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Aalborg Universitet
Analytical vs. Simulation Solution Techniques for Pulse Problems in Non-linear
Stochastic Dynamics
Iwankiewicz, R.; Nielsen, Søren R. K.
Publication date:
1997
Document Version
Publisher's PDF, also known as Version of record
Link to publication from Aalborg University
Citation for published version (APA):
Iwankiewicz, R., & Nielsen, S. R. K. (1997). Analytical vs. Simulation Solution Techniques for Pulse Problems in
Non-linear Stochastic Dynamics. Aalborg: Dept. of Building Technology and Structural Engineering. (Structural
Reliability Theory; No. 176, Vol. R9760).
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INSTITUTTET FOR BYGNINGSTEKNIK
DEPT. OF BUILDING TECHNOLOGY
AALB ORG UNIVERSITET
•
AAU
AND
•
STRUCTURAL ENGINEERING
AALBORG
•
DANMARK
STRUCTURAL RELIABILITY THEORY
PAPER NO. 176
R. IWANKIEWICZ , S.R.K. NIELSEN
ANALYTICAL VS. SIMULATION SOLUT ION TE C HNIQ U ES FOR P U LSE
PROBLEIVIS IN NON-LINEAR STO C HAS T 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 stribute 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
AALBORG UNIVERSITET
•
AAU
AND
•
STRUCTURAL ENGINEERING
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. Nielse n **
* 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 tec hniques, 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 probl ems 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 techniqu es 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 earization 3 , the improved perturbation technique to solve the generalized Fokker-Pianck equation\ techniques based on eq uations
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 dyn amical 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 ,
wh ich 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 [t 0 , 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 ) = z 0 ,
(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 cond itions , 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
~l
.t
ui
8<I>(l, Z(l)) .. (Z(· ) )) d
+~
L
c,
t , t .t
.
8 Z;
r.
•=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 in crements 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 Z 1 (t) augmented by the auxiliary variables Z 2 (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 ) = z 0
(1)
= [Z1(l)]
(Z(t) t) = [ci(Zl(/.),1)1
'
0
'
Zz(l) ' c
d(Z() l P(t)) = [d1 (Z!(t), t)g(Z 2 (t))P(t)
t ',,
,
d2(Z2(t), t)
'
Z(L)
(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(Z 2 (l)) = p(N(l)) is a .
known !"unction of the auxiliary variables.
Although the structural state vector Z 1 (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 as 11 •15
Ji.; (l) = E [c;(Z(l), t)]
~~i.i(t)
=
+ v(l)E [d;(Z(l), t, P)]
(6)
2 { E [z? (c?(Z (l), l) + 11(L)d1(Z(t), t, P))]}.
+v(t)E [d;(Z(I,), t, P)dj(Z(l), t. , P)],
0
(7)
+ ll(t)dk(Z(L), t , P))] L
(Z?d;(Z(l), t, P)dk(Z(t), l, P)]} s
3 { E [Z;0 zy (c~(Z 0 (t), t)
+3v(t) { E
+v(t)!E [d;(Z(l), l, P)d1(Z(I.), t, P)dk(Z(l), l, P)],
K.;jkt(l) =
(8)
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
L
+411(l) { E [Z?dj(Z(t), t , P)dk(Z(t) , t, P)dt(Z(t), l , P)]
+v(t)E [d;(Z(t), i, P)di (Z(t), t , P)dk(Z(t), t, P)dt(Z(t), L, P)],
(9)
where Zf(t) = Z;(t)- f.L;(t) and c~(Z 0 (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(t 0 ) = z 0 . If in the time interval [to, t[ no impulse occurred , the system has
performed the deterministic drift motion from the initial state z 0 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, t 0 ) denotes the drift from the initial state z 0 at l = L0 obtained from (1) for
= 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
dN(t)
p;(t)
= Po(Lito)c;(tlzo, to)+
4
(1- Po(tlto))p~(t),
(] 2)
( 13)
where (12) has been used. Further, the arguments of P0 (tito) and e;(tlzo, l 0 ) 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), an d 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 , t 0 ) = 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
imptdses 5 . 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]
fz(z, to) = fo(z)
fz(z, t)
, V z ESt
, V z E Sto
( 15)
= 0
where 5'1 is the solution set at the timet, bounded by the surface oS1 = 8S1(o) U ()S1( 1J U oS} 2 l.
DSP) is the non-accessible (natural) part of the boundary, whereas the accessible boundary is
1
0
made up of the exit part 8S} l and the entrance part oSf l. As indicated, absorbt.ion boundary
0
conditions must be specified on 8S} 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
p
1
where
_ d,
J- et
(I + od(a(z,p,L),l))
f)y'l"
-
Jz(z, t) J.rp(p)dp,
,
( 16)
(17)
and a = a( z, p, i) is the inverse transformation of
z=a+d(a,p,t)
( 18)
8 Y~
8
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
aJz(z,
t)
T [ fz(z, t) j =
+ Kz,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.
(20)
p
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
0
) U 88 ('2), whereas
shape functions must fulfil the boundary condition N;(z) = 0, z E
1
the weighting functions V;(z) = 0, z E ast(I}_ In contrast, for the problem (19) the Bhap<!
2
functions fulfil the boundary condition N;(z) = 0, z E 8St(I}UaS1( ), and the weighting functions
a8z
V;(z) = 0, z E 8S1( ). Further the shape and weighting functions rnu~Jt be suff-iciently smooth
that K:~.,t[ ... ] and KL[ ... ] may become adjoint operators when integrated over 8 1• 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.
2
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 lz 0 , l;) is the transition probability density function of the state vector from the
state Z(lu) = z 0 at the time i; to the state Z(l) = z at the Lime i;+ 1 and 8 1, 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
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)
where
(23)
and
(24)
is the transition probability density conditional on no impulse arrival and e( i;+ 1 Jz 0 , 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 Jz 0 , 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 Jz 0 , 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) _ ~
7fj
- L....t
k=1
Q (i) . _ 1
M
jk1fk' J- , ... , '
(26)
(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 tim es ), 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 th e 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. Th e stationary marginal displacement and velocity response
probability densities 10 •12 and reliability function 10 have been evaluated. The results obt ai ned
confirrn ec.l the congesture that t he 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 tim es, 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 sampling 12 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
Z(t)
=
Y(t) ]
_
[ Y(l) 'c (Z(t), t.)
=
[
Y(t)
]
[ o
-2(w0 Y(l)- wJY(l)- c:w5Y:~(l) 'd = J>(l) .
]
.
(29)
where ( is the damping ratio, w 0 is the natural freq uency 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: w 0 = 1s-', ( = 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[P 2 ] 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:
= 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'T 0, 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
1
between 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.
i
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, 22910
4. C .Q. Cai and Y .K . Lin (1992), Response distribution of non -linear systems excited by
non-G a ussi a n impulsive noise, lnL. .J. Non-linear Mech an 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-distribute d pulse~ tra ins , Probabilistic Engrg. Mcch. , Vol.7 , No. 3, 1:35- 148.
7. It. lwankiewicz and S.R.K . Nielsen (1991) , Dyn a mic respon se of non-linear systems t.o
ren e wal-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. Koy liioglu , 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, Ath ens, 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
Vibr a tion, Vol. 195, No. 3, 37.1-389.
15. IU wa nki e wicz and S.R. K.Niclscn ( 1997), 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 R a ndom Noise, Vol.
Cordon and Breach.
11
I, New York:
17 . S.R.K.Nielsen and R.Iwankiewicz (1997), Dy namic 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
2.5.-------------~------------~
3.5.-------------~------------.
3
2
2.5
f.r
1.5
~
,,
-~1
2::
1.5
'I
I
0.5
0.5
a)
o~------~=---~--=-------~
OL-----~~~--~--~~------~
-5
-5
0
5
yjay
0
= To.
b) Velocity response, t
Displacement response, t =To.
5
yjay
1.4~------------~------------~
2~------------~------------.
1.2
1.5
I
q..
I,,,
..
..
,.-..
"(
;t.
-,. .)
0.8
<
0.6
·;::»
~
'"..
I
I
0.4
0.5
0.2
OL-------~~--~--~------~
-5
c)
0
Displacement response, t
yjay
OL-----~~----~----~~--~
5
-5
= 2T0 .
Velocity response, t
d)
0
yjay
5
= 2To.
O.Br-------------~-----------,
0.7
0.8
0.6
-:;:' 0.5
-:;:- 0.6
;::;;
2::
·'
~
0.4
0.4
0.3
0.2
0.2
0.1
o~~-=~------~------~--~
OL-----~~----~----~------~
-5
e)
0
Displacement response, t
yjay
= 10T0 .
5
-5
f)
Velocity response, t
0
yjay
= 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 .
5
14
14
12
12
10
10
,......_
8
.i
f.
;I
J;
·I
I·
.I
I,
I
8
.,_,
·;;::,
;;::,
'--"
2:::
'-./
6
<
i
6
I
4
4
I
2
2
I
I
0
-0.5
0
Displacement response, t
a)
yjCYy
0
-1
0.5
= To.
'
-0.5
0
yjCYy
b) Velocity response, t
0.5
= T0 .
14r-------~------~~------.
.-..
....,
.
12
12
10
10
-1:
8
.-.. 8
jt
·;;::,
.i I.
H
;;::,
2:::
'-./
<
6
2
2
0~----~---d~~~--------~
0
-1
-0.5
0.5
0
Displacement response , t
,.......
....,
= 2To.
i.
I
-0.5
yjCYy
0
0.5
= 2To .
14.-------------~-----------.
14.-----------------~------~
12
12
10
10
8
8
·;;::,
'-./
6
<
6
4
4
2
2
0
-0.5
e)
I
I
I
d) Velocity response, t
;:;;
2:::
J I
4
4
c)
6
Displacement
---·0
yjCYy
response, t = 1070.
0.5
0
-1
-0.5
f) Velocity response, t
iJ/O"y
0
0.5
= 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.
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