 2ε (ε 1) Fig. 10.6, and denote (10.205) ξ = x1 /ε, η = y1 /ε. Here ξ and η are the ‘fast’ variables. For slow variables we use the following notation: x = x1 ; y = y1 . The operators ∂/∂x 1 and ∂/∂y1 applied to the function U + , U − become ∂ 1 ∂ ∂ 1 ∂ ∂ 1 ∂ ∂ ∂ ∂ + , + , + , = = = ∂x1 ∂x ε ∂ξ ∂y1 ∂y ε ∂η ∂n1 ∂n ε ∂k

(10.206)

∂ ∂ ∂ ∂ ∂ ∂ where: ∂n = ∂x cos α + ∂y cos β, ∂k = ∂ξ cos α + ∂η cos β α, β are the angels between the normal vector n, k to the boundary ∂Ω i and the co-ordinate axes. Let us represent the solution in the form of a normal expansion

U = U0 (x, y) + εU1 (x, y, ξ, η) + ε2 U2 (x, y, ξ, η) + ...,

(10.207)

Ui (x, y, ξ + 2, η + 2) = Ui (x, y, ξ, η) ,

(10.208)

where i = 1, 2, 3... .

In accordance with multiscale method we consider formally functions of two variables U i (x, y) as functions of four variables U i (x, y, ξ(x, y), η(x, y)), i = 1, 2, .... Substituting series (10.207) into the boundary value problem (10.203), (10.204), taking into account relations (10.206) and splitting it with respect to the powers of ε, one obtains the recurrent system of boundary value problem ∂2 U1± ∂2 U1± + = 0 in Ω±i , ∂ξ2 ∂η2

(10.209)


∂U0 + − (λ − 1) , −λ = [U1 = U1 ] ∂Ω , i ∂Ω ∂k ∂k ∂n i   2 + 2 2 ∂2 U1+   ∂ U1 ' ∂ U0 ∂ U0 + +(  + + + + 2  L1 U0 , U1 , U2 ≡ ∂x∂ξ ∂y∂η ∂x2 ∂y2

∂U1+

∂U1−

309

(10.210)

∂2 U2+ ∂2 U2+ + − f + = 0 in Ω+i , (10.211) ∂ξ2 ∂η2  2 −  2  ∂2 U1−   ∂ U1  ∂ U0 ∂2 U0 ' (   + + L2 U0 , U1− , U2− ≡ λ  2 + + 2  ∂x∂ξ ∂y∂η ∂x ∂y2  ∂2 U2− ∂2 U2−   − f − = 0 in Ω−i , + ∂ξ2 ∂η2 +

∂U1 ∂U2+ ∂U − ∂U − + = λ 1 + λ 2 , U0 |∂Ω = F (x, y) , [U2+ = U2− ]∂Ω , i ∂n ∂k ∂n ∂k ∂Ωi (10.212) . . . . . . . . . . The equations (10.209) with the corresponding boundary conditions (10.210) represent the local boundary value problem. For solving it in case of small inclusions (inclusions typical size a tends to zero) we use three-phase model (TPM), and in the case of large inclusions (a tends to the unity) we use a singular perturbation approach. At first replace all periodic structures with the exception of one cell by a homogenized medium Ω˜ with unknown rigidity λ˜ = q. That leads to the problem of two-phase inclusion in an infinite domain (Figure 10.7)

Fig. 10.7. Three-phase model as applied to the composite material under consideration in case of small inclusions (a → 0).

Using the method of boundary form perturbation, in the first approximation we replace square contours by circle ones, and we introduce polar coordinates

310

10 HOMOGENIZATION

for the cell problem (ξ, η → r, θ). Then the functions U i (x, y, ξ, η) transform into Ui (x, y, r, θ) , i = 1, 2, .... So in polar coordinates the local boundary value problem can be written as follows: ∂2 U1± 1 ∂U1± 1 ∂2 U1± + + = 0 in Ω˜ ±i , r ∂r ∂r2 r2 ∂θ2 1 ∂2 U˜ 1 ∂2 U˜ 1 1 ∂U˜ 1 + + = 0 in Ω˜ i , (10.213) r ∂r ∂r2 r2 ∂θ2 + ∂U1 ∂U − ∂U0 ∂U0 − λ 1 = (λ − 1) cos θ + sin θ [U1+ = U1− ]r=a , , 1 r=a ∂r ∂r ∂x ∂y 1 + ∂U0 ∂U1 ∂U1− ∂U0 + ˜ ˜ ˜ −λ = λ−1 cos θ + sin θ , [U1 = U1 ] r=a , 2 r=a ∂r ∂r ∂x ∂y 2 (10.214) ˜ 1 ∂ U U˜ 1 r→∞ → 0, → 0, ∂r r→∞ √ √ where: a1 = 2a/ π, a2 = 2/ π. Solution of equations (10.213) and (10.214) can be written in the form: U1− = A1 r cos θ + A2 r sin θ, U1+ = (B1 r + C1 /r) cos θ + (B2 r + C2 /r) sin θ, U˜ 1 = (D1 /r) cos θ + (D2 /r) sin θ,

(10.215)

where:

˜ 1 ∂U0 , B1 = − 1 + 2 (λ + 1) λT ˜ 1 ∂U0 , A1 = − 1 + 4λT ∂x ∂x 8 ˜ 1 ∂U0 , D1 = 4 1 + 2 a2 (λ − 1) + λ + 1 T 1 ∂U0 , C1 = a2 (λ − 1) λT π ∂x π ∂x −1 T 1 = λ˜ − 1 (λ − 1) a2 − λ˜ + 1 (λ + 1) , A 2 ↔ A 1 , B 2 ↔ B 1 , C 2 ↔ C 1 , D2 ↔ D1 , ∂U0 ∂U0 ↔ . ∂x ∂y

For determination of the effective rigidity λ˜ of the homogenized medium we substitute the derived expressions (10.215) into the boundary value problem (10.211) ' ( ' ( L1 U0 , U1+ , U2+ dξdη + L2 U0 , U1− , U2− dξdη+ Ω+i

Ω−i

Ω˜ i

˜ 0 , U˜ 1 , U˜ 2 )dξdη = 0 in Ω, L(U


311

˜ 0 , U˜ 1 , U˜ 2 ) = L2 (U0 , U − , U − ) for U − → U˜ 1 ; U − → U˜ 2 ; λ → λ. ˜ where: L(U 1 2 1 2 ˜ Then the unknown parameter λ may be obtained from the linear algebraic equation as follows: λ 1 + a2 + 1 − a2 q = λ˜ = ' . (10.216) ( λ 1 − a2 + 1 + a2 Expression (10.216) has been obtained under the assumption of small inclusions. However, it qualitatively represents the behaviour of the effective rigidity in the case of large inclusions too. When inclusions are large (a → 1), smallness of the parameter thickness of the wall between two neighbouring inclusions may be taking into account. Then we could construct an asymptotic solution using a singular perturbation technique (lubrication theory).

a

x

Fig. 10.8. Model of the composite material under consideration in the case of large inclusions (a → 1).

Due to symmetry we can consider each strip (see Fig. 10.8) separately and obtain solution only for that one. For instance Ω +i for it can easily be shown that ∂2 U1+ ∂2 U1+ , ∂ξ2 ∂η2

(10.217)

and the local boundary value problem can be written in the following form: ∂2 U1+ = 0, ∂η2

(10.218)

∂2 U1− ∂2 U1− + = 0, ∂ξ2 ∂η2

(10.219)

+

∂U1 ∂U1− ∂U0 + − (λ − 1) . −λ = [U1 = U1 ] η=a , η=a ∂η ∂η ∂y This condition of periodic continuity is:

(10.220)

312

10 HOMOGENIZATION

U1+ η=1 = 0.

(10.221)

Solution of the boundary value problem (10.218)–(10.221) is represented as follows: U1+ = E1 + F1 η, U1− = G1 η,

(10.222)

∂U0 ∂U0 0 where E 1 = (1 − λT 2 ) ∂U ∂y , F 1 = − (1 − λT 2 ) ∂y , G 1 = − (1 − T 2 ) ∂y , T 2 = (a + λ (1 − a))−1 . After substituting the derived expression (10.222) into (10.211) and doing homogenization (see above) we obtain: λ 1 − a2 + a3 + a2 (1 − a) q= . (10.223) λ (1 − a) + a

Fig. 10.9. Effective rigidity q as a function of inclusions concentration c = a2 and rigidity λ, according to the analytical solution (10.224).

Formula (10.223) has been obtained under assumption of large inclusion. However, it qualitatively represents the behaviour of the effective rigidity in the case of small inclusions too. Two-point Padé approximant allows us to obtain an approximate analytical expression of q, valid for all values of inclusion concentrations c = a 2 and rigidities λ. For a → 0 we use three first coefficients (k 1 = 3) of an expansion of the formula (10.216) in terms of a/(1 − a), and for a → 1 we use two first coefficients of an expansion of the formula (10.223) in terms of (1 − a)/a. The derived two-point Padé approximation (for m = n = 2) is: q=

(1 + 2a)(λ + 1) + a 2 (λ − 1) . (1 + 2a)(λ + 1) − a 2 (λ − 1)

(10.224)

10.9 Solution of cell problem: perturbation of boundary form

313

Fig. 10.10. The analytical solution (10.224) (solid curve) is compared with Bourgat’s numerical results [189] for c = 1/9, (black points).

The obtained approximate analytical solution (10.224) is in good agreement with known numerical data [189]. This fact shows that TPM allows to achieve a very accurate approximation of q in the case under consideration. However, it would not be so in other problems, which involve calculating different effective coefficients or different geometry of composite materials. The dependence of the effective rigidity q on the inclusions concentration c and rigidity λ, accordingly to the obtained solution (10.224), is shown in Figure 10.9. In Figure 10.10 formula (10.224) is compared with numerical results [189] for c = 1/9.

10.9 Solution of cell problem: perturbation of boundary form Let us investigate Poisson equation in perforated media Ω (Figure 10.11) u xx + uyy = q(x, y).

(10.225)

Let on the boundary of the holes ∂Ω k the following conditions are given: u(x, y) ¯nk |∂Ωk = 0.

(10.226)

The following boundary condition is applied on the boundary of medium: u(x, y)|∂Ω = 0.

(10.227)

We divide the perforated media into the periodic parts Ω k each with one hole, as it has been shown in Figure 10.11. Observe that a periodicity rule is violated

314

10 HOMOGENIZATION 1

Fig. 10.11. The perforated medium.

in the vicinity of the border ∂Ω. Let the dimensions of the periodical part further called “cell” be 2a × 2a, and let L be the characteristic linear dimension of whole medium (L 2a). In addition, assume that the load q(x, y) changes smoothly. Let us introduce the small parameter 1 = 2a/L ( 1 1), the fast variables ξ = x/ 1 , and y = y/ 1 , where x, y correspond to slow variables. The expressions for derivatives along co-ordinates and the normal to the contour have the form ∂ ∂ ∂ ∂ ∂ ∂ = + ε−1 ; = + ε−1 , 1 1 ∂x ∂x ∂ξ ∂y ∂y ∂η ∂ ∂ ∂ ∂ ∂ ¯ ¯ = ¯n(ξ, η) · ¯i + ¯j + j ¯ n (ξ, η) i + ε−1 , 1 ∂¯n(x, y) ∂x ∂y ∂ξ ∂η

(10.228)

where: ¯i and ¯j are the unit vectors attached to the axes ξ and η, ¯n(ξ, η) is the normal to the contour. We are going to apply two scales method. A solution to the boundary value problem (10.225), (10.226) is sought in the form of asymptotical series due to 1: u(x, y) = u0 (x, y) + ε1 u1 (x, y, ξ, η) + ε21 u2 (x, y, ξ, η) + ..., (10.229) where: u1 , u2 are the periodic functions along ξ and η with the period L. Substituting (10.228) into both equation (10.225) and the boundary conditions (10.226) and taking into account the relations (10.227), (10.228), the following recurrent system of boundary value problems is obtained: u1ξξ + u1ηη = 0, ∂ ∂ u1¯n|∂Ω1 = −¯n ¯i + ¯j u0 , ∂x ∂y u2ξξ + u2ηη + 2(u1ξx + u1yη ) + u0xx + u0yy = q(x, y), u0 |∂Ω = 0,

(10.230) (10.231) (10.232) (10.233)


∂ u2¯n|∂Ω1 = −¯n ¯i + ∂x . . . . . . .

315

¯j ∂ u1 , ∂y . . .

(10.234)

First we solve the boundary value problem (10.230), (10.231) related to a cell, and the function u 1 (x, y, ξ, η) satisfies the condition of periodicity; i.e. function values and its derivatives on opposite cell sides must be equal. Furthermore, applying into the boundary value problem (10.232)–(10.234) the homogenization operator 1 (. . .) = (. . .)dξdη, (10.235) |Ω| Ωk

we finally get the following homogenized equation 1 (u1ξx + u1ηy )dξdη + ∆u0 = q(x, y). |Ω|

(10.236)

Ωk

Now a solution to the boundary value problem (10.236), (10.233) can be easily found. A key difficulty is following of the cell problem (10.230), (10.231). Naturally, one can apply the numerical approaches like, for example, FEM. However, an analytical solution possesses many advantages. This problem is discussed below. Observe that if a cell dimension is small in comparison to the characteristic medium dimension, then one may consider in the first approximation an infinite plane. However, in this case some non-physical singularities may occur, and we need regularization procedure to handle them. Let us consider a bending of squared plate with circled hole and with the radius R subjected to an action of a uniform load q ∇4 w ≡ wxxxx + 2w xxyy + wyyyy = q/D.

(10.237)

The boundary ∂Ω is free from stresses, i.e. 6 . −(∇2 w)n + (1 − ν) 0.5(w xx − wyy ) sin 2α − w xy cos 2α |∂Ω = 0, s

∇w + (1 − ν)(w xy sin 2α − w xx sin2 α − wyy cos2 α) |∂Ω = 0,

(10.238)

where: α-angle between axis and normal ¯n to the whole contour, index s denotes derivative along the contour. Observe that the conditions on the external plate contour do not bound our considerations. Having found the particular solution w p to equation (10.237) we consider boundary value problem with a homogeneous equation and non-homogeneous boundary conditions. If a hole is small than in first approximation the problem reduces to that of plane bending with circled hole. Let us formulate the described problem using the polar co-ordinates ρ and ϕ:

316

10 HOMOGENIZATION

∇4 ρw = 0, −2

−1

(10.239) −3

−2

L1 (w) − [wρρρ + ρ wρϕϕ + ρ wρρ − 2ρ wϕϕ − ρ wρ + (1 − ν)(ρ−2 wρϕϕ + ρ−1 wρρ − 2ρ−3 wϕϕ − ρ−2 wρ )]|ρ=R = −L1 (w p ); −2

−1

L2 (w) − [wρρ + ν(ρ wϕϕ + ρ wρ )]|ρ=R = −L2 (w p ), 2

(10.240) (10.241)

2

∂ −1 ∂ −2 ∂ where: ∇2ρ = ∂ρ 2 + ρ ∂ρ + ρ ∂ϕ2 . The right sides of the boundary conditions (10.240) and (10.241) can be presented in the following form

L1 [w p ] = f0 +

∞

( f1n cos nϕ + f2n sin nϕ),

n=1

L2 [w p ] = ψ0 +

∞

(ψ1n cos nϕ + ψ2n sin nϕ).

n−1

Then, a solution to the boundary value problem (10.239)–(10.240) can be obtained as the sum w = w0 + w1 , where w0 and w1 are solutions to the following boundary value problems: (10.242) ∇4ρ w0 = 0, L1 (w0 )|ρ=R = f0 ,

(10.243)

L2 (w0 )|ρ=R = ψ0 ,

(10.244)

∇4ρ w1 L1 (w1 )|ρ=R =

∞

= 0,

(10.245)

( f1n cos nϕ + f2n sin nϕ),

(10.246)

(ψ1n cos nϕ + ψ2n sin nϕ).

(10.247)

n=1

L2 (w1 )|ρ=R =

∞ n=1

Since a solution to the boundary value problem (10.242)–(10.244) is not unique, than a problem of a proper choice of solution appears. In order to solve this problem a fictious elastic foundation with parameter k, i.e. the problem is reduced to that of a plane with a circled hole on the elastic foundation: D∇4ρ w0 = −kw1 ,

(10.248)

L1 (w0 )|ρ=R = f0 ,

(10.249)

L2 (w0 )|ρ=R = ψ0 .

(10.250)

A solution to the boundary value problem (10.248)–(10.250) has the form w0 = C1 berz + C2 beiz + C3 kerz + C4 keiz, where: z =

/4

k/Dρ; berz, beiz, kerz, keiz are the Kelvin functions.

(10.251)


317

The functions berz and beiz exponentially increase with increase of argument z. Therefore, taking into account decaying conditions in infinity, we take C 1 = C2 = 0. A solution exponentially decaying in infinity will have the form w0 = C3 keiz + C4 kerz.

(10.252)

Taking into account the expressions for the functions keiz and kerz for small z values and for only a few terms of the series expansion for k → 0, a solution to the boundary value problem (10.248)–(10.250) has the form w0 = A ln ρ + Bρ2 ln ρ.

(10.253)

The described approach is further referred as the regularization procedure. The solution w1 to the boundary value problem (10.245)–(10.247) can be found without any difficulties. The solution w = w0 + w1 governing deflection of a plane with a hole possesses discrepancies in the boundary conditions on the external plate contour, which can be compensated by a corresponding solution of the boundary value problem for the input contour without a hole. We consider one more example to avoid the infinite displacements. Since a solution referred to deformations can be defined via a solution for infinite space (observe that a problem related to deformations can possess finite solutions for infinite displacements) than also it can serve to define the displacements in the input domain taking into account given boundary conditions [542]. There exists a limited number of canonical domains, where the variable separation is possible (circle, ellipse for 2D; sphere, ellipsoid for 3D). In practice very often one must find a solution for a domain slightly different from a canonical one. For example, the following problems often appear. How to find a plate deflection slightly different from circle one or how to find plane deformations with a hole closed to circled one, etc. In this case a concept of small parameter can be used and in first approximation one can begin with canonical domain for which a solution can be found without any difficulty using variables separation. The briefly described method is sometimes called a perturbation of the boundary conditions at the surface of the body or boundary form perturbation [340]. To illustrate more clearly the introduced idea we consider a solution to Laplace equation (10.254) uρρ + ρ−1 uρ + ρ−2 uϕϕ = 0 inside of a domain bounded by the curve ρ = 1 + f (φ), boundary conditions are applied on the domain boundary u(1 + ε f (ϕ), ϕ) = ψ(ϕ).

< 1. Let the following (10.255)

Besides assume that a solution is bounded, |u(0, ϕ)| < ∞. A solution to the boundary value problem (10.254) is sought as the series of powers: (10.256) u(ρ, ϕ) = u0 (ρ, ϕ) + εu1 (ρ, ϕ) + ε2 u2 (ρ, ϕ) + ... .

318

10 HOMOGENIZATION

Let the boundary condition (10.255) holds for ρ = 1 + ε f (ϕ). Observe that after substitution of (10.256) into the boundary condition (10.255) the small parameter appears not only in the coefficients of successive series forms but also in the function argument. Typical perturbation procedure requires comparing the forms standing by the same powers, and we can not apply it until parameter will be withdrawn from the argument of function u. In order to solve the occurred problem we transfer the boundary condition from the curve ρ = 1 + ε f (ϕ) into the circle ρ = 1 with the help of Taylor series. Then the boundary condition (10.255) will have the form u(1 + ε f (ϕ), ϕ) = u(1, ϕ) +

1 uρ (1, ϕ)ε f (ϕ)+ 1!

1 (10.257) uρρ (1, ϕ)ε2 f 2 (ϕ) + ... = ψ(ϕ). 2! Substituting the series (10.256) into equation (10.254) and into the boundary conditions (10.257), and then comparing the terms standing by same powers, the following successive boundary value problems are obtained: u0ρρ + ρ−1 u0ρ + ρ−2 u0ϕϕ = 0,

(10.258)

u0 (1, ϕ) = ψ(ϕ),

(10.259)

|u0 (0, ϕ)| < ∞,

(10.260)

−1

−2

u1ρρ + ρ u1ρ + ρ u1ϕϕ = 0,

(10.261)

u1 (1, ϕ) = −u0ρ (1, ϕ) f (ϕ),

(10.262)

|u1 (0, ϕ)| < ∞,

(10.263)

u2ρρ + ρ−1 u2ρ + ρ−2 u2ϕϕ = 0,

(10.264) 2

u2 (1, ϕ) = −u1ρ (1, ϕ) f (ϕ) − 0.5u 0ρρ(1, ϕ) f (ϕ),

(10.265)

|u2 (0, ϕ)| < ∞,

(10.266)

.

.

. .

.

.

. .

.

.

Since the equations (10.258), (10.261) and (10.262) have the constant coefficients and are similar, then the corresponding boundary value problems can be easily solved using the method of variables separation. Observe also that in order to solve the original problem of an arbitrary approximation we need to find a general integral of equation (10.258) only once.

10.10 Linear vibartions of a beam with concentrated masses and discrete supports

319

10.10 Linear vibartions of a beam with concentrated masses and discrete supports The asymptotic analysis following [457] and based on the homogenization technique in frameworks of linear dynamics for arbitrary ranges of frequencies is applied in this section to the infinite 1D system which consists of elastically supported discrete masses, linked by beams. Three scale regions of eigenfrequencies are found. The first one corresponds to the continuum approach, when the system studied can be described as an effectively continuum homogeneous beam and corrections are of a higher order of magnitude. The second region corresponds to the antiphase mode where neighboring masses vibrate with slowly varying amplitudes. The highest range of frequencies reflects the short beam vibrations between neighboring masses which are immobile in the first term approach. The completeness of the spectrum analysis is shown. Dispersion relations and the peculiarities of the corresponding eigenmodes are discussed. The system studied admits generalizations and may itself serve as an adequate model for various technical applications. Let us introduce a standard measure of length, l, and study the free vibrations of an infinite beam of periodic structure (Fig. 10.12), stretched by a longitudinal force N, supported by springs of stiffness cε at a distance lε. Mε is the value of the point-like mass and ρε is the 1D density, i.e. the mass of a unit length of the beam. Let us denote transverse displacement by u and bending rigidity by EI. We assume that ε 1 and we are going to perform an asymptotic analysis in the limit ε → 0, i.e. the number of supports for l, n = l/ε, tends to infinity. N

Fig. 10.12. Scheme of beam with concentrated masses and discrete supports.

One can assume as an example that the beam analyzed is a model of a bridge, that l is the length of the bridge and that le is the distance between the supports. Clearly, it is possible to state the boundary-value problem, as discussed in Chapter 1, i.e. to assume the finite length of the beam and to set certain conditions at the ends of the bridge. However, we will not do this in order to make the analysis more clear. As will be discussed later, the boundary conditions in the case of a finite system will only reduce the low-frequency part of the spectrum. Every material constant may be chosen to be of arbitrary order in comparison with ε. Given a choice of orders of magnitude makes the spectrum analysis richer. For example, ρ (order 0) instead of ρε (order 1) for 1D density results in the absence of the highest region of frequencies, corresponding to the bending vibration of light beams between almost immobile heavy point-like masses.

320

10 HOMOGENIZATION

To exclude the length l of order 0 from further analysis let us introduce the dimensionless space variable x = x/l. However, we will not redenote the variables and new material parameters. Thus, the dimensionless standard length is assumed to be equal to 1, ε is a dimensionless small parameter and the equation for the beam system (Fig. 10.12) motion may be written as   ∞ ∞   ∂2 u ∂4 u ∂2 u  ερ + εM  (x δ − jε) 2 + EI 4 − N 2 +cεu δ (x − jε) = 0, (10.267) ∂t ∂x ∂x j=−∞ j=−∞ where t is the time, x (−∞ < x < ∞) is the 1D space coordinate, ε is the distance between neighboring masses (supports), ρ is the density of the beams, M is the point-like mass, EI is the bending rigidity, N is the constant tensile load, and c is the force constant of the immobile support. The homogenization technique implies the division of the space variable into slow, x, and fast, y = x/ε, variables and the substitution into equation (10.267) of the asymptotic series:

∞ ω (ε) l x u = exp i α t ε ul x, . ε ε l=0

(10.268)

Because of the limit ε → 0 the analyzed terms u l (x, y) should be continuous functions of restricted values. However, usually, a stronger assumption of y-periodicity is made. Mostly, the length of the period Y, |Y|, is assumed to be the same as the period of the structure, |Y| = 1. We will use a weaker assumption about the integer |Y|. It is also convenient to call a y-periodic function of period |Y| = P as Py-periodic function. The goal of this section is the study of the linear vibration problem. That is why harmonic time-dependence is assumed. Let us note that it is not the principal restriction. Otherwise (we mean in nonlinear problems), a multiscale time series should be introduced: u=

∞ l=0

t t x εl ul εα t, εk , εk 2 , . . . , x, . ε ε ε

(10.269)

To analyze an arbitrary frequency region, the order, α, of the frequencies is introduced and an asymptotic series for the frequency, ω (ε) =

∞

ωm εm ,

(10.270)

m=0

is assumed. Application to the displacement (10.268) time derivatives will change the magnitude in the following way: ∞ ∂2 u ω2 (ε) u = − u = − b m εm , ∂t2 ε2α ε2α m=0

bm =

i+ j=m

ωi ω j .

(10.271)


321

To shorten notation it is a sensible to introduce special notation for “inertial” Mm = Mbm

∞

δ (y − j),

Pm = ρbm ,

(10.272)

j=−∞

and “elastic” A−4 = EI A−1 = 4EI

∂4 ∂4 ∂4 ∂2 , A = 4EI , A = 6EI − N , −3 −2 ∂y4 ∂x∂y3 ∂x2 ∂y2 ∂y2

∞ ∂4 ∂4 ∂2 ∂2 , A = EI − N + c δ (y − j), (10.273) − 2N 0 ∂x∂y ∂x3 ∂y ∂x4 ∂x2 j=−∞

differential operators in fast and slow variables. After substitution of expression (10.268) into equation (10.267) and skipping an oscillator term we will obtain a different series of equations for u l (x, y), which will be dependent on α. Variation of α changes the asymptotic order of the inertial operators and, hence, affects the coupling between the inertial (10.272) and the elastic (10.273) terms. It will be shown that there exist only three non-empty spectrum domains. Then, nontrivial modes will be found and discussed. Empty frequency domains. Let us analyze the case of the non-semi-integer α: α n/2.

(10.274)

The chosen scale of frequency (10.274) implies an uncoupling asymptotic series of “elastic” ε−4 : ε−3 :

A−4 u0 = 0, A−3 u0 + A−4 u1 = 0,

(10.275) (10.276)

ε−2 : ε−1 :

A−2 u0 + A−3 u1 + A−4 u2 = 0, A−1 u0 + A−2 u1 + A−3 u2 + A−4 u3 = 0,

(10.277) (10.278)

A0 u0 + A−1 u1 + A−2 u2 + A−3 u3 + A−4 u4 = 0, A0 u1 + A−1 u2 + A−2 u3 + A−3 u4 + A−4 u5 = 0,

(10.279) (10.280)

ε0 : ε1 :

.

.

. .

.

.

. .

.

.

and “inertial” equations ε−2α : : ε

M0 u0 = 0, M0 u1 + M1 u0 + P0 u0 = 0,

(10.281) (10.282)

ε−2α+2 :

M0 u2 + M1 u1 + M2 u0 + P0 u1 + P1 u0 = 0,

(10.283)

−2α+1

.

.

. .

.

.

. .

.

.

The structure of the operator A −4 (see (10.273)), the requirement for u l finite values and equation (10.275) imply u 0 independence from y. Taking this into account

322

10 HOMOGENIZATION

we can conclude that u 0 = 0 from equation (10.281). Substitution of this result into equations (10.276) and (10.282) makes them similar to (10.275) and (10.281), but for u1 . Therefore we also have u1 = 0. One can repeat these arguments any number of times and conclude (10.284) u1 = 0 for arbitrary l . Let us turn to a semi-integer, but negative, α. Moreover, we will assume for the determinacy that α = −0.5. Any other semi-integer negative value can be treated in a similar way. The series of inertial equations (10.281)–(10.283) starts from order −2α = 1 in this case. Hence, equation (10.275) implies y-independence for u 0 . Subsequent substitution of u 0 , u1 , u2 and u3 into (10.276)–(10.278) and the presence of differentiation with respect to y in the operators A −4 , A−3 , A−2 and A−1 proves y-independence also for u 1 , u2 and u3 . Taking this into account, one can rewrite (10.279) as an equation for u 4 :   ∞ 2   ∂4 u0 u ∂ ∂4 u4 0 δ (y − j)u0  . EI 4 = − EI 4 − N 2 + c (10.285) ∂y ∂x ∂x j=−∞ Let us give here clear, but very important, notation, which will be frequently used later. The equation ∂4 u4 (10.286) EI 4 = f (x, y) ∂y with a y-periodic right-hand side has a finite solution if and only if the integral over the period Y of f is equal to zero: f (x, y) dy =0 . (10.287) Y

The application of condition (10.287) to the solvability of equation (10.285) gives us the equation for u 0 (x): EI

∂2 u0 ∂4 u0 − N 2 + cu0 = 0 . 4 ∂x ∂x

It has a non-zero bounded solution u0 = A exp (λx) , λ = ±

, N±

√

N 2 − 4EI · c , 2EI

(10.288)

(10.289)

√ √ if either N + N 2 − 4EI · c or N − N 2 − 4EI · c is non-positive, which is not true because N, EI and c are positive. Therefore, we have u 0 = 0. It is sensible to use here one further general notation, which corresponds to each frequency of order α. If we find rigorous arguments for the trivial main term u0, these results can be substituted into the chain of asymptotic equations and we can repeat the arguments found for u 1 , and then for u 2 , etc. Thus, it is sufficient to prove (10.284) for the main term to be sure of its validity for the whole series.


323

The cases α = 0.5, 1 and 1.5 are similar and, therefore, might be analyzed simultaneously. If α = 0.5 the equations for the terms of the asymptotic series (10.268) are of the following structure: Three highest-order equations are identical to (10.275)– (10.277). Therefore, the first main terms, u 0 , u1 and u2 , depend on x only. The requirement for u 3 bounding with the equation. ε−1 : A−1 u0 + A−2 u1 + A−3 u2 + A−4 u3 − M0 u0 = 0,

(10.290)

provides a zero u 0 and y-independence for u 3 because of the general solvability condition (10.287), with f = M 0 u0 . Therefore, (10.284) is also proved. An increase in a shifts the terms Mδ(y)b 0 u0 , Mδ(y)(b0 u1 + b1 u0 ) + ρb0 u0 , Mδ(y)(b0 u2 + b1 u1 + b1 u1 ) + ρ(b0 u1 + b1 u0 ), . . . to higher-order asymptotic equations. In particular, these terms should be used to replace equations of −1, 0, 1, . . . orders at α = 0.5 with equations of −2, −1, 0, . . . orders at α = 1 and equations of −3, −2, −1, . . . orders at α = 1.5. However, these shifts will not change the manner of the analysis as well as the consequence (10.284) relating to the trivial asymptotic solution. Finally, non-trivial solutions do not exist at high frequencies, α > 2.5. As for negative values, it is sufficient to consider a semi-integer α: α = 3, 3.5, . . .. The highest order of asymptotic equations is not less than ε −6 . Therefore, at least two of them are of purely inertial type (10.281) and (10.282). It follows from (10.281) that u0 is zero for an integer argument. Taking into account of this fact in (10.282) implies zero values of u 1 for the integer y and zero u 0 for an arbitrary value of the argument, i.e. u 0 = 0 and, hence, (10.284) is valid. Low-frequency region, α = 0. Long-wave modes. One should expect that longwave approximation can be applied in the low-frequency region and the main term of the asymptotic series can be regarded as a differential equation for a homogeneous beam with a continuous and uniform elastic support. To show this let us combine the inertial (10.272) and elastic (10.273) operators to form an asymptotic series of equations at α = 0: ε−4: A−4 u0 = 0, ε−3: A−3 u0 + A−4 u1 = 0,

(10.291) (10.292)

ε−2: A−2 u0 + A−3 u1 + A−4 u2 = 0, ε−1: A−1 u0 + A−2 u1 + A−3 u2 + A−4 u3 = 0,

(10.293) (10.294)

ε0: A0 u0 + A−1 u1 + A−2 u2 + A−3 u3 + A−4 u4 − M0 u0 = 0, (10.295) 1 ε : A0 u1 +A−1 u2 +A−2 u3 +A−3 u4 +A−4 u5 −(M0 u1 +M1 u0 )−P0 u0 = 0, (10.296) .

.

. .

.

.

. .

.

.

As already shown, equations (10.291)–(10.293) imply u 0 , u1 , u2 , and u3 are yindependent. Taking account of this independence in (10.295) transforms the last equation into   ∞   ∂4 u0 ∂4 u4 ∂2 u0 2  EI 4 = − EI 4 − N 2 + c − Mω0 δ (y − j)u0  , (10.297) ∂y ∂x ∂x j=−∞

324

10 HOMOGENIZATION

which is very similar to (10.285). Almost the same analysis for (10.287)–(10.289) leads to the following condition for the existence of a bounded solution of (10.297): EI

∂2 u0 ∂4 u0 − N + c − Mω20 u0 = 0, 4 2 ∂x ∂x

(10.298)

c − Mω20 < 0, or ω0 =

/ b0 >

c . M

(10.299)

Thus, there is a gap in the long-wave range of the spectrum. The bounded solution of equation (10.298), which provides the solvability for (10.297), is determined by the long-wave periodic function u0 = A exp (±ikx) with a wave number k equals to 2 4+ k=

(10.300)

N 2 + 4EI Mω20 − c − N 2EI

.

(10.301)

The correlation (10.301) is a dispersion relation in a low-frequency region of the spectrum. Its shape and a gap (10.299) at N = EI = c = 1 are shown in Fig. 10.13. w

3.5 3 2.5 2 .

.

.

.

.

.

.

k

Fig. 10.13. Dispersion relation in the low-frequency range of the spectrum.

Taking into account (10.298), equation (10.297) can be rewritten in the form:  ∞    ∂4 u4  (10.302) δ (y − j) − 1 u0 (x) . EI 4 = (Mb0 − c)  ∂y j=−∞


325

Let us consider the cell function χ(y)

7 χ (y) = −y2 (1 − y)2 24 at y ∈ [0, 1] ,

(10.303)

which is the y-periodic solution of the basic equation d4 χ = (δ (y) − 1) χ dy4

(10.304)

and, therefore, has a jump in the third derivative. Then u 4 can be derived as u4 (x, y) = (Mb0 − c) u0 (x) χ (y) + v4 (x) ,

(10.305)

where v4 is an unknown y-independent function. Subsequent application of the same chain of arguments to equations (10.296), . . . will give the next terms u 1 (x), u5 (x, y), . . . of series (10.268) and the corrections ω1 , . . . (series (10.270)) to the frequencies. In particular, ω1 = −

ρ ω0 . 2M

(10.306)

Medium-frequency region, α = 2. Tooth-like wave modes. In a given frequency scale the interaction between the “inertial” (10.272) and “elastic” (10.273) terms starts from the first equation of the asymptotic series: ε−4 : A−4 u0 − M0 u0 = 0, ε−3 : A−3 u0 + A−4 u1 − (M0 u1 + M1 u0 ) − P0 u0 = 0, M p uq − P p uq = 0, ε−2 : A−2 u0 + A−3 u1 + A−4 u2 − p+q=2

ε−1 : A−1 u0 + A−2 u1 + A−3 u2 + A−4 u3 −

p+q=1

M p uq −

p+q=3

ε0 : A0 u0 +A−1 u1 +A−2 u2 +A−3 u3 +A−4 u4 −

M p uq −

.

. .

.

.

. .

P p uq = 0, (10.310)

M p uq − .

P p uq = 0, (10.311)

p+q=3

P p uq = 0, (10.312)

p+q=4

p+q=5

.

(10.309)

p+q=2

p+q=4

ε1 : A0 u1 +A−1 u2 +A−2 u3 +A−3 u4 +A−4 u5 −

(10.307) (10.308)

.

Application of condition (10.287) for the analysis of the solvability of the equation (10.312) clearly gives the result: u 0 = 0. However, 1y-periodocity is not a necessary condition for the solvability of equation (10.312) in the class of bounded functions. Particularly, it is sufficient to require that j≤y+a u0 ( j) ≤ const as a → ∞ (10.313) jy−a is uniform with respect to y. In turn, condition (10.313) is true if u 0 is a Py-periodic function, P > 1, and u 0 (1)+u0 (2)+. . .+u0 (P) = 0. We will restrict our consideration to 2y-periodic functions. Let us introduce the basic one:

326

10 HOMOGENIZATION

 3  1 1    +y −4 +y for − 1 ≤ y ≤ 0, 3    2 2     3  ψ (y) =  1 1    − y − 4 − y 3 for 0 ≤ y ≤ 1,    2 2      periodically extended (with period 2) for other y.

(10.314)

The cell function ψ(y) satisfies the following conditions: 1. 2. 3. 4.

it is 2y-periodic; ψ(y) and its derivatives up to the second order are continuous functions; ψ(2n) = 1, ψ(2n + 1) − 1; ∆ψ (2n) = 48, ∆ψ (2n + 1) = −1 (∆ψ is the denotation for a jump in the third derivative); 5. d 4 ψ/dy4 = 0 for non-integer y.

The correlation between the ψ-function values and the jumps in its third derivative determine the main frequency value: 48EI ω0 = . (10.315) M If and only if the main frequency term corresponds to (10.315), the 2y-periodic solution of (10.307) exists and can be decomposed into ψ(y) and a still unknown function dependent on x: (10.316) u0 (x, y) = v0 (x)ψ(y). It should be noted that the main frequency value (10.315) is caused by the assumption of 2y-periodicity. Analysis based on Py− or quasiperiodic functions will give different values of ω 0 . Let us denote by ψk(y) the 2y-periodic integrals of ψ(y) of zero mean value over the period dk ψk (10.317) = ψ, ψk (y) dy = 0. dyk 2Y

It is easy to show that there exist unique functions satisfying the definition (10.317). In particular, 2 4   1 5 3 1    +y − +y − for − 1 ≤ y ≤ 0,    2 2 2 16  ψ1 (y) =  2 4    3 1 1 5   − −y + −y + for 0 ≤ y ≤ 1.  2 2 2 16 Let us also introduce notation: ψ k0 = ψk (0) (ψk0 = 0 for odd k). Substitution of (10.316) into (10.308) determines an equation for u 1 : EI

∞ ∞ ∂4 u1 (y = Mb δ − j)u + Mb δ (y − j)v0 ψ (y) + 0 1 1 ∂y4 j=−∞ j=−∞


ρb0 v0 (x) ψ (y) − 4EIv 0 ψ (y) .

327

(10.318)

The 2y-periodic solution of equation (10.318) can be represented as: u1 (x, y) = v1 (x) ψ (y) +

ρb0 v0 (x) ψ4 (y) − 4v 0 (x) ψ1 (y) . EI

(10.319)

Coordination between the values of the right-hand sides of (10.318) at the break points and the jumps in the third derivative of u 1 (x, y) involves the condition which determines the first correction w1 to the main frequency: ω1 = 0.5ω0 ψ40 ρ/EI .

(10.320)

Substitution of (10.316) and (10.319) into the equation (10.309) forms the equation for u 2 (x, y) similar to (10.318): EI

∞ ∞ ∂4 u2 (y)b = M δ u + M δ (y) f2 (x, y) + g2 (x, y) , 0 2 ∂y4 j=−∞ j=−∞

(10.321)

where the continuous 2y-periodic functions f 2 (x, y) and g 2 (x, y) are determined using v0 (x), v1 (x), ψ(y) and their derivatives: f2 (x, y) = b1 v1 ψ + g2 (x, y) =

ρb0 b1 v0 ψ4 − 4b1 v 0 ψ1 + b2 v0 ψ, EI

ρ2 b20 v0 ψ4 − 8ρb0 v 0 ψ1 + ρ (b0 v1 + b1 v0 ) ψ+ EI 10EIv 0 + Nv0 ψ − 4EIv 1 ψ .

(10.322)

Similarly to equation (10.318), a 2y-periodic solution of (10.321), u2 (x, y) = v2 (x) ψ +

exists if and only if with α= The function

ρ2 b20 ρb0 ρ (b0 v1 + b1 v0 +) ψ4 v ψ5 + v0 ψ8 − 8 EI 0 EI EI 2

1 10EIv 0 + Nv0 ψ2 − 4v 1 ψ1 , EI

(10.323)

v 0 + αv0 = 0

(10.324)

ρ2 b20 ψ80 2ρb1 ψ40 1 N + + + b2 . 10EI 2 ψ20 10EI ψ20 10EI 10b0 ψ20 √ v0 (x) = A exp ±i αx

(10.325) (10.326)

determines the long-wave modulations of the tooth-like term ψ(y). The requirement for v0 bounded values, α > 0, determines the region of the second correction to the frequencies, ω2 , i.e. a gap in the modulation frequencies.

328

10 HOMOGENIZATION

The next-order modulations v 1 , v2 , . . . and the next corrections to the spectrum should be determined from the next asymptotic equations (10.307)–(10.312). Thus, the modulated tooth-like mode, based on the family of 2y-periodic functions, is found. The length, λ, of the tooth-like wave is given by: λ = 2ε. It means that the wave number is equal to k = 2π/λ = π/ε. Using (10.315) and (10.320) for the frequencies, one can estimate the dispersion relation for the short waves as 48EI 24πψ40 ω0 ω1 2 = πk 1− . (10.327) Ω≈ 2 + ε M kM ε Coming back to the asymptotic series (10.268), it is convenient to divide the time dependence into the production of high-frequency, exp(iΩt), and normal-frequency, exp(iω2 t), terms as was done for u 0 (x, y) with respect to the space variable: x x ω0 + εω1 + ε2 ω2 (x) ψ = t v u x, , t ≈ exp i 0 ε ε2 ε 0 x 1 exp (iΩt) ψ exp (iω2 t) v0 (x) . (10.328) ε Representation (10.328) makes it possible to consider exp(iω 2 t)v0 (x) as the amplitude of a tooth-like vibration exp(iΩt)ψ(x/ε), which, √ in turn, is also a wave, but of a long (zero-order) wavelength, λ 0 = 2π/k0 = 2π/ α (see (10.324), (10.325) and normal (zero-order) frequency w2. High-frequency region, α = 2.5. Vibrations of the beam between immobile heavy masses. Obviously, an increase in α leads to an increase in the order of the inertial terms. In the case analyzed the highest order of the series of asymptotic equations is (−5): ε−5 : −M0 u0 = 0, ε−4 : A−4 u0 − (M0 u1 + M1 u0 ) − P0 u0 = 0, Mi u j − Pi u j = 0, ε−3 : A−3 u0 + A−4 u1 − i+ j=2

i+ j=1

ε−2 : A−2 u0 + A−3 u1 + A−4 u2 −

Mi u j −

i+ j=3

(10.329) (10.330) (10.331)

Pi u j = 0,

i+ j=2

ε−1 : A−1 u0 + A−2 u1 + A−3 u2 + A−4 u3 −

Mi u j −

(10.332)

Pi u j = 0,

(10.333)

ε0 : A0 u0 +A−1 u1 +A−2 u2 +A−3 u3 +A−4 u4 − Mi u j − Pi u j = 0,

(10.334)

i+ j=4

i+ j=3

i+ j=5

.

.

. .

.

.

. .

.

i+ j=4

.

Clearly, if one analyzes the frequency range α = 2.5, one should keep in mind ω0 0, otherwise it is another range which was already under analysis. Therefore, equation (10.329) implies fixed masses in the main order term: u0 |y=n = 0.

(10.335)


329

For every segment Y n = [n − 1, n] equation (10.330) and equation (10.335) form a homogeneous boundary value problem. Its solution, u (n) 0 (y) = u0 (y)|y∈Yn , is: sin λ (y) (2λy (2λy = a sin − 2λn + λ) − u(n) sinh − 2λn + λ) + n 0 sinh λ cos λ cosh (2λy − 2λn + λ) , bn cos (2λy − 2λn + λ) − (10.336) cosh λ where (2λ)4 = ρb0 /EI, in terms of the local variable z n = y − (2n − 1)/2 of Y n . Jumps in the third derivatives of u 0 for the integer y can be illuminated by the addi∞ tion M1 u1 (y) = Mb1 u1 (n)δ(y) in the right-hand side of (10.330). However, the j=−∞

first and second derivatives should be continuous. These requirements couple the coefficients an , bn of (10.336) by means of a matrix C: an a = C n−1 1 , (10.337) bn b1 where:

1 −2α12 α22 α11 α22 + α12 α21 , C= −2α11 α21 α11 α22 + α12 α21 α11 α22 − α12 α21 α11 = cos λ −

(10.338)

sin λ cos λ cosh λ, α12 = sin λ + sinh λ, sinh λ cosh λ α21 = sin λ, α22 = − cos λ .

The requirement for u 0 (x, y) bounding is equivalent to that for the coefficients (10.337). The necessary analysis is possible, but rather complicated. It will lead to certain corrections, but not to any conceptional conclusions. We will restrict the consideration to the y-periodic case, as was done in the previous sections. Because of representation (10.336), relations (10.337) and expression (10.338) for matrix C, the y-periodicity of u 0 implies the existence of two series of eigenmodes. The first one is: u0,m (y) = am (x) sin (λm (2y − 2n + 1)) for y ∈ Yn ,

(10.339)

where: λm = πm are the roots of the equation a21 = 0. The second, u0,m (y) = bm (x) (cos (ηm (2y − 2n + 1)) + sin ηm cosh (ηm (2y − 2n + 1))) , (10.340) corresponds to the solutions η m of the equation α 12 = 0. The requirements that u 0 are 2y−, 3y−, . . . or quasi- y-periodic will give a new series of representations for the main asymptotic term. Each of them has a structure with high-frequency vibration of the beam between neighboring immobile supports, modulated by the normal slowly changing amplitude a m (x), bm (x), etc. The equations for the amplitudes and asymptotic values of the frequencies can be found by

330

10 HOMOGENIZATION

substitution of (10.339), (10.340), etc. into the nest of equations of the asymptotic series (10.334). It is not worth delaying here with unnecessary technical details, which are similar to the techniques of the previous sections. That is why such substitutions will not be performed in the framework of this book. However, it is interesting to note that there are no non-interacting localized vibrations of the different beam sections between immobile masses M as would have been expected. The reason is that M is very large in comparison with ρ, but the masses are assumed to be point-like and, hence, can “twist” without inertia. If we introduce a high inertia moment, we will obtain independent vibration. (a)

(b)

(c)

Fig. 10.14. The sketches of long (a), medium (b), and short wave modes.

The results of the asymptotic analysis performed give evidence of the existence of three characteristic spectrum regions for the system studied. 1. Low-frequency (long-length wave) region: ω ∝ ε 0 . Long-wave mode. For the accepted orders of geometric, inertial and stiffness parameters the main term of the asymptotic series describes the dynamics of a homogeneous continuum beam on an elastic foundation, the inertial properties of which are determined


331

by the values of the discrete masses only. The next three corrections are also of a long-length wave type and account for the inertial properties of the beam itself. Only the fourth term of the asymptotic series is sensitive to the discrete nature of the system. It is important to underline that not only the displacements, but also the bending moments and forces are determined (in the main term) by a continuum approximation. A sketch of this long-wave mode is represented in Fig. 10.14a. 2. Intermediate-frequency region: ω ∝ ε −2 . Tooth-like mode. In this region of frequencies the main term of the displacements is a tooth-like antiphase vibration, modulated by a long-wave amplitude (Fig. 10.14b). The “tooth” itself, ψ(y), (10.314), and the main frequency, ω 0 , (10.315), are determined by the values of the point-like mass, M, and the beam bending rigidity, EI. The other material parameters (beam specific density, ρ, longitudinal load, N, modulus of the springs, c), affect the amplitude, v 0 (x), of the antiphase vibrations and the terms of the smaller orders, ω 1 in particular. 3. High-frequency region: ω ∝ e −2.5 . Vibration of the beam between immobile supports. At high frequency point-like masses are too heavy for moving in relation to the beam. Therefore, the corresponding mode is vibration of the light beam between immobile masses (expressions (10.339), (10.340)). Because of the zero-twisting inertia of the point-like masses, the vibrations of the neighboring segments of the beam are in agreement (Fig. 10.14c). In the case of large inertia of the masses the vibrations of the beam segments should be independent and one should observe asymptotically localized motion even in the framework of the linear approach. It should be noted in conclusion that a similar analysis is quite possible for arbitrary orders of the physical parameters of the system: a different specific density, modulus of the supports, or longitudinal load. The choice made is influenced by two goals. Clearness of the analysis is the first one. The scope of the family of the modes is the second. For example, it is clear that in the case when the specific density has a magnitude of the same order as the mass, ρ ∝ M, inertia of the beam is the same as the inertia of the point-like masses and the high-frequency mode does not exist.

11 INTERMEDIATE ASYMPTOTICS – DYNAMICAL EDGE EFFECT METHOD

Bolotin [181, 182] proposed the asymptotic approach to estimate eigenfrequencies of oscillations of elastic systems, accuracy of which increases for higher number eigenfunctions. The key idea of the method is focused on a splitting of the input equations occupied domain in two groups: a) solution in interior zone of construction, and b) dynamical edge effect localized in a neighbourhood of boundaries or along the certain lines. Note that usually both dynamical edge effect as well as a solution within an interior zone are changed fastly with respect to spatial coordinates.

11.1 Linear preliminaries In order to illustrate an application of the asymptotic approach to the earlier mentioned problem we consider a simple problem having exact solution. Namely, we consider free vibrations of a rod with length l governed by the following partial differential equation: 2 ρF ∂4 w 2∂ w , (11.1) + a = 0; a2 = EI ∂x4 ∂t2 where F is square of rod cross-section, and I is moment of inertia. We consider two variants of the boundary conditions: a) Simply supported edges for x = 0, l w = 0;

∂2 w = 0; ∂x2

(11.2)

b) clamped edges

∂w = 0. ∂x The natural oscillations are assumed to have the form for x = 0, l w = 0;

(11.3)

w(x, t) = W(x) exp(iωt), and we get the following equation d4 W − a2 ω2 W = 0. dx4

(11.4)

334


Using the boundary conditions (11.3), a solution to equation (11.4) has the form Wm = sin

mπ x, l

m = 1, 2, . . . ;

(11.5)

1 mπ 2 . (11.6) a l One can check that solution (11.5) does not satisfy the boundary conditions (11.3). However, if the eigenfunction is fastly changing along x (i.e. enough high vibration forms is considered) it is expected that approximate solution of the form (11.5) exists, and it holds for internal zone lying enough far from the boundaries. The problem can be constructed via edge effect in such a way, that it will compensate the discripancies in the boundary conditions introduced by fundamental solution (11.5) and that it will be rapidly decaying while approaching the internal zone. ωm =

Fig. 11.1. Inner solution and dynamical edge effects.

The described procedure is schematically illustrated in Figure 11.1, where curve 1 denotes an inner solution, whereas curve 2 denotes dynamical edge effect. We assume the following solution of equation (11.4) with the attached boundary conditions (11.3): π(x − x0 ) , (11.7) W0 = sin λ where: x0 is the phase shift and λ is the length of oscillation wave, and both quantities will be found during the process of an edge effect construction. The frequency ω reads π2 (11.8) ω = 2. aλ

11.1 Linear preliminaries

335

In order to realize an idea of the edge effect construction, the equation (11.4) is presented in more suitable form 2 2 d d + aω − aω W = 0. dx2 dx2 Its general solution is W = W0 + We , where W0 , We satisfy the following equations, correspondingly: d2 W0 + aωW0 = 0, (11.9) dx2 d 2 We − aωWe = 0. (11.10) dx2 For rapid changes of eigenfunctions (aω 1) the following estimations hold dW0 ∼ aωW0 ; dx

dWe ∼ aωWe . dx

However, both function have different behaviour. W 0 rapidly oscillates, whereas We is composed of the sum at two exponential functions. Observe that in this case the situation principally differs from a case of singular asymptotics, where small and large real roots of characteristic equation are separated, i.e. the corresponding fast and slow change of solution components. In our case we separate two states, one of which oscillates as fast as the edge effect decays. It means that the characteristic equation possesses real and imaginary roots of the same modulus. Let us begin with the edge effect construction governed by the equation (11.10). Taking into account the frequency (11.8), the following relations govern the edge effect in the neighborhoods of the edges x = 0 and x = l, correspondingly We(0) = C1 exp(−

πx ), λ

We(l) = C2 exp[−

π(x − l) ]. λ

(11.11)

Now we need to find x 0 , λ and arbitrary integration constants C 1 , C2 . Here various methods can be applied. For example, one may use matched asymptotic expansions [212, 213]. The being sought quantities are obtained as matching of generating solution and two boundary effects moving from two opposite boundaries. One can also use a Gol’denveizer-Vishik-Lysternik approach. For the latter case, we transform the boundary conditions (11.3) into the form: for x = 0 W0 + We(0) = 0;

d (W0 + We(0) ) = 0, dx

d (W0 + We(l) ) = 0. dx Substituting (11.7) and (11.11) into (11.12) and (11.13), one obtains πx πx 0 0 C1 − sin = 0, C1 − cos = 0, λ λ for x = l W0 + We(l) = 0;

(11.12) (11.13)

336


C2 + sin

l − x0 = 0, λ

C2 + cos

l − x0 = 0. λ

The obtained system of equations yields λ, x 0 , λ=

l , x0 = λ(0.25 + k), m = 1, 2, . . . , k = 1, 2, . . . , m + 0.5

and finally the following vibration frequency of the clamped rod is obtained ωm = π2

(m + 0.5)2 , al2

m = 1, 2, . . . .

As it has been pointed out in reference [182], the obtained formula even for the fundamental vibration frequency, gives the error of order 1%. Note that the Bolotin method has a close link to the so called “intermediate asymptotics”. It can be roughly described in the following way. Let the source differential equation have a certain family of self-similar solutions. In general, these solutions do not satisfy the given initial and/or boundary conditions. Here two cases can be encountered. In the first case, the self-similar solution gets destroyed (therefore, one can say that it is unstable). In the second case a localized state appears, which “chooses” a certain solution from the given class of self-similar solutions [130, 131, 132, 133]. The idea of “intermediate asymptotics” in combination with the concept of a boundary layer enables us to understand the role of partial asymptotic solutions in nonlinear theories. Namely, “in nonlinear problems, exact special solutions sometimes appear to be useless: since there is no principle of superposition, one cannot immediately find a solution of the problem for arbitrary initial conditions. Here asymptotic behaviour is the key that partially plays the role of the lost principle of superposition. However, for arbitrary given initial conditions this asymptotic behaviour must be proved. The problem is difficult, and in many cases numerical computations give no more than a substitute for a rigorous analytic proof” [701].

11.2 Nonlinear beam vibrations Nonlinear beam vibrations are governed by equation ∂2 w EF ∂4 w EI 4 − ' ( ∂x 2l 1 − ν2 ∂x2

l 0

∂w ∂x

2 dx + ρF

∂2 w = 0. ∂t2

(11.14)

Observe that for a simply support edges of the beam the variables can be separated [681]. We consider a case of elastic clamping: for x = 0, l, w = 0,

∂2 w ∂w = 0. −c ∂x ∂x2

(11.15)

11.2 Nonlinear beam vibrations

337

The generating solution is sought in the form W0 = A sin

π(x − x0 ) ξ(t). λ

(11.16)

Substituting (11.16) into (11.14), the following Duffing equation is obtained d2 ξ + ω2 (1 + γξ 2 )ξ = 0, dt2 + 2π(l−x ) 4 λ I 0 where: ω2 = ρπ1 λ4 , γ = (1 + δ)( Ar )2 , δ = 2πl sin λ 0 + sin 2πx , r = λ F , ρ1 = For the initial conditions for t = 0 ξ = 1, we get ξ(t) = cn(σt, k),

ρF EI .

dξ = 0, dt

/ σ = w 1 + γ,

where: cn(. . .) is the elliptic Jacobi function with the period T = 4K, K is full elliptic first order integral, π/(2w) (1 − k2 sin2 ϕ)−0.5 dϕ K= 0

/ with the moduli k = 0.5γ/(1 + γ). In result, we obtain the following solution W0 = A sin

π(x − x0 ) cn(σt, k). λ

(11.17)

The solution (11.17) satisfies the input equation (11.14), but it does not satisfy the boundary conditions (11.15). In order to construct a solution in the neighbourhoods of the edges a being sought solution can be presented in the following form w = W0 + We ,

(11.18)

where: We denotes dynamical edge effect. Substituting (11.18) into (11.14) the following equation is obtained ∂4 1 ∂2 (W0 + We ) − 2 (W0 + We ) 4 ∂x 2r l ∂x2

l 0

2 ∂W0 ∂We + dx+ ∂x ∂x

∂2 (W0 + We ) = 0. (11.19) ∂t2 Observe that contrary to the earlier considered linear case here the function W 0 and We are coupled due to non-linearities. Edge effect change has the same order as a change of the fundamental solution part. It means that the classical results of the ρ1

338


plates and shells theory in order to construct the edge effect [313] can not be applied. At the same time, both states strongly differ energetically, because the fundamental state governs the rod behaviour along its length, whereas the edge effect is localized in small neighbourhood of the rod ends. Assuming (π/λ) 1 the following estimations hold l 0

∂W0 ∂x

2 dx ∼

π 2 λ

l

π ∂W0 ∂We dx ∼ , ∂x ∂x λ

, 0

l 0

∂We ∂x

2 ∼ 1,

and (11.19) can be presented in the following way ∂4 W0 1 ∂2 W0 − ∂x4 2r2 l ∂x2 ∂4 We 1 ∂2 We − ∂x4 2r2 l ∂x2

l 0

l 0

2 ∂W0 ∂2 W0 dx + ρ1 2 + ∂x ∂t

2 ∂W0 ∂2 We dx + ρ1 2 = 0. ∂x ∂t

(11.20)

Substituting the W0 (11.16) into (11.20), the following equation is obtained ∂4 We ∂2 We ∂2We − acn(σt, k) + ρ = 0, 1 ∂x4 ∂x2 ∂t2

(11.21)

where: a = γπ2 /λ2 . Observe that the equation (11.21) is linear but with time depending coefficients. Since the space and time variables can not be exactly separated and therefore the Kantorovich method [369] is further used. Assuming the solution of the form We (x, t) S (x)cn(σt, k) and using Kantorovich method, one obtains the following ordinary differential equation with constant coefficients d 2 S π2 π2 d4S − a − + a 1 1 S = 0, dx4 dx2 λ2 λ2 √ 2 1−k2 + where: a1 = a 2k2k−1 2 2k arcsin k . Only two roots corresponding to exponents remain (the other are not taken into account, because they correspond to generating solution (11.28)), and  ,  ,     2 2   π π  S (x) = C 1 exp − + a1 x + C2 exp  + a1 x . 2 2 λ λ Since the edge effect decays with x → ∞, then C 2 = 0. For x = 0 one has

11.2 Nonlinear beam vibrations

339

W0 + We = 0, ∂W0 ∂We ∂ W0 ∂ We + + =c , ∂x ∂x ∂x2 ∂x2 2

2

which yields C1 = a sin where: x0 =

πx0 ; λ

(11.22)

πc λ arctg 0 + 1. 2 2 π λ 2 πλ2 + a1 + c πλ2 + a1

In the above the main value of function arctg is taken. Observe that for c → 0 and c → ∞ from (11.22), both the limiting cases corresponding to simply support and clamped edges can be found. One can also construct dynamical edge effect localized close to the edge x = l. The eigenforms of the non-linear oscillations can be separated into different groups because of symmetry properties. Indeed, for the symmetric (in relation to the point x = 0.5l) forms from the condition ∂W0 (0.5l, t) =0 ∂x we find l − 2xo = (2s1 + 1)π, s1 = 1, 2, ... .

(11.23)

For the anti-symmetric forms, from the condition W0 (0.5l, t) = 0 we get l − 2x0 = 2s2 π, s2 = 1, 2, ... .

(11.24)

Both equations (11.23) and (11.24) are easily reduced to the following one l − 2x0 = mπ, m = 1, 2, ... ,

(11.25)

where even (odd) m values correspond to anti-symmetric (symmetric) forms in relation to the point x = 0.5l. Therefore, equations (11.22) and (11.25) yield the being sought values of λ and x0 . The period of the considered oscillations is 4Kλ2 ρ1 T= . 1+γ π2 ∗ ∗ In Figure √ 11.2 the dependence T = T/T 10 versus amplitude A = A/h, for r = h/2 3 (h denotes cross section’s height) for different c values characterizing √ a stiffness of supports is reported, where T 10 = ρ1 l/π2 denotes the period of the fundamental mode of linear vibrations with simply supported edges.

340


.

.

.

Fig. 11.2. period versus amplitude for different support stiffnesses.

11.3 Nonlinear rectangular plate vibrations The free non-linear vibrations of the rectangular plate 0 ≤ x ≤ a, 0 ≤ y ≤ b are governed by equation Eh D∇ w − ∇2 w 2ab(1 − ∂ 2 ) 4

a b  2 2  ∂w  ∂2 w  ∂w +  dxdy + ρh 2 = 0. ∂x ∂x ∂t 0

(11.26)

0

If the plate is simply supported, one can assume wmn (x, y, t) = fmn sin

πny πmx sin ξmn (t). a b

(11.27)

Substituting (11.27) into (11.26) one obtains d 2 ξmn 2 + ω2mn (1 + γξmn )ξmn = 0, dt2 where: ω2mn Assuming for

2 π4 D m 2 n 2 = + , ρh a b

γmn =

(11.28) 2 fmn . 8h2

11.3 Nonlinear rectangular plate vibrations

t = 0, one obtains

341

dξmn = 0, dt

ξmn = 1,

/ ξmn (t) = cn (σmn t, Kmn ) , σmn = ωmn 1 + γmn , π/(2ω mn )

Kmn = 0

−0.5 0.5γ 2 2 . sin2 ϕ dϕ, kmn = 1 − kmn 1+γ

Therefore, we finally obtain wmn (x, y, t) = fmn sin

nπy mπx sin cn(σt, k). a b

So, in a case of simply support edges, the spatial and time variables are exactly separated and one can find a normal form of non-linear vibrations [656]. Consider now clamping along the plate contour x = 0, a w =

∂w = 0, ∂x

y = 0, b w =

∂w = 0. ∂y

(11.29)

A being sought deflection can be represented by the function W0 (x, y, t) = f sin k1 (x − x1 ) sin k2 (y − y1 )ξ(t),

(11.30)

where: ki , x1 , y1 are the constants to be defined. 2 Substituting (11.30) into (11.26), we get the equation ddtξ + ω2 (1 + γξ 2 )ξ = 0 with the coefficients ω2 = D(ρh)−1 (k12 + k22 )2 , 2 2 f k1 (a + A1 )(b − A2 ) + k22 (a − A1 )(b + A2 ) γ = 1.5 , h ab(k12 + k22 ) A1 = 0.5k1−1 [sin 2k1 (x − x1 )] |a0 , A2 = 0.5k2−1 sin 2k2 (y − y1 ) |b0 , which defines the time function ξ(t). Observe that the found generating solution W0 = f sin k1 (x − x1 ) sin k2 (y − y1 )cn(σt, k)

(11.31)

holds for an internal plate part located sufficiently far from its edges. Consider now a construction of dynamical edge effects localized in a neighbourhood of the plate contour. Substituting w = W0 + We into the (11.26) we obtain the following equations

342


D∇4 (W0 + We ) − N∇2 (W0 + We ) + ρh Eh N= 2ab(1 − ∂ 2 )

∂2 (W0 + We ) = 0, ∂t2

2

2  b a     ∂(W0 + We )   ∂(W0 + We )  + dxdy.       ∂x ∂y 0

(11.32)

(11.33)

0

In the next step we use the energetic relations appeared while solving the problems related to non-linear rod oscillations. Let us estimate an order of quantities occurring in the right hand side of formula (11.33) in relation to k 1 ∼ k2 1. One obtains 2 2 a b a b ∂W0 ∂W0 2 dxdy ∼ k1 , dxdy ∼ k22 , ∂x ∂y 0

0

a

b

0

0

a 0

∂W0 ∂We dxdy, ∼ k1 , ∂x ∂x b 0

∂We ∂x

2

0

0

a

b

0

0

a

∂W0 ∂We dxdy ∼ k2 , ∂y ∂y

b

dxdy ∼ 1, 0

0

∂We ∂y

2 dxdy ∼ 1.

The obtained estimations implies that in (11.33) in the first approximation the forms of the order k 12 ∼ k22 1 can be hold, and they depend only on the fundamental state. Then the equations (11.32) and (11.33) have the following form D∇4 We − N0 ∇2 We + ρh

∂2 We = 0. ∂t2

For the boundary conditions for t = 0

ξ = 0,

dξ =0 dt

one obtains ξ(t) = Cu(σt, K), 1/(2ω)

/

(1 − k sin2 ϕ)−0.5 , K 2 =

σ = ω 1 + γ, K = 1 0

N0 =

Eh 2ab(1 − ∂ 2 )

0.5γ . 1+γ

2 2  a b  ∂W0   ∂W0 +   dxdy. ∂x ∂y  0

0

Substituting the value W 0 (11.30) one finally gets: ∇4 We − Hcn2 (σt, k)∇2 We + ρhD−1

∂2 We = 0, ∂t2

(11.34)

11.3 Nonlinear rectangular plate vibrations

343

where: H = γ(k12 + k22 ). The linear partial differential equation with time dependent coefficients (11.34) can be treated as an initial step during a construction of four dynamical edge effects in the neighbourhood of the plate edges x = 0, a and y = 0, b (the interaction between the edge effects can be neglected). Let us begin with construction of the edge effect localized in a vicinity of the edge x = 0 (observe, that for other cases one can proceed in an analogical way). Taking the following form We (x, y, t) = Φ(x, t) sin k2 (y − y1 ), we obtain the following equation ∂2 Φ ∂4 Φ − Hcn2 (σt, k) + 2k22 + 4 ∂x ∂x2 ∂2 Φ k22 Hcn2 (σt, k) + k22 Φ + ρhD−1 2 = 0. ∂t Further, the Kantorovitch method [369] can be applied and we assume that Φ(x, t) ϕ(x)cn(σt, k). The following ordinary differential equation with constant coefficients is obtained 2 d4 ϕ 2 d ϕ 2 2 2 2 2 2 − (H + 2k ) + k (H + k ) − (k + k )(H + k + k ) ϕ = 0, 1 1 1 2 2 2 1 2 1 2 dx4 dx2

where:

(11.35)

  √ 2   2k2 − 1 1 − k  + H1 = H  , 2k arcsin k  2k2 2 2 d d 2 2 2 + k − k − 2k − H 1 ϕ = 0. 1 1 2 dx2 dx2

The equation (11.35) can be presented in the following way. Neglecting first multiplier term describing the fundamental state we obtain the equation governing an edge effect d2 ϕ − (k1 + 2k2 + H1 ) ϕ = 0 dx2 with the following solution + + ϕ(x) = C1 exp − k12 + 2k22 + H1 x + C2 exp k12 + 2k22 + H1 x . Finally, we obtain for the edge x = 0 0 + + 1 We(1) = C1 exp − k12 + 2k22 + H1 x + C2 exp k12 + 2k22 + H1 x × sin k2 (y − y1 )cn(σt, k).

344


Proceeding in a similar way we obtain the following edge effect for y = 0 0 + + 1 k22 + 2k12 + H1 y × We(2) = C3 exp − k22 + 2k12 + H1 y + C4 exp sin k1 (x − x1 )cn(σt, k). Let us describe the boundary condition (11.29) on the analysed edges using (11.31): ∂ (W0 + We(1) ) = 0, for x = 0 W0 + We(1) = 0, (11.36) ∂x ∂ (W0 + We(2) ) = 0. for y = 0 W0 + We(2) = 0, (11.37) ∂y The following condition of decaying holds: for x → ∞

We(1) → 0,

for y → ∞

We(2) → 0.

Satisfying the boundary conditions (11.36), (11.37) we obtain x1 = k1−1 arctg k1 (k12 + 2k22 + H1 )−0.5 −0.5 , C2 = 0, C1 = f k1 2(k12 + k22 ) + H1 −0.5 y1 = k2−1 arctg k2 (k22 + 2k12 + H1 , −0.5 , C4 = 0. C3 = f k2 2(k12 + k22 ) + H1

(11.38)

(11.39)

(11.40)

Observe that in formulas (11.39) and (11.40), the main value of the function arctg(. . .) is taken. Now we are going to separate the plate vibration forms into two groups of symmetry types in relation to central straight lines x = 0.5a and y = 0.5b. In the case of symmetric form in both directions the following relations hold: ∂W0 (0.5a, y, t) = 0, ∂x ∂W0 (x, 0.5b, t) = 0, ∂y and they yield k1 (a − 2x1 ) = (2s1 + 1)π, k2 (b − 2y1 ) = (2s2 + 1)π,

s1 , s2 = 1, 2, . . . .

(11.41)

In analogical way the relations for anti symmetric (in both direction) form are obtained: W0 (0.5a, y, t) = 0, W 0 (x, 0.5b, t) = 0,

11.4 Nonlinear shallow shell vibrations

345

k1 (a − 2x1 ) = 2s3 π, k2 (b − 2y1 ) = 2s4 π,

s3 , s4 = 1, 2, . . . .

(11.42)

The equation (11.41) and (11.42) can be reduced to the following two ones k1 (a − 2x1 ) = mπ, k2 (b − 2y1 ) = nπ,

m, n = 1, 2, . . . .

(11.43)

To conclude, in order to define the quantities k 1 , k2 , x1 and y1 the four transcendental equations (11.39), (11.40) and (11.43) are obtained. Observe that for different choice of m and n all possible oscillation forms can be obtained. The obtained system of equations can be solved using numerical subroutines without any difficulties.

Fig. 11.3. Amplitude-frequency relations for a rectangular plate.

solid curves correspond to Ω ∗ = Ω/Ω0 , where Ω = 0.25 / In Figure 11.3 2 D(1 + γ)/ρh (k1 + k22 )/(4k) is the frequency of non-linear oscillations of a squared / plate clamped along its contour, and Ω 0 = π D/ρhπ2 (a−2 + b−2 ) is the fundamental frequency of linear oscillations of this simply supported plate. The dashed curves correspond to the results given in reference [678]. Observe that a role of geometrical non-linearity essentially increases with the number of oscillation forms.

11.4 Nonlinear shallow shell vibrations Consider nonlinear vibrations of shallow cylindrical shell (0 ≤ x i ≤ ai , i = 1, 2). As the governing equations we use the approximate nonlinear equations obtained in

346


Section 8.4: D∇4 w − hR−1

∂2 F ∂2 w ∂2 w ∂2 w 2 − N∇ w + N + N + ρh = 0, 1 2 ∂t2 ∂x21 ∂x21 ∂x22 ∇4 F − ER−1

where: 6D N= 2 h a1 a2 N1 =

∂2 w = 0, ∂x21

(11.44)

2 2  a2 a1  ∂w   ∂w +  dx1 dx2 , ∂x1 ∂x2  0

12D h2 a1 a2 R

0

a2 a1 wd1 xdx2 , 0

N2 = N1 ν.

0

System (11.44) can be rewritten in the displacement form ∂w ∂2 u1 ∂2 u1 ∂2 u2 + 0.5 (1 − ν) 2 + 0.5 (1 + ν) − νR−1 = 0, 2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂2 u2 ∂w ∂2 u2 ∂2 u1 + 0.5 (1 − ν) + 0.5 (1 + ν) − R−1 = 0, 2 2 ∂x1 ∂x2 ∂x2 ∂x2 ∂x1    2 ∂2 w ∂2 w  4 −1   ∇ w − D N∇ w − N1 2 − N2 2  − ∂x1 ∂x2 ∂u2 ∂2 w ∂u1 +ν − R−1 w + ρh2 D−1 2 = 0. 12h−2 R−1 ∂x2 ∂x1 ∂t

(11.45)

(11.46)

(11.47)

Let us consider the boundary conditions as follows (clamped edges) u1 = u2 = w =

∂w =0 ∂x1

f or x1 = 0, a1 ,

(11.48)

u1 = u2 = w =

∂w =0 ∂x2

f or x2 = 0, a2 .

(11.49)

Here we shall investigate normal modes of nonlinear vibrations. For continuous systems this means that the statial and time variables may be separated in an exact or in an approximate way [656, 681]. Let us represent the interior solution of (11.44) in the form w0 (x1 , x2 , t) = f1 cos k1 (x1 − x10 ) sin k2 (x2 − x20 ) ξ1 (t) ,

(11.50)

F0 (x1 , x2 , t) = f2 cos k1 (x1 − x10 ) sin k2 (x2 − x20 ) ξ2 (t) ,

(11.51)

where k1 , k2 and x10 , x20 are unknown constants; k 1 , k2 are the wavelength and x10 , x20 are the phase shift in the x 1 and x2 direction.


347

Substituting expressions (11.50) and (11.51) into the input relations (11.44), one can obtain the ordinary differential equation for the time function ξ 1 and the relation between the functions ξ 1 and ξ2 : ∂2 ξ1 + ω2 1 + γ1 ξ1 + γ2 ξ12 ξ1 = 0, 2 ∂t −2 ξ2 = E f1 k12 (R f2 )−1 k12 + k22 ξ1 , where

(11.52) (11.53)

ω2 = Dρ−1 h−2 Ω, 2 −2 Ω = k12 + k22 + 12 1 − ν2 (hR)−2 k14 k12 + k22 , −1 γ1 = −12A3 f1 ΩRh2 a1 a2 νk12 + k22 , −1 γ2 = 1.5 f12 Ωh2 a1 a2 k12 + k22 k12 (a1 − A1 ) (a2 − A2 ) + k22 (a2 + A2 ) (a2 + A2 ) , A1 = 0.5k1−1 [sin 2k1 (a1 − x10 ) + sin 2k1 x10 ] , A2 = 0.5k2−1 [sin 2k2 (a2 − x20 ) + sin 2k2 x20 ] , A3 = − (k1 k2 )−1 [sin k1 (a1 − x10 ) + sin k1 x10 ] × [cos k2 (a2 − x20 ) − cos k2 x20 ]

Let us designate the solution of (11.52) satisfying the initial conditions ξ = 0, dξ/dt = 1 for t = 0 as ϕ(t). This solution can be represented by elliptic functions [5]. So, the radial displacement in the interior zone can be expressed in the form w0 = f1 cos k1 (x1 − x10 ) sin k2 (x2 − x20 ) ϕ (t) .

(11.54)

Using (11.45), (11.46), one finds u 10 and u20 in the interior zone to be

where:

u10 = f3 sin k1 (x1 − x10 ) sin k2 (x2 − x20 ) ϕ (t) ,

(11.55)

u20 = f4 cos k1 (x1 − x10 ) cos k2 (x2 − x20 ) ϕ (t) ,

(11.56)

f1 k1 νk12 − k22 f3 = 2 , R k12 + k22

− f1 k2 k22 + (2 + ν) k12 f4 = . 2 R k12 + k22

The constants k1 , k2 , x10 and x20 are unknown, and the boundary conditions are not yet satisfied. Consequently, one proceeds to construct corrective solutions in the narrow zone near the edges.

348


Let us introduce the new variables u 1b , u2b and wb - the components of the corrective solutions localized near the boundaries. The shell displacements can thus be expressed in the forms u1 = u10 + u1b , u2 = u20 + u2b , w = w0 + wb .

(11.57)

Substitution of expressions (11.57) into (11.45) - (11.47) yields ∂2 (u10 + u1b ) ∂2 (u10 + u1b ) (1 + 0.5 − ν) + ∂x21 ∂x22 0.5 (1 + ν)

∂2 (u20 + u2b ) ν ∂ (w0 + wb ) − = 0, ∂x1 ∂x2 R ∂x1

(11.58)

∂2 (u20 + u2b ) ∂2 (u20 + u2b ) + 0.5 (1 − ν) + 2 ∂x2 ∂x21 ∂2 (u10 + u1b ) 1 ∂ (w0 + wb ) − = 0, ∂x1 ∂x2 R ∂x2 2 a2 a1  6 6 2  ∂ (w0 + wb ) 4 + ∇ (w0 + wb ) − 2 ∇ (w0 + wb )  ∂x1 h a1 a2 0.5 (1 + ν)

0

(11.59)

0

2  ∂ (w0 + wb )  ∂2 (w0 + wb ) +  dx1 dx2 − 2 [ν  ∂x2 ∂x21

 a2 a1 ∂2 (w0 + wb )  12 ∂ (u10 + u1b )  (w0 + wb ) dx1 dx2 } − 2 [ν + 2 ∂x1 h R ∂x2

0

0

(w0 + wb ) ∂ (u20 + u2b ) ρh2 ∂2 (w0 + wb ) − = 0. + ∂x2 R D ∂t2

(11.60)

Equations (11.58)–(11.60) are complicated and cannot be solved in an explicit way without asymptotic simplifications. First of all, we must separate the interior and the corrective solutions. For this purpose one can use energy estimations. Thus, let us estimate the integral coefficients in (11.60) for large parameters k 1 ∼ k2 1 a2 a1 0

0

∂w0 ∂x1

2

dx1 dx2 ∼k12 ,

a2 a1 0

a2 a1 0

0

∂w0 ∂x1

0

∂w0 ∂x2

2

dx1 dx2 ∼k22 ,

∂wb dx1 dx2 ∼k1 , ∂x1


a2 a1 0

a2 a1 0

0

∂wb ∂x1

0

∂w0 ∂x2

349

∂wb dx1 dx2 ∼k2 , ∂x2

2

a2 a1 dx1 dx2 ∼1, 0

0

∂wb ∂x2

2 dx1 dx2 ∼1.

If we eliminate all terms of lower order in (11.60), we obtain the simplified equation 2 a2 a1  6 6 2  ∂w0 4 ∇ (w0 + wb ) − 2 + ∇ (w0 + wb )  ∂x1 h a1 a2

0

a2 a1 (w0 + wb ) dx1 dx2 } − 0

0

0

 2  2   ∂ (w0 + wb ) ∂2 (w0 + wb )  ∂w0   × +  dx1 dx2 − 2 ν ∂x2  ∂x21 ∂x22 12 ∂ (u10 + u1b ) ∂ (u20 + u2b ) ν + − h2 R ∂x1 ∂x2

w0 + wb 1 ρh2 ∂2 (w0 + wb ) = 0. + R D ∂t2

(11.61)

Substituting expressions (11.54) - (11.56) for the interior solution into system (11.58) and (11.61), one obtains approximate equations for the dynamical edge effect ∂2 u1b ∂2 u2b ν ∂2 wb ∂2 u1b (1 (1 + 0.5 − ν) + 0.5 + ν) − = 0, ∂x1 ∂x2 R ∂x21 ∂x21 ∂x22 ∂2 u2b ∂2 u2b ∂2 u1b 1 ∂2 wb (1 (1 + 0.5 − ν) + 0.5 + ν) − = 0, ∂x1 ∂x2 R ∂x22 ∂x22 ∂x21   2  ∂ wb γ1 Ω ∂2 wb  γ2 Ω 2 2 ϕ (t)  2 + ν 2  − 2 ϕ (t) ∇2 wb − ∇ wb − 2 2 k2 + νk1 ∂x2 ∂x1 k2 + k12 12 ∂u1b ∂u2b wb ρh2 ∂2 wb + − = 0. ∂ + 2 ∂x2 R D ∂t2 h R ∂x1

(11.62)

(11.63)

(11.64)

Equations (11.62)–(11.64), describing the corrective solutions, are linear differential equations with time-dependent coefficients. Spatial and time variables cannot be separated exactly in (11.62) - (11.64), but one can use the variational Kantorovitch method [369]. First of all, let us represent the solution of (11.62)–(11.64) in the form satisfying the condition of periodicity: u1b (x1 , x2 , t) ≈ U1 (x1 , x2 ) ϕ (t) , u1b (x1 , x2 , t) ≈ U2 (x1 , x2 ) ϕ (t) ,

350


wb (x1 , x2 , t) ≈ W (x1 , x2 ) ϕ (t) .

(11.65)

Now, one can substitute expressions (11.65) into (11.62) - (11.64), multiply these equations by ϕ(t) and integrate over the period τ. Then one obtains d11 U1 + d12 U2 + d13 W = 0,

d21 U1 + d22 U2 + d23 W = 0,

d31 U1 + d32 U2 + d33 W = 0,

(11.66)

where d11 =

∂2 ∂2 (1 + 0.5 − ν) , ∂x21 ∂x22

d13 = −

ν ∂ , R ∂x1

d12 = d21 = 0.5 (1 − ν)

d22 = 0.5 (ν − 1)

∂2 ∂2 + , ∂x21 ∂x22

∂2 , ∂x1 ∂x2

d23 = −

1 ∂ , R ∂x2

∂ ∂ , d32 = −c0 νR , ∂x1 ∂x2  2   ∂  ∂2 2 4 2 2  = h ∇ + c1 ν 2 + 2 + νk1 + k2  − ∂x1 ∂x2 2 2 c2 ∇ + k1 + k22 + c0 − h2 Ω,

d31 = −c0 νR d33

12 c0 = 2 , R γ2 λΩh2 c2 = 2 k1 + k22

γ1 λΩh2 c1 = − 2 νk1 + k22 τ

3

ϕ (t)dt,

λ

τ

ϕ2 (t)dt,

0

−1

τ =

0

ϕ (t)dt. 0

Now the partial differential equations (11.66) have constant coefficients, and one can use the operator method of the solution of differential equations with constant coefficients (Chapter 4.2). Then the system of equations (11.66) can be reduced to the single equation for the function Φ (11.67) D∗ Φ = 0, where:

U1 = 0.5 (1 − ν) D∗13 Φ,

U2 = 0.5 (1 − ν) D∗23 Φ,

W = 0.5 (1 − ν) D∗33 Φ.

(11.68) (11.69)

∗

Here D is a determinant of system (11.66), and are the minors of the determinant D∗ . Let us now consider the dynamical edge effect at the x i = 0 edge. In this zone we represent U 1 , U2 and W in the forms U1 = θ1 (x1 ) sin k2 (x2 − x20 ) ,

U2 = θ2 (x1 ) cos k2 (x2 − x20 ) ,

(11.70)


U1 = θ1 (x1 ) sin k2 (x2 − x20 ) ,

U2 = θ2 (x1 ) cos k2 (x2 − x20 ) .

351

(11.71)

For θ1 , θ2 and θ one obtains a system similar to system (11.66), where ∂/∂x 2 → k2 . The characteristic equation for this system is (11.72) p2 + k12 h2 p6 + a11 p4 + a12 p2 + a13 = 0, where:

a12

a11 = −h2 k12 − 4k22 + α1 ,     c0 1 − ν2 2k12 + k22   = k22 h2 2k12 + 5k22 + − 2α 1  , 2  k12 + k22     c0 1 − ν 2  4 2 2 2 − α a13 = −k2 h k1 + 2k2 + 1  , 2  2 2 k1 + k2 α1 = νc1 − c2 .

Equation (11.72) has two imaginary roots p = ±iK 1 belonging to the interior solutions (11.54) - (11.56), and they must be eliminated from (11.72). The next six roots are ω a 0.5 1 11 + , (11.73) p1,4 = ∓ −2r cos 3 3 π − ω a 0.5 1 11 p2,5 = ∓ 2r cos , (11.74) + 3 3 π + ω a 0.5 1 11 p3,6 = ∓ 2r cos , (11.75) + 3 3 where: a3 a11 a12 a13 + , ω1 = arccos qr−3 , q = 11 − 27 6 3 0.5 3a12 − a211 . r = sign (q) 3 Then, near the boundary x 1 = 0, one has Φ=

3

C1k exp (pk x1 )

k=1

(where C 1k are arbitrary constants), and the dynamical edge effect displacements may be written in the form u(1) 1b = c0 R

3 k=1

C1k pk νp2k + k22 exp (pk x1 ) sin k2 (x2 − x20 ) ϕ (t),

352


u(1) 2b = c0 Rk2

3

C1k k22 − (2 + ν) p2k exp (pk x1 ) cos k2 (x2 − x20 ) ϕ (t),

k=1

w(1) b =

3

2 C1k p2k − k22 exp (pk x1 ) sin k2 (x2 − x20 ) ϕ (t).

k=1

As described in the above, one can thus easily deal with the edge effect at the x2 = 0 boundary. Here we obtain u(2) 1b = −C0 RK1

3

C2k (s2k + ∂k12 ) exp(sk y) sin k1 (x − x0 )ϕ(t);

k=1

u(2) 2b = C0 R

3

C2k s2k [s2k − (2 + ∂)k12 ] exp(sk y) cos k1 (x − x0 )ϕt;

k=1

w(2) b =

3

C2k (s2k − k12 ) exp(sk y) sin k1 (x − x0 )ϕ(t).

k=1

Let us rewrite boundary conditions (11.48) and (11.49) in the form for x1 = 0: (1) u10 + u(1) 1b = 0, u20 + u2b = 0, w0 + wb(1) = 0, for x2 = 0:

(1) ∂w0 ∂wb + = 0, ∂x1 ∂x1

(2) u10 + u(2) 1b = 0, u20 + u2b = 0,

w0 + wb(2) = 0,

(2) ∂w0 ∂wb + = 0. ∂x2 ∂x2

(11.76) (11.77) (11.78) (11.79)

These conditions must be supplemented by (1) (1) u(1) 1b , u2b , wb → 0 for x 1 → ∞,

(11.80)

(2) (2) u(2) 1b , u2b , wb → 0 for x 2 → ∞.

(11.81)

Using conditions (11.77) and (11.79), one has   3 −1   3    2       −1 2 2 2 2 2 x10 = k1 arctg  C1k pk pk − k2  C1k pk − k2   , −       k=1 k=1 x20

  3 −1   3   2  2       = k2−1 arctan  C2k s2k − k12  C2k sk s2k − k12   . k2       k=1 k=1

(11.82)

11.5 Rayleigh-Ritz-Bolotin approach

353

Then, the arbitrary constants may be determined from conditions (11.76), (11.78), (11.80) and (11.81). The oscillation forms can be separated into symmetry types. For the symmetry in both directions, one has ∂w0 = u10 = 0 for x1 = 0.5a1, ∂x1 ∂w0 = u20 = 0 for x2 = 0.5a2. ∂x2 For the anti-symmetry in both directions one gets

(11.83)

w0 = u20 = 0 for x1 = 0.5a1 , w0 = u10 = 0 for x2 = 0.5a2 .

(11.84)

Substituting the displacement (11.54)–(11.56) into (11.83) and (11.84), one obtains the transcendental equations k1 (a1 − 2x10 ) = mπ, k2 (a2 − 2x20 ) = nπ, m, n = 1, 2, ... .

(11.85)

For m = 2k, n = 2k + 1, one has anti-symmetric (in both directions) modes, and for n = 2k, m = 2k + 1 one has symmetric (in both directions) modes.

11.5 Rayleigh-Ritz-Bolotin approach The dynamical edge effect method devoted for searching high eigenfunctions and the corresponding frequencies gives also good results for the lower eigenforms for the kinematics boundary conditions. In a case of statical boundary conditions an accuracy of estimation of lower frequencies sufficiently decreases. An application of the method to the buckling problems also does not lead to correct results. It seems that the most perspective way to increase efficiency of dynamic edge effect method is associated with energetic approaches. The latter approach leads to increase of the results accuracy as well as to extend the area of applicability of the method. Simultaneous using of edge effect method and the variational method, like for instance Rayleight-Ritz, Kantorovich or Galerkin methods, is very natural. The Bolotin method allows for a suitable approximation of a being sought eigenfunction and of its derivatives, which leads to essential accuracy increase of the variational approaches. Let us consider the problem related to free oscillations of squared plate (0 ≤ x ≤ a; 0 ≤ y ≤ a) with free edges: w xx + νwyy = w xxx − 2 (1 − ν) w xyy = 0 for y = 0, a, wyy + νw xx = wyyy − 2(1 − ν)wyxx = 0 for x = 0, a.

(11.86)

354


The variational Rayleigh-Ritz method will be used in order to find the eigenfrequency on a basis of possible deflections principle. The work of internal (U) and outer (R) forces acting on plate on the virtual displacements is equal to zero: U + V + R = 0.

(11.87)

Here V is kinetic energy. The work of internal forces or the potential energy U has the form D U= 2

a a 0

w2xx + w2yy + 2νw2xx w2yy + 2(1 − ν)w2xy dxdy,

0

and the kinetic energy corresponds to the work of inertial forces: 1 V= 2

a a 0

ρhw2t dxdy.

0

The work of external forces R is equal to zero in our case. Let w(x, y, t) = W(x, y) exp(−iωt), then from (11.87) we obtain the following formula for the dimensionless frequency λ = ωa

2

/

 a a 1/2         2 2 2 ρh/D = a  + W 2νW W + 2(1 − ν)W × W dxdy  xx yy xx yy xy       2

0 0

−1/2  a a     2 W dxdy .   0

0

The displacement obtained using the method of dynamical edge effect has the form W(x, y) = s1 (x) sin(β2 y + l2 ) + s2 (y) sin(β1 x + l1 ), where: s1 (x) = sin(β1 x + l1 ) + C11 exp(a1 x) + C12 exp(−a1 x), s2 (y) = sin(β2 y + l2 ) + C21 exp(a2 y) + C22 exp(−a2 y). Taking into account the boundary conditions (11.86) the following system of transcendental equations is obtained, which yields the wave numbers: βi a = 2li + mπ, where: βi =

iπ a , li

i = 1, 2; m = 0, 1, 2, . . . ,

(11.88)

1/2 = arctan{(βi /αi )[β2i + (2 − ν)β2k ]2 /(β2i + νβ2k )2 }, αi = βi + 2β2k ,

11.5 Rayleigh-Ritz-Bolotin approach

355

Ci1 = (α2i sin li )/(α2i − νβ2k ), Ci2 = (α2i sin(βi a + li ))/(α2i − νβ2k ), i = 1, 2; k = 1, 2; i k. Let us take the following function approximating of the displacement W(x, y): W(x, y) = s1 (x)s2 (y). Then the formulas (11.88) yield λ = a2 {[K1 + K2 − 2νK3 + 2(1 − ν)K4 ] /K0 }1/2 , where Ki are the following scalar products: ¯ A¯2 ξ), ¯ K1 = (A¯1 ¯η1 )( A¯2 ¯η2 ), K2 = (A¯1 ¯η2 )(A2 ¯η1 ), K0 = (A¯1 ξ)( K3 = (A¯1 k¯1 )( A¯2 k¯2 ), K4 = ( B¯1θ¯1 )( B¯2θ¯2 ).

(11.89)

The vector components have the following form 6 . ξ¯ = {1; 2; 1} ; ¯ηi = β4i ; −2α2i β2i ; α4i ; 6 6 . . k¯i = −β2i ; α2i − β2i ; α2i ; θ¯i = β2i ; 2αi βi ; α2i ; A¯i = {A1i ; A2i ; A3i } ; B¯i = {A4i ; A5i ; A6i } , i = 1, 2; where:

A1i = {0.5z − [0.25 sin (2(βi z + li )) /βi ]|a0 ,

a A2i = {(α2i + β2i )−1 [αi F1i (z)F 4i (z) − βi F2i (z)F 3i (z)]0 , A3i = [0.5F 5i (z)/αi + 2C1i C2i z]|a0 , A4i = [0.5z + 0.25 sin 2(β i z + li )/βi ]|a0 , 6 .a A5i = (α2i + β2i )−1 αi F2i (z)F 3i (z) + βi F1i (z)F 4i (z) , 0

A6i = [F 5i (z)/(2αi ) − 2C1iC2i z]|a0 , F1i (z) = sin(βi z + li ), F 2i (z) = cos(βi z + li ), F3i (z) = Ci1 exp(αi z) + Ci2 exp(−αi z), F4i (z) = Ci1 exp(αi z) − Ci2 exp(−αi z), 2 2 F5i (z) = Ci1 exp(2αi z) − Ci2 exp(−2αi z), i = 1, 2.

A comparison of the numerical results has been carried out for v = 0.225 using the described Rayleigh-Ritz-Bolotin method (RRBM), the Rayleigh-Ritz method (RRM), as well as using the traditional method dynamic edge effect method (DEEM) [408] is shown in Table 11.1. The results obtained by with higher approximations possess high accuracy in the domain of low eigenfrequencies. The accuracy decreases with an increase of the

356

11 INTERMEDIATE ASYMPTOTICS – DYNAMICAL EDGE EFFECT METHOD Table 11.1. Comparison of various approximations of frequencies. Mode number 1 3 5 6 7

λ, RRM [408] 14.10 35.96 65.24 74.45 109.30

λ, RRBM 14.48 36.68 66.33 75.28 109.10

Error % 2.7 2.0 1.7 1.1 0.2

λ, DEEM [408] 12.41 34.60 63.44 73.59 106.30

Error % 13.6 3.9 2.8 2.5 2.8

mode number. A comparison of the data obtained using various methods yield the following conclusions. The error introduced by RRBM (2.7%) is much more lower than that using the traditional edge effect method (13.6%) in case of computation of first eigenfrequency. Consider now stability of a square clamped isotropic plate subjected to uniform compressing load T applied to its contour in its middle surface. Stability governing equation has the following form D∇4 w + T ∇2 w = 0. We take the following boundary conditions for x = 0, a w =

∂w = 0, ∂x

∂w = 0. ∂x The variational method will be used to find a buckling load. In this V = 0 whereas the active forces work has the form for y = 0, a w =

R=−

T 2

a a 0

(w2x + w2y )dxdy.

0

The equation (11.87) yields the following formula for the dimensionless buckling load  a a −1   a 2 a a   (w2xx + 2w xx wyy + w2yy )dxdy  (w2x + w2y )dxdy , P=   π 0

0

0

0

where: P = T a2 /(Dπ2 ). Proceeding in a similar to the previous case way one obtains P=2

a2 (K1 + K3 )/(K5 + K6 ), π2

where: K1 and K3 are obtained using earlier introduced formulas (11.89), K 5 = ¯ K6 = ( B¯2 θ¯2 )( A¯1 ξ). ¯ ( B¯1 θ¯1 )( A¯2 ξ),

11.6 Parallelogram plate vibrations

357

Because we deal with the symmetric case, therefore the problem of finding wave numbers is reduced to one transcendental equation, which, due to Bolotin method, has the form: βa = 2arctg[th(βa/2)] + π. The constant li and C i j in the formulas are as follows: li = 0.5(π − βa), C i1 = Ci2 = − cos(0.5βa)/ch(0.5βa), i = 1, 2. A comparison of the computational results of the buckling load T found via an asymptotical method with those found by RRM with high approximations (P = 5.31) shows, that the error introduced by RRBM (0.8%) is essentially lower than that of DEEM (18%).

11.6 Parallelogram plate vibrations We construct now the asymptotic solution for high eigenfrequencies of parallelogram plates with an acute angle close to 90 ◦ , for different boundary conditions. The equation governing natural plate vibrations has the following form:  4   ∂ w ∂4 w ∂4 w ∂4 w  ∂4 w   − λ2 w = 0. + A 2 2 + 4 − 4 sin α  3 + (11.90) ∂η41 ∂η1 ∂η2 ∂η2 ∂η1 ∂η2 ∂η1 ∂η32 Here: η1 , η2 denote angle coordinates (0 ≤ η 1 ; 0 ≤ η2 ≤ a), which are defined by relations η1 = x1 − x2 tgα, η2 = x2 secα, and (x1 , x2 ) are Cartesian coordi2 nates; A = 2(1 / + 2w sin α); (π/2 − α) is the acute angle of the parallelogram; 2 λ = ω cos α ρh/D. A solution in the internal plate domain is assumed to have the form w (η1 , η2 ) = sin k1 (η1 − ξ1 ) sin k2 (η2 − ξ2 ) ,

(11.91)

where: ki , ξi are the unknown wave numbers and phases. Substituting (11.91) in (11.90) and using the Bubnov–Galerkin method, the first approximating formula reads λ2 = sec4 α k14 + Ak12 k22 + k24 . (11.92) In order to satisfy the boundary conditions the edge effects are introduced. A solution in the vicinity of the boundary η 1 = 0 is assumed to be w (η1 , η2 ) = W (η1 ) sin k2 (η2 − ξ2 ) .

(11.93)

Substituting (11.92) and (11.93) into (11.90) and applying the Kantorovich method, the following ordinary differential equation is obtained: d4 W1 d 2 W1 2 4 − A k2 − k1 + Ak12 k22 W1 = 0. 4 2 dη1 dη1

358


Assuming decaying in infinity, solution of this equation can be presented in the following form 0 1/2 1 W1 (η1 ) = sin k1 (η1 − ξ1 ) + C1 exp −η1 k12 + Ak22 , (11.94) where the first term corresponds to solution in the internal zone (11.91), whereas the second term describes the dynamical boundary effect. Let the edge η 1 = 0 is clamped. In this case the function W 1 should satisfy the following conditions W1 (0) = W1 (0) = 0 for η1 = 0.

(11.95)

Substituting (11.94) into (11.95), the following equations are derived sin k1 ξ1 − C1 = 0, 1/2 C1 = 0, k1 cos(k1 ξ1 ) − k12 + Ak22 0 8 1/2 1 and hence k 1 ξ1 = u1 (k1 , k2 ), where: u11 = arctan k1 k12 + Ak22 . Using symmetry, substituting η 1 by α1 −η1 in (11.94), and the following solution holds in the vicinity of η 1 = a1 : 0 1/2 1 W2 (η1 ) = sin k1 (a1 − η1 − ξ1 ) + exp (a1 − η1 ) k12 + Ak22 . (11.96) Proceeding in the similar way one finds k 1 (a1 − ξ1 ) = u12 (k1 , k2 ). Since the first terms of expressions (11.94) and (11.96) represent the same solution with respect to internal plate domain, then the following relation is obtained k1 a1 = u11 (k1 , k2 ) + u12 (k1 , k2 ) + mπ,

m = 1, 2, . . . .

(11.97)

Using again the symmetry property of the considered problem and owing to the boundary conditions on the edges η 2 = 0, the second transcendental equation with respect to wave numbers is derived: k2 a2 = u21 (k1 , k2 ) + u22 (k1 , k2 ) + nπ,

n = 1, 2, . . . .

(11.98)

Note that expressions defining u 21 , u22 for the corresponding boundary conditions can be found from the expressions u 11 , u12 by a circle permutation of the indeces k1 , k2 . The carried out computations showed, the exactness of frequency estimation defined by the obtained formula is still not enough sufficient. The following deflection function is taken into further considerations w (η1 , η2 ) =

N N

¯ mn (η2 ). Amn Wmn (η1 ) W

m=1 n=1

The components of the deflection W(η 1 , η2 ) have the following forms:

11.6 Parallelogram plate vibrations

359

Wmn (η1 ) = sin k1mn (η1 − ξ1mn ) , ¯ mn (η2 ) = sin k2mn (η2 − ξ2mn ) . W The Bubnov–Galerkin method is further applied, and the non-orthogonality of the coordinate functions is neglected. The latter is motivated by the observation that for α = 0 the coordinate functions are reduced to the asymptotical expressions for the eigenfunctions of the rectangular plate. To conclude, the problem of finding the eigenfunctions of the rectangular plate is reduced to that of finding the eigenvalues of the characteristic matrix with the following coefficients: (2) N1 υi j = C (1) pl δN2 + 4B plmnC pl ,

m, n, p, l = 1, 2, . . . , N,

where: δij is the Kronecker symbol; N 1 = (m − 1)N + n; N2 = (k − 1)N + l; 4 2 2 2 C (1) pl = k1pl + Ak1pl k2pl + k2kl ,

2 2 C (2) pl = 4 sin αk1pl k2kl k1pl + k2pl , B plmn = R (1, m, n, p, l) R (2, p, l, m, n) , sin k smn ξ smn + k spl ξ spl sin k smn ξ smn − k spl ξ spl − R (s, m, n, p, l) = − k smn − k spl k smn + k spl s = 1, 2 if m + p is even for m p; R(s, m, n, p, l) = 0 s = 1, 2 for m = p. cos k smn ξ smn − k spl ξ spl R (s, m, n, p, l) = + cos k smn ξ smn + k spl ξ spl k smn − k spl

Table 11.2. Frequencies for various angles. N 1 2 3 4 5

45◦ 33.84 32.90 61.66 53.81 77.02 75.82 88.99 79.43 121.6 102.5

α 30◦ 15◦ 0◦ 33.88 34.68 35.09 34.62 35.65 36.00 64.79 68.90 72.90 61.60 68.37 73.75 76.28 75.41 72.90 79.38 77.48 73.75 95.28 103.41 107.5 90.59 103.01 108.8 126.3 130.3 131.6 126.9 132.3 134.2

360


In the Table 11.2, the results for a 1 = a2 = a of the dimensionless eigenfrequency a 2 λ for N = 8 for the first five modes using both dynamical edge effect method (top rows) and the results reported in reference [209] are given. A good agreement has been found for the angles 0 ≤ α ≤ 30 ◦ .

11.7 Sectorial plate nonlinear vibrations Let us now consider vibrations of nonlinear elastic sectorial plate elastically clamped along its contour. The plate is strengthened by the radial ribs symmetrically situated with respect to its middle surface. The governing equations can be written as follows (see Chapter 8.5)   N−1 ρ1 F γ ∂2  ∂2 w  E1 I i ∂4 i  4 ∇ w + N∇w + γ 2 +  DRr ∂r4 + ρhR ∂t2 wδ (ϕ − ϕi ) = 0, (11.99) ∂t i=0

where: N = (12/θ)

1 θ 6 . (∂w/∂r)2 + ∂w/ (rδϕ) 2 rdrdϕ; ∇ = ∂2 /∂r2 + ∂/ (rdr) + 0 0

∂2 / (r∂ϕ)2 ; ϕ ∈ [0, θ]; R is the sector radius; ϕ i are the coordinates of the clamping places of ribs; w(r, ϕ, t) is the deflection function measured in relation to h; γ = R2 (ρh/D)1/2 , E1 I is the bending stiffness of the ith rib, (ρ 1 F)i is the unit mass of i ith rib. The following boundary conditions are taken w Γ1 ,Γ2 = u∂w/∂ϕ − (1 − c) ∂2 w/∂ϕ2 Γ1 ,Γ2 = 0, 6 . w Γ3 = c1 ∂w/∂r − (1 − c1 ) ∇2 w + (1 − ν) ∂w/ (rδr) Γ3 = 0, (i) wr (0, ϕ, t) < ∞, i = 0, 1, w (r, ϕ, 0) = 1,

wt (r, ϕ, 0) = 0,

(11.100) (11.101) (11.102)

where: (Γ1 , Γ2 , Γ3 ) are the straight line and circle contour parts, respectively; c and c1 and reduced parameters of clamping (c, c 1 ∈ [0, 1]). The deflection function is presented in the form w (r, ϕ, t) = AW (r, ϕ) cos ωt,

(11.103)

which satisfies the initial conditions (11.102) and A denotes the vibration amplitude measured in relation to h. Substituting (11.103) into (11.99) and applying the Galerkin method, one gets   1 θ     2 λ = W (r, ϕ) ∇4 + H∇2 W (r, ϕ) rdrdϕ+     0

0

11.7 Sectorial plate nonlinear vibrations

361

   1 1 θ N−1   7      4 4 y εi z2 (r, ϕ) rdrdϕ+ W (r, ϕi ) ∂ W (r, ϕi ) ∂r dr         i=0

0

0

0

−1  1  N−1    2 q ηi W (r, ϕi )dr ,     i=0

0

where: λ = γω; εi = (E 1 Ii )/(Dh); ηi (ρ1 Fi )/(ρh2 ); q = h/R; H = 1 θ 6 . (∂z/∂r)2 + ∂z/ (r∂ϕ) 2 rdrdϕ.

(11.104)

9A2 /θ

0 0

We assume that W(r, ϕ) is the spatial part of the solution of the equation governing free vibrations of smooth plate, i.e. ∇4 − H∇2 − λ˜ 2 z (r, ϕ) = 0, (11.105) where: λ˜ is the non-dimensional frequency of free vibrations of smooth plate. A solution to equation (11.105) is found assuming that λ˜ is large. Hence, a solution for away from the boundary is sought in the form W (r, ϕ) = Jk (αr) sin k (ϕ − ξ) ,

(11.106)

which satisfies (11.102) and where J k is the first order Bessel function; 9 1/2 :1/2 . α = −H/2+ (H/2)2 + λ˜ 2 Constructing dynamical edge effect near the edge r = 1, one obtains W (r, ϕ) = V (αr) sin k(ϕ − ξ),

(11.107)

1/2 where: V (αr) = Jk (αr) − CIk (βr); β = α2 + H , Ik is the modified Bessel function of the first order. A solution of dynamical edge effect on the edges ϕ = 0, θ are found using the Kantorovich method, i.e. taking W2 (r, ϕ) = Ψ (ϕ) Jk (αr) , where:

(11.108)

Ψ (ϕ) = sin k (ϕ − ξ) + C˜ 1 exp (gϕ) + C˜ 2 exp (−gϕ) , 1/2 g = s α2 + H − k2 1 s=2 0

Jk2

3

51

(αr) r dr

Jk2 (αr) rdr,

lim s = 2/3.

α→∞

0

Substituting (11.107), (11.108) into the boundary conditions (11.100), the following system of transcendental equations is obtained

362


7 αJk+1 (α) Jk (α) + βIk+1 /Ik (β) = P α2 + β2 ,

(11.109)

6 . kθ = 2 arctan ck/ ug + (1 − c) g2 + k2 + mπ, m = 1, 2, ...,

(11.110)

where: P = (1 − c1 )/[1 − v(1 − c1 )]. In order to estimate an accuracy of the proposed method the computation results for some special cases are carried out. Table 11.3. Comparison of results for circular clamped plate. A 0.2 0.4 0.6 0.8 1.0

DEEM 1.0072 1.0288 1.0640 1.1100 1.1648

λ/λ0

[245] 1.0074 1.0284 1.0624 1.1075 1.1619

In Table 11.3 the results of calculus of a lower eigenfrequency of circled clamped along a contour plate obtained using dynamical edge effect method and FEM [245] are reported (λ 0 is the frequency of linear vibrations). Table 11.4. Comparison of results for sector clamped plate. ¯n 0 0 0 1 1

m ¯ 0 1 2 0 1

DEEM 23.70 63.29 122.5 39.39 89.21

λ

[245] 23.59 64.25 128.3 38.98 –

¯n 1 2 2 2

m ¯ 2 0 1 2

DEE 158.5 57.56 117.7 197.1

λ

[466] – 54.97 – 197.0

In Table 11.4 a comparison of the obtained results with those found experimentally [466] for sectorial stiffely clamped along the contour plate (θ = π, m ¯ , ¯n number of radial and circle modes, respectively).

12 LOCALIZATION

12.1 Localization in linear chains Consider a homogeneous linear chain with one inclusion, i.e. we assume that the elastic support number n = 0 possesses a stiffness which differs from other ones: γn = γ + ∆γS 0n (S 0n is the Cronecker symbol). The following equations govern oscillations of the considered chain u¨ n − c(un−1 − 2un + un+1 ) + (γ + ∆γδ0n )un = 0.

(12.1)

Now, considering a continualization approach and looking for a boundary (with respect to n) solution of the corresponding equation un = AeΩ|n|+iωt ,

A = const,

(12.2)

2

1 one gets Ω = 2c ∆γ, ω2 = γ − 14 (∆γ) c . For ∆γ < 0 the solution (12.1) describes a localised standing wave characterized by oscillation frequency smaller than the lowest frequency of the chain without inclusion (∆γ = 0). A domain of the localisation space is estimated via the following quantity n0 ∼ 2c/ |∆γ| .

Note that in the case of two inclusions (for instance, supports n = 0 and n = p), one may proceed analogously to get (12.3) un = A eΩ|n| ∓ eΩ|n−p| eiωt , A = const, ∆γ (12.4) 1 ∓ eΩ|p| , ω2 = γ − cΩ2 . 2c More detailed analysis of the obtained expression yields the following conclusion. Two inclusions generate two spaces of localisation, which interact with each other in a way similar to that of two coupled harmonic oscillators (among others, an occurrence of sinphase and antiphase modes is observed - see Fig. 12.1). A coupling stiffness can be estimated via comparison of eigenfrequencies of coupled oscillators, defined by formulas (12.3), (12.4). For instance, if ∆γ < 0 and a distance |p| between two inclusions is large, than the equations yielding Ω possess negative roots. Ω=

364

12 LOCALIZATION

Only one step of simple iteration is required (an occurred exponent in initial approximation is neglected) to quantify the mentioned roots, i.e. Ω∼

∆γ 1 ∓ e∆γ|p|/c , 2c

|p| c/|∆γ|.

In additions, the expression defining ω 2 yields one more estimations. Namely, the coupling stiffness is proportional to the quantity (∆γ)2 /c e0.5∆γ|p|/c . (12.5) It means that the coupling stiffness exponentially decays while increasing a distance between inclusions. Consider now a higher frequency neighbourhood. Analysis of an equation governing a describing function behaviour of a continuum in the case of two inclusions yields: un = (−1)n A eΩ|n| ∓ eΩ|n−p| eiωt , A = const, ∆γ (12.6) 1 ∓ eΩ|p| , ω2 = 4 + γ + cΩ2 . 2c It means that a localisation is possible when the inclusion of stiffness are positive ∆γ > 0. The corresponding oscillation frequencies are higher than the chain’s higher frequency obtained for ∆γ = 0. Ω=−

u

n u

p

n

Fig. 12.1. Localised modes generated by two inclusions.

A mechanism of earlier described localisation is relatively simple, since it is related to ‘external’ local changes of a periodic structure. However, the introduced scheme has no physical meaning, when inclusions appear more frequently (see Fig. 12.2).

12.1 Localization in linear chains M

c

m

1

c

2

M

c

1

m 2

n

c

M

c

1

m

365

c

2

n+1

Fig. 12.2. Infinite chain with two different masses.

However, to omit the occurred problem one may apply the Green functions [461]. The Green function describing two different masses chain (Fig 12.2) is governed by the following equations 21 21 Mω2 − 2c G11 nn + cG n−1,n + cG nn = δnn ,

11 11 mω2 − 2c G21 nn + cG n,n + cG n+1,n = 0.

(12.7)

The subscripts of matrices elements of Green functions correspond to the numbers of elementary chains, whereas the superscripts correspond to cell numbers located in elementary cells. In the expressions (12.7) elements with superscripts not equal to one are neglected, and one obtains the following equations governing behaviour of a certain effective one cell chain 11 ˜ 2 − 2˜c G11 ˜ G11 (12.8) Mω nn + c n−1,n + G n+1,n = δnn , where

c2 2cm , c ˜ = − , mω2 − 2c mω2 − 2c is the mass units and coupling stiffness of the chain, correspondingly. For mω2 /2c → 0 the so called skeleton approximation is obtained. For mω 2 /2c < 1 the relations (12.8) can be substituted by the following ‘quasi-skeleton’ approximation c mω2 mω2 ˜ + . . . , c˜ = 1+ +... . M = M+m 1+ 2c 2 2c M˜ = M −

If a subchain of the chains contains of l units (atoms), the system (12.7) is composed of l equations. A reduction process yielding an effective one atom chain can be carried out in a similar way. Now we are going to address a problem of stochastic localizations in linear chains. It is worth noticing that a theoretical prediction of this problem has been given about 30 years ago [17]. In particular, Anderson showed that a probability density of an quantomechanical particle position moving in a certain probabilistic potential decays exponentially with enlarging of a distance to a center of localization space. In other words, it means that a particle is spatially localized (in the original work the phrase “absence of diffusion” is used).

366

12 LOCALIZATION

A similar like phenomena can be also observed in macrosystems. In what follows we analyse an example of one dimensional chain, governed by the equation mn u¨ n + Cn−1 (−un−1 + un ) + cn (un − un+1 ) + γn un = 0, n = 0, ±1, ±2, . . . .

(12.9)

It is assumed that the supporting springs act between each of p − 1 cells. Introducing ∞ γn = ∆γ j δn,p j , mn = m, cn = c, n = 0, ±1, ±2, . . . , j=−∞

and approaching continuum in a limit, the following equation is obtained ∞ ∆γ j n ∂2 un ∂2 un δ − j = 0. m 2 − c 2 + un p p ∂t ∂n j=−∞ Introducing a new spatial variable x = n/p and assuming a solution in the form of u = Ψ (x)eiωt the following equation governing wave mode behaviour is obtained (the so called Schrödinger equation for stationary states [419])   ∞  d 2 Ψ  2 + k − ν j δ(x − j) Ψ = 0, (12.10) 2 dx j=−∞ where:

m 2 2 p ω p , ν j = ∆γ j . c c Observe that the same equation governs behaviour of a quantum cell function, having in an external field the following potential energy k2 =

∞

ν j δ(x − j).

j=−∞

Furthermore, motivated by reference [112] we show that if the quantities ν j are stochastic, the wave function Ψ decays exponentially and in addition, we estimate a dimension of localisation space. The localisation domains correspond to chain oscillation domains. Consider now the equation (12.10). This equation, in all arbitrary interval n < x < n + 1, has the solution Ψ = An eikx + Bn e−ikx .

(12.11)

Since the function Ψ is continuous [419], taking into account the equation (12.10) one gets the equations Ψ (n + 0) = Ψ (n − 0),

12.1 Localization in linear chains

Ψ (n + 0) = Ψ (n − 0) + νn Ψ (n − 0),

367

(12.12)

governing a transition process through x = n. The obtained relations yield a coupling between coefficients of solutions (12.11) on two neighbourhood intervals An−1 An = W −n θn W n , (12.13) Bn Bn−1 where:

eik 0 W= , 0 e−ik

θn =

1−

iνn 2k

iνn 2k

− iν2kn . 1 + iν2kn

Applying sequently the transformation (12.13) on the subchain with length L, one obtains AL A0 = ML , (12.14) BL B0 ML = W −(L+1) WθL W . . . Wθ1 W.

(12.15)

Observe that a character of dependence of the matrix M L on the length L governs a behaviour of oscillation ‘amplitude’ while moving away from a certain center of excitation (in our case this is the interval 0 < x < 1). In accordance with quantomechanical language a smaller amplitude is associated with a smaller probabilisticity of a cell detection in the corresponding space domain. A cell possibility to wander between potential banners is either characterized by the transition coefficient T = |(ML )22 |−2 , or by non-dimensional damping R = T −1 − 1 = |(ML )22 |2 − 1. The localization effect correspond to the situation when the damping R increases with an increase of the distance L in an exponential manner R∼e

2 LL

0

,

which does not depend on L, so L0 ∼

2L 2L = . ln R ln I(ML )2 − 1 22

(12.16)

We explain how to find the quantity L 0 using and example of a chain with two types of barriers [119]. Let c 1 represent a concentration of the first type barriers, whereas c2 = 1−c1 corresponds to second type barriers. Therefore, the ‘unbounded’ average matrices M L can be found in result of substitution the expression c 1 θ(1) + c2 θ(2) instead of θn into (12.15), L ML = W −(L+1) W(c1 θ(1) + c2 θ(2) ) W, (12.17) where θ(i) is the matrix associated with the i-th barrier with the ‘stiffness’ ν (i) .

368

12 LOCALIZATION

The ‘unbounded’ average M L is coupled with the ‘bounded’ averaged M L N via relation L C LN c1N (1 − c1 )L−N ML N , (12.18) ML = N=0

where N is fixed number of the first type barriers. The relations (12.17), (12.18) yield M L N . Comparing the right hand sides of these relations, we get L

L C LN zN ML N = W −(L+1) W(zθ(1) + θ(2) ) W,

(12.19)

N=0

where: z=

c1 c1 = . c2 1 − c 1

Let W zθ(1) + θ(2) = AΛA−1 , where A = [a ik ] , i, k = 1, 2 is bending diagonalized matrix; A = diag [λ i ]; col(a1i , a2i ) and λi are the eigenvector and the eigenvalue correspondingly, of the matrix W(zθ (1) + θ(2) ). In addition, we take C LN = eL σN , where: σN = −cN ln cN − (1 − cN ) is the ‘entropy’ [420], whereas c N = N/L denotes first type barriers density. The equation (12.19) yields L

eL σN zN ML N = W −(L+1) AΛL A−1 W.

N=0

Hence, using the Cauchy integral formula, one obtains ? AΛL A−1 W e−L σN W −(L+1) ML N = dz. wπi zN+1 Applying the hypothesis c N = c1 , after some manipulations, the following matrix element is obtained

? ? eikL (ML N )22 = ϕ2 (z)eL(ln λ2 −c1 ln z−σN ) dz − ϕ1 (z)eL(ln λ1 −c1 ln z−σN ) dz , 2πi (12.20) where: a22 a11 a12 a21 , ϕ2 = . ϕ1 = det A det A The obtained expression is suitable for an asymptotic (for large L) estimation using the saddle-point method. The following estimation is found |(ML N )22 | ∼ √

1 2πL

|ϕ(z)| LRe f (z) e , / | f (z)|

(12.21)

12.1 Localization in linear chains

369

where: f (z) = ln λ − c1 ln z − σN ; ϕ(z) represents either ϕ 1 (z) or ϕ2 (z); λ corresponds to λi , i = 1, 2, for which Re f (z) is larger one; z denotes the corresponding stationary point being a root of the equation d f (z) 1 dλ c1 = − = 0. dz λ dz z

(12.22)

An eigenvalue of the matrix W(zθ (1) + θ(2) ) is obtained from the equation λ2 − 2(α1 z + α2 )λ + (1 + z2 ) = 0, sin k (1) V , i = 1, 2. (12.23) k Differentiating (12.23) with respect to z, and comparing the obtained result with (12.22), one gets c1 (1 + z) λ= . c1 α2 − c2 α1 z Substituting the obtained expression into (12.23) one obtains and equation with respect to z. Its roots must be chosen in a way to realize occurrence of the largest value of Re f (z). Hence, applying (12.21) and (12.16), the following estimation of the space localization dimension is obtained: αi = cos k +

L0 ∼ [Re(ln λ − c1 ln z − σN )]−1 . The transition coefficient T is proportional to second power amplitude of the wave A on the considered distance L. Hence, the following estimation holds A ∼ e−L/L0 . In Fig. 12.3, the dependence L O n the function of wave number k for different concentrations of first type barriers is reported. The computations [119] are carried out for V (1) = −1, V (2) = 0. The curves discontinuity occurred for k = π can be explained in the following way. The barriers are situated in waves nodes and don’t influence on the oscillation form. Therefore, they do not interact with chains oscillation modes. The quantum particle of the corresponding energy is not localized, and hence the point k = π is called the ‘mobility point’. In its neighbourhood the following estimation holds ) (π − k)−1/2 , π > k . L0 ∼ (k − π)−1 , k > π Note that analogous localization effect of linear modes is also possible when the barriers have the same stiffness (they are of the same type), but they are situated in a non-regular (chaotic) way. It should be emphasized, that contrary to the quantomechanical systems the described effects in real microsystems always exist with dissipation effects. The latter are exhibited via decrease of oscillation wave while amplitude is moving away from the excitation space, i.e. the process is analogous to the described stochastic localization with principally different physical nature.

370

12 LOCALIZATION L0

c1 = 0.5 c1 = 0.1

10

3

10

2

10

1

c1 = 0.9

k 1.0

2.0

3.0

4.0

Fig. 12.3. Characteristic dimension of the space localization versus wave number for various concentrations of barriers.

12.2 Localization in nonlinear chain Consider the infinite chain of masses coupled via linear and elastic springs. The barriers now are assumed to be nonlinear with the same characteristics in the form of an odd polynomial. The equation of motion have the following form u¨ n − un−1 + 2un − un+1 + γu2k−1 = 0, n

n = 0, ±1, ±2, . . . .

(12.24)

For k = 1 one deals with the linear case, where is a lack of energy localization. For k = 2, 3, . . . a frequency of the excited oscillator, which depends in nonlinear case on excitation energy, will be larger than the frequencies of non-excited oscillators. Obviously, for a given energy distribution along a chain one may find an equivalent linear chain with respect to frequencies and with appropriately chosen stiffness of the spring (note, that dynamics of both systems is different). A stochastic energy dissipation along a chain corresponds to a linear chain with stochastic inclusions with respect to barriers stiffness. An energy supply to one of the non-linear oscillators is approximated by an increase of its barrier stiffness in an equivalent linear chain. Evidently, the described equivalency with respect to a

12.2 Localization in nonlinear chain

371

chosen linear chain will be conserved in time, if in a nonlinear chain energy pumping between oscillators does not appear. Observe that the latter case is characteristic one for energy localization process. Now we are going to give a brief description of the qualitative results. In particular, for a being considered chain (12.24) an approximate solution corresponding to localized standing wave is obtained. Let ω 2 − ω2max = ε2 1, where ωmax = 2 s the maximal frequency of the free oscillators at the system (12.24) after linearization (γ = 0). Then, for k = 2 in the first approximation with respect to the terms of ε order, one obtains √ (12.25) un = (−1)n 4ε sin ωtschεn, ε = ω2 − 4. Let us analyse the expression (12.25). Observe that increasing the energy supply a oscillation frequency increases and the localization space decreases. For ω 2 a continual approximation of the describing function has no physical meaning (also a definition of a describing functions looses its sense). On the other hand, the localization effect (in a given chain) can be observed in the case of rather high energy of oscillations in order to satisfy the inequality ω2 − 4 > 0. If energy amount is sufficiently small, then orbitally stable periodic processes can be constructed by improving linear normal oscillation approach, for which energy distributed to all particles is equal. The frequencies of oscillations with localized modes lie outside of the spectrum of linear vibrations. The latter ones are responsible for energy pumping between particles. This observation leads also to conclusion that energy pumping from the localized space does not occur. The expression implies that a frequency of localized vibrations should be higher than the higher frequency of linear vibrations. In the case of frequencies zone, which lies below the lowest frequency of linear vibrations, a localization effect is also possible to observe. For example, let us consider a chain governed by the equations u¨ n − un−1 + 2un − un+1 + un − βu3n = 0,

n = 0, ±1, ±2, . . . .

In a continual approximations one gets [393] un = A(n) cos ωt + B(n) cos 3ωt, where

8 A= 3β

1/2

ε , chεn

1/2 1 8 ε3 , B= 12 3β ch3 εn

ε=

√ 1 − ω2 .

Note that this type of localization exist even for very small amplitudes although it is only slightly exhibited. However, this effect vanishes completely if non-linear terms are omitted. An existence of non-linear terms possesses a crucial meaning, since they are responsible for a satisfaction of the inequality 1 − ω 2 > 0, i.e. a transition beyond the left boundary of the spectrum of a linear system.

372

12 LOCALIZATION

Therefore, non-linearities occurred in the described systems play a role similar to that of violation a periodical structure of a chain. In the localization space the effective chains stiffness differs from a stiffness of its rest, i.e. non-excited part. This implies a frequency transition beyond a border of the spectrum of the linearized (homogeneous) chain, which resists an energy pumping from the excited space. It should be emphasized additionally that the described mechanism can be treated as a general one and does not depend on a particular chain structure.

12.3 Localization of shell buckling Let us consider a buckling of structurally orthotropic axially compressed cylindrical shell (see Chapter 5.4). We consider a case, when the edges of circular shell is simply supported, but free in circumferential direction w = M1∗ = S = T 1 (or u) = 0 for ξ = 0, d ,

(12.26)

where: M1∗ = M1 − e1 T 1 . As it has been shown [6, 317, 341, 362, 614, 618, 644], a change of the “classical” boundary condition υ = 0 for S = 0 for the isotropic cylindrical shells decreases the buckling value of the axial compression in two times. It also leads to a sufficient change of the buckling form. In this case the exact solution is obtained in the closed form. However, for the reinforced shells such possibility does not exist because of the occurrence of a large number of geometrical and stiffness parameters. Their influence on the shell’s behaviour, particularly in the relatively wide intervals, becomes complicated. In the case of conical shells difficulties increase, because the stability equations have the variable coefficients. Some of the results for the isotropic conical and ring stiffened cylindrical shells have been obtained using the numerical methods [134, 135, 689]. The analysis of the known solutions allows finding and discussing a series of characteristic properties. First of all, slow variation is observed in the ring direction. Secondly, a clear separation of a shell from the parts with high (low) stress and displacements variation in the longitudinal direction is noticed. The first zone is located in the neighbourhood of the edges, whereas the second one lies in the remaining part of the shell. Therefore, in order to describe in this case the buckling, we need to consider the equations (5.65) and (5.66). The link between them is realised using the splitting boundary conditions (12.26): for ξ = 0, d w(0) + w(1) = 0, T 1(0) + T 1(1) = 0 (or u(0) + u(1) = 0), S (1) = 0,

M1(1) = 0.

Therefore, the critical load of the axial compression is defined by solving the problem of the eigenvalues determination for the equation (5.66) for the following boundary conditions for ξ = 0 S (1) = 0, M1(1) = 0,

12.4 Localization of vibration in plates and shells

for ξ → ∞

373

S (1) → 0, M1(1) → 0.

As the result we obtain 2 −1/2 . T 10 = B2 ε1 1 + ν12 ε−2 1 ε6

(12.27)

For the isotropic case (ν = 0.3, ε 6 = 0), we obtain T 10 = 0.3Eh2R−1 , which coincides with the known solutions. For the ring-stiffened shells the expression (12.27) is verified using the numerical techniques given in reference [135, 689]. In the analogous way, the formula for the buckling axial compression for the boundary conditions (12.26) can be obtained for the conical shells , B 2 D1 ε1 , (12.28) T 10 = R0 ε21 + ν212 ε26 where: ε21 = BDR1 2 , ε6 = e1 /R0 , R0 = max {R(0), R(d)}. 2 0 We can ‘froze’ a curvature radius on the edge, because the considered state fast decaying when ξ → ∞. A comparison of formula (12.28) with known results [134] exhibits its high accuracy.

12.4 Localization of vibration in plates and shells We consider vibrations of a rectangular plate (0 ≤ x ≤ a, 0 ≤ y ≤ b), governed by the equation [102, 352] D∇4 w − ρω2 w = 0. Let the edges y = 0, b are simply supported w=

∂2 w = 0 for y = 0, y = b, ∂y2

whereas the edge x = 0 is free, i.e. ∂2 w ∂2 w ∂3 w ∂3 w + ν = 0, + (2 − ν) = 0 for x = 0. ∂x2 ∂y2 ∂x3 ∂x∂y2 Note that the boundary conditions on the edge x = a are not defined yet, since the results are do not depend on sufficiently large values of a. First, the smallest plate eigenfrequency is going to be found for a → ∞. Separating the variables and introducing the new variable via the formulas w (x, y) = w (x1 ) sin

πy πx , x1 = , b b

the following ordinary differential equation defines the function w(x 1 )

(12.29)

374

12 LOCALIZATION

d4 w d2 w − 2 2 + w − Ωw = 0, 4 dx1 dx1

(12.30)

with the attached free boundary conditions d2 w d3w dw − νw = 0, − (2 − ν) = 0 for x1 = 0. 2 dx1 dx1 dx31

(12.31)

Here

ρhω2 b4 . Dπ4 If a cylindrical bending of infinitely length in x 1 direction of the plate is analysed, then the function w does not depend on x 1 and Ω = 1. If the edges x = 0, a simply supported, then Ω=

w (x1 ) = sin

2 bx1 b2 , Ω = 1 + 2 > 0, a a

and for b/a → 0 the parameter Ω → 1. Let Ω < 1. A non trivial solution to the equation (12.30), satisfying condition w(x1 ) → 0 for

x1 → ∞,

is sought in the following form w (x1 ) = C1 e−z1 x1 + C2 e−z2 x1 , where z1 and z2 are the roots of the equation 2 z2 − 1 − Ω = 0,

(12.32) +

and C 1 and C 2 are arbitrary constants. Hence, from (12.32) we obtain z 1 = + z2 = 2 − z21 . The boundary conditions (12.31) yield the equation 2 z1 (z2 − ν)2 = z2 ν − z21 ,

1−

√ Ω,

(12.33)

which possesses only one root for Ω < 1. Taking into account that ν < 0.5, the following relations are obtained √ 2 4ν4 2ν Ω=1− , z1 = . 2 (2 − ν) (2 − ν)2 Particularly, for ν = 0.3 the following values are found: Ω = 0.9962, z 1 = 0.0436, z2 = 1.4135.

12.4 Localization of vibration in plates and shells

375

Vibration localization in edge neighbourhood exhibits a very interesting behaviour, namely the so called edge resonance occurs [307, 322, 371]. Now we are going to show that edge resonance can be also exhibited by shell vibrations. Let us consider an isotropic circular cylindrical shell. The simplified equations governing oscillations of axially compressed cylindrical shell have the following form ∂4 w1 ∂2 w1 + (1 − ω2 )w1 = 0; (12.34) a2 4 + T ∂ξ ∂ξ2 4 2 4 2 ∂ w0 ∂ ∂ w0 2 ∂ +1 +T +1 + a ∂η2 ∂η4 ∂η2 ∂ξ2 ∂η2 2 2 ∂ w0 ∂4 w0 2 ∂ −ω +1 = 0, (12.35) ∂ξ4 ∂η2 ∂η2 where: η = y/R, ξ = x/R; T is axial compressed load. Equation (12.34) corresponds to edge effect, and equation (12.35) corresponds to semi-inextensional theory. The splitting procedure of the boundary conditions (12.27) indicates two possible variants for ξ = 0, d u0 (or T 1(0) ) = w0 = 0, (12.36) or

S (1) = −S (0) , M1(1) = −M1(0) ,

(12.37)

S (1) = M1(1) = 0,

(12.38)

u0 = u1 (or

T 1(0)

=

T 1(1) ),

w0 = w1 .

(12.39)

The eigenvalue problem (12.34)–(12.37) exhibits non-decay in x - direction oscillations mode, which is usually considered. It has not been further analysed. The equation (12.34), (12.35) and the boundary conditions (12.37), (12.38), (12.39) govern not earlier analysed form of oscillations located in the vicinity of the shell edges. For a given T the equation (12.34) with the boundary conditions (12.38) is reduced to the eigenvalue problem, which yields the frequency ω. In order to define the eigenforms the equations (12.34) and (12.35) with the boundary conditions (12.38) and (12.39) must be solved simultaneously. Since we have the closed shell, the solutions to equations (12.34), (12.35) have the following form w0 = w˜ 0 (ξ) cos(nη), w1 = w˜ 1 (ξ) cos(nη). We are going to find the decay solutions for ξ → ∞. The existence condition for exponentially decay solutions to equation (12.34) reads T 2 < 4(1 − ω2 )a2 .

(12.40)

Satisfying the condition (12.40) the boundary condition for ξ = l is equivalent to the following one

376

12 LOCALIZATION

{w1 , (∂w1 /∂ξ)} → 0 for ξ → ∞.

(12.41)

The boundary value problem (12.34), (12.38), (12.41) yields ω2 = 1 − T 2 a−2 . For the decay eigenform w 0 the following inequality holds 1 − a2 n2 (n2 − 1) + T 2 [0.25n2(n2 − 1) − a2 ] < 0. For structurally orthotropic shell one has (ε1 + ν212 ε24 )

∂4 w1 ∂2 w1 + (−2ν12 ε4 + T ) 2 + (1 − ω2 )w1 = 0; 4 ∂ξ ∂ξ

∂2 +1 ∂η2

2

∂4 w0 ∂4 w0 + + (−2ν12 ε5 + T )× ∂η4 ∂ξ4 2 4 2 2 ∂ ∂ w0 ∂ w0 2 ∂ −1 −ω +1 = 0. ∂η2 ∂ξ2 ∂η2 ∂η2 ∂η2

(ε1 ε2 ε3 +

ν212 ε25 )

(12.42)

(12.43)

Use of the equation (12.42) and the conditions (12.38), (12.41) gives ω2 = 1 − (T − 2ν12 ε4 )2 (ε1 + 2ν212 ε24 )−1 . The decay condition for the function w 0 for ξ → ∞ can be presented in the following form: 1 − (T − 2ν12 ε5 )2 /4 n2 (n2 − 1) − (ε1 ε2 ε3 + ν212 ε25 ) × n2 (n − 1)− (T − 2ν12 ε4 )2 (ε1 + ν212 ε24 )−1 < 0. To conclude, the obtained results show a possibility of vibration mode localizations for conditions (12.26).

13 IMPROVEMENT OF PERTURBATION SERIES

One can calculate only a few terms of perturbation expansion, usually no more then two or three, and almost never more than seven. The resulting series is often slowly convergent, or even divergent. Yet those few terms contain a remarkable amount of information, which the investigator should do his best to extract [660,p.202].

The principal shortcoming of perturbation methods is the local nature of solutions based on them. Besides that, the following questions are very difficult for the theory: what values may ε be considered as small (large)? How can a solution for any ε be constructed if its behaviour is known for ε → 0 and ε → ∞? As the technique of asymptotic integration is well developed and widely used, such problems as elimination of the locality of expansion, evaluation of the convergence domain, construction of uniformly suitable solutions are very urgent. There exists series of methods yielding answers to the stated questions. One of the most representative seems to be Padé approximants approach.

13.1 Padé approximants (PA) In this section the approaches based on transformation of the input series of the perturbation method to a fractional rational function will be treated. Such transformation can be performed using different methods: nonlinear transformation of the input series [359], continued fraction [610, 659, 660, 661, 662], PA [121, 122]. Below we deal with PA. Let us consider PAs which allow us to perform, to some extent, the most natural analytical continuation of the power series. Let us formulate the definition. Let F(ε) =

∞

c i εi ,

i=0

Fmn (ε) =

m i=0

a i εi /

n

b i εi ,

i=0

where the coefficients a i , bi are determined from the following condition: the first (m + n) components of the expansion of the rational function F mn (ε) in a Maclaurin

378


series coincide with the first (m + n + 1) components of the series F(ε). Then F mn is called the [m/n] PA. The set of F mn functions for different m and n forms the Padé table. The coefficients a km and bkm are defined by the equation. ∞

m k=0 n

k

ck ε −

k=0

k=0

akm εk bkn εk

= O(εm+n+1 ).

Denoting by [m/n] a ratio of the corresponding powers of polynomials, one can construct the Padé table including all of the approximations [0/0] [0/1] ... [0/n] ... [1/0] [1/1] ... [1/n] ... . .. ... . [n/0] [n/1] ... [n/n] ... .

.

.

.

. .

.

.

.

The table elements [k/k], k = 0, 1, ..., are called diagonal PA. They are most widely used in practice. Let us notice that a PA is unique when m and n are specified. To construct the PAs, it is necessary to solve systems of linear algebraic equations (for optimal methods of PA coefficients determination see [121, 122]). PA performs meromorphic continuation of the function given in the form of the power series, and for this reason it allows to achieve success in the cases where analytic continuation can not be applied. If the PA sequence converges to a given function, then the roots of its denominators tend to singular points. It allows to determine sufficiently large number of the series components the occurred singularities and then to perform the analytic continuation. The data concerning PA convergence could have application in practice only as opinions which would enhance the reliability of the results. Indeed, in practice it is possible to construct only a limited number of PA while all convergence theorems require information about an infinite number of them. Gonchar’s theorem [315] states that if none of the diagonal PAs ([n/n]) has any pole in a circle of a radius R, then the sequence [n/n] converges uniformly in the circle to the input function f . Further, the lack of poles in the sequence [n/n] in a circle of radius R implies the convergence of an initial Taylor series in the circle. As the diagonal PAs are invariant to the fractional-linear transformations z ⇒ z/(αz + β), then the theorem is valid for any open circle containing the expansion point and for any domain being a union of such circles. The theorem has one important consequence for continuous fractions. Namely, the holomorphity of all suitable fractions of an initial continuous fraction inside a domain Ω implies uniform convergence of the fraction inside Ω. An essential disadvantage in practice is the necessity to verify all diagonal PAs. The point is that if inside a circle of the radius R only some subsequence of the

13.2 The effect of autocorrection

379

diagonal sequence PA has not any pole, then its uniform convergence to the initial holomorphic (in the given circle function), is guaranteed only for r < r 0 , where 0.583R < r0 < 0.584R [680]. There exists a counter-example showing that in general r < 0.8R. As in practice only a finite number of components of the series of the perturbation theory is known and there are no estimations of the convergence rate, then the above mentioned theorems could only increase the likelihood of the obtained results. This likelihood is also augmented by some known “experimental results”, since the practice of PAs application shows that the convergence domain of PA sequence is usually wider that the convergence domain of the input series [121, 122]. Let us note that widely applied continued fractions [359] form a particular case of PAs. Exactly, the suitable fractions, representing the sequence of approximations of the continued fraction, coincide with the following PA sequence: [0/0], [1/0], [1/1], [2/1], [2/2]... That is why we shall not separate the case of the continued fractions application.

13.2 The effect of autocorrection A wide application of the PA is observed due to its suitable properties. Among others, we must mention a so called effect of error autocorrection [442, 447, 448]. This effect occurs in efficient methods of rational approximation (e.g., PA, linear and nonlinear Padé-Chebyshev approximations), where very significant errors in the coefficients do not affect the accuracy of the approximation: this is because the errors in the numerator and the denominator of a fractional rational approximant compensate each other. This effect is because the errors in the coefficients of a rational approximant are not distributed in an arbitrary way, but form the coefficients of a new approximant to the approximated function. The understanding of the error autocorrection mechanism allows us to decrease this error by adapting the approximation procedure to the form of the approximant. Here we follow [442]. Let the expansion of a function f (x) into a Maclaurin series be given, f (x) =

∞

c1 xi .

i=0

The PA for f (x) is a rational function of the form R(x) =

Pn (x) , Qm (x)

(13.1)

where Pn (x) and Qm (x) are polynomials of degree n and m, respectively, satisfying the relation Qm (x) f (x) − Pn (x) = O(xm+n+1 ). (13.2) Let

Pn (x) = a0 + a1 x + · · · + an xn ,

380


Qm (x) = b0 + b1 x + · · · + bm xm . If b0 0, then (13.2) means that f (x) − Pn (x)/Qm (x) = O(xm+n+1 ), i.e., the first m + n + 1 terms of the Taylor expansion in power of x (to x m+n inclusive) of f (x) and R(x) are the same. The PA gives the best approximant in a small neighbourhood of zero; it is a natural generalization of the expansion of functions into Taylor series. One can evaluate the coefficients b j in the denominator of fraction (13.1) by solving the homogeneous system of linear equations m

cn+k+ j b j = −b0 cn+k ,

(13.3)

j=1

where: k = 1, ..., m and c l = 0 for l < 0. One can take any nonzero constant as b 0 . The coefficients a i are given by formulas i i bk ci−k = bi−k ck . (13.4) ai = k=0

k=0

For large m the system (13.3) is ill-conditioned. Moreover, the problem of computing the coefficients of PA is also ill-conditioned. Let ∆a i , ∆b j be the errors in the coefficients ai , b j which arise when numerically solving system (13.3). We shall ignore the errors of the quantities c i and x and we shall assume that, according to (13.4), the errors in the coefficients a i have the form ∆ai =

i

∆bi−k ck .

(13.5)

k=0

From (13.5) it follows that f ∆Q − ∆P =

m

∆b j x j

j=0 m ∞ i=0 i=0

∆bk ci xi+ j −

∞

ci xi −

i=0 n i

n

∆ai xi =

i=0

∆bi−k ck xi ;

i=0 i=0

the latter, after a change of indices, yields a relation similar to (13.2): ∆Q(x) f (x) − ∆P(x) = O(x n+1 ). Thus, there are reasons to expect that the error approximant indeed approximates the function f (x) and the effect of error auto-correction takes place. A natural generalization of the classical PA is the multipoint PA or PA of the second kind, i.e. a rational function of the form (13.1) whose values coincide with

13.3 Extending of perturbation series

381

values of the approximated function f (x) at some points x i (i = 1, 2, ..., m + n + 1). This definition is extended to the case of multiple points, and for x i = 0 for all i it leads to the classical PA. The calculation of coefficients in the multipoint PA can be reduced to solving a system of linear equations, and there are reasons to suppose that in this case the effect of error auto-correction takes place as well.

13.3 Extending of perturbation series As an example we consider the equations of a plate motion including a geometrical non-linearity of the form (Karman equations)

2

2

D 4 ∂2 w ∇ w = L(w, Φ) − ρ 2 , h ∂t

(13.6)

1 4 1 ∇ Φ = − L(w, w), E 2

(13.7)

2

2

2

2

∂ w ∂ Φ where: L(w, Φ) = ∂∂xw2 ∂∂yΦ2 + ∂∂yw2 ∂∂xΦ2 − 2 ∂x∂y ∂x∂y . Assume that the following boundary conditions are applied

for

x = 0, a w = 0,

w xx = 0,

for y = 0, b w = 0, wyy = 0, for x = 0, a; y = 0, b u = v = 0,

(13.8) (13.9)

where: u and v are displacements along the axes x and y, correspondingly. The following deflection function satisfies the boundary conditions (13.8) is used: πy πx sin . (13.10) w = f (t) sin a b The function Φ is defined by (13.7), and taking into account the boundary conditions (13.9), we get   2 b f 2  a 2 2πx 2πy  P x y2 Py x2 φ = E  + + , (13.11) cos cos + 32 b a a b  2 2 2

2

2

2

2

Ef +ν)π E f a where: Py = (1+νλ)π , P x = (λ8(1−ν 2 )a2 , λ = b . 8(1−ν2 )a2 Taking into account (13.11) and using the Bubnov–Galerkin method, the partial differential equations (13.6) yield the following second order ordinary non-linear differential equation: d2 f¯ + ω20 (1 + Q f¯2 ) f¯ = 0, (13.12) dt2 where:

0.75(1 − ν2 ) 4 h2 1.5 2 4 2 {[−νλ + λ + (1 + νλ )] + (λ + 1)} , Q= ' ( ( ' b2 λ2 + 1 2 λ2 + 1 2

382


f f¯ = , h

ω20 =

π4 Eh2 ρλ4 (λ2 + 1)2 . 12b4 (1 − ν2 )

Since the obtained equation is well known (it is called the Duffing equation), and since its solution, can be found using different methods, then one can estimate an efficiency of the PA. √ After applying the following variables transformation ϕ = f¯/ Q, t = τ/ω0 , the equation (13.12) yields d2 ϕ + ϕ + ϕ3 = 0. (13.13) dτ2 The exact expression for frequency of equation (13.13) has the value √ π 1 + A2 ω= √ , (13.14) 2 2 K(θ) + A2 where: θ = arctan (2+A 2 ) , K(θ) denotes the elliptic integral, A is amplitude. The series of the amplitude-frequency characteristic has the form 21 4 81 6 6549 8 3 A + A − A + ω = 1 + A2 − 8 256 2048 262144 37737 10 9636183 12 A − A + ... . (13.15) 2094152 67108864 Taking into account the first three terms of the series (13.15), the following second order PA can be constructed +

ω2 =

1 + 0.59A2 + ... . 1 + 0.22A2 + ...

(13.16)

When the terms of A 6 order are included one gets ω4 =

1 + 1.13A2 + 0.261A4 + ... . 1 + 0.756A 2 + 0.0599A 4 + ...

(13.17)

Extending further the obtained process the following series of diagonal PA is obtained 2n 1 + αi A2i i=1 ω2n = , (13.18) 2n 2i 1 + βi A i=1

where: n = 1, 2, 3, .... The coefficients αi , βi can be found by solving the linear system of algebraic equations. The computational results obtained from formulas (13.14)–(13.18) are presented in Figure 13.1. The exact solution is denoted by dashed curve, whereas the diagonal PA [2/2], [4/4] and [6/6] correspond to the curve numbers 1, 2, 3, respectively. It

13.3 Extending of perturbation series

383

is seen that the PA essentially extends an interval of application of the approximate solution. The most effective are the diagonal approximants. Another popular etalon equation which attracted an attention of many researchers is the Van der Pol equation of the form u¨ + ε(1 − u2 )˙u + u = 0.

(13.19)

It occurs that the PA allows to investigate the amplitude-frequency characteristics in this case.

Fig. 13.1. Efficiency of the perturbation series and PA for the Duffing equation.

The classical perturbation method gives the following approximation ω =1−

1 2 51 4 ε + ε + ..., 16 9216

(13.20)

and the PA gives

1 + 0.02604ε 2 + ... . (13.21) 1 + 0.08854ε 2 + ... A comparison of the results obtained using formulas (13.20) and (13.21) and the numerical results for equation (13.19) [649] is given in the Table 13.1. ω=

Table 13.1. Comparison of numerical and analytical results for Van der Pol equation. ε (13.20) (13.21) Numerical solution [649]

0 1.0 1.0 1.0

1 2 3 4 0.943 0.838 0.885 1.416 0.942 0.815 0.687 0.586 0.940 0.820 0.710 0.620

384


The PA essentially increases an exactness of the perturbation series. The described examples also give a good prognosis on application of PA for estimation of validity of the used perturbation parameter approach. A use of PA also allows to omit many problems concerning investigations devoted to obtaining the explicit estimations. They are so tedious that they are usually omitted and a rigorous proof of series convergence for sufficient small parameter is used to take the finite real numbers of this parameter. A transformation of an input asymptotic series using PA does not lead to any practical difficulties and gives estimations fully acceptable by practical calculations.

13.4 Improvement iterational procedures convergence The efficiency of application of PA essentially depends on occurrence of high terms of an asymptotic series. This fundamental drawback can be often overcome using computers, but the problem in general still remains open. It seems that more simple tools are those related to iterational processes. Consider the following iterational process T (u0 ) = 0,

un = T (un − 1),

n = 1, 2, 3, ... .

The solution u can be formally presented in a form of the series u = u0 + (u1 − u0 )ε + (u2 − u1 )ε2 + ... + (un − un−1 )εn + ... .

(13.22)

For ε = 0 we have u = u 0 , whereas for ε = 1 we have u ≈ u n . The series (13.22) can be presented as a rational function using the PA u0 + 1+

m i=1 p

j=1

a i εi

β jε j

− [u0 + (u1 − u0 )ε + ... + (uk − uk−1 )εk + ...] = O(εk+1 ).

For ε = 1 u0 + u≈

1+

m i=1 p j=1

(13.23)

ai

βj

.

As an example we consider large deflection of circled isotropic plate (R 0 ≤ r ≤ R) with a clamped contour with the surface load q ≡ const. A solution to this problem is given in reference [377] using the method of finite differences for E = 62.4 kg/cm 2 , ν = 0.335, R 0 /R = 0.1. For large loading the method of successive approximation of the non-linear algebraic equations is convergent after iterations steps of order 150 − 200 and the convergence has oscillation - like character.

13.4 Improvement iterational procedures convergence

385

Table 13.2. Results of successive approximations. Iteration number 0 1 2 3 4 5 6

N 5.27286 1.09640 4.81246 1.45039 4.55120 1.67086 4.37191

Iteration number 7 8 9 10 ... 145 146

N

Iteration N number 1.82867 147 3.02603 4.23735 148 3.11236 1.94992 149 3.02680 4.13072 150 3.11063 ... 151 3.02849 3.023320 152 3.10890 3.11416

In the Table 13.2 the results of non-dimensional radial stress N = N r R2 /D for ρ = 1 (ρ = r/R), qR4 /(2Dh) = 35 are given. An application of a generalized summation method [377] improves the results (Table 13.2). Table 13.3. Results of generalized summations. Iteration number 1 2 3 4 5

N 2.6955 3.1917 3.0140 3.0941 3.0656

Iteration number 6 7 8 9 10

N 3.0801 3.0760 3.0791 3.0789 3.0789

Lets us use PA. For our case the series (13.23) has the form (four terms are used): T = 50319 − 4.243ε + 3.794ε 2 − 3.451ε3 + ... , whereas the PA gives T=

5.319 − 284.883ε − 27.606ε 2 + ... . 1 − 52.762ε − 47.992ε 2 + ...

(13.24)

For ε = 1 the formula (13.24) gives T = 3.079. In order to illustrate the application of PA, the iterative technique results obtained in reference [655] will be also considered. In that reference the iterative technique is used to obtained an analytical approximation to the homoclinic loops of the Lorenz system dx x˙ ≡ = a (y − x) , dt dy y˙ ≡ = cx − y − zx, dt

386


dz = xy − bz, dt for the following fixed parameter values a = 10, b = 8/3 and c = 13.926. First, the local structure of the homoclinic solution for t → ±0 and t → ±∞ was analyzed and then the global solutions were constructed on the basis of quasifractional approximants [655] (concerning quasifritcional approximants see Chapter 14.2). The accuracy of the approximation is improved iteratively. Since the Lorenz system is autonomous, one can arbitrarily choose the initial conditions of the orbit to be the point where x(0) = y(0) = y 0 , z(0) = z0 . Fixing two of the Lorenz parameters to the values a = 10, b = 8/3, each iteration provides estimates for the initial conditions for the homoclinic orbit and the value c. The results for three successive iterations are given in Table 13.4. z˙ ≡

Table 13.4. Iterations and results. No· of iterations y0 0 15·08854 1 11·55446 2 11·65734

z0 c 28·92802 17·63087 19·02569 13·45975 19·27001 13·91258

The accuracy control is related to the values of the parameter c. The calculations indicate that the use of a zero order approximation for the iterational technique leads to relatively high errors. Therefore, it seems to be more efficient to begin with the first approximation u ≈ u1 + (u2 − u1 ) ε, and use PA

u21 , ε = 1. 2u1 − u2 The computational results are included in Table 13.5. u≈

Table 13.5. Results obtained using Padé approximants. y0 11·65734

z0 19·27001

c 13·9283

Following the methods of reference [655], one can estimate the accuracy of the calculated parameter c. Hassard and Zhang [333] used numerical shooting to provide improved bounds on the value c for which the homoclinic orbit exists. They showed that for a = 10 and b = 8/3, c ∈ [13 · 9265, 13 · 9270]. Accordingly, c in Table 13.5 is obtained with an error less than 0·013%, whereas the third iteration has an error of 0.096%. As it can be seen, without any additional calculations the accuracy of getting initial conditions for homoclinic orbits of the Lorenz system is increased by one order of magnitude.

13.5 Nonuniformities elimination

387

13.5 Nonuniformities elimination A transformation to a rational function allows one to describe nontrivial behaviour at infinity and to take into consideration the singular points of the sought solutions. We shall consider, as an example, the problem of flow around a thin elliptical airfoil (|x| ≤ 1, |y| ≤ ε, ε 1) by a plane stream of a perfect liquid incoming with the velocity ν. A few components of the asymptotic expansion of the relative stream velocity q on the airfoil surface are like this [660] x2 1 − ... q ∗ = 1 + ε − ε2 2 1 − x2

(13.25)

which diverges for x = 1. After replacing the expansion (13.25) by PA, the singularity for x = 1 disappears: (1 − x2 )(1 + ε) q∗ = + 0(ε4 ). (13.26) 1 − x2 + 0.5ε2 x2 Figure 13.2 presents for ε = 0.5: curve 1 is exact solution; dashed line is solution (13.25); curve 2 is PA (13.26); point line is solution according to the Lighthill method (see Chapter 2.8), which gives in this case worse results than PA.

Fig. 13.2. Illustration of nonuniformity elimination using PA.

PA does not always give such satisfactory results for all problems. It would be interesting to find general relationship (related perhaps to group features of the considered equations [312])of removing nonuniformities of asymptotic expansions with PAs.

388


13.6 Error estimation of asymptotic approaches As it has been already mentioned the following principal question related to application of perturbation techniques appears: How to estimate a magnitude of the perturbation parameter, which can be treated as a “small” one? It appears that a formal procedure to estimate the intervals of application of asymptotic parameter methods can be (in many cases) solved using the PA. Practical calculations show that an interval of convergence of the obtained series is not smaller than that of the Taylor series [121, 122]. Let us analyse the Duffing oscillator (see Chapter 13.3). It is seen that using PA a domain of application of small parameter A 2 can be estimated. In the space, where the PA, for instance [6/6], overlaps with the perturbation solution, both of them are closed to exact one. Using the technical accuracy of 5% the following estimation holds: 0 ≤ A ≤ 1.51. The Van der Pol equation is further discussed in a similar way. It will be shown that PA allows to estimate the interval of small parameter 0 ≤ ε ≤ 3.45 (see Chapter 13.3). What happens when exact (or numerically found) solution is not known? In this case we recommend to find an approximate solution using two different methods and then to find intervals, where they overlap. For instance, a perturbated solution can be estimated by its extension described earlier and then to compare it with the expression obtained using PA. PA can be also applied to construct a solution in a case of small values of ε, when for large ε values it is found numerically. Assume that we are going to solve a problem with rapidly oscillating coefficients [124]. For enough large ε values (for instance, ε = 0.5 and more) a solution can be found numerically, whereas for small ε values it is difficult to apply the numerical approaches. However since ε is small it is easy to present a solution in the form u = u0 + εu1 + ε2 u2 + ... .

(13.27)

If the numerical solution is known for ε = ε 0 , ε = ε1 , ε = ε2 , than taking into account only three terms of the series (13.27) related to ε one can find the values of u0 , u1 , u2 . Then, using PA, a solution valid for arbitrary ε can be constructed.

13.7 Localized solutions and blow-up phenomenon Localized solutions have been widely used in mechanics [1]. These are essentially nonlinear solutions which cannot be constructed using the quasi-linear approach when any number of components is conserved. It is still more interesting to note that PA allows to construct solutions of that type beginning from local (quasi-linear) expansions [416, 417]. Moreover there has appeared the term “padeon”. A model example is presented by the boundary value problem y − y + 2y3 = 0,

y(0) = 1,

y(∞) = 0,

13.8 Gibbs phenomena

which has the exact solution

389

y = ch−1 (x).

A solution in the form of the Dirichlet series y = Ce−x (1 − 0.25C 2e−2x + 0.0625C 4e−4x + ...),

C = const,

after rearrangement into PA and determination of C from the initial conditions becomes the exact solution. Solutions of nonlinear problems (e. g. calculation of nonlinear boundary layers) in the form of Dirichlet series are widespread in the contemporary mechanics. It may be expected that their combination with the PA technique should lead to interesting results. Application of PAs for jump and blow-up phenomena seems to have favourable prospects. In order to illustrate an application of PA to blow-up problems we consider the following problem dx = ax + εx2 , x(0) = 1, dt where: 0 < ε α 1. The exact solution to this boundary value problem has the following form x(t) =

α exp(αt) . α + ε − ε exp(αt)

(13.28)

For t → ln[(α + ε)/ε] the solution goes to infinity (blow-up of solution appears). A regular asymptotics of the form x(t) ∼ exp(αt) − εα−1 exp(αt)[1 − exp(αt)] + ...

(13.29)

can not describe the mentioned phenomenon, but an application of the PA using only two terms of (13.29) leads to exact solution.

13.8 Gibbs phenomena Let us consider oscillator equation with the antisymmetric constant force [316, 440, 550] x¨ + sign(x) = 0, (13.30) with the initial conditions x(0) = 0, where:

   +1, sign(x) =   −1,

x(0) ˙ = A, for

x > 0,

for z < 0.

The solution x(t) over one period can be written as follows [440]

(13.31)

390


 t   − (t − 2A)     2 x(t) =    t2    − 3At + 4A2 2

for 0 ≤ t ≤ 2A, (13.32) for 2A ≤ t ≤ 4A.

For values of t outside this interval, x(t) can be determined from the periodicity condition in the following form x(t + 4nA) = x(t),

n = 0, ±1, ±2, ... .

(13.33)

The solution (13.32), (13.33) may be represented by the Fourier series as [316] x(t) =

∞ 9 1 16A2 πt : sin (2m + 1) . 2A π3 m=0 (2m + 1)3

(13.34)

Concerning the Fourier series one can read [550]: “...It should be noted that the Fourier series form gives, in principal, an approximate solution since it is impossible to account for the infinite number of terms. As long as one can keep “any number of terms”, the above remark is not so important for the smooth time histories. However, it becomes very important when cleaning with either a discontinuous function x(t) or its discontinuous derivatives. It is known that the trigonometric series appear to be “bad working” around the discontinuities due to the Gibbs phenomenon. In terms of acceleration, the series performs an oscillating error near those points of time t at which the acceleration x(t) ¨ has the step-wise discontinuities switching its value from −1 to 1 or back as it is dictated by equation (13.30)”. The mentioned remark is true if we apply a simple summation of the Fourier series (as it is known it leads to a so called ill-posed problem). However, one can utilize the regularization properties of the PA [522, 604]. As an example we consider function sign(x) in the interval x ∈ [−π, π]. This function has the following Fourier representation f (x) =

1 1 1 1 4 (sin x + sin x + sin 5x + sin 7x + sin 9x + ...). π 3 5 7 9

(13.35)

A behaviour of f (x) in a neighbourhood of the point x = 0 is well known. Namely, a so called Gibbs phenomenon is observed. A choice of the sign (x) function is motivated by an observation that it is one of the paradigm type function exhibiting the so called Gibbs phenomenon. If the PA can be satisfactory used in this case, then also a similar like approach can be applied for other functions. In order to obtain a limiting geometrical picture of the function s n (x) (being a part of the series (13.35)) for n → ∞, one needs to extend interval of a vertical line x = 0 linking the points f (−0) and f (0) of about 18% upwards and downwards. The diagonal Padé approximant [N/N] of the series (13.35) are given in reference [604], and has the following form

13.8 Gibbs phenomena [(N−1)/2]

[N/N] =

j=0

1+ where: q2 j+1

391

q2 j+1 sin(2 j + 1)x

[N/2] j=1

,

(13.36)

s2 j cos(2 jx)

  [N/2]   s 4 1 2i  , = (2 j + 1)  + 2 2 2 π (2 j + 1) (2 j + 1) − (2i)  i=1

s2i = 2(−1)i

(N!)4 (2N + 2i)!(2N − 2i)! . (N − 1)!(N + 1)!(N − 2i)!(N + 2i)![(2N)!] 2

Fig. 13.3. The diagonal PA (dashed curve) for three term of the series (13.34). Solid curve corresponds to the exact solution (13.32).

392


The numerical results are presented in Table 13.6, where ¯x are those values of x where max[N/N] is achieved. From the Table 13.6 it is seen, that in a case of application of the trigonometric diagonal approximants the Gibbs effect does not achieve 2% (see also [604]). Results of PA application to the series (13.34) are shown in Figure 13.3. Table 13.6. PA for function sign(x) Fourier series N 2 3 4 5 6 7 8 9 10

¯x(N) 0.68 0.41 0.28 0.31 0.16 0.13 0.11 0.09 0.08

[N/N] 1.0736 1.0419 1.0301 1.0242 1.0208 1.0185 1.0166 1.0152 1.0138

13.9 Boundary conditions perturbation method It is well known that for certain boundary conditions (for instance, simple support) a solution can be found relatively simple. However, other boundary conditions (free boundaries, clamping) complicates an analysis. To overcame the occurring difficulties, the perturbation method related to boundary conditions has been proposed [252]. We briefly describe its main idea. We introduce the parameter ε into the boundary conditions in a way that for ε = 0 we obtain the problem with simple solution, whereas for ε = 1 the input boundary value problem appears. Then, a perturbation series in relation to ε is constructed and finally we take ε = 1. As an illustrative example we consider oscillations of a clamped beam (−0.5 ≤ x ≤ 0.5) starting with a simple support. We consider the following input equation and the boundary conditions: w xxxx − λw = 0, w = 0,

wx = 0

(13.37) x = ±0.5.

f or

(13.38)

Let us introduce into the boundary conditions (13.38) the parameter ε in the following way: w = 0,

(1 − ε)w xx + εw xx = 0

f or

x = ±0.5.

(13.39)

For ε = 0 we get a simple support, whereas for ε = 1 we obtain a clamping. The other ε values from the interval (0, 1) define the stiffness of support of c = ε/(1 − ε).

13.9 Boundary conditions perturbation method

393

Both displacement w and eigenvalue λ are sought in the form w = w0 + w1 ε + w2 ε2 + ...,

(13.40)

λ = λ + λε + λ2 ε2 + ... .

(13.41)

Substituting (13.40) and (13.41) into equation (13.37) and into the boundary conditions (13.39), and under splitting due to ε the following equation are obtained: w0xxxx − λ0 w0 = 0, w0 = 0,

w0xx = 0

f or

wixxxx − λ0 wi =

i

(13.42) x = ±0.5,

w jx ,

(13.43)

j+1

wi = 0,

wixx = ∓

i−1

w jx

f or

x = ±0.5,

i = 1, 2, . . . .

j=0

A zero order approximation solution has the following form: λ0 = π4 n4 , n = 1, 2, 3, . . . ,    n = 1, 3, 5, . . . , cosπnx, w0 = C    sinπnx, n = 2, 4, 6, . . . ,

(13.44)

Therefore, in the first approximation one gets the following boundary value problem    n = 1, 3, 5, . . . , cosπnx, 4 4 w1xxxx − π n w1 = λ1C  (13.45)   sinπnx, n = 2, 4, 6, . . . ,    n+1 n − 1      (−1)      2    w1 = 0, w1xx = ∓C  (13.46)  πn for x = ±0.5 .      n        (−1)n 2 The boundary value problem (13.45), (13.46) is governed by non-homogenous differential equation with non-homogenous boundary conditions. In general, it can not be solved, i.e. we can not satisfy all boundary conditions for an arbitrary value of λ1 . However, this drawback can be omitted after introduction of the so called solvability conditions. It is clear that formulation of solvability conditions finally must be reduced to that of finding λ 1 from the non-homogeneous equation (13.45). The right hand side of this equation should be matched with non-homogeneous boundary conditions (13.46). Here two approaches can be applied: 1. An addition solvability condition can be obtained. 2. The quantity λ 1 can be directly found from a boundary value problem. Let us consider the second approach. A general solution to equation (13.45) has the form

394


) wg = C1

) * * chπnx cos πnx + C2 , shπnx sin πnx

whereas its particular solution reads   λ1  − sin πnx, wp = 3 3  cos πnx, 4π n 

n = 1, 3, 5, . . . , n = 2, 4, 6, . . . .

(13.47)

(13.48)

The eigenfunction w 1 can be presented as the sum w1 = wg + w p .

(13.49)

Satisfying the boundary conditions (13.46), the following parameters are obtained:  (−1)(n−1)/2     C  ch(πn/2)   λ1 = 4π2 n2 , C1 = .    − (−1)n/2   2πn  sh(πn/2) Observe that although the constant C 2 can not be obtained . the boundary con6 πnxfrom ditions, but it can be taken as zero since the functions cos sin πnx are already taken into account in zero order approximation. Finally, first correction to the eigenfunction w has the form   (−1)(n−1)/2    chπnx − xshπnx, n = 1, 3, 5, . . . ,    2ch(πn/2)  C   w1 = (13.50)   πn   n/2   (−1)     − 2sh(πn/2) sin πnx + xchπnx, n = 2, 4, 6, . . . . One can find λ2 and w2 in a similar way. The series composed of three first approximation terms has the form 1 2 1 n πn − λ = π4 n4 + 4π2 n2 ε + 4πn πn − cth(−1) ε, 2 2 2πn

(13.51)

whereas the corresponding eigenfunction series has the form  (−1)(n−1)/2  ) *   C  cos πnx  2ch(πn/2) chπnx − x sin x   w=C + ε+    − (−1)n/2 sin πnx + x cos πnx   sin πnx πn  2sh(πn/2)

 (−1)(n−1)/2 

) *   1 C(−1)n+1  chπnx  ch(πn/2)   (−1)n πn − + πn − cth    (−1)n/2   2π2 n2  2 πn shπnx sh(πn/2) )

) * * 1 1 (−1)n πn shπnx sin πnx n+1 C − x πn − cth − (−1) − chπnx 2 2 πn cos πnx π2 n2 ) ** C cos πnx x (13.52) ε2 . sin πnx π2 n2 )


395

Observe that within the used method there exist a possibility of an asymptotical simplification of the transcendental equation in order to define the eigenvalues and then to construct the approximate analytical solution. We would like to illustrate this possibility using an example of symmetric forms. We take a solution in the form √4 λ¯ = λ. (13.53) w = C1 cosλ¯x + C2 chλ¯x, Substituting (13.53) into (13.38) leads to the following transcendental equation ¯ because of λ: ¯ ¯ ¯ ¯ ¯ ¯ ¯ λ cos λ + ε ch λ sin λ + cos λ ch λ = 0. 2(1 − ε)λch 2 2 2 2 2 2

(13.54)

In order to solve it we use the perturbation method, where λ¯ is sought as the series of perturbation parameter: λ¯ =

∞

λ¯i εi .

(13.55)

i=0

Substituting (13.55) into equation (13.54) and splitting because of ε yields the following recurrent sequence of the equations 2λ¯0 cos

λ¯0 λ¯0 ch = 0, 2 2

0 λ¯0 λ¯0 1 λ¯0 λ¯0 λ¯1 λ¯0 λ¯0 λ¯0 2 λ¯1 cos ch + cos sh − sin ch − 2 2 2 2 2 2 2 λ¯0 λ¯0 λ¯0 λ¯0 λ¯0 λ¯0 2λ¯0 cos ch + ch sin + sh cos = 0, 2 2 2 2 2 2 −1 ¯ ¯ ¯ ¯ ¯ λ λ0 λ¯0 λ2 λ0 λ0 2λ¯2 cos + 2 2 cos sh − sin ch + 2 2 2 2 2 2 01 λ¯0 λ−2 λ¯0 λ¯0 λ−2 λ¯0 λ¯0 cos − 1 sh sin − 2λ0 λ¯2 sh + 1 ch 2 2 4 2 2 2 2 2 1 −2 1 ¯ λ¯0 λ1 λ¯0 λ¯0 ch − λ2 sh + ch 2 2 4 2 2 0 λ¯0 λ¯0 1 λ¯0 λ¯0 λ¯1 λ¯0 λ¯0 λ¯0 2 λ¯1 cos ch + cos sh − sin ch = 0, 2 2 2 2 2 2 2 . . . . . . . . . . It is not difficult to solve the introduced equations and one obtains 1 1 n πn 1 1 cth(−1) − 2 2 ε2 , 1− λ¯ = πn + ε + πn πn 2πn 2 π n

n = 1, 2, 3, . . . . (13.56)

396


Observe that a truncated series (13.56) is a root of fourth power from the truncated series (13.51). It is clear that formula (13.51) is more accurate for higher forms. Therefore one can consider an estimation of first eigenvalue since it is mostly influenced by boundary conditions. Numerical solution to the transcendent equation (13.54) for ε = 1 gives the first eigenvalue of the boundary value problem λ = (1.5056) 4. The truncated perturbation series (13.51) for n = 1, ε = 1 gives λ = (1.1542) 4. The introduced error is of amount 23.33%. In order to improve the results obtained using perturbations of boundary conditions the PA can be applied. The following PA is applied to the series (13.51): λ(ε) =

a0 + a1 ε , 1 + b1 ε

where: a 0 = λ0 ,

a1 = λ1 − b1 λ0 ,

(13.57)

b1 = −λ2 λ−1 1 .

For n = 1, ε = 1 we obtain λ(1) = (1.5139) 4 (error of 0.58%). The first eigenvalue of the problem (13.37), (13.39) versus ε is shown in Figure 13.4.

Fig. 13.4. First eigenvalue of the eigenproblem (13.37), (13.39) versus the parameter ε.

It is seen that the computational results of first eigenvalue obtained using PA (13.57) practically overlap with the exact solution for all values of parameter ε. The perturbation series gives reliable results only for ε ≤ 0.4.


397

An error δ in estimation of first fifteen eigenvalues of the problem (3.100), (3.101) is illustrated in Figure 13.5, where: δ = [(λ a − λe )/λe ] · 100% and λ e (λa ) denotes exact (approximate) solution.

Fig. 13.5. Errors in estimation of first fifteenth eigenvalues to the problem (13.37), (13.38); curves correspond to formulars (13.51) for ε = 1, and (13.57) for ε = 1 respectively.

Since for low eigenvalues a difference between the asymptotical series and the PA becomes large, then their estimation can be rather oriented on application of PA (13.57). For higher eigenvalnes (n > 10) the error introduced by the asymptotical series is less than 5%. Let us turn now to more general case – oscillations of clamped plate (−0.5k ≤ x ≤ 0.5k; −0.5 ≤ y ≤ 0.5) governed by the equations ∇4 w − λw = 0, w = 0, w = 0,

w x = 0 for wy = 0 for

(13.58) x = ±0.5k, y = ±0.5.

(13.59) (13.60)

Let us transform the boundary conditions by introduction of parameter ε in the following way w = 0,

(1 − ε)w xx ± εKwx

for

x = ±0.5k,

(13.61)

w = 0,

(1 − ε)wyy ± εKwy

for

y = ±0.5.

(13.62)

398


Now we apply the perturbation method to the boundary value problem (13.58), (13.61), (13.62). As usually we present the eigenvalue λ and the eigenfunctions in the series (13.40) and (13.41). Then we substitute those series into the differential equation (13.63) and into boundary conditions (13.61), (13.62) and after splitting due to ε the following series of the boundary value problems is obtained: ∇4 w0 − λ0 w0 = 0, w0 = 0,

w0xx = 0 for

x = ±0.5k,

w0 = 0,

w0yy = 0 for

y = ±0.5,

∇4 w0i − λ0 wi =

i

(13.63)

λ j wi− j ,

j=1

wi = 0,

wiyy = ∓

i−1

w jx

for

x = ±0.5k,

j=0

wi = 0,

wixx = ∓k

i−1

w jy

for y = ±0.5,

j=0

Solution of zero order approximation (13.63) has the form ) *) * cos πm cos nπy; m, n = 1, 3, 5, .. k x w0 = X0 Y0 = C sin πm sin nπy; m, n = 2, 4, 6, ... , k x m2 + n2 , k2 whereas the first approximation gives the following problem *) * ) x cos nπy cos πm 4 k ∇ w1 − λ0 w1 = Cλ1 , sin πm sin nπy k x λ0 = π4 a2 ,

)

w1 = 0;

w1xx

n+1

(−1) 2 = ±πmC m (−1) 2 )

w1 = 0,

a=

w1yy = ±πnC

n+1

(−1) 2 n (−1) 2

*)

*)

(13.64)

* cos nπy , sin nπy

for

* cos πm k x , sin πm k x

for y = ±0.5.

A solution to (13.64) is sought in the form ) ) * * cos πm cos nπy x k , w1 = Y1 (y) + X1 (x) sin nπy sin πm k x

x = ±0.5k,

(13.65)


399

λ1 = λ1x + λ1y . After substitution of relations (13.65) into the boundary value problem (13.64) and after a variable separations the following two problems are formulated: 2 ) * mπ 2 m cos nπy 4 2 2 Y1yy − π n 2 2 + n Y1 = Cλ1y Y1yyyy − 2 , (13.66) sin nπy k k )

n+1 * (−1) 2 for y = ±0.5, n (−1) 2 ) * m2 m2 cos πm 2 k x , + 2π = λ C X X1xxxx − 2π2 n2 X1xx − π2 2 1 1x sin πm k k2 k x ) m+1 * (−1) 2 for x = ±0.5K. X1 = 0, X1xx = ∓πmC m (−1) 2

Y1 = 0,

Yyy = ∓πmC

(13.67) (13.68) (13.69)

Both problems are governed by non-homogeneous differential equation with non-homogenous boundary conditions. One can use the algorithm described in section to define the unknown parameters of the right hand side, and finally we obtain λ1x = 4π2

m2 , k2

λ1y = 4π2 n2 ,

λ1 = 4π2 a,

  ) m−1 *   (−1) 2 m ) * ) * *  )  k πm 2 (−1) C m  chπβ2 kx sin k x  cos nπy w1 = (−1)m−1 −x   ) π *  sin nπy + shπβ2 kx cos πm π ka  ch 2 β2 k x   2 sh π2 β2  ) n−1 *   (−1) 2 n ) * ) * * )  2 (−1) C π  chπβ1 y x sin nπy  cos πm k ) * − y , (−1)n−1   shπβ1 y cos nπy  sin πm π a  ch π2 β1 k x   2 sh π2 β2 m2 m2 β1 = 2 2 + n2 , β2 = 2n2 + 2 . k k Proceeding analogously for the second order approximation we obtain the eigenvalue λ with three terms:  2  n2 mk2 1 4 2 2  λ = π a + 4π aε + 4π πa + 2 2 − − πa 2π 1 m2 (−1)m π 2 (−1)n π β1 + n β2 cth β2 ε2 + ... . k β1 cth 2a k2 2k 2 Then the obtained truncated series (13.70) is transformed to the PA.

(13.70)

400

13 IMPROVEMENT OF PERTURBATION SERIES 4 2

λ=π a where: α =

4 πa ,

4 β= 3 2 π a

α + α2 − β ε α − βε

,

(13.71)

 2  n2 m πa + 2 k2 − 1 −  2π πa2

1 m2 (−1)m π 2 (−1)n π β1 + n β2 cth β2 . k β1 cth 2a k2 2k 2 Let us compare the numerical results for first eigenvalue of a squared plate (k = m = n = 1). The numerical solution gives λ = (1.9033π) 4, truncated series (13.70) for ε = 1 gives λ = (1.535π) 4 (error is %); the PA (13.71) for ε = 1 gives λ = (1.9142π) 4 (error of 0.25%).

Fig. 13.6. Eigenvalue λ versus clamping stiffness. Curves 1 and 2 correspond to formulas (13.70), (13.71), respectively.

In Figure 13.6, the eigenvalue λ versus clamping stiffness is presented illustrating the earlier observations.

13.10 Bifurcation problem

401

13.10 Bifurcation problem In bifurcation problems an application of PA requires a construction of a certain generalization, referred as Hermite-Padé approximant. Main idea of this method is illustrated using the following example [256]. Assume that we are going to study a nonlinear boundary value problem, which may be written in operator form as Φ(λ, α) = 0, (13.72) where: λ is parameter. Let us suppose that the partial sum U N (λ) =

N

Un λn = U(λ) + O λN+1

for λ → 0

n=1

represent a particular solution of equation (13.72). We sum to exploit this datum by making a plausible assumption concerning the global analytic structure of the function represented locally by the power series. We shall make the simplest hypothesis by assuming that U(λ) is the local representation of an algebraic function u of λ. Therefore, we sum a polynomial F p (λ, u) of degree p≥2 p m F p (λ, u) = fm−k,k λm−k uk , (13.73) m=1 k=0

assuming that f0,1 = 1 and taking into account the condition F p (λ, U N (λ)) = O(λn+1 ) for λ → 0.

(13.74)

Polynomial F p includes 0.5(p 2 + 3p − 2) unknowns, the requirement (13.74) reduces to a system of N linear algebraic equations for f m−k,k . Therefore, we shall take N = 0.5(p 2 + 3p − 2), so that the number of equations equals the number of unknowns. Once F p has been found, one may use standard techniques to obtain the p solution branches of the polynomial equation F p (λ, u) = 0. The bifurcation for this solution set can be then analysed locally by means of Newton’s diagram (see Chapter 3.3). It is possible to take 0.5(p 2 + 3p − 2) > N and remove a corresponding number of terms from F p . In cases where the solutions of the original problem are known a priori to possess some property, it may be possible to incorporate this information into the equations for the coefficients of F p in such a way that the roots of F p inherit that property. For instance, one can easily constrain F p to have a given singular point. One should, however, keep in mind that the imposition of such constraints may adversely affect the solvability of the equation for F p .

402


13.11 Borel summation and superasymptotics The asymptotic approach leads to infinite series F (ε) =

∞ n=0

an εn which can be di-

vergent even for all ε 0. Nevertheless we may assign a meaning to this expansion using the following idea of Borel [247, 611]. Let us consider a relationship ∞ n! =

e−t tn dt,

0

which can be obtain using n times integration by part. Then we obtain the expression F (ε) =

∞ ∞ an n!

n=0

e−t (εt)n dt.

(13.75)

0

Suppose now that an interchange of the summation and integration is performed and consider the function ∞  ∞ ∞  an  −t n (εt) dt = F˜ (ε) = e  e−t B (εt) dt. n! n=0 0

0

We call B(εt) the Borel function associated with the original series. It is worth noting that, due to presence of the denominator n! in every one of its term, this series has a better convergence than F(ε). For example, the radius of convergence for the series ∞ (−1)n n!εn n=0

is certainly zero. The corresponding Borel function B (ε) =

∞

(−1)n εn =

n=0

1 1+ε

has a singularity at ε = −1 (the radius of convergence is equal to unity). Nevertheless, for positive ε the integral F˜ (ε) =

∞ 0

e−t

1 dt 1 + εt

˜ exists and the function F(ε) may be called the Borel sum of the original series (despite the fact that the latter has no sense for arbitrary ε and the interchange of summation and integration in (13.75) can be justified only if the radius of convergence of the series for the Borel function is infinity). Let us note that for convergent

13.11 Borel summation and superasymptotics

403

series the usual sum coincides with the result of Borel summation. However, the latter can be considered as the sum even in the case when a series is non-convergent. Let us focus now on the so called superasymptotics [161, 162, 163, 164, 165], using Borel summation procedure. Let us consider the following equation d2 y = λ2 Z(z)y, dz2

(13.76)

for λ → ∞. The two lowest-order ‘wave’ solutions of (13.76) are given by WKB method (concerning WKB approach see Chapter 16.4) as    z  1   1 y(z) ≈ exp ±λ Z 2 (ζ)dζ  Z − 4 (z),   z∗

in which z∗ is an arbitrary reference point. A natural variable will be the difference between the two exponents, namely z F(z) = 2λ

1

Z 2 (ζ)dζ.

(13.77)

z∗

It is convenient to perform the analysis for the solution which is exponentially small when ReF > 0, and write this as 1 (13.78) y(z) = exp [−0.5F] Z − 4 (z) Y(F). A formal asymptotic series for Y, in descending powers of λ, can be found by substitution into (13.76) ∞ (−1)K YK (F), (13.79) Y(F) = K=0 −K

where: Y0 = 1, YK ∼ λ . Then the Y K satisfy the recurrence relation ([247], p.296) (F) = −YK (F) + G(F)Y K (F), YK+1

where primes denote derivatives with respect to F and 1 1 G(F) ≡ Z 4 Z − 4 .

(13.80)

(13.81)

Note that the large parameter λ no longer appears explicitly. Its role has been to define the terms Y K in the series representing Y. An important role is played by the transition points, that is the zeros z j of Z(z). At the corresponding points F j , the function G(F) has double poles, whose strengths depend only on the order of the zero: for an mth order zero, it follows from (13.77) and (13.81) that

404


G(F) → −m(m + 4)/4(m + 2) 2 (F − F j )2 for F → F j .

(13.82)

We consider only the generic case m = 1. When iterating (13.80) to obtain the YK , the derivatives magnify the singularities (13.82), leading as explained in ([247], p.299) to the following simple approximate formula for the late terms: YK (F) → (K − 1)!/2π(F − F 0 )K as K → ∞,

(13.83)

where F 0 denotes the transition point F j which is closest to F in the sense of having the smallest value of |F − F j |. For large |F − F 0 | (ensured by large λ), the terms decrease until K ≈ |F − F 0 | and then increase. The resurgence formula which will be central to all our subsequent analysis is a formally exact representation of Y K with (13.83) as its leading term. By direct substitution, it can be confirmed that YK (F) =

∞ 1 1 Y s [FYK − (F − F j )] s (K − s − 1)! 2π j (F − F j )K s=0

(13.84)

satisfies the recurrence relation (13.80). The function G(F) enters only through the positions F j of its poles. It should be remarked that (13.80) is formally satisfied by much more general relations, in which (i) F j are any points whatever, (ii) on the right-hand side K is replaced by K + α with α arbitrary, (iii) 1/(2π) can be any constant and (iv) the Y s can be any solutions of (13.80) (rather than those with the same integration constants as the Y K on the left-hand side). The particular choice (13.84) reproduces the limiting form (13.83), which corresponds to the term s = 0. The higher terms s > 0 provide a formal asymptotic expansion for the late Y K in terms of successive early Y K . In what follows we will use the simplified form of (13.84) obtained by neglecting all turning points other than the closest. F 0 , which we will henceforth take as the origin F = 0 (this is equivalent to choosing z ∗ = z0 in (13.77)). This gives YK (F) =

∞ 1 (K − s − 1)!(−F) sY s (F). 2πF K s=0

(13.85)

As they stand (13.84) and (13.85) are numerically meaningless because the factorials for s > K − 1 are infinite. They can, however, be made meaningful by Borel summation.

13.12 Domb–Sykes plot [340, 659] We suppose that the result of an asymptotic study can be expressed in a power series in a small positive parameter ε, f (ε) ∼

N n=0

c n εn .

13.12 Domb–Sykes plot [340, 659]

405

Although f may only make sense physically when ε is real and positive, we now extend our mathematical consideration of f onto the complex ε-plane. The radius of convergence is then the distance from the origin to the nearest singularity of f (ε) in the complex ε-plane. Thus the convergence of the expansion can be made poor by an unphysical singularity. If the nearest singularity is on the positive ε-axis, i.e. a physically real singularity, then the signs of the coefficients c n eventually become the same, for example, 1 ∼ 1 + ε + ε 2 + ε3 + . . . . 1−ε If the nearest singularity is on the negative ε-axis, then the signs eventually alternate, for example, 1 ∼ 1 − ε + ε 2 − ε3 + . . . . 1+ε The pattern of signs is usually established quickly: it will take many terms only if the amplitude of the nearest singularity is relatively weak. When there are several singularities of equal singularity and equal amplitude, as must happen for real f with complex singularities necessarily occurring in complex conjugate pairs, then a complicated pattern of singularity can emerge, for example, 1+ε ∼ 1 + ε − ε 2 − ε3 + ε4 + ε5 − ε6 − ε8 . 1 + ε2 This example has a pattern of signs + + − −. If there is a pair of singularities in the direction ±β about the real axis, then the pattern of the signs is a cycle 2π/β long. From complex variable theory, the radius of convergence ε 0 can be calculated from cn−1 ε0 = lim . n→∞ cn More information, however, can be extracted from the Domb-Sykes plot. If there is just one nearest singularity in f at ε = ε 0 and it has an index α, i.e. f has a dominant factor ) for α 0, 1, 2, . . . (ε0 − ε)α (ε0 − ε)α ln(ε0 − ε) for α = 0, 1, 2, . . . Then when this factor dominates, the coefficients c n behave like 1α 1 cn . 1− ∼ cc−1 ε0 n Thus if one plots the ration c n /cn−1 against 1/n, the intercept gives the radius od convergence ε 0 and the slope gives the index of the singularity α. Figure 13.7 gives an example for f (ε) = ε(1 + ε3 )(1 + 2ε)−1/2 ∼ 3 3 27 51 191 7 359 8 ε − ε2 + ε3 − ε4 + ε5 − ε6 + ε − ε + ... . 2 2 8 8 16 16

406


-0.5 -1 -1.5 -2

0.0

0.1

0.2

0.3

0.4

0.5

Fig. 13.7. The Domb-Sykes plot for function f (ε) = ε(1 + ε3 )(1 + 2ε)−1/2 .

By n = 7 the coefficients become dominated by the nearest singularity. If one knows the value of either ε 0 or α from the physics, then this information can be used to help the extrapolation. Some badly bent curves can be straightened by an offset in n, i.e. plotting against 1/(n − ∆). If there are several singularities at the same distance which cause the coefficients to oscillate, then one can try plotting (cn /cn−2 )1/2 . If the radius of convergence is found to be infinity, then a behaviour c n /cn−1 ∼ k/n corresponds to a factor like exp(kε), while a behaviour like c n /cn−1 ∼ k/n1/p corresponds to an integral function of order p like exp(ε p ). If the radius of convergence is found to be zero, then f has an essential singularity. This occurs with asymptotic expansions which diverge. If the coefficients like cn−1 /cn ∼ 1/kn, the further coefficients will be given by c n ∼ constant kn n! as n → ∞.

13.13 Extraction of singularities from perturbation series [340, 659] Inversion. A singularity on the positive axis usually means that f (ε) is multivalued, and that there is a maximum attainable ε. Often the inverted function ε( f ) is single valued. Thus recasting the result as a series in f for ε can give an expression which continues onto the upper branch. Figure 13.8 gives an example for f (ε) = sin−1 ε ∼ 1 3 5 7 ε + ε3 + ε5 + ε + ..., 6 40 112 with conversion

13.13 Extraction of singularities from perturbation series [340, 659]

0.0

0.5

1.0

1.5

407

e

Fig. 13.8. The continuous is f (ε) = sin−1 ε; the dotted curves are increasingly higher order expansions of f in terms of ε; the dashed curves are increasingly higher order expansions of the inversion, ε( f ).

1 3 1 5 1 7 f + f − f + ..., 6 120 5040 which gives a good approximation on to the next branch. Taking a root. If f has a mild singularity such as a branch cut, i.e., ε∼ f −

f ∼ A(ε0 − ε)α for ε → ε0 , α > 0 then the appropriate root of f , f 1/α , does not have this singularity at ε 0 . Figure 13.9 gives an example for √ f (ε) = e−ε/2 1 + 2ε ∼ 1 7 41 367 4 4849 5 1 + ε − ε2 + ε3 − ε + ε + ..., 2 8 48 384 3840 which has a radius of convergence of 0.5, whereas 3 5 7 3 f 2 ∼ 1 + ε − ε2 + ε3 − ε4 + ε5 + . . . 2 6 24 40 has an infinite radius of convergence. Multiplicative extraction. From a Domb-Sykes plot one may can know that f (ε) has a nearest singularity at ε = ε 0 with an index α. If this singularity is factored out multiplicatively, by setting f (ε) = (ε0 − ε)α f M (ε) then the new function f M should be regular at ε = ε 0 , and ought to have singularities further away from the origin, i.e. ought to converge better.

408

13 IMPROVEMENT OF PERTURBATION SERIES 1.5

1.0

0.5 0.0

0.5

1.0

1.5 e

√ Fig. 13.9. The continuous curve gives f (ε) = ε−ε/2 1 + 2ε; the dotted curves are increasingly higher order expansions of f in terms of ε; the dashed curves are increasingly higher order expansions of f 2 in terms of ε.

Additive extraction. A singular factor can be extracted additively instead of multiplicatively if the amplitude A is also known: f (ε) = A(ε0 − ε)α + fA (ε). If the singular factor is known to be additive and if α > 0, then a multiplicative extraction would fail to produce better behaviour if f M , e.g. 1 + (1 − ε)1/2 = (1 − ε)1/2 f M with f M = 1 + (1 − ε)−1/2 . Euler transformation. A non-physical singular point ε 0 can be transformed to infinity by introducing a new small parameter ε . 1 − ε/ε0 The series of f recast in terms of ε, ˜ f ∼ dn ε˜ n , has the non-physical singularity pushed out to ε˜ = ∞. Hence the new series in ε˜ should converge better on the physical real positive ε-axis. ˜ Figure 13.10 gives an example for √ f (ε) = e−ε/2 1 + 2ε. ε˜ =

This has a non-physical singularity at ε = −0.5, which can be mapped to ∞ with the Euler transform ε˜ = ε/(1 + 2ε). The expansion of f in terms of ε˜ is 1 1 31 895 4 22591 5 f ∼ 1 + ε˜ + ε˜ 2 − ε˜ 3 − ε˜ − ε˜ + . . . . 2 8 48 384 3840

13.14 Analytical continuation [407]

409

1.5

1.0

0.5 0.0

0.5

1.0

1.5 e

√ Fig. 13.10. The function f (ε) = e−ε/2 1 + 2ε is given by the continuous curve; the dotted curves are for increasingly higher order expansions of f in terms of ε; the dashed curves are for increasingly higher order expansions of f in terms of ε. ˜

While the transformed expansion does not have the sild behaviour of the original expansion at ε = 0.5, it does not provide a particularly good approximation beyond the old radius of convergence. Shanks transform [610]. This transform assumes that the partial sums of n terms, n Sn = ck εk , are in a geometric progression K=0

S n = A + BC n . The answer A can be extracted from just three partial sums by a nonlinear extrapolation S n+1 S n−1 − S n2 (S n+1 − S n )(S n − S n−1 ) . = Sn − A= S n+1 − 2S n + S n−1 (S n+1 − S n ) − (S n − S n−1 ) (The second form is more stable for computations.) This extrapolation often works better than it should do. Note that it will work in diverging series, when |C| > 1. It can be used repeatedly, i.e. the A’s produced from n = 1, 2, 3, . . . can themselves be considered a series of partial sums to which the Shanks transform can be applied resulting in new improved A’s. Such repeated applications of the transform effectively remove further geometric terms in S n = A + BC n + DE n + FGn + . . ., with |C| > |E| > |G|.

13.14 Analytical continuation [407] In order to extend an application of power series very often the so called analytical continuation is applied.

410


Let the function f (λ) is given in the form of the series f (λ) =

∞

ck λk .

(13.86)

k=0

Let it D be defined as a certain one-coupled space belonging to the space B, where the function f (λ) does not possess any singular points in spite of poles. It is assumed that the space D contains point λ = 0, and that f (x) is const in the form (13.80) in vicinity of λ = 0. On the η plane one-coupled space ∆ is chosen, which includes the point η = 0, and the function λ = ϕ(η) is constructed. The latter represents a conformal mapping of the space ∆ into space D keeping ϕ(0) = 0. Let the function λ = ϕ(η) is represented by the following series λ = ϕ(η) =

∞

an ηn .

(13.87)

n=1

In what follows the function F(η) = f [ϕ(η)] =

∞ k=0

ck [ϕ(η)]k is regular one in the

neighbourhood of η = 0, and it can be developed into the power series F(η) = f [ϕ(η)] =

∞

bn ηn ,

(13.88)

n=0

obtained via substitution of series (13.87) into series (13.86). Coefficients b n can be computed owing to the series (13.86) and (13.87), i.e. * ) n k 1 ∂n (n) (n) bn = d k ck , d k = . (13.89) ϕ(η) n! ∂ηn η=0 k=1 Let Cη be a circle with |η| < Rη , where the series (13.88) is convergent on the plane γi . Denoting dy ∆ an intersection of the space ∆ with the circle C η , and let D denotes a picture ∆ during transformation λ = ϕ(η). The following properties of the function F(η) hold. 1. If the space D transits over the boundaries of series (13.86) convergence, then the function F[ϕ −1 (λ)] defined by the series ∞ n F ϕ−1 (λ) = bn ϕ−1 (λ) ,

(13.90)

n=0

is an analytical continuation of the function f (x) from the circle C λ into the space D . 2. If the function f (λ) is regular one in the space D and ∆ is the circle |η| < 1, then F[ϕ−2 (λ)] defined by (13.90) yields analytical extension of the function f (λ) from the circle C λ into the whole space D .


411

3. If f (λ) is a mesomorphic function in the one-coupled space D and the function λ = ϕ(η) is one-leaf and conformal mapping of ∆ into D, then F(η) = f [ϕ(η)] is regular in space ∆ except of poles which are pre-pictures of the function f (λ) poles in space D during transformation λ = ϕ(η). 4. If f (x) is the rational function in plane and λ, λ 1 , λ2 , . . . , λm are its poles (λ = ∞ represents a non-singular point of f (λ)), and λ = ϕ(η) is the rational function with the poles being pre-pictures of λ = λ k of the function f (x), i.e. η = η k = ϕ−1 (λk ). Besides, a multiplicity of the pole η = η k for the function F(γ i ) does not coincide with a multiplicity of the corresponding pole of f (λ) if an only if some of the function f (x) poles overlap with the branching points of the function ϕ−1 (λ). In other words one may choose the space D and ∆ appropriately, and shift λ = λ ∗ (where a summation of the series (13.86) is carried out) into the point η = η ∗ = ϕ−1 (λ∗ ). The latter is not only located in the circle of convergence of series (13.88), but it possesses an improved characteristic. Namely, a modulo of a ration of η ∗ and the convergence radius of the series (13.88) will be less tan a modulo of a ration of λ∗ and the convergence radius of the series (13.88). In what follows examples of various choices of the space D for a certain positions of the singular points are considered. In our further considerations the circle |η| < 1 serves as the space ∆. Either point of the series summation or a pole going to be determined is denoted by λ ∗ . 1. All singular points λ i of the function f (x) are real and they have the same sign as Reλ∗ . If modulus of λ ∗ is less than all λi , than as the space D the plane with a cut along the real axis from λ = a to λ = ∞ may serve (or from λ = −a to ∞, depending on a sign of λ i ), or a plane with two cuts along the real axis from λ = a to ∞, and from λ = −a to −∞, or a half-plane Reλ ≤ a (or Reλ ≥ −a), or finally a zone −a ≤ Reλ ≤ a. In all mentioned cases o value of a is defined by inequalities |λ∗ | < |a| ≤ |λ2 |, where λ = λ1 denotes a nearest singular point of f (λ) located in vicinity of λ ∗ . If λ = λ∗ lies between two nearest singular points of the function f (x): |λ 1 | < |λ∗ | < |λ2 |, then either the plane with two cuts from λ = λ 1 to λ1 = +i∞ or form λ = a to ∞ (from λ = −a to −∞), or zone −a ≤ Reλ ≤ a with a cut along a radius from λ = λ 1 to λi + i∞ can be taken as the space D. The number a should satisfy the following condition: |λ ∗ | < |a| ≤ |λ2 |. 2. All singular points λ = λ i λ∗ of the function f (λ) are real and have the points with a sign opposed to the sign of Reλ ∗ . As the space D either the plane with a cut from a to ∞ or the half space Reλ ≤ a may serve. The number a satisfies the inequality |a| ≤ |λ1 and has the opposed sign to Reλ ∗ . In the considered case, as the space ∆ one may take the radius |η| < 1 or an angle including the points λ = 0 and λ = λ∗ inside and having a top in point λ = a. 3. All of singular points of f (x) are real and have different signs. As D plane with two cuts from a to ∞, and from −a to −∞, a plane with two cuts from a to ∞ and from −b to −∞ (a > 0, b > 0), or a plane with a horizontal cut along a real axis and with a vertical cut may serve.

412


In what follows if λ ∗ is less in modulo than all λ = λ k , then a plane with two cuts from a to ∞ and from −a to −∞ (a ≤ |λ 1 |) can be taken as D. If λ∗ lies between two singular points, say λ 1 < λk < λ2 , then a zone −λ 2 ≤ Reλ ≤ λ2 without a radius from λ 1 to λ1 + i∞ can serve as D. If λ1 > 0, λ∗ > 0 and all λi < 0 (i 1), then a plane with vertical cuts from λ 1 to λ1 + i∞ and with horizontal cut from λ 2 to −∞, can serve as D. It should be noticed that not all possible distribution of singular points of a being investigated function are considered. Any initial information on such distribution can be taken while choosing the space D. Recall, while choosing the space D one has to be motivated via easy construction of the function λ = ϕ(η). Below a brief review of some fundamental transforming functions is given. 1. Mapping of a circle |η| < 1 into a plane with a cut along a radius. a) D-plane with a cut along a positive part of the real axis from λ = a to ∞. The transforming function reads: √  √   4aη a − λ − a  −1  λ = ϕ(η) = − (λ) = η = ϕ √  √  . (1 − η)2 a−λ+ a b) D-plane with a cut along a negative real axis from −a to −∞. The transforming function reads: √  √   a + λ − a  4aη −1 λ = ϕ(η) = η = ϕ (λ) = √ √  . (1 − η)2 a+λ+ a c) D-plane with a cut along the radius arg λ = a, |λ| ≥ a. The transforming function has the form:   √ √  aeiα − λ − aeiα  4aη −1  λ = ϕ(η) = η = ϕ (λ) = √  . √ (1 − η)2  aeiα − λ + aeiα 2. Mapping of the circle |η| < 1 into a plane with two cuts. a) D-plane with two cuts along a real axis from a to ∞ and from −a to −∞. The transformation reads:   √  a − a2 − λ2  2aη −1  λ = ϕ(η) =  . η − ϕ (λ) = λ 1 + η2  b) D-plane with two cuts along the real axis from a to ∞ and from −ak to −∞; a > 0, k > 0. The transformation reads:   √ √  aeiα − λ − aeiα  4aeiαri −1  λ = ϕ(η) = − η = ϕ (λ) = √  . √ (1 − η)2  aeiα − λ + aeiα c) D-plane with horizontal and vertical cuts. Given a and b values shown in Figure 13.11, the following transformations are obtained:


-b

0

a

-b

-a

0

-a

0

b

0

413

a

b

Fig. 13.11. Transformation of the circle |η| < 1 on plane with two cross sections.

0 a 1 (ii) λ = ϕ(η) = d − + ψ(η) , d d 1 0 a 1 0a + ψ(η) , (iv) λ = ϕ(η) = −d − + ψ(η) , (iii) λ = ϕ(η) = −d d d where: d denotes a distance between cuts, and √ 2 3 3 z0 − η¯z0 1−η i ψ(η) = . 1− 16 1 − ¯η z0 − η¯z0 (i) λ = ϕ(η) = d

0a

1 + ψ(η) ,

Values of z0 is determined from the equation  d, if a and b have    √  2    different signs 3 3 1  ai = c 1− , where c =    16 z0 −d, if a and b have      same sing 3. Transformation of circle |η| < 1 into a strip (P) (Figure 13.12). The transforming function reads: λ = ϕ(η) = arctan η =

1 1 + ηi ln 2i 1 − ηi

η = ϕ−1 (λ) = tan λ .

In general, if one attempts to map the circle |η| < 1 into an arbitrary zone, first its transformation into (P) zone is required. 4. Mapping of the circle |η| < 1 into a zone with a cut out ray. The transformation shown in Figure 13.13 has the form: z0 − η¯z0 b−a a + c z0 − η¯z0 a+c ln 1 − + ln 1 + · , λ = ϕ(η) = a − i π 1−η π b−a 1−η where d denotes a halfzone width, and z 0 is defined by the equation a+c a+c b−a −ai = ln(1 − z0 ) + ln 1 + z0 . π π b−a

414


-p 4

p 4

0

Fig. 13.12. Transformation of circle |η| < 1 into a strip (P).

-c

0

b a

Fig. 13.13. Mapping of the circle |η| < 1 into a zone with a cut out ray.

5. Mapping of circle |η| < 1 into halfplane with cut out strip. The transforming function reads: z0 − η¯z0 b i /2 −1 z= − z z − 1 − cos h z , λ=d d π 1−η where d is strip width, and for z 0 we have equation + 2b π i = z0 z20 − 1 − cos h−1 z0 . a−b2 6. Mapping of circle |η| < 1 on the halfplane Reλ ≤ a. The transforming function is λ 2aη λ = ϕ(η) = − η = ϕ−1 (λ) = . 1−η λ − 2a


a

415

b

0

Fig. 13.14. Plane with cut out strip.

(D)

(D)

p n --Ö a n

0

-a

0

Fig. 13.15. Mapping of circle |η| < 1 on the halfplane Reλ ≤ a. g

c

g

c i

g

-1 b

0

1 a

g

c

-1

1 --Ö2

-g

-a

0

1 -Ö2

1

a

g

c

Fig. 13.16. Mapping of a top halfplane with a cut along imaginary axis into a top halfplane.

7. Mapping of an angle on the plane with a cut (Fig. 13.15. The transforming function is  n  1   √n  η λ n  −1   λ = ϕ(η) = −a 1 − 1 + √n  . η = ϕ (λ) = − a 1 − 1 + a  a

416


8. Mapping of a top halfplane with a cut along imaginary axis into a top halfplane (Figure 13.16). The transforming function reads η λ −1 λ = ϕ(η) = / η = ϕ (λ) = √ . 1 + η2 1 − λ2

14 MATCHING OF LIMITING ASYMPTOTIC EXPANSIONS

14.1 Two-point Padé approximants In many problems of mechanics the asymptotical series are very often obtained for various limiting values of the same parameter. An attempt to construct a uniformly suitable solution in whole interval of the parameter changes is not easy. In Chapter 6 the method of composite equations is illustrated, which in many cases yields good results. One may also apply the Bubnov–Galerkin or Ritz method, where both of the limiting states are accounted. In order to construct periodic orbits in reference [361] the following procedure is proposed. For small values of parameter ε perturbation analysis to examine periodic orbits is used. For large values of ε asymptotic analysis is used. For intermediate values of ε a computational technique is [361] reported. It appears that this problem can be solved using rational approximations, and particularly two-point Padé approximants (TPPA). First we give some definitions. The notion of TPPAs is defined in [121, 122]. Let ∞ a i εi for ε → 0, (14.1) F(ε) = i=0

F(ε) =

∞

bi ε−i

for ε → ∞.

(14.2)

i=0

The TPPA is represented by the rational function m

F(ε) =

k=0 n k=0

a k εk b k εk

,

where k + 1 (k = 0, 1, 2, ..., n + m + 1) coefficients of a Taylor expansion of F(ε), if ε → 0, and m + n + 1 − k coefficients of a Laurent series of F(ε), if ε → ∞, coincide with the corresponding coefficients of the series (14.1), (14.2). Let us investigate a model problem of vibrations of a chain consisting of n masses m, joined with springs of rigidity c. The deflection y of the k-th mass is determined by the equation

418


mÿk = c (yk+1 − yk ) − (yk − yk−1 ) , k = 1, 2, . . . , n. At the ends of the chain the boundary conditions are given yk = 0 for k < 1 or k > n. There are possible n proper forms of vibrations yk = A s sin

ksπ cos(ω s t + ϕ s ), n+1

s = 1, 2, . . . , n,

with the frequencies of natural vibrations 0.5sπ c sin . ωs = 2 m n+1

(14.3)

Let us construct the asymptotic expansions of the frequency ω ∗s in the vicinities of the points s = 0 and s = 2(n + 1), respectively. We introduce new variables ¯x =

x , 0.5π − x

x=

0.5sπ . n+1

Thus instead of the segment [0.2(n+1)] for s, we obtain the semi-infinite interval ¯x ∈ [0, ∞). The expansions as ¯x → 0 and ¯x → ∞ take the forms   π2 3 π2 4  2   ¯x − 1 − ¯x + ... for ¯x → 0, 0.5π ¯x − ¯x + 1 −   12 8 0.5π ¯x   sin =  1 + ¯x  π2 −3 π2 −4 π2 ¯x−2    + 1− ¯x − 1 − ¯x + ... for ¯x → ∞. 1−  8 12 8 A solution obtained with the TPPA method, valid for 0 ≤ ¯x < ∞, is given by c (π/2) ¯x + 0.81 ¯x2 . (14.4) ω=2 m 1 + (π/2) ¯x + 0.81 ¯x2 The results of the calculations of the frequency ω ¯ = 0.5ω(c/m)−0.5 are presented in Figure 14.1. The exact solution (14.3) is depicted by I, whereas the approximations are denoted by II and III, respectively. The rearranged TPPAs solution (14.4) practically coincides with the exact solution. Another interesting example is the Van der Pol equation. We give some necessary preliminary information according to [340]. The Van der Pol oscillator is governed by the equation x¨ + k x(x ˙ 2 − 1) + x = 0. The period of natural oscillation T is plotted in Fig. 14.2 as a function of the coefficient of the nonlinear friction k. The curve gives the numerical results obtained by means of the Runge-Kutta method [340]. The curves 1, 2 give the second order perturbation approximations

14.1 Two-point Padé approximants

419

Fig. 14.1. Frequencies of a chain obtained by various approaches.

k2 4 T = 2π 1 + + O(k ) 16

as k → 0,

T = k(3 − 2 ln 2) + 7.0143k −1/3 + ...

(14.5) k → ∞.

(14.6)

The TPPA formula constructed from two terms of the expansion (14.5) and one term of the expansion (14.6) (curve 3 in Fig. 14.2) T=

6.2832 + 1.5294k + 0.3927k 2 1 + 0.2433k

shows a good agreement with the numerical results (curve 4) for all values of k > 0. Let us consider now the Laplace transform √ F(p) = 0.5 p[H0 (p) − Y0 (p)], where H0 is Struve function, Y 0 is Bessel function, p is parameter of the Laplace transform. The exact inverse is (14.7) f (t) = (1 + t2 )−0.5 . The asymptotic inverses [142] take forms  2    1 − 0.5t + ... for t → 0, f (t)    t−1 + ... for t → ∞. By using the TPPAs one obtains

420


Fig. 14.2. period of the Van der Pol oscillator: comparison of numerical, perturbative and TPPA solutions

f (t) =

1 + 0.5t . 1 + 0.5t + 0.5t 2

(14.8)

The numerical results are plotted in Fig. 14.3. The upper curve (function (14.8)) coincides satisfactorily with the lower one (exact solution (14.7)). Hence the rational function (14.8) is an ‘asymptotically equivalent function’ in the sense of [615]. The accuracy of the TPPA solution may be improved by removing essential transform singularities [405].

f(t) 0.8 0.6

1

0.4 2

0.2 0

0

2

4

6

t

Fig. 14.3. Laplace transform inverse, the exact and TPPA solutions.


421

Elastic coefficients of composite materials may be evaluated effectively using the method of bounds. The bounds become increasingly accurate when more information on geometrical properties of the medium is known. For two-component isotropic composites, the PAs bounds for the effective constants λ e /λ1 already exists [158, 471, 491]. These bounds are usually obtained in the form of continuous fractions on the basis of the analytic properties of λ e (λ1 , λ2 ). Bergman [158] proved that λe (λ1 , λ2 )/λ1 = λe (1, λ2 /λ1 ) is a Stieltjes function of λ e (λ1 , λ2 ), analytic everywhere except on the negative real axes, satisfying λ e (λ1 , λ2 )/λ1 > 0 when λ2 /λ1 > 0. The Stieltjes functions have been extensively studied in the mathematics literature and their PAs and continuous fractions representations are well known [121, 122, 308, 359]. On the contrary, the analytic properties of TPPAs generated by two different power expansions of Stieltjes function have not been examined as deeply as the PAs. The above authors concerned themselves mostly with the TPPAs using equal number of coefficients of two power expansions at zero and infinity (‘balanced’ situation) [359]. Recently [359] investigated the TPPAs for a non-equal, finite number of terms of two power expansions of the Stieltjes functions at zero and infinity (‘unbalanced’ situation). Under some assumptions they proved that the diagonal TPPAs form sequences of lower and upper bounds uniformly converging to the Stieltjes function. The general ‘unbalanced’ situation, i.e., the TPPA corresponding to an arbitrary number of terms of power expansions at zero and infinity. They extended the fundamental inequalities derived for the PAs [639-692] to the general ‘unbalanced’ TPPAs case. We consider a Stieltjes function f (z) defined on 0 ≤ z ≤ ∞ by means of the following Stieltjes-integral representation ∞ f (z) = z 0

dγ(u) . 1 + zu

(14.9)

The spectrum γ(u) is a real, bounded, non-decreasing function defined on 0 ≤ u ≤ ∞. The asymptotic expansion of the Stieltjes function (14.9) at z = 0 takes the form f (z) ∼

∞

−cn (−z)n ,

(14.10)

n=1

where the expansion coefficients c n are given by the expression ∞ cn =

un−1 dγ(u),

n = 1, 2, . . . .

0

We also consider the asymptotic expansion at z = ∞, f (z) ∼

∞ n=0

C−n (−z)−n ,

(14.11)

422


where

∞ C−n =

u−n−1 dγ(u),

n = 0, 1, . . . .

(14.12)

0

We assume that the coefficients c n and C −n are finite. TPPA approximants calculated for series (14.10) and (14.11) have the following general form: [M/M]k =

Lk,M (z) a1k z + a2k z2 + · · · a Mk z M = . Qk,M (z) 1 + b1k z + b2k z2 + · · · b Mk z M

(14.13)

Consider the power series of (14.13) at zero, [M/M]k =

∞

−cn,k (−z)n

n=1

and infinity [M/M]k =

∞

C−n,k (−z)−n .

n=0

By definition, the rational function (14.13) is the two-point Padé approximant [M/M]k to the Stieltjes function (14.9), if cn,k = cn

for n = 1, 2, . . . , 2M − k

and C−n,k = C−n

for n = 0, 1, 2, . . . , k − 1.

According to the above definition, [M/M] 0 denotes the one Padé approximant. Two-point Padé approximants (14.13) can also be expressed in the form of an S continued fraction [121, 122]: g2M−k,k z G2M−k+1,k z G2M,k z g1,k z + ···+ + +···+ , 1 1 1 1

(14.14)

g2M−k,k G2M−k+1,k G2M,k g1,k g2,k + +···+ + +···+ , s 1 α β 1

(14.15)

[M/M]k = or alternatively, [M/M]k = where:

s = 1/z,

   1 for even k, α=  s for odd k,

   s for even k , β=  1 for odd k .

The coefficients g 1,k , . . . , g2M−k,k of a continued fraction (14.14) and (14.15) are uniquely determined by the 2M −k coefficients c n , n = 1, 2, . . . , 2M −k, of a Stieltjes series (14.10). To determine the remaining coefficients G 2M−k+1,k , . . . , G2M,k the values of k coefficients of a series (14.11) C −n , n = 1, 2, . . . , k, are additionally needed. The following theorems can be proved [637]: For real z > 0 the TPPA [M/M]1 and [M/M]2 to power series (14.10) and (14.11)

423

10 -4 0

1

2

3

4

5

6

7

f(z)

10 -3

10 -2

10 -1

10 0

10 1

[M/M] 0 , M=2,4,6,8,16

[M/M] 2 , M=2,4,6,8,16

ln((1+1000z)/(1+z))

[M/M] 1 , M=2,4,6,8,16

z


Fig. 14.4. The monotone sequences of one-point [M/M]0 and two-point Padé approximants [M/M]1 , [M/M]2 uniformly converging to the Stieltjes function.

424


form a monotone sequence of upper and lower bounds uniformly converging to the function f (z) (14.9), as M → ∞. To illustrate the obtained results we introduce the spectrum γ(u)  0 0 ≤ u ≤ 1,      u − 1 1 < u < 1000, γ(u) =      999 1000 ≤ u ≤ ∞, leading to the following Stieltjes function f (z): 1000

f (z) = z 0

1 + 1000z 1 du = ln . 1 + zu 1+z

(14.16)

The two Stieltjes power series expanded at zero f (z) =

∞ 1 − (1000n − 1)(−z)n n n=1

and at infinity f (z) = ln 1000 +

∞ 1 n=1

n

(1 − 0.001 n)(−z)−n

(14.17)

result immediately from (14.16). The Stieltjes series is convergent for |z| < 0.001, while the power expansion (14.17) is for |z| > 1. The monotone convergence of PA and TPPA to the Stieltjes functions (14.16) is shown in Figure 14.4.

14.2 Quasifractional approximants Evidently, the TPPAs is not a panacea. For example, one of the ‘bottlenecks’ of the TPPAs method is related to the presence of logarithmic components in numerous asymptotic expansions. Van Dyke wrote [660,p.207]: “A technique analogous to that of rational fractions is needed to improve the utility of series containing logarithmic terms”. This problem is the most essential for the TPPAs, because, as a rule, one of the limits (ε → 0 or ε → ∞) for a real mechanical problems gives expansions with logarithmic terms or other complicated functions. It is worth noticing that in some cases these obstacles may be overcome by using an approximate method of TPPAs construction by taking as limit points not ε = 0 and ε = ∞, but some small and large (but finite) values of ε [631]. On the other hand, Martin and Baker [465] (see also [210]) proposed the so called quasifractional approximants (QAs). The motion of QA may be formulated as follows [465]. Let us suppose that we have a powers series of ε for ε → 0 and asymptotic expansions F(ε) containing, for example, logarithm ln ε for ε → ∞. QA is a ratio R with unknown coefficients a i , bi , containing both powers of ε and ln ε. The coefficients a i , bi are chosen in such

14.2 Quasifractional approximants

425

a way, that (a) the expansion of R in powers of ε coincide with the corresponding expansion for ε → 0; and (b) the asymptotic behaviour of R for ε → ∞ coincides with F(ε). As an example a problem taken from composite theory will be further considered. One of the main tasks of the theory of dispersed media is a theoretical prediction of effective transport properties of heterogeneous media from a given microstructure and physical properties of components. That problem is formulated in a number of mathematically equivalent ways. Here we shall discuss it in the context of heat transfer. Our aim is to determine the effective heat conductivity k of infinite regular arrays of identical, perfectly conducting spheres of the volume fraction ϕ embedded in an isotropic matrix with unit conductivity. Batchelor [136] displayed eight distinct physical problems governed by identical, mathematical equations. Some of them are thermal, dielectric, magnetic, electric, elastic and other problems. Thus the prediction of the effective conductivity of a two-phase medium spans many fields. Therefore a great deal of effort has been devoted to its resolution. A detailed review one can find in [643]. In present work we shall give only a brief account of the papers. The calculation of the thermal coefficient k for an arbitrary composites was originally discussed by J. C. Maxwell and later on by many other authors. Lord Rayleigh examined the case of small spheres (ϕ tends to zero). He expressed the polarization of each sphere by an infinite set of multipole moments. Numerically, he evaluated low-order multipoles only. With the aid of modem digital computers, his evaluation has been extended to a large number of multipoles, see McPhedran and McKenzie [476, 477], Sangani and Acrivos [594]. For nearly touching spheres (ϕ tends to ϕ max ) Keller [378] derived an asymptotic solution. His results was improved by Batchelor and O’Brien [137], and Van Tuyl [666]. There still remains a certain parameter range which is covered neither by the asymptotic approach nor by the solution based on the assumption of small ϕ. The asymptotic formula for ϕ → ϕ max contains the logarithmic functions. That is a reason, why TPPA in its ‘pure form’ cannot be used in the problem under consideration. In order to overcome this obstacle the quasifractional approximants have been proposed. Here we use quasifractional approximants to derive an approximate analytical expression for effective conductivity k, valid for all the values of spheres volume fraction ϕ ∈ [0, ϕmax ]. As the bases we use the coefficients of the perturbation expansion of k at ϕ = 0 and the asymptotic formula for ϕ → ϕ max . Three different types of the spatial arrangements of the spheres arrays (simple cubic (SC), body centred cubic (BCC) and face centre cubic (FCC) arrays) are considered. Results obtained give a good agreement with the numerical data. Lord Rayleigh was analysing the case when spheres are arranged in the SC array. He developed a solution for the case ϕ → 0 by replacing spheres by dipoles and higher-order multipoles. He obtained [594]

426


k = 1 − 3ϕ

2+λ 1 − λ 10/3 +ϕ− dϕ + Oϕ14/3 1−λ 4 + 3λ

−1

,

where ϕ is the volume fraction of spheres; λ heat conductivity of spheres; d = 1.57. Table 14.1. The constants ai used in formula (14.18). SC array BCC array FCC array

a1 1.305 0.129 0.0753

a2 0.231 -0.413 0.697

a3 0.405 0.764 -0.741

a4 0.0723 0.257 0.0420

a5 a6 0.153 0.0105 0.0113 0.00562 0.0231 9.14 10−7

Meredith and Tobias [484] extended Rayleigh’s analysis and calculated the coefficient of O(ϕ14/3 ) term. The validity of Rayleigh’s method, however, was questioned because it involves the summation of nonabsolutely convergent series. McPhedran and McKenzie [476] have modified Rayleigh’s procedure in order to overcome conditional convergence. An alternative method, avoiding the difficulties encountered in Rayleigh’s original treatment, was revised by Zuzovsky and Brenner [707]. They used the method of generalized functions to develop a series expansions for k to order of O(ϕ 20/3 ) and foud that the coefficient of O(ϕ 14/3 ) reported in Meredith and Tobias [484] is not correct. The next development of that method was carried out by Sangani and Acrivos [594]. Let us start form their perturbation expansion for k in terms of ϕ 11/3 1 10/3 1 + a2 L3 ϕ + a4 L3 ϕ14/3 + k = 1 − 3ϕ − + ϕ + a1 L2 ϕ L1 1 − a3 L2 ϕ7/3 a5 L4 ϕ6 + a6 L5 ϕ22/3 + O(ϕ25/3 )−1 , where Li =

λ−1 , λ + 2i/(2i − 1)

(14.18)

i ∈ N.

The constants a1 , . . . , a6 are listed in Table 14.1. It should be stressed that for a case of large spheres (ϕ → ϕ max ) the above method fails, even if we take into account the high-order terms of the expansion (14.18). In the case of perfectly conducting large spheres (λ = ∞, ϕ = ϕ max ) the problem can be solved under the assumption that the heat flux occurs entirely in the region, where spheres nearly touch each other. For such a case, the effective conductivity is determined by the asymptotic form of the flux between two spheres represented by a logarithmically singular function. Keller [378] solved this problem with an accuracy of O(ln(χ)), where χ denotes dimensionless gap (χ → 0). Batchelor and O’ Brien [137] extended Keller’s results on the case of touching spheres and near perfect conductors. They derived the following asymptotic expansion at λ = ∞ and ϕ → ϕmax : (14.19) k = −M1 ln (χ) + M2 + O χ−1 .

14.2 Quasifractional approximants

427

Here χ = 1 − (ϕ/ϕmax )1/3 is the gap between neighboring spheres, χ → 0; M1 = 0.5pϕmax , p is the number of contact points on the surface of a sphere; M 2 is a constant dependent on the type of spheres space arrangement. The values of M 1 , M2 and ϕmax for the three cubic arrays are listed in Table 14.2. Table 14.2. The constants M1 , M2 and ϕmax . SC array BCC array FCC array

M1 π/2 √ √3π/2 2π

M2 0.7 2.4 7.1

ϕmax √π/6 √3π/8 2π/6

We now go to the problem of evaluating of the effective conductivity k in terms of quasifractional approximants. We consider a function of ϕ determined by a power series expansion (14.18) at ϕ → 0 and the asymptotic expansion (14.19) at ϕ → ϕmax . The desired solution has a logarithmic singularity as shown in the expression (14.19). In order to reproduce this singularity, the quasifractional approximant must contain a similar term. It can be written as 7 k = P1 (ϕ) + P2 ϕ(m+1)/3 + P3 ln (χ) Q (ϕ) , (14.20) where the rational functions P 1 (ϕ), Q(ϕ) and constants P 2 , P3 are determined by the following conditions: (i) the expansion of (14.20) in powers of ϕ at ϕ → 0 coincides with m leading terms of the perturbation expansion (14.18), and (ii) the asymptotic behaviour of (14.20) at ϕ → ϕ max coincides with n leading terms of the asymptotic expansion (14.19). Thus we obtain Q (ϕ) = 1 − ϕ − a1 ϕ10/3 ,

P1 (ϕ) =

m

αi ϕi/3

(14.21)

i=0

    0, n = 1. 7 P2 =    − (P1 (ϕmax ) + Q (ϕmax ) K2 ) ϕ(m+1)/3 . max Here α0 = 1, α3 = 2 − Q (ϕmax ) M1 (3ϕmax ) , Q (ϕmax ) M1 , α10 = −a1 − 10ϕ10/3 max 7 j/3 α j = −Q (ϕmax ) M1 jϕmax , j = 1, 2, 4, ..., 8, 9, 11, 12, ..., m − 1, m. The quasifractional approximant (14.20), (14.21) represents an approximate analytical expression of k, valid for all values of the spheres volume fraction ϕ ∈ [0, ϕmax ]. It should be stressed that incorporation of more terms of the ‘limiting’ expansions (14.18) and (14.20) leads to the growth of the accuracy of the obtained solution. Let us illustrate this dependence for the case of a SC array. We

428


Fig. 14.5. Effective conductivity coefficient k for regular array of spheres. Analytical results (14.20), (14.21) for different values of m, n are compared with experimental data given in [707].

Fig. 14.6. Effective conductivity coefficient k for regular array of spheres. Analytical solution is compared with experimental measurements [382].

have calculated k for different values of m and n. In Figure 14.5 our analytical results are compared with the experimental measurements. We restrict the evaluation to m = 19 and n = 2 for all types of arrays. It provides a simple analytical forms of the solution (14.20), (14.21) and satisfactory agreement with numerical data. Numerical results for BCC and FCC arrays are displayed in Figure 14.6. The obtained solution (14.20), (14.21) is compared with experimental results [477] for the BCC array with numerical data [382] for the FCC array. The discrepancy between

14.3 Post-buckling behaviour of shallow convex shell

429

our analytical solution (14.20), (14.21) and the numerical results does not exceed 3.6, 4.1 and 6.7% for the SC, the BCC and the FCC arrays, respectively.

14.3 Post-buckling behaviour of shallow convex shell In this section asymptotical method [280, 281, 282, 283, 284, 285] of integration of nonlinear equations of thin elastic shell theory is presented. Owing to application of a new perturbation parameter being proportional to the ratio of shell thickness to deflection amplitude of shell the simple analytical solution is obtained. Observe that this parameter is really small when deflections are large compared to shell thickness, especially in the case when a shell is in a post-critical equilibrium state. Asymptotical development of corresponding solution with respect to this parameter is carried out. It is shown, that first two approximations lead to a geometrical theory of shell stability formulated by A.V. Pogorelov [559]. Comparing asymptotical and numerical solutions [280], obtained for a spherical shell with axially symmetric deformation, a remarkable accuracy of the proposed method in the case of large deflections is shown. To obtain the solution for entire range of deflection amplitude two asymptotical series, where first (second) is suitable for small (large) deflections, are carried out applying PA. In the framework of the shallow shell theory a post-critical axially symmetric deformation of completed sphere under uniform external pressure q is considered. The initial governing equations have the form 1 dΦ 1 dW D d 2 (∇ W) = + h dr R dr r dr   1 dW d 2 1 (∇ Φ) = −E  + dr R dr 2r

dΦ rq + , dr 2h 2  dW  , dr 

(14.22)

where Φ is a stress function, and R is the sphere radius. System of equations (14.22) yields two obvious solutions. First of them reads W = const,

dΦ/dr = −qrR/(2h),

and it corresponds to the initial moment-less state of the shell. Second one represents an isometric transformation (bending) of sphere, obtained via mirror like reflection of a segment across its plane, and it reads W = W ◦ (1 − r 2 /(W ◦ R)). Introducing the variables of transformation z = r2 /(W ◦ R),

w = W/W ◦ .

which correspond to relation (14.23), the following equations are obtained

(14.23)

430


ε2

ε2 =

w◦

/

d2 dw dϕ dw z = 1 + 2 = q◦ , dz dz dz2 dz 2 dϕ dw dw 2 d ε 2 z =− 1+ , dz dz dz dz

(14.24)

EhW ◦ W◦ , Φ=ϕ/ , h 3(1 − ν2 ) 12(1 − ν2 ) 2 q 2E h q◦ = , q∗ = . q∗ 3(1 − ν2 ) R 2

, w◦ =

Observe that the perturbation parameter ε decreases with the increasing of the deflection amplitude W ◦ and becomes small for essentially post-buckling configurations. Limiting system of equations (for ε = 0) possesses two solutions. First one corresponds to the moment-less shell state, and the second governs isometric transformation of the middle surface, which in new coordinates has vary simple form w = 1 − z.

(14.25)

Compound solution at z = 1 exhibits a discontinuity in the first derivative, which is compensated by functions of internal boundary layer. Therefore, asymptotical series of solution of the system (14.24) for ε → 0 has the form w = εn wn−1 , ϕ = εn ϕn−1 , q◦ = q0 + εn wn , n = 1 ÷ 4, εwi = Wi (z) + ενi (t), εϕi = Φi (z) + εui (t), t = (1 − z)/ε,

(14.26)

where: νi , ui are the functions governing behaviour of internal edge effect, whereas Wi , Φi are the functions corresponding to fundamental state. One may get: dΦk dWk = 0, = −qk , z > 1, dz dz

(14.27)

dW0 dWk+1 dΦk = −1, = 0, = qk , z < 1, k = 0 ÷ 3. dz dz dz Components Wk and Φk of the solution can be presented in the form of functions depending on t. For example, for z < 1 we have W0 = 1 − z = εt. Then in the expression (14.26) w i and ϕi can be treated as functions of the variable t, which are continuous together with their derivatives. Due to relations (14.27) they should satisfy the following boundary conditions w k = 0, ϕ k = qk w k+1 = 0, ϕ k = −qk

for t → −∞; for t → +∞.

(14.28)

Applying an asymptotical analysis of equation (14.24) and taking into account series (14.26), the following equations are obtained:

14.3 Post-buckling behaviour of shallow convex shell

431

(i) fundamental approximation w 0 − ϕ0 (1 − 2w0 ) + q0 = 0, ϕ 0 + w0 (1 − w0 ) = 0,

(14.29) (14.30)

w 1 − (tw0 ) + w1 ϕ0 − ϕ1 (1 − w0 ) + q1 = 0, ϕ 1 − 2ϕ0 − tϕ0 + w1 (1 − 2w0 ) = 0,

(14.31) (14.32)

(ii) second approximation

(iii) third approximation w 2 − (tw1 ) + 2w1 ϕ1 − ϕ2 (1 − 2w0 ) + 2ϕ0 w2 + q2 = 0,

ϕ 2

−

2ϕ 1

−

tϕ 1

+

w 2 (1

−

2w 0 )

−

w 1 2

= 0.

(14.33) (14.34)

Above given equations together with boundary conditions (14.28) can be used for determination of functions w i and ϕi for any arbitrary values q i . In the second and successive approximations the equations are linear. However, the coefficients of load series development with respect to ε are not defined. To obtain these values the variational approach is applied. Let us consider the functional of full shell potential energy, which owing to asymptotical analysis and series (14.26), has the form

U = D1 J0 ε + J1 ε2 + J2 ε3 − q◦ 1 + 2ε2 ν0 dr + 2ε3 w1 dt + O(ε4 ) , (14.35) J0 = J2 =

ϕ 0 2

+

w 0 2

dt, J1 = 2 ϕ 0 ϕ 1 − tϕ 0 + w 0 w 1 − tw 0 dt,

ϕ 1 2 + 2ϕ 0 ϕ 2 − 4tϕ 0 ϕ 1 + t2 ϕ 0 2 + w 1 2 + 2w 0 w 2 − 4tw 0 w 1 + t2 w 0 2 dt,

(14.36) Notice that now and later an integration procedure is carried out on the interval of −∞ to +∞. Owing to q◦ representation in the series (14.26) form, one gets U = D1 −q0 + ε (I0 − q1 ) + ε2 (I1 + q2 ) + ε3 (I2 − q3 ) + O(ε4 ) , (14.37) I0 = J0 , I1 = J1 − 2q0

ν0 dt, I2 = J2 − 2q1

ν0 dt − 2q0

w1 dt.

Owing to relation (14.30), (14.32), (14.34), and boundary condition (14.28) variation of the full potential energy functional (14.35) with respect to functions w i and ϕi yields as Euler equations in each approximation the corresponding equilibrium equations (14.29), (14.31) and (14.33). For instance, the considered problem of functional I 2 minimum with accounting of couplings (14.30), (14.32) and (14.34) exhibits the following features. Variations with respect to functions w 2 and ϕ2 yield

432


the equation (14.29). Owing to variations with respect to functions w 1 and ϕ1 the equation (14.29) is obtained. Owing to variations with respect to functions w 2 and ϕ2 the equation (14.31) is yielded. And finally owing to variations of I 2 with respect to w0 and ϕ0 , the expression (14.33) is obtained. However, the parameter ε should be also considered as a variable since it is coupled with deflection amplitude of post-critical configuration. Owing to variation of (14.35) with respect to ε, the following relations are obtained 3 1 1 q0 = 0, q1 = J0 , q2 = J1 , q3 = J2 − q1 ν0 dt, 4 2 4 where Ji represent minimal values of these functionals. Relations (14.28)–(14.40) exhibit a symmetry because functions ϕ 0 , ϕ 2 , w 1 (ϕ 1 , w 0 , w 2 ) are even (odd). This observation yields J 1 = 0 since the corresponding integrand is an odd function. One may also check that q 2 = q4 = 0. In the relation (14.36) the terms including w 2 and ϕ 2 can be integrated. Hence, using (14.28) one gets 2 ϕ 1 2 + t2 ϕ 0 2 − 4tw 0 w 1 + t2 w 0 2 + 2ϕ 0 tϕ J2 = dt. (14.38) 1 − w1 Furthermore, in order to define q 1 it is necessary to integrate equations (14.29) and (14.30) of the fundamental approximation. The coefficient q 3 is defined also by the functions w1 and ϕ1 of second order approximation. The functional J 0 , via the corresponding change of variables, is transformed to the Pogorelov [559] functional with the minimum J ∗ = 2J0 1.12. Applying the Ritz method to solve the problem of J 2 minimum, we obtain J 2 = −0.4. Finally, the following relation (for ν = 0.3) is derived q◦ = 0.42ε + 0.26ε 3 + O(ε5 ).

(14.39)

2 1/4

The first term with accuracy up to multiplier (1 − ν ) gives known result. To conclude, the relations of geometrical theory are asymptotically exact, for ε → 0 accounting of two first approximation. The obtained result is reported in Figure ??. Curve 1 corresponds to exact solution obtained numerically [305]. Curve 2 is obtained owing to application of the fundamental approximation, which corresponds to geometrical theory. Relation (14.39) is depicted by curve 3. Comparing curves 1 and 3 one may observe that coincide for h/W ◦ ≤ 1. For ◦ W → 0 the considered asymptotical approach gives qualitatively uncorrected result. However, in this case a sufficiently good Koiter approach [203, 390, 391, 679] is applicable. Owing to this method, and applying perturbation method for small deflections, the following asymptotical formula is obtained q◦ = 1 + aw◦ + O(w◦2 ).

(14.40)

Note that in the case of axially symmetric deformation of shallow sphere a = 0. Since 1/ε is taken as perturbation parameter, then the relation (14.39) yields first series terms of the function q ◦ (ε) with respect to 1/ε.

14.3 Post-buckling behaviour of shallow convex shell q

433

o

J

1 0.6

0.2

0

2

4

wo

Fig. 14.7. Comparing of analytical and numerical results.

Two-points Padé approximation are applied for matching of asymptotic series (14.39) and (14.40). For this purpose q ◦ (ε) is sought in the form of rational functions with coefficients defined via the condition of coincidence of this function for ε → 0 and ε → ∞ with the series (14.39) and (14.40) respectively. Finally, the following formula is obtained q◦ (ε) = A(ε)/ (1 + A(ε)) , A(ε) = 0.42ε + 0.176ε 2 + 0.333ε3 + 0.4ε4

(14.41)

This result is shown in Figure 14.7 by dashed curve. A comparison with curve 1 [305] shows a good accuracy of the obtained solution.

15 COMPLEX VARIABLES IN NONLINEAR DYNAMICS

When dealing with continualization of thin-walled structures with discrete reinforcing elements (see sections 10.3–10.5), a long wavelength asymptotics is usually implied. As this takes place, short wave length processes are excluded from consideration. Meantime, such processes can be important in many cases, especially for high frequency excitations. Similar situation was described in section 10.10 in application to a beam with concentrated masses and discrete supports. Here we focus an attention on much more complicated nonlinear problems. It is shown that adequate approach to short wave length nonlinear dynamics of weightless strings and beams with concentrated masses and discrete supports may be developed using complex representation of equations of motion. The complex representation of classical equations of motion for a system of linear oscillators was first used in quantum mechanics and for the analysis of so-called coupled oscillations and waves in mechanics, electronics and solid state physics. In this representation, the complex conjugate linear combinations ν j +iu j and ν j −iu j of displacements u j and velocities ν j of oscillators can be visually presented as vectors of equal length rotating in opposite directions. Actually it is enough to find only one complex function for each oscillator thus completely determining both displacement and velocity. Such a choice of variables in particular leads to a very simple and natural procedure of quantization, complex conjugate functions become operators of creation and annihilation, and their squared moduli - number of elementary excitations. In case of coupling between oscillators, both complex conjugate functions are included in each equation of motion. However, in the theory of coupled oscillations it is supposed that the unidirectional rotations of oscillators are connected more strongly than the vectors with opposite directions. This simplification reduces the order of equations of motion by the factor of two and ensures right expressions for the first two terms in the expansion of an exact solution of the linearized equations in powers of the coupling parameter. For a coupled system of nonlinear oscillators, the possibility of comparison with exact solutions is absent. In this connection the equations of motion for unidirectional rotations are usually treated phenomenologically as the simplest mathematical model of the nonlinear oscillatory system. Its validity is confirmed by comparison with the averaged equations for complex amplitudes, qualitative reasons and asymptotic estimations. However, the scheme of perturbation theory is expected not only to justify the domination of coupling be-

436


tween unidirectional rotations but also to provide a possibility of constructing higher approximations. The main goal of this part is to demonstrate the possibility of crucial simplification of nonlinear problems by means of the complex representation.

15.1 Nonlinear oscillator with cubic anharmonicity Free oscillations Let’s first consider nonlinear oscillatory system with one degree of freedom described by the equation mU¨ + 2nU˙ + c1 U + c3 U 3 = 0

(15.1)

with the initial conditions: t = 0, U = U 0 , U˙ = V0 .√After introduction of dimensionless variables: τ = ω0 t, u = √EU/c , where ω0 = c1 /m, E 0 initial energy of an 0 1 oscillator, we have d2 u du + 2εγ + u + 8εαu3 = 0, (15.2) 2 dτ dτ du = v0 . τ = 0; u = u0 ; dτ Here: n c3 εγ = √ ; 8αε = 2 · E0 . c1 m c1 Let’s write the equation of motion (15.2) as a system of two first order equations dv = −2εγv − u − 8αεu3 , dτ du =v. dτ Multiplying the second equation on imaginary unit and adding it with first, and then deducting from first we obtain dψ − iψ + εγ (ψ + ψ∗ ) + iεα (ψ − ψ∗ )3 = 0, dτ dψ∗ + iψ∗ + εγ (ψ + ψ∗ ) + iεα (ψ∗ − ψ)3 = 0, (15.3) dτ where: ψ = v + iu, ψ∗ = v − iu. Obviously, the second equation can be obtained from the first one by operation of complex conjugation. Therefore only first equation will be considered later on. Let’s change the dependent variable in (15.3): ψ = eiτ ϕ(τ) .

15.1 Nonlinear oscillator with cubic anharmonicity

437

Then we obtain dϕ +εγ ϕ + e−2iτ ϕ∗ +iαε e2iτ ϕ3 − 3|ϕ|2 ϕ + 3e−2iτ |ϕ|2 ϕ∗ 2 − e−4iτ ϕ∗3 = 0 . (15.4) dτ Hereinafter we assume that ε 1. The direct expansion of the solution of the equation (15.4) by small parameter ε should lead to the appearance of secular terms. Alternatively we introduce a “slow” time (15.5) τ1 = ετ 1 + ξ1 (τ1 )ε + ξ2 (τ1 )ε2 + . . . , along with time τ where ξ i (τ1 ) yet not defined functions, and we will consider required complex function as function of two variables ϕ(τ 1 , τ1 ). Let’s present this function as expansion by a small parameter ε: ϕ = ϕ0 + εϕ1 + ε2 ϕ2 + . . . .

(15.6)

With allowance for (15.4)–(15.6) we obtain ∂ ϕ0 + εϕ1 + ε2 ϕ2 + . . . + ∂τ εL (τ1 , ε) ·

' ( ∂ ϕ0 + εϕ1 + ε2 ϕ2 + . . . + εM ϕ1 , ϕ∗1 , ε = 0, ∂τ1

(15.7)

where: L(τ1 , ε) =

1 + ξ1 (τ1 )ε + ξ2 (τ1 )ε2 + . . . dξ (τ ) dξ (τ ) , 1 1 2 1 1 − ετ1 dτ + dτ ε + . . . · 1 − ξ1 (τ1 )ε − . . . 1 1

M(ϕ1 , ϕ∗1 , ε) = γ ϕ0 + εϕ1 + ε2 ϕ2 + . . . + e−2iτ ϕ∗0 + εϕ∗1 + ε2 ϕ∗2 + . . . + 0 3 iα e2iτ ϕ0 + εϕ1 + ε2 ϕ2 + . . . − 3ϕ0 + εϕ1 + ε2 ϕ2 + . . .2 ϕ0 + εϕ1 + ε2 ϕ2 + . . . + 3e−2iτ ϕ0 + εϕ1 + εϕ2 + 3 1 . . .2 ϕ∗0 + εϕ∗1 + ε2 ϕ∗2 + . . . − e−4iτ ϕ∗0 + εϕ∗1 + ε2 ϕ∗2 + . . . = 0, (15.8) l = 0, 1, . . . . Now we equate to zero the coefficients at each from growing powers of the parameter ε ∂ϕ0 = 0. Therefore ϕ 0 = ϕ0 (τ1 ), i.e. the principal approximation is the 1. ε0 : ∂τ function ϕ0 depends only on “slow” time. 2. ∂ϕ0 ∂ϕ1 −γe−2iτ ϕ∗0 −iα e2iτ ϕ30 − 3e−2iτ ϕ20 ϕ∗0 + e−4iτ ϕ∗3 +γϕ0 −3αϕ20 ϕ0 = − ε1 : 0 : ∂τ1 ∂τ . . . . . . . . . . .

(15.9)

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The left part of this equation depends only on slow time τ 1 . The right part contains a derivative of function ϕ 1 (τ, τ1 ) on fast time and function of slow time with fast varying exponential coefficients. integrating both parts of the equation (15.9) by fast time in limits from 0 to π, we will obtain in a principal approximation with respect slow time −∆ϕ1 ∂ϕ0 , (15.10) + γϕ0 − 3iα|ϕ0 |2 ϕ0 = ∂τ1 π where: ∆ϕ! = ϕ1 (π, τ1 ) − ϕ1 (0, τ1 ). As is shown lower the function ϕ 1 can be defined in such a manner that ∆ϕ 1 0; then the principal approximation is described by the equation: ∂ϕ0 + γϕ0 − 3iα|ϕ0 |2 ϕ0 = 0 . (15.11) ∂τ1 We pay attention to that important circumstance, that in a main approximation for ϕ0 (τ), in difference from phenomenological model, the terms describing inertial, dissipative and nonlinear elastic forces have the same (zero) order on a parameter ε, the linearly-elastic force in the equation is absent. To obtain in the explicit form a solution of the equation (15.11) we introduce a new change of a dependent variable ϕ0 = e−γτ1 Φ0 .

(15.12)

After a substitution (15.12) in (15.11) we obtain dΦ0 − 3iαe−2γτ1 − Φ20 − Φ0 = 0. ∂τ1

(15.13)

At γ = 0 the equation (15.13) and equation conjugate to it, can be written in the hamiltonian shape: dΦ∗0 dH dH dΦ0 = ; =− ; (15.14) dτ1 dΦ∗0 dτ1 dΦ0 where: H = 32 iα|Φ0 |4 , and, hence, admit the first integral H = const. However, in general case there is another integral of the equation (15.13) also, which can be found multiplying this equation and conjugate to it accordingly on Φ ∗0 and Φ0 , and then summing the obtained expressions. As a result, it appears that after transition to variable Φ 0 , despite of a dissipation presence, |Φ 0 |2 = N = const. Hence, the equation (15.13) is the integrable system and its solution has a form: √ 3iα −2γτ1 Φ0 = Ne− 2γ N (e −1) . The appropriate functions ϕ 0 , ψ0 are written as follows: √ √ 3i 3α −2γτ1 −2γετ ϕ0 = Ne−γτ1 e− 2γ (e −1) ; ψ0 = Ne−γετ ei 1− 2γ N (e −1) .

(15.15)

After selection of real and imaginary parts this outcome completely coincides with a solution obtained in the 1-st approximation on the basis of the real equations by an averaging method [180].


439

The advantages of the complex equations, noticeable already at this stage (reducing of the system order by factor two), especially are exhibited at a construction of higher approximation. For the determination of function ϕ 1 we return to the equation (15.8). After a substitution of expressions for ϕ 0 in the left (15.8) the latter has a form ∂ϕ1 = −γe−2iτ ϕ∗0 − eα e2iτ ϕ30 + 3e−2iτ ϕ20 ϕ∗0 + e−4iτ ϕ∗3 . 0 ∂τ π 1 Then ∂ϕ ∂τ dτ = 0 (i.e. ∆ϕ1 = 0 really) and the function ϕ 1 (τ, τ1 ) can be written as 0

follows:

1 1 1 −4iτ ϕ1 = − iγϕ∗0 e−2iτ − α ϕ30 e2iτ − 3ϕ20 ϕ∗0 e−2iτ − ϕ∗3 e . 2 2 2 0

(15.16)

Let’s consider still equation for the function ϕ 2 (τ, τ1 ), equating zero coefficients at ε2 in (15.7)

∂ϕ2 ∂ϕ1 dξ1 dϕ0 =− − τ1 + ξ1 (τ1 ) − γϕ1 − γe−2iτ ϕ∗1 − 3iαe2iτ ϕ20 ϕ1 + ∂τ ∂τ1 dτ1 δτ1 −4iτ ∗2 ∗ 3iα 2|ϕ0 |2 ϕ1 + ϕ20 ϕ∗1 − 3iαe−2iτ 2|ϕ0 |2 ϕ∗1 + ϕ∗2 ϕ0 ϕ1 . (15.17) 0 ϕ1 + 3iαe In the right part of the equation (15.17) all magnitudes except for function ξ 2 (τ1 ) are known. It contains the resonance terms which lead to solution for ϕ 2 (τ, τ1 ) growing in fast time. However due to presence of not yet defined function ξ 1 (τ1 ), it is possible to convert in zero a sum of resonance terms. In limiting cases α = 0 and γ = 0 the solutions for function ξ 1 (τ1 ) are degenerated in constants, in particular ξ 1 (τ1 ) = 12 iγ for linear system. Generally function ξ 1 (τ1 ) satisfies to the linear differential equation of the first order: 1 Q(τ1 ) dξ1 + ξ1 = − dϕ , dτ1 τ1 τ1 dτ10 which particular solution has a form: 1 ξ(τ1 ) = τ1

Q(τ 1)

dτ1 , dϕ0 dτ1

where: Q(τ1 ) - the sum of resonance terms (i.e. terms, not containing fast time, in the equation (15.17)), which we do not write out here because of its unwieldy form. Nonlinear oscillator in a harmonic external field Let oscillator, considered in item 1, is now in a periodic external field F(t) = 2ε f cos ωt. By introducing of dimensionless variables τ = ω 0 t, u = f U/c1 we rewrite the equation of motion as

440


iωτ du2 du −i ω τ + 2εγ + u + 8αε − u3 = ε e ω0 + e ω0 , 2 dτ dτ where: εγ =

n , −c1 m

8αε =

(15.18)

c3 2 f . c21

Let’s assume that ε 1 and lω − mω 0 = β1 εω0 , where l and m are integer mutually prime numbers β 1 = O(1). Case 1) l = m = 1 corresponds to a principal resonance; 2) m = 1, l > 1; 3) l = 1, m > 1 and 4) l > 1, m > 1 – sub-harmonic, ultra-harmonic and sub-ultra-harmonic resonance accordingly. Then the complex representation of the equation of motion has a form: 1 1 dψ − iψ + εγ(ψ + ψ∗ ) + iαε(ψ − ψ∗ )3 = ε ei l (m+β1 ε)τ + e−i l (m+β1 ε)τ . dτ After the change of a variable

ψ = eiτ ϕ(τ)

(15.19)

(15.20)

we obtain dϕ ∗ + eγ ϕ + e−2iτϕ + iα e2iτ ϕ3 − 3|ϕ|2 ϕ + 3e−2iτ |vp|2 ϕ∗ − e−4iτ ϕ∗3 = dτ m m ε ei( l −)τ eiβ1 ετ + e−i( l +1)τ e−iβετ ;

β1 . (15.21) l As well as in case of a free oscillator we introduce now a slow time (15.5). Also search for a solution as an expansion on a small parameter ε (15.6). Substituting (15.5), (15.6) in the equation (15.21) we have β=

∂ ∂ ϕ0 + εϕ1 + ε2 ϕ2 + . . . + ϕ0 + εϕ1 + ε2 ϕ2 + . . . + εL(τ1 , ε) · ∂τ ∂τ1 ' ( εM ϕ1 , ϕ∗1 , ε = εr (τ, τ1 , ε) , (15.22) where: m m r(τ, τ1 , ε) = ε ei(−1+ l )τ eiβ(1−ξ1 (τ1 )ε+...)τ1 + e−(1+ l )τ e−iβ(1−ξ1 (τ1 )ε+...)τ1 .

(15.23)

1. Main resonance. If ml = 1 we come to case of a main resonance. Then, equation zero coefficients at various powers of a small parameter ε we obtain: ∂ϕ0 = 0, therefore ϕ 0 = ϕ0 (τ1 ), ∂τ dϕ0 ε1 : + γϕ0 − 3iαϕ20 ϕ0 − eiβτ1 = dτ1 ∂ϕ1 −2iτ −iβτ1 −γe−2iτ ϕ∗0 − iα e2iτ ϕ30 + 3e−2iτ −ϕ20 ϕ∗0 −e−4iτ ϕ∗3 e (15.24). − 0 +e ∂τ1

ε0 :


441

Integrating both parts of the equation (15.24) on a fast variable τ from 0 up to π we obtain in principal approximation on “slow” time τ 1 (as well as in the case of free oscillations, the function ϕ 1 is determined in such a manner that ∆ϕ 1 = 0): dϕ0 + γϕ0 − 3iα|ϕ0 |2 ϕ0 = e−iβτ1 . dτ1

(15.25)

Let’s introduce in (15.25) the change of variable ϕ0 = iΦ0 (τ1 )eiβτ1 .

(15.26)

Then we come to the autonomic equation: dΦ0 + (γ + iβ) Φ0 − 3iα|Φ0 |2 Φ0 + i = 0, dτ1 which at γ = 0 together with conjugate equation form the integrable Hamiltonian system: dΦ∗0 dΦ0 ∂H ∂H = , =− , dτ1 ∂Φ∗0 dτ1 ∂Φ0 admitting the first integral: 3 H = iβ|Φ0 |2 + iα|Φ0 |4 − i Φ0 + Φ∗0 = const. 2 In the general (γ 0) the representation of required function as Φ0 = a(τ1 )eiδ(τ1 ) , where: a(τ1 ) and δ(τ1 ) – are real functions, leads to the system of the nonlinear equations: da + γa − sin δ = 0, dτ1 dδ + βa − 3αa3 + cos δ = 0 . a dτ1 The stationary points of this system corresponding to steady-state oscillations satisfy to cubic equation for the squared amplitude: b3 − α2 β2 + α1 β − α0 = 0, where:

1 1 2 2β 2 , , α = + β γ , α2 = 1 3α 9α2 9α2 and also to transcendental equation: b = a2 , α0 =

sin δ = γa, determining the phase δ through amplitude.

442


The simple analysis shows, that if a ≤ 2/3γ 3 , the condition of uniqueness of the real solution of the given cubic equation is fulfilled irrespective of magnitude of frequency parameter β. If α > 2/3γ 3 , at each fixed pair of α and γ, thee are regions of rather small and rather large values β in which the real solution of a cubic equation also is single, but between these regions the values β corresponding to three real roots, i.e. three steady state regimes (resonance, unstable and nonresonance), are located. Thus the maximal amplitude of a resonance mode is determined by relation a = 1/γ. If γ → 0 the value β corresponding to upper (at α > 0) and low (at a < 0) boundaries of nonuniqueness region go to ±∞, as well as amplitude of a resonance mode. The lower (at α > 0) and upper (ar α < 0) boundaries of this region are determined by values of frequency parameter β = ∓(9/2) 2/3α (if α < 0 the boundary value b is also negative). At γ = 0, which the equation concerning function Φ 0 represents integrable system, the phase of the stationary solution is determined by a condition δ = kπ (k = 0, ±1, . . .), so the amplitude satisfies to a cubic equation: 3αa3 − βα ∓ 1 = 0 . Obviously, in this case the stable stationary points on a phase plane (a, δ), due to presence of first integral H = const, are enclosed by a set of trajectories, which equation has a form 3 βa2 − αa4 − a cos δ = C . 2 The regions close to various unstable points, corresponding to some critical values of a constant C, are divided by separatrixes transiting through unstable stationary points. The analysis of a main resonance in the principal approach is in a full accordance with the results for phenomenological model [393]. However, within the framework of the asymptotic approach the opportunity of construction of higher approaches is unclosed also. For determination of the first correction to principal approximation, determined by the function ϕ 1 , let us consider the equation (15.24). After substitution in it the expression for ϕ 0 one can obtain: ∂ϕ1 = −γc∗ e−2iτ e−βτ1 − iα e2iτ c3 e3iβτ1 + 3e−2iτ |c|2 ce−iβτ1 − ∂τ e−4ıτ c∗3 e−3iβτ1 + e−2iτ e−βτ1 . From here the first correction to a principal approximation can be found by integration on a fast variable τ: 1 α ∗3 2iτ 3iβτ1 ϕ1 (τ, τ1 ) = − iγc∗ e−2iτ e−iβτ1 − − 3e−2iτ |c|2 ce−iβτ1 + c e e 2 2


443

1 −4iτ ∗3 −3iβτ1 1 e c e + ie−2iτ e−iβτ1 . 2 2 Thus the value ∆ϕ 1 really is equal to zero. At a determination second and consequent corrections there are resonance terms which compensating is achieved by an appropriate choice of function ξ i (τ1 ), similarly to case of a free oscillator. 2. Secondary resonances. At m/l 1 the resonant effect consists in excitation of steady-state oscillations with frequency, close to a natural frequency of linearized system, under action of external force, far of frequency. As the analysis shows it is possible only under condition that the intensity of external action has zero, and dissipative force – at least, second order on parameter ε. Really, in such case of the equations for the first two approaches look like: m m ∂ϕ0 = ei(−1+ l )τ eiβτ! + e−i(1+ l )τ e−iβτ1 , ∂τ

(15.27)

dϕ0 ∂ϕ1 3 − γe−2iτ ϕ∗0 − iα e2iτϕ0 + 3e−2iτ |ϕ0 |2 ϕ∗0 − e−4iτ ϕ∗3 − 3iα|ϕ0 |2 ϕ0 = − . 0 dτ1 ϕτ (15.28) The general solution of the linear nonhomogeneous equation (15.27) is written as follows: 1 i(−1+ m +εβ)τ 1 −i(1+ m +εβ)τ l l e + e . (15.29) ϕ0 = ϕ˜ 0 + i 1 − ml 1 + ml Substituting this expression in the equation (15.28) and fulfilling integration on fast variable τ in limits from 0 up πl we obtain in main approach on slow variable τ1 : 24iα dϕ˜ 0 2 − (15.30) ϕ˜ 0 − 3iα|ϕ˜ 0 | ϕ˜ 0 = 0, dτ1 1 − (m/l)2 2 (as well as in case of a main resonance it is possible to be convinced, that ∆ϕ 1 = 0). 24iα

2 2 After transformation ϕ˜ 0 = e [1−(m/l) ] Φ0 we come to equation

dΦ − 3iα|Φ0 |2 Φ0 = 0 dτ1 with the stationary solution: Φ0 = Then ϕ˜ 0 = if

√

  i

24α

2 1−(m/l)2

Ne [

]

  +3αN  τ1

N=

√

Ne3iαNτ1 .

, and the requirement of periodicity is satisfied, 8 β − . 3α 1 − (m/l)2 2

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At realization of this requirement in a steady-state mode the combination of oscillations with frequency, close to a natural frequency of linear system and frequency ofexternal excitation is implemented though the period of external excitation in ml times more (less) than a period of natural linear oscillations. Or else, as well as in case of a main resonance the speech goes about “supported” natural oscillations, by accompanied in this case by forced oscillations with frequency far from a natural frequency of linear system. The analysis is valid so long as the magnitude ml itself does not become a small or large parameter. In the latter case it should be taken into account from the very beginning at the asymptotic analysis. As for existence of combination resonances it is required still, as it was underlined above the smallness of dissipative forces on a comparison with driving forces (influence former one should be exhibited only in the equation concerning the second order corrections ϕ 2 ), practically observable there can be only the resonance of low order. Let’s note, that using the complex representations the research of resonance conditions becomes considerably simpler on a comparison with a standard averaging method [180], due to reducing of the equation order by coefficient two. Especially the treatment of combination resonances becomes noticeably simpler. Parametrical excitation of a nonlinear system Let’s consider now case of parametrical excitation of a nonlinear oscillator, when the equation of motion in dimensionless variables has a form: mU¨ + 2nU˙ + (c1 + p cos ωt) U + c3 U 3 = 0 .

(15.31)

Again we introduce the change of independent and dependent variables, U0 1 1 1 u = , E0 = mU˙ 02 + c1 U02 + c3 U04 , 2 2 4 E0 c1 ω du d2 u + 1 + 4ε cos + 2εγ τ u + 8εαu3 = 0, dτ ω0 dτ2

τ = ω0 t,

(15.32)

nω0 p c3 E 0 ; 4ε = , 8εα = . c1 c1 c1 We write now the equation for complex functions: ψ(τ), ψ ∗ (τ):

where: εγ =

i ωτ ∂ψ −i ωτ − iψ + εγ (ψ + ψ∗ ) + iαε (ψ − ψ∗ ) − iε e ω0 + e ω0 × (ψ − ψ∗ ) = 0 . (15.33) ∂τ Let lω − mω0 = β1 εω0 where l and m – integer mutually prime numbers, ε 1, β1 = O(1). As well as earlier we introduce the change of a dependent variable ψ = eiτ ϕ(τ). Then


445

dϕ + εγ1 ϕ + e−2iτ ϕ∗ + iαε e2iτ ϕ3 − 3ϕ2 ϕ + 3e−2iτ ϕ2 ϕ∗ − e−4iτ ϕ∗3 − dτ m m iε ei(−1+ l )τ e−iβετ + e−i(1+ l )τ eiβετ · ϕeiτ − e−iτ ϕ∗ = 0; Again we introduce a “slow” time τ1 = τε 1 + ξ1 (τ1 )ε + ξ2 (τ1 )ε2 + . . . .

β=

β1 . l

(15.34)

(15.35)

Also we present a solution as ϕ(τ, τ1 ) = ϕ0 + εϕ1 + ε2 ϕ2 + . . . .

(15.36)

With allowance for (15.34), (15.35) and (15.36) we obtain ' ( ∂ ∂ ϕ0 + εϕ1 + ε2 ϕ2 + . . . +εM ϕ1 , ϕ∗1 , ε − ϕ0 + εϕ1 + ε2 ϕ2 + . . . +εL(τ1 , ε)· ∂τ ∂τ1 εr(τ, τ1 , ε)× eiτ ϕ0 + εϕ1 + ε2 ϕ2 + . . . − eiτ ϕ∗0 + εϕ∗1 + εϕ∗2 + . . . = 0 . (15.37) The main parametrical resonance will be realized at m/l = 2. As well as in the previous cases ϕ0 = ϕ0 (τ1 ), and the equation of a principal approximation in slow time has a form: dϕ0 + γϕ0 − 3iα|ϕ0 |2 ϕ0 + ieiβτ1 ϕ∗0 = 0 . (15.38) dτ1 By the change of variables ϕ 0 = Φ0 e(−γ+iβ)τ1 we come to the equation: β dΦ0 + i Φ0 + Φ∗0 − 3iαe−2γτ1 |Φ0 |2 Φ0 = 0 . (15.39) dτ1 2 Let’s assume Φ0 = a(τ1 )e−iδ(τ1 ) , where a(τ1 ) and δ(τ1 ) – real functions. Substituting this expression in the equation of motion (15.39) after simple transformations we obtain the system of real equations: da = −a sin 2δ , dτ1 β dδ = − − cos 2δ + 3αa2 e−2γτ1 . dτ1 2

(15.40)

At γ = 0 the equation (15.39) and equation conjugated to it form the Hamiltonian system: dΦ∗0 dΦ0 dH dH = ; =− ; δtau1 dΦ∗0 dτ1 dΦ0 where:

i −β|Φ0 |2 + 3α|Φ0 |4 Φ20 + Φ∗2 0 2 and, hence, admit first integral H = const. H=

446


Hence, variable a(τ 1 ) and δ(τ1 ) are connected by a relation: −βa2 + 3αa4 − 2a2 cos 2δ = const, representing the equation of a phase trajectory on a plane (a, δ). The stationary points on a phase plane are determined as follows: a2 = ±

δ = 0, ±1, ±2, . . . ;

β 1 + . 3α 6

It is obvious that for a 0 all trajectories on a phase plane are restricted. The conditions for determination of stationary regimes are 1. δ = 0; a2 =

β 1 + . 3α 6

2. δ = π;

β 1 + . 3α 6 β 1 The first type regime is stable (it exists when < ), but the second type is unstable 2 α β 1 (it appears when > ). 2 α The close equation is postulated (without a derivation) in [393], where its full analysis is performed. Obviously, it is applicable only in a neighbourhood of a main resonance. The asymptotic approach allows to obtain the equation for secondary parametrical resonance also. a2 = −

Self-excited oscillations We will show now how it is possible to obtain the complex equation in slow time for a self oscillating system described in a dimensionless form by the Van der Pol equation: du d2 u − 2ε(1 − 2bu 2 ) + i = 0, b > 0 . (15.41) dτ dτ Let’s introduce the complex functions ψ and ψ ∗ again. Then

1 ∂ψ ∗ ∗ 2 − iψ − ε(ψ + ψ ) 1 + b (ψ − ψ ) = 0 . (15.42) ∂τ 2 After a change of dependent variable ψ = e iτ ϕ(τ) we obtain 2 1 ∂ϕ − ε ϕ + e−2iτ ϕ∗ 1 + b eiτ ϕ − eiτ ϕ∗ = 0 . ∂τ 2

(15.43)

Introducing of “slow” time τ1 = ετ 1 + ξ1 (τ1 )ε + ξ2 (τ1 )ε2 + . . .

(15.44)

15.2 System of two weakly coupled nonlinear oscillators

447

and considering the function ϕ as function of two times we search for a solution as ϕ = ϕ0 + εϕ1 + ε2 ϕ2 + . . . .

(15.45)

After a substituting (15.44) and (15.45) in the equation (15.43) we obtain: ∂ ∂ ϕ0 + εϕ1 + ε2 ϕ2 + . . . − ϕ0 + εϕ1 + ε2 ϕ2 + . . . + εL (τ1 , ε) × ∂τ ∂τ1 ε ϕ0 + εϕ1 + ε2 ϕ2 + . . . + e−2iτ ϕ∗0 + εϕ∗1 + ε2 ϕ∗2 + . . . × ) 2 * 1 1 + b eiτ ϕ0 + εϕ1 + ε2 ϕ2 + . . . − e−iτ ϕ∗0 + εϕ∗1 + ε2 ϕ∗2 + . . . = 0. 2 ∂ϕ0 = 0 and ϕ0 = ϕ0 (τ1 ). A condition of equality to zero of As well as earlier ∂τ coefficient at the first power of a parameter ε leads to equation dϕ0 ∂ϕ1 + e−2iτ ϕ∗0 · b e2iτ ϕ30 + e−2iτϕ20 ϕ∗0 + e−4iτ ϕ∗3 − ϕ0 1 − bϕ20 = − . (15.46) 0 dτ1 ∂τ Integrating (15.44) on fast time in limits from 0 up to π we come to the equation of a principal approximation in slow time: dϕ0 − ϕ0 1 − b|ϕ0 |2 = 0 dτ1

(15.47)

with a particular solution ϕ 0 = b−1 e−θ0 (limit cycle), where θ 0 – is arbitrary phase. Returning to the variable ψ we have ψ0 = b−1 ei(τ−θ0 ) . The non-stationary solution (approaching to the limit cycle) is also obtained from (15.45). Let’s note that in the equation close to (15.47) is obtained as a result of transformation of some modified system which is distinct from the equation (15.41). It appears however that (15.45) is a main approximation to the Van der Pol equation in “slow” time.

15.2 System of two weakly coupled nonlinear oscillators Let us consider a system of two identical weakly coupled nonlinear oscillators with asymmetric cubic potentials. Their dynamics is described by the equations, m m

d2 U1 dU 1 + c1 U1 + c2 U12 + c3 U13 + c12 (U1 − U2 ) = 0, + 2n dt dt2

dU 2 d2 U2 + c1 U2 + c2 U22 + c3 U23 + c12 (U2 − U1 ) = 0, + 2n 2 dt dt

(15.48)

448


where U j ( j = 1, 2) are displacements of the oscillators. We rewrite these equations using the complex representation, 2 dψ j − iψ j + ε2 γ ψ j + ψ∗j − α1 ε ψ j − ψ∗j + dτ 3 iα2 ε2 ψ j − ψ∗j − iβε2 ψ j − ψ∗j − ψk − ψ∗k = 0,

(15.49)

j = 1, 2, k = 3 − j,

√ where: ψ j = ν j + = ν j − iu j ( j = 1, 2); ν j = du j /dτ; i = −1; τ = ω0 t; ui = Ui /r0 ; r0 is the distance between particles in the undisturbed state; and iu j , ψ∗j

/ c3 r02 n c2 r0 c12 ; 4α1 ε = ; 8α2 ε2 = ; 2ε2 β = ; ε 1; ω0 = c1 /m. ε2 γ = √ c1 m c1 c1 c1 Introducing the new variables φ = ψ j e−iτ and “slow” times τ1 = ετ0 ;

τ2 = ε2 τ0 ...,

where τ0 = τ,

(15.50)

we consider ϕ j as functions of many times τ 0 , τ1 , τ2 ,... . Let us present a solution in a form of power expansion as, ϕ j (τ0 , τ1 , τ2 , ...) = ϕ j,0 + εϕ j,1 + ε2 ϕ j,2 + ... .

(15.51)

The substitution of the above expressions into equation (15.49) leads to the equation ∂ ∂ (φ j,0 + εφ j,1 + ε2 φ j,2 + ...) + ε (ϕ j,0 + εϕ j,1 + ε2 ϕ j,2 + ...)+ ∂τ0 ∂τ1 ∂ ε2 (ϕ j,0 + εϕ j,1 + ε2 ϕ j,2 + . . .) + . . . + εM ϕ j,0 , ϕ∗j,0 , ϕ j,1 , ϕ∗j,1 , ..., ε − ∂τ2 iε2 β ϕ j,0 − ϕk,0 − e−2 iτ ϕ∗j,0 − ϕ∗k,0 + . . . = 0 (15.52) where M φ j,0 , φ∗j,0, φ j,1 , φ∗j,1 , . . . , ε = γε (φ j,0 + εφ j,1 + . . .) + e−2iτ (φ∗j,0 + εφ∗j,1 + . . .) − 2 α1 eiτ0 (φ j,0 + εφ j,1 + . . .)2 − 2e−iτ0 φ∗j,0 + εφ∗j,1 + . . . + e−3iτ (φ∗j,0 + εφ∗j,1 + . . .)2 + 2 iα2 ε [ e2iτ (φ j,0 + εφ j,1 + . . .)3 − 3 φ j,0 + εφ j,1 + . . . (φ j,0 + εφ j,1 + . . .)+ 2 3e−2iτ φ j,0 + εφ j,1 + . . . (φ∗j,0 + εφ∗j,1 + . . .) − e−4iτ (φ∗j,0 + εφ∗j,1 + . . .)3 ] , (15.53) j, k = 1, 2,

k = 3 − j.

Now we equate to zero the coefficients at each growing power of the parameter ε:


449

∂ϕ j,0 = 0. Therefore, ϕ j,0 = ϕ j,0 (τ1 , τ2 , ...), i.e. in the principal approxima∂τ0 tion the functions ϕ j,0 depend on “slow” times only; (2) ε0 : 2 ∂φ j,0 ∂φ j,1 =− + α1 eiτ0 φ2j,0 − 2e−iτ0 φ j,0 + e−3 iτ0 φ∗j,0 2 . ∂τ0 ∂τ1

(1) ε0 :

The condition of the absence of secular terms leads to equation ∂ϕ j,0 = 0, ∂τ1

so ϕ j,0 = ϕ j,0 (τ2 , τ3 , ...).

(15.54)

Integrating equations for ϕ j,1 with respect to variable τ 0 and taking into account equation (15.54), we obtain 2 1 φ j,1 = −α1 i eiτ0 φ2j,0 + 2e−iτ0 φ j,0 − e−3 iτ0 φ∗j,0 2 . (15.55) 3 Now we consider the terms of the second order: (3) ε2 : ∂φ j,0 ∂φ j,1 ∂φ j,2 =− − + γ φ j,0 + e−2iτ0 φ∗j,0 + ∂τ0 ∂τ2 ∂τ1 2α1 eiτ0 φ j,0 φ j,1 − e−iτ0 φ j,0 φ∗j,1 + φ j,1 φ∗j,0 + e−3iτ0 φ∗j,0 φ∗j,1 − 2 2 iα2 e2iτ0 φ3j,0 − 3 φ j,0 φ j,0 + 3e−2iτ φ j,0 φ∗j,0 − e−4iτ0 φ∗j,0 3 + (15.56) iβ φ j,0 − φk,0 − e−2iτ φ∗j,0 − φ∗k,0 = 0, form which we obtain the following conditions for the absence of secular terms: 2 ∂ϕ j,0 + γϕ j,0 − αi ϕ j,0 ϕ j,0 − iβ ϕ j,0 − ϕk,0 = 0, ∂τ2

(15.57)

2 where j, k = 1, 2, k = 3 − j, α = 3α 2 − 20 3 α1 , which is the system of equations of motion in the principal approximation.

The system which is close to 15.57 (at γ =)) was considered in [393] as an example of a phenomenological model (‘a discrete model with self-localization’). It was shown that this system is integrable. Besides, at some energy of oscillations the localized normal modes arise in addition to in-phase and out-of-phase normal modes. Their appearance is a consequence of instability of in-phase or out-of phase normal modes (dependent on the sign of parameter α). Let us discuss the influence of damping forces in more details. Introducing in (15.57) the new variables, φ j,0 = e(iβ−γ)τ2 Φ j,0 , we obtain

j = 1, 2,

450


2 ∂Φ1,0 + iβΦ2,0 − iαe−2γτ2 Φ1,0 Φ1,0 = 0, ∂τ2 2 ∂Φ2,0 + iβΦ1,0 − iαe−2γτ2 Φ2,0 Φ2,0 = 0. (15.58) ∂τ2 We note that the Lagrangian corresponding to the equations of motion (15.56) at γ = 0 has the form,  2  2 1 ∂Φ∗j,0 ∂Φ j,0  1  ∗ Φ j,0  − β Φ1,0 Φ∗2,0 + Φ2,0 Φ∗1,0 + α L= i − Φ j,0 Φ4 2 j=1 ∂τ2 ∂τ2 2 j=1 j,0 and the integral of energy is written as 4 ' ( 1 Φ j,0 = H(τ3 , τ4 , . . .). β Φ1 Φ∗2 + Φ2 Φ∗1 − α 2 j=1 2

(15.59)

Exact integrability of a system (15.58) at γ = 0 follows from the existence of energy integral (15.55) and integral 2 Φ 2 = N(τ , τ , . . .). 3 4 j,0

(15.60)

j=1

Returning to the full system (15.58) (γ 0) we conclude that the relation (15.60) remains valid in this case also. Indeed, multiplying first and second equations (15.58) respectively by Φ ∗1,0 and Φ∗2,0 , making the operation of conjugation and combining all four equations we obtain ∂N = 0, ∂τ2

or

N = (τ3 , τ4 , ...).

(15.61)

Therefore, as well as in the case of zero damping [393], the unknown functions Φ1,0 and Φ2,0 can be presented as Φ1,0 =

√ θ N cos eiδ1 (τ2 ) ; 2

Φ2,0 =

√

θ N sin eiδ2 (τ2 ) . 2

Substituting these expressions in (15.53) we obtain 1 θ ∂θ θ ∂δ1 θ − itg − − βtg ei(δ2 −δ1 ) + αNe−2γτ2 cos2 = 0, 2 2 ∂τ2 ∂τ2 2 2 1 θ ∂θ θ ∂δ2 θ ictg (15.62) − − βctg ei(δ1 −δ2 ) + αNe−2γτ2 sin2 = 0. 2 2 ∂τ2 ∂τ2 2 2 Equating the real and imaginary parts of relations (15.62) to zero we come to the equations for real functions θ(τ) and ∆δ(τ) = δ 1 (τ) − δ2 (τ) ∂θ = 2β sin ∆, ∂τ2


∂∆ = 2βctgθ cos ∆ + αNe−2γτ2 cos θ. ∂τ2

451

(15.63)

The stationary points in which right sides of the equations (15.63) are equal to zero are π (1) ∆ = 0, θ = ; 2 π (2) ∆ = π, θ = . 2 They correspond to the in-phase and out-of-phase cooperative normal modes which also survive in the presence of energy dissipation. In the case the additional out-of-phase (α > 0) or in-phase (α > 0) normal modes appear [393]. They correspond to values ∆ = π (α > 0); ∆ = 0 (α > 0), N0 2β 2β = , where N0 = , αN N α and describe the nonlinear oscillations which are localized more and more (with increasing N) in the vicinity of a single particle. Indeed, the ratio of complex functions Φ1,0 and Φ2,0 corresponding to additional modes is written as + / 1 + 1 − ρ2 Φ1,0 N0 =−+ , where ρ = . / Φ2,0 N 1 − 1 − ρ2 sin θ =

Obviously such modes arise only at ρ < 1, and at ρ → 0 (that, naturally, falls outside the limits of applicability of the principal approximation) the full localization of excitation upon a single particle would be reached. The consideration of a dissipation reveals once more important aspect of the problem. In this case at N·e −2γτ2 ≥ N0 the energy of the system can also be originally localized on a particle. But because of presence of an exponential coefficient in the second equation of motion, the lo1 calization of excitation becomes impossible in the instant τ 2 = 2γ ln NN0 and the energy should be redistributed between the both particles. This important effect can be observed in computer experiment (in some instant the domination of one mass is replaced by a redistribution of the energy between masses). Let us note that the determination of higher approximations can also be realized without difficulty, if the principal approximations for localized normal modes are known. For example, the functions φ j,2 can be determined from the equations of first approximation after substitution of the functions ϕ j,0 , ϕ j,1 . Direct computation of localized normal modes becomes difficult if the number of degrees of freedom is more than two. Therefore generalization for more complicated cases requires a more universal approach. The existence of localized modes can be associated with bifurcations and instability of cooperative modes. For the system treated the bifurcation analysis becomes more simple with the use of the equations (15.62), i.e. in terms of real functions θ(τ 2 ), δ(τ2 ) [393]. However, the appropriate extension on the systems with many degrees of freedom becomes also

452


impossible. In this connection let us perform a bifurcation analysis using the complex equations (15.53) at γ = 0. 2 ∂Φ1,0 + iβΦ2,0 − iα Φ1,0 Φ1,0 = 0, ∂τ2 2 ∂Φ2,0 + iβΦ1,0 − iα Φ2,0 Φ2,0 = 0. ∂τ1

(15.64)

Let us focus on the analysis of the stability of out-of phase collective mode Φ2,0 = −Φ1,0 . We consider the perturbations of this mode w 1 and w2 , Φ1,0 + w1

and

− Φ1,0 + w2 .

(15.65)

Substituting expressions (15.65) in the equations of motion and taking into account relations (15.64), we obtain 2 ∂w1 + iβw2 − iα Φ21,0 w∗1 + 2 Φ1,0 w1 + 2Φ1,0 |w1 |2 + w21 Φ∗1,0 + |w1 |2 w1 = 0, ∂τ2 2 ∂w2 + iβw1 − iα Φ21,0 w∗2 + 2 Φ1,0 w2 − 2Φ1,0 |w2 |2 − w22 Φ∗1,0 + |w2 |2 w2 = 0, ∂τ2 (15.66) where N i(β+α N )τ2 2 Φ1,0 = e . 2 By combining these equations and by denoting through W 1 and W2 the sum and the difference of w 1 and w2 accordingly, we get the system of two nonlinear equations respect to new variables. Linearization of this system leads to two independent equations of parametrical oscillations for functions W 1 and W2 , describing perturbation of an initial mode with respect to in-phase and out-of-phase modes accordingly, 0N 1 N ∂W1 e2i(β+α 2 )τ2 W1∗ = 0, + i (β − αN) W1 − iα ∂τ2 2 0 N 2i(β+α N )τ2 1 ∗ ∂W2 2 e − i (β + αN) W2 − iα W2 = 0. ∂τ2 2

(15.67)

At the value N ≥ 2β α , the conditions of main parametrical resonance are satisfied and the periodic solution W 1 = Aeiωτ2 exists. Here A is a real constant and ω = β + αN. This solution corresponds to the boundary between the stability and instability regions for the first equation (15.67). On this boundary and for N ≥ 2β α the intensive transfer of energy to in-phase mode becomes possible and localized normal modes exist. In the second equation (15.67), as it is easy to be convinced, the instability

15.3 Nonlinear dynamics of an infinite chain of coupled oscillators

453

occurs at infinitesimal amplitudes, but it is realized as a phase shift of out-of-phase oscillations. The exact solution of the equations (15.64) gives the same value of critical amplitude. But as it is shown below, the applied approach allows us to find the conditions of the energy localization not only in finite-dimensional systems of the high dimensionality, but also in infinite-dimensional models. We consider subsequently the weightless string and beam with concentrated masses on discrete nonlinear supports.

15.3 Nonlinear dynamics of an infinite chain of coupled oscillators The equations of motion for an infinite weightless string with equal concentrated masses on uniformly situated nonlinear elastic supports have the form m

dU j d2 U j + c1 U j + c2 U 2j + c3 U 3j + c˜ (2U j − U j−1 − U j+1 ) = 0, + 2˜γ 2 dt dt −∞ < j < ∞.

(15.68)

Here m is magnitude of discrete mass, c is tensile force, c 1 , c2 , c3 are characteristics of nonlinear supports, j is damping coefficient. We introduce a change of variables Uj c1 t; uj = , (15.69) τ= m r0 where r0 distance between elastic supports (and concentrated masses). Then we have d2u j 2 du j 2 2 3 2 + u + 2ε γ + 4α εu + 8α ε u + 2ε β 2u − u − u = 0, (15.70) j 1 2 j j−1 j+1 j j dτ dτ2 where 4α1 ε =

c2 c3 2 · r0 , 8αε2 = ·r , c1 c1 0

7 γε2 = γ˜ √c1 m, 2 ε2 β = c˜/c1 ,

ε 1.

Let us now use the complex representation of the equations of motion, 2 3 dψ j − iψ j + ε2 γ ψ j + ψ∗j − iα1 ε ψ j − ψ∗j + iα2 ε2 ψ j − ψ∗j − dτ ε2 iβi 2 ψ j − ψ∗j − ψ j−1 − ψ∗j−1 − ψ j+1 − ψ∗j+1 = 0, where ψ j = ν j + iu j ; ψ∗j = ν j − iu j . For new variables

ψ j = eiτ ϕ j ,

454


we come to a set of equations 2 dφ j + ε2 γ(φ j + e−2 iτ φ∗j ) − iα1 ε eiτ0 φ2j − 2e−iτ0 φ j + e−3 iτ0 φ∗j 2 + dτ 2 2 iα ε2 (e2iτ φ3 − 3 φ φ + 3e−2iτ φ φ∗ − e−4 iτ φ∗ 3 )− 2

j

j

j

j

j

j

iε2 β 2φ j − φ j−1 − φ j+1 − e−2 iτ 2φ∗j − φ∗j−1 − φ∗j+1 = 0.

(15.71)

We introduce now the ‘slow’ times τ1 = ετ0 , τ2 = ε2 τ0 , τ3 = ε3 τ0 , ..., where τ0 = τ.

(15.72)

and consider ϕ j as the functions of the “fast” time τ 0 and the ‘slow’ times τ1 , τ2 , .... By presenting the unknown variables as expansions on the parameter ε, ϕ j = ϕ j,0 + εϕ j,1 + ε2 ϕ j,2 + ...

(15.73)

and substituting (15.73) in the equations of motion we obtain ∂ ∂ (φ j,0 + εφ j,1 + ε2 φ j,2 + ...) + ε (φ j,0 + εφ j,1 + ε2 φ j,2 + ..)+ ∂τ0 ∂τ1 ∂ ε2 (φ j,0 + εφ j,1 + ε2 φ j,2 + ..) + +εM ϕ j,0 , ϕ∗j,0 , ϕ j,1 , ϕ∗j,l ..., ε − ∂τ2 6 2 iε β 2 ϕ j,0 + εϕ j,1 + ... − ϕ j−1,0 + εϕ j−1,1 + ... − φ j+1,0 + εφ j+1,1 + ... − . 2 φ∗j,0 + εφ∗j,1 + ... − φ∗j−1,0 + iφ∗j−1,1 + ... − φ∗j+1,0 + εφ∗j+1,1 + ... e−2 iτ0 = 0, (15.74) where the expression for M is presented above. Equating to zero the coefficients at various degrees of the small parameter ε we have: (1) ε0 : (2) ε1 :

∂φ j,0 ∂τ0

= 0, therefore ϕ j,0 = ϕ j,0 (τ1 , τ2 , ...), 2 ∂φ j,0 ∂φ j,1 =− + α1 eiτ0 φ2j,0 − 2e−iτ0 φ j,0 + e−3 iτ0 φ∗j,0 2 . ∂τ0 ∂τ1

(15.75)

The condition of the absence of secular terms leads to the equation ∂ϕ j,0 = 0, ∂τ1

or ϕ j,0 = ϕ j,0 (τ2 , τ3 , ...).

(15.76)

Taking into account (15.76) and integrating (15.75) with respect to variable τ 0 we obtain 2 1 φ j,1 = −iα1 eiτ0 φ2j,0 + 2e−iτ0 φ j,0 − e−3 iτ0 φ∗j,0 2 . (15.77) 3 Let us select now the terms corresponding to second order of the small parameter:


(3) ε2 :

455

∂ϕ j,2 ∂ϕ j,0 ∂ϕ j,1 =− − + ∂τ0 ∂τ2 ∂τ1 γ ϕ j,0 + εϕ j,1 + ε2 ϕ∗j,2 + .. + e−2iτ0 (ϕ∗j,0 + εϕ∗j,1 + ε2 ϕ∗j,2 + ..) + 2α1 i eiτ0 ϕ j,0 ϕ j,1 − eiτ0 ϕ j,0 ϕ∗j,1 + ϕ j,1 ϕ∗j,0 + e−3iτ0 ϕ∗j,0 ϕ∗j,1 − 2 2 iα2 e2iτ0 φ3j,0 − 3 φ j,0 φ j,0 + 3e−2iτ φ j,0 φ∗j,0 − e−4iτ0 φ∗j,0 3 + (15.78) iβ 2φ j,0 − φ j+1,0 − φ j−1,0 − e−2iτ0 2φ∗j,0 − φ∗j+1,0 − φ∗j+1,0 .

The condition of the absence of secular terms lead to the system of equations of motion in principal approximation 2 ∂φ j,0 + γφ j,0 − iα φ j,0 φ j,0 − iβ 2φ j,0 − φ j+1,0 − φ j−1,0 = 0, ∂τ2 −∞ < j < ∞.

(15.79)

We introduce now the new change of the variable ϕ j,0 = e−γτ2 Φ j,0 .

(15.80)

Then the system of equations in the principal approximation is presented as 2 ∂Φ j,0 − αie−2γτ2 Φ j,o Φ j,0 − iβ 2Φ j,0 − Φ j+1,0 − Φ j−1,0 = 0. ∂τ2

(15.81)

Let us seek the functions Φ j,0 in the form Φ j,0 = f0 (τ2 )ei(kr0 j+θ0 ) ,

(15.82)

where r0 the distance between particles, θ 0 the phase corresponding to j = 0. Substituting (15.82) in (15.81), we obtain kr0 ∂ f0 · f0 = 0. − iαe−2γτ2 | f0 |2 f0 − 4iβ sin2 ∂τ2 2

(15.83)

After the change of variable 2 kr0 2

f0 = e4iβ sin

τ2

w0 (τ2 ),

we come to the equation ∂w0 − iαe−2γτ2 |w0 |2 w0 = 0, ∂τ2 which coincides with the equation for a system with one degree of freedom. Its solution has the form

456


w0 = so that Φ j,0 =

√

√

iα N exp − N(e−2γτ2 − 1) , 2γ

kr0 N −2γτ2 τ2 − α N exp i 4β sin2 − 1 + kr0 j + θ0 , e 2 2γ

(15.84)

√ where N is the modulus of the complex amplitude. Returning to the initial complex variables we obtain

√ N −2γε2 τ kr0 2 − 1 + kr0 j + θ0 . ψ j,0 = Ne−γε τ exp i τ + 4ε2 βτ sin −αε2 e 2 2γ (15.85) Thus, the analysis of the principal approximation in slow time leads to the conclusion that in an infinite string with concentrated masses on nonlinear elastic supports the quasi-harmonic waves with exponentially decreasing amplitude and variable frequency can spread. At γ = 0 and α = 0 they turn out to be usual harmonic waves √ (15.86) ψ j,0 = Nei(ωτ+kr0 j+θ0 ) , where ω = 1 + 4ε2 β sin2 kr20 , with a spectrum of wave numbers 0 ≤ kr 0 ≤ π, spectrum of frequencies 1 ≤ ω ≤ 1 + 4ε 2 β and a constant amplitude. The spectrum of simple harmonic waves determined by expression (15.86) is characterized by double degeneration of all modes, except of corresponding to wave numbers and k = 0 and k = π/r0 . It means that the modes with different phases θ 0 having identical frequencies and wave numbers can be obtained by a superposition of two such modes. Introduction of nonlinearity removes the degeneration [393]. Actually the dependence of normal oscillations frequencies on distribution of the energy between the particles arises. And, generally speaking, such a distribution depends on the value θ0 . Let us note that the obtained solution turns out to be asymptotically valid if the lengths of considered waves are not too large as compared with intersupport distances. Further we accept r 0 as a unit of length, then this condition will be noted as k ∼ 1. In the case k 1 there is one more small parameter in the problem. Its appearance requires a revision of the overall procedure of asymptotic expansion. In the case of strong coupling c˜ /c 1 = O(1) the complex representation becomes justified (i.e. the rotations in opposite directions are separated in the principal approximation) if the coupling between oscillators is “effectively weak” because of relative smallness of the second differences in the equations of motion. It means that the field of applicability of complex representation in that case coincides with the sphere of applicability of the continuum approximation. The equations of motion (15.52) take in this case the form du j d2 u j + u j + 4εu2j + 8α2 ε2 u3j + 2β 2u j − u j−1 − u j+1 = 0, + 2ε2 γ 2 dτ dτ

(15.87)


457

where β = ¯c/c1 = O(1). Now it is expedient to measure the distances between particles in the units of ε −1 r0 , so we have for the second differences in (15.87), after introduction of the continuum approximation, the expansion ∂ 2 u ε4 ∂ 4 u − 2u j − u j−1 − u j+1 = ε2 2 + −··· , 12 ∂ζ 4 ∂ζ where ζ is corresponding space coordinate. Thus, in spite of the fact that β ≈ O(1), the derivatives terms contain now a small parameter and the complex representation turns out to be justified when using the continuum approximation. The similar situation arises in the case of short waves. As a result the equations in the principal approximation take the form (if γ 0) (1) k 1: i

∂2 Φ0 ∂Φ0 − β 2 + αe−2γτ2 |Φ0 |2 Φ0 = 0. ∂τ2 ∂ζ

(15.88)

i

2 ˜0 ˜0 ∂Φ ∂2 Φ ˜ 0 Φ ˜0 = 0 + β 2 + αe−2γτ2 Φ ∂τ2 ∂ζ

(15.89)

(2) π − l 1:

Relations (15.87) and (15.88) at γ = 0 become the nonlinear Schrodinger equation (NSE), which is the integrable system [691]. Its solution in a wide class of the initial conditions can be obtained by the inverse scattering method. It is known [691] that, if α > 0, at the relation of coefficients signs which has place in the case of the equation (15.89), all periodic wave packets are unstable. As a result, the localized soliton-like waves (‘envelope solitons’ or ‘bright solitons ’ ) exist:  1/

  2 1 ˜ (ζ, τ2 ) =  2S  ei(kζ−ωτ2 ) sec h S /2 (ζ − ντ2 ) , Φ α/ β where k=

ν 2β

;ω =

(15.90)

ν2 − S. 4β2

Here amplitude and velocity of the soliton are independent parameters. At α < 0 (“soft” nonlinearity) the periodic wave packets described by the equation (15.89) are stable and bright solitons do not exist. At signs of coefficients in the NSE corresponding to the equation (15.88), on the contrary, the wave packets are unstable that leads to the formation of solitons. If α > 0 the wave packets are stable and solitonic solutions are absent. Thus if the coupling between oscillators is not weak, the conditions of localized excitations formation are well known and are formulated in terms of the continuum approximation. How is the matter in the case of the weak coupling, when it is necessary to use a discrete description?

458


As well as in the system with two degrees of freedom, in the infinite chain of nonlinear oscillators apart from cooperative modes the waves with some space localization can be realized. They result from the instability of the cooperative modes. Considering small perturbations of cooperative nonlinear normal modes, we substitute in the equations (15.81) instead of functions Φ j,0 the expressions Φ j,0 + w j,0 . Then, taking into account that the functions Φ j,0 satisfy the equations (15.81) and |w j,0 | |Φ j,0 |, supposing γ = 0, we obtain 2 ∂w j,0 − iβ 2w j,0 − w j+1,0 − w j−1,0 − iα Φ2j,0 w∗j,0 + 2 Φ j,0 w1,0 = 0, ∂τ2

(15.91)

where the cooperative mode is determined by expression (15.84). Let us assume now that instead of the infinite string we consider a finite system with number of masses N˜ = 2n and the conditions of periodicity are satisfied: Φ0,0 = ΦN,0 ˜ ,

Φ−1,0 = ΦN−1,0 , ..., ˜

ω0,0 = ωN,0 ˜ ,

ω−1,0 = ωN−1,0 , ..., ˜

(15.92)

j = 0, 1, 2, ..., 2n − 1. Then the wave number can have the magnitudes l = 2π m where m = 0, 1, 2, ..., n. N˜ Let us consider solutions of the system (15.91) satisfying to the relations w j−1,0 = e−ik1 w j,0 ,

w j+1,0 = eik1 w j,0 , k1 =

2π m1 , N˜

˜ m1 = 0, 1, 2, ..., N/2 .

(15.93)

The system (15.91) after substitution (15.93) becomes uncoupled: ∂w j,0 2 k1 2ik j 2 k = 2i 2β sin + αN w j,0 + iαNe exp 2i αN + 4β sin τ2 w∗j,0 . ∂τ2 2 2 (15.94) Thus, stability analysis of the cooperative mode characterized by a wave number m, with respect to the mode with wave number m 1 is reduced to the equation of parametrical oscillations considered above. The criterion of instability of this cooperative mode (i.e. of reaching the boundary between regions of stability and instability) is determined by the condition 4β sin2

k k1 + αN = 3αN + 4β sin2 . 2 2

(15.95)

From here for the magnitude of squared complex amplitude corresponding to instability of the collective mode with a wave number m, with respect to the mode with a wave number m 1 , we obtain the expression k β 2 k 2 1 − sin N = 2 sin , (15.96) α 2 2


459

where

2π 2π m, k1 = m1 . ˜ N N˜ As follows from (15.96), at α > 0 each mode corresponding to a certain value m with increasing intensity of excitations becomes unstable sequentially with respect to the modes with m 1 = m − 1, m − 2, ..., smaller than the wave number of the considered nonlinear normal mode. The threshold excitation energy increases with decreasing m1 at fixed m. It is necessary to focus especially on the case m 1 = m, when the critical value of excitation energy is equal to zero. In this case instability at no matter how small amplitudes is nothing but the Lyapunov instability, and actually leads only to a phase shift of an analyzable mode (similar situation arose in the system with two degrees of freedom - section II). The most important is the first nontrivial instability with respect to the mode with m 1 = m − 1, corresponding to the minimal (if to eliminate the case m 1 = m) excitation energy. The matter is that at superposition of modes with close wave numbers m and m 1 (space beatings!) there is a tendency to localization of excitation, which becomes more and more noticeable with an increase of the energy. With a decrease of m 1 , when the critical energy increases, the localization becomes weaker. Let us consider, ˜ for example, stability of the out-of-phase mode of minimum length (m = N/2). Then at the excitation energy corresponding to squared modulus of complex amplitude, ˜   2π N2 − 1  β   N = 2 1 − sin2  N˜ α k=

2

the instability is realized. This instability leads to the localization of oscillations practically on one mass, just as it happens in a system with two degrees of freedom. It is clear that the formed mode is multiply degenerated, as the localization of excitation can be realized on any from the particles. The comparison with models consisting of three and four particles, allows to conclude that all other critical values of energy correspond to formation of unstable modes. Now, returning to the initial model, we will consider the limiting case N → ∞. With increasing N the arguments of trigonometric functions for the nearest m and m 1 in (15.96) differ less and less, so that in the limit the energetic barrier, corresponding to the instability, tends to zero. But it means that all cooperative modes in an infinite chain are unstable (except for a homogeneous mode for which there are no values m1 , satisfying to the condition m < m 1 ) and spatially localized oscillations turn out to be the unique elementary excitations. If anharmonicity is negative (α < 0), the situation is quite similar. In this case all cooperative modes, except the out-of-phase mode with the minimal wavelength, are unstable in an infinite limit with respect to the modes with wave numbers m 1 > m. As a result of this instability the localized modes arise again. The complex representation of dynamics of coupled nonlinear oscillators turns out to be efficient in the asymptotic analysis of the systems with weak as well as strong coupling. In the first case the equations of the principal approximation re-

460


main discrete and in the second case the asymptotic approach leads to continual description in terms of a nonlinear Schrödinger equation and its generalizations.

15.4 Nonlinear dynamics of an infinite chain of coupled particles Let’s consider now an infinite chain of atoms, the interaction between which being described by a gradient-type symmetric potential U=

∞ ∞ 2 1 4 1 c1 U j − U j−1 + c3 U j − U j−1 . 2 j=−∞ 4 j=−∞

(15.97)

Let’s begin with a brief discussion of the case of long waves when the complex representation in not adequate. The system of equation of motion after transformation to dimensionless variables accepts to the form: 0 9 2 1: d2 u j + 2u − u − u − u + u − u − u 1 + εα u u = 0, j j−1 j+1 j+1 j−1 j j+1 j j−1 dτ2 (15.98) where: c3 r02 Uj c1 t, u j = , εα = ; ε1 . τ= m r0 c1 Let’s measure a distance between atoms in terms ε −1/2 · r0 and introduce the appropriate space coordinate ζ. Then, representing the difference expressions in (15.98) by their Taylor expansions, we obtain one equation with partial derivatives instead of infinite system of the ordinary differential equations:   2  ε2 ∂4 u ∂2 u ∂2 u  ∂u − ε 2 1 + 3αε + . . . − + . . . = 0, (15.99) 2 ∂ζ 12 ∂ζ 4 ∂τ ∂ζ where: u = u(ζ, τ), and “. . . ” corresponds to terms of the higher order of smallness on the parameters ε. Here, unlike the case of long waves in the chain of coupled oscillators, there are no reasons for using of complex representation, as the equation (15.98) contains only gradient terms. Natural way of the use of the smallness of the parameter ε consists in the introduction of new space and time variables: η = ζ − ε1/2 · τ;

τ1 = ε3/2 · τ .

(15.100)

After substitution (15.100) in (15.99) we have 2 ∂2 u 3 ∂2 u ∂u 1 ∂4 u + α 2 + + ... = 0. ∂τ1 ∂η 2 ∂η ∂η 24 ∂η4 The new space coordinate η is counted from the front of linear wave, and the new time τ1 is slow in comparison with τ.

15.4 Nonlinear dynamics of an infinite chain of coupled particles

461

∂u = w and limiting by the principal approximation, we ∂η come to the modified Korteveg-de Vries (mKdV) equation Introducing a notion

∂w 3 2 ∂w 1 ∂3 u + + αw = 0. ∂τ1 2 ∂η 24 ∂η3

(15.101)

In this equation all terms have the same order on the parameter ε and it ensures the most simple exposition of nonlinear dynamics for a chain of couple particles in the case of symmetric anharmonicity. As it is known, mKdV equation has localized solutions (solitons, breathers and multisoliton waves). In the case of short waves the situation drastically changes: the complex representation becomes adequate again. The equations of motion (15.98) have exact solution in the form of nonlinear standing wave with the minimal wavelength: u j = (−1) j w(τ);

−∞ < j < ∞ .

Meaning the analysis of short-wave modes we use, following to [393], the change of variables u j = (−1) jw j . Then the equations of motion are written as follows: 0 9 2 1: d2 w j + 2w +w +w −w + w +w +w 1+εα w w = 0. j j−1 j+1 j+1 j−1 j j+1 j j−1 dτ2 Let’s introduce the function of two variables w(ζ, τ) describing in a continual limit the nonlinear dynamics of the chain of coupled particles in the supposition that the modulations of a nonlinear mode with a minimal wavelength have a characteristic space scale essentially exceeding the distance between atoms, which in selected units is equal ε. Thus w j±1 ( j, τ) = w j (ζ, τ) ± ε

∂w 1 2 ∂2 w ±... + ε ∂ζ 2 ∂ζ 2

(15.102)

where: jε = ζ. With (15.102) we obtain the following continual equation of motion: ∂2 w ∂2 w 3 + 4(w + 4αεw ) + ε + . . . = 0, ∂τ2 ∂ζ 2 where “. . . ” corresponds to terms of higher powers of the parameter ε. Now, as distinct from the case of long waves, the equation of motion contains a non-gradient term. Therefore coupling between atoms along a chain described by gradient term becomes “effectively weak” and the use of complex representation is justified. The similar situation arose at intermediate coupling between oscillators in section III, when the complex representation was introduced in a continual model of a system of oscillators. Thus, from a mathematical point of view the problem of nonlinear

462


short-wave dynamic for a chain of coupled particles is equivalent to the dynamical problem for a system of coupled oscillators in the case of intermediate coupling. After a change of variables τ 1 = 2τ we use the complex functions ψ(ζ, τ1 ) =

∂w + iw; ∂τ1

ψ∗ (ζ, τ1 ) =

∂w − iw, ∂τ1

for which two conjugate equations are obtained. Let’s consider one of them: 2 ∂ ψ ∂2 ψ∗ 1 1 ∂ψ + αiε(ψ − ψ∗ )3 + . . . = 0 . − iψ − iε − ∂τ1 8 ∂ζ 2 ∂ζ 2 2 By introducing of new function ϕ, so that ψ = eiτ1 (ζ, τ1 ) and slow time

τ2 = ετ1 1 + ξ1 (τ2 )ε + ξ2 (τ2 )ε2 + . . .

we are seeking a solution in the form: ϕ = ϕ0 + εϕ1 + . . . , where: ϕ = ϕ(ζ, τ1 , τ2 ). We come in the principal approximation to NSE in slow time i

∂ϕ0 1 ∂2 ϕ0 3 α + + − ϕ0 = 0 . ∂τ1 8 ∂ζ 2 2 ϕ20

The sings of coefficients in this equation are the same as for a system of oscillators with intermediate coupling between them in short wave approximation. Therefore the same conditions for existence of localized excitations are preserved.

16 OTHER ASYMPTOTICAL APPROACHES

16.1 Matched asymptotic procedure We consider pendulum oscillations with small mass governed by the equation ε x¨ + x˙ + x = 0,

ε 0,

(16.1)

with the following initial conditions x(0) = 0,

x(0) ˙ = 1.

(16.2)

The limiting equation (ε = 0) x˙0 + x0 = 0

(16.3)

can not satisfy all initial conditions (16.2). Observe that during limiting approach ε → 0 we ‘have lost’ rapidly decaying state, localized close to the motion beginning in the neighbourhood of t = 0. It is clear, since we implicitly assumed that x˙ ∼ x, i.e. differentiation do not change the function order. However, this is not true in the vicinity of t = 0, where the differentiation sufficiently increases the function dx ∼ ε−1 x. dt The corresponding rapidly changed state has the form εx p + x˙ p = 0.

(16.4)

Observe that now we have two equations, (16.3) and (16.4), governing the system behaviour in whole considered domain and the initial conditions (16.2). How to match them? There exist various approaches, but two of them are most popular, i.e. matched asymptotic expansions [237, 349, 380, 660, 663] and boundary layer method [645, 667, 672, 673, 674]. The first has found wide applications in hydrodynamics, whereas the second in mechanics of a deformable body. We briefly discuss advantages and disadvantages of both of them. We begin with the matched asymptotic expansions. A general solution to equations (16.3) and (16.4) has the form x0 = Ce−t ,

(16.5)

464

16 OTHER ASYMPTOTICAL APPROACHES −1

x p = C1 + C2 e−ε t .

(16.6)

The solution (16.5) is called external (fundamental) one, whereas the solution (16.6) is called internal one. The internal solution (16.6) can satisfy the given initial conditions, assuming that C2 = −ε, C1 = ε. Let us proceed with a matching procedure. It is known that a success of the matching depends on that if an overlapping domain, where both internal and external solutions are suitable, exists. An occurrence of the overlapping domain gives rights to assume that in this domain the external solution already has achieved its internal limit (i.e. for t → 0), whereas the internal solution has already achieved its external limit (i.e. for t → ∞). In other words we can simply formulate the following matching principle: internal limit of external series must be equal to external limit of internal series, i.e. lim x0 = lim x p . t→0

(16.7)

t→∞

Application of the introduced principle gives C = C 1 = εa for our case. The described simple rule of matching can be generalized describing more exactly behaviour of being matched quantities [660]. Instead of simple limits we take the asymptotical representation. It gives the following matching principle: internal representation of external representation is equal to external representation of internal representation. By external (or internal) representation we mean a first non-zero term of the asymptotic series in external (or internal) variables. The principle can be extended into high order approximation terms including more terms of asymptotic expansions. We can use the principle when the terms number is different in internal and external series. Therefore, the principle of asymptotic matching can be formulated in more general way: m term internal series of n terms external series is equal to n terms external series of m terms internal series. Usually we choose m = n or m = n + 1 [660]. Note that in practice two essentially different methods can be used. The first is referred as the additive composition. A sum of external and internal series can be improved by extracting their general part, which can be found as internal series of external series. A second method is called as the multiplicative composition. The result of external series is multiplied by certain improving multiplier being the ratio of internal over external series of this internal series (or a similar like approach can be applied to internal series). Both additive and multiplicative laws are coupled because a ratio of two quantities close to one can be extended into the series due to binom formula. Observe that although the additive principle is in general more simple in applications, but the multiplicative one sometimes give more simple results. In our case the additive matching gives −1

−1

x ≈ x0 + x p − lim x0 = εe−t − εeε + ε − ε = ε(e−t − ε−ε t ), t→0

(16.8)

16.2 Hilbert transform

465

whereas the multiplicative principle yields the following solution x=

x0 x p −1 = εe−t (1 − ε−ε t ). lim x0

(16.9)

t→0

It can be checked that both solutions (16.8) and (16.9) are equivalent. Matching of external and internal solutions definition of their domains and a construction of uniformly suitable solution applicable in a whole considered space belong to difficult problems of matched asymptotic expansions procedure. The difficulties arise when it is impossible to find a general analytical solution or when one deals with equations of high order, which is typical for theory of shells. On the other side, the matching procedure, contrary to the Vishik-Lyusternik approach, does not need to be regular. We have to note than Western scientists prefer to use matched asymptotic expansions, whereas scientists from the forme Soviet Union were more oriented on boundary layer method. In fact, a choice depends on the problem properties, as well as tradition also.

16.2 Hilbert transform In reference [657] it has been shown that sometimes an application of the Hilbert transformation is more effective than an averaging method while separating fast and slow components. We briefly describe the Hilbert approach. Let 1 u(t) = π

∞ [Vc (ω) cos ωt + V s (ω) sin ωt]dω, 0

and the function v(t) which is adjoint to u(t) in the Hilbert sense has the form 1 v(t) = H[u] = π

∞ [Vc (ω) sin ωt − V s (ω) cos ωt]dω. 0

One may also use the formula 1 v (t) = V.P. π

∞

−∞

u (s) ds, t−s

where V.P. denotes the main integral value in the Cauchy sense. The Hilbert transformation is linear,   n n    H  Ci ui  = Ci H[ui ], Ci ≡ const, i=1

i=1

466


and it commutes to other homogenous transformations including differentiation, du d H[u] = H . dt dt However, it should be emphasized that a ‘slow’ multiplier x, whose spectrum does not intersect a spectrum of ‘fast’ multiplier, can be take out of integral operator action, H[xu] = xH[u], i.e. the slow multipliers can be ‘frozen’. For example, H[cos 99t cos 100t] =

1 H[cos 199t + cos t] = 2

1 (sin 199t + sin t) = cos 99t sin 100t = cos 99tH[cos 100t]. 2 Using the adjoint functions u(t) and v(t) the following function (so-called analytical signal) can be constructed 1 w(t) = u(t) + iv(t) = π

∞

U(ω)eiωt dt,

(16.10)

0

where the complex spectral amplitude of the real input function has the form U(ω) = U0 (ω) − iU s (ω). Observe that a transition to analytical signal due to formula (16.10) is analogous to of transition of a harmonic a cos(ωt +φ) to the complex exponent a exp[i(ωt +φ)]. It is not difficult to check that a complex spectral amplitude of analytical signal W(ω) = 2U(ω) for ω > 0 and W(ω) = 0 for ω < 0. In addition, the condition ω < 0 is necessary and sufficient one in order for W(t) to be an analytical signal. This property is used during analysis of non-linear oscillations. Let us consider the Duffing equation u¨ + u = εu3 . For ε = 0 we get

u¨ 0 + u0 = 0.

In the next approximation we get u¨ 1 + u1 = εu30 .

(16.11)

Now, instead of using Fourier series approximation for the right hand side of (16.11), we use the analytical signals w (1) and w1 of the form u(1) = u1 =

1 (1) (w + w ¯(1) 1 ), 2 1 1 (w1 + w ¯1 ), 2

16.3 Normal forms in non-linear problems

467

where a bar denotes complex conjugated quantities. Because in equation (16.11) only analytical signal must appear, we include only terms with positive spectrum: w¨ 1 + w1 =

3 ε 2 (1) (3a w + w(1) ), 4

(16.12)

¯1 . where: a2 = w1 w The right hand side of (16.12) includes first (w (1) ) and third (w (1) )3 harmonics of a solution. The left hand side is sought in the form w 1 = w(1) +w(3) and a comparison of harmonics gives

3 2 (1) (1) (16.13) w¨ + 1 − εa w = 0, 4 ε 3 w¨ (3) + w(3) = w(1) . (16.14) 4 From (16.13) one obtains 3 ω2 = 1 − εa2 , 4 whereas from (16.14) we obtain the third harmonics of the form w(3) = −

ε (1)3 w . 32

Then the iterational procedure can be extended. The described method is similar to that of harmonic balance. It possesses a particular efficiency while constructing higher order approximation [657].

16.3 Normal forms in non-linear problems It seems that the fundamental drawback of perturbation method relies on same treatment of both small linear and nonlinear terms. This seriously complicates the calculations of higher order approximations. We have illustrated the method [564], where all linear conservative effects are considered separately from non-linear perturbations. Using a suitable change of variables one can consider normal waves which are uncoupled in linear approximation, but they are coupled only due to non-linearities. This leads to maximal simplification of linear part of the problem and gives a good starting point for efficient and unique further simplifications of equations using the approximate methods of theory of non-linear waves with both strong and weak dispersion. The idea of normal waves is similar to that of normal form oscillations in multibody discrete systems. It is well known that in linear systems one can introduce the normal coordinates, where oscillations can appear only along one of them. Consider the following system of equations

468


∂ ∂ ¯u ∂ ∂ +B ¯u = F¯ ¯u, , , ∂t ∂x ∂t ∂x

(16.15)

where: ¯u is N-dimensional vector of physical variables; B(∂/∂x) is linear operator ¯ u, ∂/∂t, ∂/∂x) denotes vector of non-linear and non-conservative terms. matrix; F(¯ The left hand side terms describe linear conservative part of the problem. Observe that a transition from (16.15) into equation of coupled normal waves is related to diagonalisation of the operator matrix B(∂/∂x) in order to obtain its eigenbasis using the following change of co-ordinates ¯u(x, t) =

N

¯rk (∂/∂x)U k (x, t),

(16.16)

k=1

where: U k (x, t) are new wave co-ordinates; ¯rk are the right hand side eigenvectors of the matrix B (i.e. B¯rk = λk ¯rk ), and λk (∂/∂x) are the corresponding eigenvalues. Substituting (16.16) into equation (16.15) and multiplying them by left eigenvectors l¯j (∂/∂x) we get the following equations of coupled normal waves dU k ¯ + λk (∂/∂x)U k = (l¯k ¯rk )−1 [l¯k F(∂/∂t, ∂/∂x)], k = 1, 2, . . . , N, dt

(16.17)

where the following orthogonalization conditions have been applied: l¯j ¯rk = 0 for j k. The values λk define the various branches of dispersive equation of the linear system (16.18) (iωk + λk )(iωk − λk ) = 0. Observe that a certain arbitrarity in the eigenvectors choice to can be used to simplify right hand side of equation (16.17). Observe also that equation (16.17) are equivalent to the input system (16.15). We have not assumed either small non-linearity or a weak dispersion. The smallness of the mentioned quantities can be used further in order to simplify the obtained equations. During analysis of wave processes in non-homogeneous and non-stationary systems during a transition to equations of coupled normal waves (CNW), the corresponding terms must be transferred to the right hand side of equation. As an examply, dynamics of a rod governed by the equation. 2 2 1 2 ∂ ∂w ∂2 v 2∂ v −c = c , ∂t2 ∂x2 2 ∂x ∂x  2  4  ∂v ∂w  ∂2 w 2 ∂  2 2∂ w  + − c r = 0, (16.19) + c     ∂x ∂x ∂x ∂t2 ∂x4 / where: v(x, t), w(x, t) are longitudinal and transvere displacements, c = E/ρ is √ velocity of a longitudinal wave in the rod material; r = I/E is the radius of inertia of the rod’s cross-section.

16.3 Normal forms in non-linear problems

The equation (16.19) is transformed to the following matrix form ∂ ¯ ¯ ∂ ∂ ¯u +B , ¯u , U=F ∂t ∂x ∂x

469

(16.20)

¯ where: ¯u(x, t) = (vt , v x , w, wt ) is the vector of physical variables, F(∂/∂x, ¯u) is the vector of non-linear quantities, B(∂/∂x) is the linear operator matrix 4 × 4. Further we will use operator formalism, taking ∂/∂t → p and ∂/∂x → q. Then  2  0 −c q 0  −q 0 0 B(q) =  0  0 0 0 0 c 2 r 2 q2

 0   0  , −1  0

   F¯ =  

 0.5c2 q(qw)2   0  .  0  2 2 c q v + 0.5(qw) qw

In order to transform the system (16.20) to the equations of coupled normal waves one needs to reduce the matrix B(q) to a diagonal form using left hand side eigenvectors

q2 , q, 0, 0 , ζ¯3,4 = (0, 0, p3,4(q), −1). ζ¯1,2 = − p1,2 (q) The above two vectors correspond to the eigenvalues p 1,2 = ±cq, related to longitudinal excitations, whereas the last two eigenvalues p 3,4 = icrq2 describe dispersional branches of bending waves. The second step includes introduction (instead of physical variables ¯u(x, t)) the wave variables Vm (x, t) and U m (x, t), m = 1, 2, for each of dispersion branch of the linear system ¯u(x, t) =

2

ψm (q)Vm(x, t) +

m=1

2

ψm+2 (q)U m (x, t),

m=1

where: ψm (q) is the vector of operator coefficients of splitting of normal wave being the right eigenvector of the matrix B(q). The equations of coupled normal waves have the following form: ∂V1,2 ∂V1,2 c ∂ ± c = ± 2 (U1 − U2 )2 , ∂t ∂x 2r ∂x i

∂U1,2 1 ∂U1,2 v − ϕ1 + a(U 1 − U2 )2 − (V1 + V2 )(U 1 − U2 ) = 0, ∂t 2 ∂x ν1

where: ϕ1 is group velocity of normal form, ϕ 1 = 2crk, a = 0.25/(cr −3). A transition from physical quantities into V m and U m yields the main wave part of the problem from the details of its physical statement. Physical and wave variables are coupled via the following relations: vt (x, t) = c[V2 (x, t) − V1 (x, t)],

v x = [V2 (x, t) + V1 (x, t)],

470


wt = −

∂U2 ∂U1 + , ∂x ∂x

wx =

2i [U2 − U1 ]. ϕ1

In the one wave approximation, V 2 = U2 = 0, V1 = V, U 1 = U the perturbations moving in the positive direction of the x axis are governed by the equations ∂V ∂V c ∂U 2 +c = 2 , ∂t ∂x 2r ∂x i

∂U 1 ∂2 U c − ϕ1 2 + a |U|2 U − VU = 0. ∂t 2 ∂x ϕ1

Observe that deflection waves occurring in rods are governed by nonlinear Schrödinger equation which appears in various branches of physics and describe the modulated waves in strongly disspersive media.

16.4 WKB - approach As an example a longitudinal autonomous vibrations of a rod (0 ≤ x ≤ l) with the variable plane EFϕ 1 (x) and the plane ρFϕ 2 (x), where E, ρ, F ≡ const, is considered [138]. The governing equation and the boundary conditions have the following form

d du EF (16.21) ϕ1 (x) + ρϕ2 (x)Fω2 u = 0, dx dx u(0) = u(l) = 0.

(16.22)

After an introduction of the non-dimensional coordinate one obtains d2 u du + q + q1 λ2 u = 0, 2 dξ dξ

(16.23)

where: q(ξ) = ϕ 1 (ξ)/ϕ1 (ξ), q1 = ϕ2 (ξ)/ϕ1 (ξ), λ2 = ρω2 /(El2 ). Observe that the following condition holds for high frequency vibrations, λ 2 1. For q = 0, q 1 = 1 a solution of the equation (16.23) has the form u = C1 exp(−iλξ) + C 2 exp(iλξ). Therefore, one may transform the equation (16.23) with variable coefficients using the following change of variables u = exp(λg(ξ)).

(16.24)

Substituting (16.24) into (16.23), we get λ2 g 2 + λg g + λqg + q1 λ2 = 0.

(16.25)

A solution of the equation (16.25) can be sought as a function of the following series

16.4 WKB - approach

471

g(ξ) = g0 + λ−1 g1 + λ−2 g2 + . . . , whose successive terms are defined by the following set of recurrent equations g 2 0 + q1 = 0,

λ2 : λ: .

.

2g 1 g 0 + g 0 g 0 = 0,

. .

.

.

Solving the equation (16.26) we obtain g0 = ±i

ξ

. . /

0

.

(16.26) (16.27)

.

q1 (τ)dτ.

Therefore, an approximate solution of the equation (16.21) has the form ξ ξ / / q1 (τ)dτ + C2 cos q1 (τ)dτ . u ≈ C1 sin 0

0

Owing to the boundary conditions (16.22), we obtain 1 / C2 = 0, sin λ q1 (τ)dτ = 0, 0

and hence

λ

0

1

/

q1 (τ)dτ = nπ,

n = 1, 2, 3, . . . .

The being sought frequency is estimated by the formula √ nπl E ω= √ 1 / . ρ 0 q1 (τ)dτ The obtained approximation is called the WKB - approach. Using the WKB method as an example, it is interesting to see how difficult it often is to identify the author of one or another asymptotic method. Indeed, as far as known, this method was first used by Francesco Carlini in 1817 for the investigation of the elliptical motion of planets around the Sun. But this paper was not paid proper attention, although it was reprinted in a German translation by Jacobi in 1850. In 1837 Jacques Liouville and George Green discovered this method again. Later on this method was improved by Lord John W. Rayleigh in 1912 and in 1915 by the German physicist Richard Gans. The most systematic results were obtained in 1924 by Harold Jeffreys. However, all these papers remained unnoticed, in particular a quite general solution by Jeffreys. Jeffreys himself noted in 1956 that “he missed the earlier investigations by Gans”. Finally, the method received its name after the publication in 1926 of papers by Gregor Wentzel, Hendrik A. Kramers, and Leon Brillouin. As E.J. Hinch [340] mentioned: “While everyone agrees that Messrs. W, K, B and

472


J did not invent this (WKBJ) method, there is little agreement over who did. Certainly the following all contributed important developments: Liouville, 1837, Green, 1837, Jakob Horn, 1899, Rayleigh, 1912, Gans, 1915, Jeffreys, 1923, Wentzel, 1926, Kramers, 1926, Brillouin, 1926, Rudolph E. Langer, 1931, Frank W. J. Olver, 1961, and Richard E. Meyer, 1973.” However, a solution to the equation (16.21) (and similar like equations) can be also sought in a different manner. For this purpose we consider autonomous vibrations of a beam with a variable transversal cross-section, governed by the following equation

d2 w d2 (x) (16.28) EIϕ − ω2 ρFϕ2 (x)w = 0. 1 dx2 dx2 The following boundary conditions are attached w=

dw = 0 for dx

x = 0, l.

(16.29)

Introducing a new variable via relation ξ = x/l, one gets

d2 d2 w ε4 2 ϕ1 (ξ) 2 − ϕ2 (ξ)w = 0, dξ dξ where: ε3 = EI/(ω2 ρFl4 ). A solution of the equation (16.30) is sought in the following form ξ g(τ)dτ u0 (ξ) + εu1 ( xi) + ε2 u2 (ξ) + . . . . w = exp ε−1

(16.30)

(16.31)

0

Substitution (16.31) into (16.30) and after a splitting with respect to ε one gets (ϕ1 g4 − ϕ2 )u0 = 0,

(16.32)

(ϕ1 g4 − ϕ2 )u1 + 4ϕ1 g3 u 0 + 6g2 g u0 = 0,

(16.33)

.

.

Therefore

. .

g=

ϕ2 ϕ1

.

.

. .

,

u0 =

l/4

.

.

1 g3/2 ϕ1/2 1

,

and a general solution to the equation (16.30) in the first approximation has the form ξ ξ −1 −1 w(ξ) = C1 u0 ε g(τ)dτ + C2 u0 u ε g(τ)dτ + 0

0

C3 exp −ε−1 g(0)ξ u0 (0) + C 4 exp ε−1 g(1)(1 − ξ) u0 (1). Observe that variable components are ‘frozen’ on the edges, since on the beam boundaries these functions are fastly damped and their influence on the results is of minor magnitude.

16.5 The WKB method and turning points

On the other hand, the boundary conditions (16.29) yield l −2 2 , 1 ϕ2 EI 2 dx , ω=π n+ 2 ρFl4 0 ϕ2

473

(16.34)

n = 0, 1, 2, . . . . It is worth noting that for ϕ 2 = ϕ1 = 1 the formula (16.34) overlaps with that obtained using the Bolotin method (see Chapter 11.1). This observation yields also an additional conclusion. Namely, both WKB and Bolotins approaches can be matched. In words, and oscillatory solution part can be found applying the WKB-approach, whereas the boundary layer problems are solved using the Bolotin’s method.

16.5 The WKB method and turning points The turning points occurred in WKB method are defined by sign changes of a coefficient standing by the unknown function ϕ(η), i.e. in the equation d2 v − ϕ(η)v = 0, dη2 ϕ(η) > 0 for η > 0,

(16.35)

ϕ(η) < 0 for η < 0.

It is clear, that applying a suitable change of variables one may transform this problem to the coordinates with an origin being a turning point. We use further McLaurin series of the form ϕ(η) = a0 + a1 η + a2 η2 + . . . . Observe that for small η the first two terms of the McLaurin series will play a fundamental role. A simple change of variables reduces the equation (16.35) to the following form d2 v − ηv = 0. (16.36) dη2 These are the Airy’s equations with knows solutions. Namely, two Airy’s functions Ai(η)) and Bi(η) are the solutions, and they are real for real η. The corresponding McLaurin series read [138]: √ Ai(η) = a1 f1 (η) − a2 f2 (η), Bi(η) = 3(a1 f1 (η) + a2 f2 (η)), (16.37) where: f1 = 1 +

∞ k=1

f2 = 1 +

∞ k=1

dk η3k+1 , dk =

bk η3k ,

bk =

bk−1 , 3k(3k − 1)

dk−1 3−2/3 3−1/3 , a1 = , a2 = , 3k(3k + 1) Γ(2/3) Γ(1/3)

474


and Γ(z) denotes Euler gamma-function. Recall that √ Bi(0) = Ai(0) 3.

(16.38)

The asymptotic expansions of the Airy’s functions for η → ∞ are defined by formulas (see Chapter 1.6). (−1)k ck 1 , √ η−1/4 e−ζ ζk 2 π k=0 ∞

Ai(η)

| arg η| < π;

∞ 1 ck π Bi(η) √ η−1/4 eζ , | arg η| < ; k 3 ζ π k=0 0 1 2π π π 1 Ai(−η) √ η−1/4 D1 (ζ) sin(ζ + ) − D2 (ζ) cos(ζ + ) , | arg η| < ; 4 4 3 π 0 π π 1 2π Bi(−η) = η−1/4 D1 (ζ) cos ζ + ; + D2 (ζ) sin ζ + , | arg η| < 4 4 3

D1 (ζ)

∞ (−1)k c2k

ζ 2k

k=0

,

D2 (ζ)

∞ (−1)k c2k+1

ζ 2k+!

k=0

,

Γ(3k + 1/2) . 5.4k k!Γ(k + 1/2) Consider now the following linear second order differential equation ζ=

2 3/2 η , 3

µ2

ck =

d2 y − q(x, µ)y = 0, dx2

(16.39)

(16.40)

where the function q(x) if holomorphic one for x ∈ (x 1 , x2 ), 0 < µ 1. The points x = x∗ , where q0 (x∗ ) = 0, are called turning points. If q (x∗ ) 0, then the turning point x = x ∗ is simple. The asymptotic series of solutions to equation (16.40) in the neighbourhood of a simple turning point x = x ∗ for q (x∗ ) > 0 has the form y(x, µ) = a(0) (x, µ)v[η(x, µ)] + µ1/3 a(1) (x, µ)

dv , dη

(16.41)

where: v(η) is the Airy function, η(x, µ) = µ−2/3 ξ(x), a( j) (x, µ)

ξ(x) = 1.5 ∞

a(k j) (x)µk ,

x

/

2/3 q(x)dx

x∗

j = 0, 1.

, (16.42)

k=0

The function ξ(x) and coefficients a (k j) (x) are obtained by substitution of solutions (16.41) into (16.42) and comparison of coefficients standing by µ k v and


475

µk (dv/dη). If there are not other turning points, the functions ξ(x) and a (1) k (x) are holomorphic, including the point x = x ∗ . If |η| 1, then the asymptotic expansions of the Airy’s functions can be used. Let the function q(x, µ) is real for real x and let the following conditions are satisfied q(x∗ ) = 0, q (x∗ ) > 0, q(x) < 0 for x < x ∗ ,

q0 (x) > 0 for x > x ∗ .

Therefore, the following asymptotic series hold 1 y1 (x, µ) = √ Ai(η)(1 + O(µ)) + Ai (η)O(µ4/3 ), ξ 1 x/ q0 (x)dx (1 + O(µ)), x > x∗ , y1 (x, µ) = 0.5a exp − µ x∗

x∗ / π 1 + O(µ) , x < x∗ . y1 (x, µ) = a sin −q0 (x)dx + µ x 4

(16.43) (16.44) (16.45)

1 y2 (x, µ) = √ Bi(η)(1 + O(µ)) + Bi (η)O(µ4/3 ), (16.46) ξ x / 1 y2 (x, µ) = a exp q0 (x)dx (1 + O(µ)), x > x∗ , (16.47) µ x∗ x∗

/ π −q0 (x)dx + (16.48) y2 (x, µ) = a cos + O(µ) , x < x∗ , 4 x √ / √ where: a = 6 µ/( 4 |q0 | π). Formulas (16.43), (16.43) are uniformely suitable for x 1 ≤ x ≤ x2 , formulas (16.44), (16.47) – for x > x ∗ , whereas formulas (16.45), (16.48) are uniformely suitable for x < x∗ . For x > x∗ the function y ‘ exponentially decays, whereas the function y2 increases. For x < x∗ both functions oscillate. In next example, free vibrations of a string with variable density is analysed. y + Ω2 ρ(x)y = 0,

(16.49)

y(0) = y(l) = 0.

(16.50)

Here ρ(0) = 0; ρ (0) > 0; ρ(x) > 0 for 0 < x ≤ l. We are going to construct a solution for high frequencies by taking Ω = µ −1 . Therefore, the equation (16.49) is reduced to the equation (16.40), which has the solution (16.41). The asymptotic splitting yields 1 y(x, µ) = √ v0 (η)(1 + O(µ2 )) + v 0 (η)O(µ4/3 ), η = −µ−2/3 ξ, ξ 2/3 x/ √ 3 v0 (η) = 3Ai(η) − Bi(η), ξ = ϕ(x) , ϕ(x) = ρ(x)dx. 2 0

(16.51)

476


We have v0 (0) = 0. A being sought Ω n can be computed from condition v0 [η(l)] = 0. In result, we get Ωn =

2|η n |3/2 + O(n−1 ), 3ϕ0

n → ∞,

l / where: ϕ0 = ϕ0 = ϕ(l) = 0 ρ(x)dx, η n denotes n-th zero of the function v 0 (η). Substituting η n by its asymptotic representation, we obtain Ωn =

π(n − 1/12) + O(n−1 ), n → ∞. ϕ0

Now we are going to find the asymptotic series of the eigenvalues Ω n assuming that ρ(0) = 0, ρ (0) > 0, ρ (l) < 0, ρ(x) > 0 for 0 < x < l. In the considered problem two turning points exist: x = 0 and x = l, and therefore it is impossible to construct asymptotic series uniformly suitable on the whole interval 0 ≤ x ≤ l using the Airy’s functions. Therefore two different solutions y(1) (x) and y(2) (x) will be constructed. The solution y (1) (x) is uniformely suitable for 0 ≤ x ≤ l − ε (ε > 0) and satisfies the condition y (1) (0) = 0, whereas the solution y(2) (x) is uniformly suitable for ε ≤ x ≤ l and satisfies the condition y (2) (l) = 0. The eigenvalues Ωn are defined through the condition y(1) (x) = Cy(2) (x),

C = const,

ε ≤ x ≤ l − ε.

(16.52)

Owing to formulas (16.43), (16.51) and for ε ≤ x ≤ l − ε, the following estimation is obtained 1 1 sin(µ−1 ϕ(x) + π/12), y(1) (x) / v0 (η1 ) / ξ1 πρ(x) −2/3

η1 = −µ

2 ξ1 , ξ1 = ϕ1 (x) 3

3/2 ,

1 1 y(2) (x) / v0 (η2 ) / sin(µ−1 ϕ(x) + π/12), ξ2 πρ(x) 3/2 l/ 2 η2 = −µ−2/3 ξ2 , ξ2 = ϕ2 (x) , ϕ1 (x) = ρ(x)dx. 3 x The identity (16.52) is satisfied only if µ−1 ϕ(x) + π/12 + µ−1 ϕ1 (x) + π/12 = nπ, which for n → ∞ yields the being sought asymptotic formula Ωn =

π(n − 1/6) + O(n−1 ). ϕ0


477

Finally, a string lying on elastic foundation, which vibrations are governed by the equation d2 y (16.53) T 2 − c1 (x1 )y + ω2 ρ1 (x1 )y = 0, dx1 y(0) = y(l) = 0,

(16.54)

is considered. After a transition to non-dimensional variables x = x 1 /l, c1 = c0 c(x), ρ1 = ρ0 ρ(x) (c(x), ρ(x) ∼ 1) the problem (16.53), (16.54) can be reduced to the standard form µ2 y − q(x)y = 0, (16.55) where: q(x) = q(x, λ) = c(x) = λρ(x), µ 2 = T/(c0 l2 ), λ = ω2 ρ0 /c0 . Assuming that µ λ, the frequency spectrum will be further analysed. In addition, we assume that c(x), ρ(x) > 0 for 0 ≤ x ≤ 1. In order to proceed smoothly with further calculations the following function and quantities are introduced z(x) =

c(x) , ρ(x)

λ− = min z(x),

λ+ = max z(x).

x

x

For λ < λ− the function q(x) > 0 for all x, and the problem (16.55) does not possess non-zero solutions. For λ > λ+ the function q(x) < 0 for all x, and for λ > λ + + ε, ε > 0, a solution has the form 1/ ϕ0 (λ) = µnπ + O(µ2 ) = −q(x)dx. (16.56) 0

For λ− < λ < λ+ the integration interval [0, 1] includes the turning points, which move along the axis x with a change of λ ∈ [λ − , λ+ ]. Only the case when z (x) > 0, x ∈ [0, 1] will be considered. An integration interval for all λ ∈ [λ − , λ+ ] possesses only one turning point x ∗ (λ), where λ = z(x∗ ), q (x∗ ) > 0. During a construction of asymptotic series of the solutions of equation (16.56) only the main term of (16.41) is taken into considerations. Then x/ 1 −q(x)dx, (16.57) y(x) = √ (C1 Ai(η) + C 2 Bi(η)), ϕ = ξ x∗ where: C 1 , C2 are arbitrary constants, η = µ 2/3 ξ, ξ = (3ϕ/2)2/3, where η < 0 for x < x∗ and η > 0 for x > x ∗ . After substitution of solution (16.57) into boundary conditions (16.54), the following approximate equation is obtained Ai(η0 ) − γBi(η0 ) = 0, where:

γ=

Ai(η2 ) , Bi(η1 )

(16.58)

478


2/3 3ϕ1 η0 = − < 0, 2µ 2/3 3ϕ2 η1 = > 0, 2µ

ϕ1 =

0

ϕ2 =

x∗

1

/ −q(x)dx, /

q(x)dx.

x∗

Further, two particular cases will be considered. (i) If the turning point x ∗ lies far from the edge x = 1, then η 2 1, γ 1, and the equation (16.58) is simplified, and either has the form Ai(η 0 ) = 0, or 2µ|ηn |3/2 µπ(n − 1/4), 3

n = 1, 2, . . . ,

(16.59)

where ηn is the n-th zero of the function Ai(η). The eigenfunction oscillates to the left of the turning point x ∗ and expotentially decays from the right hand side of this point. (ii) If the turning point x ∗ lies in the neighbourhood of the edge x = 1, then the equation (16.58) can not be applied. In particular, √ for x ∗ = 1, i.e. when the turning point and the edge overlap (for γ = 1/ 3), the equation (16.58) is substituted by the equation (16.49) ϕ 1 µπ(n − 1/12). Let us assume tan(απ) = γ. Then the equation (16.58) yields the expression ϕ1 µπ(n + α − 1/4).

(16.60)

For λ > λ+ the formula to obtain η 1 is not appropriate one. Assume, that the functions c(x) and ρ(x) can be analytically extended to the right of the point x = 1. Hence, for x ∗ > 1 and in order to calculate η 1 in (16.58) the following formula is used 1 2/3 x∗ / 3ϕ η1 = − , ϕ2 = −q(x)dx. 2µ 1

16.6 A distributional approach Notice that the function exp(−ε −1 t) can not be developed into Taylor series for ε → 0, if typical smooth function are applied. However, this possible to carry out if theory of distributions are used. Namely [272] (p.245), H(α) exp(−ε−1 t) =

∞

(−1)nεn+1 δ(n) (t),

(16.61)

n=0

where: H(α) is Heaviside function, δ(t) is Dirac’s δ-function, δ (n) (t), n = 1, 2, ..., are derivatives of δ-function. Now a process of series (16.61) constructions is addressed, since it contains an interesting idea on transition from initial problem into certain conjugated space.

16.6 A distributional approach

Applying Laplace transformation ¯x(p) = one gets ∞

0

∞ 0

479

e−pt x(t)dt to the function exp(−ε −1 t),

e−pt exp(−ε−1 t)dt =

ε . εp + 1

Applying Maclaurin series with respect to ε, and then obtain step by step inverse transform the series (16.61) is obtained. It is worth noticing that now singularly perturbed problem can be considered as regular one [250, 272, 273, 668]. Let for example εy + y = 0, y = 1 for t = 0. Singularity occurs for ε = 0, since a smooth solution y = 0 appears which does not satisfy the given initial condition. However, one may represent a solution in the form of non-smooth function. Substituting z(t) = H(t)y(t), the input Cauchy problem yields (16.62) εz = −z + εδ(t). A solution to equation (16.62) can be sought in the form of regular series z=

∞

zn εn .

n=0

In the result, one obtains z0 = 0,

z1 = δ(x),

and finally z(x) =

zn+1 = (−1)n δ(n) (x), ∞

n = 1, 2, ...,

(−1)n εn+1 δ(n) (x).

(16.63)

n=0

Note that using expression (16.63) it is not difficult to get a result expressed via smooth functions. In this aim one may apply first Laplace transformation, then Padé approximation is used in the space p, and finally an inverse transform. Let us consider also the following Cauchy problem x¨ + 2k x˙ + x + cxn = 0, x(0) = 1,

n = 2p + 1, p = 0, 1, 2, ..., x(0) ˙ = 0.

(16.64) (16.65)

In a first step, the damping term is excluded from equation (16.24) using the following standard change of variables: x = y exp(−kt). In result one gets the following Cauchy problem y¨ + a2 y + exp(−k(n − 1)t)y n = 0, y(0) = 1,

y˙ (0) = k,

(16.66) (16.67)

480


where: a2 = 1 − k + k2 . As it has been shown in many references [546, 547, 548, 549, 550, 551, 552, 553], a construction of asymptotics to the system with high power form non-linearity n → ∞ is very important, and can be effectively applied to analysis of nonlinear dynamical systems. In equation (16.66) two parameters occur, k and n. The traditional asymptotical methods are oriented rather to use of the parameter k, suppose k 1. We deal with asymptotics n → ∞, k ∼ 1. Let us apply the distributional approach. For our considered case we get exp (−k(n − 1)t) =

δ(t) δ (t) δ (t) − 2 2 + 3 3 + ... . kn k n k n

(16.68)

Substituting the series (16.68) into both equation (16.66) and the initial conditions (16.67), we get in the first order approximation (for n → ∞): y¨ 0 + a2 y0 = 0, y0 (0) = 1;

(16.69)

y˙ 0 (0) = k.

(16.70)

A solution to the Cauchy problem (16.69), (16.70) has the form y0 = C1 cos(at) + C 2 sin(at),

(16.71)

where: C 1 , C2 are the constants going to be estimated further. We choose solution to the Cauchy problem (16.66), (16.67) in the form (see Chapter 2.14) y1 y2 + + ... . (16.72) y = y0 n 1 + y0 y0 Therefore, one obtains y = y0 +

y1 + ..., n

yn = yn0 +

yn−1 0 y1 + ... . n

In the first order approximation, the following Cauchy problem is obtained 1 y¨ 1 + a2 y1 = − δ(t)yn0 (0), k

(16.73)

y1 (0) = y˙ 1 (0) = 0.

(16.74)

The equation (16.73) is transformed to the form y¨ 1 + a2 y1 = −

C1n δ(t). k

(16.75)

A solution to the Cauchy problem (16.71), (16.75) can be found using the Laplace transformation kC1n sin(at). (16.76) y1 = a


481

Substituting (16.71) and (16.76) into (16.72) and satisfying the conditions (16.67), we get kn , C1 = 1, C2 = a(n + 1) and finally

n kn sin(at) + a(n + 1)

n *1/n k kn sin(αt) . cos(at) + a a(n + 1) )

x = exp(−kt)

cos(at) +

AFTERWORD

Asymptotics and computers Ironically, as numerical analysis is applied to larger and more complex problem, non-numerical issues play a larger role. John Guckenheimer [326]

The reader would probably ask the question: is it worth using asymptotic methods if computers have developed so much in recent time? Maybe it is simpler to program the input problem in full generality and solve it with universal numerical methods? We would reply to the reader as follows. First, the application of asymptotic methods is a very useful preliminary stage of the problem analysis even in cases when the final results are obtained numerically. They allow one to choose the best numerical approach and to deal extensive but not well organized numerical data. “Effective computer timesaving numerical methods must always use information about the analytical nature of the problem” [495]. Second, asymptotic methods work well in regions of extreme parameter values, i.e., in regions where numerical methods fail completely or meet great difficulties. Laplace said not without reason that the asymptotic methods are “the more exact, the more necessary.” One should note the role of asymptotic methods in many-dimensional problems. “Though many people are of the opinion that any problem can be solved with the aid of a good computer, this is in fact far from being the case. For instance, as far as eigenvalue problems are concerned, numerical calculations give reliable results so far only in the one-dimensional case. Even a two-dimensional problem seems to be very difficult from the standpoint of numerical calculations” [650]. “One of the more conspicuous properties of nature is the great diversity of size or length scales in the structure of the world. An ocean, for example, has currents that persist for thousands of kilometers and has tides of global extent; it also has waves that range in size from less than a centimeter to several meters; at still finer resolution seawater must be regarded as an aggregate of molecules whose characteristic scale of length is roughly 10 −8 centimeter. From the smallest structure to the largest is a span of some 17 orders of magnitude” [693].

484


At the same time, there exists a well founded expectation for the effective investigation of such systems with the help of, e.g., renormalization group. Hawking [334] wrote about solving one of the problems he has dealt with: “Even with a computer, it was reckoned it would take at least four years, and the chances were very high that one would make at least one mistake, probably more. So one would know one had the right answer only if someone else repeated the calculation and get the same answer, and that did not seem very likely!” Meanwhile, the application of approximate analytical approaches allowed this difficulty to be overcome. At present it is possible to build algorithms which determine numerically the smooth parts of solutions and use asymptotic approaches for the regions where there is fast change (for example, boundary layers). Last but not least, asymptotic methods develop our intuition and play a great role in formulation of concepts and development of thinking of contemporary natural scientists and engineers. Therefore, asymptotic and numerical methods should be considered not as competing but as complementary. By the way, the development of computers has greatly stimulated the development of asymptotic methods. For example, one of the most difficult stages of asymptotic approaches is the construction of higher approximations. As a rule, for complicated problems it is possible to construct “by hand” a maximum of three approximations. Now it is possible to assign this routine job to computers. This has already been done in some cases. True, it is worth noting that this problem is not simple even for a computer since the number of terms of asymptotic expansion N grows fast with the increase of the approximation order n. In many cases N is determined by the Catalan numbers: Nn = (2n)!/ [n! (n + 1)!] , so that N1 = 1, N2 = 2, N3 = 5, N4 = 14, N5 = 42, and so on. As a particular example of interaction between numerical and analytical methods we would like to consider the mechanics of deformable solids. “Perhaps, it is difficult for an amateur to imagine to what degree physicists (as, probably, the representatives of other sciences as well) are subject to fashion” [582]. The fashion in large-scale application of numerical methods captured the mechanics of deformable solids approximately 25 years ago. This involved especially the finite element method, which is quite simple from ideological point of view and exceptionally effective. The lectures given at conferences and symposia, in which preference was given to analytical approaches, looked like anachronisms. Timid attempts on the part of their authors to remind their listeners that the aim of science is mainly “to replace calculations by ideas” (N. Bourbaki) [187] were not heard in the chorus of sceptics. Scientists joked that popular titles of reports about “the newest, the very-very general” numerical method such as “On the usual approach...” sounded like “On the only approach...” It is not strange that, the less the speaker is aware of how to handle computers, the more he treats numerical methods as a certain universal panacea and the more


485

he criticizes the “antiquated” and “outdated” analytical approaches. At the same time, researchers, who had to compute much and seriously, well understood the role of asymptotic methods, especially in cases when, using V.I. Feodos’ev’s [290] expression, “the computer must be taken by the hand.” Now, one is inclined to believe that this “childhood illness” is mostly overcome, and that analytical methods – “an old but strong weapon” – have mainly regained their importance.

Are asymptotic methods a panacea? To the person with only a hammer, Everything looks like a nail. Proverb

Let us formulate briefly the main merits and demerits of asymptotic approaches indexapproach!asymptotic. The following merits need to be listed first: 1. Essential simplification of the solution process. A solution can often be obtained in analytic form. 2. Asymptotic methods can be easily combined with other approaches: numerical, variational, and so on. Thus, after simplifying the input boundary value problem it is possible to use effectively finite element or boundary element methods. The asymptotic method allows one to obtain the solution structure, and therefore the form of suitable approximating functions for the Bubnov–Galerkin, Ritz, Trefftz, Kantorovich method, and other variational approaches. 3. Asymptotic methods are closely connected with the physical essence of the problem and, at the same time, allows one for deeper understanding. 4. Asymptotic methods often allow one to clear up the mathematical and physical foundations of essentially approximate engineering methods, to make them more exact and to raise the reliability of solutions obtained on their basis. 5. Asymptotic methods often give the opportunity to use a uniform approach to problems which at first are quite different. This allows one to understand of discover their hidden intrinsic sameness or similarity. At the same time, naturally, asymptotic methods are not a panacea. Their main demerit is that the first approximation does not always give the necessary accuracy, while the construction of higher approximations is often a very laborious task. Next, estimation of the accuracy of asymptotic approximations and the limits of applicability of solutions obtained with them is a very nontrivial problem. Finally, a purely subjective obstruction to the application of asymptotic methods may consist, in our opinion, in the following. Imagine a situation when a researcher has an alternative: to use a conventional program tool, which is based, for example, on the finite element method, or to attempt to really analyze the input problem first

486


and to try to simplify it. Which way will he choose? Using a program, at first at least, does not need much in the way of brains. (True, “insight” often comes after great expenditures of time, effort, and means, and then, anyway, one has to resort to analytical investigation). Let us emphasize once more a simple thought: the main ideas of asymptotic simplification are frequently used by scientists and engineers, explicitly or, more often, implicitly. The choice of a method of asymptotic investigation and the question of how to introduce small parameters into a system is a stage of investigation which, in principle, cannot be formalized. Here experience, intuition, understanding of the physical essence of a problem, and analysis of experimental and numerical results must bring help. But after the introduction of (the right) small parameters and after the choice of an investigation method, it is not necessary to “reinvent the wheel” - it is better to use some well known and well worked out approach. Let us say a few words about small parameters. They may be present in the system from the very beginning or be introduced artificially. For example, the following values play the role of natural small parameters in the theory of shells: h/R, the ratio of the shell thickness to its radius; a/b, the ratio of characteristic sizes (for example, of the plate length to its width); ω−1 , where ω is frequency of oscillations; A, the dimensionless amplitude of oscillations; w/R, where w is normal displacement; the ratio of bending of a structurally-orthotropic shell, or the ratio of shear rigidity of such shell to its membrane rigidity. A parameter ε 1 may also characterize a small deflection of the initial region from the circular shape; or of initial variable thickness from a constant; or the ratio of a shallow shell rising to its curvature radius R, and so on. There can exist different possibilities when a certain parameter may be either small or large. If a “natural” small (large) physical parameter is not found, one may try to introduce it into the equations artificially. Here there is “An essential problem”, which is typical of applied mathematics. It is the problem of the zero-th approximation. It consists in the choice of a certain object, or a family of such objects, such the required solution will appear close to one of them. This choice can be based on the expected character of the solution and is often done at the stage of construction of the mathematical model. To look successfully for something, it is always desirable to know, even if only approximately, what one is looking for. The family just mentioned may be constructed by different nonequivalent methods, which essentially affect the simplicity and accuracy of the further procedure. Here analogies and intuition play a great role” [170]. The simplest method is to introduce a parameter ε in such a way that at ε = 0 some simplified problem is obtained, and at ε = 1 the input one. However, here one encounters a serious problem of convergence of perturbation theory series at ε = 1. In this case the methods of analytical and meromorphic continuation, the method of generalized summation, and so on, may turn out to be useful.


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Finally, the following aspect of application of asymptotic methods has thus far not received enough attention. If it is possible to obtain an asymptotics at ε → 0, is it possible to obtain reasonable simplifications at ε → ∞? This question is often physically grounded. Next one may try to match two limit solutions and to construct expression, which would describe the solution at any ε (for example, using twopoint Padé approximation, see Chapter 14).

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Subject index

Airy stress function, 45, 82, 258, 473–476 amplitude, 73, 85, 87, 105, 107–109, 111, 115, 116, 215, 243, 255, 260, 319, 328, 329, 331, 339, 340, 360, 367, 369, 371, 382, 405, 408, 435, 441, 442, 453, 456–459, 466, 486 amplitude-frequency dependence, 71, 73, 84, 345, 382, 383 approach – asymptotic, 15, 16, 21, 23, 27, 71, 98, 104, 105, 107, 115, 122, 333, 388, 402, 425, 442, 446, 460, 463, 484 – numerical, 388, 483, 485 – quasi-linear, 75, 388 approximant, 16, 93, 94, 97, 264, 265, 306, 307, 312, 377, 379, 380, 383, 386, 390, 401, 417, 422, 425 – quasifrictional, 386, 424, 425, 427 approximation, 15, 21–23, 30, 48, 49, 53, 54, 56, 57, 62, 66, 67, 69, 70, 73, 85, 87–101, 106, 108, 109, 111, 112, 114–117, 120, 124, 127, 133, 140, 141, 143–149, 153, 154, 169, 178, 197, 198, 201, 206–210, 234, 246, 248, 259, 260, 262–265, 276, 279, 283, 294, 295, 300, 304, 305, 307, 309, 312, 313, 315, 317, 318, 342, 353, 355–357, 364, 365, 378, 379, 383, 385, 386, 393, 394, 398, 399, 407, 409, 417, 418, 437–439, 442, 445, 447, 449, 451, 455–457, 459, 461, 462, 466, 467, 470–472, 479, 480, 484–487 – asymptotic, 29, 485 – continual, 265, 331, 456, 457 – higher–order, 49, 100, 107, 141, 206, 209, 247, 262, 272, 436, 464, 467, 484, 485 – long–wave, 323 – short-wave, 462

– successive, 49, 58, 86, 98, 145, 149, 197, 283, 384, 385 asymptotic reduction, 172 averaging, 16, 36, 122, 241, 244, 245, 248–250, 287, 444 – method, 74, 241, 244, 246, 256, 259, 438, 465 – operator, 293, 298, 299 – procedure, 107, 241, 248, 249 beam, 61, 105, 115, 135, 152, 319, 320, 323, 328–331, 336, 392, 435, 453, 472 Berger, 256 – equation, 256–258 Bessel – function, 39, 45, 297, 361, 419 bifurcation, 101–103, 401, 451, 452 body, 15, 141, 317, 425 – deformable, 463 – rigid, 276 Bolotin – approach, 333, 473 – method, 336, 353, 357, 473 boundary, 15, 16, 26, 40, 48–50, 52, 53, 56–62, 92, 99, 101, 102, 105–107, 111–113, 115, 117–119, 121, 122, 129–131, 133, 135–139, 141, 142, 144– 149, 152–154, 157, 160, 161, 164–167, 169, 170, 174–176, 178, 179, 182–185, 189–193, 195, 197, 199, 201–203, 206, 212, 216, 222–224, 227, 229, 235, 237–240, 257, 258, 260–265, 267–275, 277, 279–284, 287–291, 293–301, 303, 304, 306–319, 329, 333–336, 342, 344, 346, 347, 351–354, 356–358, 360, 361, 363, 371–376, 381, 388, 389, 392–394, 396–399, 401, 418, 442, 452, 458, 470–473, 477, 485

530

Subject index

– layer, 15, 25, 27, 31, 117–125, 127, 133, 136, 137, 139, 140, 279, 280, 282, 284, 288, 291, 336, 389, 463, 473, 484 Bubnov–Galerkin method, 82, 107, 244, 357, 359, 381, 417, 485 case – isotropic, 141, 174, 177, 190, 193, 226, 227, 373 – limit, 257 – limiting, 87, 115, 116, 160, 167, 202, 203, 211, 266, 277, 278, 306, 339, 439, 459 – nonlinear, 184, 256, 370 – singular, 29, 117 Cauchy problem, 72, 85, 92–95, 479, 480 chain, 86, 256, 259, 260, 262, 263, 265, 322, 325, 363, 365–367, 369–372, 417–419, 453, 458–462 complex – amplitude, 255, 435, 456, 458, 459, 466 components – fast, 241, 290, 335, 465 – slow, 119, 241, 268, 283, 288, 290, 335, 465 condition – initial, 251, 436 condition(s) – boundary, 26, 48–50, 52, 53, 57, 58, 60–62, 80, 99, 101, 102, 106, 112, 117–119, 121, 122, 131, 133, 136–139, 141, 142, 144–149, 152–154, 160, 164–167, 170, 174–176, 178–185, 189–193, 195, 197, 203, 206, 216, 222, 224, 227, 229, 235, 237–240, 257, 258, 262–264, 267–272, 274, 275, 277, 281–284, 287–291, 293, 295–301, 309, 313, 314, 316–319, 333–337, 342, 344, 346, 347, 352–354, 356–358, 360, 361, 372–375, 381, 392–394, 396–399, 418, 470–473, 477 – – homogeneous, 139, 315 – – non-homogeneous, 315 – initial, 26, 73, 75, 78, 83–86, 91, 95, 97, 105, 109, 111, 116, 118, 121, 123–125, 147, 182, 212, 216, 222, 235, 237–239, 259, 265, 292, 336, 337, 347, 360, 386, 389, 457, 463, 464, 479, 480 continualization, 16, 259, 263, 264, 363, 435

continuation – analytic, 42, 377, 378, 409, 410 – meromorphic, 378, 486 – periodic, 294, 298, 303 continuous fraction, 378, 421 continuum, 15, 266, 319, 330, 331, 364, 366, 456, 457 convergence, 23, 34, 78, 88–90, 99, 198, 199, 378, 379, 384, 388, 402, 405–407, 409, 410, 424, 426, 486 – domain, 377, 379 – theorems, 378 coordinate, 40, 65, 68, 70, 79, 101, 103, 117, 122, 139, 141, 144–147, 150, 259, 264, 273, 294, 309, 310, 320, 333, 357, 359, 360, 457, 470, 473 damping, 367, 449, 450, 479 – coefficient, 453 dimensionality, 28, 453 domain, 26, 30, 31, 42, 75, 77, 88, 104, 117, 144, 149, 256, 292–295, 299, 309, 317, 355, 357, 363, 366, 378, 379, 388, 463–465 Domb–Sykes – plot, 404 Domb–Sykes plot, 405–407 Duffing – equation, 26, 71, 72, 78, 241, 245, 337, 382, 383, 466 – oscillator, 388 effect – dynamic edge, 213, 215, 218, 333, 334, 337, 339, 341, 343, 349–351, 353–355, 361, 362 – edge, 120, 135, 162, 164, 165, 173, 177–179, 183, 184, 189, 192, 197, 201, 203, 205–210, 215–217, 219, 221, 224, 226, 227, 229, 234, 236, 237, 239, 334, 335, 337, 338, 343, 344, 352, 353, 356, 357, 360 eigenfunction, 56, 58, 62, 63, 101, 154, 271, 297, 333–335, 353, 359, 394, 398, 478 equation – autonomic, 441 – averaged, 243, 247, 248, 304, 306, 435 – biharmonic, 141

Subject index – differential, 26, 49, 50, 52, 53, 56, 58, 59, 67, 83, 85, 104, 117, 119, 121–123, 127, 129, 131, 153, 159, 167, 171, 201, 229, 243–246, 249, 252, 256, 259, 263, 266, 267, 276, 280, 291, 323, 333, 336, 343, 347, 349, 350, 357, 373, 381, 393, 398, 399, 474 – Hausdorff, 251 – homogeneous, 134, 137, 138, 171, 204 – homogenized, 270, 275, 279, 297, 298, 300–302, 315 – integro-differential, 244, 245 – nonlinear, 49, 58, 73, 94, 190, 219, 220, 256, 258, 269, 345, 441, 452 – recurrent, 108, 124, 242, 267, 269, 271, 274, 286, 471 – transcendental, 98, 138, 345, 353, 354, 357, 361, 395, 396 – transcendetal, 358 – trasncendental, 395 Euler transformation, 408 expansion, 25, 29, 49, 73, 75, 76, 88, 105–108, 110, 112, 113, 141, 145, 146, 254, 284, 293, 296, 298, 301, 308, 312, 317, 377–380, 387, 388, 402, 405, 407, 409, 417–419, 421, 424–427, 435, 437, 440, 448, 454, 457, 460 – Asymptotic, 474 – asymptotic, 15, 22, 75, 76, 106, 108, 111, 256, 335, 387, 404, 406, 421, 424, 426, 427, 456, 463–465, 475, 484 – WKB, 256 Fourier – method, 261, 284 – series, 91, 265, 295, 390, 392, 466 frequency – natural, 297, 443, 444 function – bounded, 249, 325 – complex, 435, 437, 444, 446, 451, 462 – deflection, 358, 360, 381 – delta, 478 – elliptic, 67, 72, 73 – gamma, 474 – periodic, 267, 281, 286, 287, 290, 293, 314, 320, 324, 325, 327, 328 – rational, 377, 379, 380, 384, 387, 411, 420, 422, 427

531

– smooth, 85, 479 Galerkin – functional, 297 – method, 146, 147, 296, 353, 360 – procedure, 297, 298, 301, 302 homogenization – approach, 284, 302 – method, 276 infinite limit, 459 instability, 449, 451, 452, 458, 459 integral – elliptic, 337, 382 – energy, 450 invariant, 256, 257, 378 Jacobi – function, 337 Kantorovich – method, 139, 140, 280, 283, 291, 338, 357, 361, 485 Laplace – transform, 419, 420 – transformations, 263, 479, 480 Laurent – series, 417 Lie group, 249 Lighthill method, 81, 387 linearization, 190, 371, 452 load – external, 190, 267, 277, 281, 292 localization, 16, 35, 40, 41, 125, 363, 365, 367, 369–373, 375, 376, 449, 451, 453, 458, 459 Maclaurin series, 378, 379, 479 matching, 16, 122, 140, 193, 201, 202, 211, 220, 221, 224, 229, 234, 236, 266, 335, 464, 465 mechanics, 128, 388, 389, 417, 435, 463, 484 – celestial, 19, 104 Mellin transformation, 139 membrane, 65, 99, 100, 129, 135, 157, 158, 163, 169, 171, 183, 189, 190, 273, 276, 298–300, 302, 486

532

Subject index

– perforated, 292, 298, 300 method – approximate, 49, 467 – asymptotic, 76, 294, 357, 388, 471, 483–485, 487 – asymptotical, 119, 480 – finite element, 485 – large δ, 91 – multiscale, 104, 122, 125, 302, 308 – numerical, 372, 483, 484 – perturbation, 60, 73, 83, 91, 98, 104, 154, 299, 303, 377, 383, 392, 395, 398, 467 – Runge–Kutta, 92–94 – saddle-point, 39 – small δ, 90, 91, 93 – stationary phase, 38 mode – normal, 104, 346, 449, 451, 452, 458, 459 model – continuous, 15, 262, 263 – discrete, 264, 449 – mathematical, 435, 486 modulation, 327, 328 Newton method, 104, 197–199 nonlinear dynamics, 16, 252, 435, 453, 461, 480 nonlinearity, 65, 70, 90, 456, 457 operator, 48, 50–52, 57, 62, 102, 152, 159, 171, 190, 195, 204–206, 212, 227, 229, 260, 264, 267, 287, 293, 298, 299, 308, 321–323, 350, 401, 435, 466, 468, 469 – differential, 50, 52, 159, 212, 260, 263, 264, 302, 321 – homogenization, 268, 271, 272, 315 – inertial, 321 – Laplace, 139, 147 orthogonality, 359 oscillations, 38, 71, 98, 111, 199, 265, 284, 285, 292, 333, 339, 345, 363, 371, 375, 392, 397, 435, 449, 459, 463, 467, 486 – free, 260, 262, 264, 353, 436, 441 – low-frequency, 245 – nonlinear, 339, 342, 345, 451, 466 – normal, 456 – out-of-phase, 453 – parametric, 452, 458

oscillator, 259, 321, 363, 370, 371, 388, 389, 418, 420, 435, 448, 453, 456, 457, 473 – harmonic, 88, 363 – nonlinear, 370, 435, 447, 458, 459 – one-degree-of-freedom, 77, 109 Padé – approximants, 16, 89, 93, 94, 97, 377, 386, 417, 422, 425 – approximation, 90, 143, 263, 265, 479 – mulit-point approximants, 307 – multi-points approximants, 306 – table, 378 – three-point approximants, 307 – two-point approximants, 264, 265, 312, 417, 422, 487 – two-point approximation, 312 parameter – asymptotic, 213, 388 – large, 98, 128, 348, 403, 444 – perturbation, 26, 65, 153, 197, 201, 273, 305, 384, 388, 395 – small, 15, 16, 21, 47, 51, 59, 60, 66, 73–75, 83, 88–90, 95, 98, 99, 102, 105, 107–110, 117, 120, 123, 125, 129, 141, 143, 145, 172, 174, 253, 257, 277, 281, 292, 314, 317, 318, 320, 384, 388, 408, 437, 440, 454, 456, 457, 486 pendulum, 71, 72, 119, 120, 122, 463 period, 72, 78, 85, 93, 95, 111, 243, 267–269, 274, 277, 281, 286, 288, 292, 293, 302, 314, 320, 322, 326, 337, 339, 340, 350, 389, 418, 420, 444 periodicity, 78, 88, 270, 320, 329, 443 – condition, 70, 71, 84, 268, 272, 295, 306, 315, 326, 349, 390, 458 – rule, 313 perturbation, 15, 19, 26, 47, 49, 54, 56, 58, 60, 79, 80, 83, 88, 91, 98, 104, 108, 117, 143, 150, 154, 294, 299, 303, 317, 318, 377, 381, 383, 392, 395, 396, 398, 406, 417, 418, 467 – classical, 383 – equation, 63 – expansion, 377, 425–427 – parameter, 26, 47, 65, 153, 201, 273, 305, 313, 384, 388, 395 – small, 56 – solution, 388

Subject index – technique, 59, 71, 72, 89, 99, 105, 149, 155, 305, 311, 388 – theory, 104, 152, 379, 486 – thoery, 435 phase – difference, 456 – shift, 334, 346, 453, 459 phenomenon, 111, 260, 390 plate, 47, 103, 120, 131, 132, 135–137, 147–150, 153, 154, 163, 167, 174, 178, 179, 184, 188, 193–195, 201, 203, 213–217, 220, 221, 223, 224, 227–229, 234, 236, 244, 256, 272, 273, 276, 281, 284, 285, 287, 291, 315, 317, 338, 340, 341, 344, 345, 354, 357, 358, 360–362, 373, 374, 381, 397, 486 – circular, 47 – isotropic, 356, 384 – orthotropic, 185, 188 – rectangular, 256, 258, 284, 340, 345, 359, 373 – square, 315, 345, 353, 400 – stringer, 178, 186 – viscoelastic, 244 Poincaré-Lindstedt method, 78, 249 point – saddle, 40, 42–44, 368 – singular, 378, 387, 401, 408, 410–412 – stationary, 369, 441, 442, 446, 451 points – saddle, 39 – stationary, 38, 39 Poisson – coefficient, 82, 143, 151, 158, 170 – ratio, 144 – ration, 65 potential, 105, 141, 159–163, 166, 171, 173, 177, 179, 190, 206, 208, 213, 214, 265, 365, 367, 447, 460 – energy, 256, 354, 366 problem – boundary, 15, 59, 122, 138, 139, 164, 238 – boundary layer, 291, 473 – boundary value, 16, 26, 49, 52, 92, 102, 105, 107, 111–113, 115, 129, 137, 141, 147, 152, 157, 160, 161, 167, 169, 182, 201, 202, 206, 212, 216, 223, 229, 237, 240, 260, 261, 263, 265, 271–273, 283,

533

287, 288, 293–296, 298, 299, 303, 304, 306, 308–312, 314–319, 329, 376, 388, 389, 392, 393, 396, 398, 399, 401, 485 – cell, 313 – initial value, 75 procedure – dequantization, 129 – homogenization, 169, 268, 272, 273 – matching, 122, 464, 465 – quantization, 129 – regularization, 315, 317 QA, 424 recurrent – sequence, 111, 137, 293, 395 – set, 303 renormalization, 484 resonance, 103–105, 107, 108, 110, 111, 114, 116, 249, 255, 375, 439, 440, 442–444, 446, 452 rib, 82, 151, 169, 170, 192, 193, 195–197, 279–282, 284, 285, 292, 360 rigidity, 15, 48, 51, 57, 65, 171, 190, 231, 307, 309–313, 319, 320, 331, 417, 486 Ritz method, 104, 353–355, 417, 485 rod, 48, 50, 51, 53, 54, 57, 65, 101, 103, 105, 109, 111, 257, 267, 307, 333, 336, 338, 342, 468, 470 – flexible, 66 Runge-Kutta method, 92–94, 418 Schrödinger equation, 366, 470 secular term, 26, 78, 79, 106, 112, 113, 124, 125, 242, 249, 437, 449, 454, 455 self-adjoint, 50–54, 56–59, 62 self-localization, 449 separatrix, 442 series, 19–26, 29, 30, 33, 41, 47, 49, 51, 52, 58–61, 71, 73, 76–80, 83, 86–91, 95, 96, 102, 104–106, 111, 117, 119, 125, 126, 137, 143, 146, 175, 197, 249–251, 253, 254, 259, 261–265, 267, 270, 274, 277, 280, 284, 286, 289, 291–293, 295, 296, 301, 303, 308, 317, 318, 320–322, 325, 328, 329, 372, 377–380, 382–384, 390–392, 394–396, 398–404, 406, 408–411, 417, 422, 424, 426, 427, 464, 466, 470, 473, 475, 478–480, 486

534

Subject index

– asymptotic, 19, 20, 23–25, 27, 29–31, 71, 73, 77, 80, 99, 110, 123, 145, 160, 314, 320, 321, 323, 325, 328, 330, 331, 384, 397, 403, 417, 464, 474, 476, 477 – divergent, 143, 409 – polynomial, 25 shell, 58, 67, 82, 150, 151, 154, 164, 165, 169, 170, 174, 183, 190, 193, 195, 197, 216, 218–220, 222, 230, 231, 237, 256, 272, 345, 348, 372, 375, 486 – cylindrical, 65, 68, 150, 154, 157, 160, 169, 197, 212, 216, 218–220, 234, 345, 372, 375 – isotropic, 198 – orthotropic, 81, 151, 182, 195, 376 – reinforced, 172 – shallow, 150, 152, 167, 198, 202, 258, 486 – spherical, 153 – stringer, 190, 232, 233 singularity, 25, 75, 80, 122, 387, 402, 405–408, 427, 479 soliton, 457, 461 solution – approximate, 26, 77, 81, 117, 150, 154, 196, 197, 217, 248, 313, 334, 371, 383, 388, 390, 393, 397, 471 – periodic, 83, 95, 104 – perturbed, 388 – unperturbed, 78 spectrum, 48, 49, 135, 150, 185, 186, 222, 241, 260, 261, 265, 319, 321, 328, 330, 371, 372, 421, 424, 456, 466, 477 stability, 50, 53, 60, 61, 104, 172, 190, 191, 193–195, 239, 240, 356, 372, 452, 458, 459 – investigation, 189 stress-strain state, 15, 120, 133, 142, 146, 201, 203, 210, 213, 276 string, 435, 453, 456, 458, 475, 477 structure, 15, 26, 27, 29, 40, 71, 72, 103–105, 109, 111, 115, 116, 129, 250, 293, 321, 323, 329, 372, 483, 485 – period, 292, 308, 309, 319, 320, 364, 372 – thin-walled, 15, 16, 435 superposition, 105, 128, 336, 456, 459 support – elastic, 129, 363, 453, 456

– nonlinear, 453 – stiff, 150, 153, 154 symmetry, 15, 131, 297, 305, 311, 339, 344, 353, 358 system – continuous, 260–262, 346 – discrete, 128, 260–262, 467 – homogeneous, 159, 205, 380 – integrable, 438, 442, 457 – limiting, 65–69, 71, 131–135, 145, 161, 174, 183–188, 191, 220, 258, 279 – linear, 113, 371, 382, 439, 443, 444, 467–469 – non-autonomous, 249 – nonlinear, 69, 107, 110, 112, 244, 256, 444, 480 – vibro-impact, 91 Taylor series, 30, 105, 263, 318, 378, 380, 388, 478 tensor – strain, 256, 257 – stress, 141 theory – linear, 69, 140 – of shallow shell, 203, 209 – of shallow shells, 192, 213 time – variable(s), 346 TPPA(s), 417–422, 424 trajectory, 446 transformation, 37, 42, 51, 65, 68, 69, 71, 72, 95, 127, 139, 143–145, 159, 248–255, 263, 297, 367, 377, 378, 382, 384, 387, 408, 410–414, 443, 445, 447, 460, 465, 466, 479, 480 Van der Pol – equation, 383, 388, 418, 446, 447 – oscillator, 418, 420 variable – fast, 123–125, 267, 268, 273, 277, 279, 281, 282, 288, 293, 302, 303, 308, 314, 320, 441–443 – slow, 121, 123, 256, 267, 269, 281, 282, 302, 308, 314, 320, 321, 443 – small, 119 – spatial, 366 wavelength, 328, 346, 435, 459, 461

Subject index Young modulus, 82, 140, 151, 158, 170, 284 zone, 32, 33, 130, 133, 162, 183, 186, 187, 189, 192, 201–204, 206–208, 210,

535

215, 217, 219, 220, 225–227, 229, 234, 236–238, 278–280, 305, 333, 334, 347, 350, 358, 372, 411–414

Asymptotical Mechanics of Thin-Walled Structures.

Asymptotical Mechanics of Thin-Walled Structures.

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