a new direct time integration scheme for the nonlinear

10th HSTAM International Congress on Mechanics Chania, Crete, Greece, 25 – 27 May, 2013

A NEW DIRECT TIME INTEGRATION SCHEME FOR THE NONLINEAR EQUATIONS OF MOTION IN STRUCTURAL DYNAMICS John T. Katsikadelis School of Civil Engineering National Technical University of Athens e-mail: [email protected], web page: http://users.ntua.gr/jkats Keywords: equations of motion, structural dynamics, integral equations, analog equation method, differential equations, numerical solution, nonlinear. Abstract. A new direct time integration method is presented for the solution of the nonlinear equations of motion in structural dynamics. It applies also to any large systems of second order differential equations with fully populated, non symmetric coefficient matrices as well as to equations with time dependent coefficients. The proposed method is based on the principle of the analog equation, which converts the coupled N equations into a set of single term uncoupled second order ordinary single term linear quasi-static differential equations under fictitious loads, unknown in the first instance. The fictitious loads are established from the integral representation of the solution of the substitute single term equations. The method is simple to implement. It is self starting, unconditionally stable, accurate and conserves energy. It performs well when large deformations and long time durations are considered and it can be used as a practical method for integration of the equations of motion in cases where widely used time integration procedures, e.g. Newmark, become unstable. Several examples are presented, which demonstrate the efficiency of the method. The method can be straightforward extended to evolution equations of order higher than two.

1

INTRODUCTION

In dynamic analysis the equations of motion are obtained by considering the dynamic equilibrium of the external, internal and inertia forces, namely   fD (u )  fS (u)  p(t ) Mu

(1)

 are the inertia forces, fD (t )  fD (u ) the damping forces, fS (t )  fS (u) the elastic forces and where fI (t )  Mu p(t ) are the external excitation forces; u  u(t ) is the displacement vector. The problem consists in establishing the time history u  u(t ) , where t  [0,T ], T  0 , satisfying Eq. (1) with the initial conditions

u(0)  u 0 ,

u (0)  u 0

(2)

The forces fD (u ) and fS (u) are in general non linear functions of their arguments. For linear problems they are given as fD (t )  Cu and fS (t )  Ku and Eq. (1) becomes   Cu  Ku  p(t ) Mu

(3)

where M , C and K are the mass, damping and stiffness matrix of the structure, respectively. In the last fifty years, significant advances have been made in developing numerical methods for the solution of the equations of motion. Many methods have been proposed either in the time or in frequency domain. The reader is advised to relevant literature, where extensive surveys of the various numerical solution methods for linear and nonlinear structural systems under dynamic loads are presented, detailed analyzed and critically discussed, e.g. [1-5]. The numerical damping of the proposed schemes is a major drawback. An in depth investigation of this problem, especially in nonlinear systems, has been studied by Tama and coworkers [6-7], Cristfield et al. [8] and Bathe [9], where several remedies for the problem are presented and discussed. All these techniques apply to the dynamics of inertial systems, stemming from the three fundamental principles: the Principle of Virtual Work in Dynamics, Hamilton’s Principle and as an alternate Hamilton’s Law of Varying Action, and the Principle of Balance of Mechanical Energy.

John T. Katsikadelis.

In this paper a new direct time integration method is presented. In contrast to the conventional methods, which start by assuming a variation of the displacements, velocities and accelerations within each time interval, with the finite differences playing a dominant role, the present method is based on the concept of the Analog Equation introduced by Katsikadelis [10,11], according to which the system of the N coupled equations of motion, linear or non-linear, is replaced by a set of uncoupled linear single term quasi-static equations each of which includes only one unknown displacement and is subjected to appropriate unknown fictitious external loads. These fictitious loads, which actually in dynamics represent the accelerations, are established numerically from the integral representation of the solution of the substitute equations and the requirement that the equations of motion are satisfied at discrete times. The method is self-starting, second order accurate for liner systems, unconditionally stable and it does not exhibit amplitude decay or period elongation. In contrast to the reported methods, the present scheme does not use any remedy to overcome the drawback of energy dissipation, because it conserves energy, even when motions of long duration are studied. The solution algorithms are simple. The method applies to the case of time dependent mass, damping or stiffness. Moreover, it applies to any second order differential equations, no matter where they originate. Numerical examples are presented to illustrate the method and demonstrate its efficiency and accuracy. The so called “soft” pendulum [8,9], which is used as a benchmark problem, is studied to show that the method performs well when large deformations and long time durations are considered. Finally, for the convenience of the reader an Appendix is provided, where the solution algorithms are stated as they might be implemented on the computer. 2 THE AEM SOLUTION 2.1 The one-degree of freedom system The method is illustrated with the linear one-degree-of-freedom mu  cu  ku  p(t ) u(0)  u 0 ,

(4)

u(0)  u 0

(5)

Let u  u(t ) be the sought solution. Then, if the operator d 2 / dt 2 is applied to it, we have u  q(t )

(6)

where q(t ) is a fictitious source, unknown in the first instance. Eq. (6) is the analog equation of (4). It indicates that the solution of Eq. (4) can be obtained by solving Eq. (6) with the initial conditions (5), if q(t ) is first established. This is achieved as following. Taking the Laplace transform of Eq. (6) we obtain 1 1 1 U (s )  u(0)  2 u(0)  2 Q(s ) s s s

where U (s ),Q(s ) are the Laplace transforms of u(t ), q(t ) . The inverse Laplace transform of the above expression yields the solution of Eq. (6) in integral from as t

u(t )  u(0)  u(0)t   q( )(t   )d

(7)

0

Thus the initial value problem of Eqs (4), (5) is transformed into the equivalent Volterra integral equation for q(t ) . un u (t )

u1

u0

uN

u2

u N -1

u3

t h

h

h

h

h

h h T = Nh

h

h

h

h

Figure 1. Discretization of the interval [0,T ] into N equal intervals h  T / N

John T. Katsikadelis.

Eq. (7) can be solved numerically within a time interval [0,T ] following a procedure analogous to that for fractional differential equations [12]. Thus, the interval [0,T ] is divided into N equal intervals t  h , h  T / N , in which q(t ) is assumed to vary according to a certain law, e.g. constant, linear etc. In this analysis q(t ) is assumed to be constant and equal to the mean value in the interval h . That is q rm 

q r 1  q r 2

(8)

Hence, Eq. (7) at instant t  nh can be written as h 2h nh u n  u 0  nhu 0   q 1m  (nh   )d  q 2m  (nh   )d    q nm  (nh   )d  0 h (n 1)h  

(9)

which after evaluation of the integrals yields n 1

u n  u 0  nhu 0  c1   2(n  r )  1  q rm  c1q nm r 1 n 1

 u n 1  hu 0  2c1  q rm  c1q nm

,

c1 

h2 2

(10,11)

r 1

The velocity is obtained by direct differentiation of Eq. (7) making use of the Leibnitz rule. Thus we have t

u(t )  u(0)   q( )d 0

(12)

Using the same discretization for the interval [0,T ] to approximate the integral in Eq. (12), we have n 1

un  u 0  c2  q rm  c2q nm r 1

(13)

 un 1  c2q nm

c2  h Solving Eq. (13) for

n 1

q

m r

(14)

and substituting in Eq. (10) gives

r 1

u n  u n 1  hun  c1q nm

(15)

By virtue of Eq. (8), Eqs (15) and (13) can be further written as c1 c q n  hun  un   1 q n 1  un 1 2 2 

c2 c q n  un  un 1  2 q n 1 2 2

(16) (17)

Moreover, Eq. (4) at time t  nh is written as mq n  cun  kun  pn

(18)

Eqs (16), (17) and (18) can be combined as  m   1 c1  2  1   c2  2

k   qn  h 1   un    un 1 0  c

 0   1      c1 2     1  2 c2

 0 0 1    q n 1    0 1   un 1    0  pn   u n 1      0 1 0  

(19)

Since m  0 , the coefficient matrix in Eq. (19) is not singular for sufficient small h and the system can be solved successively for n  1,2,  to yield the solution un and the derivatives un , un  q n at instant t  nh  T . For n  1 , the value q 0 appears in the right hand side of Eq. (19). This quantity can be readily

John T. Katsikadelis.

obtained from Eq. (4) for t  0 . This yields q 0   p 0  cu 0  ku 0  / m

(20)

Eq. (19) can be also written as

Un  AUn 1  bpn ,

n  1, 2,  N

(21)

in which   1  qn   1   Un   un  , A   c1  2  un   1c   2  2

 2  2   h 1   1 0  

1

  0  1   c1  2  1c  2 2

  0 0  1   0 1  , b   1 c1   2  1c 1 0   2 2 

 2    h 1   1 0   2

1

1    0 , 0  

pn  p n / m

(22a,b,c,d)

  m / k is the eigenfrequency and   c / 2m the damping ratio. The recurrence formula (21) can be employed to construct the solution algorithm. However, the solution procedure can be further simplified. Thus, applying Eq. (21) for n  1,2,  we have U1  AUo  bp1 U2  AU1  bp2  A(AUo  bp1 )  bp2  A2 Uo  Abp1  bp2    Un  An Uo  (An 1 p1  An 2 p2   A 0 pn )b

(23)

Apparently, the last of Eqs (23) gives the solution vector Un at instant tn  nh using only the known vector U 0 at t  0 . The matrix A and the vector b are computed only once. In the following subsections proofs of the stability and convergence of the scheme for the linear systems are presented. 2.2 Stability of the numerical scheme The matrix A is the amplification matrix. In order that the solution is stable, An must be bounded. This is true if

(A)  max  1 , 2 ,  3

1

(24)

Setting s  h , we obtain the eigenvalues of A as

1  0 ,

2 

4s (1   2 ) 4  s2 , i 2 4  s  4s 4  s 2  4s

3 

4s (1   2 ) 4  s2 i 2 4  s  4s 4  s 2  4s

(25a,b,c)

Which implies

1   2 

(s 2  4)2  16 2s 2

 (s 2  4)  4 s 

2

1

(26)

The equality holds for   0 . We see that for   0 it is (A)  1 , thus the scheme is strongly stable. 2.3 Error analysis and convergence The error is due to the approximation of the integrand in the integral of Eq. (7) in the n integration interval [(n  1)h, nh ]



t1

t0

f ( )d ,

t 0  (n  1)h, t1  nh

(27)

John T. Katsikadelis.

where f ( )  q( )(t1   )

(28)

f( )  q m (t1   )

(29)

which in this analysis is approximated as

Expanding f ( ) and f( ) in Taylor series at   0 and evaluating the integral of f ( )  f( ) over the interval [t 0 , t1 ] we find 2 ( )]d  (q 0  q 0m )h 2  (q 1  q 0  q 0m ) h  3 q 0m h 2 [ ( ) f   f t0 2 2 t1

(30)

Therefore the convergence of the algorithm is O(h 2 ) . 2.4 Accuracy For free vibrations, the numerical solution can be written in terms of the eigenvalues u n  c1  2n  c2  3n

( 1  0)

(31)

or u n  r n (c1 sin n  c2 cos n )  r n (c1 sin tn  c2 cos tn )

(32)

where r  a 2  b 2 ,   tan 1 (b / a ) , a  Re(  2 ) , b  Im(  2 ) ,    / h tn  nh . The corresponding exact solution is u n  e tn (c1 sin D tn  c2 cos D tn ) , D 

1   2 , tn  nh

(33)

Comparison of Eqs (32) and (33) could show the accuracy of the numerical scheme. Thus, if T and T are the exact and the approximate periods, respectively, we define the period elongation -3

2 =0 =0.1 =0.2

0.2

x 10

 =0  =0.1  =0.2

0

0.15

-2 

period elongation %

0.25

0.1

-4

0.05

-6

0 0

0.05

0.1

0.15

0.2

0.25

0.3

h/T

-8 0

0.05

(a)

0.1

0.15 h/T

0.2

0.25

0.3

(b)

Figure 2. (a) Period elongation (b) Amplitude decay      versus h / T for different values  . Figure . PE 

T T s 1   2  1 T 

(34)

For the amplitude decay we can define an equivalent damping ratio  from the relation r n  e tn  e  n

(35)

   ln r / 

(36)

which gives

John T. Katsikadelis.

The difference      can be employed as a measure for the amplitude decay. The dependence of the period elongation and amplitude decay on h / T is shown in Fig. 2 and Fig. 3, respectively. Apparently, for small values of h / T the scheme is accurate. Note that for   0 it is  2   3  r  1 and Eq. (36) yields

  0 . That is, there is no amplitude decay.

3

MULTI-DEGREE-OF-FREEDOM SYSTEMS

3.1 Linear systems The solution procedure described in Section 2 may be applied to systems of N linear equations of motion provided that the coefficients m, c, k and the quantities u 0 , u 0 , u n , un , q n , pn are replaced with the coefficients matrices M, C, K and the vectors u 0 , u 0 , u n , u n , q n , pn , respectively, and the scalar operations with matrix operations. Thus Eqs (19) and (20) become   C K 0 0 0  M  I    qn     q n 1    c c   1 1  I hI I   u n     I 0 I   u n 1    0  pn  2    2  u     c2   u n   c2   n 1   0   I I 0 I I 0    2   2  q 0  M 1 (p 0  Cu 0  Ku 0 ),

det(M)  0

(37)

(38)

Eq. (37) is solved for n  1,2,  3.2 Nonlinear systems The solution procedure developed previously for the linear equations can be straightforward extended to nonlinear equations of multi-degree of freedom systems described as   F(u , u)  p(t ) Mu

(39)

u(0)  u 0 , u (0)  u 0

(40)

where M is N  N known coefficient matrix with det(M)  0 ; F(u , u) is an N  1 vector, whose elements are nonlinear functions of the components of u, u . The solution procedure is similar to that for the linear systems. Thus Eq. (39) for t  0 gives the initial acceleration vector q 0  M 1 [p 0  F(u 0 , u 0 )],

 q0  u

(41)

Subsequently, we apply Eq. (39) for t  tn Mq n  F(u n , u n )  pn

(42)

Apparently, the second and third of Eqs (37) are valid also in this case and can be written in the form  c1   c1   I I   hI I   u n   0 I   u n 1    2    q   u    u    n  2  q n 1  I 0   n   I 0   n 1    c2 I   c2 I   2   2 

(43)

Eqs (42) and (43) are combined and solved for q n , u n , u n , n  1,2,  ..Note that Eq. (43) are linear and can be solved for u n , u n . Then substituting into Eq. (42) results in a non linear algebraic equation, which can be solved to yield q n . In our examples the function fsolve of Matlab has been employed for the solution of the nonlinear algebraic equations.

John T. Katsikadelis.

4

EXAMPLES

Example 1. The simple pendulum The equation of the simple pendulum (Fig. 3) has been chosen as a single-degree-of-freedom non linear system, namely the initial value problem g l

  sin   0  (0)   0 ,

(44)

(0)  0

(45)

 (t ) represents the angle of the pendulum from the vertical position; l its length and g the acceleration of the gravity y x

O

q (t ) l

y

m

x

Figure 3. Simple pendulum Eq. (44) admits an exact solution [13]

 (t )  2 sin 1



k sn  g / l (t  T0 ); k 



(46)

1.5 1.25 1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 -1.5 0

-29

computed exact

-29.5 E(t)

(t)

where k  sin 2 ( 0 / 2) and T0 the quarter of the period. The response of the pendulum for l  g and  0  0.40 is shown in Fig. 4a, while the energy variation  / m  gy   2l 2 / 2 is presented in Fig. 4b. Apparently, the method conserves the energy, thus it exhibits no numerical damping (amplitude decay).

E(t)=-29.7386

-30

-30.5

20

40

60

80

100

-31 0

20

40

(a)

60

80

100

t

t

(b)

Figure 4. (a) Response of the simple pendulum (b) Energy variation in simple pendulum

Example 2. The elastic pendulum The elastic pendulum, also called “soft pendulum”, is chosen as a two degree of freedom nonlinear system. In this pendulum the rod is assumed elastically extensible with a stiffness k  EA / l ; A is the area of the cross section and E the modulus of elasticity. Referring to Fig. 3 the total energy is expressed as

John T. Katsikadelis.

  A  K U

(47)

where 1 K  m(x 2  y 2 ) , 2

A  mgy ,

1 U  ke 2 , 2

e

x 2  y2  l

(48)

are the potential of the conservative force (gravity), the kinetic energy and elastic energy, respectively. Using the Lagrange equation method we obtain the equations of motion as [14] mx 

my 

EA  1 L 

EA  1 L 

 x  0  x 2  y2 

(49a)

  y  mg x 2  y2 

(49b)

L

L

with the initial conditions x (0)  x 0 , x(0)  x 0 ,

y(0)  y 0 ,

y(0)  y0

(50)

4 0< t