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Analytical vs. Simulation Solution Techniques for Pulse Problems in Non-linearStochastic DynamicsIwankiewicz, R.; Nielsen, Søren R. K.

Publication date:1997

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Citation for published version (APA):Iwankiewicz, R., & Nielsen, S. R. K. (1997). Analytical vs. Simulation Solution Techniques for Pulse Problems inNon-linear Stochastic Dynamics. Aalborg: Dept. of Building Technology and Structural Engineering. StructuralReliability Theory, No. 176, Vol.. R9760

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INSTITUTTET FOR BYGNINGSTEKNIK DEPT. OF BUILDING TECHNOLOGY AND STRUCTURAL ENGINEERING AALB ORG UNIVERSITET • AAU • AALBORG • DANMARK

STRUCTURAL RELIABILITY THEORY PAPER NO. 176

R. IWANKIEWICZ, S.R.K. NIELSEN ANALYTICAL VS. SIMULATION SOLUT ION TECHNIQU ES FOR P ULSE PROBLEIVIS IN NON-LINEAR STOCHAST IC DYNAMICS DECEMBER 1997 ISSN 1395-7953 R9760

Th STRUCTURAL RELIABILITY THEORY papers are issued for early di s ·emina­tion of r search result s from the Structural Reliability Group at th Depar tment of Building Technology and Structural Engin ering , Univ rsity of AaJborg. These papers are generally submitted to scientific meetings, con£ rences or journals and should there­for not be wid ly di str ibute l. Whenever possible reference should be given to the final publications (proc edings, journals, etc.) and not to the Structural Reliability Theory papers.

I PrinLecl a.L Aalborg Universi Ly I

; .

INSTITUTTET FOR BYGNINGSTEKNIK DEPT. OF BUILDING TECHNOLOGY AND STRUCTURAL ENGINEERING AALBORG UNIVERSITET • AAU • AALBORG • DANMARK

STRUCTURAL RELIABILITY THEORY PAPER NO. 176

R. IWANKIEWICZ, S.R.K. NIELSEN ANALYTICAL VS. SIMULATION SOLUTION TECHNIQUES FOR PULSE PROBLEMS IN NON-LINEAR STOCHASTIC DYNAMICS DECEMBER 1997 ISSN 1395-7953 R9760

ANALYTICAL VS SIMULATION SOLUTION TECHNIQUES FOR PULSE PROBLEMS IN NON-LINEAR STOCHASTIC DYNAMICS

Radoslaw lwankiewi cz * and S0ren R.K. Nielsen **

* lnsl. of M alerials Science and A pplierl M cchanics, WTOclaw University of Technology, Poland

*• Dept. of Building 'l'cchnology and Slrur:luml 8nginecr·ing, A albory University, Denmar·k

Abstract

Advantages and disadvantages of available analytical and simulc1tion techniques for pulse prob­lems in non-linear stochast.ic dynamics arc discussed. First, random pulse problems, both those which do ;u1d do not. lead to Markov theory, a.rc presented . Next, t.he analytical and analytically-numerical techniques suitable for Markov response problems such as moment equa­tions, Pctrov-Galerkin and cell-to-cell mapping techniques are briefly discussed. Usefulness of these techniques is limited by the fad that effectiveness of each of them depends on the mean rate of impulses. Another limitation i::; t.he size of the problem, i.e. the number of state vari­ables of the dynamical system. In contrast., the applicability of the simulation techniques is not. limited to Markov problems, nor is it dependent on the mean rate of impulses. Moreover their use is straightforward for a large class of point processes , at least for renewal processes.

1 Introduction

J\n often posed question is : whether or not. it. is of general interest. to consider the problems of rcspoiJSf~ of dynamical systems to random pulse trains. In order to justify such an interest kt. us realize the fact. that any excitation to the dynamical mechanical system may be d!"ectu<ttcd i11 either of two ways: as a continuous function of t.ime , or hy impulses (jumps in the velocity response process). J\ class of practical engineering problems leading to the random impulse process representation can be listed , which embraces all kinds of tra ins of shocks and impacts .

lf it is of interest, which has been rather commonly accepted for a long time, t.o consider problems of response to random excitations with time continuous sample paths, it. is of equally general interest to consider problems of response to impulse process exc.itations. Howf~ver, the la.Ucr excita.tions cannot be treated by usuctl techniques, because th ey revea.l inherently non­Caussia.n nature. A Gaussian process is only an asymptotic special ca.se of a. stochast.ic impulse process. Alt.hough as a basic model an external impulse process excitation rna..y lw assumed, a.n insight int.o the physi cal nature of many problems reveals that the jump of the velocity response is U1f~ result. of collision of two bodies . Conscquen1.ly, the t.ot.a.l impulse equal to the incr<~rnent of t.he momentum of the system under consideration depends on the vc~ locities of both colliding bodies prior to t.he collision a.nd hence it depe!nds on the stat.e variables of the problem. This leads to the multiplicative noise (pa.rarnetric excitation) problcm 1

. llencc ev<'JI for a silllplc rnechanicc1l mode l, quite advanced techniques may be required.

One of the earliest papers dealing with the response of a non-linear dynamical system to a random train of impulses is certainly due to Roberts. 2 Next different a.pproatlles to the problem were developed, such as the equivalent lin earization3 , the improved perturba­tion technique to solve the generalized Fokker-Pianck equation\ techniques based on equations for moments,5

-7

•11 Petrov-Galerkin method,8 cell-to-cell mapping technique.9 •10•12 Recently Di

Paola and co-workers 13 as well as Grigoriu 14 have tackled the problem .

2 Markov and non-Markov response problems

Consider a general multi-degree-of-freedom non-linear dynamical system under a random train of general pulses driven by a stochastic point process {N(t), t E [lo , oo[}, Pr{N(lo) = 0} = I , which is uniformly regular and has a finite number of points in a finite time interval. The counting process gives the number of time points in the interval [t0 , t[. The state vector of the system, Z(t), consisting of the structural generalized displacements and velocities augmented possibly by the state variables of the auxiliary filter, is governed by the set of equation~> of motion

dZ(t) = c(Z(l),l)dt + d(Z(l), t, P(t))dN(l), Z(l 0 ) = z0 , (J)

where dN(t) = N(t + dt) - N(t) and JJ(i) assumes the values P(l;) = fJ; (random mark variableo) at the times l ; of the impulses occurrences. If N( l) iH a Poisson counting process, independent of the random mark variables which also are mutually independent. , and both counting process and mark variables are independent of the initial conditions , the state vector Z(l) is a Markov process. It is not for all other point processes, e.g . renewal processes.

Nevertheless the differential of the function <I>(t, Z(l)) of the process governed by equations ( 1) can be evaluated from the following differential rule, 15

d<I>(t, Z(l)) u<I>(l, Z(t))d ~ 8<I>(l, Z(l)) .. (Z(· ) )) d ~l .t + L

8 r. c, t , t .t

ui . Z; •=1

+ [<t>(t ,Z(l)+d(Z(t) ,t,P(t))) -<t>(t,Z(l))] dN(t), (2)

where n is the number of state variables . This differential formula is the startpoint to derive the differential equations for moments.

However since the response process Z(t) is the functional of the random mark variables P(i;) =

P;, l; < t and of the counting process N(T), T < t, this formula can only be used effectively, yielding explicit moments at the right-hand sides of equations, if the averaging of the last term is feasible, i.e. if the correlation between the expression in square brackets and dN(l) can be split. This is possible if the increments of the stochastic point process are independent , the random mark variables arc independent and these two arc also mutually independent. This is so if t.he underlying process is a. compound Poisson process, i.e. if t.hc augmented dynamical system is Poisson-driven. However if the increments of the regular counting process N(t) can be expressed as

dN(t) = p( N(t)) dN( t ),

2

where N(t) is a homogeneous Poisson counting process and p(N(t)) is a suitably chosen zero­memory transformation, the problem can be converted to a Poisson-drivcn one at the cxpE!rlsc of introducing a number of auxiliary state variables. Such a transformation, being a linear function of the auxiliary state variables, each of whose is an exponential transformation of N(t), has been found for a class of Erlang renewal processes. 11

•12

•17 Then the structural sta.tc

vector Z1 (t) augmented by the auxiliary variables Z2(t) is governed by the st.ochastic differential equations

dZ(t) = c(Z(t), t.)di + d(Z(l), l, P(t))dN(I.), Z(t 0 ) = z0 ( 1)

Z(L) = [Z1(l)] (Z(t) t) = [ci(Zl(/.),1)1 Zz(l) ' c ' 0 '

d(Z() l P(t)) = [d1 (Z!(t), t)g(Z2(t))P(t) t ',, , d2(Z2(t), t) '

(El)

where P(t) assumes the values P(i;) = P; at the times l; of the Poisson events and P; an~

mutually independent and identically distributed as P. Further, g(Z2(l)) = p(N(l)) is a . known !"unction of the auxiliary variables.

Although the structural state vector Z1 (t) is a non-Markov process, the augmented state vector Z(l) becomes a non-diffusive Markov process.

3 Analytical solution techniques I

3.1 Equations for moments and modified closure approximations

For a general Poisson-driven pulse problem governed by equations ( 1) and ( 5) the equations for t.he mean values, the second-, third- and fourth-order joint central moments of the response, arc obtained as11

•15

Ji.; (l) = E [c;(Z(l), t)] + v(l)E [d;(Z(l), t, P)]

~~i.i(t) = 2 { E [z? (c?(Z0 (l), l) + 11(L)d1(Z(t), t, P))]}. +v(t)E [d;(Z(I,), t, P)dj(Z(l), t. , P)],

3 { E [Z;0 zy (c~(Z0 (t), t) + ll(t)dk(Z(L), t , P))] L +3v(t) { E (Z?d;(Z(l), t, P)dk(Z(t), l, P)]} s

+v(t)!E [d;(Z(l), l, P)d1(Z(I.), t, P)dk(Z(l), l, P)],

K.;jkt(l) = 1 {E [z?zJzf(c?(Z 0 (t),t)+v(t)dt(Z(t),t,P))]L

+6v(t) { E [z? ZJdk(Z(I. ), t, P)dt(Z(t), t, P)] L +411(l) { E [Z?dj(Z(t), t , P)dk(Z(t) , t, P)dt(Z(t), l , P)] L +v(t)E [d;(Z(t), i, P)di (Z(t), t , P)dk(Z(t), t, P)dt(Z(t), L, P)],

(6)

(7)

(8)

(9)

where Zf(t) = Z;(t)- f.L;(t) and c~(Z0 (t), t) = Cj(Z(t), t)- E [cj(Z(l), t)] denote the components of the zero-mean (centralized) state vector and drift vector, and { .. . }s denotes the Stratorwvich symmetrizing operator. 16

It is no doubt that the accuracy of the results obtained with the help of a closure technique depends on the ability of the tentative probability density function for the evaluation of the unknown expectations entering the moment equations, to qualitatively model the actual density function, i.e. it should have freedom to represent possible multimodal or multipeak shapes and discrete probability components.

Consider the dynamical system subjected to a random train of impulses and to initial conditions Z(t0 ) = z0 . If in the time interval [to, t[ no impulse occurred , the system has performed the deterministic drift motion from the initial state z0 at the time t 0 to the state z(t) = e(tlzo, t 0 ) at the timet, or it has been at rest , in the case of zero initial conditions. Notice that e(tolzo, la) = zo.

If the train of impulses is driven by a homogeneous Poisson process, the probability P0 of no impulse occurrence in the time interval [to, t[ is expressed as

Pa(tlto) = Pr{N(t) = OIN(to) = 0} = exp( -v(t- to)). (10)

The probability P0 (tlt 0 ) may be high, close to the unity, if the length t - t 0 of the time interval is small, i.e. at the early transient stage, especially if <1lso the mean arrival rat.e v is small.

.Joint probability density function of the state vector Z(t) can be represented in form of the sum of the continuous and discrete parts as ·

fz(z, t) fz(z, t I N(t) = 0) Pr{N(t) = 0} + Jz(z, t I N(t) > 0) Pr{N(t) > 0} n

Pa(tlta) IT o(z;- e;(tlzo,to)) + (1- Po(tlta))J~(z,t), ( 11) i=l

where e;(llzo, t0 ) denotes the drift from the initial state z0 at l = L0 obtained from (1) for dN(t) = 0. Hence the system must be at the position z; = c;(tlzo, t0 ) with probability one as specified by the delta spikes, if no impulses have arrived. So, f~(z, t) = fz(z, t I N(L) > 0) denotes the continuous joint probability density function on condition t.hat at least one impulse has occurred during the preceding time interval [to, t[. Expectations evaluated with respect to n(z, t) are denoted as E[· · ·]0 . In particular, the conditional mean value function and the conditional joint central moments of the order r are denoted as p.?(t) and K?1 ;2

... ;r (l). The

relationships between unconditional and conditional moments read 15

p;(t) = Po(Lito)c;(tlzo, to)+ (1- Po(tlto))p~(t), (] 2)

4

( 13)

where (12) has been used. Further, the arguments of P0 (tito) and e;(tlzo, l0 ) have been omitted for ease of notation. The inverse relationship can be similarly derived in few steps

( 11)

For syi:itcms with polynomial urift vectors the following modified cumulant neglect closure i:iclteme may be used. In case of closure at the order N all centralized moments K~1 ; 2 .. ;T(l)

, T > N with respect to the continuous joint probability density function J£(z, t) are first. expressed in terms of corresponding centralized moments of the order j ~ N by means of th e ordinary cumulant neglect closure approximations. This will work if J~(z, t) is monomodal and not deviating too much from a multivariate normal distribution, since the joint cumulants are ~cro in the latter case. Then, the corresponding unconditional moments K;

1;:r·ir(t), r· > N

rnay he expressed in terms of the centrali~ed moments x:?1

;2

... ;Jl), j ~ N by means of (13) .

Finally, all joint. moments x:?1

;2

... ;j(t), j ~ N within this expression can be expressed in terms

of x:;1

;2

... ;i(i), j ~ N by means of (111), and the required closure scheme is obtained. In case of closure at th e order N = 4 the explicit closure approximations for the .St.h and 6th order joint central ized moments have been derived for the case e;(t lzo , t0 ) = 0. 5

•15

The modified curnulant-neglect closure technique proved to be effective in the case of eval­uating transient response moments for low mea.n rates of impulses, i.e. for spa.rse tra.ins of imptdses5

. Stationary moments, even though the mea.n rate is low , can often be cva.l uatcd

.s

with the help of ordinary cumulant-neglect closure approximations , because for very l~ng time intervals the spike of the probability density function becomes rather insignificant. However, as the experience of the authors shows, numerical integration scheme combined with ordinary cumulant-neglect closure approximations runs into instability at the early transient stage if the mean rate of impulses is low.

3.2 Petrov-Galerkin method to solve the forward and backward Kolmogorov-Feller equations

The forward Kolmogorov- :F'eller equation for the joint probability density .fz(z, t) of the state vector Z(t) reads in the case of absorbtion on a part of the boundary 15

•11

;

~Jz(z, t) = Kz ,t[f(z , t)] , V t E]io, t1] , V z ESt

fz(z, to) = fo(z) , V z E Sto ( 15)

fz(z, t) = 0

where 5'1 is the solution set at the timet, bounded by the surface oS1 = 8S1(o) U ()S1(1J U oS}2l.

DSP) is the non-accessible (natural) part of the boundary, whereas the accessible boundary is

made up of the exit part 8S}1l and the entrance part oSf0 l. As indicated, absorbt.ion boundary

conditions must be specified on 8S}0 l. The forward integro-differential Kolmogorov-Feller operator is given by 15

•18

Kz,t[fz(z, t)] =

-2.::: a:i [c;(z, t)Jz(z, t)J + v(t) j [~z(a(z,p, t), t) 1 ~ 1 - Jz(z, t) J .rp(p)dp, 1 p

( 16)

where _ d, (I od(a(z,p,L),l))

J- et + f)y'l" , (17)

and a = a( z, p, i) is the inverse transformation of

z=a+d(a,p,t) ( 18)

88Y~ is the gradient of d(y, p, t) with respect to y and f p(p) is the probability density of the mark variable P.

The backward Kolmogorov-Feller equation with absorbtion boundary conditions is 15

8 T [ j aJz(z, t) + Kz,t fz(z, t) = 0 , ViE [to,LJ[ , V z ESt

( 19)

fz(z,t) = 0

6

where I1(z) is the terminal value of the unknown function .fz(z, t) . In t.his case the absorb1jon

boundary condition is specified on as?l. The backward integro-differential Kolmogorov-Feller operator is given by

K~, 1 [fz(z, t)] =

"'£ c;(z, t) a~Jz(z, t) + v(t) j [.rz ( z + d(z, p, t)) - .rz ( z, l) J fp(p)dp. ' p

(20)

The Galerkin method for solving the boundary and initial value problems (15) ami (19) consists in expanding the unknown function fz(z,t) in series of approximating shape functions and expanding the variational field in series of weighting functions. For the problem (15) the

shape functions must fulfil the boundary condition N;(z) = 0, z E a8z0) U 88

1('2), whereas

the weighting functions V;(z) = 0, z E ast(I}_ In contrast, for the problem (19) the Bhap<!

functions fulfil the boundary condition N;(z) = 0, z E 8St(I}UaS1(2

), and the weighting functions

V;(z) = 0, z E 8S1(2

). Further the shape and weighting functions rnu~Jt be suff-iciently smooth that K:~.,t[ ... ] and KL[ ... ] may become adjoint operators when integrated over 81• In order to achieve numerical stability due to the large Courant number in part of the mesh an upwind

-differencing in the weighting function becomes necessary as performed in the Petrov-Galerkin variational method .

11. has been found that for a. two-dimensional problem, i.e. the Duffing oscillator under trains of impulseB with high to moderate mean rate, the Petrov-Galerkin method provides very accurate solution of the backward Kolrnogorov-Feller equation 8

. Unfortunately ii. has not been possible to devise the Petrov-Galerkin technique suitable for sparse trains of puiBes. J\notlwr drawback of the method is that, despite todays technology, the solution is not feasible for sLate vectors of dimension larger than 4 or 5, not even with parallellization of the calculations.

3.3 Cell-to-cell mapping technique

The problem iB discretized in time and space. The Lime axis is divided into srnctll Lime intervals 6.l and fz(zo, i;) denotes the first order probability density function at. the Lime l; = l 0 + ·i.6.l, ·i. = 0, 1, 2, .... The probability density function at the Bubsequent instant li+I, is given hy the convolution integral

fz(z, ti+I) = j qz(z, t;+IIzo, i;).fz(zo, t;)dzo, (21)

s,,

where q(z, l;+ 1 lz0 , l;) is the transition probability density function of the state vector from the state Z(lu) = z0 at the time i; to the state Z(l) = z at the Lime i;+ 1 and 81, is the sample space.

If the time interval 6.t is short enough and if the mean rate is low, it follows from the Poisson law that the probability of occurrence of more than one impulse in this time interval rnay be neglected and the following asymptotic form of the transition probability density may be a.ssumed, 10

7

where

and

qz(z, ti+1lzo, ti) = Po(ii+1ltJ)q~)(z, ti+1lzo, i;)+

(1 - Po(ti+1Jt;))q~)(z, ti+llzo, t;) + 0 ((vlltt) , (22)

(23)

(24)

is the transition probability density conditional on no impulse arrival and e( i;+1 Jz0 , i;) has the same meaning as in equation (11). Surely, (22) is fulfilled at best for sparse pulse trains where v is small. Hence the method is expected to work at best in this case. The transition probability density q~)(z, i;+ 1 Jz0 , t;) conditional on one impulse arrival is of continuous type . Hence the expansion (22) is based on pretty much the same idea as the modified cumulant neglect closure

scheme derived from (11). Algorithms have been devised for evaluation of q~)(z, l;+1 Jz0 , l;) .10

Especially a method based on a Taylor expansion in the impulse magnitude P is tractable:9•12

The sample space is divided into a finite number M of small volumes (cells) 19, where the

volume llzk of the mesh element is centered at Zk . Assume that llzk is sufficiently small for qz(Zj, i; + lltJzk, t;) and fz(zk, t;) to be approximately constant throughout the cell. The probability of being in the kth cell at the time l; is

(25)

The probability 7fy+1) of being in the jt.h cell at the time i;+1 is then given by

M (i+1) _ ~ Q (i) . _ 1 M 7fj - L....t jk1fk' J- , ... , ' (26)

k=1

(27)

where Qjk is the the element in the jth row and kth column of the transition probability matrix. The transition probability Qik will not depend on time l; if the Markov process is stationary.

The transition of states can be represented by the matrix equation

(28)

where Qi = Q · · · Q (Q multiplied by itself i times ), 7l"(i) is an M -dimensional vector of the

sl.a.tc probabilities 7l"~i) after ith transition, and 7l"(o) denotes the initial distribution at the time t 0 . The stationary distribution 7l"( oo) may be obtained after infinite many transitions as i -t

oo . Obviously, 7l"( oo ) = Q7l"( oo), determining 7l"( oo ) as the normalized eigenvec1.or rela ted l.o the eigenvalue ,\ = 1 of the matrix Q .

The cell-to-cell mapping tech nique described above has been applied to a Dufiing oscillator under Poisson 10 and renewal impulses 12

, hence for the problems two and three (one auxil­iary) state variables, respectively. The stationary marginal displacement and velocity response probability densities10

•12 and reliability function 10 have been evaluated. The results obtai ned

confirrn ec.l the congesture that the method is highly effective for sparse pulse trains. 9•10

•11

8

4 Simulation technique

The sample path of the random train of impulses is obtained by generating the tlequencc of interarrival times and the sequence of impulses magnitudes. The interarrival times may be sampled directly from the given probability distribution, or in the case of an Erlang procestl, with the help of a generated train of Poisson distributed points . The impulses magnitudes arc generated as sample values of a random variable P with a prescribed probability distribution.

The response sample curve is next obtained by numerical integration, of the homogeneoutl governing equation of motion between the impulses arrival times, whereas at. every time point. of an impulse occurrence the velocity response is increased by a jump, which gives the updated initial condition for the next interarrival time interval. Usually a standard 1th order Runge­Kutta technique may be used for numerical integration.

Utiually an ensemble of 50 000 response sample curves was generated for the problems considered by the authors, and in the case of ergodic sampling12 the generated response sample curve had a length of 4 000 000 natural periods of a comparative linear oscillator.

The simulation technique is quite straightforward to apply for a large class of regular point processes if the probability distribution of the interarrival times is specified. This clastl certainly embraces all the renewal processes.

Potlsible extensions of the simulation technique can be performed for the problems in which the interarrival times and the impulse magnitudes are correlated random variables.

I

5 Illustrative numerical results

Consider a Duffing oscillator under a Poisson driven train of impulses. Let Y(t) a.nd Y(t) denote, respectively, the displacement and velocity responses of the Duffing oscillator. Then Z(t), c (Z(t), t) and d(Z(t), t) of equation (4) arc given by

[ Y(t) ] _ [ Y(t) ] [ o ]

Z(t) = Y(l) 'c (Z(t), t.) = -2(w0Y(l)- wJY(l)- c:w5Y:~(l) 'd = J>(l) . . (29)

where ( is the damping ratio, w0 is the natural frequency of the linea.r oscillator corresponding to the Duffing oscillator and E is the non-linearity parameter. The data assumed for the Duffing oscillator is: w0 = 1 s-', ( = 0.05, E = 0.5.

Computations have been performed for two different values of the mean arrival rates fo impulses: v = O.lw0 and v = O.Olwo.

The random magnitudes of impulses have been assumed as centralized, Rayleigh distributed random variables, calibrated in such a way that in both cases vE[P2

] has the same value and that the variance of the stationary response of a comparative linear oscillator has unit value (d. e.g. 12) .

A uniform ,r')Q x 50 mesh has been used with the limits [-6ay,6ay] x [-6ay-,6ay-], where ay

and ay- are standard deviations of the stationary response of a comparative linear oscillator. The length of a transition time interval has been assumed as 6t = Tu/2, where T'o = 27r /wu is the natural period of the comparative linear oscillator. The transient marginal probability density functions of the displacement and the velocity response evaluated at the time points:

i = T0 , t = 2T0 and i = l0T0 for v = O.lw0 and v = O.Olw0 are shown in Figures 1. and 2, respectively. The solid line(--) and dashed line(----) represent, respectively, the analytical and simulation results obtained in the case of non-zero initial conditions: Y(O) = l, Y(O) = 0. The dotted line ( .. .... ) and the dashed-dotted line (-· -· -· -·) represent , respectively, the analytical and simulation results obtained in the case of zero initial conditions.

The simulation results have been based on the ensemble of 100 000 of the responsr. sample functions obtained by numerical integration, with the help of 4th order Runge-Kutta technique, of the homogeneous governing equation of motion (1) in the time intervals between the impulses arrivals, with the updated initial conditions for each time interval, due to the jump in the velocity response process at each impulse arrival time.

It is seen that the agreement between the analytical and simulation results is certainly very good in the case of zero-mean initial conditions and in both cases v = 0.1w0 and v = 0.01w0 .

It. is so in the very early transient stage, i.e. at i = T0 , l = 2T0 as well as at l = ] O'T0, in which case the response is almost stationary as shows the zero-mean value of the velocity response. The discrepancy of the peak values in the case of very sharp spikes of the density curves, should be attributed to the fact that in these cases the mesh is not fine enough.

In the case of non-zero mean initial conditions, for v = 0.1w0 , the analytically predicted probability density curves are also sufficiently close to the simulated curves except only, as before, for the peak values. For v = O.Olw0 the analytical prediction is only accurate <!nougb

in the nearly stationary case of the displacement response (Fig. 2 e)). Significant discrepancy between

1 the analytical nad simulation results in other cases may be explained by the fact that

analytical, piece-wise linear, probability density functions have been obtained with a mesh too coarse to idealize sharp spikeli of the probability density. Unfortunately assuming a finer rnesb, e .g. 100 x 100, resulting in a 10~ x 10~ transition probabilities matrix, turned out to be <1

computational task excessively large for the computers available.

6 Conclusions

A brief review of analytically -numerical techniques, having been developed by the authors and eo-workers, for non -linear dynamical systems under random impulses is done. Their advan­tages and shortcomings are discussed. Concluding, authors wish to cxpresli the opinion that the avenue of further development of the solution techniques for pulse problems in stochast.ic clynarnics should be directed onto optimization of the simulation techniques.

References

1. A. Tylikowski (1!182), Vibration of a harmonic oscillator due to a sequence of random collisions, Proc . of !.he Inst. of Machines Construction Foundation , Technical UIJiversity of Warsaw, No. 1, (in Polish).

2. J .B . H.oberts (1972), System res ponse to random impulses, J. Sound and Vibration, Vol. 21!, 23-:l4.

3. A. Tylikowski and W. Ma.rowski (1986), Vibration of a non-linear single-degree-of-freedo!1l system due to Poislionian impulse excitation, lni. .J. Non-linear Mechanicli, Vol. 21, 229-

10

4. C .Q. Cai and Y.K . Lin (1992), Response distribution of non -linear systems excited by non-G a ussi an impulsive noise, lnL. .J. Non-linear Mechan ics, Vol. 27, 9!J!}-967.

f>. Il. lwankicwicz , S.R.K. Nielsen and P. Thoft-Christcnsen (1990), Dynamic response of non-linear systems to Poisson-distributcd pulse trains: Markov approach, Strudura.l Safety, Vol. 8, 223-238 .

6. lUwankicwicz and S.R.K.Nielscn (1992), Dynamic response of hysterctic systems to Poisson-distributed pulse~ tra ins , Probabilistic Engrg. Mcch. , Vol.7 , No. 3, 1:35- 148.

7. It. lwankiewicz and S.R.K . Nielsen (1991) , Dyn amic response of non-linear systems t.o ren ewal-driven random pulse trains , In!.. .J. Non-linear Mechanics, Vol. 29, 555- .167.

8. H.U. Koyluoglu, S.R.K. Nielsen and H.. lwankiewicz (1994) , Reliability of non-linear os­cillators subject to Poisson-driven impulses , J . Sound and Vibration, Vol. 176, No. J, 19-3 :~ .

9. 11 . U. Koyliioglu , S.R. K. Nielsen and A .~ . (Jakmak (1994) , Fast. Cell-to-Cell Mapping (Path Integration) with Probability Tr1ib for the Stochastic Response of Non-linear White Noise, and Poisson Driven Systems, Proc. of 2nd lnt. ConL on Cornp. St.ochastic Mechan ics, Athens, Greece, .June 1:3-1[} , ;{61-:HO.

10. H .lJ . Koyliioglu, S.R. K. Nielsen and H .. lwankiewicz; ( 199.5) , H.csponse and H.eliabilit.y of Poisson Driven Systems by Path Integration, J. Engng Mech., ASCE, Vol. 121, No. I, 117-130.

11. S.H..K .Nielsen, IUwankiewio:: and P.S. Skjaerbaek (1995), Moment equations for non­linear systems under renewal-driven random impulses with gamma-distributed intcrarrival t.irncs, IUTAM Symposium on Advances in Nonlinear Mechanics, Trondheirn, Norway,

.July 1995, Eds. A . Nacss and S. Krc nk , I<luwcr Academic Publishing, :~:n-:340.

12. IUwankicwicz and S.lLK.Nielsen (1996) , Dynamic response of non-linear systems to re­newal impulses by path integration , .J . Sound and Vibration , Vol. 19.1, No. 2, 17fi-19:l.

1 :~ . M.Di Paol a and G .Falsonc (1993) , lto and St.ratonovich integrals for delta-corrcdatcJ processes, Probabilistic Engrg. Mcch ., Voi.S, 197-208.

11 . M.G rigoriu ( 1996), Response of dynami c systems to Poisson white noise, .J. Sound and Vibra tion, Vol. 195, No. 3, 37.1-389.

15. IU wa nkie wicz and S.R. K.Niclscn ( 1 997), Vibration Theory, Vol. 1, Advanced Met. hods in St.ochastic Dynamics of Non -linea r Systems , University of Aa.lborg.

16. ILL .S tratonovich ( 1963), Topics ir1 the Theory of Random Noise, Vol. I, New York:

Cordon and Breach.

11

17 . S.R.K.Nielsen and R.Iwankiewicz (1997), Dynamic systems driven by non-Poissor;ian im­pulses: Markov vector approach, to be presented a.t ICOSSAR'97, Nov. 21-28, 1997, Kyoto, Japan.

18. A . Renger (1979), Equation for probability density of vibratory systems subjected to continuous and discrete stochastic excitation , Zeitschrift fur Angewandtc Ma.t.lJcrnc-tLik und Mechanik, Vol. 59, 1-13 , (in German).

19 . .J.Q. Sun and C.S. Hsu (1988), First-Passage Time Probability of Non-Linear Stocha.stic Systems by Generalized Cell Mapping Method , .J. Sound and Vibration, Vol. 124, No.

2, 233-248.

12

3.5.-------------~------------.

3

2.5

2:: 1.5

0.5

f.r ~ -~1 ,, 'I I

o~------~=---~--=-------~ -5 0 yjay 5

a) Displacement response, t =To.

2~------------~------------.

1.5

0.5

.. .. "( ;t. ~

OL-------~~--~--~------~

-5 0 yjay

c) Displacement response, t = 2T0 .

0.8

-:;:- 0.6 ;::;;

2:: 0.4

0.2

5

OL-----~~----~----~------~

-5 0 yjay 5

e) Displacement response, t = 10T0 .

2.5.-------------~------------~

2

1.5

0.5

OL-----~~~--~--~~------~

-5 0 yjay 5

b) Velocity response, t = To.

1.4~------------~------------~

1.2

I

q ,.-.. -,...) 0.8

I .. ,,, '" .. · ;::»

I

< 0.6 I

0.4

0.2

OL-----~~----~----~~--~

0 yjay

d) Velocity response, t = 2To.

-5 5

O.Br-------------~-----------,

0.7

0.6

-:;:' 0.5

·' 0.4 ~

0.3

0.2

0.1

o~~-=~------~------~--~

-5 0 yjay 5

f) Velocity response, t = lOTo.

Fig . 1. Transient marginal probability densities jy(y, t) of the displacement response and JJ'(iJ, t) of the velocity response of a Duffing oscillator to a Poisson impulse process with mean rate v = O.lw0 .

14 14

12 12 .i f. ;I

10 J; · I 10 I·

8 . I ,......_ I, .,_,

I ·;;::, i '-./

6 < I

4 I

8

;;::, 6 '--"

2::: 4

I ' 2 2 I

0 0 -0.5 0 yjCYy 0.5 -1 -0.5 yjCYy 0 0.5

a) Displacement response, t = To. b) Velocity response, t = T0 .

14r-------~------~~------.

12 12

10 10 1: -H

. .-.. 8 .-.. 8 jt ...., ;;::,

2::: 6

·;;::, i I '-./

< 6 . . J I

4 4 I i.

I I

2 2

I 0~----~---d~~~--------~

-0.5 0 0

0.5 -1 -0.5 yjCYy 0 0.5

c) Displacement response, t = 2To. d) Velocity response, t = 2To.

14.-------------~-----------. 14.-----------------~------~

12 12

10 10

,....... 8 8 ...., ;:;; · ;;::,

2::: 6 '-./ 6 < 4 4

2 2

0 ---·- 0 -0.5 0 yjCYy 0.5 -1 -0.5 iJ/O"y 0 0.5

e) Displacement response, t = 1070. f) Velocity response, t = 10T0 .

Fig. 2. Transient marginal probability densities fy(y, t) of the displacement response and fy(iJ, t) of the velocity response of a Duffing oscillator to a Poisson impulse process with mean rate v = O.Olwo.

STRUCTURAL RELIABILITY THEORY SERIES

PAPER NO. 148: H. U. Koyli.ioglu, S. R. K. Nielsen & A. ~ - <;;:akmak: Hyste1·etic MDOF Mod el to Q1w.ntijy Damage for TC Shear Fra.mes subjec t to Ea.rthquak es. ISSN 1395-7953 R.9601.

PAPER NO. 149: P . S. Skj a:-rbcek, S. R. K . Nielsen & A. ~- (~akmak: Da.mnge Loca­tion of Severely Damag ed RC-StructuTes based on MerLS7t.red Eigenpe1·iods jTOm a. Single Response. ISSN 0902-7513 R9518.

PAPER NO . 150: S. R. K. Nielsen & H. U. Koyliioglu: Path Integration a.zlplied to Struct-uml Systems vn:th Uncertain Propert?:es. ISSN 1395-7953 R9602.

PAPER NO . 151: H. U. Koyli.ioglu & S. R. K. Niels n: Syst em DynrLmics and Modified Cwm1,lant Neglect Closure Sch emes . ISSN 1395-7953 R9603 .

PAPER NO. 152: R. C. Mic<J.letti, A . ~ - <;;:akmak, S. R. K. Nielsen, H. U. Koyli.ioglu: Ap­pToximnte Ana.lyt1:cal S ol11.tion for the 2nd- Order moments of n SD 0 F H:lJsteretic Oscilla ­tor with Low Yi eld Levels Excited by Stationary Ga.v.ss1:o.n Whit e Nois e. ISSN 1395-7953 R9715.

PAPER NO. 153: R. C. Micaletti, A.~ · <;;:a.kmak, S. R. K. Nielsen & H. U. Koyllioglu: A Sol·ution Met.hoa joT Linea.T and Geometrically Nonlinea.r MDOF Syst.ems with Random PropeTti es subject to Random Excitation. ISSN 1395-7953 R.0G32.

PAPER NO. 154: J. D . S0rensen, M. H. Faber, I. B. Kroou: Optimal Relio.bility-Bn.s ed Planning of Experiments for POD Curve.s. ISSN 1395-7953 R9542.

I

PAPER NO. 155: J. D. S0rf' nsen , S. Engelun l: Stoclwstic Finite Elements 1:n Relinbility-Bas erl Strv.cturo.l Optimization. ISSN 1395-7953 R9543 .

PAPER NO . 156: C. Pedersen , P. Thoft-Christensen : Gnidelines for Interactiv e Relialn:lity­Ba.s ed Stntctural Optimization nsing Q7J.asi-Newton Algorithms. ISSN 1395-7953 R9615.

PAPER NO. 157: P . Thoft-Christ HScn, F. M .. Jensf'n, C. n. Midclleton, A. Bluclonorf': Assessment of the Relia.bih:ty of Concret e Slab Bridge.~. ISSN 1395-7953 R9616.

PAPER NO . 158: P. Thoft-Christensen: Re- A.ssessment of Concrete Bridges. ISSN 1305-7953 R9605.

PAPER NO. 159: H. l. Hansen P. Thoft-Christensen: W1:nrl Tnnnel Testing of Active Contr-ol System for Bridg es . ISSN 1395-7953 R9662.

PAPER NO 160: C. Peelers n : Interactive Relia.bility-Based Optimization of Struci:uml Syst ems. Ph.D.-Tbesis. ISSN 1395-7953 R9638.

PAPER NO . 161: S. Engelund, J. D. S!llrensen: Stocha .. st?:c Mod els for- Chloride -initinted Corrosion in Reinforced Concrete.ISSN 1395-7953 R9608.

PAPER NO . 162: P. Thoft-ChrisLf'Hseu, A. S. Nuw11k: Principles of Bridge Reliability - Application to Design a.nd Assessment Codes. ISS N 1395-7953 R9751.

PAPER NO. 163: P. ThofL-Christenseu, F.M. J ensen, C. Middleton, A. Blackmore: Re'l1ised Rules joT ConcTete Bridges . ISSN 1.395-7953 R9752.

STR U CT URAL RELIA BILITY THE ORY SE R IES

PAPER NO . 164: P . T hoft -Christens -n : Bridge Managem ent Systems. Presfnt and Hlt·ure. ISSN 1395-7953 R9711 .

PAPER. NO . 165: P . H. Kirkega.ard , F . M. J ensen P . Thoft -CbrisLensen : Modellin-ig of Sv.rface Ships using A rtifi cial N e1tml N etwork.s. ISSN 1593-7953 R9625 .

PAPER NO . 166: S. R. K. Nielsen , S. Kr nk: Stochastic Response of Energy Balnncecl M oriel for Vo rtex-Inclttced Vibmtion. ISS N 1395-7953 R9710 .

PAPER NO. 167: S.R.K. Nielsen , R. Iwanki~wi c:z : Dyna.m1:c .s yst ems Driven by No n­Pois sonia:n Impnl.ses: Ma.rko v Vector Approach. ISSN 1395-7953 R9705.

PAPER NO . 168: P. Thoft-Christ nsen : Lifetime Relia.bildy Assessm ent of Concrete Slnb Bridg es. ISSN 1395-7953 R 9717.

PAPER NO . 169: P. H. Kirkega.ard , S. R. K . Nielsen , I. Enevoldsen : Hea·uy Vehicl e .~ on Minor Highway Bridges - A Lit.eratttre Review. ISSN 1395-7953 R9719.

PAPER NO. 170: S.R..K. Nielsen , P.H. Kirkega.ard , I. Enevoldsen : Heavy Vehicles on Minor Highway Bridg e .~ - Stochastic M orlelling of Surface fTregtdnriti es. ISSN 1395-7953 R9720.

PAPER NO. 171: P . H. Kirkegaarcl , S. R . K . Nielsf'u . I. Enevoldscn: Heavy Vehicles on Minor HighwnJJ Bridges - Dynnmic Modelb:ng of Vehicles and Bridges. ISS N 1395-7953 R9721.

PAPER. NO. 172: P. H. Kirkega.anl, S. R.. K. Nielsen , I. Enevoldsen : Heavy Vehicles on Minor Highwny Bridges - Calcu.la.tion of Dynamic Impa ct Factors jTOm Selected CT'ossing Scenarios. ISSN 1395-7953 R9722.

PAPER NO. 173: P. H. Kirkegaarcl , S. n. K. Nielsen , I. Enevoldsen : DynrLmic Vehicle Impa ct for Safety Assessment of BT'irlg e.s . ISSN 1395-7953 R9810.

PAPER NO. 175: C. Frier , J .D. S11jrensen : St.ochastic P T'operties of Plasticity Based Constitv.tive Law joT' Concrete. ISSN 1395-7953 R9727 .

PAPETI. NO. 176: R. Iwa.nkiewicz, S.R .K. Nielsen: A nalytica.l vs Simulation Solution Techniques for Pnlse problems in Nor~;-Lin car Stocha.stic Dynamics . ISSN 1395-7953 R9760.

PAPER NO. 177: P. Thoft- Christenscn : Review of Industria.[ AzJplications of Strv.ct·uml Relia.bility Th eor-y. ISSN 1395-7953 R9750.

PAPER NO . 178: P. Thoft-Chri stPnsen , C . R . Middleton : Reliab ility A sse .ssm ent of Concrfi e Bridg es . ISSN 1395-7953 TI0755.

PAPER NO. 179: C. R. Middleton, P . Thoft.-Chri stem; JJ : A ssessm ent of the Reliab ility of Con r:Tet.e Bn:dg es. ISSN 1305-7953 n 9756.

PAPER NO . 180: P. Thoft-Christenseu: Reliability Ba.8erl Opt?:miza.tion of Fire Protec ­tion. ISSN 1395-7053 R9757.

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