Efficient analysis of dynamic response of linear systems

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 12, 521-536 (1984)

EFFICIENT ANALYSIS OF DYNAMIC RESPONSE OF LINEAR SYSTEMS

ANESTIS s. VELETSOS* AND CARLOS E. V E N T U R A ~

Department of Civil Engineering, Rice University, Houston, Texas 77251, U.S.A

SUMMARY

After reviewing briefly a recently proposed procedure for evaluating the dynamic transient response of a classically damped linear system from its corresponding steady-state response, a modified procedure is presented which also appears to be highly efficient for non-classically damped systems of the type encountered in studies of soil-structure interaction. The concepts involved are developed by reference to viscously damped single-degree-of-freedom systems, and numerical solutions are included to illustrate the accuracy and efficiency of the proposed procedure and its superiority over the classical Discrete Fourier Transform approach.

INTRODUCTION

The Discrete Fourier Transform (DFT) in combination with any of the Fast Fourier Transform (FFT) algorithms provide a powerful means for evaluating the dynamic response of linear structural systems. This approach may be used for viscously damped systems, for which the classical modal superposition method of analysis is applicable, as well as for non-classically damped systems of the type encountered in studies of soil- structure interaction.

While highly efficient for evaluating the steady-state response of systems to periodic excitations, the DFT approach may still entail excessive computational effort for evaluating the corresponding transient response, and it may lead to significant errors if it is not properly,implemented. For example, significant errors have been reported for systems whose natural periods are long compared to the dominant period of the excitation.' The resulting errors may be particularly large for lightly damped systems, as the method is inapplicable to truly undamped systems. For a more detailed account of the reasons for these possible inaccuracies the reader is referred to References 1, 2 and 3.

A procedure of high efficiency has been proposed recently4 for calculating the transient response of a linear structure from its corresponding steady-state response to a periodic extension of the excitation. The procedure involves the superposition of a corrective, free vibrational solution which effectively transforms the steady-state response to the desired transient response. In the form in which it has been presented, however, the procedure can be efficiently applied only to classically damped systems. Most problems of soil-structure interaction are not of this type.

The objectives of this paper are ( 1 ) to review briefly the salient features of the previously reported procedure for extracting the transient response of a linear system from its corresponding steady-state response; (2) to present a modification which permits the procedure to be applied efficiently to non-classically damped systems as well; and (3) with the aid of judiciously selected numerical solutions, to illustrate the superiority of the proposed procedure over the classical DFT approach.

The relevant concepts are developed and the method is illustrated by reference to single-degree-of-freedom systems. The extension of these concepts to more complex, multi-degree-of-freedom systems is in the process of being studied, and the results of this effort will be reported as they become available. The corrective

* Brown & Root Professor. f Graduate Student.

0098-8847/84/04052 1-1 6$01.60 @ 1984 by John Wiley & Sons, Ltd.

Received 30 August I983 Revised 30 January I984

522 A. S. VELETSOS AND C. E. VENTURA

solution in the original procedure4 is obtained by use of transient response concepts, whereas in the proposed modification it is implemented by use of steady-state response concepts.

Notation

extensively are summarized in Appendix 11. The symbols used in this paper are defined when first introduced in the text, and those used most

STATEMENT OF PROBLEM

Consider a viscously damped single-degree-of-freedom system subjected to the force, P(t) , shown in Figure l(a), in which td is the duration of the excitation and t o is the time interval during which the response is to be evaluated. To ensure that the maximum value of the response during free vibration is computed, to should exceed t d by at least one-half the natural period of vibration of the system.

Let x(t) be the desired displacement of the system at time t , and y ( t ) be the corresponding steady-state displacement to a periodic extension of the excitation. The period of the latter excitation and of the displacement y( t ) is, of course, to. In comparing the two displacements, the origin of y(t) is taken at t = 0. Possible histories for x( t ) and y ( t ) are shown in Figure l(b). The initial values of x(t) in this plot are considered to be different from zero in the interest of generality.

Starting with the steady-state response of the system over the period to, it is desired to evaluate the corresponding transient response for a specified set of initial conditions. It should be re-emphasized that the steady-state response, which is presumed to be known in this treatment, can be computed quite efficiently by means of Fourier series techniques and any of the FFT algorithms.

FUNDAMENTAL CONCEPT

The response of a system to a prescribed excitation depends on the characteristics of the forcing function, the properties of the system and the initial state of the motion. If the same system is subjected (a) to a transient excitation, and (b) to a periodic version of it, the difference between the resulting responses is due to the differences in the initial states of the two motions. Accordingly, if one response is known the other may be determined by the superposition of a corrective, free vibrational solution which ensures that the initial state of the desired motion conforms to the prescribed state.

P to

(b) -4

Figure 1, Comparison of transient and steady-state responses

DYNAMIC RESPONSE OF LINEAR SYSTEMS 523

REVIEW OF ORIGINAL PROCEDURE

The transient displacement, x(t) , may be related to its steady-state counterpart, y(t), by the equation

x(t) = Y( t ) + (1)

in which ( ( t ) is the corrective displacement representing the effect of the unsatisfied initial conditions. The corrective displacement may, in turn, be expressed as

t(t) = 5(0) + 4m h(t) (2)

in which g(t) and h(t) represent the displacement at time t produced by a unit initial displacement and a unit initial velocity, respectively; and ((0) and l(0) are constants that represent the yet undetermined initial values of t ( t ) and of its first derivative.

The unit response functions, g(t) and h(t), are given by

and

(4) 1

P h(t) = -=e-cP'sinpt

in which p is the circular natural frequency of the system without damping; damping; and

is the fraction of critical

P = P JCl -('I is the damped circular natural frequency of the system.

The initial values of the corrective solution, ((0) and 4(0), are determined simply from

t(0) = 4 0 ) - Y(0)

i (0) = i(0) - j(0)

(54

(5b)

in which x(0) and i(0) are the prescribed initial displacement and initial velocity of the desired transient solution, and y(0) and j (0) are the corresponding values of the known steady-state solution.

Applicability of' procedure The procedure can also be applied to multi-degree-of-freedom systems for which the classical modal

superposition method of analysis is applicable. The prescribed steady-state response in this case is expressed in terms of its modal components; the transient response of each mode is computed from the corresponding steady-state response by repeated application of equations (2) and (l), making use of the impulse response functions g(t) and h(t); and the total transient response is obtained by superposition of the corresponding modal responses. The details are available in Reference 4.

In principle, the procedure can also be extended to non-classically damped systems, but this requires the evaluation of the counterparts of the impulse response functions g(t) and h(t) for the entire system. The precise evaluation of these functions entails a great deal of computational effort and the procedure loses its attractiveness in such cases. With the modification described in the following sections, the procedure may be implemented much more efficiently.

MODIFIED PROCEDURE

The corrective solution, ( ( t ) , in the original procedure is expressed in terms of the transient unit response functions, h(t) and g(t) . This solution can also be expressed in terms of steady-state response components. To this end, let g(t) be the steady-state displacement at a time 0 -= t < t o due to a periodic set of unit displacement changes applied at intervals to, and &(t) be the corresponding displacement due to a periodic set of unit


velocity changes applied at the same intervals. It should be clear that &t) and the first derivative of h(t) are discontinuous at t = 0 and t = to . Specifically,

and a ( O + ) - g ( O - ) = g(O)-g(t,) = 1

$(O+)- i (o- ) = i(0)-L(to) = 1

in which the plus and minus superscripts denote times slightly greater than and smaller than zero, respectively. The evaluation of these functions is considered later. Unless specifically otherwise indicated, t = 0 in the following expressions will refer to a time slightly in excess of zero, and t = t o will refer to a time slightly less than t , . The same interpretation is also implied in equations (5) and (6).

The corrective solution, ( ( t ) , can now be expressed as a linear combination of .g(t) and h(t) as

t(t) = ag(t)+ bT;(t) (7)

in which a and b are constants that may be determined from the solution of the following system of equations:

Equation (8) is obtained by requiring that 4 0 ) = Y(0) + t(0)

i(0) = jt(0) + &O) (9b)

and determining ( (0) and i(0) from equation (7) and its first derivative. With the values of u and b established, the corrective displacement C(t) may be determined from equation

(7), and the desired x(t) may be determined from equation (1). I t is instructive to identify here the physical meanings of the factors a and b in equation (7). If this equation

is evaluated for t = 0 and t = l o , and use is made of equation (6a) and of the fact that &(to) = h(O), one obtains

a = ((0) - &o) (10a)

Similarly, if the first derivative of equation (7) is evaluated for t = 0 and t = to, and use is made of equation (6b) and of the fact that g(0) = g( to) , one finds

( 1 Ob) b = 4(0) - 4@o)

The right-hand member of equation (loa) represents the total discontinuity of ( ( t ) at t = 0, and the right- hand member of equation (lob) represents the corresponding discontinuity of i(t). If these discontinuities are denoted by A((0) and A&O), respectively,equation (7) can be rewritten in a form analogous to equation (2) as

C(t) = W O ) G(t) +A&O) h(t) (1 1)

EVALUATION OF FUNCTIONS y(t) AND h(t) Any one of a number of different approaches may be used to evaluate the periodic response functions g ( t ) and & ( t ) . Two of these are described in the following sections.

The first approach starts from the general solution for free vibration of the system, and leads to closed- form expressions similar to those presented earlier for the corresponding transient response functions, y(t) and !I([). In addition to being applicable to single-degree-of-freedom systems, the resulting expressions, like those of y(t) and h(t) , may be applied to the analysis of multi-degree-of-freedom systems for which the classical modal superposition method of analysis is valid. The second approach makes use of the frequency response function of the system and leads to series expressions for the desired responses. While less direct, the second approach is the more general of the two and can be applied with equal ease to non-classically as well as classically damped multi-degree-of-freedom systems, and to systems with frequency-dependent para-

(81)

SN3ISAS XVBNII dO 3SNOdS3X 3INVNAa szs


E

a 01 fb = 0.5 a u 2- 01 N I

I I I

.-

-2- ft, = I .9 4 fb.3

I S ' * ' : " ' " t ' * ' ' : ' a . " 0.5 t / to I 0 0.5 t/to ' 0

Figure 2. Representative histories of a ( t ) and h(t) for systems with (' = 0.05

0

Figure 3. Alterna!ing step force and associated responses

in which .xs, = P o / k is the static displacement induced by the peak value of the applied force, Po; and k is the stiffness of the system. For td/2 6 t 6 t,,

(19) A t ) = - Y ( t -


The initial values of the steady-state motion are

y(0) = 0 and j(0) = J3pxs,

Accordingly, the values of ((0) and ((0) needed in the original procedure are

((0) = x(0) - y(0) = 0

((0) = i ( O ) - j ( O ) = -J3pxs,

(20a)

(20b)

and

and the values of a = A((0) and b = A((0) needed in the modified procedure are determined from the following expressions, obtained from the solution of the system of equations (8):

( 2 1 a) CW) - L(0)l A((0) = (1 - cos pio) [x(O) - y(O)] - sin pt,,

P and

The results are

and

The corrective solution, ( ( t ) , may now be expressed either in terms of ((0) and ((0) by use of equation (2), or in terms of A((0) and A((0) by use of equation (11). The desired transient displacement is then obtained from equation (1). The results are displayed graphically in Figure 3(b). In particular, the value of <(t) for t = tJ4, determined by either approach, is - 1 . 5 ~ ~ ~ ; the value of y( t ) at that instant is 3x4 and the corresponding value of x(t ) is 1 . 5 ~ ~ ~ .

Solution from frequency response function Let H ( o ) be the frequency response function of the system, defined as the complex-valued amplitude of

the steady-state displacement induced by a harmonic force of unit amplitude and circular frequency o. Further, let H , = H(w,) be the value of H(w) at w = w,, in which w, = mAw; m is an integer; and Aw = 2n/t0.

It can then be shown (see Appendix I) that the periodic response functions &(t) and a(t) for any time t between 0 and to are given by

and

in which 1: = k , is a dimensionless measure of H,, defined by equation (A6) in Appendix I; k is the stiffness of the system; 6, = wJp; and M is the largest value of m considered. The latter value must be such that the associated frequency, w,, is well above the natural frequency of the system and it includes all the frequencies that may contribute materially to the response.

The computation of g( t ) by direct application of equation (24) presents the following difficulties which may adversely affect the efficiency of the procedure: (1) because of the presence of the factor 6, in the numerhtor, equation (24) does not converge rapidly; (2) because g(t) is discontinuous at the ends of the period to, the

528 A. S. VELETSOS AND C . E. VENTURA

series converges to the mean values of the discontinuities at the ends rather than to the desired values on either side; and (3) the values of g(t) near the ends of the period also are affected by the Gibb’s phenomenon. These difficulties may be overcome, however, by rewriting equation (24) in the following form (see Appendix I for derivation):

The discontinuities of g(t) in this expression are incorporated in the first term in parentheses. The end values of the function represented by the series are now continuous, and by virtue of the term dm in the denominator, the series converges quite rapidly, in fact more rapidly than the expression for h(t).

It can further be shown (see Appendix I) that the derivatives of B(t) and h(t) can effectively be computed from

and g(t) = - p2 h(t)

f ( t ) = g(t) - 2[ph(t)

The values of &(O), g(O), g(0) and f (0) needed in the solution of the system of equations (8) may then be determined from equations (23), (25), (26) and (27), respectively, by letting t = 0.

Improved evaluation ofh(O), g(0) and h(0). Since the rate of convergence of equation (23) is not as rapid as that of equation (25), the values of h(t) obtained with a finite number of terms in the relevant series will generally be less accurate than the corresponding values of g ( t ) . The same also is true of the values of i ( t ) and h(t) computed from equations (26) and (27), since they depend on h(t). The solution of the system of equations (8) for the factors a and b in the expression for the corrective solution is sensitive to the initial values of g(t), h(t) and their first derivatives. It is important, therefore, that a special effort be made to improve the accuracy of h(O), g(0) and f(0). A technique for accomplishing this is presented in the following paragraphs.

Let hM(0) be the truncated value of A(0) computed from equation (23) by considering the group of terms through m = M , and let AhM(0) be the contribution of the terms from m = M + 1 to infinity. Then

h(0) = h M ( 0 ) + AhM(0) (28)

The correction AhM(0) is determined from equation (23) to be

in which Re stands for the real part of the quantity that follows. This correction may be approximated as follows.

For large values of m, the dimensionless frequency response function, H:, is given approximately by (see equation (A6) in Appendix I)

Substituting this expression into equation (29), one obtains

to “ 1 2x2 m = M + l m

A h M ( 0 ) = -- C

and noting that the first series within the parentheses is given by n2/6, equation (31) may be rewritten as

Tabl

e I.

Four

ier

seri

es a

ppro

xim

atio

n an

d ex

act

valu

es o

f Q

(t) a

nd p

h(t)

for

syst

ems

with

i =

0.0

5

Dim

ensi

onle

ss t

ime,

f/to

Solu

tion

O*

0 0.

1 0.

2 0.

3 0.

4 0.

5 0.

6 0.

7 0.

8 0.

9 1

1*

(a) V

alue

s of

g(t)

ft,

= 0.5

M=

4

0.53

9 0.

513

0'438

0.

323

0.18

0 0.

025

-0.1

28

-0.2

64

-0.3

70

-0.4

38

Exac

t 0.

539

0.51

3 0.

438

0.32

3 0.

181

0.02

5 -0

.128

-0

.264

-0

.370

-0

.438

ft

o =

1

M=

4

3.70

3 3.

058

1.31

5 -0

.804

-2

.499

-3

.164

-2

.617

-1

.136

0.

678

2.12

6 M

=8

3.

703

3.05

5 1.

316

-080

5 -2

.499

-3

.164

-2

.618

-1

.136

06

77

2.12

8 Ex

act

3.70

3 3.

056

1.31

6 -0

.805

-2

'499

-3

.164

-2

.618

-1

.136

0.

677

2.12

8 ft,

= 3

M

=4

1.

631

-0.3

35

-1.1

72

0958

0.

419

-1.0

18

0.21

9 0.

717

-0.5

71

-0.2

95

M=

8

1.63

6 -0

'366

-1

,150

09

48

0.42

9 -1

.021

0.

215

0.72

1 -0

.586

-0

.273

M

=1

6

1.63

7 -0

.364

-1

.152

09

45

0.42

7 -1

.021

0.

217

0.72

5 -0

.584

-0

.275

Ex

act

1.63

7 -0

.365

-1

.152

0.

945

0.42

6 -1

.021

0.

217

0.72

4 -0

.584

-0

.274

(b) V

alue

s of

$(t

)

ft, =

0.5

M

= 4

0.

036

0.00

1 0.

158

0.31

3 04

13

0.48

6 0.

495

0.47

1 0.

387

0.28

5 0.

138

M =

8

0.02

0 O

QO

1 0.

168

0310

04

17

0482

04

98

0467

0.

392

0282

01

49

M =

16

0.01

1 0.

001

0.16

6 0.

308

0,41

7 0.

482

0.49

8 0.

467

0.39

2 0.

280

0.14

6 Ex

act

0.00

1 0.

165

0.30

8 04

17

0.48

2 0.

499

0.46

7 0.

392

0.28

0 0.

146

ft, =

1

M =

4

-0-0

07

-0.0

78

2.03

5 3.

300

3-22

9 1.

997

0072

-1

.734

-2

.817

-2

.760

- 1.

724

M =

8

-0.0

41

-0.0

79

2.05

6 3'

294

3.23

8 1.

987

0.07

8 -1

.744

-2

.808

-2

.767

- 1.

702

M =

16-

0060

-0

.079

2.

051

3.29

0 3.

238

1.98

9 0.

079

-1.7

42

-2.8

08

-2.7

71

- 1,

707

Exac

t -0

.079

2.

050

3.29

1 3.

237

1.98

9 0,

080 - 1.

742

-2.8

08

-2.7

70 - 1.

708

Po =

3 M

= 4

0.

228

0.01

7 1.

361

-0.7

31

-078

8 1.

099

-0.0

10

-0.8

55

0.43

4 0'

524

-0.7

32

M =

8

0.09

3 -0

.020

14

48

-0.7

65

-074

6 1.

056

0.02

0 -0

.899

0.

478

0.48

5 -0

.639

M

= 1

6 0.

034

-0.0

24

1.43

1 -0

.780

-0

.748

1.

062

0.02

6 -0

.893

0.

477

0.47

2 -0

.656

M

= 3

2 0.005

-0.0

25

1.42

6 -0

.777

-0

,749

1.

061

0.02

7 -0

.894

0.

477

0.47

4 -0

.661

Ex

act

-002

5 1.

427

-0.7

77

-0,7

49

1.06

1 0.

028

-0.8

94

0.47

7 0.

474

-0.6

60

* Res

ults

obt

aine

d fr

om e

quat

ion

(23)

. Rem

aini

ng d

ata

of h

(t) fo

r t =

0 a

nd t

= to

com

pute

d fr

om e

quat

ion

(28)

.

- 0.

46 1

- 0.

46 1

2.70

3 2.

703

2.70

3

0.63

1 0.

636

0.63

7 0.

637

0.00

1 0.

001

0.00

1

0.00

1

- 00

78

- 0.

079

- 00

79

- 0.

079

0017

- 0.

020

- 0.0

24

- 0.

025

- 00

25

0.03

6 00

20

0.01

1

- 0~

007

-0.0

41

- 0.

060

0.22

8 0.

093

0.03

4 0.005

530 A. S. VELETSOS A N D C . E. VENTURA

The values of i ( 0 ) and h(0) should be computed from equations (26) and (27) using the improved value of h(0). The same technique may also be used to improve the values of ;(to) and &(to), keeping in mind that &(to) is the same as h(0).

Discussion. The period to of the functions g7t) and h(t) is the same as the time interval over which the response of the system to the prescribed excitation is to be evaluated. The longer this period, the smaller Am, and hence the larger is the number of terms that must be considered to cover the frequency range of interest.

In principle, equations (23) and (25) may also be used to evaluate the transient response functions, g(t) and h(t) , needed in the implementation of the original p r ~ c e d u r e . ~ However, to compute the latter functions accurately, it will in general be necessary to consider for to a value to* which may be several times greater than the duration over which the response of the system is desired. Specifically, to* must be long enough such that the response of the system at the end of the perod is negligible and there is no substantial difference between the transient and steady-state responses. The longer the natural period of the system or the smaller its damping, the larger will generally be the required to*. It is in this respect that the original procedure is inferior to the modification presented.

As an illustration of the accuracy with which the function g(t) and 6(t) may be evaluated from the frequency response function of the system, in Table I are presented the results obtained for systems with [ = 0.05 and several different values offto using progressively more terms in equations (25) and (23). Included for purposes of comparison are also the exact solutions obtained from equations (15) and (17). Finally, to demonstrate the effectiveness of equations (28) and (32), the values of &(t) for t = 0 and t = to were computed both from these expressions and by direct application of equation (23).

It can be seen that both functions, particularly g( t ) , converge to the exact answers rather rapidly, and that the smaller the value offt,, the smaller is the number of terms that must be considered to limit the error to a prescribed tolerance. Asfi, tends to infinity, g( t ) and 6(t) tend to g(t) and h(t), respectively, and there is no special advantage, as previously intimated, in using the modified procedure over the original one. Note further that the terminal values of h(t) may be significantly in error unless the improved computational scheme is used.

EFFICIENCY OF MODIFIED PROCEDURE

As a measure of the computational efficiency of the modified procedure, the maximum responses of a number of systems subjected to the sinusoidal force pulse shown in Figure 4 were evaluated, and they are compared in this section with those obtained by the classical DFT approach. Three different values of the frequency parameter, f t d , and two different functions of critical damping, [ = 0.01 and 5 = 0.05, were considered.

The forcing function in these solutions was sampled at a constant time interval, At, for a total of Nd intervals as shown in Figure 4, and several different values of Nd were used. Strictly speaking, the response of the system is desired over a period to = N A t which exceeds the duration of the excitation by only one-half the natural period of the system, T = llf. In the solutions by the classical DFT approach, however, a much longer period must be employed. The steady-state displacement of the system, y(t) , was evaluated by (a) determining the DFT of the forcing function; (b) multiplying the resulting sequence by the corresponding ordinates of the frequency response function of the system, H(m); and (c) taking the inverse DFT of the resulting product.

Let P(tn) be the ordinate of the discretized forcing function at t , = nAt, in which n is an integer and At is a constant time increment. When expressed in a discrete Fourier series, P(t,) is given by

in which r is an integer ranging from 0 to N - 1, and I N - i

DYNAMIC RESPONSE OF LINEAR SYSTEMS 53 1

Figure 4. Sinusoidal force pulse considered

The complex-valued amplitude of the steady-state displacement of the system due to the rth component of the exciting force is then given by

= H, F, (34) in which Hr = H ( o r ) ;

for O<r ,<N/2 - ( N - r ) A w for N / 2 < r , < N - 1

and Am = 2n/to = 2n/(NAt). Finally, the steady-state displacement of the system at t, is given by the series

Equations (34) and (35) can be evaluated efficiently by any of the available FFT In this study, the Cooley-Tukey algorithm,’ which requires that N be a power of 2, was used.

With y(t) established, the desired transient response x(t) in ihe proposed procedure was obtained by superimposing the corrective solution, t(t). The latter solution was expressed in terms of the periodic response functions, y(t) and h(t), determined from equations (25) and (23), respectively.

The transient response of the system was also evaluated by the classical DFT approach. To ensure that the results were reasonable approximations of the desired transient response values, x(t) , rather than of their steady-state counterparts, y(t), the period of definition of the exciting force, to, was augmented by the addition of a grace band of zeros at the end, and the DFT of the augmented force sequence was evaluated. Increasing to naturally increases the number of required operations.

The results are presented in Table 11, along with the corresponding exact responses. The solutions by the proposed procedure were obtained for a duration of free vibration equal to the duration of the forcing function, N = 2N,, using three different values of N,. The solutions by the classical DFT approach were obtained for the same three values of Nd using six different values of N / N , in the range between 2 and 64. For systems withft, = 0 5 , a value of N = 2Nd ensures that the response is computed for a period of free vibration equal to one-half the natural period of the system, as actually desired. For values of N > 2N, and for systems with f td>O*5, the free vibrational period was substantially larger. For example, a value of N = 32Nd corresponds to a free vibrational period of 15.5 times the natural period of the system whenft, = 0 5 , and of 93 times the natural period whenft, = 3.

The data in Table I1 clearly show that the accuracy of the proposed procedure depends exclusively on the number of points, Nd, used to discretize the forcing function, and that good accuracy can be achieved with a relatively small number of points. The period of free vibration that must be considered in this approach is determined by the time over which the response is desired, subject, of course, to the restriction on N imposed by the FFT algorithm being used.

By contrast, the accuracy of the solution by the classical DFT approach depends importantly on the duration of the grace band of zeros considered. The lower the value of the frequency parameter, ffd,


Table 11. Values of I x,,, l/xs, for systems subjected to sinusoidal force pulse

N -

Nd

~~~

2 4 8

16 32 64

2

2 4 8

16 32 64

2

f t d = 0.5 ._

Nd = 32 N d = 6 4 Nd = 128 Nd = 32 N d = 6 4 N d = 128

jid = 3

21.167* 10.91 6* 5.80 1 * 3,263* 2.033* 1.488*

1.290*

4.228* 2.444* 1.595* 1.242* 1.150* 1.142*

1.143*

21.206* 10*936* 5.81 1* 3.269* 2.037*

1.291*

1.292*

4.239* 2.450* 1.598* 1.244* 1.152*

1.144*

1.145*

(a) Systems with [ = 0.01 Classical DFT approach

21.215* 25.785 25.785 25.785 10.94 l* 13-702 13.702 13.702 5.814* 7.707 7.707 7.707 3.270* 4.802 4802 4.802

3.516 3.516 3.101

Proposed procedure 1.292* 3.045 3.045 3.045

Exact solution 3.045

(b) Systems with [ = 0.05 Classical DFT approach

4.24 1 * 5.780 5.780 5.780 2.45 1 * 3.770 3.770 3.770 1.599* 2.936 2.936 2.936 1.245* 2.717 2.717 2.717

2.699 2699 2.699

Proposed procedure 1.145* 2.699 2.699 2.699

Exact solution 2.699

1.326 1.322 1.321 1.424 1419 1-417 1.470 1.464 1.462 1.488 1.481 1.479 1.492 1.485 1.492

1.487 1.484 1.483

1.483

1.341 1.337 1-336 1.415 1.409 1.408 1.425 1.419 1.419 1.426 1.420 1.419 1.426 1.420 1.426

1.421 1.419 1-419

1.419

*Maximum occurs during free vibration. Remaining maxima occur during forced vibration.

or the lower the damping factor of the system, i, the longer is the required duration of this band; therefore, the greater is the computational effort involved.

As an illustration, note that, for both values of damping considered, the peak responses computed by the proposed procedure forft, = 0-5 using Nd = 32 and N = 2Nd agree to three significant figures with the exact responses. To obtain the same agreement with the classical DFT approach using the same value of Nd, it is necessary to consider a total of N = 2,048 points for systems with = 005 and a much larger number of points for systems with { = 0.01. It should be clear that the proposed procedure is significantly more efficient than the classical DFT approach, particularly for low-frequency and lightly damped systems.

The efficiency of the proposed procedure can be further improved by evaluating the combination of the steady-state response and the corrective solution concurrently rather than sequentially. Specifically, the components in the frequency domain of the elements of the corrective solution must be added to the corresponding components of the response due to the forcing function, and the inverse DFT of the combination must be taken rather than of its component parts. This technique will reduce the computational effort by about 30 to 40 per cent and will significantly decrease the required computer memory. The solutions presented in Table I1 were ,obtained in this manner.

CONCLUSION

The limitations of a recently proposed, efficient procedure for evaluating the transient response of a viscously damped linear system from its corresponding steady-state response have been identified, and a modified procedure has been presented which significantly extends the range of appliability of the procedure. In


particular, the modified procedure appears to be capable of extension to non-classically damped systems of the type encountered in studies of soil-structure interaction. The basic concepts involved have been introduced by reference to single-degree-of-freedom systems, and the procedure has been shown to be considerably more efficient than the classical DFT approach, particularly for low-frequency and lightly damped systems.

ACKNOWLEDGEMENTS

This study was motivated by a research project on the dynamics of offshore structures which is sponsored at Rice University by Brown & Root, Inc., and it was partly supported by that project. The project has been under the administrative oversight of Dr. Damodaran Nair, whose co-operation is acknowledged with thanks. Appreciation also is expressed to the reviewers for their constructive comments.

APPENDIX I

Derivation of expressions for g ( t ) and h(t) Consider a viscously damped single-degree-of-freedom system excited by periodic sets of displacement

changes, Ay, and velocity changes, Aj, applied concurrently at intervals, to. Let y = y( t ) be the resulting steady-state displacement of the system at any time t during a typical period, say 0 < t < to. The equation of motion of the system is then

r n o j + c j + k y = 0 ('41)

in which rno, c and k are the mass, coefficient of damping and stiffness of the system, respectively, and a dot superscript denotes differentiation with respect to t .

Consider now a periodic fhnction z = z(t) defined in the interval 0 G t < to such that

in which the plus and minus superscripts denote values slightly greater than and less than zero, respectively. The derivatives of z at any time 0 < t <to may then be expressed as

in which d(t) is the unit impulse function, and 6(t) is its derivative, the so-called unit doublet. When expressed in terms of z and its derivatives, equation (Al) becomes

m 0 Z + c i + k z = Q(t) 043)

(A4)

in which Q(t) is an equivalent set of periodic lateral forces given by

Q(t) = rno{AyCh(t) + 2bYt) l+ ALd(t))

It can be seen that the velocity changes are equivalent to a set of periodic impulses, each of magnitude rn, A j , and that the displacement changes are equivalent to a set of periodic doublets of magnitude rno Ay combined with a set of impulses of magnitude 2(prn0Ay. The quantities p and 5 in these expressions represent the circular natural frequency of the system, J(k/rn,), and the fraction of critical damping, c/(2rn0 p ) , respectively.

With the initial conditions thus replaced by equivalent lateral forces, the steady-state response y(t) = z( t ) for 0 < t < to can be determined by application of well-established Fourier series techniques. For a more detailed description of this method of replacing the prescribed initial conditions by equivalent lateral forces, the reader is referred to Reference 8.


Let H(wm) be the frequency response function of the system, defined by

1 H(wm) = - H ; k

where HZ is a complex-valued dimensionless function given by

and

The steady-state response of the system may then be expressed as M

z ( t ) = y( t ) = 1 Q(w,) H(wm) ei"-' for 0 < t < to m = - M

in which M is an integer and I rto

Q(wm) = Q(t)e-i"mtdt to 0

On substituting equation (A4) into equation (A9), replacing the result obtained after integration into equation (AS), and making use of the expression p = ,/(k/mo), one obtains the following expression for y( t ) within 0 < t < to:

in which w, = mAw = 2nm/to. The response function g(t) may now be obtained from equation (A10) by letting A y = 1 and A y = 0, and

the response function h(t) may be obtained by letting A y = 0 and A j = 1. The results are presented as equations (24) and (23) in the body of the paper.

Improved expression for g(t)

l/(i&) is added and subtracted from the remainder of the series, one obtains If the term corresponding to m = 0 is extracted from the series expression in equation (24) and the term

(A1 la)

Now, on noting that H,* = 1 and making use of equations (A6) and (A7), one can rewrite equation (A1 la) as

(A1 lb)

Finally, the first series on the right-hand member of this expression can be expressed as ( i - t / to ) to yield equation (25) presented in the body of the paper.

Expressions for g(t) and h(t) The derivative of y( t ) at any time O < t < to is obtained from equation (A2b)

j ( t ) = i ( t ) - AyS(t) (A121

in which i(t) may be determined by differentiation of equation (A10). On substituting the result of this differentiation into equation (A12), expressing S(t) by its corresponding Fourier series, taking A y = 1 and


A j = 0, and replacing j ( t ) by g( t ) , one obtains

1 M l M g(t) = - 1 (i2r4, - 6;) H: e'"*' - - 1 ei"*' (A 13)

t o m = - M t o m = - M

This expression may now be simplified to the following by adding and subtracting unity inside the first series, making use of equation (A6), and cancelling common terms:

Finally, on comparing equation (A14) with equation (23) in the body of the paper, one establishes the relationship between g(t) and A(t) given as equation (26).

The expression for h(t) is obtained in a similar manner from equations (A12) and (A10) by taking Ay = 0 and A j = 1. The result is

which. on substituting into equation (24) and making use of equation (23), leads to equation (27).

APPENDIX I1

constants in equation (7) constants in equation (12) coefficient of viscous damping for system Discrete Fourier Transform natural frequency of system Fast Fourier Transform displacement of system due to a unit initial displacement change steady-state displacement of system due to a periodic set of unit displacement changes applied at intervals to displacement of system due to a unit initial velocity change steady-state displacement of system due to a periodic set of unit velocity changes applied at intervals to complex-valued frequency response function of system H(w,) = value of H(w) at o = w, kHm = dimensionless measure of Hm

stiffness of system variable integer mass of system number of terms considered in a series variable integer number of time or frequency intervals number of intervals in which forcing function is discretized circular natural frequency of system without and with damping, respectively instantaneous value of exciting force P(t,) = value of P ( t ) at t , peak value of P ( t ) variable integer time

J-1

A. S. VELETSOS AND C. E. VENTURA

duration of excitation nAt = discretized time period during which the response is to be evaluated; also period of periodic extension of excitation transient displacement of system static displacement induced by Po steady-state displacement of system value of y( t ) at t = t , = nAt periodic response defined for 0 < t < to

unit impulse function derivative of d(t) or unit doublet t , /N = time increment 2 4 t , = frequency increment periodic displacement and velocity changes, respectively fraction of critical damping corrective displacement

circular frequency of excitation mAw = sample of u

O m l P = m/!o

REFERENCES 1. 1. W. Meek and A. S. Veletsos, ‘Dynamic analysis by extra fast Fourier transform’, J. eng. mech. diu. ASCE 98, 367-384 (1972) 2. G. D. Bergland, ‘A guided tour of the fast Fourier transform’, I E E E spectrum 41-52 (1969). 3. S. C. Liu and L. W. Fagel, ‘Earthquake interaction by fast Fourier transform’, J . eng. mech. diu. ASCE 97, 1223-1237 (1971). 4. A. S. Veletsos and A. Kumar, ‘Steady-state and transient responses of linear structures’, J. eng. mech. diu. ASCE 109, 1215-1230

(1983). 5. K. W. Dotson, A. S. Veletsos and C. E. Ventura, ‘Dynamic response of simple linear systems to periodic forces’, Structural Research

at Rice, Report No. 26, Department of Civil Engineering, Rice University, Houston, Texas, 1983. 6. E. 0. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, New Jersey, 1974. 7. J. W. Cooley and J. W. Tukey, ‘An algorithm for the machine calculation of complex Fourier series’, Mathematics of Computation,

8. R. Bracewell, The Fourier Transform and its Applications, McGraw-Hill, New York, 1978, pp. 232-236. 19, 297-301 (1965).

Efficient analysis of dynamic response of linear systems

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