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Simulation of Multivariate and Multidimensional Random Processes

M. SHINOZUKA*

Columbia University, New York, New York 10027

Efficient and practical methods of simulating multivariate and multidimensional processes with specified cross-spectral density are presented. When the cross-spectral density matrix of an n-variate process is specified, its component processes can be simulated as the sum of cosine functions with random frequencies and random phase angles. Typical examples of this type are the simulation, for the purpose of shaker test, of a multivariate process representing six components of the acceleration (due to, for example, a booster engine cutoff) measured at the base of a spacecraft and the simulation of horizontal and vertical components of earthquake acceleration. A homogeneous multidimensional process can also be simulated in terms of the sum of cosine functions with random frequencie. s and random phase angles. Examples of multidimensional processes considered here include the horizontal component fo(t,x) of the wind velocity perpendicular to the axis (x axis) of a slender structure, the vertical gust velocity field fo(x,y) frozen in space, and the boundary- layer pressure field fo(x,y,t). Also, a convenient use of the present method of simulation in a class of nonlinear structural vibration analysis is described with a numerical example.

INTRODUCTION

The purpose of this study is to develop a method by which a class of multivariate and multidimensional random processes with specified cross-spectral densities can be simulated with the aid of a digital computer within a reasonable amount of time.

The motivation of the present effort includes the necessity of simulating a multivariate random process which statistically represents multicomponent measure- ments of the acceleration excitation at the base of a spacecraft due to, for example, a booster engine cutoff. The random processes thus simulated can then be used to perform a shaker test for a spacecraft of similar design to guarantee its dynamic integrity in a similar flight.

Also, such a simulation plays an essential role in a general class of the "numerical probabilistic mechani cs" which includes the following problems: (a) numerical analysis of nonlinear structure subjected to multidimensional random excitatitn (e.g., dynamic analysis involving large deflection of a thin plate acted upon by a turbulent flow whose pressure on the plate can be represented by a multidimensional random process) and (b) numerical stress analysis or eigenvalue computation of a structure consisting of a material whose thermomechanical properties are random functions of space variables.

I. SIMULATION OF A STATIONARY RANDOM

PROCESS WITH A SPECIFIED MEAN-

SQUARE SPECTRAL DENSITY

As a preliminary study, consider a Gaussian random process fo(t) with mean zero and the mean-square spectral density function So(co). This process can be simulated by way of the following series'

where

f(t) =a E cos(cokt-}- qok), (la)

.= s0½)&

is the standard deviation of the process f0(t); co• (k=l,2,...,N) are independent random variables identically distributed with the density function g(co)---- g(co•) obtained by normalizing So(co);

g(co)= So(co)/a•'; (2)

and q• are independent random variables identically distributed with the uniform density 1/(2r) between 0 and 2r. Note also that cok and qt (k,l,= 1,2,...,N) are independent.

The Journal of the Acoustical Society of America 357

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M. $HINOZUKA

It is easy to show that the ensemble average E[-f(t)• of f(t) is zero'

a(2N)'f/fo•COS(Oot+•)g(oo)d•doo=O. (3) E[' f(t)-l= 2;r •o ' The autocorrelation function R(r) of 'f(t) becomes

R (r) = E[f(t q- r) f(t)-]

-- Z Z •{cos½•t+•) cos[•(t+•)+•]} X k==l

ø g( ) f•ø -- •' COStoT co rico -- coscorSo(co)dco. (4)

The last member of Eq. 4 is identical to the autocorrelation function Ro(r) of fo(t) because of the Wiener-Khintchine relationship'

e0(•) = s0½)•'•'a• =

and

s0½) cos•a• (s)

So½) =• e0(•)•-'•'& =-- R(r) cosw-dr, (6)

where i is the imaginary unit. This indicates that, when the ensemble average is

considered, the simulated process f(t) possesses the autocorrelation R(r) and the mean-square spectral density S(co) which are identical, respectively, with the (target) autocorrelation Ro(r) and the (target) spectral density So(co).

The temporal mean {f(t)} is zero since it can be shown that

, cos (cokt -}- •k)dt I (f(t))[ = }•m• To\N/ •--• •_vo/•. 2a(2N)t

__< lim ---0, (7) •'0-•o Tow*

where co* is the smallest of cok (k- 1,2,..-,N). The temporal autocorrelation function R*(r)

= (f(tq- r)f(t) ) becomes

R*(r) = (f(t+r)f(t))

2a • •r •v ;Vo/•. = lim E E cos(cokt+w)

x cos[-.,,(• + •-) + 0 -2 N

=-- 52 cos•. (8) 2V

By applying the Wiener-Khintchine relationship, one can obtain the corresponding mean-square spectral

density S*(co) as

s*½) =2•; E co• co•a• 0 -2 N

= E 2N •=•

(9)

where 5(.) is the Dirac delta function. Equation 9 produces the mean-square spectral distribution function r*(•)-

r*½) = s*½')•'=--•v½). (lo) • 2N

In Eq. 10, N(co) is the number of cok (among the sample of size N) that are less than co or larger than --co if co-<0. If, however, co>__O, N(co)=Nffco)q-N2(co)q-Na(co), in which N•(co) equals twice the number of cok such that -co_-<co•-<co; N2(co) equals the number of co• such that co•-<-co, and Ns(co) equals the number of co• strclx that cok = co.

It is important to note that, when dealing with the' temporal averages, cog and qg are not considered as random variables but are treated as sample values of these random variables. This convention has been used

for simplicity of notation. It can be shown (Appendix A) that F*(co+

converges in the mean-square sense to corresponding target value r0(w+aw)-r0(w), which is in fact equal to S0(co)Aco for small values of Aco, as N approaches infinity.

1.i.m. [r*½+a•)-r*½)3=ro½+a•)-ro½), (11)

where F0(co) is by definition

r0½) = s0½')a•'. (12)

It is in this sense that, as N approaches infinity, the: process f(t) becomes ergodic in the mean value and the mean-square spectral density which, in the present case,. are respectively identical with the corresponding target values. The central limit theorem assures the normality of f(t).

It can be shown that similar series proposed for simulation of multivariate and multidimensional processes in the following sections have the asymptotic ergodicity in the same sense. In applications, however, as large a value of N as practical is used to obtain an approximate ergodicity.

This particular method of simulation has been used z to reproduce the road-surface roughness of a bridge in a study where the dynamic interaction among the moving mass, the surface roughness, and the bridge is investigated for the purpose of estimating the fatigue life of simple bridges, Figure l(a) shows a typical

358, Volume 49 Number ! (Part 2) 1971


SIMULATION OF RANDOM PROCESSES

spectral density function used in Ref. 1 in a solid line and the spectral density values in open circle (computed from a simulated process of length 50 sec of the form of Eq. 1 with N=200). In this case, the idealized spectral density defined over the wavenumber of surface roughness based on a measurement is transformed into the spectral density over the frequency domain assuming that the vehicle moves with a speed of 40 mph. Figure 1 (b) illustrates a typical section of the simulated surface.

It appears that Eq. 1 was first used for simulation purposes by Goto and Toki. 2 In Ref. 2, however, no discussion is given on the interrelationship between the temporal average and the ensemble average of the series f(t). The present method of simulation possesses the following advantage of great practical significance. If f(t) given in Eq. 1 is used as an input to a linear system with the frequency response function K(oo), the output q(t) is "immediately" obtained (without per- forming the integration in time domain) as

q(t)=a E [K(•o•)lcos(•oktq -qkq-a•), (13a)

where

ti•= tan-•[ImK(oo•)/ReK(oo•)•, (13b)

with ImZ and ReZ standing for imaginary and real part of Z, respectively.

Such an advantage can also be expected of the similar series proposed for multivariate and multidimensional processes in what follows. This advantage is unique to the present method, not being shared by any other means (for example, Refs. 3 and 4) of simulation.

II. SIMULATION OF A MULTIVARIATE STA- TIONARY RANDOM PROCESS WITH

A SPECIFIED CROSS-SPECTRAL DENSITY MATRIX

Consider a set of - stationary random processes fo•(t) (i= 1,2,.-. ,n) with mean zero and with a specified cross-spectral density matrix S/ø(•o) = [Sø/i//(•o)-], where Sø•,•/(•o) are mean-square spectral densities of fo•(t) if i= j, and cross-spectral densities of foi(t) and fo•(t) if i• j. A set of n functions of time f•(t) (i= 1,2,... ,n) with the target cross-spectral density matrix Sø•(•o), which can be simulated within a reasonable amount

of time, are to be found. To this end, consider a set of n stationary random

processes f•(t) defined by

f ,(t) = E hq(t--r)ny(r)dr, (14)

where hii(t) are "real" functions of time possessing Fourier transforms and ns(t) (j= 1,2,...,n) are white noises I-n•(t) and hi(t) are independent for i• j-] with

O.1

0.01

OOOl

o o ø o

ø ø

i.o IO

f (cps)---

Fro. 1. (a) Target and simulated spectra. (b) A section of road roughness.

the unit spectral density. It is evident that the mean values (ensemble average) of fi(t) are all zero.

It can be shown a that the cross-spectral density matrix Si(•o) of fi(t) (i= 1,2,...,n) is

(15)

where S,(w) is the cross-spectral density matrix of n•(t) (i= 1,2,---,n), which is the identity matrix of order n in the present case. H(w) is the matrix obtained from the matrix h(t)=[hq(t)• by taking the Fourier transform. The overbar indicates the complex con- jugate, and the prime the transpose.

From Eq. 15, it follows that i--j element St•/•(w) can be written as

Sq(oo) = E H,k(oo)•y•(oo), (16)

where So(oo) is written for S/• i [similar notation for

If, therefore, Ho(oo) are so chosen that

S?(•o) = E H,•(•o)_•i•(•o), (17)

then a set of n stationary processes f•(t) have the same mean values and the same cross-spectral matrix as those associated with fo•(t).

Equation 17, in fact, provides a set of n 2 equations for n • unknown functions H•(•o). However, considering

The Journal of the Acoustical Society of America 35 c )


M. SHINOZUKA

the well-known property of cross-spectral densities --

s ø,•(•) = •5,(•), (18)

one can easily conclude that the number of unknown functions //ij(w) should be reduced by (n"-n)/2. In this connection, an interesting procedure proposed by Borgman 4 is described here.

Consider, in Eq. 14, that

hij(t)=0, for j>i, (19) and therefore,

His(•o) = 0, for j> i. (20)

This evidently means that the upper off-diagonal elements of the matrices h(t) and H(•o) are zero. Hence, Eq. 17 becomes

Sø11(lo) -- [ Hn(•o) i ", (2ia)

Sø=•(•) = •øn(•)= H=•(•)•n(•), (2lb) Sø•(•) = [H•i(•)[ •+ [H22(•) [ •, (21c)

S ø a• (•) = g0 •a(•) = H a• (•)• n (•), (21 d)

Søa=(•)=Sø=a(•)=Ha•(•)•=•(•)+Ha=(•)•==(•), (21e)

Søaa(•) = [Hal(•)! =+ IH•=½)! =+ IH•½) I =, (21f)

S 041(•) = •014(•) = H 41(•)Hll(•), (2 lg)

Søt=(•)=•ø=t(•) = Ht•(•)•=i(•)+H4=(•)•==(•), (21h) s 0t•½)=•t(•)=•1½)•1½)+•

+H4a(•)•aa(•), (21i)

Søtt(w) = IHtl(w)I =+ [Ht2(w)l =+ I u4,½)I +1•,½)1 =, (21j)

etc. These equations can sequentially be solved for

H n (•) = IS ø n (•) •t, (22a)

H • (•) = S ø • (•)/IS o n (•) •t, (22b)

Ha•(w) = S øa•(w)/ES øn(•) •t, (22d)

Ha•(•) = ESøa•(•)--Ha•(•)•(•)•/H•(•), (22e)

•½) = ES0•½)-I •1½)I =- I •=½) I =•, (22f)

H t•(•) = S øt•(w)/ES ø n(•) •t, (22g)

H4•(•)= ESøi=(•) -Hi•(•)•=•(•)•/H==(•), (22h)

•4,½) = Esot•½)-•t1½)•1½) -H4=(•)•a=(•)]/Haa(•), (22i)

•4t(•)= Es0tt½)- -I•=½)1 =-1•½)1=•. (22j)

etc. Note that Hii(•) are real and Borgman reco•ends in effect that, after H?(•) is

thus found, the numerical Fourier transforms be per-

formed on H•j(•o) to evaluate ho(t) in approximation (such numerical Fourier transforms are no longer considered intractable; see Ref. 6, for example). Upon evaluating his(t), Eq. 14 is to be used to produce fi(t) as simulated functions of fo•(t).

In the following, however, an alternative way of simulating foi(t) with the aid of the technique described in the preceding section is presented.

For this purpose, introduce q/o(•o) as

ß •s½)- I•,s½)I/l•ss½) I. (23)

Note that 7o(•o)= 1 for i= j, and 7o(•)=0 for i<j. Using 7o(•), write Ho(•) in the polar fore'

with

1[ImH"(•)• 0o(•) = tan- LRe•J' (25) where 0ii= 0.

Since ho(t) is assumed to be real,

Re•(•)= Re.,•(-•) (26) and

I• •(•) = -- ImH o(--•). (27 )

It follows from Eqs. 25-27 that

0•(•) = - 0•(-•). (28)

Construct then the following series corresponding to Eq. 14 with the restriction i• j as introduced in Eq. 19'

j=l

(29)

where •osk (k= 1,2,.--,N) are random variables identically and independently distributed with the density function

with

•/-- I •½)1 •d•, (30b)

and •k (k= 1,2,..-,N) are also identically and independently distributed with the uniform density 1/(2•r) between 0 and 2•r.

It then follows that, for i__> j,

x cos[•.•t + •.• + 0,. (•.•) ]

xcos[•½+ •+0•(•)q

360 Volume 49 Number 1 (Part 2) 1971


SIMUI. ATION OF RANDOM PROCESSES

Taking into consideration that I Ho(co)[ is an even function of co (see Eqs. 26 and 27) and that the argument of cosine function in the above integration is an odd function of co (see Eq. 28),

Ro(r) = /5 ø •. I//•(•o){ I//•(•o)[ Xexp{

Using Eq. 24,

=

Finally, with the aid of Eq. 17,

f/ 0 i•or __ = 6> j).

Equation 31 indicates that the series f•(t) as constructed in Eq. 29 possesses the target cross-correlation functions (and therefore target cross-spectral densities) on the ensemble average.

The construction of the series fi(t) requires simulation of (a) n independent sequences of N numbers coj• (j= 1,2,. ß .,n; k= 1,2,. ß .,N) taken from the distribution with the density gj(co); (b) n independent sequences of N numbers qo• (j= 1,2,. ß .,n; k= 1,2,...,N) taken from the uniform distribution between 0 and 2;r; and (c) computation of the moduli [H•(co•)l, from which 'ro(co•,) = I H•j(co•,)l/IH•(co•,) can directly be evaluated, and the angles 0o(coi•) of Ho(coi•).

Simulation of independent sample values from a specified distribution is routine and the computation involved in (c) above can be performed within a computer if the cross-spectral densities are given in analytical form. However, since the measured cross- spectral densities are usually given numerically in terms of the real and the imaginary parts or the moduli and the angles, the computational work associated with item (c) above can be minimal. Therefore, the method of simulation presented here appears more practical than that proposed in Ref. 4, which requires (a) the inverse Fourier transformation of N(N+ 1)/2 functions of co, Ho(co ) and (b) the same number of integrations in the time domain. Also, as pointed out in the preceding section, the present form of the simulated function consisting of a sum of cosine functions can be proved ex-

tremely advantageous, when used as an input, in evaluating the corresponding output of a linear system.

The present method produces Gaussian processes because of the central limit theorem.

III. SIMULATION OF HOMOGENEOUS

MULTIDIMENSIONAL PROCESSES

The method of simulation for multivariate processes described in the preceding section is of extreme im- portance in engineering application. In fact, this method can in principle be applied to simulation of multidimensional processes such as wind velocity fo(t,x) (or a component wind velocity to be precise) along x axis, transverse pressure fo(t,x,y) on a plate, etc.; for example, consider a simulation of set of processes f o(t,x), f o(t,x+ Ax), f o(t,x+ 2Ax), --..

The present section shows first that when two- dimensional processes with homogeneity are considered, a drastically simpler method of simulation is possible.

Consider the following series f(t,x) for fo(t,x) with mean zero:

f(t,x) =• 5'. COS(cokt+•2kX+ •). (32) \N/ •=•

In Eq. 32, co• and •2k are random variables jointly distributed with the joint density function g•.(co,•2) ----g•.(co•,•2•) and independent of cot and •2• if k•l, qo• are random variables identically distributed with the density 1/(2;r) between 0 and 2;r, and • is the standard deviation defined by

E[f'(t,x) •= (r •', (33)

where the fact that

E[f(t,x)•=O (34)

has been taken into consideration. Also, note that •k is independent of • (k•l) and of cos and •2• (/= 1,2,...,N). Equation 32 suggests that f(t,x) is Gaussian, again owing to the central limit theorem.

Construct the cross-correlation function R(r,•) from Eq. 32'

R(r,•) =EEf (t+r, x+ •) f(t,x)]

=(r •' cos(cor+•)ge(co,•)dcod•. (35)

Write •,(•2[co) and •(co), respectively, for the conditional probability density function of f• given co and for the marginal density function of co [hence, •,(•21 co )-> 0 and ß (co) >__ 0-]. Then,

R(r,•) =•r•' f_ • [c(co,//) coscor--q(co,//) sincot]

X'I'(•o)&o, (•6) where c(co,•) and q(co,•) are, respectively, the conditional



M. SHINOZUKA

expectations of cosfl•/and sinfl•/, given co'

and

c(.,,e) = cosUe.v•(u ] .,)an (37)

q(co,•/) = sin•,(• I •)d•. (38)

If an even function of •, •(•lco) and an odd function of f•, •.(•[ co) are so constructed that

w,(•l•)- •Ew,(•l•)+w,(-•l•)], (39)

then

Therefore,

and

c(•,t/) = 2

(40)

(41)

(42)

q(co,//)-2 sin •//,I,,(•21 co) d•2. (43)

The cross-spectral density S(co,//) can then be obtained from Eq. 36 with the aid of the Wiener- Khintchine relationship:

S(co,•) =-- R(r,•)e-'•'dr 2r •o

[c(co',•) cosco'r-q(co',•) sinco'r]

X •(co')e-i•dco'dr

oo 2

+iv' q(co',•) • -oo 2

0-2

=-It (.,, •) .v (.,) +4 -.,, •) .v(-4 ] 2

-+- i--[q (co,//) xI, (co) -- q (-- co,//) ,I, ( -- co) ]. 2

(44)

From the preceding definitions of 0-z, R(r,•), and it follows that

0-2 =R(0,0) = S(co,O)dco, (45)

where S(co,O) is the mean-square spectral density function of f(t,x) with x being kept fixed. Let •(co) be defined as

ß ½)=s½,0)/•t (46)

This definition of •(co) makes it possible to interpret it as a probability density function. In fact, in this case, the fact that xI,(co)=xI,(--co)_>0 and f_•xI,(co)dco= 1 follows respectively from the well-known property of the mean-square spectral density, i.e., S(co,0)= S(-co,0) => 0 and from Eq. 45.

Because •(co) is an even function of co, Eq. 44 can be written as

with

and

S(co,•) = C(co,•)-iQ(co,•), (47)

0-2

=--'I'(co) [c (co, •/) + c ( -co, t/)-] (48) 2

0-2

----xI,(co) [q (co,//) _q(_co,//)-], (49) 2

where C(co,//) and Q(co,•) are, respectively, the cospectral density function and the quadrature density function. Equations 48 and 49 are consistent with the fact that the cospectral density and the quadrature (of real processes) are, respectively, even and odd in co. It is pointed out that Eqs. 48 and 49 are also even and odd in //, respectively, which is true for a real stationary process.

Equations 48 and 49 identify c(co,//) and q(co,//) up to an arbitrary function in terms of cross-correlation functions. To do so, define

e*(•,e)= «[e(•,•)+e(-•,e)3, (50)

•-(,',•) = «[•(,-,•)-•(-,-,•)3, (5•) and

R(r,•) = R*(r,i)+ R-(r,i). (52)

Evidently, R+(r,•) is an even function of r, whereas R-(r,•) is an odd function of r. Therefore,

C(w,•) =-- R*(r,•) coswrdr (53) 2• •

and

Q(w,•) =-- R-(r,•) sinwrdr. (54) 2r •

It then follows that

c½,e) ---• - 2r•'I,(o•) •o

and

q(co'•/) 2•ra•.,i,(co) •

n*(•,e) cos•a•+c0½,•) ($5)

R-(r,•) sincordr+qo(co,•), (56)

where Co(co,//) and q0(co,//) are assumed to possess inverse Fourier transforms and to satisfy

c0½,•)= c0½, -•)= -c0(-•, •) (57) and

q0(co,//) = --q0½, - •/) = q0(--co,//). (58) 362 Volume 49 Number I (Part 2) 1971



Except for these restrictions, c0(co,•/) and q0(co,//) are arbitrary. This arbitrariness makes it possible to find c(co, t/) and q(co,t/) which, upon inversion of Eqs. 42 and 43, produce xI, fff•lco) and xI,•(f•]co), satisfying the condition that their sum is non-negative and equal to unity when integrated over the entire domain of f•.

It is noted that Eqs. 55 and 56 can also be written as

and E

q(co, •) = -- [ 1/a•',I, (co) •Q(co, •) + qo(co, •).

(59)

(60)

The functions xI, fffllco ) and ,I,•.(fllco) can be found from Eqs. 42 and 43 with the aid of the sine and cosine Fourier inversion formulas'

and lf0• =- cosna (61)

1; © ,I,2(f• [ co) =- q(co,•) sin•l•,t•. (62) •rd0

In summary, the following steps have to be taken to simulate fo(t,x) in terms of f(t,x)' (a) Identify C(co,//), Q(co,t/), and q,(co) associated with the target cross- spectral density function So(co,t/); (b) construct c(co,•) and q(co,t/) using Eqs. 59 and 60 with appropriate functions c0(co,•/) and q0(co,•/) [-note that c0(co,•/) = q0(co,•) =0 is a possible choice-]; (c) evaluate xI,•(fllco) and xI,•.(ftlco ) with the aid of Eqs. 61 and 62; (d) make sure that q,c(ftlco)--XI, l(ftlco)-+-q,•.(ftlco) is non-negative and satisfies the conditionf_• © xI'c(ftlco)df•= 1; (e) simulate cok (k= 1,2,...,N) according to the density function xI,(co) and also •k (k= 1,2,..-,N) from the uniform distribution between 0 and 2•r; (f) simulate (k= 1,2,...,N) using the conditional density and finally (g) construct the series f(t,x) as in Eq. 32.

If the independent variables t and x in f(t,x) are replaced by space coordinates x and y, then f(x,y) can represent, for example, a random fluctuation of a material property within a thin plate where such a fluctuation along thickness can be disregarded, or a random pattern of a gust vertical velocity frozen in space.

In this case, however, situations may arise where a circular symmetry exists'

R0(•/,•/) = E[fo(x,y)fo(x+ •, y+•)']= Ro*(r), (63)

where r= (•'+ •') •. Define the two-dimensional spectral density function

$0(co,u) as the double Fourier transform of R0(t/,•)'

= (2•r) •' R0(t/,•) X exp[-- i(co• + •)d•d•, (64)

where co/2•r and u/2•r, respectively, represent wave- numbers in x and y directions.

By introducing the polar coordinate r and 0,

t/= r cos0, • = r sin0, (65) one can write 7

=

with

lfo•fo•'•exp[-ir(cocosOq-usinO)• (2•r) •- X Ro* (r)rdrdO

fo © 2•r (66)

p= (co•q- u•-) t, (67)

where the integral representation of Bessel function of order zero has been used;

lf0• Jo(,'p) =-- e-'r, cøsødO. (68) 2•r

By means of Hankel inverse transformation,

Ro(r) = 2•r Jo(rp)(I)o* (p)pdp, (69)

where •0*(p)= •0(co,u). Equations 66 and 69 indicate that, if R0(•,•)---- Ro*(r),

then •0(co,u)=--•0*(p) and vice versa. Evidently, the method of simulation described in

relation to fo(t,x) can also apply to the present case where the two-dimensional spectral density becomes a function of p. All that is to be done is to evaluate S0(co,•). Considering the fact that •0(co,u) has a circular symmetry,

So(oo,,) = = •0(co,u) cosu•du. (70)

The rest of the procedure is exactly the same as that for the simulation of fo(t,x) described previously.

If, however, $0(co,u) is non-negative [-$0(co,u) is real since Ro(t/,•) has a circular symmetry-], the inversion of Eq. 64 provides a much simpler method of simulation:

Ro( •,V) = •I,o(co,u)e +i ("'•+•'") dcodu, (71)

which, because of the circular symmetry of $0(co,u), becomes

Ro(•,n) = $0(co,u) cos(cot/q-un)&odu. (72)

Therefore, to simulate fo(x,y) with the (target) two- dimensional spectrum $0(co,u) or the (target) cross-



M. SHINOZUKA

correlation Ro(r,•i), construct a series

f(x,y) =a COS(cokXq-Ukyq- •k), (73) k•l

with a2=Ro(O,O), •o• and u• being random variables jointly distributed according to the joint density function

g2(w,u) = $0(w,u)/a 2, (74)

and q• a random variable uniformly distributed between 0 and 2,r, where co• and u• are independent of wt and ut(l•k) and of v• (/= 1,2,...,N), and also q• is independent of qt(k•l). Then, f(x,y) has the two- dimensional spectrum that is identical with the target spectrum $0(w,u).

It is noted that Eqs. 71 and 72 are also valid if Ro(r,•) has a double symmetry with respect to r=0 and •=0. Equation 73 can then be used again for simulation of fo(x,y) if its two-dimensional spectral density $0(w,u), which is real and has a double s•- metry with respect to w=0 and u= 0, is non-negative.

Extension of the preceding discussion to the case of a general multidimensional homogeneous processes fO(Xl,X•,''',Xn) appears quite possible and is an interesting subject of future study.

In this case, define R0(•,•,- --,•) and $0(Wl,W•,'' ',w•), respectively, as

Ro(•l,•2,''' ,•n) = E[f o(xl,x•, " ' ,xn)

X fo(xl+•l,X•+•,' ",xn+•n)•, (75) n-fold

''' Ro(•l,•,'",•.) -. _. x exp[- ß ß ß

Xd•ld•2...d•n. (76)

In Eqs. 75 and 76, if R0(•x,•2,-'' ,•n) is an even function of all of its arguments •,•2,''' ,•n, then •0(COl,CO2,..- ,con) is real and even with respect to all of its arguments COl,CO•,. ß .,con and vice versa. If, furthermore, ß 0•1,w2,'' .,wn) is non-negative, s then fo(xl,x2," ',xn) can be simulated in the form

f(Xl,X2, ' ' ' ,Xn)

= ff COS(001kXl'Jf"O.)9. kX2-Jl -''' q-oo,,•x• q- •'k ) , (77)

with qok being uniformly distributed between 0 and 2•r and independent of qot for kyal and with (k= 1,2,...,N) being random variables independent of w•t,w2•,...,w•t for kyal and distributed according to the joint density function

gn(cOl,CO•., . " ,con) -- •o(cobco•., ' " ,con)lit •', (78)

where

n-fold

0 '2= ]__ ''' ri>0((.01,(.02,''' ,con)dcoldoo•... do•,,. (79)

IV. EXAMPLES

A. Wind Velocity

Consider a wind velocity distribution due to atmo- spheric turbulence along the axis of a slender line-like structure lying on a horizontal plane. Taking the structural axis as the x direction, the horizontal component f(t,x) of the fluctuating part of the wind velocity perpendicular to the x axis is considered.

Following the observations, 9.•ø it is assumed that the quadrature density Q(w,•) is of negligible magnitude and that the coherence •2(co,•) is a negative exponential function of [ •1,

I ...... e -•"1•11•1 (80) '

where a is a positive constant. The cospectral density can therefore be written as

C(w,•i)=S(w,O)e-"lo'lltil=tr•",I,(w)e-"10'11•il. (81)

By setting Co(W,•)=0 and qo(w,•)=0 in Eqs. 59 and 60, one obtains

c(w,•)=e-,l•ll•l (82) and

q(•,•) =0. (83)

It then follows from Eqs. 61 and 62 that

ß •(• I w) =- e -"• cos•a• = (84) • •(.•+•)

and

=0, (85)

and, therefore,

(86)

which can easily be shown to be a non-negative function of •2 and equal to unity when integrated over --o. to q-o. with respect to 9.

Use of xI,•(•g[w) (Eq. 86), together with xI,(w) and a uniform distribution between 0 and 2r for phase angle, makes it possible to use the series f(t,x) in Eq. 32 for simulating fo(t,x) with mean zero and with the specified coherence function.

B. Gust Vertical Velocity

Consider a random gust vertical velocity frozen in space. Suppose that the two-dimensional spectral den-

364 Volume 49 Number i (Part 2) 1971



sity is given u by

3a•L • L•[-(co/2a.)•+(t•/2a.) •--] •o

(I,o(w,u) = 4a- { l+L•'[(oo/2r)•+(u/2r)•-I}•' (87) where a 2 is the variance of the gust vertical velocity and L is the scale of turbulence.

Since •0(co,u) in Eq. 87 is non-negative, Eqs. 73 and 74 can be used to simulate the corresponding gust vertical velocity.

For an airplane in high-speed flight moving along the negative x direction, therefore, one should use f(x,y) in Eq. 73 with x-Xo--vt, where x0 is the initial position of the plane on the x axis and y is the convection speed.

C. Boundary-Layer Turbulence

For a simulation of boundary-layer turbulence, consider the following form of the cross-correlation function of the boundary-layer pressure field fo(x,y,t) (Ref. 5, p. 219):

{ Ro(•i,rt,r) =a 2 exp -----[-(•-vr)•+n'ø] - , (88) 4L •

where •=Xx-X2, rt=yx-y2, r=tx-t2, and L is the scale of turbulence, v the convection speed (in the x direction), 0 the life expectancy of turbulence eddies, and a 'ø' the variance of pressure field.

The Fourier transform (Eq. 76) of Eq. 88 with v=0 becomes

L•.O

2•.•.

L2 +(•) wa•], (89) Xexp_ [•(w•+w22) which can be written as

with ½,o(cox,co.o.,cos) -- a'ø'f x(cox) f•.(co•) f s(coa),

L

fl((.01) =-- exp[-- (L2/•)oo12],

(90)

and

L

/•.(w•) =- exp[--- (L2/ •-)wo. 23, (91)

0

fa(wa) = L exp[-_(O/2)%•2-].

Equations 90 and 91 indicate that (I,o(Wl,W•.,wa) is non- negative and therefore, the pressure field can be simulated in the form of Eq. 77 with n= 3 and xl, x•., and xa being replaced respectively by x-vt, y, and t (recall the last paragraph of Example B). The joint density function for (-01k, 002k, and wak is independent of k and given by

ga(wl,CO2,coa)-- fl(wl)f2(co2)fa(coa), (92)

where fx(cox), f•.(cou), andfa(wa) are, respectively, Gaussian density as indicated in Eq. 91. Hence, cob}, and coa} to be simulated are "independent" Gaussian variables with densities given in Eq. 91.

D. Nonlinear Structural Response

One of the most interesting and significant applications of the proposed method is the simulation of random generalized forces. The necessity of simulating random generalized forces arises when the dynamic response analysis is performed in time domain either for the purpose of obtaining information beyond the second-order statistics (such as the first passage time distribution) or when the structure is nonlinear and therefore an approximate random response is sought by simulating the excitation. An example of the latter case is well illustrated in Ref. 12 where a dynamic analysis of "large" deflection is considered, with the aid of Galerkin's method, of a simply supported rectangular plate subjected to turbulence pressure field on one side and to cavity pressure on the other with the interaction of the plate and external and/or internal (cavity) air flow being taken into consideration. For a simply supported plate, the generalized force Qm,•(t) due to the fluctuating part of the turbulence can be written as

Q•,•(I) = f(x,y; l) sin•x sin•ydxdy, (93) a b

where f(x,y;t) denotes the fluctuating part of the turbulent pressure acting on the plate at time t, and a and b are lengths of the plate in x and y directions.

Assuming that the convection speed is v, f(x,y;t) can be written as

f(x,y; t) =a Y] cos[-w•(x-vt)d--•yd- •-], K•I

O<__x_< a. (94)

By substituting Eq. 94 into Eq. 93 and expressing the sine and cosine functions in terms of the exponential function, Q•,,(t) becomes

where

with

and

X exp['-i(- 1)"+ •(w•vl- •)-I, (95a)

(--1)i+• A•gi=•--(1 -- e•) (1-- egO•i), (95b)

a•ibz, t•i

ai•,i = ( -- 1)•'+'ookaq- (-- 1)i+ imw

b&t,5 = (- 1)•+I/.t•b-{- (-- 1)/+Inw.

(95c)

(9Sd)

The computation of Q•,•(t), therefore, can be performed without any difficulty with the aid of a corn-



M. SHINOZUKA

1.8

1.6

• 1.o ._•

õ o8 g

._

o

• 0.4

0.2

o

Linear Nonlinear

T=5 , N= 100 o ß

T = 50, N = 500 A /• Linear

/•/ j'"•Nonlinea/ J& r

0.2 0.4 06 08 10 1.2 1.4

rrns of nond•mens•onal excitation o'

FIO. 2. Root-mean-square response of a string to random excitation.

purer. The time required for this is practically nothing compared with the time that would be needed for simulating Q=,•(t) with the direct use of the double integration in Eq. 93 (such a direct integration is extremely time consuming and cannot be avoided if other means of simulation is used).

Similar closed form expressions can be obtained if the modes consist of sine, cosine, sine- and cosine-hyperbolic functions (their sum and/or product).

A note of caution is added by stating that the computer program should include alternative expressions to Eq. 95b to accommodate the situation where akh• or bkhi is close to zero. This can be done by simply expand- ing the corresponding exponential function;for example,

(1--e•akh•)/a•--i-[-a•, if a•<<l. (96)

In order to assert the validity of the preceding discussion, consider the problem of the nonlinear vibration of a string in the time domain simulating the random generalized forces. The governing differential equation is

02u Ou [- AE [Z' (Ou•'dxlO2u + p---t-c--=LTo-t-•J ø -- -- fo(x,l), (97) Ot •' Ot \Ox/ JOx •'

where t• is mass per unit length, c is linear viscous damping, To is initial tension, fo(x,t) is force per unit length, u is the lateral displacement of string, L is the length of string, A is the cross-sectional area of string, E is the elastic modulus of string, and the boundary conditions are u(O,t)-u(L,t)-O. Evidently, Eq. 97 is nonlinear, taking into consideration the change in

tension due to elongation of the string (but not the variation of tension along the string).

Assuming an approximate solution of the following form•

u(x,t) = E o.(t) si x, ,,=• n-• (98) one obtains a set of M ordinary differential equations for b,,(t) (n= 1,2,.- .,M)'

E 4T0 •=•

f0 •' L mr =8 •oofO(X,t) sin--xdx, (99) L

where/•= c/(2t•). In deriving Eq. 99, it is also assumed that the fundamental frequency of the corresponding linear string is 1.0 Hz or To/(i•L •) = 4 sec-L

The right-hand side of Eq. 99 is the random generalized force Q,•(t) to be simulated. In the present example, the forcing function fo(x,t) has the coherence given in Eq. 80 and the power spectral density function

S(c0,0)- Ea/(a•'-kco•')•(az•'/•r). (100)

Once the generalized force is simulated, Eq. 99 can be integrated numerically. A numerical example is carried out for the case where •5=0.1X2,r sec -• (Eq. 99), a=4•r sec -• (Eq. 100), aL=0.7 sec (Eq. 80), and the initial strain To/ (A E) - O.05. The assumption aL=0.7 sec implies that the space correlation in terms of the coherence at the fundamental frequency is exp(--4.4x/L).

The root mean square aa of nondimensional response

L L L

at the midspan is plotted as a function of the root mean square a? of nondimensional forcing function

in Fig. 2. The solid straight line indicates the (exact) analytical solution for the corresponding linear string, and the dashed curve is the solution for the nonlinear string obtained by means of interpolation through the simulated points. In order to examine accuracy, the present method is also applied to the linear string. The result shows that the method is reasonably accurate if the simulation of fo(x,t) is based on 500 cosine functions (N-500 in Eq. 32) and the computation of the root mean square of the response is performed over the time interval equivalent of 50 fundamental periods

•366 Volume 49 Number 1 (Part 2) 1971


SIMULATION OF RANDOM I•ROCESSES

(T= 50), whereas the solution is unreliable if T and N are as small as 5 and 100, respectively. On the basis of this observation, the interpolation for the nonlinear solution is performed through the simulated points based on T= 50 and N- 500.

It is to be noted, however, that the result of the simulation for the nonlinear string behaves much less violently even for T= 5 and N= 50 because of the type of nonlinearity considered. This suggests that smaller values of T and N than those needed for the corresponding linear string may usually serve the purpose. In th;s example, the analysis has considered up to the third mode (M-3 in Eq. 98). The time required to produce five simulated points with T= 50 and N= 500 for the nonlinear problem is of the order of 150 sec on an IBM 360/91 system.

This method of solution is superior to other methods such as equivalent linearization ta in two points: (1) it does not require that the deviation from th elinearity be small; (2) the accuracy of the solution can be confirmed in terms of the stability of the simulated points.

A more detailed discussion of the present numerical example together with the result of the study of the nonlinear plate vibration currently being undertaken will be published in a separate paper.

V. CONCLUSION

Efficient and practical methods of simulating multivariate and multidimensional processes with specified cross-spectral density are presented.

When the cross-spectral density matrix of an n- variate process is specified, its component processes can be simulated as the sum of cosine functions with random

frequencies and random phase angles. The random frequencies are simulated from distribution functions derived from the cross-spectral density matrix, whereas the random phase angles are from the rectangular distribution between 0 and 2•r. Typical examples of this type are the simulation, for the purpose of shaker test, of a multivariate process representing six components of the acceleration (due to, for example, a booster engine cutoff) measured at the base of a spacecraft and the simulation of horizontal and vertical components of earthquake acceleration.

It is shown that a homogeneous multidimensional process can also be simulated in terms of the sum of cosine functions with random frequencies drawn from

the joint density function associated with the generalized spectrum of the process and with random phase angles uniformly distributed between 0 and 2•r. Ex- amples of multidimensional processes considered here include the horizontal component fo(t,x) of the wind velocity perpendicular to the axis (x axis) of a slender structure, the vertical gust velocity field fo(x,y) frozen in space, and the boundary-layer pressure field fo(x,y,t). Also, a convenient use of the present method of simulation in a class of nonlinear structural vibration analyses is described with a numerical example.

ACKNOWLEDGMENTS

This paper presents the results of one phase of re- search carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract, and sponsored by the National Aeronautics and Space Administration. Also, this work is partially supported by the Department of Civil Engineering and Engi- neering Mechanics, Columbia University.

* Consultant, Jet Propulsion Lab., Pasadena, Calif. 91103. 1 M. Shinozuka and T. Kobori, "Fatigue Life of Simple Bridges;

A Sensitivity Study," Columbia University, Tech. Rep. No. 8, NSF-GK 3858.

"H. Goto and K. Toki, "Structural Response to Nonstationary Random Excitation," Proc. World Conf. Earthquake Eng., 4th, Santiago, Chile (1969).

a j. Laning and R. Battin, Random Processes in Automatic Control (McGraw-Hill, New York, 1956).

4 L. E. Borgman, "Ocean Wave Simulation for Engineering Design," J. Waterways Harbors Div., Proc. Amer. Soc. Civil Eng. 95, No. WW4, 557-583 (Nov. 1969).

5 ¾. K. Lin, ,Probabilistic Theory of Structural Dynamics (Mc- Graw-Hill, New York, 1967), p. 158, Eq. 6-13.

6 M. Shinozuka and J.-N. Yang, "Numerical Fourier Transform in Random Vibration," J. Eng. Mech. Div., Proc. Amer. Soc. Civil Eng. 95, No. EM3, 731-746 (June 1969).

7 M. S. Bartlett, An Introduction to Stochastic Processes (Cam- bridge U. P., Cambridge, England, 1962).

8 After this paper had been processed for publication, it came to the author's attention that Eq. 76 was in fact non-negative, owing to Bochner's theorum.

9 H. E. Cramer, "Use of Power Spectra and Scales of Turbulence in Estimating Wind Loads," Meteorol. Monogr. 4, 12-18 (1970).

10 H. E. Cramer, "Measurement of Turbulence Structure near the Ground with the Frequency from 0.5 to 0.01 cycle sec-1," in Advances in Geophysics 6 (At•nospheric Diffusion and Air Polution) (Academic, New York, 1959), pp. 75-96.

n G. K. Batchelor, Theory of Homogeneous Turbulence (Cam- bridge U. P., Cambridge, England, 1953).

1,. E. H. Dowell, "Transmission of Noise from a Turbulent Boundary Layer through a Flexible Plate into a Closed Cavity," J. Acoust. Soc. of Amer. 46, 238-252 (1969).

•a For example, T. K. Caughey, "Response of a Nonlinear String to Random Vibration," J. Appl. Mech. 26, 341-344 (1959).



M. SHINOZUKA

Appendix A

For positive co and Aco, a random variable ['*(co+ Aco) --F*(co) can be written as

if2

[,¾(•+a•)-•½)] 2N

0 ,'2 N

= 22 z,•, (A•) 2N z•-•

r*(•+a•)-r*½)=

where Ik are indicators such that

Ik= 1, if co=<co•-<coq-Aco, or -- co -- A co = co • _<_ -- co , = 0, elsewhere. (A2)

Then, the probability that co, will take values that satisfy co < co• < co+ Aco or --co-- Aco < cok < --co is, for a small value of Aco,

The second moment. of r*(co+ Aco) -- r*(co) becomes

•,[r*½+a•)- r*(•)

•4 N N

E EEI•']+ 5-] E EEI•3E[I,3(1-•sk,) 4N •' z•=t z•-.t

a'x { 2NSø(•)Aco - +(,v'-N) - 4N 2 a 2 a 4 /'

(A6)

where/i•t is Kronecker's delta.

Therefore, the variance of r*½+a•)-r*½) is

Var[r*½+a•)-- r*½)]

= So(•)a•--So'-(•) (a•)'-, 2N N

which indicates

(A7)

P E ( co --< co • =< co + A co ) U ( -- co -- A co _< co • _< --co)]

fo• ø•+Aø• /•-• = g(co')&o'+ g(co')dd

2So(co)ZXco ---- 2g(co)Aco-- , (A3)

where ExUE,. indicates the union of two events Ex and E•., and PIE] the probability of occurrence of event E. In deriving Eq. A3, a use is made of Eq. 2 together with the fact that g(co) is an even function of co.

It then follows that

Elis] = E[Ik 23 =/'[(•__< •__< •+ a•) r/(-•- a•=< •__< -•)3 = 2S0(co) zXco/a'-, (A4)

and therefore

•[r* ½+ a,o) - r*½)3 = s0½)a•. (A5)

lim Var[r*(o,+a.,)-- r*½)] =0. (A8)

Because of symmetry, the expected value and the variance of r*(-•)-r*(-•-a•) are respectively identical with Eqs. A5 and A7. Therefore, the statement in Eq. 11 follows.

Incidentally, the coefficient of variation of r* (co+ Aco) -- r* (co) is, from Eqs. A5 and A7,

c.v.[r*½+a•)- r*½)]= F. V2•a•s0½)]•, (A9)

where the second term of the right-hand side of Eq. A7 has been disregarded.

It is interesting to note that the coefficient of variation of the mean square of those components of an ergodic Gaussian process that are associated with frequencies between co and co+Aco is proportional to 1/(TZXco)t when measured from a sample function of duration T with the aid of a narrow-band filter at co with the band width Aco. The coefficient of variation in

the present case is proportional to 1/(NAco)t as in Eq. A9. Equation A9 also indicates that larger values of coefficient of variation are expected as I col gets larger. This can be considered as a part of the reason why the discrepancy between the target value and the simulated value is rather conspicuous for larger values of f=co/2•' in Fig. l(a).

368 Volume 49 Number 1 (Part 2) 1971


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