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ZAMM Z. Angew. Math. Mech., 1 18 (2013) / DOI 10.1002/zamm.201200245

A new direct time integration method for the equations of motionin structural dynamicsJohn T. Katsikadelis

Institute of Structural Analysis and Aseismic Research, School of Civil Engineering, National Technical University ofAthens, Zografou Campus, 15773, Athens, Greece

Received 11 November 2012, revised 26 March 2013, accepted 8 April 2013Published online 16 May 2013

Key words Equations of motion, structural dynamics, integral equations, analog equation method, differential equations,numerical solution.

A direct time integration method is presented for the solution of the equations of motion describing the dynamic response ofstructural linear and nonlinear multi-degree-of-freedom systems. It applies also to large systems of second order differentialequations with fully populated, non symmetric coefficient matrices as well as to equations with variable coefficients. Theproposed method is based on the concept of the analog equation, which converts the coupled N equations into a set ofsingle term uncoupled second order ordinary quasi-static differential equations under appropriate fictitious loads, unknownin the first instance. The fictitious loads are established from the integral representation of the solution of the substitutesingle term equations. The method is simple to implement. It is self starting, unconditionally stable and accurate andconserves energy. It performs well when large deformations and long time durations are considered and it can be usedas a practical method for integration of the equations of motion in cases where widely used time integration procedures,e.g. Newmarks, become unstable. Several examples are presented, which demonstrate the efficiency of the method. Themethod can be straightforward extended to evolution equations of order higher than two.

c 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 IntroductionIn dynamic analysis the equations of motion are obtained by considering the dynamic equilibrium of the external, internaland inertia forces, namely

fI(t) + fD(t) + fS(t) = p(t)

which may be written as

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

u(0) = u0, u(0) = u0. (2)The forces fD(u) and fS(u) are in general non linear functions of their arguments. For linear problems they are given asfD(t) = Cu and fS(t) = Ku and Eq. (1) becomes

Mu + Cu + Ku = p(t) (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 the development and application of numerical methodsto the solution of the equations of motion governing the dynamic behavior of structural systems. Many methods havebeen proposed either in the time or in the frequency domain. The reader is advised to relevant literature, where extensivesurveys of the various numerical solution methods and computational procedures for linear and nonlinear structural systems

E-mail: [email protected]

c 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

2 J. T. Katsikadelis: A new direct time integration method for the equations of motion in structural dynamics

subjected to dynamic loads are presented, detailed analyzed and critically discussed, e.g. Hughes [1], Bathe [2], Karabalisand Beskos [3], Dokainish and Subbaraj [4, 5]. The energy dissipation in 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 techniquesapply to the dynamics of inertial systems, stemming mainly from the three fundamental principles: the Principle of VirtualWork in Dynamics, Hamiltons Principle and as an alternate, Hamiltons Law of Varying Action, and the Principle ofBalance of Mechanical Energy.

In this paper a new direct time integration method is presented. In contrast to the conventional methods, which start byassuming a variation of the displacements, velocities and accelerations within each time interval, with the finite differencesplaying 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 ofuncoupled linear single term quasi-static equations each of which includes only one unknown displacement and is subjectedto appropriate unknown fictitious external loads. These fictitious loads, which actually here represent the accelerations, areestablished numerically from the integral representation of the solution of the substitute equations and the requirement thatthe equations of motion are satisfied at discrete times. Numerical examples, including linear as well as non linear systems,are treated by the proposed method and the results are compared with those obtained by exact or other popular timeintegration methods. The method is self starting, second order accurate for liner systems, unconditionally stable and it doesnot exhibit amplitude decay or period elongation. In contrast to the previously reported methods, the present scheme doesnot use any remedy to overcome the drawback of energy dissipation, because it conserves energy, even when motions oflong duration are studied. The solution algorithms are simple. The method applies also to the case of time dependent mass,damping or stiffness, i.e. to equations of variable coefficients. Moreover, it applies to any second order differential equations,no matter where they originate. The scheme has been employed in [1215] to crosscheck the solution of several largesystems of linear and nonlinear semidiscretized equations of motion obtained by another method and it has performed well.The rest of the paper is organized as follows. In Sect. 2, the method is illustrated with the linear equation of motion of thesingle degree-of-freedom system. In Sect. 3, the method is applied to equations with time dependent coefficients. In Sect. 4,the linear multi-degree-of-freedom systems are treated, while in Sect. 5 the method is applied to nonlinear systems. The socalled soft pendulum [8,9], which is used as a benchmark problem, is studied to show that the method performs well whenlarge deformations and long time durations are considered. In Sect. 6, the scheme is applied to large linear and nonlinearsystems, which validate the efficiency of the method and its suitability to solve large systems of semidiscrized equations ofmotion resulting from FEM, BEM, Meshless or other modern computational methods. A summary of conclusions is givenin Sect. 7. Finally, for the convenience of the reader an Appendix is provided, where the solution algorithms are stated asthey might be implemented on the computer.

2 The one-degree-of-freedom system2.1 The AEM solution

For the linear one-degree-of-freedom system the initial value problem (2), (3) becomes

mu + cu + ku = p(t), (4)

u(0) = u0, u(0) = u0. (5)Let u = u(t) be the sought solution. Then, if the operator d2/dt2 is applied to it, we have

u = q(t) (6)

where q(t) is a fictitious source, unknown in the first instance. Equation (6) is the analog equation of Eq. (4). It indicatesthat the solution of Eq. (4) can be obtained by solving Eq. (6) with the initial conditions (5), if the q(t) is first established.This is achieved as follows.

Taking the Laplace transform of Eq. (6) we obtain

s2U(s) su(0) u(0) = Q(s)

or

U(s) =1su(0) +

1s2

u(0) +1s2

Q(s)

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ZAMM Z. Angew. Math. Mech. (2013) / www.zamm-journal.org 3

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

u(t) = u(0) + u(0)t + t0

q()(t )d . (7)

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

( )u t

t

1u2u

3u0u

Nu

T Nhh h h h h h h hh h h

1Nu

nu

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

Equation (7) is solved numerically within a time interval [0, T ]. Following a procedure analogous to that for fractionaldifferential equations [16], the interval [0, T ] is divided into N equal intervals t = h, h = T/N , in which q(t) is assumedto vary according to a certain law, e.g. constant, linear etc. In this analysis q(t) is assumed to be constant and equal to themean value in the interval h. That is

qmr =qr1 + qr

2. (8)

Hence, Eq. (7) at instant t = nh can be written as

un = u0 + nhu0

+

[qm1

h0

(nh )d + qm2 2h

h

(nh )d + + qmn nh(n1)h

(nh )d]

(9)

which after evaluation of the integrals yields

un = u0 + nhu0 + c1n1r=1

[2(n r) + 1] qmr + c1qmn

= un1 + hu0 + 2c1n1r=1

qmr + c1qmn (10)

where

c1 =h2

2. (11)

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

u(t) = u(0) + t0

q()d . (12)

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

un = u0 + c2n1r=1

qmr + c2qmn

= un1 + c2qmn (13)

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where

c2 = h. (14)Solving Eq. (13) for n1r=1 qmr and substituting in Eq. (10) gives

un = un1 + hun c1qmn . (15)By virtue of Eq. (8), Eqs. (15) and (13) can be further written as

c12

qn hun + un = c12 qn1 + un1, (16)

c22

qn + un = un1 +c22

qn1. (17)Moreover, Eq. (4) at time t = nh is written as

mqn + cun + kun = pn . (18)Equations (16), (17), and (18) can be combined as

m c k12 c1 h 1 12 c2 1 0

qn

un

un

=

0 0 0 12 c1 0 112 c2 1 0

qn1un1un1

+

100

pn. (19)

Since m = 0, the coefficient matrix in Eq. (19) is not singular for sufficient small h and the system can be solved succes-sively for n = 1, 2, . . . to yield the solution un and the derivatives un, un = qn at instant t = nh T . For n = 1, the valueq0 appears in the right hand side of Eq. (19). This quantity can be readily obtained from Eq. (4) for t = 0. This yields

q0 = (p0 cu0 ku0) /m. (20)Equation (19) can be also written as

Un = AUn1 + bpn, n = 1, 2, . . .N (21)in which

Un =

qn

un

un

, A =

1 2 212 c1 h 1 12 c2 1 0

1

0 0 0

12 c1 0 112 c2 1 0

, (22a,b)

b =

1 2 212 c1 h 1 12 c2 1 0

1

100

, pn = pn/m. (22c,d)

=

m/k is the eigenfrequency and = c/2m the damping ratio. The recurrence formula (21) can be employed toconstruct the solution algorithm. However, the solution procedure can be further simplified. Thus, applying Eq. (21) forn = 1, 2, . . . we have

U1 = AUo + bp1,

U2 = AU1 + bp2,

= A(AUo + bp1) + bp2

= A2Uo + Abp1 + bp2 = Un = AnUo + (An1p1 + An2p2 + . . . A0pn)b. (23)

Apparently, the last of Eqs. (23) gives the solution vector Un at instant tn = nh using only the known vector U0 at t = 0.The matrix A and the vector b are computed only once. In the following subsections proofs of the stability and convergenceof the scheme for the linear systems are presented.



2.2 Stability of the numerical schemeThe matrix A is the amplification matrix. In order that the solution is stable, An must be bounded. This is true if the spectralradius (A) satisfies the condition

(A) = max {|1| , |2| , |3|} 1. (24)If (A) < 1 the method is strongly stable.

Setting s = h, we obtain the eigenvalues of A as

1 = 0, (25a)

2 =4 s2

4 + s2 + 4s+ i

4s

(1 2)4 + s2 + 4s

, (25b)

3 =4 s2

4 + s2 + 4s i 4s

(1 2)

4 + s2 + 4s, (25c)

and thus

|1| = |2| =

(s2 + 4)2 162s2[(s2 + 4) + 4s]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 convergenceThe error is due to the approximation of the integrand in the integral of Eq. (7) in the n-th integration interval [(n1)h, nh]

t1t0

f()d , t0 = (n 1)h, t1 = nh (27)

where

f() = q()(t1 ) (28)which in this analysis is approximated as

f() = qm(t1 ). (29)

Expanding f() and f() in Taylor series at = 0 and evaluating the integral of f() f() over the interval [t0, t1] wefind

t1t0

[f() f()]d = (q0 qm0 )h2 + (q1 q0 + qm0 )h2

2

=32qm0 h

2 (30)

Therefore the convergence of the algorithm is O(h2).

2.4 AccuracyFor free vibrations, the numerical solution can be written in terms of the eigenvalues

un = c1n2 + c

2

n3 (1 = 0) (31)

or

un = rn(c1 sinn + c2 cosn)

= rn(c1 sin tn + c2 cos tn) (32)



where r =

a2 + b2, = tan1(b/a), a = Re(2), b = Im(2), = /h, tn = nh.The corresponding exact solution is

un = etn(c1 sinDtn + c2 cosDtn), 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 theapproximate periods, respectively, we define the period elongation

PE =T T

T=

s

1 2

1. (34)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

h/T

perio

d el

onga

tion

%

[=0[=0.1[=0.2

0 0.05 0.1 0.15 0.2 0.25 0.3-8

-6

-4

-2

0

2x 10-3

h/T

'[

[=0[=0.1[=0.2

Fig. 2 Period elongation versus h/T for different values . Fig. 3 Amplitude decay = versus h/T for differentvalues of .

For the amplitude decay we can define an equivalent damping ratio from the relation

rn = etn

= en (35)

which gives

= ln r/. (36)

The difference = can be employed as a measure for the amplitude decay. The dependence of the period elongationand amplitude decay on h/T is shown in Fig. 2 and Fig. 3, respectively. Apparently, for small values of h/T the scheme isaccurate. Note that for = 0 it is |2| = |3| = r = 1 and Eq. (36) yields = 0. That is, there is no amplitude decay.

2.5 Numerical examples

Following the steps of the previous solution procedure a MATLAB program has been written and various example problemshave been solved.

Example 1. Free undamped vibrations

Equation (4) has been solved with data: m = 1, = 0, = 5, u0 = 1, u0 = 0. The response of the system for longduration is shown in Fig. 4. The scheme shows no amplitude decay and negligible period elongation (PE = 0.02% forh = 0.01 and PE = 0.00021% for h = 0.001).



Example 2. Free and forced damped vibrations. Error discussionEquation (4) has been solved with data: m = 1, = 0.07, = 2, u0 = 1, u0 = 20, T = 25, h = 1/500. Case (i) p(t) = 0;case (ii) p(t) = p0H(t), p0 = 10; H(t) is the Heaviside function.

The analytical solution is obtained as

u(t) =[u(0) + u(0)

DsinDt + u(0) cosDt

]et +

p0k

[1

(cosDt +

1 2 sinDt

)et

]

(37)

where D =

1 2. Figure 5 and Fig. 6 show the obtained solution together with the error e = u uexact for the twoload cases. Moreover, in Fig. 7 the variation of the computed error e = e(h), h(k) = 1/500k, k = 1, 2, . . . , 8 has beenplotted, which verifies that the convergence is of O(h2).

90 92 94 96 98 100-1

-0.5

0

0.5

1

t (h=0.001)

u(t)

computedexact

990 992 994 996 998 1000-1

-0.5

0

0.5

1

t ( h=0.001)

u(t)

computedexact

Fig. 4 Free undamped vibrations in Example 1.

0 5 10 15 20 25-10

-5

0

5

10Displacement u(t)

t

computed exacterror x 105

0 5 10 15 20 25-20

-15

-10

-5

0

5

10

15

20Velocity u,t(t)

t


0 5 10 15 20 25-40

-30

-20

-10

0

10

20

30Acceleration u,tt(t)

t


Fig. 5 Displacement, velocity, acceleration and respective errors inExample 2, Case (i).



0 5 10 15 20 25-8

-6

-4

-2

0

2

4

6

8

10

12Displacement u(t)

t


Fig. 6 Displacement and error in Example 2, Case (ii).

1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5 x 10-5 case (i)

k

erro

r

computedexpected e=c*h2

1 2 3 4 5 6 7 80

0.5

1

1.5

2 x 10-5

case (ii)

k

erro

r

computedexpected e=c*h2

Fig. 7 Computed and expected error e = e(h) in Example 2; h(k) = 1/500k, k = 1, 2, . . . , 8, e = ch2, c = e(1)/h(1).

3 Variable coefficients

So far we have developed the method for the solution of the equation of motion with constant coefficients. Obviously, if thecoefficients m, c, and k are functions of the independent variable t, the previously described solution procedure remainsthe same except that the elements m, c, k in the first row of the coefficient matrix in the left hand side of Eq. (19) dependon time. Therefore, this coefficient matrix in the respective solution algorithm must be reevaluated in each step. In thefollowing, the efficiency of the method is demonstrated by solving an equation with variable coefficients. Note that theexact solution in the example below is obtained using the inverse method. That is, a solution is assumed, which yields thecorresponding excitation force after inserting it in the equation. This technique has been used in all examples where noreference is made for the exact solution.

Example 3. Variable coefficientsWe consider the initial value problem

(1 + t2)u + tu + e1/(1+t)u = p(t), (38a)

u0 = 1, u0 = 0.1 (38b)

with p(t) = 0.01e0.1t[(99+10t+99t2) cos t+(20+100t20t2) sin t100e1/(1+t) cos t)]. The equation admits anexact solution uexact(t) = e0.1t cos t. The computed solution for T = 20 is shown in Fig. 8 as compared with the exactone.



4 Multi-degree-of-freedom systemsThe developed solution algorithm can be applied to systems of N linear equations of motion describing the response ofmulti-degree-of-freedom systems, namely to Eq. (3) with initial conditions (2).

Apparently, the solution procedure described in Sect. 2 may be also applied to this case provided that the coeffi-cients m, c, k and the quantities u0, u0, un, un, qn, pn are replaced with the coefficient matrices M, C, K and the vectorsu0, u0, un, un, qn, pn, respectively, and the scalar operations with matrix operations. Thus Eqs. (19) and (20) read

M C Kc12 I hI I

c22 I I 0

qnun

un

=

0 0 0 c12 I 0 I

c22 I I 0

qn1un1un1

+

I00

pn, (39)

q0 = M1(p0 Cu0 Ku0), det(M) = 0. (40)Equation (39) is solved for n = 1, 2, . . ..

Example 4. System of equationsWe consider the initial value problem for the system of three equations

52 10 2010 150 3020 30 441

u1

u2

u3

+

34.3199 27.9975 89.461527.9975 172.1296 50.424989.4615 50.4249 770.3077

u1

u2

uu3

+

1472 407 5553407 1001 41545553 4154 43516

u1

u2

u3

=

111

10H(t) (41)

with initial conditions

u0 = {10 1 5}T , u0 = {0 50 20}T . (42)

The damping matrix has been constructed as a Caughey type proportional damping matrix C =2

q=0 aqM(M1K)q =

a0M + a1K + a2KM1K with modal damping coefficients 1 = 0.3, 2 = 0.05, 3 = 0.09. This ensures orthogonalityof the modes i of the free undamped vibrations with respect to the damping matrix and the exact solution can be obtainedusing the modal superposition method. Results for 0 t 5 are shown in Fig. 9.

0 5 10 15 20-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Displacement u(t)

t


0 1 2 3 4 5-15

-10

-5

0

5

10

15

20

u1

u2

u3

t

computed exact

Fig. 8 Solution u and error u uexact in Example 3. Fig. 9 Solution u = {u1 u2 u3}T in Example 3.



5 Nonlinear equations of motionThe solution procedure developed previously for the linear equations can be straightforward extended to nonlinear equa-tions.

The nonlinear initial value problem for multi-degree-of-freedom systems is described as

Mu + F(u, u) = p(t), (43)u(0) = u0, u(0) = u0 (44)

where M is N N known coefficient matrix with det(M) = 0; F(u, u) is an N 1 vector, whose elements are nonlinearfunctions of the components of u, u; p(t) is the vector of the N given load functions and u0, u0 given constant vectors.

The solution procedure is similar to that for the linear systems. Thus Eq. (43) for t = 0 gives the initial accelerationvector

q0 = M1[p0 F(u0, u0)], q0 = u. (45)

Subsequently we apply Eq. (43) for t = tnMqn + F(un, un) = pn. (46)

Apparently, the second and third of Eqs. (39) are valid in this case and can be written as[hI I

I 0

]{un

un

}=

[0 II 0

]{un1un1

}+

[c12 I

c22 I

]qn +

[ c12 I

c22 I

]qn1. (47)

Equations (46) and (47) are combined and solved for qn, un, un with n = 1, 2, . . .. Note that Eq. (47) are linear and can besolved for un,un. Then substitution into Eq. (46) results in a non linear equation, which can be solved to yield qn. In ourexamples the function fsolve of Matlab has been employed to obtain the numerical results.

Example 5. The Duffing equationThe numerical scheme is employed to solve the initial value problem for the Duffing equation

u + 0.2u + u + u3 = p(t), (48)u(0) = 0, u(0) = 1. (49)

For p(t) = e0.1t[(0.01 sin t 0.2 cos t sin t) 0.2(0.1 sin t cos t) + sin t + e0.2t(sin t)3], Eq. (48) admits an exactsolution uexact(t) = e0.1t sin t. The solution with t = 0.01 is shown in Fig. 10.

0 5 10 15 20 25-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t

ComputedExacterror X103

y

x

( )tR l

O

mx

y

Fig. 10 Solution u and error u uexact in Example 4. Fig. 11 Simple pendulum.



Example 6. The simple pendulumThe solution of the simple pendulum equation has been chosen as a second one-degree-of-freedom non linear problem,namely the initial value problem

+g

lsin = 0, (50)

(0) = 0, (0) = 0. (51)(t) represents the angle of the pendulum from the vertical position, l is its length (Fig. 11), and g the acceleration of thegravity.

Equation (50) admits an exact solution

(t) = 2 sin1{

k sn[

g/l(t + T0); k]}

(52)

where k = sin2(0/2) and T0 the quarter of the period [17]. The response of the pendulum for l = g and 0 = 0.40is shown in Fig. 12. Moreover, the energy variation E(t) = gy + 2l2/2 is presented in Fig. 13. Apparently, the methodconserves the energy, thus it exhibits no numerical damping (amplitude decay).

0 20 40 60 80 100-1.5

-1

-0.5

0

0.5

1

1.5

t

(t)

computedexact

0 20 40 60 80 100-31

-30.5

-30

-29.5

-29

t

E(t)

E(t)=-29.74

Fig. 12 Response of the simple pendulum. Fig. 13 Energy variation in simple pendulum.

Example 7. The elastic pendulumThe elastic pendulum, also called soft pendulum, is chosen as a two-degree-of-freedom nonlinear system. In this pen-dulum the rod is assumed elastically extensible with a stiffness k = EA/l; A is the area of the cross section and E themodulus of elasticity. Referring to Fig. 14 the total energy is expressed as

= V + K + U (53)

y

x

( )u t/k EA l

O

mx

y

Fig. 14 Elastic pendulum.

where

V = mgy, K =12m(x2 + y2), U =

12ke2, e =

x2 + y2 l (54)



are the potential of the conservative force (gravity), the kinetic energy, and elastic energy, respectively.Using the method of Lagrange equations we obtain the equations of motion as

mx +EA

L

(1 L

x2 + y2

)x = 0, (55a)

my +EA

L

(1 L

x2 + y2

)y = mg (55b)

with the initial conditions

x(0) = x0, x(0) = x0, y(0) = y0, y(0) = y0. (56)

This problem, in absence of gravity (V = 0), has been used as a benchmark problem by earlier investigators [8,9] to checkthe performance of their method in the effort to overcome the instability of the Newmark method arising when long durationmotions are considered in nonlinear structural dynamics. The pendulum is studied using the scheme with l = 3.0443 m,

0 1 2 3 4 5-4

-2

0

2

4

t

x-di

spal

cem

ent

0 1 2 3 4 5-2

0

2

4

6

8

t

y-di

spal

cem

ent

0 1 2 3 4 5-10

-5

0

5

10

t

x-ve

loci

ty

0 1 2 3 4 5-10

-5

0

5

10

t

y-ve

loci

ty

0 1 2 3 4 5-40

-20

0

20

40

t

x-ac

cele

ratio

n

0 1 2 3 4 5-40

-20

0

20

40

t

y-ac

cele

ratio

n

0 1 2 3 4 50

0.01

0.02

0.03

t

axia

l stra

in

Fig. 15 Elastic pendulum using the schemewith t = 0.01.



EA = 104 N, x0 = 0, x0 = 7.72 ms1, y0 = y0 = 0, and m = 6.667 kg, which are the data employed in [9]. Therod has been treated as truss element with consistent mass having A = 6.57 kg/m. The response of the system obtainedwith t = 0.01 is presented in Fig. 15, which is identical with that in [9]. In Fig. 16 the x-displacement in the intervals0 t 5 and 990.71 t 995.71 has been plotted. This demonstrates that the response remains unchanged aftera long duration of motion. Figure 17 shows that the total energy of the system is conserved. Apparently, the proposedscheme exhibits no period elongation or amplitude decay in analyzing nonlinear dynamic systems. Finally, Fig. 18 showsthe response of the elastic pendulum obtained using the current scheme with t = 0.01 and the Newmarks trapezoidalscheme with t = 0.00001. Obviously, the present scheme performs well for a relatively large time step, while Newmarksscheme becomes unstable even for a very small time step.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-4

-3

-2

-1

0

1

2

3

4

t

x-di

spal

cem

ent

991 992 993 994 995

0< t


l2 = mgy

2 1

2m(x2 + y2), S = 2l (58a,b)

with initial conditions

x(0) = x0, x(0) = x0, y(0) = y0, y(0) = y0 (59)

where is the Lagrange multiplier and S the axial force of the rod. Figure 19 shows the response of the pendulum withl = 3.0443 m, x0 = 0, x0 = 7.72 ms1, y0 = y0 = 0, and m = 6.667 kg. The response resulting from Eqs. (55) with alarge rod stiffness, EA = 107 N, is superimposed in the figure. Apparently, both solutions coincide.

0 1 2 3 4 5-4

-2

0

2

4

t

x-di

spal

cem

ent-c

constraint'stiff'

0 1 2 3 4 50

2

4

6

8

t

y-di

spal

cem

ent-c

constraint'stiff'

0 1 2 3 4 5-8

-6

-4

-2

0

2

4

6

8

t

x-ve

loci

ty-c

constraint'stiff'

0 1 2 3 4 5-8

-6

-4

-2

0

2

4

6

8

t

y-ve

loci

ty-c

constraint'stiff'

Fig. 19 Stiff pendulum using the scheme with t = 0.01.

6 Large systems

In this sections the scheme is applied to the solution of large linear and nonlinear systems of equations, which result assemidiscretized equations of motion from the analysis of continuous systems. The results demonstrate that the developedscheme can efficiently solve large systems of equations of motion originating from the application of modern computationalmethods, such as FEM, BEM, and Meshless methods.

Example 9. Linear equations

In this example the numerical scheme is employed to solve a large system of non symmetric linear equations of motion. Theequations represent the semi-discretized equations of motion resulting from the analysis of the simply supported rectangularplate of Fig. 20. The analysis is performed using the AEM [13]. The response of the plate is governed by the followinginitial boundary value problem

D4w + hw = p(x, y, t) in , t > 0, (60)

w = Mn = 0 on , (61a,b)



w(x, y, 0) = w(x, y, 0) = 0 in (62a,b)where w is the transverse displacement, D = Eh3/12(1 2) the plate stiffness , h the plate thickness, E the elastic

modulus, the Poisson ratio, the mass density, p(x, y, t) the excitation force, and Mn the normal bending moment on theboundary.

For p(x, y, t) = w0[D

((m/a)2 + n/b)2

)2 h2]W (x, y) sin t the problem admits the exact solution

wexact(x, y, t) = w0W (x, y) sin t, W (x, y) = sin(m

ax) sin(

n

ay). (63)

6

y

x

4 ss

ss

ss

ss

Fig. 20 Rectangular simply supported plate.

The numerical results are obtained with: h = 0.1 m, = 2.4 kg/m3, = 0.25, E = 2.9 107 kN/m2, = 10,m = n = 1, w(x, y, 0) = 0, w(x, y, 0) = w0W (x, y). The AEM is employed with 400 constant boundary elementsand 524 domain linear triangular elements. This procedure yields a system of 240 equations of motion in terms of thenodal displacements with non symmetric fully populated coefficient matrices. The equations are solved using the presentnumerical scheme and Newmarks trapezoidal rule. Since it is max = 66240.74, the central difference method appliesonly if t < 2/max = 3 105. In Fig. 21, the time history of the deflection w(1.5, 1, t) obtained with t = 0.01 isshown as compared with the exact solution. Moreover, Fig. 22 presents the error e = max

t|w wexact|, 0 < t 200,

at point (1.5, 1) in the present scheme and the Newmarks trapezoidal rule. Apparently, the error is the same for the twoschemes, since both schemes are second order accurate.

180 185 190 195 200

-5

0

5

x 10-3

t

w(1

.5,1

,t)

computedexact

0.05 0.1 0.15 0.20

1

2

3

4 x 10-4

h

e=m

ax|w

-wex

|

present scheme Newmark's rule

Fig. 21 Time history of w(1.5, 1, t) in Example 9(t = 0.01).

Fig. 22 Error e = maxt

|w wexact| versus time steph = t.

Example 10. Nonlinear equationsIn order to test the efficiency of the scheme with regard to the time cost, Eq. (3) has been solved for various values of thenumber of the degrees of freedom N and the total time of the motion ttot. The employed matrices M, C, K are symmetricand fully populated and they are randomly generated for each set of values (N, ttot). The computation time for each caseis shown in Table 1. The scheme was programmed in Matlab (R2009a) and the solution with t = 0.01 was obtained on aFujitsu Celsius Series 700 notebook.



Table 1 Computation time in Example 10. Upper row: Newmarks trapezoidal rule. Lower row: present scheme.

ttotsecN

20 40 60 80 100 120 140 160 180 20025 0.439

0.0201.0910.031

1.8790.079

2.9370.146

3.6670.219

4.5320.249

5.2270.309

5.9580.402

6.7520.527

7.5100.652

50 2.1380.039

5.4710.108

8.8210.160

11.9010.260

14.8450.366

17.9040.434

21.7560.594

25.5070.781

27.5830.971

30.6671.148

100 10.7860.091

23.3650.142

35.8280.305

48.1010.435

60.8080.623

72.7600.817

85.3061.114

98.6101.477

116.0441.801

121.5442.110

Example 11.

In this example the numerical scheme is employed to solve a large system of non linear equations of motion, which representthe semi-discretized equations of motion resulting from the analysis of the nonlinear wave equation

u,xxuu,xu = e2x cos2 t + 2ex cos t, 0 x 1, 0 < t (64)

u(x, 0) = ex, u(x, 0) = 0, (65a,b)

u(0, 0) = 1, u(1, 0) = e1. (65c,d)

The AEM is employed with 31 constant elements [19]. This procedure yields a system of 31 nonlinear equations of motionof the form (43) in terms of the nodal values, which are solved using the present numerical scheme. The problem admits anexact solution uex = ex cos t. Figure 23 shows the solution at x = 0.5 as compared with the exact one together with theerror = |u(0.5, t) uexact(0.5, t)|

0 10 20 30 40 50-1

-0.5

0

0.5

1

t

u(0.

5,t)

computedexact

error*103

Fig. 23 Solution at x = 0.5, t = 0.1.

7 Conclusions

An integral equation method has been developed for the numerical solution of second order linear and nonlinear differentialequations. The coefficient matrices may be fully populated, non symmetric and time dependent, i.e. variable. The resultingnumerical scheme is applied to the solution of the equations of motion arising in structural dynamics. The method is simpleto implement. It is self starting, second order accurate for liner systems, unconditionally stable and it does not exhibitamplitude decay or period elongation. On the base of the solution algorithms the method can be categorized as explicit.Several examples are presented, which illustrate the method and demonstrate its efficiency. It is shown that the proposedintegration performs well when large deformations and long time durations are considered in the transient response analysesof structures. It can be used as a practical method for integration of the equations of motion in cases where widely usedtime integration procedures, e.g. Newmarks, do not conserve energy and momentum, and become unstable. The methodcan be readily extended to equations of order higher than two.



Appendix: Pseudo-codes of the algorithmsFollowing the derivation of the solution schemes described in Sections 4 and 5, the algorithms are stated in a pseudo-codetype notation. In this way the reader interested in implementing the method can easily do so in the computer language ofhis preference.

Algorithm 1: Linear equations of motion

A. DataRead: M, C, K, u0, u0, p(t), ttotB. Initial computations

1. Choose : h := t and compute ntot2. Compute : c1 := h2/2 c2 := h q0 := M1(p0 Cu0 Ku0)3. Formulate : U0 :=

{q0 u0 u0

}T

4. Compute : A :=

M C Kc12 I hI I

c22 I I 0

1

0 0 0 c12 I 0 I

c22 I I 0

, b:=

M C Kc12 I hI I

c22 I I 0

1

I0

0

C. Compute solutionfor n := 1 to ntol

Un = AUn1 + bpnend

Algorithm 2: Nonlinear equations of motion

A. DataRead: M, F(u, u), u0, u0, p(t), ttot

B. Initial computations1. Choose : h := t and compute ntot2. Compute : c1 := h2/2 c2 := h q0 := M1[p0 F(u0, u0)]:

C. Compute solutionfor n := 1 to ntol solve for {qn un un}T the system of the nonlinear algebraic equations :

Mqn + F(un, un) = pnhI I

I 0

un

un

=

0 I

I 0

un1

un1

+

c12 I c22 I

qn +

c12 I

c22 I

qn1

end

References[1] T. J. R. Hughes, The Finite Element Method (Prentice Hall, New Jersey, 1987).[2] K. J. Bathe, Finite Element Procedures (Prentice Hall, New Jersey, 1996).[3] D. L. Karabalis and D. E. Beskos, Numerical Methods in Earthquake Engineering, In: Computer Analysis and Design of Earth-

quake Resistant Structures. A Handbook, edited by D. E. Beskos and S. A. Anagnostopoulos (Computational Mechanics Publica-tions, Southampton, 1997).

[4] M. A. Dokainis and K. Subbaraj, A survey of direct time-integration methods in computational structural dynamics I. Explicitmethods, Comput. Struct. 32(6), 13711386 (1989).



[5] K. Subbaraj and M. A. Dokainish, A survey of direct time-integration methods in computational structural dynamics II. Implicitmethods, Comput. Struct. 32(6), 13871401 (1989).

[6] K. K. Tamma, J. Har, X. Zhou, M. Shimada, and A. Hoitink, An overview and recent advances in vector and scalar formalisms:Space/time discretizations in computational dynamics A unified approach, Arch. Comput. Methods Eng. 18, 119283 (2011).

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[8] D. Kuhl and M. A. Crisfield, Energy-conserving and decaying algorithms, Int. J. Numer. Methods Eng. 45, 569599 (1999).[9] K. J. Bathe, Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme, Comput. Struct.

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[15] J. T. Katsikadelis, The BEM for numerical solution of partial fractional differential equations, Comput. Math. Appl. 62, 891901(2011).

[16] J. T. Katsikadelis, Numerical solution of multi-term fractional differential equations, Z. Angew. Math. Mech. 89(7), 593608(2009).

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