Applications of fractional calculus to dynamic ...

Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids Yuriy A Rossikhin and Marina V Shitikova Department qf Theoretical Mechanics. Voronezh State Academy of Construction and Architecture ul Kirova 3-75. Voronezh 394018. Russia The aim o f this r e v i e w article is to collect together separated results o f research in the application o f fractional derivatives and other fractional operators to p r o b l e m s c o n n e c t e d with vibrations and w a v e s in solids h a v i n g hereditarily elastic properties, to m a k e critical evaluations, and thereby to help m e c h a n i c a l engineers w h o use fractional derivative models o f solids in their work. Since the fractional derivatives used in the simplest viscoelastic models ( K e l v i n - V o i g t , M a x w e l l , and standard linear solid) are e q u i v a l e n t to the weakly singular kernels o f the hereditary theory o f elasticity, t h e n the papers wherein the hereditary operators with w e a k l y singular kernels are harnessed in d y n a m i c p r o b l e m s are also included in the review. Merits and demerits o f the simplest fractional calculus viscoelastic models, which manifest t h e m s e l v e s during application o f such m o d e l s in the p r o b l e m s o f f o r c e d and d a m p e d vibrations o f linear and nonlinear hereditarily elastic bodies, propagation o f stationary and transient w a v e s in such bodies, as w e l l as in other d y n a m i c problems, are demonstrated with numerous examples. As this takes place, a c o m p a r i s o n b e t w e e n the results obtained and the results found for the similar p r o b l e m s using viscoelastic m o d e l s with integer derivatives is carried out. The methods o f Laplace, Fourier and other integral transforms, the approximate methods based on the perturbation technique, as well as num e r i c a l m e t h o d s are used as the methods o f solution o f the e n u m e r a t e d problems. This r e v i e w article includes 174 references.

3. FORCED VIBRATIONS OF THE HEREDITARILY ELASTIC OSCILLATOR WITH THE RABOTNOV KERNEL ................. 47 4. NONSTATIONARY WAVES IN"A HEREDITARILY ELASTIC ROD WITH A WEAKLY SINGULAR KERNEL OF HEREDITY ........................................................................... 48 4.1. Shock waves ...................................................................... 49 4.2. Stress waves ........................................................................ 50 4.3. Impulse load propagation in a hereditarily elastic rod with a singular kernel of heredily ................................. 51 4.4. Impact of a rod against a rigid barrier ................................. 52 5. HARMONIC WAVES IN A 3D HEREDITARILY ELASTIC MEDIUM WITH THE RABOTNOV KERNEL ......................... 54 5.1. Volume waves ................................................................... 55 5.2. Rayleigh waves ................................................................... 57 6. NONLINEAR WAVES IN ID HEREDITARILY ELASTIC MEDIA WITH FRACTIONAL OPERATORS ........................... 59 7. BOUNDARIES FOR THE FRACTIONAL CALCULUS APPLICATIONS ......................................................................... 61 8. CONCLUSION............................................................................ 63 REFERENCES .................................................................................. 64

CONTENTS 1. INTRODUCTION ....................................................................... 15 1.1. Definition of a fractional derivative ................................... 16 1.2. Simplest models of viscoelastic media with fractional derivatives and other fractional operators .......................... 18 1.3. Equivalence of viscoelastic models with fractional derivatives to the Volterra hereditarily elastic models with singular kernels of heredity. Distribution functions .......... 20 1.4. Some features of the fractional exponential function and its integral operators ................................................... 24 1.5. Behavior of hereditarily elastic models with weakly singular kernels under harmonic excitation ........................ 25 1.6. Application of numerical methods .................................... 27 1.7. Experimental data substantiating the harnessing of fractional derivatives and other fractional operators ......... 28 2. DAMPED VIBRATIONS OF VISCOELASTIC OSCILLATORS WITH FRACTIONAL DERIVATIVES AND OTHER FRACTIONAL OPERATORS ........................... 29 2.1. Generalized standard linear solid and MaxweU models .... 31 2.2. Generalized Kelvin-Voigt model ....................................... 32 2.3. General-purpose oscillator whose model contains fractional derivatives with two different fractional parameters ........................................................................... 35 2.4. First model with the fractional operator involving two independent fractional parameters .............................. 37 2.5. Second model with the fractional operator involving two independent fractional parameters .............................. 40 2.6. Effect of the fractional operator parameter on the region of dissipative processes with significant intensily .............. 41

INTRODUCTION The theory o f linear heredity put f o r w a r d by V o l t e r r a (1909, 1913) finds applications not only in m e c h a n i c s o f d e f o r m able solids, but in other divisions o f m a t h e m a t i c a l physics as well.

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16

Rossikhin and Shilikova: Applications of fractional calculus to dynamic problems

It is known that the integral relationships of hereditary elasticity are equivalent to the linear differential relationships with constant coefficients if the kernel is defined as the sum of exponential functions. At the same time during experimental data handling it turns out that the kernels involving one or several exponential terms are poorly appropriate for describing the features of actual materials. Certainly, inserting discrete creep and relaxation spectra, an experimental curve may be approximated with any degree of exactness (Bland, 1960; Ferry, 1961). However, it seems likely that a set of elastic and viscous elements is not a suitable model for solids with imperfect elasticity, and other functions should be chosen to describe its behavior (Scott Blair, 1944; Scott Blair and Caffyn, 1949; Stiassnie, 1979; Rogers, 1983; Makris and Constantinou, 1993). Boltzmann (1876), who was the first to formulate the principle of heredity, proposed the hereditary kernel in the form ct-', where t is the time and c is a constant value, which Possesses a peculiarity at t = 0. In experiments on creep, usually at the beginning of the process, ie directly after the loads have been imposed, the velocity of creep is found to be quite large. Moreover, it is too large to be measured without difficulty, so the process of loading should be considered as a dynamic process. The Boltzmann kernel gives an infinitely large deformation velocity at the moment of load application, however the peculiarity turns out to be unduly strong, since not only the deformation velocity, but as well the deformation itself become infinitely large. This flaw can be remedied by using the function cF -Y , where 0 < 7 < 1, proposed by Duffing (1931) ~s the creep kernel. This function is often written in the Abel form putting c = F-~(y), where F(y) is the gamma-function. The Volterra integral equations with the kernels having a weak peculiarity of the Abelian type have been used by Bronski (1941) and Slonimski (1939, 1961, 1967 I) for describing aftereffects in rubber and polymers. The simplest kernel of the Abelian type reproduces well the character of the material behavior at the first period after load application; however, damping of the deformation velocity sometimes occurs faster than it is observed in experiments. The quest for the integration of the features of the kernels with a weak singularity and the features of the exponential kernels has led the above mentioned authors to the construction of the following kernel: cg Y exp(-t'-Y). A somewhat different kernel of the same character has been proposed by Rzhanitsyn (1949), and its resolvent has been constructed by Wolfson (1960) and Koltynov (1966). The kernels retaining the peculiarity of the Abel kernel type, but allowing for greater pliability when describing properties of actual materials, have been suggested by Rabotnov (1948) and have been named by him as fractional exponential functions. The idea to insert the integral operators with kernels of the Abelian type which have a weak singularity into the relationships of the heredity theory may be interpreted as a generalization of the idea of viscoelasticity. This generalization is based on the fact that the integral relationships of hereditary elasticity with certain weakly singular kernels are equivalent to linear differential relationships with constant

Appl Mech Rev vol 50, no 1, January 1997

coefficients, wherein fractional derivatives are used instead of integer derivatives (Meshkov et al, 1967; Meshkov, 1967). On the other hand, the application of weakly singular kernels of different constructions in the Volterra integral equations allows one to generalize the idea of fractional differentiation and integration and to introduce into consideration new operators involving one or several fractional parameters, which may be called fractional operators. The mathematical aspects of the theory of fractional integro-differentiation of functions of one or several variables and applications of this theory to integral and differential equations are covered adequately in monographs by Oldham and Spanier (1974), McBride (1979), Ross (1975), Miller and Ross (1993), and Kiryakova (1994). In Russia one of the first books devoted to fractional calculus is the monograph by Samko et al (1987) translated into English in 1993, wherein the classical and advanced results of the theory of fractional calculus are systematically presented. This book has an encyclopedic character, encloses a great variety of known forms of fractional integration and differentiation, and contains a large number of references in this area up to 1986. ttowever, the questions connected with the application of fractional calculus to the dynamic problems of mechanics of solids have not been adequately addressed in the literature and are presented in not easily accessible publications on frequent occasions. Later sections are devoted to the application of fractional derivatives and other fractional operators in the problems connected with vibrations and waves in solids with hereditarily elastic properties. 1.1 Definition o f a fractional derivative

Fractional differentiation and fractional integration of functions are the generalization of common differentiation and common integration. These extensions were proposed by Liouville (1832) and Riemann (1876), and are performed easily and correctly in the space of generalized functions. For this purpose we introduce the generalized functionf~ (x) dependent on the real-valued parameter ct: [ H!~!X~-I,ot>O. 'fa = / r(~)

(1.1.1)

[ fi+' '°t- -or and n > - 13, we obtain

D~+u= F ( 1 ; y )

f(m) , .,~+, r(") =(so+~'s~+.) _ . . ~(m+.)= f(,,+,) •f 0 (as well as the Riemann-Liouville operator). Let us obtain the expression for the fractional derivative o f a certain function u to be convenient for appl!cations using the definition (1.1.1) and formulas (1.1.2). To accomplish this, we put the parameter ot = 1 - ~, > 0 (0 < y < 1) in (1.1.1). Yhenf~ =.ft-~ and

~ u(x- y)dy f

Approximating the function fi on the s e g m e n t j h _< x _< 0" + 1)h by the finite difference

~(t-~)

~ u"-s - hu._.;_l '

(1.1.11)

and substituting (1.1.11) into (1.1.10), from (1.1.9) we obtain

DVu =

1

d ' ,(x-.,,)a.,,

(1.1.10)

hvr( 2 -Y)

{(1-Du(o/+2(,._,_u._,_, " nr

U+I/'-'-/-'

j=o

1}.

(1.1.12)

or

d

' .(. :.,9+

D ~'u = -~x ! -~( i _ TTV r

For the higher-order fractional derivatives, when

(1.1.4)

7 = [Y] + {Y},

(1.1.13)

where D v u =.f_~ * u is the derivative of the order y of the function u. Formula (1.1.4) is the desired expression for the fractional derivative.

where [y] is the integral part of the number y, and {y} is the fractional part of the number Y, 0 < {Y} < I, we have

If the function is given within the segment [a,b], then in parallel with formula (1.1.4) the following relationships can be introduced:

¢ d ]" fx. u(y)dy ,o (x _ y),-.+, '

(1.1.14)

u(y)dy

(1.1.15)

D~+u-

1

r(1-r)

d ix ~_y)dy

(1.1.5)

_

O~u

1

(-1)" ( d ' ] " f b

= r ( . - r ) t d x ) * 1,

ie. for a standard linear solid, the distribution of the relaxa-

0.25

tion and retardation times is described by the Dirac 8function. By this means the weakly singular distribution functions (1.3.7) correspond to the weakly singular Rabotnov kernels (1.3.2). This is clearly evident from Fig 6a, wherein figures near curves indicate values of the magnitude y.

0.5

~/~i

1

1.5

2

b) t.o

l.O -0.3 0.75

0.75

0.5

0.5

0.25

0.25

D

I

0

0.5

I

Z/Za

1

!

1.5

2

Fig 5. Distribution function Ao(x,%)of the creep times for the Abelian kernel at % = 1. Figures near curves indicate the magnitudes of the parameter y.

0

I

I

I

0.5

1

1.5

2

Fig 6. Distribution function (a) A,(x,%) of the relaxation (creep) times and (b) A,(s) of logarithms of the relaxation (creep) times for the Rabotnov relaxation (creep) kernel at x, = 1


Appl Mech Rev vo150, no 1, January 1997

Rossikhin and Shitikova: Applications of fractional calculus to dynamic problems

22

The first formula from (1.3.10) was obtained by Cole and Cole (1941) when investigating dispersion of dielectric permeability. Reference to Fig 6b, where figures near curves indicate values of the parameter y, shows that plot of the function Ai(s) represents a symmetric curve which peaks to A~ ax(0) = tan(ny/2)/2n at s = 0. If now we turn from images to pre-images, then as a result we obtain the Volterra relationships with singular kernels of heredity

a) 1.0

[

,

]

~(t)= J= ~(t)+ v a SK~(t-t')a(t')dt' , --oo

(1.3.11)

a(t)=Eoo[~(t)-ve!=Ke(t-t')~(t')dt'], where the Rabotnov's 9t-functions (1.3.2) are used as the resolvent kernels Ko(t) and Ke(t), and the integral operators entering into Eqs (1.3.11) are also called resolvent ones. Note that the creep and relaxation functions for the hereditarily elastic models with the Rabotnov's kernels have been obtained by Caputo (1966, 1967) and Caputo and Mainardi (1971). These functions can be expressed in terms of the Mittag-Leffler functions. From formulas (1.3.1) one can obtain the expression linking the Laplace transforms of the creep kernel Ka(P) and the relaxation kernel ffTe(p), namely, [veK~(p) ] - [ v o K o ( p ) l

,.,0.75 Oi5

]- "t'+

-v t" ( t -t') "¢-' a(t'ldt'], (1.3.13)

~=.,~=,~==~=~=,~,i

I

~0"1

I

0.25

0

0.5

I

' ( t-t'~.,., ~=Jo!~-l,-~-~(t) dt ,

0.75

x/%

b)

(1.3.12)

In the specific cases when the material behavior is given by Eqs (1.2.6) and (t.2.7), the linkage between the values o and ~ or a and o is written, respectively, as

~0.5

O.25

=I.

(1.3.14)

1.o

;(,-,')-'

,,-

,l

Let us now consider the after effect kernel suggested by Rzhanitsyn (1968) whose Laplace transform has the form

0.75

ga (p)= (1 + p%)-~. 0.5

(1.3.15)

Considering that 0.5..

d'K'o = (_ 1)" z:y(y + 1)...(y + n - 1X1 + p%)-(~+")

dp"

0.25

o/ ~.~.__.~.~.i~ 0.25

s

,

,

0.5

0.75

j

Fig 7. Distribution function (a) A,,(x,%)of the creep times and (b) of the creep times for the Rzhanitsyn creep kernel

and using the formula (Widder, 1941)

we obtain the expression for the original of this kernel

Ads) of logarithms




l T-1

K~(t) = zv~y ) e-t/~°.

(1.3.16)

23

like sequence. Really, using approximation of the function

A~(z,x~) at T ~ 1 The rheological Eq (1.2.11) can be written for the kernel (1.3.16), but the rheological Eq (1.2.12) is associated with the kernel

K~(t)=tv-lx~v[F(y)]-lexp(-t/xE).

AE(z,%)_ X/~'H(% - x)

x ~ ' r c ( 1 - y)

(r-vo)2 +Tvo 2(1- )

(1.3.17) and going to limit at T ~ 1 in this expression yields

Applying the procedure described above in detail for finding distribution functions to the distribution function of the retardation spectrum corresponding to the kernel (1.3.16) yields (Davidson and Cole, 1951) A=(z,%) = sin rot

17Y-1 _ z)v_l H ( Z o - z).

(1.3.18)

a)

4

The behavior of Ao(z,z~) and A~(S) at various magnitudes of the parameter ~¢is shown in Figs 7a, b. According to Eq (1.3.12) the resolvent kernel of relaxation in the space of transforms has the form -1 Ve

(1.3.19)

,E

1+ v;'(1 + pZo) r"

2

Using the shift theorem and the definition of a fractional exponential function, one can find (Wolfson, 1960; Koltynov, 1966)

10.7

g,(t) = v~t exp|---/-7-:7 E x (I.3.20)

oxp 0.25 Integral presentation of fractional exponent (1.3.5) enables to determine the form of the distribution function for the corresponding relaxation spectrum (Meshkov, 1974) b)

~Ve, 2COS~,+Vc(.~n,~-I_i) -'¢ + v~l(,o.~ ~-1_1)'"

0.5

0.75

~/~

1.5

(1.3.21) For small values of z, the function A~ possesses Abelian asymptotics defined by Eq (1.3.8). Plots of the functions A~(T,%) and At(s) = x At(z,%) for the case vo = 1 are presented in Figs 8a and b, respectively, for various magnitudes of the parameter T. The distribution function of logarithms of relaxation times peaks to A~aX =

1. tan ~ under the condition vo = ~ 2rtv~ 2 z

1

0.5

1,

It can be shown that when T ~ 1 the distribution function (1.3.21), along with the distribution function (1.3.7), is a 8-

I

0

0.25

S

0.5

!

,

0.75

1

Fig 8. Distribution function (a) AE(z,z~) of the relaxation times and of logarithms of the relaxation times for the Rzhanitsyn relaxation kernel at v,~= z~= 1

(b) At(s)



24

the Volterra integral operators. To decipher them, the simplest algebraic operations on the operators with the fractional exponential kernels should be established. 9~(f~) signifies the Volterra integral operator with the

, ~/Tvo , , lim d~(~,x~) = H ( z o - x ) - - 8 ( T - v o ) = y-~l ~V~

V~o

kernel

where T = ~x ~ - 1. Thus, in the limiting case, the distribution of relaxation times is described by the 8-function with the evident fulfillment of the normalization condition, since at T= v~ or ~ = % we obtain that H(~¢~ - %) = 1, v,~v~-' %~o-~ = 1. The generalized models (1.2.9) and (1.2.10) can be represented in terms of the expressions (1.3.1t) as well, if we apply the Laplace transformation to Eqs (1.2.9) and (1.2.10) and use the inversion theorem for the first sheet of Riemannian surface (Fig 4). As a result, we obtain the relation (1.3.11) with the kernels expressed in terms of the distribution functions 1 ~' 1:-2 sin 13q9i t Ki(,) = ~ ' ~e x p ( - ; ) ~ ¢ , (i =~.,~),


( 1.3.22)

where

~n ( l -- ,t;)n('+°)

9a (IB,t - x) = (t - x)a n~o~[~-~'l~ T~-)], - 1 < ct < 0,(1.4.1)

ie. t

~,~

(f~)u = Ig,~(~,t-~)u(~)d~.

(1.4.2)

0

Then, as it has been shown by Rabotnov (1948), the relationship ~

* (x) ' ~a ()

(y)

y =

x-y

,

(1.4.3)

fulfills, what constitutes the multiplication theorem for the operators ~a • If we put x = y in Eq (1.4.3), then we obtain the operator •2

0 9~ (x)

0x tancPi

=sinfirc[cos6~z+('c/'~i-'.)~]

At ~5= I and 13 = y, formulas (1.3.22) transform into formulas (1.3.6) with the distribution function (1.3.18) both for the creep and relaxation kernels (for the relaxation kernel, the value to is needed to be replaced by % in formula (1.3.18). Thus, it may be generally deduced that all kernels discussed above possessing the peculiarity of the Abelian type at t = 0 are consistent with the functions of the relaxation and creep times distribution, which also possess the peculiarity of the Abelian type but now with respect to the relaxation and creep times at ~ = 0.

1.4 Some features of the fractional exponential function and its integral operators Reference to Eqs (1.3.2) shows that if the integral operator in the Volterra equation of second kind (1.3.10) is the operator for whose kernel use has been made of one of the functions (1.3.3), then the resolvent of the integral equation will be one of the functions (1.3.3) as well. Exponential kernels possess the same feature. Hence the fractional exponential function combines flexibility of the exponential function and weak singularity of the Abel function which is observed at the initial stage of deformation of actual viscoelastic materials. That is why the fractional exponential function is of frequent use when solving various problems of hereditary elasticity. The problems, which are solved using the Volterra principle to the effect that the solution to a hereditarily elastic problem is obtained from the solution of the corresponding elastic problem with the Volterra integral operators in place of elastic moduli, could result in some combinations of

(1.4.4)

'

which has as its kernel a new transcendental fimction resulting from differentiating of the series (1.4.1) with respect to the parameter l~. The extension of the formula (1.4.4) obtained by Rozovsky (1963) has the form 3,-1 * •.

1

9a ( x ) = ( n - l ) !

% (x)

0x"- 1

(1.4.5)

We shall now highlight one more formula useful for calculations (Rabotnov, 1980)

l+~t2 9~ (22) =1+ 2, -x2 +~2

(2,)(1.4.6)

,2(2, -22 + ' 2 - ' , )

(22_,2)

21 -- 22 q"~2 Algebra of the operators 9~ (1~) based on the relationships (1.4.3)-(1.4.6) has been developed by Rozovsky (1959, 1961, 1967) and his collaborators (Krush, 1965; Kmsh and Rozovsky, 1965; Rozovsky and Sinaisky, 1966) and has been successfully applied for solving numerous static and quasistatic problems of the theory of hereditary elasticity. Here, these papers will not be considered, since their review is available in Rozovsky (1967). Ifi conclusion of this paragraph it can be noted that the tables allowing one to calculate the fractional exponential function and its integral are presented in Rabotnov (1980). These tables represent a contraction of more extended tables




(Rabotnov et al, 1969). They contain the magnitudes of two auxiliary functions

1

l!

where x = 13t'÷~, J3 > 0. Series for the fractional exponential function converge rather slowly, so for the preparation of tables the representation in the form of a series was used only for the magnitudes x < 1. When x > 4, the asymptotic formulas (Annin, 1961) nt

l+~

)

25

Following Meshkov (1967), let us carry out calculations for the complex compliance of weakly singular kernels. Choose the Abelian kernel as the creep kernel. Substituting Fourier transform of the Abelian kernel into formulas (1.5.1), we obtain the following magnitudes for the real and imaginary parts of the compliance, respectively:

J'= J~ + A/(0~*o) -~ cosvL, (1.5.3) J"= &,./(oyez)-~ sinvL, VL = rty/2.

-n

The vector diagram of the complex compliance in terms of relative units f ' =f'(j'), f ' = J"/A/, j' = (J'- Joo)/A/

t

is the straight line emerging from the origin of coordinates at the angle ~L

"f'"o gave a good approximation with rather small number o f terms taken into account. Finally, for the interval 1 _< x _< 4,

f' =ftan~v The loss tangent is equal to

use was made of rearrangement of series improving their convergence. 1.5 Behavior of hereditarily elastic models with weakly singular kernels under harmonic excitation To investigate the behavior of hereditarily elastic media under periodic excitation, it will suffice to put the lower limit of the integral from relations (1.3.11) equal to -oo, and to substitute the value Ooe'°'' instead of o(t) and the value c,e"" in place of e (t) in the first and second equations of (1.3.11), respectively. As a result we obtain

~(t)=aoJe i'°',

a(t)=%

J = J~[1 + voKo(im)] =

lg imt z~e ,

tan 8

.

( 1.5.5)

J'= Jo~ + ,5.I(1 + (02T~)-Y/2COSq./, (1.5.6) J"= ,5./(1 + m 2X2oj~-'/2sin V, V = Yarctan(mXo). The tangent of the mechanical loss angle is equal to tan 6 =

A/sin ~

(1.5.7)

•& ,(1+c02~~)~a + ~ c o s v

E=E~[I-v~K~(im)]=E'+iE", where K~(im) and J~(im) are Fourier transforms of the creep and relaxation kernels. Hence, the compliance J and the elastic modulus E are defined by complex numbers. After their multiplication by o0 and ~0 the real terms o,,J'and ~oE' represent those parts of the strain and the stress which change cophasally, but the imaginary terms are displaced from them in phase by 90 °. The presence of the terms displaced in phase governs dissipative processes in a hereditarily elastic medium. The magnitude of energy losses (internal friction) is determined in terms of the loss angle tan 8 =E'/E' = d"/J'.

A/sin ~ L

For the Kelvin-Voigt and Maxwell models, the complex compliance and the loss tangent are given in Smit and de Vries (1970). Using the kernel (1.3.16) in formulas (1.5.1) yields

(1.5.1)

J'-iJ",

(1.5.4)

Figure 9 presents the vector diagram of the complex compliance j" = f'(/") for the Rzhanitsyn kernel which is determined by the equation f ' = f tan[y arctan(0)~o)].

(1.5.8)

0.5 .-0.25

(1.5.2)

More detailed data on the application of weakly singular kernels for the description of internal friction are presented in Meshkov et al (1966), Meshkov (1970a), and Meshkov and Pachevskaya (1967).

0

j,

0.5

1

Fig 9. Vector diagram of the complex compliance •i" =.1.tt,-U.. t ) for the Rzhanitsyn kernel


26

Rossikhin and Shitikova: Applications of fraclional calculus to dynamic problems

The value y is chosen as a parameter. As it is seen from Fig 9, each curve is the asymmetric half-circle, which is held up against the j'-axis as V decreases. In the region of high frequencies, Eq (1.5.8) goes over into Eq (1.5.4). At ¥ = l relations (1.5.6) describe a standard linear solid whose vector diagram has the form 2 (j,__21__)+ j,,2=l. (1.5.9) Figure 10 illustrates the frequency dependence of the real j' and imaginary j" components of the compliance. It is seen that decreasing in the parameter y decreases the peak of the value j" and displaces it in the direction of high frequencies ms compared with the value for a standard linear solid which reaches its peak of the magnitude of 0.5 at ooxa = 1. A decrease in y results in more smooth changes of the dynamic compliance. For the Rabotnov's aftereffect kernel (1.3.3), it is an easy matter to obtain from (1.5.1) the following

J'=Joo+&l

(s,_½) 2


1

2

+ (J"+'2 c°t~l/) = 1--csc2 VL4

(1.5.11)

with the central angle of?~ and the radius r = 1/2 csc(ny/2),

(1.5.12)

that is the radius of the circle diagram is governed by the fractional parameter ? only. Further it is not difficult to find that the angle ~L = ~ / 2 controls the slope of the tangent to each arc about the abscissa axis at the points 0 and 1. The tangents in themselves in the diagram correspond to the Abelian kernel. From Fig 12, wherein the frequency dependencies of the real j' and imaginaryj" parts of the compliance are depicted for the Rabotnov creep kernel, it is seen that the parameter 3' causes fuzzifying of the retardation (relaxation) spectrum. The complex modulus has been investigated in Bagley (1987), wherein the frequency dependence of its real and imaginary parts is given. Finally, for the creep kernel (1.3.22) from (1.5.1) we have

(OXn)-'¢ + COSq/L (m%)v + ( o z ~ ) - r + 2COSVL ' sin~L

s"=

+ 2cos L

0.5

, ~L = n?/2, (1.5. lO)

0.25

AJsin~L

tan5 =

Jo(o*oV +

+(+o + J : ) c o s v /

Full symmetry of formulas (1.5.10), which at ? = 1 transform to common relationships for a standard linear solid, is worthy of notice. Figure 11 presents the diagrams j" = j"(f) for various which are the circle diagrams by Cole and Cole (1941)

j

0.25

j,

0.5

0.75

1

Fig 11. Vector diagram of the complex compliancej" =f'(f) for the Rabotnov creep kernel

!

O.75

0.75

0.5

0.5

0.25

0.25

0 -,

-2

0

2

4

Fig 10. Frequency dependence of the real.[ and imaginary./" components of the compliance for the Rzhanitsyn kernel

0 -4

-2

0

2

4

ln(c0%) Fig 12. Frequency dependence of the compliance components




J ' = J~ + zk/r -n cos13~, J"= zk/r -~ sin 13~, tan 8 =

(1.5.13)

,SJr -Is sin 13~¢ J~ + z~Jr-1~cos13~¢ '

where 8

' g"'

"~8

r = ~l+2(ox~) cosS-~+(ox~,)- , •

~

8

tan~ = ( o x ~ ) ~ s i n S ~ [ l + ( o % ) c o s S ~ ] -1. The diagram f ' = j"(/') which is determined by the equation { (t°%)~sinS~" -,]l j " = j ' t a n 13arctan l+(¢o,,,)~cosfi~.]/

(1.5.14)

is presented in Fig 13 for the parameter magnitudes indicated in the caption to the figure. The shape of the curve obtained is as follows: in the range of high frequencies, it is linear and is described by the equation n J" = J "tan VL, VZ = 8 13~-,

(1.5.15)

which is similar to Eq (1.5.4) for the Abelian kernel, but in the range of low frequencies, it is the segment of arc whose equation is similar to Eq (1.5.8). At fi = 1 and 13 = 1, Eqs (1.5.13) and (1.5.14) goes over into Eqs (1.5.6) and (1.5.8), respectively. Thus, the examples discussed above point to the fact that the peculiarity of the kernels of the type (t - t3v' (0 < ~, < 1) governs the angle % = ~ / 2 or % = 813n/2 according to which the curve of the vector diagram at to ~ oo cuts the abscissa axis, where the real magnitudes of the compliance (modulus) are plotted. When ,/= 1 or 8 = 13= 1, the peculiarity is absent, and intersection occurs at a right angle. 1.6 Application of numerical methods Numerical methods are of frequent use in the analysis of damped vibrations of an oscillator. Bagley and Torvik (1983) were among the first investigators to apply numerical methods when considering dynamic problems of viscoelasticaUy damped structures. The authors note that the use of numerical methods in the transform domain to describe the frequently0.3 dependent mechanical properties of viscoelastic materials is cumbersome at 0.2 best. The major drawback is the substantial effort required to calculate nu- "~ merically the transport inversion inte0.1 gral for every point in time at which the response of the structure is needed. Finite element method

0

27

viscoelastic model when considering equations of motion of a two degree-of-freedom system. The finite element representation of this system is used after transforming to the Laplace domain. The fifth-order eigenvalue problem was solved resulting in 10 complex eigenvalues (resonances) and I0 complex eigenvectors (mode shapes) of the equations of motion. Bagley and Torvik (1985) used the same approach to determine the transient response of a simply supported damped beam with a constrained layer. The satisfactory agreement between the results obtained through continuum and finite element formulation confirms the effectiveness of the special numerical procedure developed for the solution of the matrix equations. Tsai (1994) has investigated vibrations of the construction based on viscoelastic dampers whose elastic properties are temperature dependent, but viscosity is described by the fractional derivative with the fractional parameter equal to 0.6. The elastic and viscous elements are connected in parallel what corresponds to the generalized Kelvin-Voigt model. The finite element method is used as the method of solution, resulting in close agreement between the theoretical and experimental data. Morgenthaler (1991) has developed a procedure and numerical algorithms which can be used in the design and analysis of structures with materials described by fractional derivative models, and has provided an accurate reducedorder state-space form which can be used to design highauthority modern control systems and predict system performance. The method proposed relies on an iterative solution of the differential equations of motion in the Laplace domain. This method is used in conjunction with the subspace iteration eigensolution procedure to develop an efficient numerical algorithm for the solution of large fractional derivative eigenproblems typical of those which may be encountered in realistic structural applications with vibration control requirements. The author based in his procedure on the results byBagley and Torvik (1986), Bagley and Calico (1989), and Gaudreault and Bagley (1989). Boundary element method

Gaul et al (1989) have investigated free damped vibrations of the oscillator described by the generalized Kelvin-Voigt model, in so doing the use was made of the Fourier transformation method. The solution obtained in the form of im-

!

0.2

0.4

j,

0.6

0.8

1

Bagley and Torvik (1983) used the fractional derivative standard linear Fig 13. Vector diagram of the complex compliancef' =.f'(/") for the model (1.2.9) at 8 = 0.8 and 13= 0.3 Downloaded 06 Aug 2009 to 140.118.196.65. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm


28

proper integral was analyzed then by the numerical method. Based on this result, further Gaul and Schanz (1994) and Schanz (1995) develop the boundary element method for the calculation of elastodynamic response in time domain for constructions whose viscoelastic behavior is described by taking fractional order time derivatives into account. Viscoelastic properties are introduced after Laplace transformation by means of an elastic-viscoelastic correspondence principle. The transient response is obtained by inverse transformation in each time step. As an example, the wave propagation in a viscoelastic bar is calculated numerically.


1.7 E x p e r i m e n t a l data s u b s t a n t i a t i n g the h a r n e s s i n g o f f r a c t i o n a l d e r i v a t i v e s a n d other f r a c t i o n a l o p e r a t o r s

As it has been noted, the weakly singular kernels of heredity provide an infinitely large creep velocity at the moment of the application of loading and are presumably bound to describe adequately the initial stage of this process. A large body of experimental data presented by various authors is evidence in favor of this assumption. Thus, for example, Rabotnov (1980) presents the experimental data relevant to creep of a fiber glass reinforced laminate and compares them with the theoretical data obtained with the formula

Finite d~erence method

t

Currently the fractional difference method is gaining acceptance. Thus, Koh and Kelly (1990) explored the problem on free damped vibrations of the oscillator described by the generalized Kelvin-Voigt model involving different numerical methods for its solution. The authors develop efficient numerical multi-step schemes based on the central finite difference method and the shifted Ll-algorithm. In doing so, as the comparative analysis shows, the solutions obtained by numerical methods on the basis of the Laplace and Fourier transformation methods are in good agreement with each other. When applied to shaking table tests of a base-isolated bridge desk, the fractional derivative model is found to agree with the experimental results. The same procedure was used by Suarez and Shokooh (1995b), wherein the numerical results were compared with those obtained by the Laplace and Fourier transform methods. The fractional difference method has been advantageously applied by Podlubny (1995) for solving boundary-value problems of hereditary mechanics with weakly singular kernels o f heredity. a)

1.5

-- 1.25

1.0

i

0

20

0

40

,

,

40 60 t, hours

,

80

I00

b) 1.0

d~

0.75

0.5

i

|

80

i

120 t, hours

i

160

200

c(t) - ° °

l+k

~

(1.7.1)

at 5 = -0.7. The findings of a comparison are given in Figs 14a, b, where solid lines are constructed in terms of the formula (1.7.1) at E = 3.46. 106 Mpa, 13 = 1.096, k = 0.905 (Fig 14a) and E = 2.34 • 106 MPa, 13 = 0.540, k = 0.405 (Fig 14b), but dots have been obtained experimentally. From Figs 14a, b it is seen that the experimental and theoretical data are in good agreement. Alternatively, the presence of the fractional parameter offers additional possibilities for more flexible approximation of experimental findings by theoretical calculations (Scott Blair et al, 1947; Scott Blair and Caffyn, 1949; Scott Blair and Coppen, 1942; Scott Blair and Reiner, 1950; Nutting, 1921, 1943, 1946; Gemant, 1936, 1938). Moreover, as investigations show, the value 5 plays the role of a certain structural parameter responsive to the change in crystal lattice of a material owing to the processes connected with machining of this material. Thus, as an example, Merrill et al (1960) have shown that during X-ray exposure of a polyethylene the vector diagrams of its complex modulus E' + iE" are the arcs of the circles whose radius is dependent on a dose of irradiation, such that in appropriate deciding on the magnitude of the fractional parameter the experimental dots fall on the circles of the radius (1.5.9) (Figs 15a, b, and c). In studies of dispersion in low-molecular organic or inorganic compounds by the dielectric method, complex dielectric permittivity is customarily measured as a function of frequency under constant temperature. Further the temperature is taken, and the cycle of the frequency measurements is repeated. For each set of the isothermic measurements, the real part of the complex dielectric constant is constructed as a function of its imaginary part for all frequencies (the construction is conducted in the complex plane). In the construction of this sort, the two characteristic forms of dependencies - the form of an arc segment (1.5.11) or the form of an asymmetric half-circle (1.5.8) - are observed. Then depending on the form of a curve, the dispersion parameters are determined and their association with the peculiarities of molecular structure of the material is estimated.

Fig 14. Comparison between the experimental and theoretical data relevant to creep of a fiber glass reinforced laminate Downloaded 06 Aug 2009 to 140.118.196.65. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Appl Meeh Rev vol 50, no 1, January 1997


Although this method of the notion of experimental data is widely applied for simple low-molecular systems, it is rather rare in use for the analysis of dispersion data for polymers on two counts. First, the dispersion in polymer systems extends into a wide region of frequencies, therefore when deciding the dispersion there is a need to combine the data obtained at different temperatures. Second, the form of the dispersion obtained in a complex plane is rare in agreement with the simplest characteristics inherent in low-molecular compounds what renders the evaluation of the dispersion data impossible. However, Havriliak and Negami (1966, 1969) show that for experimental data handling during the investigation of the a-dispersion for five polymers (polycarbonate and polyisophthalat on the basis of bisphenol A, isotactic polymethyl methacrylate, polymethylacrylate, and copolymers of phenylmethacrylate and acrylonitrile) the relationship (1.5.15) can be used, which constitutes the general formula for the three known dispersions: the Debye dispersion (1.5.4), the arc segment and the asymmetric half-circle. The complex dielectric constant for poly(n-octyl methacrylate) calculated on the basis of this relationship and experimental data are presented in Fig 16a-c, from which it is seen that the theoretical and experimental findings agree closely with each other. Other examples showing good agreement between theoretical and experimental results obtained by the use of weakly singular kernels of heredity can be found in Rabotnov (1966, 1980), and in the book by Tseitlin and Kusainov (1990), where many significant results by Rabotnov are presented.

29

2 DAMPED VIBRATIONS OF VISCOELASTIC OSCILLATORS W I T H FRACTIONAL DERIVATIVES AND O T H E R FRACTIONAL OPERATORS The renewed interest to the viscoelastic models and their application to dynamic problems (Suarez and Shokooh, 1995a, b; Rossikhin and Shitikova, 1995, 1996a-e, Rossikhin et al, 1996a, b) centers around the elaboration of new damping systems in engineering and technology based on continuum of damping elements distributed uninterruptedly throughout the relaxation or creep times instead of the discrete system of damping elements (Mace, 1994). Free damped vibrations of a hereditarily elastic oscillator, whose hereditary properties are described by a rheological model containing fractional operators, have been treated first by Rozovsky and Sinaisky (1966). The model of a standard linear solid has been used as the generalized model. The authors sought the solution by the Laplace transformation method, in so doing inversion of the Laplace transform was based on a knowledge of the roots of a characteristic equation with fractional powers. Rationalization of the characteristic equation was made through substitution for the parameter of the integral transformatio n , whereupon the Laplace transform was decomposed into common fractions. As a result of inversion, the solution was represented in

a) ~,0.5[

a) E"

2.5

3.0

c'(o0)

3.5

4

b) 0

1500

2OO0

E,

0.5

b) 2.5

3.0

3.5

c'(co) I

0

1000

e)

E'

1500

c)

0 2.5

1000

1500

2000

E'

Fig 15. Vector diagrams of the complex modulus E' + iE" for a radiated polyethylene at (a) "~= 0.39, (b) 3' = 0.26, and (c) y = 0.18

I

3.0

~'(~0)

3.5

4

Fig 16. Complex plane plot of the complex dielectric constant for poly(n-octyl methacrylate) at three temperatures: (a) T = 6.1 °C,/5 = 0.808, 13= 0.426, (b) T = 21.5°C, 6 = 0.843, 13= 0.455, and (c) T = 57.6°C, 6 = 0.91, 13= 0.49


30

Rossikhin and Shitikova: Applications of fractional calculus Io dynamic problems


terms of linear combination of Rabotnov's fractional expo- be written in the two equivalent forms in terms of the renential functions which were dependent upon the time and laxation kernel K~(t) and the creep kernel K~(t), respectively, the real and complex roots of the rationalized characteristic .~+~0® x - v e Ke(t-t'xt')dt' =F8 t, (2.1) equation. The solution constructed in such a manner is poorly amenable to analytical treatment, since for its realization tabulation of fractional exponential functions of complex variable, which has not jet been carried out, is required, although the tables for fractional exponential functions of real variable have long been in existence (Rabotnov et al, Here x is the coordinate, F is the amplitude of force im1969). pulse per unit mass, and o)~ is the frequency of elastic vibraFree damped vibrations of hereditarily elastic bodies pos- tions corresponding to the nonrelaxed magnitude of the elassessing finite or infinite degree of freedom with the use of tic modulus. the generalized Kelvin-Voigt and standard linear solid modApplying the Laplace transformation to Eqs (2.1) and els have been studied by Bagley and Torvik (1979, 1983b, (2.2) yields 1985), Torvik and Bagley (1984, 1985), Suarez and Shokooh (1995a, b) by the Laplace transformation method. To con+ p 2 [ l + v o g o ( p ) ] " (2"3) struct the solution, the authors also used replacement of the parameter of the integral transformation for rationalization of the characteristic equation with fractional powers. However, The solution in the space of inverse transforms is deterafter determining the roots of the rationalized equation the mined according to the Mellin-Fourier inversion formula solution has been constructed in terms of the theory of resix(t) = 2,r--7 l lc+'°° dues on the first sheet of Riemannian surface (the complex c-;~, ~(P)eptdp" (2.4) plane with the cut made along the negative real semiaxis). A limitation of this method is that almost for all magnitudes of To calculate the integral in (2.4), it is necessary to find all the fractional parameter the order of the rationalized equa- singular points of the complex function Z(p). The weakly tion is rather high, so that the analysis of the roots of this singular kernels K~(t) and Ko(t) discussed above have the equation is found to be hampered. Thereafter in the abovebranch points p = -s., s. >_ 0 and p = -oo and the ordinary enumerated papers, the order of the fractional derivative was poles at the same magnitudes o f p which vanish the denomichosen equal to 0.5 resulting in rather simple rationalized nator in the formula (2.3), ie they are the roots of the equacharacteristic equation. Suarez and Shokooh (1995b) point tions: out that when the fractional derivative order is distinct from p2 + 0~2[1_ v~g~(p)] = 0, 0.5, then calculating difficulties become insuperable. A radically new way of looking at the analysis of free (2.5) damped vibrations of oscillators described by the fractional 2 +p211+ v e g a ( p ) ] ,-~ 0. 0)oo calculus Maxwell and standard linear solid models, which is devoid of the enumerated limitation, has been proposed by For multivalued functions possessing branch points, the Rossikhin (1970) and extended by Zelenev et a! (1970, inverse transform theorem is applicable only for the first 1971, 1972) and Meshkov et al (1971). The Laplace transsheet of Riemannian surface, ie when -r~ < arg p < ~. Thus formation method was also applied in these papers, but the the closed contour should be chosen in the form presented in nonrationalized characteristic equation, ie the equation with Fig 4. Due to the Jordan lemma, the curvilinear integrals fractional powers, was used for calculating the roots. It has taken along the arcs c R tend to zero at R ~ oo. For weakly been shown (Sinaisky, 1969; Rossikhin, 1970) that the charsingular kemels, the integral taken along cp also tends to acteristic equations for the above-enumerated generalized zero when p ~ 0. models have no real roots but possess two complex conjuUsing the main theorem of the theory of residues, the sogate roots located in the left half-plane of the complex plane. Upon determining the roots of the characteristic equation, for lution to Eqs (2.1) and (2.2) may be written as whose calculation a highly efficient method was developed x(t) = by Rossikhin (1970), the solutions were constructed on the ~ I o [~(se-'~ ) - ~(se"~)]H(s - s*)e-s' ds + Zres[2(Pk )eP~t]' first sheet of Riemannian surface with the use of the theory of residues. Below, when considering free damped vibrations of sin(2.6) gle-mass systems described by the generalized viscoelastic models, we shall primarily follow the papers by Rossikhin where the summation is taken over all isolated singular and Shitikova (1995, 1996a-e) which generalize the ap- points (poles). proach proposed by Rossikhin in 1970. Due to the hereditarily elastic Volterra relationships (1.3.10), the equation of motion of a single-mass system can 1

oo

.

.

k




2.1 Generalized standard linear solid and Maxwell models

For the fractional derivative standard linear solid model (1.2.5) (the Volterra relationship (1.3.11) with the kernel (1.3.3), the formula (2.3) takes the form (Gonsovski et al, 1972a)

F ( p ' +1, I r ( p ) = pr+2 + r,p2 + to~,pV + too21¢'

1¢ = ~e ~', (2.1.1)

where too is the frequency o f elastic vibrations corresponding to the relaxed magnitude o f the elastic modulus. To define the poles o f the function (2.1.1), we find the roots o f the equation (2.1.2)

pr+2 + top2 + ¢o2p r + c02x = 0.

To calculate the roots o f the set o f Eq.(2.1.4), we intro2 duce the new variables x~ = r , x 2 = 1¢r2v, and x~ = ~Kr~. Then for every fixed angle n/2 < x¢ < r~ at given ,[ and ~, we obtain the following system o f equations: x I c o s 2 ~ + x 2 cos(2 - 'y)W + x 3 cos'~w + 1 = 0,

xlsin2~g+x2sin(2-y)~g-x3siny

]-'

--0,

and put p = re "v (0 < ~ < 1/2 rr). As a result, separating real and imaginary parts, we obtain r2 c o s 2 ~ + to2~(1- v ~ R ~ c o s * t ) = 0, (2.1.3) r2 sin2x¢ + to2~v~R~l sinO~ = O, where

R~=[l+2rVx:cos3'xc+r

tanO~ =

2

Vx2~v]

1/2

,

rVx~sinyv 1 + r V ~ cos'y~¢"

Since tan O~ < tan ~ and hence O~ < 3,~ < rq,/2, then the second equation o f (2.1.3) is different from zero at every magnitude o f r and 0 < ~ < ~/2, which proves the given proposition. To define the complex roots o f Eq (2.1.2) for r~/2 < W < r~, we reduce its left side by pV and put p = re 'v in the resulting equation. Then separating real and imaginary parts yields r 2 cos2 v + Kr 2-v cos(2 - y ) ~ + {~.-v cos'y~ + 1 = 0, (2.1.4) r E sin 2W + r r 2-~ sin(2 - y)~g - ~lcr -v sin y~g = 0, where ton = 1, ~ = Eo/g,~ = 1 - v~.

(2.1.5)

First we express x, and x 2 in terms o f x, from the first two equations o f the system (2.1.5) Xl=

sin(2 - y)~¢ sin2~ sink + xssin~' (2.1.6)

x2=

what is contradictory to the input assumption. To prove the fact that Eq (2.1.2) lacks complex conjugate roots in the right half-plane of the complex plane, we rewrite it in the form

~ =0,

XiX 3 = ~X 2.

It may be shown that Eq (2.1.2) has not any real negative root. Really, putting p = -y, y > 0 and separating real and imaginary parts yields y ' ( . v 2 + to~)sinrry = 0 ,

31

sin2~ siny~g

xs

sin(2 + 3,)~ sin'Bg '

and after substituting (2.1.6) into the third equation of the system (2.1.5), we are led to the quadratic equation in x3:

x32 + Ax 3 + ~ = 0, A = sin(2 - )')V + ~sin(2 + Y)V

(2.1.7)

sin2~ As calculations show, Eq (2.1.7) defines two real positive roots, for which there are two magnitudes o f x~ and x 2 according to Eqs (2.1.6) and hence there are two magnitudes o f r = X~/2 and x = x2rY-2, ie for each fixed magnitude o f the parameter K there exists only one set o f the values r and satisfying the system o f Eqs (2.1.5). In other words, the characteristic Eq (2.1.2) in the left half-space o f the complex plane has two complex conjugate roots, whose behavior as a function of the parameter K is shown in Figs 17a-d for ~ = 1 and four magnitudes o f ~ = 0, 1/50, 1/9, and 1/6, respectively, where figures near curves denote the magnitudes o f the value y . It is seen that the K-dependence o f the two complex conjugate roots P~.2 = -ct 5: ito o f the characteristic equation at 1, ~ 1 leave the points 5: i and converge in the points 5: i~ ''~ when K changes from 0 to ~; in so doing it does not meet the real negative semiaxes and remains inside the curves for the K-dependencies o f three roots o f the characteristic equation with y = 1 (the ordinary standard linear solid model). The behavior of the two roots o f three at y = 1 (the third root is the real root all the time and changes from 0 to oo as x varies from 0 to oo) essentially depends on the magnitude of the value ~. Thus, at the values ~ taken from the interval (0, 1/9) the two complex conjugate roots first become real as 1 1/9, the domain of aperiodicity completely disappears (Fig 17d). Thus, for the fractional derivative standard linear solid, the domain of aperiodic motions does not arise, ie the frequency 0~ ¢ 0 at all magnitudes 0 _< K < oo and 0 < { < 1. Knowing the behavior of the roots of the characteristic equation and considering that the branch points are s~ = 0 and -0% the solution (2.6) can be written in the form x ( t ) = Ao(t) +

A = 2F

Aexp(-ott)sin(ot-(p)

(2.1.81)

[(

a 2 +b2)-a(K: 2 + r 2"t + 2K:r"t cosyv

tang =

K c ° s l 3 + r r c°s(lg-YV) Ksinl3 + r r s i n ( p - 3n¥) '

,

(2.1.9)

tanl3 = b--,(2.1.10) a

a ---

The In % dependence of the values In et and co is presented in Fig 18 for ~ = 1/50, from which it is evident that at every magnitude y # 1 as % varies from 0 to 0% the damping factor passes a maximum and the vibration frequency increases monotonically only when y < 0.85. With y > 0.85 as % increase, the vibration frequency first decreases from ~,,2 to some value and then increases upto 1 with further increase in %. At y ¢ 1 and y = 1 the behavior of the temperature dependence of these and other values was studied in Rossikhin and Shitikova (1996a) and in Meshkov et al (1964), respec= tively. Note that at ~ = 0 (Eo = 0) the generalized standard linear solid model transforms into the generalized Maxwell model (1.2.6) or (1.3.13), for which the behavior of the roots of the characteristic equation as function of K is shown in Fig 17a. The solution for this model is written in the form (2.1.8), where

F[

(2 + y)r l+v cos(l+ Y)V + 2K:r c o s y + Y0)Zrv-1 c o s ( y - 1)V, A=

Kr-~(1-1/4y)+cospg] -uz,

(2.1.13)

1 - y _----ra 7 ~_~-_,r.--7 +--2cosy-"--~"

b= tang =

(2 + y ) r '+r sin(1 + y)~¢ + 2,(r sin v + yo)~,rr-1 sin(y -1)V,

2(Io "-r + K-lr ~ + 2COSgt)COSW - y[Kr -~ cos w + cos(1 - y)gt] where r ~ = o 2 + a~, tan W = - ooC ~. The function Ao(t) describing the drift of the equilibrium position of the single-mass system may be represented in the form (1,3.6), ie

Ao(t) = If'z-IB ("~,1¢ ) e-t/~d'r.

(2.1.11)

2(va "-v + K-lr ~ + 2 COSW)sin W - y [Kr -v sin W + sin(1 - 7)W] ' (2.1.14)

Fo,)L~3 [ 0 ~ ('ff)] -1

sinyr~ g

Y

-1

(2.1.15)

+2cos

Here the function B(~,K) = 2

siny~

1

2.2 Generalized Kelvin-Voigt model

-1 3

v~o,,F[0~(z)]- [0o(z)] z 1

7~

=

-y

+ l,

00(,)

-1

y

'

+ i.

For the fractional calculus Kelvin-Voigt model (1.2.7) or (1.3.14), the formula (2.3) takes the form 2"(p) =

(2.1.12) gives us the distribution of the relaxation parameters of the dynamic system. Note that in the quasi-static case, when % ---) oo and 0)~ ---) oo, the distribution function (2.1.12) transforms into the distribution function (1.3.7) which corresponds to the Rabotnov's kernel. Thus, relationships (2.1.8)-(2.1.1 0) define the damped vibrations around the drifting equilibrium position with the natural frequency o and the damping factor cc Let us trace the temperature dependence of the values ct and co. Since for the majority of the relaxation processes z~ = %exp(U/RT), where U is the activation energy, R is the characteristic gas constant, and T is the absolute temperature, then the In % dependence of physical values is equivalent to the temperature dependence.

F

2 ~

p2 +,~pr +O)Zo' ~.=Oo%.

(2.2.1)

It can be shown, as it has been done in Sec 2.1, that the characteristic equation p2 + KpV + 0)2 = 0

(2.2.2)

lacks real negative roots as well as complex roots in the fight half-plane of the complex plane. To find the complex roots belonging to the left half-plane, we put p = re'v in Eq (2.2.2). Then separating the real and 2 imaginary parts, and introducing the new variables x, = r and x: = v,.rv, for every fixed angle r:/2 < ~ < rt at given y and o2o we obtain a system o f two linear equations in two unknowns x, and x 2 x I cos2w + X 2 cosytl/.-t- (020 = 0,

(2.2.3)

x I s i n 2 ~ + x 2 s i n ' w = O.




After finding the magnitudes x, and x~, the values r

=

X~/2

and K = X~I=v are calculated, which together with the chosen govern one root o f the characteristic equation. With -V in place o f tg , one obtains its complex a) conjugate root. The behavior o f roots PJ.2 = -or 5: i o in the complex plane as function o f the parameter K at co02= 1 is presented in Fig 19, where the magnitudes o f the value 3' are indicated by figures. It is seen that as the value K changes from 0 to o% the curves for the K-dependence o f the roots P,,2 issue out o f the points :t: i and grow monotonically and indefinitely for 'y _< 0.85, in so doing rapidly come close to the corresponding asymptotes leaving the origin o f the coordinates under the angles ~L = -+ n/(2-

I

33

and hence the period o f vibrations decreases, although for viscoelastic systems with ordinary viscosity the reverse picture is evident. The solution for this model is written in the form (2.1.8),

b)

0

1

°i

i

3')- At y < 0.85 the d) curves increase mo- e) -ot_~../o -~_t_/00 notonically, but when 3' > 0 . 8 5 , as lc increases, the curves for the 1c-dependence o f P~.2 first envelop the arcs o f the unit circumference (the curves for the ordinary Kelvin-Voigt model), and then 0 1 during further increase in K the curves p~,~ rapidly come close to the corresponding asymptotes. Thus, some anomaly is observed in the behavior o f the roots o f the characteristic Eq (2.2.2), namely, the vibration frequency increases -i ~ -i with increase in the Fig 17. Behavior of the complex conjugate rootsp,. 2 = --a ::t ioo for a single-mass system based on the fractional damping coefficient, calculus standard linear solid model: (a) ~ = 0, (b) ~ = 1/50, (c) ~ = 1/9, and (d) ~ = 1/6

1 . 0 ~

1.q


34


.(~.,.)

where A = 2 F 4 r z + y2K2r z(Y-1) + 4y~:r ~ cos{2 - Y)V

=

sin ~y

, (2.2.4)


F~[0 0 (1:)]-1

(2.2.6)

[Oo(~)]-' ~ - ' c ,-2 + Oo(~)~ 2-, ÷ 2cos,~," tan~0 =

2 r c o s w + yva-r-I cos(1 - T)V 2 r s i n v - yIc:rv-I sin(1 - y)W '

a)

(2.2.5)

For the generalized K e l v i n - V o i g t model, the b e h a v i o r o f the main characteristics o f the vibrating m o t i o n can be found in Rossikhin and Shitikova (1996a) and Suarez and S h o k o o h (1995a, b).

1I

-1

0.5 -2

-3

1.0 -4

0.8 /

0.91 1.0

-5

I

-6

~

t

/

-5

(

I

-4

I

I

-3

-2

I

I

-1

0

~t

I

1

2

ln~

b)

1.0 1

0.98

0.75

0.5

0.25

0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 -6

I -5

I -4

- - ~ -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I -3

I -2

-1

I 0

I 1

2

In xe Fig 18. The In xE dependence of the values (a) In e~ and (b) o7 for a single-mass system based on the fractional calculus standard linear solid model at ~ = 1/50



Rossikhin and Shitikova: Applications of fractional calculus 1o dynamic problems

2.3 General-purpose oscillator whose model contains fractional derivatives with two different fractional parameters

Ib = COo,

+ ~D~

= E0(~ +x~D I~ ~ ) , 0 < ~,13 _< 1,

{:'

(2.3.1)

which is the immediate generalization of the standard linear solid model (1.2.5). For the model (2.3.1), the formula (2.3) takes the form

~(P)=

pZ[1+(p,~)n]+~Oo~[l+(p,~.)l ]" ~

(2.3.2)

To obtain the poles of the function bar x[p), it is necessary to find the roots o f the characteristic equation 2.fl.,fl+o)20=0. p 2+6 r~&+ p 2 + ~OO,,rF

~0 = r~/2,

12o= 0, I~ ~5 the curves representing the xo (%) dependence o f the characteristic equation roots have asymptotes emerging from the origin of the coordinates at the angles Woo> n/2 to the horizontal axis, to which these curves approach without bound at ~ ( % ) oo. When I3 < 8, the lines issuing out the origin under the angles ~ < n/2 to the horizontal axis are the tangents to the curves presenting the %(,~) dependence of the roots at the point 0, to which these curves approach from the right halfplane. For calculating the complex conjugate roots o f the characteristic equation (2.3.3a), l e t p = re'v in it, separate the real and imaginary parts, and introduce new variables x 1=r

(2.3.3a)

2+8 ~5

%,

x2

= r2

, x3=rl3~.

(2.3.7)

Then for the each fixed magnitude o f 0 < Ivl < n at o% = 1 and given 6, 13 and ~ = ~8z o-~5 , we obtain the following set o f

or

p2[1

a

I~ -~

equations in x~, x z, and x 3

One can show that Eq (2.3.3a) has no real negative roots, if put p = -y, y > 0 in it. Then for the imaginary part we obtain the equation

4/ ± i~

'" ,

0.4

y2+8.~~ sin r~6 + o ~ 2 y l ~ sin r~13= 0,

3i

whose left side vanishes at none of the magnitudes y > 0, x~ and ~,, > 0, 0 < 8, 13 < 1. For the qualitative analysis of the complex conjugate roots o f the characteristic equation, letp = re~' in Eq (2.3.3b) and divide the real and imaginary parts. As a result we obtain

r z R~R2' cos(2 v

',

~'~.~.~.~.9

+ * , - • 2 ) + o~2 = 0,

~

2

i

""-..

(2.3.4)

rZR~R2~sin(2 v

0.98

+ qb~ - O2) = 0, 5

where

&=

~

2

~

......................................... _..d ............

3"

0.98 r

~

, . . e -~-

,-i

//"

0.9

R E = 41 + 2rl~x~ cos13v + r2~x~13,

,~

.1 ~

/"

z""

r~¢~sin 6W tan@ l = l + r ~ ¢ ~ c o s 6

r~x~ sin 13W

.J

, t a n O 2 = 1 + rl3x~ cos13v

,"

,J

/

-2i

/

#

t -3i

It follows from the set o f Eqs (2.3.4) that

r 2 = o~R2R~-1,

2V + * 1 - • 2 = ±re.

(2.3.5) 0.4

Tending % (To) to 0 or oo in Eqs (2.3.5), we obtain the corresponding limiting values of ~ and

-4i

Fig 19. Behavior of the complex conjugate rootsp, a = -e~ =1:io~ for a single-mass system based on the fractional calculus Kelvin-Voigt model



36

Table

~, 1/50

13 0.98 0.8 0.8 0.5 0.5 0.5 0.98 0.8 0.8 0.5 0.5 0.5 0.98 0.8 0.8 0.5 0.5 0.5

1/9

1/6

1. The limiting values of % and % 8 ~_, s ~, s 1 4.722 236.1 41.37 0.764 0.98 34.65 0.693 1 70.202 0.528 0.8 19.33 0.357 0.98 17.4 0.348 1 88.43 9.825 1 l 1.84 1.258 0.98 10.1 1.122 1 9.4 0.603 0.8 3.67 0.39 0.98 3.44 0.382 1 66.87 11.145 1 8.25 1.325 0.98 7.07 1.179 1 5.35 0.57 0.8 2.37 0.381 0.98 0.373 2.24 1

co. s' 6.569 4.325 4.192 2.766 2.375 2.35 2.838 2.184 2.151 1.706 1.638 1.637 2.328 1.868 1.844 1.539 1.516 1.517

x 1cos(2 + 8)V + x 2 cos2~g + x 3 cos13~g + 1 = 0, x 1 sin(2 + 8)V +

xz s i n 2 v

+ x3 sin13v = 0,


From Figs 20, it is seen that at 13 > 8 the roots o f the characteristic equation with the change in % behave similarly to the roots of the characteristic equation for the generalized Kelvin-Voigt model (see See 2.2), ie for the model lacking simultaneous elasticity, such that the curves increase without bound at % ~ oo rapidly approaching its asymptotes. When 13 = 8, the roots of the characteristic equation with change in % from 0 to a certain magnitude r~ behave similarly to the roots of the characteristic equation for the generalized Maxwell and standard linear solid models (see Sec 2.1), ie for the models possessing simultaneous elasticity. In so doing r~ = oo when fi = 13 = y andr~ is a finite magnitude when 13 < 8 (see Table 1). I f % > r~, then at 13 < fi the roots of the characteristic equation fall into the right complex haft-plane and the solution loses its meaning. In other words, this model under certain magnitudes of the fractional parameters 8 and 13 may describe both the behavior of viscous fluids and the behavior of viscoelastic solids, ie it is a general-purpose model. Knowing the behavior of the roots o f the characteristic equation, the solution (2.6) can be written in the form (2.1.8), where

(2.3.8)

A=

2F~(h 2 + q2)-'(1 + 2 z ~ ' cosS~ + z2~'r 2') (2.3.11)

=

tango :

From the first two equations of (2.3.8) we express first x~ and x~ in terms o f x,

sinz + ,~d"~ sin(z + By) h cosz + % r 8 cos(z + B y ) ' tanz = --, q

sin2v sin(2 - 13)V xl - sin&g + sin&g x3'

Ao(t ) : IoSB( S) e x p ( - St)dS,

(2.3.9) x2 = -

sin(2+8)~ sin&g

sin(2+6-~)V sin 8~

x3'

~-IFR1 *-1S-3 s i n ( ~ - ~ )

and then substituting (2.3.9) into the third equation of (2.3.8), we are led to the equation in the real positive x 3 s i n 2 v + X3 sin(2 -13)V + ~xg/13

B(S) =

o);2S2Rl*R; -1 + o)o2S-2Ra*-IR; + 2co

s(

*:~ - O?

)'

where (2.3.10)

Ri* = R1 v=~,

[sin(2 + 6 ) ~ + x3 sin(2 + 6 - 1 3 ) ~ ] = 0. Determining the value x~ from Eq (2.3.10) and substituting it into formulas (2.3.9), we find the corresponding values x, and x r Thereafter from (2.3.7) we define r =x~/2 and =

The behavior of the roots of the characteristic equation in the complex plane as function of the parameter % is presented in Figs 20a-c for three magnitudes of the value ~ = 1/50, 1/9 and 1/6, respectively. Figures near curves denote the magnitudes of the fractional parameter 6.

h = (2 + 8)rl+~¢~cos(1 + 8)V + 2 r c o s v + (o~z~r13-'13cos(1 - 13)V, q = (2 + 8)r1+8 @~sin(l + 8)V + 2r sinv - co~x~rO-113sin(1 - [3)V. Expressions (2.3.11) determine damped vibrations around the drifting position of equilibrium A,(t) with the natural frequency co and the damping coefficient ct. The character of the solution x(0 behavior is presented in Fig 21, where the figures near curves denote the magnitudes of the fractional parameter 8.


Appl Mech Rov vol 50, no 1, January 1997

Rossikhin and Shilikova: Applications of fractional calculus to dynamic problems

that the characteristic equation

2.4 First model with the fractional operator involving two independent fractional parameters

For the model whose behavior is described by the rheological Eq (1.2.9) or by the Volterra relationships with the hereditary creep kernel (1.3.22), formula (2.3) takes the form

~(p)-

F{[l+(PX~)~]f~+v~} p2+

2

(2.4.1)

~ fl

37

-

(p2 + m2~)[1+ (px~)~ ]1~+ p2v~ = 0

(2.4.2,

do not possess real negative roots and complex roots lying in the fight half-plane of the complex plane. For finding the complex roots located in the left halfplane of the complex plane, we setp = re'v (n/2 < IV[ < ~) in Eq (2.4.2), and separating real and imaginary parts yields

As is the case with the previous models, it can be shown a)

14

~

/ /

12

,]

/

/ /

'""""'",

'""""

/'

//

6i

10

/ ,,'

/

10

b)

I

'/

6

"

'"

4

i

"'",,

'"

2 -4

-3

-2

-1

0

I

2

0.3

"'"""'''I1 -2.5

-2

-1.5

-1

-0.5

0

c) -

i - 2

-

2

0

J

-1.5

-1

-0.5

0

0.5

Fig 20. Behavior of the complex conjugate roots -et+io~for a general-purpose oscillator model at % = 1 for: (a) ~ = 1/50, 13= 0.8, (b) ~ = 1/9,13 = 1, and (c) ~ = 1/6, 13= 0.8 Downloaded 06 Aug 2009 to 140.118.196.65. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

38


Rossikhinand Shitikova: Applications of fractional calculus to dynamic problems

tour -2 cos2~+

vcrRJ~cosfl~cr+l = 0,

where R~ =Rolv= ~, O ~ = ~ v = ~ ,

(2.4.3)

(o2~r-2 sin 2 7r +

vcrRJ3sin ~ r = 0,

where

a)

Ra=~] l+ 2r~r~c°s&qt+r2t~t2~' tanOa=

0.35 0.3

r,~t~sin&w 1 + r 6 r 6 o c o s 61~' '

Further we multiply the first equation of the system of Eqs (2.4.3) by sin2~ and the second one by cos2~, and subtract the second equation from the first one. Introducing the new variable x = (rxo)8 and putting ¢ooo= 1 yields

0.25 ~'~ 0.2 ~ 0.15 0. 1

sinZw + v~ (1 + 2x cosSw + X 2 ) -13/2

0.05 sin[2u/-13arctan(-xsinS~ )]=0. 1 + xcosS~

0

(2.4.4)

k./ i

-0.05 0

From Eq (2.4.4) at every fixed angle r~/2 < IWI < ~ and given magnitudes of vo, 8, and 13, we determine the value x. Then substituting the found magnitude o f x into the second equation of (2.4.3), first we find

r=

5

t.lT~

i

t

10

15

10

15

~_

J

10

15

20

b) 0.4 0.5 0.3

I _ R o ~ sin2v vo sin13~ '

0.2

and thereafter we can calculate the value

"7

0.1

~cr = X n/ar-n.

The behavior of the roots in the complex plane as function, of the parameter ~o is presented in Figs 22 for three magnitudes of vo = 49, 8 and 5, respectively, when 13= 0.98. Knowing the behavior of the roots of the characteristic equation (2.4.2), the solution (2.6) can be written in the form (2.1.8), where

0.98 0.8

-0.1 0

5

-t

20

e)

0.4

0,3 (2.4.5)

tan q~

I

v°sinz+R~sin(130~+Z) tanz h vo co~z + Ro0~o~(rSOo + Z)' =--, q

0.2

'\

'7

(2.4.6)

0.1

e(~:o)-FTt-' to2x5[0~,(x)]-1sin [~ ~, 0 o o ( ~ ) ~ , , ;,

+[o~o(~)]-1 ~ .~ vo +2~os~o; , (2.4.7)

0".5 -0.1

,

- -

0

5

t~t

20

Fig 21. Character of the solution x(t) behavior for a general-purpose model at (a) ~ = 1/50, (b) ~ = 1/9, and (c) ~ = 1/6 for 13= 0.8 Downloaded 06 Aug 2009 to 140.118.196.65. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm


h = 2rR cos(v


The investigation of the behavior o f the values determined by formulas (2.4.5)-(2.4.7) can be found in Rossikhin and Shitikova (1996c, e). The character of the solution x(t) behavior is presented in Fig 23, where the figures near curves denote the magnitudes of the fractional parameter 6. As it has been noted in See 1.2, when 6 = 1 and 13 = y, the model under consideration goes over into the first model (1.2.11) with the Rzhanitsyn operator for which the behavior of the characteristic equation roots resembles the character o f the roots behavior for the generalized standard linear solid model (see Fig 19) with the only difference that in this case

+

+ rl+Srl~z~ R~ -1 cos[03 -1)O a + (6 + 1)~g] 2 6 6-1 13-1 +cooo613z~r R. c o s [ ( ~ - l ) O ~ + ( 6 - 1 ) t g ] + 2 r v , costg,

q = 2rR~ sin(~ +130.) + •~ , + ~ u pn~.c~r ~. Pa- ,

39

sin[(13-1)cl),~ +(~5 + 1)V]

2 6 5-1 +oJ=613x~r R~-1 sin[(13-1)cl)~ + ( 6 - 1 ) ~ g ] + 2 r v ~ s i n v .

a)

b) 1

1

0.8

0.8

/

0.6 S 0.4

0.2

I

0

-1

-0.8

0.6

.8

.8 0.4

1

0.2

I -0.6

I -0.4

I -0.2

I

-1

l

-0.8

-0.6

P

-0.4

I

-0.2

0

c) 1

0.8 0.8 0.6

0.5

1

S

°~

0.4

0.2

0 -1

I -0.8

I

I

-0.6

-0.4

-0.2

0

Fig 22. Behavior of the complex conjugate roots -et + io) for the first model with the fractional operator involving two independent fractional parameters at m~ = 1 for: (a) v~ = 49, (b) v,~ = 8, and (c) v~ = 5 when I3 = 0.98



40

the noticeable asymmetry is observed in the behavior of the roots. This circumstance is pictorially illustrated by asymptotics of the characteristic equation (2.4.2) at ~5= 1 and 13= y p2+r+p2vo.~;v+o)2pV=0

(%>>0'

yp3+p2x;,(l+vo)+ypc02+0020x~,=0

A=2F~(h~+q2)-'(Ro2V+2Ro~vocosycDo+v~o) , vosinz+R~sin(Tvo+Z)

tanz

v o cosx + Rovcos(y~Vo + Z)' B(z,~o) = sinn3'

2.5 Second m o d e l with the fractional operator involving two i n d e p e n d e n t fractional p a r a m e t e r s

For the model whose behavior is described by the rheological Eq (1.2.10) or by the Volterra relationships with the hereditary relaxation kernel (1.3.22), formula (2.3) takes the form

(% + oo, but on the left and right boundaries of the aperiodicity region it goes uninterruptedly into the amplitudes H 3 and H~, respectively. Within the aperiodicity domain, the amplitudes H 3 and Ht approach asymptotically the right and left bounda-

(2.6.10)

0.355 < % < 0.599

0.98

8/9 or 5/6

% = 1.955 0 < x~ < 0.543 0.819 < % < oo

0.885 < x~ < ~o 0.5

% = 0.881

(2.6.9)

~ = 0.885

0.8

0.881 < % < 0.885

(2.6.8)

0 _> 1),

where

(5.2.3)

(i = 0 , ~ ) ( 5 . 2 . 9 )

.2 -2 vary in the The constants N, > 0, since the values cacti

and

(5.2.4) Knowing the complex function k(im), one can determine the propagation velocity c and the damping coefficient a in the direction of the Rayleigh wave propagation from the formulas:

c(m)=m/~k(im),

or(m) = Uk(im).

(5.2.5)

Let us take the Rabotnov fractional exponential functions (1.3.3) as the kernels of the bulk and shear relaxation, give due consideration for Eq (5.1.7) in formulas (5.2.3) wherein put % = z~ = % for simplicity, and then substitute c,2 and ct2 for

(5.2.6)

>> 1),

(5.2.7) and the wave number k is ex-

The value r I must be real and positive one, with q < 1, such that n, and n, are to be real values. Equation (5.2.2) has only one root satisfying these conditions. For a hereditarily elastic medium, the form of the relations (5.2.1) and (5.2.2) remains the same. However the values entering in these relations will take on a some what different meaning: the squared velocities c 2 and c 2 and hence

1

No~(io)xe) -v (mx~

c, are the velocities of

pressed in terms o f some intermediate value rl(k = m/c,q) which in its turn obeys the algebraic equation 'q6--8114+8"q2 3

(oo~ 0 from a nonlinear hereditarily elastic material, for which the association between the response o(t) and excitation ~(t) is written in the form of a series of the Volterra-Frech6t multiple-integral relationships

0.8

0.6 "~ 0.4

0 0.5 0.2 -0.015

0 -0.01

-2

,

0.925

0193

0.935

0'.94

0.945

r I' Fig 36. Vector diagram for the complex function rl(ico). The figures near curves denote the magnitudes of the fractional parameter 3'

0

2

4

6

In (mx~) Fig 37. Frequency dependence of the amplitude of forced vibrations for a standard linear solid (~, = 1) at the fixed value ofx. The figures near curves denote the magnitudes of %


60


given by the formula

( A / A o ) 2 = {1+ ~otAo 3 2 [1- exp(-13x)]} -1 exp(- 13x), ('~-v 13= ¢0c-1v~,o0z~!

sinai/L ( y ~ l ) ,

(6.5)

13= v~(cx~)-' (y ¢-1), where d 0 = r~0E~ 1 is the initial amplitude. From (6.5) it follows that at y = 1 the amplitude A tends to a constant value with increase in the frequency m, but at y , 1 it asymptotically tends to zero. This fundamental difference in the behavior of the amplitude agrees with the results presented in Section 5. The character of the amplitude behavior as a function of the dimensionless parameter 00% for various magnitudes of the fi'actional parameter (Fig 37) confirms the asymptotic character of the amplitude behavior at y = 1 and 3' ~ 1 when o) ----~ oo,

A similar problem for the nonlinear hereditarily elastic medium, which is described by the following rheological equation (Rabotnov, 1966): O3

cy(x,t) = I G(t').f[~(t-t')]:h',

(6.6)

o


tem's dynamic characteristics, such as amplitude, phase, dynamic rigidity, square of hysteresis loop and quality factor, was investigated. Among other things it has been established that the reciprocal of the quality factor, which is taken as the measure of internal friction, is independent of the vibration amplitude and coincides with the result by the linear theory. Other dynamic characteristics are found to be sensitive to the nonlinear features of the given system. The Rabotnov fractional exponential functions have been chosen as the concrete hereditary functions. Byrdin and Rozovski (1984) considered the same problem as Astafiev and Meshkov (1970), but as the method of solution the authors used the Fourier transformation method developed for double and triple integrals by Nakada (1960) and generalized for n-fold integrals by Byrdin and Rossikhin (1979). The weakly singular Rzhanitsyn kernel was taken as the relaxation kernel. The pursued investigations testify that the asymptotic behavior of the damping coefficient of the traveling wave is similar to that of the traveling wave in the linear hereditarily elastic medium with weakly singular kernels, ie at y = 1 the damping coefficient tends to a finite value when m ~ oo and tends to infinity at 3' # 1. Sugimoto et al (1983, 1984a, b) and Sugimoto and Kakutani (1985) investigated waves in a one-dimensional nonlinear hereditarily elastic medium whose behavior is described by the following system of equations:

G(t) = Eoo[8(t ) - v~K~(t)], was considered by Karkovski and Meshkov (1976b). For the method of solution, the asymptotic method was used with the assumption of smallness of nonlinear and inelastic effects. The calculations have been carried out under the assumptions that the relaxation kernel is the Rabotnov fractional exponential function and the nonlinear function f has the form (Blitshtein et al, 1970) f(~) = (l+cl~ a-') ~

(6.7)

where c > 0 and ~t > 0 are the parameters of the system's nonlinear properties. Calculations showed that the asymptotic character of the amplitude behavior at m --~ oo and various magnitudes of the parameter y is analogous to the character of the amplitude behavior for the same boundary-value problem but with nonlinearity in the form of the series of the Volterra-Frech4t multiple-integral relationships (6.1). Thus, in the first approximation of the solution to the problem under consideration, the medium models due to the Volterra and Rabotnov theories produce similar results. Stationary forced vibrations of a single-mass system executing motion under the action of a monoharmonic force were considered by Meshkov (1970b). The role of a restoring force was played by a nonlinear function having regard to the heredity effect in the form of the series (6.1). The problem was solved by the method of equivalent linearization due to Krylov-Bogolubov (Bogolubov and Mitropolski, 1963), in so doing triple integral of a hereditary type was taken into account. The influence of nonlinearity on the sys-

K = ( L + 2 " ) E + I E 2 + O ( E 3 ) + v O V E / O t r +O(vE2)

(6.9)

(0 0 , K(t) is the kernel of heredity, 13ois the density, and y is a small parameter taking into consideration the effect of viscosity. Beginning from the moment t = 0 the boundary x = 0 of the half-space x > 0 is loaded so that its initial velocity and acceleration are equal to v0 and a0, respectively. The ray method has been used as the method of solution. As it has been shown, impact loading of the hereditarily elastic half-space boundary gives rise to the shock wave in the half-space, ie the geometric surface on which stresses and strains have a discontinuity, in so doing the shock wave velocity is as follows G = c(1 + btq),

K, = o0{ 1-(aob + 13)t},

(7.3)

where ~:, = [Ou/Ot] = (Ou/Ot)+- (3u/30 is the discontinuity of the value au/Ot, b = (or - k)k"(2e)", e 2 = k9o 1, the signs + and denote that the value Ou/Ot is calculated immediately ahead of and behind the wave front, respectively, and [3 = K(0)(Zk)". Reference to Eq (7.3) shows that at [3 < -ha o (b < 0 ) the value v:, increases with time, but when 13 > -ba o the value v:, is damped out to zero in the finite time lapse between 0 and t", where t' = ([3 + bao)" . If the kernel of heredity possesses weak singularity, ie, I~ --+ co, then damping of the discontinuity K, takes place in an infinitely short time interval, and in that case G = c. Really, in the nonlinear hereditarily elastic media with weakly singular kernels of heredity, the shock waves cannot propagate in the form of geometric surfaces of strong discontinuity. It seems likely that the impossibility of the propagation of strong discontinuity geometric surfaces in the linear and -


nonlinear hereditarily media with weakly singular kernels of heredity is associated with the idealization of the experimental fact, which lies in the fact that at the moment of the application of loading the strain velocity is assumed to be not simply large but infinitely large. Certainly, the impossibility of the existence of discontinuities and hence the impossibility of the application of such an effective method of solving dynamic problems as the ray method severely narrows the circle of the resolving problems wherein the fractional calculus models are used. However, if to obtain further insight into the problem, then it turns out that the presence of fractional operators in rheological models does not annihilate but only fuzzifies the wave front, which in media with integer operators propagates as the geometric surface of discontinuity. In order to demonstrate this fact, we turn to the problem considered above (Burenin and Rossikhin, 1990), but contrary to the assumptions made by Burenin and Rossikhin (1990) we suppose that in such a medium the shock layer of thickness h propagates with the velocity G = c, in so doing within the layer the functions a, au/& and au/at change monotonically and uninterruptedly from the magnitudes a ÷, (au/&) ÷ and (Ou/&) + to the magnitudes c , (au/&) and (au/at). To construct the solution within the shock layer, eliminate the stress ci from Eqs (7.1) and (7.2). As a result, we obtain O2U OU ¢~2U 020 _2¢~2U OU O2U I ~+2K-- ? 0 - - ~ - T = c /-77T + - OX2 OX OX2 t Ol Ol Ot~X )'

(7.4)

where yo = ?k", • = Jo'K(t - t')u(t')dt', K(0) = 0% and vc= o.k". Using the conditions of compatibility which link derivatives in the two coordinates systems: the stationary coordinate system and the moving coordinate system traveling together with the layer

Ou

l

=

Ot

Ou

1 C [

8u

~

On

,

fit

Ou

Ou

O2u

On

~x 2

= l ,

Ox

=

O2u On2 '

(7.5)

axat = - c an--T + & L an )'

02u

02u

5 ¢Ou~ ~2u

Ot 2 = c 2712-2c-~7~,~n)-~1 / 8t 2, where O/On is the derivative with respect to the normal to the wave layer, and 6/8t is the Thomas &derivative (Thomas, 1961), from Eq (7.4) we obtain

02U

32u

Ou

On2

0#1 Oil2

020 0#12 -- c

-2f 2 O2u - 2 - 8 (Ou'~ ~c On2

2(_cOU+fiUy_cOZU+

+t

__

62u 8t 2

6 Ou)l

a,, 8t Jr, a,? ~-gT,,jj (7.6)

Let us reduce Eq (7.6) to the dimensionless form introducing the following dimensionless values: w = %2aou, 0 = -I -3/2 -1/2 1/2 -112 oo aot, Z = Oo c aon, and e = uo c . A s a r e s u l t , wehave



OZ `92.2

`gZ2

80 ~ `gZ )

_ 2a 2 _~__8 ( 5~_l _ 2e 2 8w `92w ÷ 2a3 8w

5o t `gz j

Rossikhin and Shitikova: Applications of fractional calculus 1o dynamic problems

802

`gZ `9.7,2

8 (owl C77J'

5o `9z

(7.7) where O' = ~o°K(0 - 0')~(0')d0', v = 3'02, and z = Ooao". Considering that e is the small value, we seek the solution of Eq (7.7) in the form (7.8)

w = wo+ ewe+ e~w2+ ...

Substituting Eq (7.8) into Eq (7.7) and limiting ourselves by the zeroth term of the series (7.8) yields

l) `gwo `92wo + ± ( `gwol _ az 2

&

• 02

60 t & )

(7.9)

e0

where v' = v(2e)". Carrying out the change o f the variable o = (x - 1)`gwJ3z in Eq (7.9) yields `9 o V*~zI0 K(0-0')u(0')d0'..

(7.10)

Equation (7.10) describes variations in the value v within the shock layer which moves with the constant speed c. Integrating Eq (7.10) with respect to the normal to the wave surface from -1/2n = o o3,2c ,aao" h/2 to 1/2n = 3/2 1/2 -1 o o c a 0 h/2 and passing on to the limit at n ~ 0 , we obtain

(7.1 l) v * ~0° K ( 0 - 0*)[u + (0') - u - (0')]d0', where (6o/60)' is a certain magnitude of the value 6u/60 within the length between -n/2 and n/2, ~+ and u" are the magnitudes o f the function u calculated immediately ahead o f and behind the shock layer, respectively. Knowing the magnitudes of the values u + and u" at the instant of time 0, as well as previous history of variations in these values on the time interval between 0 and 0, and supposing that (60/80)" = 112 6(0++ 0)/80, one can approximately determine the time dependence of the shock layer thickness from Eq (7.11). If we suppose that the hereditary properties of the medium are governed by a fractional derivative (Rossikhin and Shitikova, 1996f), that is o=

`gx

ot

+ 3'-07c 0t-----7-,

then as a result we are led to the equation

&

02 f0

t~(0')d0'

,90 v"& TJ° (0-0')rr(1-3')

,

(7.13)

where v. = 3'0xv(2ez)"t, which is the generalized Burgers' equation and goes over into the Burgers' equation at 3' = 1. It can be shown in the same manner as it has been done in Sugimoto and Kakutani (1985) that Eq (7.13) describes evolution of a smooth wave profile which determines fast and continuous variation of the value to be found within a finite length. Thus, the fractional calculus hereditarily elastic models can be used for investigating shock waves as well if by a shock layer is meant a certain layer, within which the desired values change continuously from some magnitudes to other, instead of a geometric surface o f strong discontinuity. 8 CONCLUSION

,

v - 7 - j 0 K(0 - 0')w0(0')d0',

Ou 8t~ o - - + - - =60 0z

0t~ 0o t~--+-=

63

(7.12)

The majority of the investigations presented in this state-ofthe-art paper have been pursued in Russia and Ukraine beginning in 60 years within the framework o f the school of thought headed in due course by an academician Yuriy N. Rabotnov. The investigations concerned essentially hereditarily elastic media, wherein the Rabotnov's fractional exponential function (Rabotnov, 1948) was used as the relaxation kernel or as the creep kernel, ie, the generalized standard linear solid model was examined. For this model, it has been shown that when solving stationary dynamic problems the Abelian singularity o f a kernel at t = 0 is associated with the Abelian singularity of the corresponding distribution function of the relaxation (creep) times at z = 0 . In the Fourier space, this singularity is realized when co ~ oo, in so doing it determines at this point the slope of the tangent to the vector diagrams o f a complex modulus or a complex compliance which is equal to roy/2 . Hence the deduction can be drawn regarding the character o f the behavior of stationary solutions. If the solutions are dependent on the dimensionless value orq; (i = e, o), then the Abelian singularity manifests itself only through the mediation of the fractional parameter 3' which is the exponent o f a power for the value o0z~ or the argument o f trigonometric functions. In those cases when m and ~ enter independently from each other, the Abelian singularity shows itself in an explicit form at ~ = 0 or to --+ oo as demonstrated by the behavior of the damping coefficients for volume and surface harmonic waves. When investigating nonstationary dynamic problems it has been shown that the Abelian singularity causes the impossibility of the aperiodic regime existence under nonstationary vibrations of mechanical systems, as well as the impossibility of the propagation of strong and weak discontinuities in a continuous medium, the waves with a fuzzified front travel in their stead and in the process the fractional parameter governs the power of fuzzifying. As for other fractional operator models which have been considered in the reviewed papers, their applications to dynamic problems have more likely an occasional character. While many o f characteristics in the behavior of mechanical


64

Rossikhin and Shilikova: Applications of fractional calculus 1o dynamic problems

systems are retained w h e n e m p l o y i n g other models, this nevertheless d o e s not p r e c l u d e the occurrence o f s o m e fund a m e n t a l differences. As an e x a m p l e , during the study o f free d a m p e d vibrations o f a hereditarily elastic oscillator with the R z h a n i t s y n kernel used as the relaxation kernel, the feasibility o f the transition o f the system into the aperiodic r e g i m e has b e e n discovered, so that it has b e e n possible to trace the influence o f the fractional parameter y on the region o f dissipative processes with significant intensity. In studies o f free d a m p e d vibrations o f an oscillator with two different fractional parameters (the general-purpose model), it has b e e n possible to trace the t o p o l o g i c a l b e h a v i o r o f the characteristic e q u a t i o n roots in the c o m p l e x plane as the g i v e n m o d e l passes uninterruptedly f r o m the generalized K e l v i n V o i g h t m o d e l to the g e n e r a l i z e d M a x w e l l or standard linear solid m o d e l . F r o m the aforesaid it m i g h t be c o n c l u d e d that in spite o f the w i d e and d e e p m a t h e m a t i c a l treatment o f fractional differentiation and integration, its application to d y n a m i c p r o b l e m s o f m e c h a n i c s leaves m u c h to be desired. Few, i f any p r o b l e m s c o n c e r n i n g the d y n a m i c b e h a v i o r o f beams, plates and shells h a v e b e e n solved, and practically there are f e w i n v e s t i g a t i o n s d e v o t e d to the d y n a m i c b e h a v i o r o f spatial bodies. T h e r e f o r e the authors l o o k forward to future investigators w h o c o n c e i v a b l y m i g h t c o m p e n s a t e for this gap in the study o f fractional operators.

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Rossikhin and Shitikova: Applications of fractional calculus 10 dynamic problems

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Rossikhin and Shilikova: Applications of fractional calculus 1o dynamic problems

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Yuriy A Rossikhin is a Professor o f the Department o f Theoretical Mechanics at the Voronezh State Academy o f Construction and Architecture in Russia. He t'eceived his MSc degree in Applied Mechanics from the Voronezh State University in 1966, a PhD degree in Mechanics and Physics o f Solids in 1970, a Habilitation Docent Degree in 1972, and an Associate Professorship in 1974,fi'om the Voronezh Polytechnic Institute. He received a DSc degree in Solid Mechanics from the Chyvash UniversiO~ in 1991 and a full Professorship in 1992from the Voronezh Civil Engineering Institute (which was renamed the Voronezh State Academy o f Construction and Architecture in 1993). Since 1993, he has been a Full Member o f the Acoustical Socie O, o f America, in 1994 he was elected as an Active Member of the New York Academy o f Sciences. Since 1995 he has been a Member o f the EUROMECH (European Mechanical Society). GAMM (Gesellschafi far Angewandte Mathematik und Mechanik) and ASME (American Society o f Mechanical Engineers). His area o f research as well as teaching is theoretical mechanics and applied mathematics and mechanics. He has published about 150papers in journals and proceedings dealing with wave dynamics, vibrations, acoustics, problems on dynamic contact interaction, and mechanical and thermal shock.

Marina V Shitikova is a Professor o f the Department o f So'uctural Mechanics at the Voronezh State Academy of Construction and Architecture in Russia. She received her MEng in Civil Engineering in 1982. a PhD degl~e in Structural Mechanics in 1987.from the Voronezh Civil Engineering Institute, a DSc degree in Solid Mechanics in 1994 fi'om the Belarus State Polytechnic Academy and a Professorship in 1995 fi'om the Voronezh State Academy o f Construction and Architecture. Since 1994, she has been an Associate Member o f the Acoustical SocieOp o f America, since 1995 she has been a Member o f the EUROMECH, GAMM and ASME. She has published about 80papers dealing with structural mechanics, vibrations, wave dynamics, and acoustics.
