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Random vibration of hinged elastic shallow arch
Journal
of Sound
and Vibration (1989) 132(2), 299-315
RANDOM VIBRATION OF HINGED ELASTIC SHALLOW ARCH J.-Q. Department
of Mechanical Engineering, (Received
SUN
AND
c.
s.
Hsu
University of California, Berkeley, California 94720, U.S.A.
19 May 1988, and in revisedform
9
January 1989)
The response to Gaussian white noise excitation of a hinged elastic shallow arch is studied. The steady state solutions are obtained by means of the Fokker-Planck equation. A class of transient solutions, viz. the first-passage time probability for snap-through of a sinusoidal shallow arch, is studied numerically by the generalized cell mapping method and direct Monte Carlo simulation. In obtaining the global information of first-passage time probability of the system, there are large computational advantages of the generalized cell mapping method over direct Monte Carlo simulation.
1.
INTRODUCTION
Arches are probably the simplest structural members which are inherently non-linear and whose snap-through global instability has important practical implications. There have been many studies of the response of arches to deterministic excitation [ 1,2]. However, the study of the response of arches to random excitation appears to be less extensive. To the authors’ knowledge, the first-passage problem of shallow arches capable of exhibiting snap-through has not been studied. The present paper deals with the response to Gaussian white noise excitation of a hinged elastic shallow arch. Specifically, the joint steady state probability density function for the first N modes of the arch is obtained by means of
the Fokker-Planck equation. This probability density function is then used to evaluate the steady state mean and mean squared response of the arch. A class of transient solutions, viz. the first-passage time probability of a hinged sinusoidal arch, is studied by the generalized cell mapping (GCM) method and direct Monte Carlo simulation. The effects of several parameters on the first-passage time probability are examined in the numerical examples. In references [3,4] a similar study of the first-passage time for snap-through of a cylindrical shell was carried out. Therein, the motion of the shell was approximated by the first mode. The finite difference method and the simulation method were applied. In the present paper, the one- and two-mode approximations of the arch are examined, and the GCM method is applied to obtain the global information of first-passage time probability of the arch. There are large computational advantages of the GCM method over direct Monte Carlo simulation. To analyze first-passage problems for highly nonlinear structural members such as arches is an extremely difficult task. This paper is a first attempt to show that for problems of this kind, the GCM method could perhaps offer a new method of attack which has advantages over other methods. 2. GENERAL BACKGROUND The deterministic analysis of a shallow arch in reference [5] is briefly reviewed here to provide the background for the subsequent discussions. Consider an elastic shallow 299 0022-460X/89/
140295 + 17 $03.0@/0
@ 1989 Academic
Press Limited
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arch with’both ends hinged. Assume that the arch is initially thrust free. The equation of motion is given by
(1) where every term in equation (1) has been non-dimensionalized [5]. With the assumed end conditions, the displacement U(X, t) of the arch, the initial configuration uo(x), and the load distribution Q(x, t) may be expressed as u(x, t)=
O” 1
1 y[A,+a,(t)]sin n=l n
nx,
Uo(x) = “E, -$ A, sin nx,
(2)
06xSn;
0 C x c T,
(3)
OSXGT.
(4)
Substituting equations (2)-(4) in equation (l), one can obtain the equations governing the generalized displacements a,(t) and the generalized velocities fin(t) in the functional phase space as da,/dt=n2/3.,
n=l,2
,...,
da,/dt=-n’~,+G(A,+a,)-can-Qn,
(5)
n=1,2 ,...,
where G=
f ~A:-n~l~(A.+an)2. n=1 n
(6)
A dimensionless total energy H of the arch can be expressed as H=
; [a:+&+fG’ “=,
The equilibrium states of the free undamped algebraic equations P. =O,
n=l,2,...,
(7)
system (5) can be found by solving the
-n2an+G(A,,+~n)=0,
n=l,2
,....
(8)
Consider a sinusoidal shallow arch specified by A,ZO;
A,=O,
n=2,3,...,
(9)
Some results of the equilibrium states of free undamped motion of the sinusoidal arch are listed below. Notice that /3. = 0 for all n 3 1 is always true for the equilibrium states. (a) ForO‘, Pi;‘, j = 2,3, . . . , m. A typical pair of P’,f’and P’,:] are located at (Y,= -j’A,/( j2 - l), Cyj= *j’[( j2 - 2)A:/( j2 - 1)’- l]l”, (Y~=O, kfl, j.
HINGED
SHALLOW
ARCH
RANDOM
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It can be shown that POis always stable, and so is Pi(-) when it exists; other equilibrium states are unstable. Hence, for Ai > 2, the sinusoidal arch may jump from the preferred equilibrium state POto another stable equilibrium state Pi-’ under external disturbances. This phenomenon is known as the snap-through instability of the arch. A sufficient condition of stability against snap-through can be derived by using a Lyapunov function (10)
V(P) = H(P) - H(P,), where P is any state in the phase state PO. If P is moved gradually energy surface V(P) first meets sinuoidal arch, the region defined
space. V(P) defines an energy surface enclosing the away from PO, the energy surface expands. When the another equilibrium state P* of the free undamped by
S(P,)
= {P: V(P) < v(P*)}
(11)
is then a sufficient region of stability against snap-through in the sense that if an initial state P is located inside S(P,), the trajectory of free undamped motion of the arch will remain in S(P,-,), and no snap-through can occur. The critical energy level V(P*) with regard to the preferred state PO can be found as V(P*)=&[-3A,+(A;-4)1’2]2{8+[h,+(A:-4)”2]2}, V(P*)
= 16[(A:/3)
2sA,
-f],
(12)
When the sinusoidal arch has a Kth order harmonic imperfection (A, # 0, AK # 0), the critical energy level V(P*) may be reduced. Let V(P*), denote the critical energy level of the sinusoidal arch with a Kth harmonic imperfection. Evaluation of V(P*), involves the real roots of a fourth order polynomial. For example, when K = 2, the equilibrium states of the arch of the type with (Y~Z 0, CQ# 0, (rk = 0, k > 2, are determined by and
A,a;+A3a;+A2~;+Ala:+AO=0
(~2= [A2/(2a,
+‘%)I~, ,
(13914)
where AJ=204AI,
A4 = 36,
A1=384A;+15A1A;+96A,,
A2=424A;+7A;+36,
A0=128A4+8A;A;+64A:. 1
(15)
According to Hsu [5], when A2;l(4A,) is small so that (A,, A2) falls in the region in the Al-A2 plane as delineated in Figure 4 of reference [5], there may be two or four real roots of equation (13), and snap-through is possible. V( P*)2 is determined by one of the roots which represents a unstable equilibrium state with the lowest energy level among other unstable equilibrium states. 3. STEADY STATE SOLUTIONS To simplify the analysis, let us consider a random excitation (4) such that
E[Qn(f)l=O, E[On(t)Om(t')l=2Dqs,,S(t-t'), 1cn,
rns N,
Q.(t)=O,
(16)
n> N
where S,, is the Kronecker delta function, and 6(t) the Dirac delta function. We further assume that the arch has an initial configuration such that A,=O,
for n > N,
(17)
and the arch is initially at rest. Under these conditions, it can be shown that &)=P,(r)=O,
for any t > 0,
andn>
N.
(18)
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-Hence, we need to take only the first N modes of the arch into account. The Fokker-Planck equation for the first N modes of the arch can be written as
where p = p( (Y,p, t) is the joint probability density function of cy= {czr , a2, . . . , aN} and . . . , PN}. The steady state solution of equation (19), when $/at = 0, can be obtained as [6-91
P={P,,P2,
Aa, PI = G exp where C, is a normalization constant. In references [ 10,l l] solutions similar to expression (20) for a non-linear string and a non-linear elastic beam were obtained. One interesting aspect of the problem is to examine the manner by which the bifurcation phenomenon of the deterministic system exhibits itself when the arch is subjected to random excitations. Consider the sinusoidal shallow arch specified by equation (9). As discussed previously, the arch has a bifurcation at A, = 2. It is interesting to note that the probability density function (20) of the stochastic system has a “bifurcation” at Al = 2, and it changes from a function with one peak to another with two peaks as A, changes. To make the discussion transparent, we take N = 1. Then j’(ar,Pr)=C,exP The
marginal probability P(R) = R, ~(a~, I
(21)
{-(C/~DQ)[~:+P:+~(~A~(Y,+(~:)~I}.
density function of (Y, can be obtained as PI)
d/% = G
-(C/20Q)[(l+2h:)(Y:+2A,~~+~~~]
exp
.
1
{
(22)
In equations (21) and (22) C, and C, are the normalization constants for the corresponding probability density functions. p( LYJ is plotted in Figures l(a, b) for some values of hi. The appearance of the second peak of p(~r) at Pi-’ is clearly seen as A, crosses the bifurcation value 2 from below. Also, there is a valley at the point Py’. The height of the peak at Pi-'is small, indicating that P,(-) has a much higher potential level than PO. Hence, the more stable ‘a state is, the more probable it will be.
(b)
0
Figure 1. (a) The variation of the probability density function p(c~,) with A, the probability density function p(q) in Figure l(a).
; (b) enlargement of left tail of
HINGED
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With equation (20), the steady state response of the arch can be readily evaluated. Let us again consider the sinusoidal arch, and denote the mean and variance of the steady state displacement u and velocity u of the arch by E[ u], var [u], E[ u] and var [u]. It can be shown that N 1 var [u] = C avar n=l n
E[u]=(A,+E[(~,])sinx,
[a,] sin’ nx,
and var[u]=c
N n=,
sin (2N+ 1)x 2sinx
C sin*nx=z DQ N+$-
DQ
E[u]=O,
(23-25)
T, (26) 1’ xzo,
where E[ CY,]and var [ cy,] can be evaluated numerically by using equation (20). In deriving equations (23)-(26), we have used the following results: E[cw,]=O,
na2, E[JI,]=O,
E[wd
= ‘4 EM&l = 0, n + S
var[Pn]=DQ/c,
(27)
nil,
and a formula 2 z sin’nx=N+f“=,
sin (2N + 1)x 2 sin x ’
XfO,
77.
(28)
It is noted that the limits of equations (26) and (28) exist as x+0 and 7~.The variance of displacement of the arch converges very fast as N increases, while the variance of velocity grows unbounded with N. This suggests that as N + cc in equations (16) and (17), every mode of the arch is in resonance due to the flat bandwidth of the excitation. This phenomenon is commonly observed in linear and nonlinear structures subjected to wide band noise excitation [lo-131. Hence, if the objective of the analysis is only to study the displacement of the arch under wideband noise excitation, by retaining a few modes, one may obtain satisfactory results because the displacement is dominated by lower frequencies of the system. If, on the other hand, the second or higher order spatial derivatives of the displacement, or the velocity and acceleration of the arch are sought, one has to take a sufficient number of modes into account, depending upon the bandwidth of the random excitation under consideration [ 131. 4. FIRST-PASSAGE
TIME
PROBABILITY
Next, we study a class of transient solutions, viz. the first-passage time probability. First, we discuss the domain of safe operation of a shallow arch. To take advantage of the results of the deterministic analysis of the arch, we restrict the discussion to the sinusoidal shallow arch specified by equation (9). It has been pointed out that for Al > 2, the free undamped sinusoidal shallow arch can exhibit snap-through instability under external disturbances. When snap-through occurs, the arch is considered unsafe. The necessary and sufficient condition under which snap-through will or will not happen has not been available. A sufficient condition against snap-through of the free undamped sinusoidal shallow arch has been obtained by Hsu [5] and is given by equations (11) and (12). We shall take the domain in the phase space defined by equations (11) and (12) as the domain of safe operation of the sinusoidal shallow arch. Assume that the arch is initially in the domain of safe operation. Our interest is in the probability that the response of the arch leaves the domain of safe operation for the first time during a given period of time. It is noted that the sufficient condition against snap-through discussed previously
304
J.-Q.
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-. is
sometimes conservative. The first-passage time probability based upon this sufficient condition may be considered as a conservative probabilistic measure of reliability against snap-through instability of the arch under random excitation. In this sense the first-passage time probability under consideration may be called the first-passage time probability for snap-through of the arch [3,4]. The domain of safe operation of the sinusoidal shallow arch in the phase space can be written explicitly as [~yZ,+/32,]+$G*< V(P*),
;
(29)
?I==1
where V(P*) is given by equation (12). It is evident from equation (29) that the first-passage problem of the sinusoidal shallow arch discussed above is a study of energy level crossing. If the total energy exceeds the value V(P*), snap-through may occur and, according to our definition of the domain of safe operation, first-passage failure has taken place. The mathematical formulation of the first-passage problem may be found in references [ 14,151. In reference [ 161 this problem is studied from the viewpoint of generalized cell mapping. A very early and important study of the first-passage problem by using a method similar to generalized cell mapping can be found in reference [17] by Crandall, Chandiramani and Cook. However, the exact solution to this problem is not available even for simple linear oscillators of one degree of freedom. Here, we shall attack this problem by using the GCM method and direct Monte Carlo simulation. In order to make the paper reasonably self-contained, the GCM method is briefly described next. For a general and complete description of the GCM method, the reader is referred to reference [18]. 5. THE GCM METHOD
In the GCM method the state space is discretized into small cells. When a finite number of cells are dealt with, each cell is denoted by an integer. The dynamical evolution of the system is then described by a finite Markov chain: Pitn
+
1)= ?
n=0,1,2
PqPj(n),
,...,
(30)
j=l
where p,(n) is the probability that the state of the system is located in cell i at time t = to+ nr, or at the nth mapping step with r being the mapping step time, Pv the one step transition probability from cell j to cell i and N, the total number of cells under consideration. For the first-passage problems studied in the present paper, the cells, either being completely outside the domain of safe operation or at least having one point on the boundary of this domain, form a failure set denoted by F. Since first-passage failure occurs when the response of the system crosses the boundary from inside the domain of safe operation, the one step transition probability for a cell in F is determined as Pfi = 1, Pij = 0,
ifjE Fandj#
i.
(31)
The one-step transition probability for the regular cells in the domain of safe operation can be in general computed by the Monte Carlo simulation method [ 19-221. A statistical error analysis of the GCM method via Monte Carlo simulation is available [22,23]. Let P(n) be the first-passage time probability of the system at the nth mapping step. Assume that the system is initially at some cell k with probability one. Then
P(n)=
c PAn),
jcF
(32)
HINGED
SHALLOW
ARCH
RANDOM
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305
with PJc(O)=l, The first-passage time probability evaluated as
forj#
Pj(")=os
k.
(33)
density function p(t) at time t = to+ (n + l/2)7 can be
p(t)=[P(n+l)-P(?I)]/T.
(34)
When an asymptotic form of p(t) exists in the form p(t) = DA em*‘,
for t >>1,
(35)
the constants D and A can be computed as
Q(n)
*=lim.!
,,-.a,~ Q(n+l)’
Q"+'(n) Q"(n+l)'
D=f+?
(36)
where Q(n)=l-P(n). Once P-,(i,j=1,2 ,..., N,) are computed and pi(O) (i = 1,2,. . . , N,) are specified, the solutions to the first-passage problem can be obtained from equations (32), (34) and (36) simply by iterating equation (30). 5.1.
NUMERICAL
5.1.1.
EXAMPLES
Example 1
Consider the sinusoidal shallow arch subjected to a random excitation given by
Qk t) = Ql(t) sin x,
(37)
where E[Q,(t)l
= 0,
E[Q,(t)Q,(t’)l= W&t - 0.
(38)
Assume that the arch is initially at rest. In other words, the system is initially in the preferred equilibrium state PO. Under these conditions, all the modes remain quiescent except the first mode. Hence, it is sufficient to retain only the first mode for the study of first-passage time probability. The equations of motion for the first mode are darldr = P1,
d/3,/dt = -&a1 - 3hla: - a: - 20,@, - Q, ,
(39)
and the domain of safe operation is defined by (a:+fJf)+$[2hla1+af]2<
V(P*),
(40)
where 0: = 1 + 2h:, c = 2~~5. The GCM method is applied to study the first-passage time probability of the arch. In the computation reported hereafter we have set D, =2&o:,
l=O*Ol.
(41)
Cell sizes are taken to be uniform throughout the cell state space in this paper. Figure 2 shows a typical coverage of the domain of safe operation by a finite number of rectangular cells. The cell sizes are set to be 0.15 x 0.15, and the mapping step time is chosen to be T = I’,/& where T1 = 2a/w,. Three hundred sample functions were used to compute the transition probability matrix by Monte Carlo simulation. The results of first-passage time probabilities of the arch initially at rest are plotted in Figures 3-5 for Al = 2.1, 2.3 and 3.1. The corresponding results by direct Monte Carlo simulation are also plotted in the figures. The agreement is excellent. The accuracy of all the results in this paper by direct Monte Carlo simulation is within 2%, based upon a similar analysis to the one in reference [24].
J.-Q.
306 5 4
0 -I -2 -3 -4 g
-
-4
-3
-2
-I
0
Figure 2. Regular cells in the domain of safe operation center of a cell. A, =2-l; cell size: 0.15X0.15.
Figure 3. The first-passage + + +, direct simulation.
Figure 4.. The first-passage
C. S. HSU
... ..... ...... ........ ......... .......... ........... ............ ............ .............. .............. ............... ............... ................ ................ ................. .................. ................... .................... .................... ..................... ...................... ...................... ....................... ........................ ........................ ........................ .:::::::::::::::::::::: ...................... ..................... ................... ................... .................. .................. ................. ................ ................ ............... ............... .............. .............. ............ ............ ........... ........... ......... ........ ...... ..... ...
2
-5
AND
,,(,,,,,,
3
a
SUN
time probability
time probability
I
2
3
4
of the sinusoidal
5
shallow
arch. A dot represents the
of the sinusoidal
shallow
arch. h, =2*1. -,
GCM results;
of the sinusoidal
shallow
arch. A, = 2.3. Key as Figure 3.
HINGED
SHALLOW
ARCH
RANDOM
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N
Figure 5. The first-passage
time probability
of the sinusoidal
shallow
arch. A, = 3.1. Key as Figure 3.
As has been commented in reference [ 161, one of the advantages of the GCM method is that the global picture of first-passage time probability as a function of initial conditions can be readily obtained by the GCM method. Here, we present such a study for Al = 2.1 and 3.1. Fourteen initial conditions of ((Y,, PI) are chosen as shown in Figure 6. The
/
-2_
-3
-2
I
I
-I
0
1
I
2
01 Figure
6. Initial conditions
selected to study the first-passage
problem.
first-passage time probability, and the mean and standard deviation of first-passage time are computed for each initial condition. The results of first-passage time probabilities for several initial conditions are shown in Figures 7 and 8. The mean and standard deviation of first-passage time for all 14 initial conditions are listed in Tables 1 and 2. A comment on the computation of the mean and standard deviation of first-passage time is in order. Let fi and a[N] be the mean and standard deviation of first-passage time in cycles of a linear oscillator with period 7’r. Let p(N) be the first-passage time probability density function at the Nth cycle. Then m Nr A= Np(N) dN= I+(N) dN+ NP(N) dN, (42) I NL I0
308
J.-Q. SUN AND C. S.
HSU
Figure 7. The first-passage time probability of the sinusoidal shallow arch for initial conditions: (l), (3), (4), (9) and (14) in Figure 6; A, =2-l; GCM results.
where NL is a large number such that for N 2 NL, the following asymptotic form of p( N) holds: p(N) = DA7’, emATtN,
Nz= NL>>l,
(43)
where D and A, the limiting decay parameters, are part of the GCM results. Hence, R can be evaluated, by using the trapezoidal rule, as m-l
C (iAN)p(iAN)++p(NL) i=l
AN+~e-AT~N~(~7’lNL+l), I
I.0
(4)
1
_
Figure 8. The first-passage time probability of the sinusoidal shallow arch for initial conditions: (l), (3). (4). (9) and (14) in Figure 6; A, =3.1; GCM results.
HINGED SHALLOWARCH RANDOMVIBRATION
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TABLE 1
TABLE 2
The mean and standard deviation (in cycles) of the first-passage time of the sinusoidal shallow arch; A, = 2.1
7’he mean and standard deviation (in cycles) of the first-passage time of the sinusoidal shallow arch; A 1= 3 *1
Point/ no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14
8.2845 7.3117 3.2152 6.2094 8.1418 7.6969 7.0326 7.1672 4.7512 7.0102 8.1443 7.7005 6.2367 2.3982
Point/no.
N
1 2 3 4 5 6 7 8 9 10 11 12 13 14
12.0878 11.1185 7.2799 10.1422 12.0055 11.7942 10.7439 10.8271 7.6719 10.7482 12.0108 11.8014 10.1616 3.5012
6.0866 6.0436 4.8145 58910 6.0861 6.0760 6.0313 6.0450 5.5525 6.0198 6.0865 6.0748 5.9006 4.2133
UN
9.0718 9.0445 8.3477 8.9582 9.0716 9.0702 9.0234 9.0256 8.4808 9.0153 9.0708 9.0683 8.9648 6.3380
where AN = NJ m. Similarly, we have l/2
(I
m[N-N]2p(N)dN 0 >,
a(N]=
m-1
=
,z, (iAN)2p(iAN)
+$
p(N,)]
AN
D -e-Ar~N~[(AT,NL)2+2(ATINL)+2]-~2 + (ATJ2
In order to check the GCM results, the mean and standard deviation of first-passage time for the initial condition (0,O) were also computed by direct Monte Carlo simulation, and the results of the mean and standard deviation of first-passage time are 8.6099 and 6.4185 (cycles) for A1= 2.1, and 12.2652 and 9.3800 (cycles) for Ai = 3.1. These four numbers deviate from the corresponding ones in Tables 1 and 2 by 3*9%, 5*5%, 2.4% and 3*0%, respectively. We now add a comment on the computational efficiency of the GCM method. It is found that the CPU time of one GCM run in which the results of the first-passage time probabilities for 14 initial conditions are obtained is less than 1.4 times that of a direct Monte Carlo simulation run in which the first-passage time probability for only one initial condition is obtained. This means that more than ten folds of computational efficiency of the GCM method versus direct Monte Carlo simulation are achieved. The task of obtaining more detailed information on the effect of initial condition over the domain of safe operation becomes rapidly prohibitive to direct Monte Carlo simulation. This task, however, is still tractable by the GCM method because the GCM method always deals with a finite number of cells, and, for each initial condition, the GCM method does not need much extra effort other than a limited amount of computation for the results of first-passage time probability by iterating equation (30). Generally speaking, when only a single initial condition is of interest, the GCM method and the method of direct
310
J.-Q.
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simulation are about equally efficient, provided that a comparable accuracy is maintained for both methods. The advantage of the GCM method over simulation lies in its ability to yield global results when all possible initial conditions in the domain of interest are to be investigated. The advantage also becomes apparent for any comprehensive study where by using the GCM method different types of results can be obtained in one computer run. 5.1.2. Example 2 Consider the sinusoidal shallow arch under a random excitation given by Q(x, t) = Q1( t) sin x+ Qk( t) sin IQC,
(46)
where
E[Oi(t)l=o,
E[Qi(t)Oj(t’)l=2D,6,6(t-rr),
i,j=l,k.
(47)
Once again we assume that the arch is initially at rest. From equations (5), it is seen that under the excitation (46) all the modes will remain quiescent for all t > 0 except the first and kth modes. In the study of first-passage time probability these two modes must be retained. We take k = 2 in this example. The equations for the first and second modes are da,ldt
= I%, da,/dt
da,/dt=-o:al-3hla:-cu:-(cw:/4)(Al+a,)-201SB,-Q,, d82/dt=-4~z-$cu:-(2hl(Y1+(Y:)(YZ-20,3P2-QZ,
= 4&,
(48) (49)
and the domain of safe operation is defined by V(P*).
; (crZ,+p2,)+$[2h,(~~+(~:+(y3/4]~< ?I=1
(50)
The parameters D, and 5 are given by equations (41). The state space is four-dimensional in this example. The cell sizes for GCM are set to be O-4 x 0.4x 0.4 x O-4 for A1= 2.1, and O-5 x 0.5 x O-5 x 0.5 for A, = 2.5. The mapping step time is T,/8. Fifty samples were used to compute the transition probability matrix by Monte Carlo simulation. The domain of safe operation defined by equation (50) contains 7839 regular cells for Al = 2.1, and 9421 regular cells for A, = 2.5. Because of the large cell sizes used, the failure set F of GCM was also used in the direct Monte Carlo
1
t z.
I.0 -
/+-f--+-f--f--+
-
f
z
$
0.8
if
P
E
‘Z
0.6
_
8
0 x
0.4-
5 Ii
0
s
IO
I5
20
N
Figure 9. The first-passage time probability of the sinusoidal shallow arch under random excitation (46); k=2.-, GCM results; + + +, direct simulation.
A,=2.1,
HINGED
SHALLOW
ARCH
RANDOM
VIBRATION
311
N
Figure 10. The first-passage time probability of the sinusoidal shallow arch under random excitation (46); h,=2.5, k=2, Key as Figure 9.
simulation for these cases in order to compare the two methods on an equal basis. In the direct Monte Carlo simulation, a trajectory was terminated as soon as it goes into the failure set F. As has been commented in reference [ 161, the first-passage time probability obtained based upon this failure set F is slightly higher than the true value. However, it can be shown that this difference diminishes rapidly as the cell sizes for GCM are reduced. The results for the first-passage time probability for A, = 2.1 and 2.5 obtained by both the GCM method and direct Monte Carlo simulation are plotted in Figures 9 and 10. The agreement between the results by these two methods is fairly good. It is remarked that, for higher-dimensional spaces, the complexity of computational procedures and computer programs for the GCM method does not increase much. In fact, there is virtually no change in the iteration procedure and its computer programs. However, the memory need of the GCM method increases rapidly with the dimension of the state space. 5.1.3. Example 3 Next, we consider a sinusoidal shallow arch with the second harmonic imperfection such that h,#O,
Az#O;
h,=O,
for n > 2.
(51)
Assume that the arch initially at rest is excited by the random excitation given by equation (37). It is evident that despite the zero initial condition for all the modes, the second mode will no longer remain quiescent for t > 0. Only direct Monte Carlo simulation has been applied to study the first-passage time probability of the arch for the cases Ai = 3-l and AZ/4 = 0.0,0~125,0~25. The critical energy level V( P*)* was determined by the method outlined earlier in the paper. It is found that V(P*), = 39446 for A2/4=0*125, and V(P*), = 365095 for AZ/4 = 0.25. Upon comparing these V(P*), values with V(P*) = 43.2533 for Al = 3.1 and A2= O-0, it is seen that the second harmonic imperfection indeed causes the reduction in the critical energy level of the arch. The results for the first-passage time probabilities for the above three cases are plotted in Figure 11. It is seen from the figure that the second harmonic imperfection causes higher first-passage failure after about five cycles of the motion.
312
J.-Q. SUN AND C. S. HSU
N
Figure 11. The first-passage time probability ofthe sinusoidal shallow arch with the 2nd harmonic imperfection under random excitation (37); A, = 3.1 and A,/4= 0.0, 0,125, 0.25.
5.1.4. Example 4 Finally, we revisit Example 1 with an initial condition specified by cQ(O)# 0,
a,(O) = 0,
n # 2,
&(O)=O,
n91.
(52)
In this case the second mode is no longer identically zero for t > 0. We examine the effect of initial condition (52) on the first-passage time probability of the sinusoidal shallow arch. Extensive Monte Carlo simulation studies were carried out. Some of the results are shown in Figures 12 and 13 and in Tables 3 and 4. By examining these results, it is
Figure 12. Effect of initial condition (52) on the first-passage time probability of the sinusoidal shallow arch under random excitation (37); A, = 3.1; direct simulation. -. -, ar(0) = 4.0, a,(O) = 0, n f 2; -, a,(O) = 0, n 2 1; - - -, a,(O) = 0.01, a,(O) = 0, n # 2.
HINGED
SHALLOW
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RANDOM
313
VIBRATION
Figure 13. Effect of initial condition (52) on the first-passage time probability of the sinusoidal shallow arch under random excitation (37); A, = 4; direct simulation. Key as Figure 12.
observed that the first-passage time probability for a small (~~(0) (>O) is lower than the corresponding probability for q(O) = 0. This difference for a fixed (Ye, say LYE = O-01, is not large when AI is small, but is significant when AI is large. This observed difference exists even for simulation runs with a very large number of 2000 samples. Corresponding to the lower first-passage time probability, the mean of first-passage time for a small a,(O) (>O) is larger than the corresponding mean value for ~(0) = 0. On the other hand, for a large (~~(0) (>O), it is observed that the first-passage time probability is higher than the
TABLE
TABLE
3
The mean (in cycles) of the first-passage time of the arch shallow sinusoidal under initial condition (52); A, =3-l
04000 04008 0*0004 O-0081
oTMo3 O-0806 o-1210 0.1613 0.2016 0.2419 0.2823 O-3226 O-3629
12.27 12.71 12.83 13.00 13.48 13.49 13.64 13.10 12.30 11.83 10.52 7.99 5.87
4
The mean (in cycles) of the first-passage time of the sinusoidal shallow arch under initial condition (52); Al =4-o %(0)/4A*
A
04000 04006 0*0031 om63 0.0313 o-0625 O-0938 0.1250 O-1563 0.1875 0.2188 0*2500 0.2813
14.34 15.53 1559 1.586 15.98 15.83 15.72 15.53 15.38 14.76 13.77 13.17 10.93
314
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corresponding probability for cuz(0)= 0, and that the mean of first-passage time is smaller than the corresponding mean value for (Y>(O)= 0. Here, the difference of the probability for a fixed a,(O), say (~~(0)= 4.0, is larger when A1 is small than that when Ai is large. The above discussion seems to suggest that there may exist a crossover value of (~~(0) depending on A1 such that the first-passage time probability for (~~(0)# 0 is equal to that for (~~(0)= 0. It is found from a series of computations that when (~~(0)/(4A,) = 0.20, this crossing over occurs for all the tested values of A,. The mechanism of the above observations is not completely clear at the present time. More study is needed in order to understand these observations and to delineate the precise parameter range in which the qualitative and quantitative changes of the firstpassage time probability can be predicted. 6. CONCLUSIONS In this paper the steady state solutions of a hinged shallow arch have been studied by means of the Fokker-Planck equation. The first-passage time probability of a hinged sinusoidal shallow arch has been studied numerically by the GCM method and direct Monte Carlo simulation. It has been demonstrated that the GCM method has large computational advantages over direct Monte Carlo simulation when the global information of first-passage time probability of the system is the objective of the study. ACKNOWLEDGMENT This material is based upon work supported by the National Science Foundation under grant No. MSM-8519950. REFERENCES
1. R. SCHMIDTand D. A. DADEPPO 1971 ZndustriuZMathematics 21,91-114. A survey of literature on large deflection of non-shallow arches. Bibliography of finite deflections of straight and curved beams, rings and shallow arches. 2. P. A. A. LAURA and M. J. MAURIZI 1987 Shock and Vibration Digest 19(l), 6-9. Recent research on vibrations of arch-type structures. 3. H. N. PI, S. T. ARIARATNAM and W. C. LENNOX 1970 Boceedings ofthe 5th South-Eastern Conference on Zheoreticul and Applied Mechanics, Raleigh-Durham, 4. 5. 6. 7. 8. 9.
10. 11. 12.
North Curolinu, U.S.A.
Numerical solution for the mean first-passage time for snap-through of shells. H. N. PI, S. T. ARIARATNAM and W. C. LENNOX 1971 Journal of Sound and Vibration 14, 375-384. First-passage time for the snap-through of a shell-type structure. C. S. Hsu 1967 Journal of Applied Mechanics 34, 349-358. The effects of various parameters on the dynamic stability of a shallow arch. T. K CAUGHEY and H. J. PAYNE 1967 International Journal of Non-Linear Mechanics 2, 125-151. On the response of a class of self-excited oscillators to stochastic excitation. T. K. CAUGHEY and F. MA 1982 Journal of Applied Mechanics 49, 629-632. The steady-state response of a class of dynamical systems to stochastic excitation. H. RISKEN 1984 The Fokker-planck Equation-Methods of Solution and Applications. Springer Series in Synergetics 18. New York: Springer-Verlag. Y. YONG and Y. K. LIN 1987 Journal of Applied Mechanics 54, 414-418. Exact stationaryresponse solution for second order nonlinear systems under parametric and external white-noise excitations. A. C. ERINGEN 1957 Journal of Applied Mechanics 24, 46-52. Response of beams and plates to random loads. S. T. ARIARATNAM 1962 Journal of Applied Mechanics 29, 483-485. Response of a loaded nonlinear string to random excitation. R. E. HERBERT 1964 Journal of the Accoustical Society of America 36( 1l), 2090-2094. Random vibrations of a nonlinear elastic beam.
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VIBRATION
315
13. S. H. CRANDALL 1979 Developments in Statistics (P. R. Krishnaiah, editor) 2, l-82. New York: Academic Press. Random vibration of one- and two-dimensional structures. 14. D. A. DARLING and A. J. F. SIEGERT 1953 Annals of Mathematical Statistics 24, 624-639. The first passage problem for a continuous Markov process. 15. W. C. LENNOX and D. A. FRASER 1974 Journal of Applied Mechanics 41, 793-797. On the first-passage distribution for the envelope of a nonstationary narrow-band stochastic process. 16. J.-Q. SUN and C. S. HSU 1988 Journal of Sound and Vibration 124,233-248. First-passage time probability of nonlinear stochastic systems by generalized cell mapping method. 17. S. H. CRANDALL, K. L. CHANDIRAMANI and R. G. COOK 1966 Journal of Applied Mechanics 33, 532-538. Some first-passage problems in random vibration. 18. C. S. HSU 1987 Cell-to-Cell Mapping-A Method of Global Analysis for Nonlinear Systems. New York: Springer-Verlag. 19. H. M. CHIU 1984 Ph.D. Thesis, Mechanical Engineering, University of California, Berkeley. Random vibration analysis by cell mapping methods. 20. C. S. Hsu and H. M. CHIU 1986 Journal of Applied Mechanics 53, 695-701. A cell mapping method for nonlinear deterministic and stochastic systems-Part I: the method of analysis. 21. H. M. CHIU and C. S. l-&U 1986 Journal of Applied Mechanics 53, 702-710. A cell mapping method for nonlinear deterministic and stochastic systems-Part II: examples of application. 22. J. Q. SUN 1988 Ph.D. Thesis, Mechanical Engineering, University of California, Berkeley. Random vibrations of nonlinear systems based upon the cell state space concept. 23. J.-Q. SUN and C. S. HSU 1988 Journal of Applied Mechanics 55, 694-701. A statistical study of generalized cell mapping. 24. J. B. ROBERTS 1969 Journal of Sound and Vibration 10, 42-61. Estimation of the probability of first-passage failure for a linear oscillator.
of Sound
and Vibration (1989) 132(2), 299-315
RANDOM VIBRATION OF HINGED ELASTIC SHALLOW ARCH J.-Q. Department
of Mechanical Engineering, (Received
SUN
AND
c.
s.
Hsu
University of California, Berkeley, California 94720, U.S.A.
19 May 1988, and in revisedform
9
January 1989)
The response to Gaussian white noise excitation of a hinged elastic shallow arch is studied. The steady state solutions are obtained by means of the Fokker-Planck equation. A class of transient solutions, viz. the first-passage time probability for snap-through of a sinusoidal shallow arch, is studied numerically by the generalized cell mapping method and direct Monte Carlo simulation. In obtaining the global information of first-passage time probability of the system, there are large computational advantages of the generalized cell mapping method over direct Monte Carlo simulation.
1.
INTRODUCTION
Arches are probably the simplest structural members which are inherently non-linear and whose snap-through global instability has important practical implications. There have been many studies of the response of arches to deterministic excitation [ 1,2]. However, the study of the response of arches to random excitation appears to be less extensive. To the authors’ knowledge, the first-passage problem of shallow arches capable of exhibiting snap-through has not been studied. The present paper deals with the response to Gaussian white noise excitation of a hinged elastic shallow arch. Specifically, the joint steady state probability density function for the first N modes of the arch is obtained by means of
the Fokker-Planck equation. This probability density function is then used to evaluate the steady state mean and mean squared response of the arch. A class of transient solutions, viz. the first-passage time probability of a hinged sinusoidal arch, is studied by the generalized cell mapping (GCM) method and direct Monte Carlo simulation. The effects of several parameters on the first-passage time probability are examined in the numerical examples. In references [3,4] a similar study of the first-passage time for snap-through of a cylindrical shell was carried out. Therein, the motion of the shell was approximated by the first mode. The finite difference method and the simulation method were applied. In the present paper, the one- and two-mode approximations of the arch are examined, and the GCM method is applied to obtain the global information of first-passage time probability of the arch. There are large computational advantages of the GCM method over direct Monte Carlo simulation. To analyze first-passage problems for highly nonlinear structural members such as arches is an extremely difficult task. This paper is a first attempt to show that for problems of this kind, the GCM method could perhaps offer a new method of attack which has advantages over other methods. 2. GENERAL BACKGROUND The deterministic analysis of a shallow arch in reference [5] is briefly reviewed here to provide the background for the subsequent discussions. Consider an elastic shallow 299 0022-460X/89/
140295 + 17 $03.0@/0
@ 1989 Academic
Press Limited
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arch with’both ends hinged. Assume that the arch is initially thrust free. The equation of motion is given by
(1) where every term in equation (1) has been non-dimensionalized [5]. With the assumed end conditions, the displacement U(X, t) of the arch, the initial configuration uo(x), and the load distribution Q(x, t) may be expressed as u(x, t)=
O” 1
1 y[A,+a,(t)]sin n=l n
nx,
Uo(x) = “E, -$ A, sin nx,
(2)
06xSn;
0 C x c T,
(3)
OSXGT.
(4)
Substituting equations (2)-(4) in equation (l), one can obtain the equations governing the generalized displacements a,(t) and the generalized velocities fin(t) in the functional phase space as da,/dt=n2/3.,
n=l,2
,...,
da,/dt=-n’~,+G(A,+a,)-can-Qn,
(5)
n=1,2 ,...,
where G=
f ~A:-n~l~(A.+an)2. n=1 n
(6)
A dimensionless total energy H of the arch can be expressed as H=
; [a:+&+fG’ “=,
The equilibrium states of the free undamped algebraic equations P. =O,
n=l,2,...,
(7)
system (5) can be found by solving the
-n2an+G(A,,+~n)=0,
n=l,2
,....
(8)
Consider a sinusoidal shallow arch specified by A,ZO;
A,=O,
n=2,3,...,
(9)
Some results of the equilibrium states of free undamped motion of the sinusoidal arch are listed below. Notice that /3. = 0 for all n 3 1 is always true for the equilibrium states. (a) ForO
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It can be shown that POis always stable, and so is Pi(-) when it exists; other equilibrium states are unstable. Hence, for Ai > 2, the sinusoidal arch may jump from the preferred equilibrium state POto another stable equilibrium state Pi-’ under external disturbances. This phenomenon is known as the snap-through instability of the arch. A sufficient condition of stability against snap-through can be derived by using a Lyapunov function (10)
V(P) = H(P) - H(P,), where P is any state in the phase state PO. If P is moved gradually energy surface V(P) first meets sinuoidal arch, the region defined
space. V(P) defines an energy surface enclosing the away from PO, the energy surface expands. When the another equilibrium state P* of the free undamped by
S(P,)
= {P: V(P) < v(P*)}
(11)
is then a sufficient region of stability against snap-through in the sense that if an initial state P is located inside S(P,), the trajectory of free undamped motion of the arch will remain in S(P,-,), and no snap-through can occur. The critical energy level V(P*) with regard to the preferred state PO can be found as V(P*)=&[-3A,+(A;-4)1’2]2{8+[h,+(A:-4)”2]2}, V(P*)
= 16[(A:/3)
2sA,
-f],
(12)
When the sinusoidal arch has a Kth order harmonic imperfection (A, # 0, AK # 0), the critical energy level V(P*) may be reduced. Let V(P*), denote the critical energy level of the sinusoidal arch with a Kth harmonic imperfection. Evaluation of V(P*), involves the real roots of a fourth order polynomial. For example, when K = 2, the equilibrium states of the arch of the type with (Y~Z 0, CQ# 0, (rk = 0, k > 2, are determined by and
A,a;+A3a;+A2~;+Ala:+AO=0
(~2= [A2/(2a,
+‘%)I~, ,
(13914)
where AJ=204AI,
A4 = 36,
A1=384A;+15A1A;+96A,,
A2=424A;+7A;+36,
A0=128A4+8A;A;+64A:. 1
(15)
According to Hsu [5], when A2;l(4A,) is small so that (A,, A2) falls in the region in the Al-A2 plane as delineated in Figure 4 of reference [5], there may be two or four real roots of equation (13), and snap-through is possible. V( P*)2 is determined by one of the roots which represents a unstable equilibrium state with the lowest energy level among other unstable equilibrium states. 3. STEADY STATE SOLUTIONS To simplify the analysis, let us consider a random excitation (4) such that
E[Qn(f)l=O, E[On(t)Om(t')l=2Dqs,,S(t-t'), 1cn,
rns N,
Q.(t)=O,
(16)
n> N
where S,, is the Kronecker delta function, and 6(t) the Dirac delta function. We further assume that the arch has an initial configuration such that A,=O,
for n > N,
(17)
and the arch is initially at rest. Under these conditions, it can be shown that &)=P,(r)=O,
for any t > 0,
andn>
N.
(18)
302
J.-Q. SUN
AND
C. S. HSU
-Hence, we need to take only the first N modes of the arch into account. The Fokker-Planck equation for the first N modes of the arch can be written as
where p = p( (Y,p, t) is the joint probability density function of cy= {czr , a2, . . . , aN} and . . . , PN}. The steady state solution of equation (19), when $/at = 0, can be obtained as [6-91
P={P,,P2,
Aa, PI = G exp where C, is a normalization constant. In references [ 10,l l] solutions similar to expression (20) for a non-linear string and a non-linear elastic beam were obtained. One interesting aspect of the problem is to examine the manner by which the bifurcation phenomenon of the deterministic system exhibits itself when the arch is subjected to random excitations. Consider the sinusoidal shallow arch specified by equation (9). As discussed previously, the arch has a bifurcation at A, = 2. It is interesting to note that the probability density function (20) of the stochastic system has a “bifurcation” at Al = 2, and it changes from a function with one peak to another with two peaks as A, changes. To make the discussion transparent, we take N = 1. Then j’(ar,Pr)=C,exP The
marginal probability P(R) = R, ~(a~, I
(21)
{-(C/~DQ)[~:+P:+~(~A~(Y,+(~:)~I}.
density function of (Y, can be obtained as PI)
d/% = G
-(C/20Q)[(l+2h:)(Y:+2A,~~+~~~]
exp
.
1
{
(22)
In equations (21) and (22) C, and C, are the normalization constants for the corresponding probability density functions. p( LYJ is plotted in Figures l(a, b) for some values of hi. The appearance of the second peak of p(~r) at Pi-’ is clearly seen as A, crosses the bifurcation value 2 from below. Also, there is a valley at the point Py’. The height of the peak at Pi-'is small, indicating that P,(-) has a much higher potential level than PO. Hence, the more stable ‘a state is, the more probable it will be.
(b)
0
Figure 1. (a) The variation of the probability density function p(c~,) with A, the probability density function p(q) in Figure l(a).
; (b) enlargement of left tail of
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With equation (20), the steady state response of the arch can be readily evaluated. Let us again consider the sinusoidal arch, and denote the mean and variance of the steady state displacement u and velocity u of the arch by E[ u], var [u], E[ u] and var [u]. It can be shown that N 1 var [u] = C avar n=l n
E[u]=(A,+E[(~,])sinx,
[a,] sin’ nx,
and var[u]=c
N n=,
sin (2N+ 1)x 2sinx
C sin*nx=z DQ N+$-
DQ
E[u]=O,
(23-25)
T, (26) 1’ xzo,
where E[ CY,]and var [ cy,] can be evaluated numerically by using equation (20). In deriving equations (23)-(26), we have used the following results: E[cw,]=O,
na2, E[JI,]=O,
E[wd
= ‘4 EM&l = 0, n + S
var[Pn]=DQ/c,
(27)
nil,
and a formula 2 z sin’nx=N+f“=,
sin (2N + 1)x 2 sin x ’
XfO,
77.
(28)
It is noted that the limits of equations (26) and (28) exist as x+0 and 7~.The variance of displacement of the arch converges very fast as N increases, while the variance of velocity grows unbounded with N. This suggests that as N + cc in equations (16) and (17), every mode of the arch is in resonance due to the flat bandwidth of the excitation. This phenomenon is commonly observed in linear and nonlinear structures subjected to wide band noise excitation [lo-131. Hence, if the objective of the analysis is only to study the displacement of the arch under wideband noise excitation, by retaining a few modes, one may obtain satisfactory results because the displacement is dominated by lower frequencies of the system. If, on the other hand, the second or higher order spatial derivatives of the displacement, or the velocity and acceleration of the arch are sought, one has to take a sufficient number of modes into account, depending upon the bandwidth of the random excitation under consideration [ 131. 4. FIRST-PASSAGE
TIME
PROBABILITY
Next, we study a class of transient solutions, viz. the first-passage time probability. First, we discuss the domain of safe operation of a shallow arch. To take advantage of the results of the deterministic analysis of the arch, we restrict the discussion to the sinusoidal shallow arch specified by equation (9). It has been pointed out that for Al > 2, the free undamped sinusoidal shallow arch can exhibit snap-through instability under external disturbances. When snap-through occurs, the arch is considered unsafe. The necessary and sufficient condition under which snap-through will or will not happen has not been available. A sufficient condition against snap-through of the free undamped sinusoidal shallow arch has been obtained by Hsu [5] and is given by equations (11) and (12). We shall take the domain in the phase space defined by equations (11) and (12) as the domain of safe operation of the sinusoidal shallow arch. Assume that the arch is initially in the domain of safe operation. Our interest is in the probability that the response of the arch leaves the domain of safe operation for the first time during a given period of time. It is noted that the sufficient condition against snap-through discussed previously
304
J.-Q.
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AND
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-. is
sometimes conservative. The first-passage time probability based upon this sufficient condition may be considered as a conservative probabilistic measure of reliability against snap-through instability of the arch under random excitation. In this sense the first-passage time probability under consideration may be called the first-passage time probability for snap-through of the arch [3,4]. The domain of safe operation of the sinusoidal shallow arch in the phase space can be written explicitly as [~yZ,+/32,]+$G*< V(P*),
;
(29)
?I==1
where V(P*) is given by equation (12). It is evident from equation (29) that the first-passage problem of the sinusoidal shallow arch discussed above is a study of energy level crossing. If the total energy exceeds the value V(P*), snap-through may occur and, according to our definition of the domain of safe operation, first-passage failure has taken place. The mathematical formulation of the first-passage problem may be found in references [ 14,151. In reference [ 161 this problem is studied from the viewpoint of generalized cell mapping. A very early and important study of the first-passage problem by using a method similar to generalized cell mapping can be found in reference [17] by Crandall, Chandiramani and Cook. However, the exact solution to this problem is not available even for simple linear oscillators of one degree of freedom. Here, we shall attack this problem by using the GCM method and direct Monte Carlo simulation. In order to make the paper reasonably self-contained, the GCM method is briefly described next. For a general and complete description of the GCM method, the reader is referred to reference [18]. 5. THE GCM METHOD
In the GCM method the state space is discretized into small cells. When a finite number of cells are dealt with, each cell is denoted by an integer. The dynamical evolution of the system is then described by a finite Markov chain: Pitn
+
1)= ?
n=0,1,2
PqPj(n),
,...,
(30)
j=l
where p,(n) is the probability that the state of the system is located in cell i at time t = to+ nr, or at the nth mapping step with r being the mapping step time, Pv the one step transition probability from cell j to cell i and N, the total number of cells under consideration. For the first-passage problems studied in the present paper, the cells, either being completely outside the domain of safe operation or at least having one point on the boundary of this domain, form a failure set denoted by F. Since first-passage failure occurs when the response of the system crosses the boundary from inside the domain of safe operation, the one step transition probability for a cell in F is determined as Pfi = 1, Pij = 0,
ifjE Fandj#
i.
(31)
The one-step transition probability for the regular cells in the domain of safe operation can be in general computed by the Monte Carlo simulation method [ 19-221. A statistical error analysis of the GCM method via Monte Carlo simulation is available [22,23]. Let P(n) be the first-passage time probability of the system at the nth mapping step. Assume that the system is initially at some cell k with probability one. Then
P(n)=
c PAn),
jcF
(32)
HINGED
SHALLOW
ARCH
RANDOM
VIBRATION
305
with PJc(O)=l, The first-passage time probability evaluated as
forj#
Pj(")=os
k.
(33)
density function p(t) at time t = to+ (n + l/2)7 can be
p(t)=[P(n+l)-P(?I)]/T.
(34)
When an asymptotic form of p(t) exists in the form p(t) = DA em*‘,
for t >>1,
(35)
the constants D and A can be computed as
Q(n)
*=lim.!
,,-.a,~ Q(n+l)’
Q"+'(n) Q"(n+l)'
D=f+?
(36)
where Q(n)=l-P(n). Once P-,(i,j=1,2 ,..., N,) are computed and pi(O) (i = 1,2,. . . , N,) are specified, the solutions to the first-passage problem can be obtained from equations (32), (34) and (36) simply by iterating equation (30). 5.1.
NUMERICAL
5.1.1.
EXAMPLES
Example 1
Consider the sinusoidal shallow arch subjected to a random excitation given by
Qk t) = Ql(t) sin x,
(37)
where E[Q,(t)l
= 0,
E[Q,(t)Q,(t’)l= W&t - 0.
(38)
Assume that the arch is initially at rest. In other words, the system is initially in the preferred equilibrium state PO. Under these conditions, all the modes remain quiescent except the first mode. Hence, it is sufficient to retain only the first mode for the study of first-passage time probability. The equations of motion for the first mode are darldr = P1,
d/3,/dt = -&a1 - 3hla: - a: - 20,@, - Q, ,
(39)
and the domain of safe operation is defined by (a:+fJf)+$[2hla1+af]2<
V(P*),
(40)
where 0: = 1 + 2h:, c = 2~~5. The GCM method is applied to study the first-passage time probability of the arch. In the computation reported hereafter we have set D, =2&o:,
l=O*Ol.
(41)
Cell sizes are taken to be uniform throughout the cell state space in this paper. Figure 2 shows a typical coverage of the domain of safe operation by a finite number of rectangular cells. The cell sizes are set to be 0.15 x 0.15, and the mapping step time is chosen to be T = I’,/& where T1 = 2a/w,. Three hundred sample functions were used to compute the transition probability matrix by Monte Carlo simulation. The results of first-passage time probabilities of the arch initially at rest are plotted in Figures 3-5 for Al = 2.1, 2.3 and 3.1. The corresponding results by direct Monte Carlo simulation are also plotted in the figures. The agreement is excellent. The accuracy of all the results in this paper by direct Monte Carlo simulation is within 2%, based upon a similar analysis to the one in reference [24].
J.-Q.
306 5 4
0 -I -2 -3 -4 g
-
-4
-3
-2
-I
0
Figure 2. Regular cells in the domain of safe operation center of a cell. A, =2-l; cell size: 0.15X0.15.
Figure 3. The first-passage + + +, direct simulation.
Figure 4.. The first-passage
C. S. HSU
... ..... ...... ........ ......... .......... ........... ............ ............ .............. .............. ............... ............... ................ ................ ................. .................. ................... .................... .................... ..................... ...................... ...................... ....................... ........................ ........................ ........................ .:::::::::::::::::::::: ...................... ..................... ................... ................... .................. .................. ................. ................ ................ ............... ............... .............. .............. ............ ............ ........... ........... ......... ........ ...... ..... ...
2
-5
AND
,,(,,,,,,
3
a
SUN
time probability
time probability
I
2
3
4
of the sinusoidal
5
shallow
arch. A dot represents the
of the sinusoidal
shallow
arch. h, =2*1. -,
GCM results;
of the sinusoidal
shallow
arch. A, = 2.3. Key as Figure 3.
HINGED
SHALLOW
ARCH
RANDOM
307
VIBRATION
N
Figure 5. The first-passage
time probability
of the sinusoidal
shallow
arch. A, = 3.1. Key as Figure 3.
As has been commented in reference [ 161, one of the advantages of the GCM method is that the global picture of first-passage time probability as a function of initial conditions can be readily obtained by the GCM method. Here, we present such a study for Al = 2.1 and 3.1. Fourteen initial conditions of ((Y,, PI) are chosen as shown in Figure 6. The
/
-2_
-3
-2
I
I
-I
0
1
I
2
01 Figure
6. Initial conditions
selected to study the first-passage
problem.
first-passage time probability, and the mean and standard deviation of first-passage time are computed for each initial condition. The results of first-passage time probabilities for several initial conditions are shown in Figures 7 and 8. The mean and standard deviation of first-passage time for all 14 initial conditions are listed in Tables 1 and 2. A comment on the computation of the mean and standard deviation of first-passage time is in order. Let fi and a[N] be the mean and standard deviation of first-passage time in cycles of a linear oscillator with period 7’r. Let p(N) be the first-passage time probability density function at the Nth cycle. Then m Nr A= Np(N) dN= I+(N) dN+ NP(N) dN, (42) I NL I0
308
J.-Q. SUN AND C. S.
HSU
Figure 7. The first-passage time probability of the sinusoidal shallow arch for initial conditions: (l), (3), (4), (9) and (14) in Figure 6; A, =2-l; GCM results.
where NL is a large number such that for N 2 NL, the following asymptotic form of p( N) holds: p(N) = DA7’, emATtN,
Nz= NL>>l,
(43)
where D and A, the limiting decay parameters, are part of the GCM results. Hence, R can be evaluated, by using the trapezoidal rule, as m-l
C (iAN)p(iAN)++p(NL) i=l
AN+~e-AT~N~(~7’lNL+l), I
I.0
(4)
1
_
Figure 8. The first-passage time probability of the sinusoidal shallow arch for initial conditions: (l), (3). (4). (9) and (14) in Figure 6; A, =3.1; GCM results.
HINGED SHALLOWARCH RANDOMVIBRATION
309
TABLE 1
TABLE 2
The mean and standard deviation (in cycles) of the first-passage time of the sinusoidal shallow arch; A, = 2.1
7’he mean and standard deviation (in cycles) of the first-passage time of the sinusoidal shallow arch; A 1= 3 *1
Point/ no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14
8.2845 7.3117 3.2152 6.2094 8.1418 7.6969 7.0326 7.1672 4.7512 7.0102 8.1443 7.7005 6.2367 2.3982
Point/no.
N
1 2 3 4 5 6 7 8 9 10 11 12 13 14
12.0878 11.1185 7.2799 10.1422 12.0055 11.7942 10.7439 10.8271 7.6719 10.7482 12.0108 11.8014 10.1616 3.5012
6.0866 6.0436 4.8145 58910 6.0861 6.0760 6.0313 6.0450 5.5525 6.0198 6.0865 6.0748 5.9006 4.2133
UN
9.0718 9.0445 8.3477 8.9582 9.0716 9.0702 9.0234 9.0256 8.4808 9.0153 9.0708 9.0683 8.9648 6.3380
where AN = NJ m. Similarly, we have l/2
(I
m[N-N]2p(N)dN 0 >,
a(N]=
m-1
=
,z, (iAN)2p(iAN)
+$
p(N,)]
AN
D -e-Ar~N~[(AT,NL)2+2(ATINL)+2]-~2 + (ATJ2
In order to check the GCM results, the mean and standard deviation of first-passage time for the initial condition (0,O) were also computed by direct Monte Carlo simulation, and the results of the mean and standard deviation of first-passage time are 8.6099 and 6.4185 (cycles) for A1= 2.1, and 12.2652 and 9.3800 (cycles) for Ai = 3.1. These four numbers deviate from the corresponding ones in Tables 1 and 2 by 3*9%, 5*5%, 2.4% and 3*0%, respectively. We now add a comment on the computational efficiency of the GCM method. It is found that the CPU time of one GCM run in which the results of the first-passage time probabilities for 14 initial conditions are obtained is less than 1.4 times that of a direct Monte Carlo simulation run in which the first-passage time probability for only one initial condition is obtained. This means that more than ten folds of computational efficiency of the GCM method versus direct Monte Carlo simulation are achieved. The task of obtaining more detailed information on the effect of initial condition over the domain of safe operation becomes rapidly prohibitive to direct Monte Carlo simulation. This task, however, is still tractable by the GCM method because the GCM method always deals with a finite number of cells, and, for each initial condition, the GCM method does not need much extra effort other than a limited amount of computation for the results of first-passage time probability by iterating equation (30). Generally speaking, when only a single initial condition is of interest, the GCM method and the method of direct
310
J.-Q.
SUN
AND
C. S. HSU
simulation are about equally efficient, provided that a comparable accuracy is maintained for both methods. The advantage of the GCM method over simulation lies in its ability to yield global results when all possible initial conditions in the domain of interest are to be investigated. The advantage also becomes apparent for any comprehensive study where by using the GCM method different types of results can be obtained in one computer run. 5.1.2. Example 2 Consider the sinusoidal shallow arch under a random excitation given by Q(x, t) = Q1( t) sin x+ Qk( t) sin IQC,
(46)
where
E[Oi(t)l=o,
E[Qi(t)Oj(t’)l=2D,6,6(t-rr),
i,j=l,k.
(47)
Once again we assume that the arch is initially at rest. From equations (5), it is seen that under the excitation (46) all the modes will remain quiescent for all t > 0 except the first and kth modes. In the study of first-passage time probability these two modes must be retained. We take k = 2 in this example. The equations for the first and second modes are da,ldt
= I%, da,/dt
da,/dt=-o:al-3hla:-cu:-(cw:/4)(Al+a,)-201SB,-Q,, d82/dt=-4~z-$cu:-(2hl(Y1+(Y:)(YZ-20,3P2-QZ,
= 4&,
(48) (49)
and the domain of safe operation is defined by V(P*).
; (crZ,+p2,)+$[2h,(~~+(~:+(y3/4]~< ?I=1
(50)
The parameters D, and 5 are given by equations (41). The state space is four-dimensional in this example. The cell sizes for GCM are set to be O-4 x 0.4x 0.4 x O-4 for A1= 2.1, and O-5 x 0.5 x O-5 x 0.5 for A, = 2.5. The mapping step time is T,/8. Fifty samples were used to compute the transition probability matrix by Monte Carlo simulation. The domain of safe operation defined by equation (50) contains 7839 regular cells for Al = 2.1, and 9421 regular cells for A, = 2.5. Because of the large cell sizes used, the failure set F of GCM was also used in the direct Monte Carlo
1
t z.
I.0 -
/+-f--+-f--f--+
-
f
z
$
0.8
if
P
E
‘Z
0.6
_
8
0 x
0.4-
5 Ii
0
s
IO
I5
20
N
Figure 9. The first-passage time probability of the sinusoidal shallow arch under random excitation (46); k=2.-, GCM results; + + +, direct simulation.
A,=2.1,
HINGED
SHALLOW
ARCH
RANDOM
VIBRATION
311
N
Figure 10. The first-passage time probability of the sinusoidal shallow arch under random excitation (46); h,=2.5, k=2, Key as Figure 9.
simulation for these cases in order to compare the two methods on an equal basis. In the direct Monte Carlo simulation, a trajectory was terminated as soon as it goes into the failure set F. As has been commented in reference [ 161, the first-passage time probability obtained based upon this failure set F is slightly higher than the true value. However, it can be shown that this difference diminishes rapidly as the cell sizes for GCM are reduced. The results for the first-passage time probability for A, = 2.1 and 2.5 obtained by both the GCM method and direct Monte Carlo simulation are plotted in Figures 9 and 10. The agreement between the results by these two methods is fairly good. It is remarked that, for higher-dimensional spaces, the complexity of computational procedures and computer programs for the GCM method does not increase much. In fact, there is virtually no change in the iteration procedure and its computer programs. However, the memory need of the GCM method increases rapidly with the dimension of the state space. 5.1.3. Example 3 Next, we consider a sinusoidal shallow arch with the second harmonic imperfection such that h,#O,
Az#O;
h,=O,
for n > 2.
(51)
Assume that the arch initially at rest is excited by the random excitation given by equation (37). It is evident that despite the zero initial condition for all the modes, the second mode will no longer remain quiescent for t > 0. Only direct Monte Carlo simulation has been applied to study the first-passage time probability of the arch for the cases Ai = 3-l and AZ/4 = 0.0,0~125,0~25. The critical energy level V( P*)* was determined by the method outlined earlier in the paper. It is found that V(P*), = 39446 for A2/4=0*125, and V(P*), = 365095 for AZ/4 = 0.25. Upon comparing these V(P*), values with V(P*) = 43.2533 for Al = 3.1 and A2= O-0, it is seen that the second harmonic imperfection indeed causes the reduction in the critical energy level of the arch. The results for the first-passage time probabilities for the above three cases are plotted in Figure 11. It is seen from the figure that the second harmonic imperfection causes higher first-passage failure after about five cycles of the motion.
312
J.-Q. SUN AND C. S. HSU
N
Figure 11. The first-passage time probability ofthe sinusoidal shallow arch with the 2nd harmonic imperfection under random excitation (37); A, = 3.1 and A,/4= 0.0, 0,125, 0.25.
5.1.4. Example 4 Finally, we revisit Example 1 with an initial condition specified by cQ(O)# 0,
a,(O) = 0,
n # 2,
&(O)=O,
n91.
(52)
In this case the second mode is no longer identically zero for t > 0. We examine the effect of initial condition (52) on the first-passage time probability of the sinusoidal shallow arch. Extensive Monte Carlo simulation studies were carried out. Some of the results are shown in Figures 12 and 13 and in Tables 3 and 4. By examining these results, it is
Figure 12. Effect of initial condition (52) on the first-passage time probability of the sinusoidal shallow arch under random excitation (37); A, = 3.1; direct simulation. -. -, ar(0) = 4.0, a,(O) = 0, n f 2; -, a,(O) = 0, n 2 1; - - -, a,(O) = 0.01, a,(O) = 0, n # 2.
HINGED
SHALLOW
ARCH
RANDOM
313
VIBRATION
Figure 13. Effect of initial condition (52) on the first-passage time probability of the sinusoidal shallow arch under random excitation (37); A, = 4; direct simulation. Key as Figure 12.
observed that the first-passage time probability for a small (~~(0) (>O) is lower than the corresponding probability for q(O) = 0. This difference for a fixed (Ye, say LYE = O-01, is not large when AI is small, but is significant when AI is large. This observed difference exists even for simulation runs with a very large number of 2000 samples. Corresponding to the lower first-passage time probability, the mean of first-passage time for a small a,(O) (>O) is larger than the corresponding mean value for ~(0) = 0. On the other hand, for a large (~~(0) (>O), it is observed that the first-passage time probability is higher than the
TABLE
TABLE
3
The mean (in cycles) of the first-passage time of the arch shallow sinusoidal under initial condition (52); A, =3-l
04000 04008 0*0004 O-0081
oTMo3 O-0806 o-1210 0.1613 0.2016 0.2419 0.2823 O-3226 O-3629
12.27 12.71 12.83 13.00 13.48 13.49 13.64 13.10 12.30 11.83 10.52 7.99 5.87
4
The mean (in cycles) of the first-passage time of the sinusoidal shallow arch under initial condition (52); Al =4-o %(0)/4A*
A
04000 04006 0*0031 om63 0.0313 o-0625 O-0938 0.1250 O-1563 0.1875 0.2188 0*2500 0.2813
14.34 15.53 1559 1.586 15.98 15.83 15.72 15.53 15.38 14.76 13.77 13.17 10.93
314
J.-Q. SUN
AND
C. S. HSU
corresponding probability for cuz(0)= 0, and that the mean of first-passage time is smaller than the corresponding mean value for (Y>(O)= 0. Here, the difference of the probability for a fixed a,(O), say (~~(0)= 4.0, is larger when A1 is small than that when Ai is large. The above discussion seems to suggest that there may exist a crossover value of (~~(0) depending on A1 such that the first-passage time probability for (~~(0)# 0 is equal to that for (~~(0)= 0. It is found from a series of computations that when (~~(0)/(4A,) = 0.20, this crossing over occurs for all the tested values of A,. The mechanism of the above observations is not completely clear at the present time. More study is needed in order to understand these observations and to delineate the precise parameter range in which the qualitative and quantitative changes of the firstpassage time probability can be predicted. 6. CONCLUSIONS In this paper the steady state solutions of a hinged shallow arch have been studied by means of the Fokker-Planck equation. The first-passage time probability of a hinged sinusoidal shallow arch has been studied numerically by the GCM method and direct Monte Carlo simulation. It has been demonstrated that the GCM method has large computational advantages over direct Monte Carlo simulation when the global information of first-passage time probability of the system is the objective of the study. ACKNOWLEDGMENT This material is based upon work supported by the National Science Foundation under grant No. MSM-8519950. REFERENCES
1. R. SCHMIDTand D. A. DADEPPO 1971 ZndustriuZMathematics 21,91-114. A survey of literature on large deflection of non-shallow arches. Bibliography of finite deflections of straight and curved beams, rings and shallow arches. 2. P. A. A. LAURA and M. J. MAURIZI 1987 Shock and Vibration Digest 19(l), 6-9. Recent research on vibrations of arch-type structures. 3. H. N. PI, S. T. ARIARATNAM and W. C. LENNOX 1970 Boceedings ofthe 5th South-Eastern Conference on Zheoreticul and Applied Mechanics, Raleigh-Durham, 4. 5. 6. 7. 8. 9.
10. 11. 12.
North Curolinu, U.S.A.
Numerical solution for the mean first-passage time for snap-through of shells. H. N. PI, S. T. ARIARATNAM and W. C. LENNOX 1971 Journal of Sound and Vibration 14, 375-384. First-passage time for the snap-through of a shell-type structure. C. S. Hsu 1967 Journal of Applied Mechanics 34, 349-358. The effects of various parameters on the dynamic stability of a shallow arch. T. K CAUGHEY and H. J. PAYNE 1967 International Journal of Non-Linear Mechanics 2, 125-151. On the response of a class of self-excited oscillators to stochastic excitation. T. K. CAUGHEY and F. MA 1982 Journal of Applied Mechanics 49, 629-632. The steady-state response of a class of dynamical systems to stochastic excitation. H. RISKEN 1984 The Fokker-planck Equation-Methods of Solution and Applications. Springer Series in Synergetics 18. New York: Springer-Verlag. Y. YONG and Y. K. LIN 1987 Journal of Applied Mechanics 54, 414-418. Exact stationaryresponse solution for second order nonlinear systems under parametric and external white-noise excitations. A. C. ERINGEN 1957 Journal of Applied Mechanics 24, 46-52. Response of beams and plates to random loads. S. T. ARIARATNAM 1962 Journal of Applied Mechanics 29, 483-485. Response of a loaded nonlinear string to random excitation. R. E. HERBERT 1964 Journal of the Accoustical Society of America 36( 1l), 2090-2094. Random vibrations of a nonlinear elastic beam.
HINGED
SHALLOW
ARCH
RANDOM
VIBRATION
315
13. S. H. CRANDALL 1979 Developments in Statistics (P. R. Krishnaiah, editor) 2, l-82. New York: Academic Press. Random vibration of one- and two-dimensional structures. 14. D. A. DARLING and A. J. F. SIEGERT 1953 Annals of Mathematical Statistics 24, 624-639. The first passage problem for a continuous Markov process. 15. W. C. LENNOX and D. A. FRASER 1974 Journal of Applied Mechanics 41, 793-797. On the first-passage distribution for the envelope of a nonstationary narrow-band stochastic process. 16. J.-Q. SUN and C. S. HSU 1988 Journal of Sound and Vibration 124,233-248. First-passage time probability of nonlinear stochastic systems by generalized cell mapping method. 17. S. H. CRANDALL, K. L. CHANDIRAMANI and R. G. COOK 1966 Journal of Applied Mechanics 33, 532-538. Some first-passage problems in random vibration. 18. C. S. HSU 1987 Cell-to-Cell Mapping-A Method of Global Analysis for Nonlinear Systems. New York: Springer-Verlag. 19. H. M. CHIU 1984 Ph.D. Thesis, Mechanical Engineering, University of California, Berkeley. Random vibration analysis by cell mapping methods. 20. C. S. Hsu and H. M. CHIU 1986 Journal of Applied Mechanics 53, 695-701. A cell mapping method for nonlinear deterministic and stochastic systems-Part I: the method of analysis. 21. H. M. CHIU and C. S. l-&U 1986 Journal of Applied Mechanics 53, 702-710. A cell mapping method for nonlinear deterministic and stochastic systems-Part II: examples of application. 22. J. Q. SUN 1988 Ph.D. Thesis, Mechanical Engineering, University of California, Berkeley. Random vibrations of nonlinear systems based upon the cell state space concept. 23. J.-Q. SUN and C. S. HSU 1988 Journal of Applied Mechanics 55, 694-701. A statistical study of generalized cell mapping. 24. J. B. ROBERTS 1969 Journal of Sound and Vibration 10, 42-61. Estimation of the probability of first-passage failure for a linear oscillator.