The covariance response of linear systems to non-stationary random excitation

J. Sound Vib. (1971) 14 (3), 385-400

THE COVARIANCE

RESPONSE

NON-STATIONARY

OF LINEAR

RANDOM

SYSTEMS

TO

EXCITATION”f

J. B. ROBERTS National Physical Laboratory, Teddington, Middlesex, England (Received 4 December 1969, and in revisedform 3 August 1970)

An approximate method of calculating the covariance response of linear, time-invariant systems to a class of non-stationary excitation processes is given, which takes full advantage of frequency domain relationships. It is shown that the covariance response can be built up from harmonic components, in a simple way, to any required degree of accuracy. As an illustration of the method, it is used to determine the response of an idealized model of a ten-storey building to non-stationary ground acceleration.

1. INTRODUCTION Barnoski and Maurer [l] have recently studied the variance response of a simple, linear oscillator to non-stationary random excitation of the form Y(t) = A(t)Z(t),

(1)

where A(t) is a modulating function and Z(t) is a stationary process with zero mean. By working in the frequency domain, rather than adopting the conventional time domain approach (e.g. see [2]), they were able to obtain exact solutions for a few simple cases. It is clear from this work, however, that the analytical solution of more complicated problems, where A($) is not a simple function of time, or the system has more than one degree of freedom, will involve considerable algebraic difficulties. In addition, it seems difficult to compute the covariance response by extending the exact method of [l], except in the simplest cases. In many practical applications, such as earthquake design and gust response prediction, where it is reasonable to model the excitation process by equation (I), the appropriate A(t) function is effectively zero except in a short time interval; in other words, the excitation consists of a short burst of random excitation. This paper describes an approximate technique for calculating the covariance of the displacement response of linear systems to excitation of the form of equation (I), which is useful when A(t) has such a “pulse shape.” It is shown that, by suitably approximating the excitation process, the covariance response of a simple oscillator to any pulse of excitation of the form of equation (1) can be easily built up in stages, to any required degree of accuracy. Furthermore, the introduction of a modal approximation leads to a general method of dealing with multi-degree-of-freedom systems. As in reference [ 11, a frequency domain method is used, but by adopting an approximate rather than an exact approach, full advantage is taken of working in this domain. For simplicity, the case where Z(t) is a white noise will be considered first; later on, it will be shown that the method can be extended to situations where Z(t) is a correlated process. t This work was conducted while the author was at the Department of Mechanical Engineering, University College London, Gower Street, London, W.C. 1. 385

386

J. B.

ROBERTS

2. UNCORRELATED

EXCITATION

It is convenient to begin with the following well-known input-output linear system excited by a non-stationary process (e.g. see [3]) : FAN,, fz> = f f M, - 4 4tz - 4 -co -co

relationship for a

wy(r,, 4 d7, h.

(2)

Here, h(t - T) is the impulse response function of the system, giving the response at time t to a unit impulse applied at time T. w,(t,, t2) and w,,(tl, t2) are, respectively, the covariance functions for the displacement response process, X(t), and the excitation process, Y(t). If Z(t) is a white noise, of unit “strength”, the covariance function for Y(t) is, from equation (l), w,(t,, tz) = E(A(t,)A(t,)Z(t,)Z(t,)} =

Z(t).

qt,

-

f2),

(3)

where Z(t) = P(t)

(4)

is the so-called “strength function” of Y(t). When this expression is substituted into equation (2) we find that W&l 2f2)

h(t, - T) h(t2 - T) z(T) dT.

=

(5)

-co

Since we are concerned with situations where A(t), and hence Z(t), is zero except in a short interval of time, we can represent Z(t) in the form Z(t) = f C(Q)eintdQ. -m

(6)

On substituting the above expression into equation (5), and rearranging, we obtain w,(t,, f2) =

C(Q) ei*C(t1+t2)‘21j?(Q, T) dQ,

f

(7)

-CC where p(fi,

T) =

f

h(U

-

e+o”du

T/2) h(24 + T/2)

(8)

-Co and T=

Furthermore,

t2 -

t,.

(9)

since we have the relationship m

h(u) =

&

s

a(w) eiwu dw

(10)

-m

between h(u) and the frequency response function, a(w), of the system, it is easy to show that /3(9,T), which we shall call the “harmonic covariance function”, can be expressed in the alternative, and generally more useful, form

B(fi, T) =

&f -cc

a*(w -

5212)a(w

f

Q/2) eiwr dw.

(11)

COVARIANCE

RESPONSE OF LINEAR SYSTEMS

387

It is noted that in the special case of stationary excitation, where Z(f) = Z,a constant, we have C(Q) = 16(Q), and equations (7) and (11) reduce to (12) which is a well-known expression [4]. The frequency domain relationship given by equation (7), together with equation (1 l), offers several advantages over the conventional time domain relationship given by equation (5). (i) It is possible to deal directly with the frequency response function of the system, a(w), which is a more convenient function to measure, interpret and analyse than its Fourier transform, h(u). (ii) The powerful technique of contour integration can often be used to evaluate /3(Q, 7) analytically, as will be shown later. (iii) In most cases of practical concern, C(Q) and /~(Q,T) are well-behaved, smooth functions, the moduli of which tend to zero rapidly as Q becomes large. The integral in equation (7) is, therefore, usually rapidly convergent. A simple approximate method of evaluating the integral in equation (7), which has a direct physical significance, is obtained from the following reasoning. Suppose that, instead of considering a single pulse of white noise, we conceive of a train of such pulses, each stat-. istically identical to the single pulse, and spaced at regular intervals apart. If the spacing between these pulses is sufficiently great, the system will respond to each pulse in the train in a manner almost identical to that in which it would respond to an isolated pulse. Thus the “overlapping error” will be small and the statistics of the response to one pulse in the train. will approximate to the statistics of the response to an isolated pulse. The degree of accuracy of this approximation will obviously improve as the spacing between the pulses is increased, provided that the system is damped to some extent. For a train of regularly spaced pulses of uncorrelated excitation, the strength function Z(t) will be periodic, with period T, say. Z(t) can therefore be decomposed into harmonic components, as follows : Z(t) = 2 Cneinpr, *=-co

(13)

p = 297/T.

(14)

where Equation (13) can be regarded as an approximation to equation (6) whereby the continuous function C(Q) is replaced by a “Dirac comb”, concentrated at the frequencies 2rrn/T (n = 0, 1, . . .). On substituting equation (13) into equation (5) and proceeding as before, we find that w,(t,, f,) =

2

Cneinp(tl+t~)2)/3(np, T). “=-CC

(15)

The problem of computing w,(c,, t2) has thus been reduced to a simple summation, which is a convenient approximation to equation (7). In any application of this technique one cannot, of course, include all the harmonic components in the computation: one sums from n = -m to n = +m, say. However, as we have already remarked, [C(Q)1 and ]/3(Q,T)] tend to zero, usually quite rapidly, as Sz becomes

388

J. B.

ROBERTS

large, so that /C,l and 1/3(rzp, 7)I will be small when n is large. There is usually, therefore, little error in neglecting higher frequency components in the summation given in equation (15). The accuracy of the harmonic technique improves as the spacing of the pulses in the train increases. Such an increase will bring in more low frequency harmonics and lead to an increase in computational effort. Any required degree of accuracy can be obtained by suitably choosing the pulse spacing, and by taking into account a sufficient number of components. For the purpose of assessing failure probabilities, it appears [5,6] that a knowledge of the following second moments is important: w,,(t,, tJ = E{X”-“(t,)

P-“(t,)

(r, s =

1,2,3).

(16)

Here, X(‘)(t) denotes the rth derivative of X(t) with respect to time (assuming that this derivative exists, in some sense). These moments can be derived from w,(tl, t2) [= w1I(t,, tz)] by using the relationship

To avoid awkward numerical differentiation techniques, it is convenient to compute w,,,(t,, tz) directly. Again, the harmonic technique can be used. The appropriate harmonic covariance function for w,,(t,, f2), denoted /$&), T), is given by a&?, 7) = 2’r;1 [--i(o - n/2)1’-’ [i(o + G/2)1”-’ E*(O - Q/2) CL(W + Q/2) eior dw -CO

(18)

and we have P&z 4 = &(Q, -4 = /W-Q, 4.

(1%

The following simple relationship is useful if wI2(t, t) is required : /%z@,O)

2.1.

SINGLE-DEGREE-OF-FREEDOM

=

wY2)/3,d.n,O).

Gw

SYSTEMS

Consider a linear, single-degree-of-freedom, mechanical system with viscous damping, A conceptual model of such a system is shown in Figure 1. If X(t) is the relative displacement between the mass and the vehicle, and Y(t) is the random acceleration of the vehicle, the appropriate equation of motion is X(t) + 259 k(t) + OJ:X(t) = -Y(t),

(21)

where 5 is the damping ratio and w1 is the natural frequency. For this system, we have E(X)w; = -l/(1 - x2 + 2i
(22)

A= O/O*.

(23)

where

Figure 1. A simplelinear oscillator.

COVARIANCE

RESPONSE

OF LINEAR

389

SYSTEMS

On substituting the above expression for IX(W)into equation (18), and integrating by residue theory, we obtain, for T 2 0 [see equation (19)] -ie-wlr(5+iy/2) u;+“-~[Q& - 5’cos WT+ R,, sin &T] (24) kLs(Qn, 4 = 22/1-52(y-2i5)(1+i5y-y2/4) ’ where Y=J+I (25) and

QII = 1,

RI I = (5 + bd2),

QU = iyP,

R,2 = (1 + 3iy5/2 - y2/2),

Q21 =

W,

R2, = (1 + i&/2), R22 = (5~12 - i - i c2 y - iy/2).

Q22 = (1 + i5r - y2/2), cz is the damped natural frequency

&=2/l-&U,.

(26)

On setting D = 0 in equations (24) to (26), we recover the well-known stationary results:

(27:) WI 2(T)

=

W2, (T)

=

(28)

0,

The harmonic variance functions, obtained by setting Bri(sz,o) = [2wj(l + icy -&)(y /%Z(~,O) = /32l(.n,O) =

0,

have a fairly simple form:

- 2i5)]’

[40:(1 + i5y _Yr2/4)(y - 2i5)]’ + icy -

-i(l

is22(sz70)=[2w,(l

T =

+ icy-

y2/2) y2/4)(y-2i5)]’

(30)

(3 1)

(32:)

These complex functions have some interesting properties, which are fully described in [7] [see also [3] for a discussion of /3,,(1;2.O)]. 2.2. TWO-DEGREE-OF-FREEDOM SYSTEMS An exact determination of the harmonic covariance function for the displacement response of a two-degree-of-freedom linear system seems to be prohibitively difficult. It is possible, however, to obtain an exact expression for /3i,(L?,O) [and hence /3,2(sZ,0), see equation (20)] by using a result due to Phillips [8]. This expression is a generalization of the corresponding stationary result [i.e. B, ,(O,O)] deduced by Crandall and Mark [4] (see also [9]). The general form of the frequency response function E(W) for the displacement response of a two-degree-of-freedom system can be written as cl(W) =

w2 B2 + ioB, + B, W4A4+iW3A3+W2A2+iwA,+Ao’

(33)

390

J. B. ROBERTS

where the coefficients A, and B, are dependent on the system parameters. On substituting the above expression for a(o) into equation (1 l), and employing Phillip’s result, we obtain a result in the form p

((J

11

0)

=

7

4Q) + WJ)

(34)

c(i-2)+ id(Q) ’

where a(Q), b(Q), c(Q) and d(G) are related to A, and B, in the manner given in Appendix I. 2.3.

MULTI-DEGREE-OF-FREEDOM

SYSTEMS

It is difficult to evaluate /3,&J, T) exactly for a system with many degrees of freedom because of the complicated form which K(W) can assume. In principle, the technique of contour integration offers a solution but the formidable algebra required will, in most cases, make this approach intractable. A suitable approximate technique is needed. By expressing the equations of motion in terms of principal co-ordinates, and neglecting the cross-coupling terms due to damping, any frequency response function for the displacement of an n-degree-of-freedom system can be approximated as

4W>= r=l i src&J),

(35)

where Q,(W)= l/&J,2 - w2 + 2i<,w,w).

(36)

5, is the rth modal damping factor and w, is the rth natural frequency of the undamped system. S, are coefficients which are related to the system parameters. On substituting the above expression for CC(W) into equation (18) we obtain

rBr.~(~,~) = j=l 2

Ii

k=l

Sj

skZ!,k(s2,7),

(37)

where m z;‘,(Q, T) = 2; / [-i(W - Q/2)1’-’ [i(W + Q/2)1”-’ ,:(0J-C0

G/2) ‘+‘(w+ fi/2)eiw’dw.

(38)

The above integral can be readily evaluated by residue theory, and the resulting expression is given in Appendix II. In the stationary case (0 = 0) we recover the results given by Robson

WI.

When j = k we have MQ, 7) = L(Q, T),

(39)

as defined by equation (18) [setting cc(o) = ad]. The expressions given previously for a single-mode system [equations (24) to (26)] are therefore applicable to the n “direct terms” in equation (37). The cross-modal terms (j # k) are such that I;‘,@, T) = z;#2, -7) = I$(-Q,

T).

(40)

Some idea of the accuracy of the modal solution can be gained by comparing the exact values of /3, i(Q,O) for a two-degree-of-freedom system, given previously, with values obtained from the corresponding modal approximation. For the system shown in Figure 2 (also considered in [4] and [9]) the author has examined two fi,,(G,O) functions, one relating to the relative displacement between the mass and the vehicle, and the other relating to the relative displacement between the masses. The exact variation with LJ of these two harmonic variance functions is discussed in [7] and compared with results from the modal analysis. The accuracy of the modal approximation was found to be remarkably good over a wide range of system

391

COVARIANCE RESPONSE OF LINEAR SYSTEMS Vehicle c

Figure 2. A linear, two-degree-of-freedom

system.

parameters. This limited evidence suggests that the modal approximation may be satis factory, for engineering purposes, for systems with many degrees of freedom. 2.4. AN EXAMPLE To illustrate the harmonic technique, we will consider the linear system shown in Figure 3, which is an idealized model of a ten-storey building. Each floor has the same mass, M. The interfloor stiffness, defined in terms of resistance to horizontal displacement, is taken to vary linearly, as shown, from k at the top to 1Ok at the bottom. If Y(t) is the random horizontal acceleration of the ground and xi(t) is the displacement of the ith floor (i = 1,2,. . ., IO), relative to the ground, the stochastic differential equations of motion for the undamped system may be written as Mfi(t)

MX,(t) + 19kX,(t) - 9kX,(t) =-Y(t), - (10 - i)kX,+,(t) =-Y(r) (i = 2,3,. . .) 9), =-Y(t). M&,(t) + kX,,(t) -k&(t)

+ (21 - 2i)kX,(t)

- (11 - i)kX,_,

(41) i

We will focus attention on the relative horizontal displacement of the top floor, X,,(t), in future referred to simply as X(t). M L

M

k

M

2h 3k

M

L r

1

M

4k

M

5k

M

6h

M

7k

M

0h

M

9k . IOh ,I-

,,

.:

Figure 3. An idealized model of a ten-storey building.

It will be assumed that the building is viscously damped. In general, such damping will lead to cross-coupling terms, when the equations of motion are expressed in principal co-ordinates. For simplicity, these coupling terms will be ignored and a modal damping factor ci = < (i= 1,2,..., 10) of O-1will be assigned to each mode. The frequency response function, a(w), relating X(t) to Y(t) can then be expressed in the series form given by equations (35) and (36). 26

J. B. ROBERTS

392

An eigenvalue analysis of equations (41) yields the values shown in Table 1 (correct to four significant decimal places) for the natural frequencies ol, and the coefficients S, (i = 1,2,. . ., lo), where w. = l/k/M. (42) TABLE

Mode

1

no.

i

1 2 3 4 5 6 I 8 9 10

(w/w)*

Sl

0.1378 0.7295 1.8083 34014 5.5525 8.3301 11.8438 16.2793 21.9966 29.9207

-1.4961 0.7415 -0.3473 0.1351 -0*0414 oGO95 -0*0015 omO2 O@OOO 00300

We will consider the case where the modulating function, A(t), is a “half sine” pulse i.e. it has the form O
(b) Figure 4. Variation of A(f) with r.

this train symmetric about the time origin, so that the C,, coefficients in the Fourier expansion of Z(t) are all real. For this periodic wave form we find that c

sin(dr)

=

n=0,%1,*2

2fln(l - nz/r2)

n

)...)

(44)

where r = T/T’.

On substituting equation (44) into the generalized form of equation (15), we have 1 %O I 3f2)

-

y$

m

2

n=_-m

sin (m/r) 41

-

_ e1nP(tI+tZ)/2

flr,(fp,

T)

n21r2)

for the practical, truncated series expression, where &JsZ, T) is given by equations (37) and (38) (see also Appendix II). In Figure 5(a) we show the initial stages of building up the response statistic, E(X2(t)}, according to equation (46), taking into account only the first term in the modal expansion

COVARIANCE RESPONSE OF LINEAR SYSTEMS

393

for a(~). The horizontal axis is scaled, in this figure, so that the non-dimensional time, o0 T, is equal to 2 units. In this case, r = 2 and N, defined by

periodic

N = w. T’/2n is equal to 10. The curves numbered 0, 1 and 2 show the results of taking m = 0, 1 and 2, respectively. Figure 5(b) shows the result of adding further harmonics up to m = 5 (note that Ck4, G6,. . ., are all zero in this case, since r = 2). It was found that the addition of further harmonic components produced a negligible effect. Furthermore, the addition of other terms in the modal expansion for E(W) did not affect the result shown in Figure 5(b) significantly. Since the overlapping error here is clearly very small, we can conclude that the curve

(b) 80

60

40

20 PUkC start 0

I -08

-06

-0-4

-02

0

02

04

06

08

Figure 5. Variation of E{_??(t)} with t. Uncorrelated excitation. <= 0.1, N = 10,r = 2.

First mode only.

Sine pulse. (a) m = 0,1 and 2. (b) m = 5.

in Figure 5(b) is a good approximation to the mean square response of the building to an isolated pulse. The rapid convergence of the series solution, in this example, is readily explained. We note from equation (30) that, for a single-mode system, /3, i(np,O) falls off like nw3when n is large. Furthermore, from equation (44), C, also falls off like n-), when n is large. The terms in the summation of equation (46) therefore fall off like nm6. Figure 6(a) shows the initial stages of calculating the covariance function, w,,(t,, tz), of X(T), where t, is chosen to be the mid-point of the pulse. The horizontal axis is scaled as before. Once again a rapid convergence is achieved and a good approximation to the required covariance function is obtained with m = 5; the result is shown in Figure 6(b). Again, it was found that only the first term in the modal expansion of ,X(W)was significant. The rapid convergence achieved in this example is partly due to the smooth shape we have chosen for the modulating function A(t). If A(t) changes rapidly with time, then a slower

394

J. B. ROBERTS

degree of convergence is obtained. To illustrate this point, let us consider the “rectangular pulse” of the form A(t) = 1 O
=Isin+n

C,

m?

n=O,11,&2

)....

Figure 7(a) shows the first stages in building up the response statistic, E{X2(t)}, for this kind of pulse. It is clear that a rough approximation to the variation of ,5(X2(t)) with time is rapidly 60

40

20

0

-20

“9 ?

-40

P g

P

lb) 60-

40-

20-

0

-20

-

-40

-

tFuse start

I

I -08

-06

I

I -0-4

-02

I 0

02

04

06

0%

‘2

Figure 6. Variation of ~,(t~, tZ) with t2. tI at pulse mid-point. r = 2. First mode only. Sine pulse. (a) m = 0 and 1. (b) m = 5.

Uncorrelated

excitation.

5 = 0.1, N = 10,

obtained. There is, however, in this case a minor ripple in the response curve which is only revealed by taking m = 17 [see Figure 7(b)]. Once again we find that only the first mode is significant. The ripple in the curve shown in Figure 7(b) can be explained in terms of the characteristics of the /3,I(Q,O) function, given by equation (30). The curve of j/3,,(sZ,O)l vs. !Z shows that the lowest harmonics contribute most to the variance response [3]. There is, however, a peak in this curve at 52 = 20,, which is reflected in the small amplitude ripple in the E{X2(t)} vs. t response curve. To reproduce this minor ripple, using the harmonic technique, enough harmonics must be included to cover the peak in the I~,,(sZ,O)l - Sz curve. In other words,

395

COVARIANCE RESPONSE OF LINEAR SYSTEMS

harmonics from n = --it’ to n = n’ must be included, where pn’ > 2w,, or IZ’> 2rNw,/ql,. In the present example we find that m must be at least 15. When n is large, we find from equation (48) that C,, falls off like n-‘; thus, for the square pulse, terms in the harmonic series

% ii

,. ,

I ICO-

m=l7 SO-

6X3-

40-

20n

O-’

7%

I__

I -no -VP

_nc -vo

-04

-0.2

Figure 7. Variation of E{X*(r)} with f. Uncorrelated Square pulse. (a) m = 0, 1 and 3. (b) m = 17.

0

02

excitation.

04

06

08

5 = 0.1, N = 10, Y= 2. First mode only.

expansion for +vll(tl,fl) fall off like ne4, as opposed to the nPj decay in the case of the sine pulse. Because of the greater convergence for the sine pulse, the ripple in the mean square response to this pulse is much smaller than in the corresponding response to a square pulse;, in fact, we find that it is negligibly small, as Figure 5(b) indicates.

3. CORRELATED

EXCITATION

If Z(r) is a non-white, stationary process we can write

where W,(T) is the covariance function of Z(t). From equations (2) and (49) the covariance function of the response is given by M’,(t,,tz) = 7 J’ h(t, -T,)& -a, -co

- T~)A(T,)A(T~)&(Q - T,)dT, dT2.

(50)

J. B. ROBERTS

396

In this case, it is convenient to express the modulating function A(t) in the form A(t) = f c(Q) eint d0. -CO

(51)

On substituting the above expression into equation (50), and rearranging, using equation (17), we obtain (52) where the harmonic covariance function is given by ,BrS(Q1,!&, T) = 7 S,(o) [-i(w - sZ,)l’-’ [i(w + sZ,)Y-’ a*(~ - Q,) c+ + Q) eiwr do, -cc

(53)

and m

s

w,(r) emfwTd7

(54)

-m

is the spectral density function for Z(t). The harmonic covariance function has the property that L(Q,P Q,, 7) = MQ,, 01, -7) = M-Q,, -Q,, 7). (55) We note that in the special case of stationary excitation, where A(t) = A, a constant, we have c(Q) = A. S(Q), and equations (52) and (53) reduce to wr,(tl, ?J = j?JO, 0,~) =

J’

S,(o) [--iw]‘-’ [zi.o]s-’ 1a(w)12 eiwr dw, -CO

(56)

which is a well-known result for stationary processes. The advantages of using the frequency domain relationship, given by equations (52) and (53), rather than the equivalent time domain expression, given by equation (50), for correlated excitation, are comparable with the advantages gained in the uncorrelated excitation case, by using equations (7) and (1 l), rather than equation (5). As before, a convenient, approximate method of performing the necessary integration over frequency is obtained from the consideration of a train of pulses, rather than an isolated pulse. Here we assume A(t) to be periodic, with period T, and represent this function as A(t) = The following useful approximation w,,(t,, t2) =

2

2 c,einpt. PI--m

(57)

to equation (52) is then obtained :

z c,,

PI,=-*“2=-.X

cn2eip(nlrl+n2rz)/3&z, p, n2p,

T).

(58)

The integration in equation (53) can often be performed analytically, if S,(w) is expressible as the ratio of two polynomials in w. Once the roots of the denominator of the integrand in equation (53) have been found, it is a simple matter to evaluate the integral by means of the contour integration technique. It is worth remarking that much of the tedious algebra

COVARIANCE

397

RESPONSE OF LINEAR SYSTEMS

involved in separating out the real and imaginary parts of residues can be avoided by using the complex arithmetic facilities available on most digital computers. A convenient simplification can sometimes be obtained by expressing S,(W) in the series form

where n is some positive integer and u,, and c, are constants. Each term in the series is a “Markov spectrum”. On substituting equation (35) and (59) into equation (53), &(X2, ,Q2, 7) is obtained as the sum of integrals of the form

m [-i(w

- fi,)r-’

[i(w f sz2)]“-’ aT(w-

‘1)

%cw +‘2)

eiw7

dw.

(a))

(co2+ a2) -CO

This integral is readily evaluated analytically (see Appendix II). It is noted that J!W,

>Q,, d =

wQ2,

J-J, 3 -7)

=

W-Q,

9 -Q2,4,

(61)

and that Z#J, 3.1.

T) = limJR(Q/2, Q/2,7). 0-+m

(62)

AN EXAMPLE

To illustrate the use of the harmonic technique for correlated excitation, we return to the system shown in Figure 3 and assume, as before, that A(t) is a “half sine” pulse, as defined by equation (43). For simplicity we will take the spectrum of Z(t) to be of the form

where a is a constant and attention will be focused, as before, on the relative displacement of the top floor, X&t) = X(t). For a train of half-sine pulses we find that 2 cos (7rn/r) c”=%‘(l -4n2/r2)’

(64)

On substituting this result into equation (58), we have m

W&l,

t2) =

&

cos (7rnJr) cos (7rnJr) (65) n,=_m n2=_m(1 _ 4n;/r2)(1 _ 4n~~r2)e~p~“‘r’~n*r*~~~~(~~~~~~2~~~) Fiz

for the practical, truncated series expression, where /$,(s2, ,L$, 7) is given by equation (53). Figure 8 shows the result of calculating the mean square displacement response, as a function of time, for the case where r = 2, N = 10 and & = 5 = 0.1. As before, it was found that, for this response statistic, only the first mode contributes significantly. In the case where a = 4w0, the response curve is found to be virtually identical to that shown in Figure 5(b), showing that, for an excitation of this bandwidth, an uncorrelated model of the process is satisfactory. The effect of reducing the excitation band width is shown by the other curve in Figure 8, which is for a value of a = 0.5~0,. For both values of a, the value of m in equation (65) was set to 6, since it was found that the addition of further harmonic components produced no discernible effect. Figure 9 shows the result of calculating the mean square acceleration response, as a function of time, for the same values of N, r and 5 as before, and a value of a = 40,. It was found that

J. B.

398

ROBERTS

the first five modes contributed significantly to this response statistic, as Figure 9 indicates, but that only harmonics up to m = 6 could not be considered as negligibly important.

ao“0

360-

”

R k z

40-

20-

0

-08

-06

-04

-02

0

02

04

06

O-8

Figure 8. Variation of E{X*(t)} with 1. Correlated excitation. 5 = 0.1, N = 10, r = 2. First mode only. Sine pulse. a = 0.5 and 4~~. m = 6. ”

5-

I-

OL

I -00

-06

-04

-02

0

0.2

04

06

00

Figure 9. Variation of E(X*(t)] with i. Correlated excitation. 5 = 0.1, N = 10, r = 2. First five modes. Sine pulse. a = 4w0. m = 6.

REFERENCES 1. R. L. BARNOSW and J. R. MAURER 1969 J. appl. Mech. 36,221. Mean-square response of simple mechanical systems to nonstationary random excitation. 2. T. K. CAUGHEY and H. J. STUMPF1961 J. appl. Mech. 28,563. Transient response of a dynamic system under random excitation. 3. J. B. ROBERTS 1965 J. Sound Vib. 2, 336. On the harmonic analysis of evolutionary random vibration. 4. S. H. CRANDALL and W. D. MARK 1963 Random Vibration in Mechanical Systems. New York: Academic Press. 5. J. B. ROBERTS1966 J. mech. Eng. Sci. 8, 392. Structural fatigue under non-stationary random loading. 6. J. B. ROBERTS1968 J. Sound Vib. 8, 301. An approach to the first-passage problem in random vibration. 7. J. B. ROBERTS1968 Ph.D. Thesis, University oflondon. Techniques for determining the behaviour of mechanical systems subjected to non-stationary random excitation. 8. H. M. JAMES,N. B. NICHOLS and R. S. PHILLIPS1947 Theory of Servomechanisms. New York: McGraw-Hill.

COVARIANCE RESPONSE OF LINEAR SYSTEMS

399

9. A. J. CURTIS and T. R. BOYKIN 1961 J. acoust. Sot. Am. 33, 655. Response of two degree of freedom systems to white noise base excitation. 10. J. D. ROBSON1966 Aeronaut. Q. 17, 21. The random vibration response of a system having many degrees of freedom. Il. J. L. B~GDANOFFand J. E. GOLDBERG 1959 Fourth Midwestern Conf: on Solid Mechanics, University of Texas p. 488. On the transient behaviour of a system under a random disturbance. 12. J. B. ROBERTS 1965 J. Sound Vib. 2, 375. The response of linear vibratory systems to random impulses. APPENDIX

I

Let F. = A,fi4/16

+ AzQ2/4

F, = -A,@/2

- A2Q,

+ A,,

Fz = 3A, Q2/2 + A2, F3 = -2A,Q, F4

=

A,,

Go = -A3 fi3/8 - A, 5212, G, = 3A3Q2/4+

A,,

Gz = -3A3 i-2, Gj = Aj, K,, = (B, + B2 522/4)2 - Bj a2/4, K, = 2B,, B2 - B; i-i’*/2 + B:, Kz=B: Lo = -B, i&B0 + B2 Q2/4), L, = B, B,i2, then, a(~)=-F,FjL1-FoG~K,$FoG,K2-F3GoK,+GoG,L,+F,GoK2-F4G,Ko$ tF)GzKo~F2G3Ko-F,F4L,-tFiF3LO-G?G3L0, b(Q)=-F,F,K,

+F,,G,L,+F,,F,K2+F3G11L,

+GoG3K,

-G,G,Kz--F,F,K,,+

+FlFF,K,,-G2G,Ko+F4G,Lo-F3G2Lo-F2G,Lo, c(.n)=2{F,F~F4-F,F4G~+F~F~-F~G~-2F,F,GoG,-F,F,F2F3+FOF,GZG,+ +F~F,G,G2+F~F2G,G3-2F,F4G~G,-2FoF3GOG3-G~F~+G~G~+ +F,F,GoG2+F,F2G,G~+FzF~GoG,-G,,G,GzGj), d(a) = 2(-2F,

Fl F4 G, - 2F; F, Gj - F,, F: G,, + F,, G,, G: + F. F, F, Gz + F. F, F2 Gj +

+FoFzFjG,-FoG,GZG,-F::F4Go+F4GoGf-FoF~Go+FoGoG~+ +2F,G~G~+F,F~F3GO-F,GOG2G~-F~GOG,G2-F~GOG,G3). APPENDIX

Using residue theory, J{:(Q, ,Q2, r) [see equation

II

(60)] can be expressed

J:‘:(sZ,, Q2, T) = ia2(R< + Ri + R<),

as

400

J. B. ROBERTS

where R:, R< and R: are, respectively, the residues corresponding to the poles at o1 =-Qz+wk~+i~k~kr wz=-~n2-~kZ/I-5~+i5k~kr co3= iu.

We have

and R,=-

[-i(iu - Q,)r-’ [i(ia+ sZ,)~-’ e-UT (ia J2# 2icj(iu - a,) wj]. [cot - (iu + Qn,)? + 2ic,(iu + Sz,) co,] 2iu[wj -

Equation (62) can be used to find Z/j@, T). The result is Zii(Q, T) = i(R: + R:), where

and