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Dynamic response of linear structures to correlated random impulses
Journal of Sound and Vibration (1983) 86(3), 303-317
DYNAMIC
RESPONSE
TO CORRELATED
OF LINEAR RANDOM
STRUCTURES IMPULSES
R. IWANKIEWICZ Znstituteof Civil Engineering,
Technical University of Wrocfaw, Wybrzeie Wyspian’skiego 27, SO-370 Wrocfaw, Poland AND
K. SOBCZYK Znstitute of Fundamental
Technological Research, Polish Academy 00- 049 Warszawa, Poland
of Sciences, Swietokrzyska 21,
(Received 5 October 1981, and in revised form 31 March 1982)
A systematic analysis of the dynamic response of linear continuous structures to randomly arriving impulses is presented. A counting (or point) process characterizing the considered stream of impulses is described by the product density functions of degree one and two. By making use of a normal mode approach and assuming specific forms for the product densities (characterizing the expected arrival rate of the impulses and their correlation) the formulae for the variances and cross-covariances of modal responses are derived. The variance of the plate response is obtained and discussed for different practical situations and the results are shown graphically.
1. INTRODUCTION In many problems of structural dynamics an excitation acting on a specific structure consists of a series of impulses occurring at random times and with random strength. For example, among the many stochastic processes that have been suggested for modelling vehicular traffic flow and the corresponding loading of highway bridges such a process of randomly arriving impulses has been successfully used [l], [2]. Another example is an excitation due to earthquake; a number of authors have simulated earthquake process as a series of random pulses [3-51. Lin [4] indicated the conditions under which there may be valid reasons for modelling an earthquake excitation by uncorrelated and correlated random impulses, whereas Verne-Jones [S], using earthquake data from New Zealand, focused his attention on modelling an earthquake excitation by a general point process (stream of correlated random events). The analysis of the response of vibratory systems to random impulses has received much attention in the past [6-91. The basic assumption usually made in the literature is that the number of impulses occurring in any finite collection of non-overlapping time intervals form a set of independent random variables or-in other words-that the points on the time axis corresponding to the times of occurrence of impulses are distributed in accordance with the Poisson law with constant (stationary stream of impulses) or with variable (non-stationary) arrival rate [6,8,9]. As the examples indicated above show, the assumption about independence of impulses may not always be adequate; most frequently the Poisson process is introduced because of its inherent simplicity and the possibility of obtaining more effective results. It is, therefore, of importance to obtain information about the response of engineering systems subjected to correlated random 303 0022-460X/83/030303
+ 15 $03.00/O
@ 1983 Academic Press Inc. (London) Limited
304
R. IWANKIEWICZ
AND
K. SOBCZYK
impulses. Various aspects of this problem have been considered in the literature [4,7,10]. From the point of view of structural dynamics the analysis presented in the works by Lin is of special methodological interest. In the present paper we wish to extend the existing methodology to a specified practical problem in order to obtain results useful in structural engineering. A systematic analysis of the dynamic response of linear (continuous) elastic structures to randomly arriving impulses with special regard to the effect of correlation of impulses is provided. A counting (or point) process characterizing a series of pulses is described by the product densities of degree one and two. By assuming specific forms for these functions and performing numerical calculations the variance of the deflection of an elastic plate has been obtained for different practical situations. The figures show the results in the case of uncorrelated and correlated impulses.
2. RESPONSE Consider
vibrations
TO RANDOM
of an arbitrary L[w(r,
IMPULSES;
linear continuous
t)]+cti(r,
t)+mti(r,
GENERAL structure
FORMULAE governed
t)=p(r)F(t),
by the equation (1)
where L is a linear operator in the spatial variables, and m and c are the mass density and the damping coefficient, respectively, which can be, in general, functions of the spatial co-ordinates, Here the excitation is assumed to be of separable form, where p(r) denotes the deterministic function of the spatial co-ordinates and F(t) the stochastic process. Using a normal mode approach [4,11], i.e., substituting in equation (1) w (r. r) = C
(2)
qj(r)fi(r),
where qj(t) andfj(r) denote the generalized co-ordinate and the normal mode, respectively, and performing appropriate operations, one obtains the set of uncoupled equations qj +2’YjWjQj +O?qj = Yj(t), Yj(t)
=
pj
PjFCt),
=
j=1,2,...,
JR f;(r)p(r) dr/JR4
(3)
(r) dr7
where wj is the natural frequency of vibration, a, is the damping ratio and R is an appropriate measure (length, area, etc.) of the structural element considered. Note that since it is assumed that damping in the different modes is decoupled, the normal modes must be orthogonal also with respect to the damping coefficient c, which condition is satisfied, for example, when c is proportional to m. This is assumed to be the case here. Then 2ajwj = 2aiwi = const. = c/m. The subscription of the normal modes must be, of course, double if a two-dimensional structure is considered (e.g., a plate). The excitation considered herein is the train of Dirac delta function impulses with random amplitudes and occurring at random times. Thus, the excitation F(t) is represented as N(t) F(t)
=
c i=l
fiS(t
-&),
(3
where Fi is the random amplitude of the pulse and N(t) is a general counting process. If the pulses’ arrivals ti are regarded as random points on the time axis, then the process N(t) gives the number of points in the time interval (0, t]. In what follows, it is continuously assumed that the amplitudes Fi and the process N(t) are statistically independent.
RESPONSE
TO CORRELATED
RANDOM
305
IMPULSES
As in reference [4] it is assumed that the probability of occurrence of one impulse in the infinitesimal time interval dr is proportional to dr, and the probability of occurrence of more than one impulse in this interval is negligibly small: i.e., P&V(r) = 1) =fi(r)
dr,
P{dN(r) > 1) = u(dr).
(6 7)
From the above assumptions it follows that E[dN(r)] =f1(7) dr and also E{[dN(7)12} =f1(7)
dr,
(89)
where fl(~) is called the first-order product density and is the expected arrival rate of impulses. The correlation between arrival times is characterized by [5] E[dN(rl)
dN(r2)l
=f2(71,72)
dT1
(10)
dT2
when the intervals do not overlap; in the opposite case E[dN(rl)
CW(72)lci,l=dq = E{[cWh)12)
=~I(TI)
(10')
dT1.
The function f2(7i, r2) is called the second-order product density of the point process N(t). The first two correlation functions of the sequence of random times are defined as [4,51
g1(7)=f1(7),
g2(71,72)=f2(n,
72)-fl(dfl(72).
(11)
If the impulses arrive at uncorrelated times, then g2(T1, r2) = 0. The functions defined above characterize the basic probabilistic features of the point stochastic process N(t). Complete information about the probabilistic structure of N(t) is contained in the characteristic functional [4]
(12) where I belongs to a suitable class of functions, functional [5,12] which is defined by
GkWl=E{w
or in the probability
generating
1 log50)dN(O],
(13)
where t(t) belongs, for example, to the class of functions of bounded variation, equal to unity outside a finite interval and satisfying everywhere O<[(t) G 1. To obtain the distribution of the number of events in a given interval (or collection of intervals) from the functional (13) it is necessary only to take a suitable choice of t(t). The probability generating functional is related to the product densities in the following way: G[1+6]
= 1 +,I
$, j j . . . j [(TI) . ’ * ‘$(Tk)fk(Tl,
. . . Tk)
dT1
where fk (TV,.. . , Tk) is the product density of order k, defined analogously to f2(71r
(14)
. . . dTk,
f1(7)
and
72).
Probability generating functionals constitute an important tool in the theory of point stochastic processes [5, 121; however it is difficult to handle them in applications to system dynamics (except when the process considered is a simple Poisson process). For this reason in the characterization of the response of a structure we shall restrict ourselves to the product densities fl(r) and f2(Tl, TV). The modal response or generalized co-ordinate I, as a solution of equation (3), can be written in the form of a stochastic integral with respect to the counting process N(t)
306
R. IWANKIEWICZ
AND
K. SOBCZYK
[13] as 4jtc)=Pj
o*F(~)hj(~-~)
w(T),
(1%
J
where hj(t -7) = SF' exp [-ajWj(t - r)] sin [j(t - 7) is the impulse response function and [j = mjJ1-(wi2 is the damped natural frequency. The mean value (denoted by E) of the modal response is given by E[qj(t)] = Pj
J’ -7)flCT) dT* 0E[F(T)Ihj(t
In the general case of correlated arrival times and correlated mean square of the modal response is
E[d WI = Pi” Jo' Jo'hj(t
(16)
random amplitudes the
-T~)~~(~-T~)E[F(T~)F(T~)I E[~(TI)
PI*
(17)
dT2,
(17’)
Taking into account formulae (10) and (10’) one obtains
=&
E[qf(t)]
J’hi(r
-7)
E[F’(T)&(T)
0 f
+&
JJ 0
dr
f
0
hj(t_7l)hj(f-~*)kFf2(~1r
TZ)~TI
where kF(rl, r2) = E[F(T~F(T~)] ch aracterizes the correlation of the amplitudes. The expression for variance of the modal response in the general case has the form
~~j(~)=E[4?(~il-{E[~j(t)g2=pf~‘~~(f-~)E[~2(~)lf,(~)d~ 0
I +P;
xbf2(w
JJ 0
t
0
hj(t-Tl)hj(t-T*)
72)-
JW(dl
W(T2)lf1(T1)f1(T2)~dT1
dT2.
(18)
The second term in equation (18) accounts for the influence of the correlation of impulses on the response of the structure. In the simplest case when the arrival times are uncorrelated f2(r1, 72) =f1(~1)f1(T2) and the amplitudes of the impulses are uncorrelated as well (,&(T~, T2) = E[F(T~)] E[F(rz)]) only the first term in formula (18) remains. The corresponding general formulae (correlated arrival times and correlated random amplitudes) for the cross-covariances of the modal responses can be derived in a similar way. If the arrival times are correlated, but the amplitudes are assumed to be mutually independent and identically distributed, then cr;,(t)=&
E[F2] Jc/z;p~)g~(r)d~ 0
+PjPk{E[F]}2
J’J’~j(~-~l)hk(~-~2)g2(~1~ 0 0
72) dT1 d72*
(20)
RESPONSE
TO
CORRELATED
RANDOM
307
IMPULSES
The expression for the variance of the total structural response w (r, t) takes the form &r,
t) = E[w’(r, t)l-{E[w(r,
= c d,(f)f: tr)+ 1 j
t)]}2
CoVqjqk(f)f;(dfk(r)e
(21)
j,k j#k
The question which arises at this stage is that of the determination of the probability distribution of the displacement field w(r, t), in the general case. Since the random function w (r, t) is given by expression (2), the basic problem is associated with obtaining the probability distribution of qj(t), represented by the stochastic integral (15). This goal, however, is in general difficult, if not impossible, to achieve. If the counting stochastic process N(t) is the Poisson process (impulses are independent) then the probability density function of qj (t) can be determined as a solution of a certain differential equation [13,14]. In the case of a Poisson stream of pulses one can also find the conditions under which qj(t) tends to a normally distributed process [6]; roughly speaking the excitation must consist of a very dense stream of impulses whose strength approaches zero. If the process N(t) in the integral (15)-considered as a random process whose realizations are step functions (which change discontinuously at the instants when impulses occur)-is a purely discontinuous Markov process of the Feller-Kolmogorov type then the vector process [41(t), N(t)] is Markovian and its joint probability density function satisfies an integro-differential equation [13]. In more general cases, some results in the asymptotic theory of stochastic differential equations are likely to be promising [ 151. 3. RESPONSE
TO THE SPECIFIED
RANDOM
IMPULSES
By making use of formula (21) with expressions (19) and (20)-or expression (18) and the corresponding formula for cov,,,(t)-it is possible to determine the variance of the response for any specified product densities fl(rj and f 2(71, TV)(or correlation functions gi(7) and g2(Ti, 72)) and for an arbitrary linear continuous structure, if the normal modes fj(r) are known. In this section we specify expressions (19) and (20) by assuming particular forms of g*(T) and g2(T1, 72). As far as gi(7) is concerned, we assume it in stationary form to be gi(T) = A = const, and in non-stationary form to be g1(7) = A +B sin PT(A aZ3);the latter form seems to be interesting since the structure excited by a train of impulses with such a characteristic exhibits a kind of “mean square resonance” as was observed by Roberts [6]. The statistical dependence between arrival times at instants T1 and T2 should decrease as the time lag (TI - 721 increases; hence it is reasonable to assume the second order correlation function in exponential form: i.e., g2(T1, 72) = D exp (-bITI - 721) [4]. 3.1.
TRANSIENT
RESPONSE
Case I: g,(r) = A = constant, gz(Ti, respectively,
TV) =
0. From equations (19) and (20) one obtains,
c:,(t) = (pj/lj)2A E[F2](1/2w~){(~~/2ajwj)[l -exp (-2CXjWjt) sin COVq,qk (t)
=
(Pj@k/lj6k)A +%~j~j+~kwk)[Ll(@j,
cjl(ajWj
sin
ljf
+lj
-exp (-2ajojt)] COS lit)},
ELF21 ’{!iexP[-(‘“jwj+ ~k@k)f]~2(-~jWj wk)+LZ(wj,
wk)l),
(22) -akwk,
6j,
lh,
t) (23j
308
R. IWANKIEWICZ
K. SOBCZYK
exp (xt) sin yt sin zt dt
12(x, y, z, t) = 2 exp (-xt) =
AND
xcos(y-z)t+(y-z)sin(y-z)t x2+(y
xcos(y+z)t+(y+z)sin(y+z)t
_
-2)2
x2+(y
Ll(wj,Wk)=1/[((YJ~j+~~WL)2+(5j-5k)21,
>
+z)*
L2(Wjy @Jk) = l/[(aJwj
+~!~~k)~+
(5, +6121. (24)
In the case of an undamped
system,
applying
c+$(t) = (pj/oj)2A (t)
covq,qk
=
2
A
the limiting
E[F2](1/4wj)(2wjt
and =A and one
-sin
(all oj + 0), one obtains
2wjt),
(25)
Ep?-1; ( sinE;;;k)r_sin;;;k)r). I
I
The results (22) Case II: gr(r) expressions (19) to gr(r) = const.,
process
(26)
J
(25) are the same as those of Roberts [6]. +B sinpr, g2(r1, TV)= 0. Inserting gr(r) into the integrals given in (20) and omitting for the sake of brevity the terms corresponding obtains, after integrating,
aij(t) = ($2B
E[F2]${ exp (-2aj~j~)[2ajwj
~~f~~~~~~“,~““” J I I
25it
2p - 2CYjWjsin 2rjt + (2f; +p) COS 2ljt + 4cY;w; +p2 I 4oTWT + (2[j +p)2 + hj6Jj
sin Pt - 2p
-
2ajWj sin pt - (p - 2lj)
COS Pt
+p2
4a;w;
-
2ajOj sin pt - (p + 253)
COS pt
4Ly~W~ + (p -21j)2
COS Pt
(27)
I’
40!fWT + (p + 2[j)*
A3+&K-_k) +
[(CujWj
+
akok)2+(~j-_k+p)21(~j-_k) A3-AzKj
+lk)
+R ~Jw,+(YkOk)2+(5j+5k-P)21(5j+5k) A3+A2(lj
+Lk)
Al(x,
y) =
hwi+(Y,&)x
A2 = (aj0Jj +CUkWk)Sh
sin xc - [(CljWj + (Y,&)‘+ [(ajWj
Pf i-p
(28)
I’
-[(~i~j+~k~k)2+(~j+~kkP)21(~j+~k)
+ (Ykwk)2
COS Pt,
y (x + y)]
+ (X + Y)~]x
A3=(~j~j+ak~k)~+p~COSPt.
cos
xl
, (2%
The expressions (27) and (28) consist both of transient terms (decaying exponentially with time) and of steady state terms. The steady state term in expression (27) is the same as that given by Roberts [6]. It is seen that the jth modal response is amplified when p = 24’+ The contribution of cross-covariances in the total response depends not only on the lag wk -q, but also on the relation between p and both wk -wj and ok +q. This contribution is increased when p = uk -wj or p = wk +q. For an undamped system, one
309
RESPONSETO CORRELATED RANDOM IMPULSES has 2c_of(COSpt-1)+p2SiI12wjt
(30) p(p-20j)(p+2wj)
covqiqk
Lwk ---I3
(t) =
WjWk
'
E[~zJ~[(wj+w”~~~(~~~~k +p)
COS (Oj -Wk)f -
(wj -@k
-p)(wj
-wk
+p) 4WjWk
1
COS Pt
‘(Wj-Wk-p)(Wj-“k+p)(WjfWk-P)(Wj+Wkfp)
(31)
’
It can be shown, after some rearranging, that there is no singularity in expression (30) p = 2Oj and in eXpreSSiOIl (31) fOI’ p = uk -0j and p = Oj +ok. Case III: gz(ri, 72) # 0. If the impulses arrive at correlated times then the double integrals in expressions (19) and (20) also have to be evaluated. In what follows only additional terms resulting from these integrals will be given. When the second-order correlation function is g2(r1, Q) = D exp (-f~ 171 - 72 I), then the variance and cross-covariante function of the modal responses are, respectively,
for
a*.(t) = 41
4 li
+
D
E2[F]
02
(b -ajLdj)2
[exp {-(b
COS2 [jt -
+ajWj)t}([j
-iCl(Wk) d~l(Wjy
+ ‘Yj exp (-2CYjWjt){(d1_
1 f CY;-
sin
2ljt
+ 2ffj
sin2 ljt)}
2&jt - ‘Yj COS 25jt))
21j (b +ajUj)2+lT
COS ljt + (b + ajWj)
+akWk)t}I2(-ajWj
X [i eXp {-((ujwj
-$cl(wj)
sin
sin 5jt)-[jl),
(32)
E2F’I{[C3(-~i)+c3(-Ok)1
COVq,4k (t) = (pj@k/ljck)D
1 +dajWj
-exp (-2CXjUjt){(q
2Wj
T{l - exp (-2ajUjt)}]
+exp (-2cUjWjf) X
-b
{ [
+ [f
+akuk)(Ll(Wj, exp
Wk )-L2(Wjv
[-(@jWj
exp
-ffkukr
lj,
lk,
Wk))]
+~kWk)tl~l(-ajWj_“kWk,
[-(QjWj
t)
+~kWk)t]~l(-ajWj
lkv lj, -akWk,
Wk)(~j-~k)[cl(~j)-c1(~k)1-~~2(Wj,
t)
lj? 6kp t) Wk)(lj+6k)[Cl(Wj)+Cl(Wk)l
+cl(Wj)[-exp{-(b+~kWk)t}S(~k)+C2(~k)I+Cl(~k) X
[-exp {-(b
1i(x, y, z, t) = 2 exp (-xt)
+ajWj)cIS(Wj)
I
(33)
+ c2(wj)lI,
exp (xt) sin yt cos zt dt
xsin(y-z)t-(y-z)cOS(y-r)t+xsin(y+z)t-(y+r)cos(y+r)t ZZ x2+(y -# x2+(y +2)2 Cl(Wj) C3(Wj)
= (b + ajWj)/[(b
= ljl[(b
-“jWj)2
+cYjWj)2
+ lf],
+
Lf19 S(Wj)
CZ(Wj)
= cj/[(b
= C3(Wj)
sin
+ajoj)2 ljt
+ C2(Wj)
,
+ &Ti’l, COS ljt*
(34)
When b + co, the impulses arrival times become uncorrelated and both expressions (32) and (33) accounting for the influence of this correlation tend to zero. On the other hand,
R. IWANKIEWICZ
310
AND
K. SOBCZYK
if b + 0 then the impulses are correlated regardless of the length of time lag 1TV--n[, and this may be referred to as the case of infinitely strongly correlated impulses. In the case of an undamped system
COVqflk
(t) =
&[exp
zDE’[F]{
[CCJ(Wj)
+
(-bt)(b sin C,(Wk)][
Wjt
+ Wj COS Wjt)
+A cos (wj -Wk)t
+ci(Wj)Ci(W-exp Cl(Wj)=Wj/(b’+Wf),
1
,
I
1 [-cl(oj)
Wj-Wk
Wj]
sin~~~~k”sin~~~k’t] I
2
-
ccl(@k)I+-
COS (Wj
2
+Wk)t [Cl(Wk)+Cl(Wk)l
wi+wk
(36)
(-bf)[Cl(Wj)s(Wk)+Cl(Wk)s(Wj)I]t S(Wj)
C3(Wj)=b/(b2+Wf),
=Cj(Wj)
Sin Wjt+Cl(Wj)
COS Wjt.
(37) The variance of the total structural response exhibits fluctuations, which are of rather a complicated nature. Each jth transient modal response fluctuates with frequency 251 (equations (22), (27) and (32)). The coupling of each pair of normal modes contributes additional fluctuations with frequencies [j + f;c and & - f; (equations (23), (28) and (33)). The influence of this coupling depends, of course, on the lag 1[k -(,I. In Case II also fluctuations with frequency p are present. 3.2.
STEADY
STATE
RESPONSE
The expressions for variances and cross-covariances state are obtained by applying the limiting process transient quantities. In Case I this yields ai, = (pj/lj)‘A
E[F2](l
of the modal responses in steady t+oo to the expressions for the
-~~)/4~jwj,
(38)
2ljCk(ffjWj+akWk)
(39)
[(ajWj+LukWk)2+(~j-~k)2I[(~jWj+akWk)2+(f;.flk)2I’
In Case II, for the steady
-
2ajwj
state one obtains
sin pt - (p - 21,) 40ZFWf +(p-21j)2
COS pt
2ajWj
-
sin Pt - @ + 25,) 4CYyi2WT + (p +
COS Pt
21j)2
1 ’
(40)
(41)
RESPONSE
TO
CORRELATED
RANDOM
311
IMPULSES
For an undamped system (in Case II) ati
=&B
CO”qiq* (t) = PjPkB E[F*l
E[F2]
2 cos pt
(42)
P(P - 2Oi)(P + 2Wi)’ 2p cospt
(43)
(Oj-Wk-P)(Wj-Wk+P)(Oj+Ok-P)(WjfOk+P)’
The results (38), (40) and (42) are identical with those obtained by Roberts [6]. There is a singularity in expression (42) for p = 0 and p = 2Wj and in expression (43) for p = wk - wi and p = Uj + wk, but this observation is of no practical importance, since in a real excitation the stationary component of g1(7) must always exist, and hence the variance of the steady state response of the undamped system is infinite (see equation (38)). In Case III (the influence of the correlation of impulses), one has 2°F -- b 2cUjwj 1f(b+ajaj)2+[f
2DE2[F](1-~j)2 (+“= 0 z (b-ajwj)2+[T 2
CO”w?k=
fij
(~j~k/~j~k)~~2[~1{2~j~k~l(~j,
mk)L2(wj,
1
(44)
’
~k)[c3(-Wj)+c3(-Wk)l
+S[~,(wj,Ok)(5j_5k){Cl(Wj)-C1(Wk)}-L2(0j,
~k)(f;++k){cl(~j)+cl(~k)}l
(45)
+cl(~j)c2(~k)+cl(~k)c2(~j)}.
Expressions (44) and (45) also could be obtained by using a spectral method, with the excitation considered as a continuous stochastic process with correlation function K (W 72) = D exp (-b I 71 - 72 I). This is justified by the analogy between the integrals in such case and the double integrals in formulae (19) and (20). 4. EXAMPLE
PROBLEM:
RESPONSE OF A RECTANGULAR
PLATE
The response of an uniform rectangular elastic plate, simply supported at all edges, can be examined as an example. As is well known, in this case the operator is L = N(d4/ax4 + 2a4/ax2ay2 + a4/ay4) and the normal modes have the trigonometric form fi,j(x, y) = sin (irx/ll) sin (j7ryIE2). Natural frequencies can be computed by using the formula (see, e.g., reference [16]) for the thin elastic plate Wi,j
=
T2[(i2/l:)
+
(46)
(i’/li)]d$G,
where N, rn, I1 and l2 denote plate bending rigidity, mass density and side lengths, respectively. The expression (21) for the variance of the plate response (deflection) takes the form c2w(X,Y, t)=C~2,,,(t)ftj(X, i,j
C
Y)+ i#k
co”q~.~q~,, (t)fi,j(X,
Y)fk,l
(x3
Y 1.
(47)
i.j.k.1 and/or j#l
If the excitation is spatially uniform, i.e., p(x, y) = const. = po, then .
sin2 7
sin2 ‘7 dx dy = 1
2
The form of @i,jindicates that for this type of excitation the contribution of succeeding normal modes in the total structural response diminishes as the indices of the mode increase. Thus the contribution of modal cross-covariances depends here not only on the spacings between the natural frequencies but also on the indices of modes. In other
R. IWANKIEWICZ
312
AND
K. SOBCZYK
words the contribution of cross-covariance between two high modes is rather small even if they are close together. With attention confined to the deflection in the middle of the plate, one need take into account only symmetrical mode shapes (strictly-bisymmetrical, i.e., for odd i and j), In the example under consideration nine normal modes have been taken into account: that is fi,i(x, y) for i = 1,3,5 and j = 1,3,5. The nine corresponding natural frequencies of the example plate with side lengths ratio f1/12 = 2/3 and fundamental frequency Wl,J = 10 s-l are listed in Table 1. It is seen that they are fairly well separated. TABLE
1
First nine natural frequencies rectangular plate
of a
i
i
%
1 1 3 1 3 3 5 5 5
1 3 1 5 3 5 1 3 5
10.0 34.615 65.385 83,846 90.0 139.231 176.154 200.769 250.0
The contribution of higher modes variances and cross-covariances appeared to be negligible, which can be explained by the form of modal variances and cross-covariances, as well as by the form of the the character of the spectrum of the frequencies. The approximate variance of the deflection (within an accuracy not less than 3%) is under consideration as c&11/2,
I*/&
t) =(+~l.l(t)-2rCOV91.191.3(f)
in expression (47) the expressions for coefficients pi,j and expression for the found for the plate
+COv,I.,,,.I(t)l. (4%
At present the nature and numerical values characterizing real random impulsive excitations are not well recognized; hence some hypothetical reasonable data are assumed. The constants A, B and D are taken equal to 5 s-l, 5 s-l and 25 SC*, respectively. To avoid difficulties in adopting the data referring to the impulse strength F, the nondimensional quantity mt(11/2,12/2, t) w :.I /p :,1 E2[F] has been computed and it has been assumed that E[F*] = 1.1 E*[F]. The variance of the plate deflection in the transient state, in the case of uncorrelated impulses and for various damping ratios, is plotted in Figure 1. When the expected arrival rate is constant (g*(r) = const.), variance of the transient response fluctuates with a frequency which is double the fundamental natural frequency 01,1 of the system [6]. Variance of the transient response to non-stationary excitation (gl(T) = A + B sin pi) fluctuates mainly with a frequency p (here p = wlJ, though in the case of a lightly damped (al,l=O~O1) and an undamped (a 1,1= 0) system fluctuations with frequency 20r,~ can also be observed (the effect of transient terms in expressions (27) and (28)). The transient response of the plate in the “resonant” case of uncorrelated impulses is presented in Figure 2. Here the variance of the response fluctuates only with the frequency 201,i. It is seen that peak values of the variance are much greater than in the case of stationary excitation (constant expected arrival rate).
RESPONSE
0
TO
CORRELATED
Zn
4n ?,I
Figure 1. Transient variance damping ratios. -, gl(t)=A;
Figure 2. Transient damping ratios. -,
RANDOM
313
IMPULSES
6rr
f
of the plate deflection in the case of uncorrelated -, gl(t)=A+B sinoI,It.
variance of the plate deflection in “resonant” gI(t) = A; -, g,(t) =A +B sin 2wl,,t.
case of uncorrelated
impulses,
impulses,
for different
for different
The variance of the steady state response to the stationary excitation is, of course, constant (see expressions (38) and (39)). The variance of the steady state response to non-stationary excitation (variable expected arrival rate) oscillates about a certain fixed level with frequency p and amplitude depending on this frequency, as shown in Table 2 [6]. The variance of the transient response to stationary excitation in the case of correlated impulses, (i.e., the sum of the appropriate expressions in Cases I and III) is plotted in Figure 3. Here also the variance fluctuates with frequency 2~~,~. Only in the case of infinitely strongly correlated impulses (b = 0) does the variance fluctuate with frequency wi,i. It is to be supposed that for certain value of the correlation parameter b (undoubtedly between 2 and 30) the influence of the correlation of impulses attains a maximum. Straight lines in Figure 3 denote the asymptotes corresponding to the steady state. The response of the structure to correlated impulses, for different damping ratios, in the case of stationary excitation is shown in Figure 4 and in the case of non-stationary
314
R. IWANKIEWICZ
AND K. SOBCZYK
TABLE
2
Characteristics of the variance of the steady state response to nonstationary excitation
al.1
Fixed level
Amplitude for P = w1.1
0.01 0.05 0.2
13.75 2.743 0.688
0.367 0.360 0.331
Amplitude
for
P = 2Wl,,
6.86 1.37 0.349
Figure 3. Variance of the plate deflection for a~.~ = 0.05 and different correlation parameters b.
7
6
5
4
3
2
I
0 0
477
wI,I
-,
f
Figure 4. Variance of the plate deflection in transient and steady state for different damping ratios (gl(l) = A). For uncorrelated impulses; -, for correlated impulses (b = 10 s-l).
excitation in Figures 5 and 6. The variance of the response fluctuates in the same manner, regardless of the fact of whether the impulses are correlated or not. The influence of this correlation appears to be strongly affected by damping. Variance of the steady state response to non-stationary excitation, including the correlation of impulses, also oscillates about the fixed level, represented by the asymptotes in Figure 4.
RESPONSE
TO CORRELATED
Y
RANDOM
IMPULSES
315
t
Figure 5. Variance of the plate deflection in transient state for different dam ing ratios P for correlated impulses (6 = 10 s- ). B sin wi.it). -, For uncorrelated impulses; -,
Figure 6. Transient variance of the plate deflection in “resonant” B sin 2o,,ir), for different damping ratios. -, For uncorrelated (b = 10 s-i).
(g,(t) = A +
case of correlated impulses (gi(r) = A + impulses; -, for correlated impulses
The contribution of the terms accounting for the correlation of impulses arrival times in the steady state response is presented in Figure 7. It is seen that this contribution attains a maximum, which for lightly damped systems is in the vicinity of b = 10 s-l, and shifts to the left as damping ratio increases. When the impulses arrival times are infinitely strongly correlated (b = 0), the influence of this correlation is constant, regardless of the damping ratio of the system.
316
R. IWANKIEWICZ
AND
b
Figure7. Influence
of the correlation
of impulses
K.SOBCZYK
Is-'1 arrivals
on the steady
state response
variance.
It should be also noted that the influence of the correlation of impulses depends, in the case of uncorrelated random amplitudes, on the variance of these amplitudes (T: = E[F’]-E2[F]. Reviewing formulae (19), (20) and subsequent detailed formulae one can observe that as this variance increases the relative contribution of terms accounting for the correlation of impulses decreases. The prevailing contribution of the response in the first mode in the total structural response is due to the uniform spatial distribution of the excitation and to the character of the spectrum of the frequencies (frequencies corresponding to symmetrical mode shapes of the example plate are not closely spaced). However, in structures in which the natural frequencies are more closely spaced the contribution of modal cross-covariances may prove very high (as, for example, in shells [17]). REFERENCES 1. TUNG CHI CHAO 1967 Journalof theEngineering Mechanics Division, American Society of Civil Engineers 93, 79-94. Random response of highway bridges to vehicle loads. 2. TUNG CHI CHAO 1969 Journal of the Engineering Mechanics Division, American Society of Civil Engineers 95, 41-57. Response of highway bridges to renewal traffic loads. 3. C. A. CORNELL 1964 Department of Civil Engineering, Stanford University, Technical Report 34. Stochastic process models in structural engineering. 4. Y. K. LIN 1967 Probabilistic Theory of Structural Dynamics. New York: McGraw-Hill Book Company. 5. D. VERNE-JONES 1970 Journal of the Royal Statistical Society Series B, 32, l-62. Stochastic models for earthquake occurrence. 6. J. B. ROBERTS 1965 Journal of Sound and Vibration 2, 375-390. The response of linear vibratory systems to random impulses. 7. Y. K. LIN, 1965 Journal of the Acoustical Society of America 38, 453-460. Non-stationary excitation in linear systems treated as sequences of random pulses. 8. S. K. SRINIVASAN, R. SUBRAMANIAN and S. KUMARASWAMY 1967 Journal ofSound and Vibration 6, 169-179. Response of linear vibratory systems to non-stationary stochastic impulses. 9. A. RENGER 1979 Zeitschrift fiir angewandte Mathematik und Mechanik 59, 1. Equation for probability density of vibratory system subjected to continuous and discrete stochastic excitation (in German). 10. P. MAZZE~I 1964 Nuovo Cimento 31, 88. Correlation function and power spectrum of a train of non-independent overlapping pulses having random shape and amplitude.
RESPONSE 11. I. I. GIKHMAN
12. 13. 14. 15. 16. 17.
TO CORRELATED
RANDOM
IMPULSES
317
1977 Applied Mechanics 13(11), 18. Asymptotic behaviour of solutions of a mixed problem associated with random vibrations (in Russian). I. ~.KUZNIETSOV and R.L.STRATONOVICH 1956 Izuesfya AkademiiNauk USSR,Series Mathematics 20, 167-178. On the mathematical theory of correlated points. S. K. SRINIVASAN 1978 Solid Mechanics Archives 3, 325-379. Stochastic integrals. A. RAMAKRISHNAN 1956 Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 59, 121-127. Processes represented as integrals of a class of random functions. G. C. PAPANICOLAU and R. HERSH 1972 Indiana University Mathematics Journal 21, 815. Some limit theorems for stochastic equations and applications. W. NOWACKI 1972 Dynamics of Elastic Structures. Warsaw: Arkady (in Polish.) I.ELISHAKOFF 1977 Journal of Sound and Vibration 50, 239-252. On the role of crosscorrelations in the random vibrations of shells.
DYNAMIC
RESPONSE
TO CORRELATED
OF LINEAR RANDOM
STRUCTURES IMPULSES
R. IWANKIEWICZ Znstituteof Civil Engineering,
Technical University of Wrocfaw, Wybrzeie Wyspian’skiego 27, SO-370 Wrocfaw, Poland AND
K. SOBCZYK Znstitute of Fundamental
Technological Research, Polish Academy 00- 049 Warszawa, Poland
of Sciences, Swietokrzyska 21,
(Received 5 October 1981, and in revised form 31 March 1982)
A systematic analysis of the dynamic response of linear continuous structures to randomly arriving impulses is presented. A counting (or point) process characterizing the considered stream of impulses is described by the product density functions of degree one and two. By making use of a normal mode approach and assuming specific forms for the product densities (characterizing the expected arrival rate of the impulses and their correlation) the formulae for the variances and cross-covariances of modal responses are derived. The variance of the plate response is obtained and discussed for different practical situations and the results are shown graphically.
1. INTRODUCTION In many problems of structural dynamics an excitation acting on a specific structure consists of a series of impulses occurring at random times and with random strength. For example, among the many stochastic processes that have been suggested for modelling vehicular traffic flow and the corresponding loading of highway bridges such a process of randomly arriving impulses has been successfully used [l], [2]. Another example is an excitation due to earthquake; a number of authors have simulated earthquake process as a series of random pulses [3-51. Lin [4] indicated the conditions under which there may be valid reasons for modelling an earthquake excitation by uncorrelated and correlated random impulses, whereas Verne-Jones [S], using earthquake data from New Zealand, focused his attention on modelling an earthquake excitation by a general point process (stream of correlated random events). The analysis of the response of vibratory systems to random impulses has received much attention in the past [6-91. The basic assumption usually made in the literature is that the number of impulses occurring in any finite collection of non-overlapping time intervals form a set of independent random variables or-in other words-that the points on the time axis corresponding to the times of occurrence of impulses are distributed in accordance with the Poisson law with constant (stationary stream of impulses) or with variable (non-stationary) arrival rate [6,8,9]. As the examples indicated above show, the assumption about independence of impulses may not always be adequate; most frequently the Poisson process is introduced because of its inherent simplicity and the possibility of obtaining more effective results. It is, therefore, of importance to obtain information about the response of engineering systems subjected to correlated random 303 0022-460X/83/030303
+ 15 $03.00/O
@ 1983 Academic Press Inc. (London) Limited
304
R. IWANKIEWICZ
AND
K. SOBCZYK
impulses. Various aspects of this problem have been considered in the literature [4,7,10]. From the point of view of structural dynamics the analysis presented in the works by Lin is of special methodological interest. In the present paper we wish to extend the existing methodology to a specified practical problem in order to obtain results useful in structural engineering. A systematic analysis of the dynamic response of linear (continuous) elastic structures to randomly arriving impulses with special regard to the effect of correlation of impulses is provided. A counting (or point) process characterizing a series of pulses is described by the product densities of degree one and two. By assuming specific forms for these functions and performing numerical calculations the variance of the deflection of an elastic plate has been obtained for different practical situations. The figures show the results in the case of uncorrelated and correlated impulses.
2. RESPONSE Consider
vibrations
TO RANDOM
of an arbitrary L[w(r,
IMPULSES;
linear continuous
t)]+cti(r,
t)+mti(r,
GENERAL structure
FORMULAE governed
t)=p(r)F(t),
by the equation (1)
where L is a linear operator in the spatial variables, and m and c are the mass density and the damping coefficient, respectively, which can be, in general, functions of the spatial co-ordinates, Here the excitation is assumed to be of separable form, where p(r) denotes the deterministic function of the spatial co-ordinates and F(t) the stochastic process. Using a normal mode approach [4,11], i.e., substituting in equation (1) w (r. r) = C
(2)
qj(r)fi(r),
where qj(t) andfj(r) denote the generalized co-ordinate and the normal mode, respectively, and performing appropriate operations, one obtains the set of uncoupled equations qj +2’YjWjQj +O?qj = Yj(t), Yj(t)
=
pj
PjFCt),
=
j=1,2,...,
JR f;(r)p(r) dr/JR4
(3)
(r) dr7
where wj is the natural frequency of vibration, a, is the damping ratio and R is an appropriate measure (length, area, etc.) of the structural element considered. Note that since it is assumed that damping in the different modes is decoupled, the normal modes must be orthogonal also with respect to the damping coefficient c, which condition is satisfied, for example, when c is proportional to m. This is assumed to be the case here. Then 2ajwj = 2aiwi = const. = c/m. The subscription of the normal modes must be, of course, double if a two-dimensional structure is considered (e.g., a plate). The excitation considered herein is the train of Dirac delta function impulses with random amplitudes and occurring at random times. Thus, the excitation F(t) is represented as N(t) F(t)
=
c i=l
fiS(t
-&),
(3
where Fi is the random amplitude of the pulse and N(t) is a general counting process. If the pulses’ arrivals ti are regarded as random points on the time axis, then the process N(t) gives the number of points in the time interval (0, t]. In what follows, it is continuously assumed that the amplitudes Fi and the process N(t) are statistically independent.
RESPONSE
TO CORRELATED
RANDOM
305
IMPULSES
As in reference [4] it is assumed that the probability of occurrence of one impulse in the infinitesimal time interval dr is proportional to dr, and the probability of occurrence of more than one impulse in this interval is negligibly small: i.e., P&V(r) = 1) =fi(r)
dr,
P{dN(r) > 1) = u(dr).
(6 7)
From the above assumptions it follows that E[dN(r)] =f1(7) dr and also E{[dN(7)12} =f1(7)
dr,
(89)
where fl(~) is called the first-order product density and is the expected arrival rate of impulses. The correlation between arrival times is characterized by [5] E[dN(rl)
dN(r2)l
=f2(71,72)
dT1
(10)
dT2
when the intervals do not overlap; in the opposite case E[dN(rl)
CW(72)lci,l=dq = E{[cWh)12)
=~I(TI)
(10')
dT1.
The function f2(7i, r2) is called the second-order product density of the point process N(t). The first two correlation functions of the sequence of random times are defined as [4,51
g1(7)=f1(7),
g2(71,72)=f2(n,
72)-fl(dfl(72).
(11)
If the impulses arrive at uncorrelated times, then g2(T1, r2) = 0. The functions defined above characterize the basic probabilistic features of the point stochastic process N(t). Complete information about the probabilistic structure of N(t) is contained in the characteristic functional [4]
(12) where I belongs to a suitable class of functions, functional [5,12] which is defined by
GkWl=E{w
or in the probability
generating
1 log50)dN(O],
(13)
where t(t) belongs, for example, to the class of functions of bounded variation, equal to unity outside a finite interval and satisfying everywhere O<[(t) G 1. To obtain the distribution of the number of events in a given interval (or collection of intervals) from the functional (13) it is necessary only to take a suitable choice of t(t). The probability generating functional is related to the product densities in the following way: G[1+6]
= 1 +,I
$, j j . . . j [(TI) . ’ * ‘$(Tk)fk(Tl,
. . . Tk)
dT1
where fk (TV,.. . , Tk) is the product density of order k, defined analogously to f2(71r
(14)
. . . dTk,
f1(7)
and
72).
Probability generating functionals constitute an important tool in the theory of point stochastic processes [5, 121; however it is difficult to handle them in applications to system dynamics (except when the process considered is a simple Poisson process). For this reason in the characterization of the response of a structure we shall restrict ourselves to the product densities fl(r) and f2(Tl, TV). The modal response or generalized co-ordinate I, as a solution of equation (3), can be written in the form of a stochastic integral with respect to the counting process N(t)
306
R. IWANKIEWICZ
AND
K. SOBCZYK
[13] as 4jtc)=Pj
o*F(~)hj(~-~)
w(T),
(1%
J
where hj(t -7) = SF' exp [-ajWj(t - r)] sin [j(t - 7) is the impulse response function and [j = mjJ1-(wi2 is the damped natural frequency. The mean value (denoted by E) of the modal response is given by E[qj(t)] = Pj
J’ -7)flCT) dT* 0E[F(T)Ihj(t
In the general case of correlated arrival times and correlated mean square of the modal response is
E[d WI = Pi” Jo' Jo'hj(t
(16)
random amplitudes the
-T~)~~(~-T~)E[F(T~)F(T~)I E[~(TI)
PI*
(17)
dT2,
(17’)
Taking into account formulae (10) and (10’) one obtains
=&
E[qf(t)]
J’hi(r
-7)
E[F’(T)&(T)
0 f
+&
JJ 0
dr
f
0
hj(t_7l)hj(f-~*)kFf2(~1r
TZ)~TI
where kF(rl, r2) = E[F(T~F(T~)] ch aracterizes the correlation of the amplitudes. The expression for variance of the modal response in the general case has the form
~~j(~)=E[4?(~il-{E[~j(t)g2=pf~‘~~(f-~)E[~2(~)lf,(~)d~ 0
I +P;
xbf2(w
JJ 0
t
0
hj(t-Tl)hj(t-T*)
72)-
JW(dl
W(T2)lf1(T1)f1(T2)~dT1
dT2.
(18)
The second term in equation (18) accounts for the influence of the correlation of impulses on the response of the structure. In the simplest case when the arrival times are uncorrelated f2(r1, 72) =f1(~1)f1(T2) and the amplitudes of the impulses are uncorrelated as well (,&(T~, T2) = E[F(T~)] E[F(rz)]) only the first term in formula (18) remains. The corresponding general formulae (correlated arrival times and correlated random amplitudes) for the cross-covariances of the modal responses can be derived in a similar way. If the arrival times are correlated, but the amplitudes are assumed to be mutually independent and identically distributed, then cr;,(t)=&
E[F2] Jc/z;p~)g~(r)d~ 0
+PjPk{E[F]}2
J’J’~j(~-~l)hk(~-~2)g2(~1~ 0 0
72) dT1 d72*
(20)
RESPONSE
TO
CORRELATED
RANDOM
307
IMPULSES
The expression for the variance of the total structural response w (r, t) takes the form &r,
t) = E[w’(r, t)l-{E[w(r,
= c d,(f)f: tr)+ 1 j
t)]}2
CoVqjqk(f)f;(dfk(r)e
(21)
j,k j#k
The question which arises at this stage is that of the determination of the probability distribution of the displacement field w(r, t), in the general case. Since the random function w (r, t) is given by expression (2), the basic problem is associated with obtaining the probability distribution of qj(t), represented by the stochastic integral (15). This goal, however, is in general difficult, if not impossible, to achieve. If the counting stochastic process N(t) is the Poisson process (impulses are independent) then the probability density function of qj (t) can be determined as a solution of a certain differential equation [13,14]. In the case of a Poisson stream of pulses one can also find the conditions under which qj(t) tends to a normally distributed process [6]; roughly speaking the excitation must consist of a very dense stream of impulses whose strength approaches zero. If the process N(t) in the integral (15)-considered as a random process whose realizations are step functions (which change discontinuously at the instants when impulses occur)-is a purely discontinuous Markov process of the Feller-Kolmogorov type then the vector process [41(t), N(t)] is Markovian and its joint probability density function satisfies an integro-differential equation [13]. In more general cases, some results in the asymptotic theory of stochastic differential equations are likely to be promising [ 151. 3. RESPONSE
TO THE SPECIFIED
RANDOM
IMPULSES
By making use of formula (21) with expressions (19) and (20)-or expression (18) and the corresponding formula for cov,,,(t)-it is possible to determine the variance of the response for any specified product densities fl(rj and f 2(71, TV)(or correlation functions gi(7) and g2(Ti, 72)) and for an arbitrary linear continuous structure, if the normal modes fj(r) are known. In this section we specify expressions (19) and (20) by assuming particular forms of g*(T) and g2(T1, 72). As far as gi(7) is concerned, we assume it in stationary form to be gi(T) = A = const, and in non-stationary form to be g1(7) = A +B sin PT(A aZ3);the latter form seems to be interesting since the structure excited by a train of impulses with such a characteristic exhibits a kind of “mean square resonance” as was observed by Roberts [6]. The statistical dependence between arrival times at instants T1 and T2 should decrease as the time lag (TI - 721 increases; hence it is reasonable to assume the second order correlation function in exponential form: i.e., g2(T1, 72) = D exp (-bITI - 721) [4]. 3.1.
TRANSIENT
RESPONSE
Case I: g,(r) = A = constant, gz(Ti, respectively,
TV) =
0. From equations (19) and (20) one obtains,
c:,(t) = (pj/lj)2A E[F2](1/2w~){(~~/2ajwj)[l -exp (-2CXjWjt) sin COVq,qk (t)
=
(Pj@k/lj6k)A +%~j~j+~kwk)[Ll(@j,
cjl(ajWj
sin
ljf
+lj
-exp (-2ajojt)] COS lit)},
ELF21 ’{!iexP[-(‘“jwj+ ~k@k)f]~2(-~jWj wk)+LZ(wj,
wk)l),
(22) -akwk,
6j,
lh,
t) (23j
308
R. IWANKIEWICZ
K. SOBCZYK
exp (xt) sin yt sin zt dt
12(x, y, z, t) = 2 exp (-xt) =
AND
xcos(y-z)t+(y-z)sin(y-z)t x2+(y
xcos(y+z)t+(y+z)sin(y+z)t
_
-2)2
x2+(y
Ll(wj,Wk)=1/[((YJ~j+~~WL)2+(5j-5k)21,
>
+z)*
L2(Wjy @Jk) = l/[(aJwj
+~!~~k)~+
(5, +6121. (24)
In the case of an undamped
system,
applying
c+$(t) = (pj/oj)2A (t)
covq,qk
=
2
A
the limiting
E[F2](1/4wj)(2wjt
and =A and one
-sin
(all oj + 0), one obtains
2wjt),
(25)
Ep?-1; ( sinE;;;k)r_sin;;;k)r). I
I
The results (22) Case II: gr(r) expressions (19) to gr(r) = const.,
process
(26)
J
(25) are the same as those of Roberts [6]. +B sinpr, g2(r1, TV)= 0. Inserting gr(r) into the integrals given in (20) and omitting for the sake of brevity the terms corresponding obtains, after integrating,
aij(t) = ($2B
E[F2]${ exp (-2aj~j~)[2ajwj
~~f~~~~~~“,~““” J I I
25it
2p - 2CYjWjsin 2rjt + (2f; +p) COS 2ljt + 4cY;w; +p2 I 4oTWT + (2[j +p)2 + hj6Jj
sin Pt - 2p
-
2ajWj sin pt - (p - 2lj)
COS Pt
+p2
4a;w;
-
2ajOj sin pt - (p + 253)
COS pt
4Ly~W~ + (p -21j)2
COS Pt
(27)
I’
40!fWT + (p + 2[j)*
A3+&K-_k) +
[(CujWj
+
akok)2+(~j-_k+p)21(~j-_k) A3-AzKj
+lk)
+R ~Jw,+(YkOk)2+(5j+5k-P)21(5j+5k) A3+A2(lj
+Lk)
Al(x,
y) =
hwi+(Y,&)x
A2 = (aj0Jj +CUkWk)Sh
sin xc - [(CljWj + (Y,&)‘+ [(ajWj
Pf i-p
(28)
I’
-[(~i~j+~k~k)2+(~j+~kkP)21(~j+~k)
+ (Ykwk)2
COS Pt,
y (x + y)]
+ (X + Y)~]x
A3=(~j~j+ak~k)~+p~COSPt.
cos
xl
, (2%
The expressions (27) and (28) consist both of transient terms (decaying exponentially with time) and of steady state terms. The steady state term in expression (27) is the same as that given by Roberts [6]. It is seen that the jth modal response is amplified when p = 24’+ The contribution of cross-covariances in the total response depends not only on the lag wk -q, but also on the relation between p and both wk -wj and ok +q. This contribution is increased when p = uk -wj or p = wk +q. For an undamped system, one
309
RESPONSETO CORRELATED RANDOM IMPULSES has 2c_of(COSpt-1)+p2SiI12wjt
(30) p(p-20j)(p+2wj)
covqiqk
Lwk ---I3
(t) =
WjWk
'
E[~zJ~[(wj+w”~~~(~~~~k +p)
COS (Oj -Wk)f -
(wj -@k
-p)(wj
-wk
+p) 4WjWk
1
COS Pt
‘(Wj-Wk-p)(Wj-“k+p)(WjfWk-P)(Wj+Wkfp)
(31)
’
It can be shown, after some rearranging, that there is no singularity in expression (30) p = 2Oj and in eXpreSSiOIl (31) fOI’ p = uk -0j and p = Oj +ok. Case III: gz(ri, 72) # 0. If the impulses arrive at correlated times then the double integrals in expressions (19) and (20) also have to be evaluated. In what follows only additional terms resulting from these integrals will be given. When the second-order correlation function is g2(r1, Q) = D exp (-f~ 171 - 72 I), then the variance and cross-covariante function of the modal responses are, respectively,
for
a*.(t) = 41
4 li
+
D
E2[F]
02
(b -ajLdj)2
[exp {-(b
COS2 [jt -
+ajWj)t}([j
-iCl(Wk) d~l(Wjy
+ ‘Yj exp (-2CYjWjt){(d1_
1 f CY;-
sin
2ljt
+ 2ffj
sin2 ljt)}
2&jt - ‘Yj COS 25jt))
21j (b +ajUj)2+lT
COS ljt + (b + ajWj)
+akWk)t}I2(-ajWj
X [i eXp {-((ujwj
-$cl(wj)
sin
sin 5jt)-[jl),
(32)
E2F’I{[C3(-~i)+c3(-Ok)1
COVq,4k (t) = (pj@k/ljck)D
1 +dajWj
-exp (-2CXjUjt){(q
2Wj
T{l - exp (-2ajUjt)}]
+exp (-2cUjWjf) X
-b
{ [
+ [f
+akuk)(Ll(Wj, exp
Wk )-L2(Wjv
[-(@jWj
exp
-ffkukr
lj,
lk,
Wk))]
+~kWk)tl~l(-ajWj_“kWk,
[-(QjWj
t)
+~kWk)t]~l(-ajWj
lkv lj, -akWk,
Wk)(~j-~k)[cl(~j)-c1(~k)1-~~2(Wj,
t)
lj? 6kp t) Wk)(lj+6k)[Cl(Wj)+Cl(Wk)l
+cl(Wj)[-exp{-(b+~kWk)t}S(~k)+C2(~k)I+Cl(~k) X
[-exp {-(b
1i(x, y, z, t) = 2 exp (-xt)
+ajWj)cIS(Wj)
I
(33)
+ c2(wj)lI,
exp (xt) sin yt cos zt dt
xsin(y-z)t-(y-z)cOS(y-r)t+xsin(y+z)t-(y+r)cos(y+r)t ZZ x2+(y -# x2+(y +2)2 Cl(Wj) C3(Wj)
= (b + ajWj)/[(b
= ljl[(b
-“jWj)2
+cYjWj)2
+ lf],
+
Lf19 S(Wj)
CZ(Wj)
= cj/[(b
= C3(Wj)
sin
+ajoj)2 ljt
+ C2(Wj)
,
+ &Ti’l, COS ljt*
(34)
When b + co, the impulses arrival times become uncorrelated and both expressions (32) and (33) accounting for the influence of this correlation tend to zero. On the other hand,
R. IWANKIEWICZ
310
AND
K. SOBCZYK
if b + 0 then the impulses are correlated regardless of the length of time lag 1TV--n[, and this may be referred to as the case of infinitely strongly correlated impulses. In the case of an undamped system
COVqflk
(t) =
&[exp
zDE’[F]{
[CCJ(Wj)
+
(-bt)(b sin C,(Wk)][
Wjt
+ Wj COS Wjt)
+A cos (wj -Wk)t
+ci(Wj)Ci(W-exp Cl(Wj)=Wj/(b’+Wf),
1
,
I
1 [-cl(oj)
Wj-Wk
Wj]
sin~~~~k”sin~~~k’t] I
2
-
ccl(@k)I+-
COS (Wj
2
+Wk)t [Cl(Wk)+Cl(Wk)l
wi+wk
(36)
(-bf)[Cl(Wj)s(Wk)+Cl(Wk)s(Wj)I]t S(Wj)
C3(Wj)=b/(b2+Wf),
=Cj(Wj)
Sin Wjt+Cl(Wj)
COS Wjt.
(37) The variance of the total structural response exhibits fluctuations, which are of rather a complicated nature. Each jth transient modal response fluctuates with frequency 251 (equations (22), (27) and (32)). The coupling of each pair of normal modes contributes additional fluctuations with frequencies [j + f;c and & - f; (equations (23), (28) and (33)). The influence of this coupling depends, of course, on the lag 1[k -(,I. In Case II also fluctuations with frequency p are present. 3.2.
STEADY
STATE
RESPONSE
The expressions for variances and cross-covariances state are obtained by applying the limiting process transient quantities. In Case I this yields ai, = (pj/lj)‘A
E[F2](l
of the modal responses in steady t+oo to the expressions for the
-~~)/4~jwj,
(38)
2ljCk(ffjWj+akWk)
(39)
[(ajWj+LukWk)2+(~j-~k)2I[(~jWj+akWk)2+(f;.flk)2I’
In Case II, for the steady
-
2ajwj
state one obtains
sin pt - (p - 21,) 40ZFWf +(p-21j)2
COS pt
2ajWj
-
sin Pt - @ + 25,) 4CYyi2WT + (p +
COS Pt
21j)2
1 ’
(40)
(41)
RESPONSE
TO
CORRELATED
RANDOM
311
IMPULSES
For an undamped system (in Case II) ati
=&B
CO”qiq* (t) = PjPkB E[F*l
E[F2]
2 cos pt
(42)
P(P - 2Oi)(P + 2Wi)’ 2p cospt
(43)
(Oj-Wk-P)(Wj-Wk+P)(Oj+Ok-P)(WjfOk+P)’
The results (38), (40) and (42) are identical with those obtained by Roberts [6]. There is a singularity in expression (42) for p = 0 and p = 2Wj and in expression (43) for p = wk - wi and p = Uj + wk, but this observation is of no practical importance, since in a real excitation the stationary component of g1(7) must always exist, and hence the variance of the steady state response of the undamped system is infinite (see equation (38)). In Case III (the influence of the correlation of impulses), one has 2°F -- b 2cUjwj 1f(b+ajaj)2+[f
2DE2[F](1-~j)2 (+“= 0 z (b-ajwj)2+[T 2
CO”w?k=
fij
(~j~k/~j~k)~~2[~1{2~j~k~l(~j,
mk)L2(wj,
1
(44)
’
~k)[c3(-Wj)+c3(-Wk)l
+S[~,(wj,Ok)(5j_5k){Cl(Wj)-C1(Wk)}-L2(0j,
~k)(f;++k){cl(~j)+cl(~k)}l
(45)
+cl(~j)c2(~k)+cl(~k)c2(~j)}.
Expressions (44) and (45) also could be obtained by using a spectral method, with the excitation considered as a continuous stochastic process with correlation function K (W 72) = D exp (-b I 71 - 72 I). This is justified by the analogy between the integrals in such case and the double integrals in formulae (19) and (20). 4. EXAMPLE
PROBLEM:
RESPONSE OF A RECTANGULAR
PLATE
The response of an uniform rectangular elastic plate, simply supported at all edges, can be examined as an example. As is well known, in this case the operator is L = N(d4/ax4 + 2a4/ax2ay2 + a4/ay4) and the normal modes have the trigonometric form fi,j(x, y) = sin (irx/ll) sin (j7ryIE2). Natural frequencies can be computed by using the formula (see, e.g., reference [16]) for the thin elastic plate Wi,j
=
T2[(i2/l:)
+
(46)
(i’/li)]d$G,
where N, rn, I1 and l2 denote plate bending rigidity, mass density and side lengths, respectively. The expression (21) for the variance of the plate response (deflection) takes the form c2w(X,Y, t)=C~2,,,(t)ftj(X, i,j
C
Y)+ i#k
co”q~.~q~,, (t)fi,j(X,
Y)fk,l
(x3
Y 1.
(47)
i.j.k.1 and/or j#l
If the excitation is spatially uniform, i.e., p(x, y) = const. = po, then .
sin2 7
sin2 ‘7 dx dy = 1
2
The form of @i,jindicates that for this type of excitation the contribution of succeeding normal modes in the total structural response diminishes as the indices of the mode increase. Thus the contribution of modal cross-covariances depends here not only on the spacings between the natural frequencies but also on the indices of modes. In other
R. IWANKIEWICZ
312
AND
K. SOBCZYK
words the contribution of cross-covariance between two high modes is rather small even if they are close together. With attention confined to the deflection in the middle of the plate, one need take into account only symmetrical mode shapes (strictly-bisymmetrical, i.e., for odd i and j), In the example under consideration nine normal modes have been taken into account: that is fi,i(x, y) for i = 1,3,5 and j = 1,3,5. The nine corresponding natural frequencies of the example plate with side lengths ratio f1/12 = 2/3 and fundamental frequency Wl,J = 10 s-l are listed in Table 1. It is seen that they are fairly well separated. TABLE
1
First nine natural frequencies rectangular plate
of a
i
i
%
1 1 3 1 3 3 5 5 5
1 3 1 5 3 5 1 3 5
10.0 34.615 65.385 83,846 90.0 139.231 176.154 200.769 250.0
The contribution of higher modes variances and cross-covariances appeared to be negligible, which can be explained by the form of modal variances and cross-covariances, as well as by the form of the the character of the spectrum of the frequencies. The approximate variance of the deflection (within an accuracy not less than 3%) is under consideration as c&11/2,
I*/&
t) =(+~l.l(t)-2rCOV91.191.3(f)
in expression (47) the expressions for coefficients pi,j and expression for the found for the plate
+COv,I.,,,.I(t)l. (4%
At present the nature and numerical values characterizing real random impulsive excitations are not well recognized; hence some hypothetical reasonable data are assumed. The constants A, B and D are taken equal to 5 s-l, 5 s-l and 25 SC*, respectively. To avoid difficulties in adopting the data referring to the impulse strength F, the nondimensional quantity mt(11/2,12/2, t) w :.I /p :,1 E2[F] has been computed and it has been assumed that E[F*] = 1.1 E*[F]. The variance of the plate deflection in the transient state, in the case of uncorrelated impulses and for various damping ratios, is plotted in Figure 1. When the expected arrival rate is constant (g*(r) = const.), variance of the transient response fluctuates with a frequency which is double the fundamental natural frequency 01,1 of the system [6]. Variance of the transient response to non-stationary excitation (gl(T) = A + B sin pi) fluctuates mainly with a frequency p (here p = wlJ, though in the case of a lightly damped (al,l=O~O1) and an undamped (a 1,1= 0) system fluctuations with frequency 20r,~ can also be observed (the effect of transient terms in expressions (27) and (28)). The transient response of the plate in the “resonant” case of uncorrelated impulses is presented in Figure 2. Here the variance of the response fluctuates only with the frequency 201,i. It is seen that peak values of the variance are much greater than in the case of stationary excitation (constant expected arrival rate).
RESPONSE
0
TO
CORRELATED
Zn
4n ?,I
Figure 1. Transient variance damping ratios. -, gl(t)=A;
Figure 2. Transient damping ratios. -,
RANDOM
313
IMPULSES
6rr
f
of the plate deflection in the case of uncorrelated -, gl(t)=A+B sinoI,It.
variance of the plate deflection in “resonant” gI(t) = A; -, g,(t) =A +B sin 2wl,,t.
case of uncorrelated
impulses,
impulses,
for different
for different
The variance of the steady state response to the stationary excitation is, of course, constant (see expressions (38) and (39)). The variance of the steady state response to non-stationary excitation (variable expected arrival rate) oscillates about a certain fixed level with frequency p and amplitude depending on this frequency, as shown in Table 2 [6]. The variance of the transient response to stationary excitation in the case of correlated impulses, (i.e., the sum of the appropriate expressions in Cases I and III) is plotted in Figure 3. Here also the variance fluctuates with frequency 2~~,~. Only in the case of infinitely strongly correlated impulses (b = 0) does the variance fluctuate with frequency wi,i. It is to be supposed that for certain value of the correlation parameter b (undoubtedly between 2 and 30) the influence of the correlation of impulses attains a maximum. Straight lines in Figure 3 denote the asymptotes corresponding to the steady state. The response of the structure to correlated impulses, for different damping ratios, in the case of stationary excitation is shown in Figure 4 and in the case of non-stationary
314
R. IWANKIEWICZ
AND K. SOBCZYK
TABLE
2
Characteristics of the variance of the steady state response to nonstationary excitation
al.1
Fixed level
Amplitude for P = w1.1
0.01 0.05 0.2
13.75 2.743 0.688
0.367 0.360 0.331
Amplitude
for
P = 2Wl,,
6.86 1.37 0.349
Figure 3. Variance of the plate deflection for a~.~ = 0.05 and different correlation parameters b.
7
6
5
4
3
2
I
0 0
477
wI,I
-,
f
Figure 4. Variance of the plate deflection in transient and steady state for different damping ratios (gl(l) = A). For uncorrelated impulses; -, for correlated impulses (b = 10 s-l).
excitation in Figures 5 and 6. The variance of the response fluctuates in the same manner, regardless of the fact of whether the impulses are correlated or not. The influence of this correlation appears to be strongly affected by damping. Variance of the steady state response to non-stationary excitation, including the correlation of impulses, also oscillates about the fixed level, represented by the asymptotes in Figure 4.
RESPONSE
TO CORRELATED
Y
RANDOM
IMPULSES
315
t
Figure 5. Variance of the plate deflection in transient state for different dam ing ratios P for correlated impulses (6 = 10 s- ). B sin wi.it). -, For uncorrelated impulses; -,
Figure 6. Transient variance of the plate deflection in “resonant” B sin 2o,,ir), for different damping ratios. -, For uncorrelated (b = 10 s-i).
(g,(t) = A +
case of correlated impulses (gi(r) = A + impulses; -, for correlated impulses
The contribution of the terms accounting for the correlation of impulses arrival times in the steady state response is presented in Figure 7. It is seen that this contribution attains a maximum, which for lightly damped systems is in the vicinity of b = 10 s-l, and shifts to the left as damping ratio increases. When the impulses arrival times are infinitely strongly correlated (b = 0), the influence of this correlation is constant, regardless of the damping ratio of the system.
316
R. IWANKIEWICZ
AND
b
Figure7. Influence
of the correlation
of impulses
K.SOBCZYK
Is-'1 arrivals
on the steady
state response
variance.
It should be also noted that the influence of the correlation of impulses depends, in the case of uncorrelated random amplitudes, on the variance of these amplitudes (T: = E[F’]-E2[F]. Reviewing formulae (19), (20) and subsequent detailed formulae one can observe that as this variance increases the relative contribution of terms accounting for the correlation of impulses decreases. The prevailing contribution of the response in the first mode in the total structural response is due to the uniform spatial distribution of the excitation and to the character of the spectrum of the frequencies (frequencies corresponding to symmetrical mode shapes of the example plate are not closely spaced). However, in structures in which the natural frequencies are more closely spaced the contribution of modal cross-covariances may prove very high (as, for example, in shells [17]). REFERENCES 1. TUNG CHI CHAO 1967 Journalof theEngineering Mechanics Division, American Society of Civil Engineers 93, 79-94. Random response of highway bridges to vehicle loads. 2. TUNG CHI CHAO 1969 Journal of the Engineering Mechanics Division, American Society of Civil Engineers 95, 41-57. Response of highway bridges to renewal traffic loads. 3. C. A. CORNELL 1964 Department of Civil Engineering, Stanford University, Technical Report 34. Stochastic process models in structural engineering. 4. Y. K. LIN 1967 Probabilistic Theory of Structural Dynamics. New York: McGraw-Hill Book Company. 5. D. VERNE-JONES 1970 Journal of the Royal Statistical Society Series B, 32, l-62. Stochastic models for earthquake occurrence. 6. J. B. ROBERTS 1965 Journal of Sound and Vibration 2, 375-390. The response of linear vibratory systems to random impulses. 7. Y. K. LIN, 1965 Journal of the Acoustical Society of America 38, 453-460. Non-stationary excitation in linear systems treated as sequences of random pulses. 8. S. K. SRINIVASAN, R. SUBRAMANIAN and S. KUMARASWAMY 1967 Journal ofSound and Vibration 6, 169-179. Response of linear vibratory systems to non-stationary stochastic impulses. 9. A. RENGER 1979 Zeitschrift fiir angewandte Mathematik und Mechanik 59, 1. Equation for probability density of vibratory system subjected to continuous and discrete stochastic excitation (in German). 10. P. MAZZE~I 1964 Nuovo Cimento 31, 88. Correlation function and power spectrum of a train of non-independent overlapping pulses having random shape and amplitude.
RESPONSE 11. I. I. GIKHMAN
12. 13. 14. 15. 16. 17.
TO CORRELATED
RANDOM
IMPULSES
317
1977 Applied Mechanics 13(11), 18. Asymptotic behaviour of solutions of a mixed problem associated with random vibrations (in Russian). I. ~.KUZNIETSOV and R.L.STRATONOVICH 1956 Izuesfya AkademiiNauk USSR,Series Mathematics 20, 167-178. On the mathematical theory of correlated points. S. K. SRINIVASAN 1978 Solid Mechanics Archives 3, 325-379. Stochastic integrals. A. RAMAKRISHNAN 1956 Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 59, 121-127. Processes represented as integrals of a class of random functions. G. C. PAPANICOLAU and R. HERSH 1972 Indiana University Mathematics Journal 21, 815. Some limit theorems for stochastic equations and applications. W. NOWACKI 1972 Dynamics of Elastic Structures. Warsaw: Arkady (in Polish.) I.ELISHAKOFF 1977 Journal of Sound and Vibration 50, 239-252. On the role of crosscorrelations in the random vibrations of shells.