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Dynamic response of hysteretic systems to Poisson-distributed pulse trains
Probabilistic Engineering Mechanics 7 (1992) 135-148
Dynamic response of hysteretic systems to Poisson-distributed pulse trains Radoslaw lwankiewicz Institute of Materials Science and Applied Mechanics, Technical University of Wroclaw, Wybrze~e Wyspimlskiego 27, PL-50-370 Wroctaw, Poland
& Soren R. K. Nielsen Institute of Building Technology and Structural Engineering, University of Aalborg, Sohngaardsholmswej 57, DK-9000 Aalborg, Denmark (Received January 1990; accepted August 1991) A single-degree-of-freedom hysteretic system subjected to a specific non-Gaussian random excitation in the form of a Poisson-distributed train of random pulses is considered. The hysteretic behaviour is described by the Bouc-Wen model of smooth hysteresis. The total hysteretic energy dissipation is assumed as a cumulative damage indicator. The state variables of the system together with the damage indicator form in that case a Poisson-driven, non-diffusive Markov vector process. Two equivalent systems are introduced by substituting the original non-analytical, non-algebraic non-linearities by equivalent linear and cubic forms in the pertinent state variables. Equations for mean responses are obtained by direct averaging of the governing equations, whereas the equations for second- and higher-order joint central moments are derived from the equivalent systems with the help of a generalized Ito's differential rule. Appearing in the equations for mean responses and for equivalent coefficients, expectations of non-algebraic functions of state variables are performed with respect to a non-Gaussian joint probability density function assumed in the form of a generalized, bivariate Gram-Charlier expansion. In the case of equivalent cubic non-linearities, the equations for moments form an infinite hierarchy which is truncated with the help of a cumulant-neglect closure technique. Mean values and variances of the response variables are evaluated by the numerical integration of equations for moments for two equivalent systems. For the sake of comparison the excitation process is also substituted by a Gaussian white noise and the usual equivalent linearization technique, combined with the Gaussian closure, is applied. The cases of non-zero-mean as well as zero-mean excitation processes are included. The case of general pulses is dealt with by a suitable augmentation of the state vector. The accuracy of the analytical techniques developed is verified against Monte Carlo simulations.
1 INTRODUCTION
to analytical treatment, has fast progress of the studies and development of the methods been accomplished. Different methods were proposed for the analysis of hysteretic systems under r a n d o m excitations, for example methods based on the F o k k e r - P l a n c k - K o l m o g o r o v equation, such as a direct linearization of this equation for the system with bilinear hysteresis 2 or a generalized Galerkin solution to the F o k k e r - P l a n c k - K o l m o g o r o v equation for the system with smooth hysteresis. 3 Also the stochastic averaging method ~ and a Gaussian closure of
Considerable attention has been given in the literature over the last two decades to the dynamic response of inelastic, hysteretic systems to r a n d o m excitations. In the earliest studies a bilinear hysteresis model was used (e.g. Refs 1, 2). However only since Wen introduced the Bouc model of smooth hysteresis, 3 which lends itself very well
Probabilistic Engineering Mechanics 0266-8920/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 135
136
R. lwankiewicz, S.R.K. Nielsen
moment equations 7 have been used to study the response of hysteretic systems. The equivalent linearization technique was applied for the first time to the analysis of a hysteretic system by Wen, 8 and ever since it has perhaps become the most widely used technique for this kind of problem. It has been used not only for single-degree-of-freedom systems but also for the analysis of hysteretic structures, e.g. frames. 9-~2 Recently the technique of the equivalent polynomial expansion has been developed for hysteretic systems. 13't4 This technique consists of considering an equivalent nonlinear system obtained by substituting the original nonanalytical, non-algebraic non-linear functions of the state variables by the polynomials in these variables. This technique has been shown to improve the results over those obtained from the equivalent linearization technique. A crucial assumption which is made in all the methods mentioned above is that the excitation is characterized by a Gaussian random process (a Gaussian white noise or a filtered white noise). However, in structural dynamics there are important excitation processes for which a Gaussian assumption is not justified. Some of them, e.g. wave-induced forces acting on structures, can be represented as non-linear functions of a Gaussian process.~5 Another class of non-Gaussian excitations may be idealized as a random train of pulses. Such an idealization is especially suitable for traffic loads on a bridge. ~6 However it also finds applications in other problems such as, for example, a random 'down-wash' exciting intermittently the tail of an airplane, ~7 randomly occurring wind gusts associated with eddies, 18't9 or in some cases ground motion acceleration due to earthquakes. 2° Another example is the behaviour of the vehicle travelling over rough ground, where the vehicle structure is subjected to shocks produced by sudden bumps in the ground surface) ~ Finally, the dynamic loading due to wave slamming can also be idealized as a random pulse train. Wave slamming loading has been known for a long time to act upon ship hull structures, and recently it has been observed to be of relevance to off-shore structures, in particular to horizontal members of trussed towers, z2 Inelastic effects are known to occur in these structures. Under the assumption that the pulses are independent, i.e. that they obey a Poisson distribution, a random pulse train is a filtered Poisson process, or a Poisson-driven shot noise. The subject of the present paper is the dynamic response of a single-degree-of-freedom system to Poissondistributed trains of random pulses. The main objectives of the study are to develop and verify the appropriate approximate analytical techniques and to gain an insight into the dynamic behaviour ofhysteretic systems subjected to this particular type of non-Gaussian excitations.
Pulses of two types are taken into account: Dirac delta impulses and general pulses. Two analytical approaches are developed; a version of the equivalent linearization technique and the technique of an equivalent system with cubic non-linearities. The version of the equivalent linearization technique developed herein has two specific properties. The first is that prerequisite equations for mean responses are formulated by direct averaging of the governing stochastic equations, and next the equations for zero-mean responses are substituted by linear ones. Such a procedure is used in the case of asymmetrical non-linearities, ~3 since it allows preservation of the property of a non-zero-mean stationary response to zero-mean excitation, in the case of the equivalent linearization technique. Such a procedure is also used for non-zero-mean problems. 24 In the present case the non-linearity is symmetrical but the similar effect should be predicted because of the asymmetrical distribution of the response probability density. The second property of the version of the equivalent linearization technique used in the present paper is that the exact form of the joint probability density function of the response of the equivalent linear system is not known, hence in order to perform the pertinent expectations of the non-algebraic functions of the response processes, this joint density function is assumed in an approximate form of a truncated bivariate Gram-Charlier series. The expansion coefficients of this series (joint cumulants of the response processes) are calibrated from the moments and hence the equations for moments of order higher than two have to be derived. The required number of the equations for moments is determined by the truncation level of the Gram-Charlier expansion. In the case of the equivalent non-linear system the equations for moments form an infinite hierarchy, and the cumulant-neglect closure technique is used to truncate it. Also in this case the bivariate GramCharlier expansion is assumed to perform the pertinent expectations of non-algebraic functions of response processes. The equations for moments are integrated numerically, which gives transient response moments. The computed time histories of the response mean values and variances are compared with those obtained from the Monte Carlo simulations.
2 STATEMENT OF THE PROBLEM: STOCHASTIC EQUATIONS GOVERNING THE POISSON-DRIVEN MARKOV VECTOR RESPONSE PROCESS Consider a single-degree-of-freedom hysteretic system under a Poisson-distributed train of random pulses. The response is governed by the equation
D y n a m i c Response o f H y s t e r e t i c S y s t e m s to Poisson-distributed Pulse Trains
,~ + 2¢coo,¢ + ~O~o~X+ (1 - ~)~oo~Z N(O
=
(1)
E e i w ( t -- ti) i=I
where w(t - t~) is the pulse shape function, usually defined in the interval t E (t;,t~ + T) and equal to zero outside this interval, T being the pulse duration. The hystereticcomponent Z of the restoring force is given by the Bouc-Wen model) which can be represented in the form 2
=
a:(-
[3l:?lZlZI "-I -
~:?lZl n =
Governing equations can be converted into the following set of first-order differential equations d Nft) d-~X(t) = a(X(t)) + e ~ P~6(t - t;) (6a) i=l
where in the case of Dirac delta impulses
x(t) X(t) X(t) =
D(t)
[
(2)
=
a(X) =
g(.¢, z) h(:?, z )
eIi
(6b)
and in the case of general pulses
-x(t) f((t) z(t)
X(t) =
D(t) U(t)
~, P : ( t -- ti) i=l
is, by virtue of the linear superposition principle, the response of the auxiliary filter to the train of Dirac delta impulses, governed by the equation
_ (J(t)
~'(t)
0 + 2Zl'~00 + D.~U
=
"
-~co0~x- 2~co02- (1 - ~)~0Z
:¢(0
U(t)
,
z(t)
g(:?, z )
In eqn (2) ¢ denotes the damping ratio, COois the natural frequency of the corresponding linear system, a is the fraction of the restoring force made up of the linear elastic component, and a, t, 7 and n are constant parameters which describe the hysteresis loop and which must be calibrated to empirical data. If the excitation is a train of Dirac delta impulses, then w(t - t~) = 6(t - t~). In the case of general pulses let us consider the situation when the pulse shape has the form of, or may be idealized as, the impulse response function s(t - t~) of an auxiliary linear filter with properly chosen eigenfrequency and damping ratio. Indeed, if the filter damping ratio is high, close to 1, then the impulse response function becomes the single pulse with duration equal to half of the filter natural period and with a rapidly decaying with time, insignificant tail. Then the train of general pulses
137
E P~c~(t - t,)
-~co02X - 2~co0~: - (1 - a)co~Z + U
(3)
g(:/', z) h(:?, z)
i=1
where ~ and ;f are the filter eigenfrequency and damping ratio, respectively. Evolution of damage of the system due to the plastic deformations can be described by an extra equation. If the damage is modelled as the hysteretic energy, this equation has the following form /~ = )?Z = h(~', Z) (4) In eqn (1), N ( t ) is the Poisson counting process, which gives the number of impulse arrivals in the time interval [to, t). The expected arrival rate of events (or the expected number of events per unit time) is v(t), i.e. E[N(dt)] = v(0dt + o(dt) (5) where N(dt) is the random number of impulses in the infinitesimal time interval [t, t + dt). Random magnitudes P~ of the impulses are assumed to be mutually independent random variables, which are also independent of the counting process N ( O , and are identically distributed as a random variable P.
a(X) =
0
-o~v- 2z~0 -0 0 0 e
=
(6c)
0 0 _I with the general initial conditions X(t0) = X0.
(6d)
As is seen the case of general pulses is dealt with by the suitable augmentation of the state vector.
R. lwankiewicz, S.R.K. Nielsen
138
Stochastic integral equations are obtained from the governing equations (6a) by the integration over t and substitution of the compound Poisson process by the counting integral (cf. Refs 25, 26)
fying eqn (6a), the following differential rule is valid25'31 dV(X(/), t) = + ~, a,(X(t))
N(O
X P/ -- f:0 fa, tiM(dz, dti)
where M(dz, dr/) is the Poisson random measure, 25'27 which specifies the increment of the compound Poisson process in the infinitesimal time interval IT, z + dz) and denotes the whole sample space of the random variable P. The Poisson random measure has the following properties N(dt)
t i> to (10)
where dX(t) = X(t + dt) - X(t) denotes the infinitesimal increment in X(t) during the time interval [t, t + dO. In what follows, zero initial conditions are assumed. The process X(t) which satisfies eqn (10) is the Poissondriven Markov vector process25 whose joint probability density fx(x, t) is governed by the following integrodifferential partial equation, an analogue of the FokkerPlanck-Kolmogorov equation for diffusion processes (cf. Ref. 28) ~fx(X, t) + ~ ~Xi [fx(X, t)a,(x)] + v(tff'x(X, t) -
where V(X(t), t) must be bounded for all t and (X~. . . . . X,) finite and must be once continuously differentiable with respect to all its arguments. Performing averaging of both sides of eqn (12) yields the equations for moments which have the general form d E[V(X, t)] = E [dV(X, t)] d-t [ ~-t
(9)
where fp(ti) is the probability density of the random variable P. The resulting stochastic integral equations can be represented in an equivalent differential form
- v(t) f, fx(x
~ V(X(t), t) OX/ dt
+ f~, [V(X(t) + tle, t) - VX(t), t)]M(dt, dr/) (12)
(8)
E[n(dt, dr/)] = v(t)fe(ti)dtdti
dX(t) = a(X(t))dt + e f tiM(dt, dr/),
Ot
(7)
i~l
f~ M(dt, dti) -
aV(X(t), t) dt
tie, t)fp(ti)dti = 0
(11)
Direct solution to this equation is, unfortunately, unknown. Instead the application of a Fourier transform to eqn (11) yields the first-order partial differential equations for the characteristic function. 2s-3° In the case of a linear system, the equation for the characteristic function can be solved to give an explicit expression for the response characteristic function. 29'3° An alternative approach is to derive the equations for moments of the Poisson-driven Markov vector process x(t).
3 EQUATIONS FOR RESPONSE MOMENTS 3.1 General form of the equations for moments For the Poisson-driven Markov vector process X(t) satis-
+ ~.E . [ai(X)c3V(X'~3ji~/t)] + v(t) f, EtV(X + tie, t) - V(X, t)lfp(ti)dti (13) Equations for the response nth order moments may be obtained from eqn (13) by substituting V(X, t ) = X~,X ~ . . . X~,,where ir = 1. . . . . 4 or 1 , . . . , 6 for each
i,. Due to the non-linearity of the drift terms a3(X) = g()/', Z) and a4(X) = h(8, Z), the typical problem for non-linear systems arises, i.e. to close the set of equations for moments by evaluating the expectations of non-linear functions of the state variables. Since the function g(.Y, Z) is non-algebraic the pertinent expectations only can be performed as mult-fold integrals with respect to the joint probability density functions of the relevant processes. Such operations are often performed in non-linear problems. It is obvious that the quality (or accuracy) of the results obtained depends on how close the assumed, or tentative, form of this probability density function is to the true one. Often the tentative joint probability density function which must reveal the non-Gaussian behaviour is assumed in the form of the truncated GramCharlier expansion, e.g. Refs 32-34. A similar approach was used by Minai & Suzuki 35'36 who considered the response of an elasto-plastic structure to a Gaussian excitation and used a modified Gram-Charlier form of the tentative probability density function. The modification consisted of including a couple of Dirac delta spikes, which correspond to the fact that the sample space of the hysteretic component is in that case the finite interval between the two yield points, which are related with finite probabilities. In the present case the joint multidimensional probability density functions of all the four or six state variables X,. would be required to perform the expectations. Because the evaluation of integrals with respect to such density functions may be cumbersome, and for a largedimensional state vector it may even become prohibitive,
Dynamic Response of Hysteretic Systems to Poisson-distributed Pulse Trains another approach is developed. Instead of dealing with the original system, another physical system is considered in which the non-linear function is substituted by the equivalent polynomial form (in particular a linear form and the cubic form) in the pertinent state variables. It should be emphasized that since the original non-linear function g(,~', Z) is non-analytical, the polynomial form cannot be obtained as a truncated Taylor expansion (which does not exist). The coefficients of the equivalent system are determined by the least squares criterion and updated in each time point of the assumed discrete set (in each numerical integration step). Then it can be proven ~3 that if the polynomial expansion of order M in the centralized state variables is considered, the mean values and joint central moments up to and including the order M + 1 obtained from the equivalent polynomial system are identical with those obtained from the original system. The main advantage is that only joint bivariate probability density function of the state variables ,~" and Z is then required.
3.2 Equivalent Hnearization technique
139
(10), as Ela,(X)] + v(t)E[P]e,
/~i =
(14)
The specific form of the equations for means in the case of impulses is (cf. eqn (6b)) ti,
=
/~2
/J2 =
-otc°2PJ - 2(c°0;h - (1 - ~t)t02#3 + v(t)E[P]
/i~
E[g(X~, X~)]
=
/~, = E[h(X2 ' X3)]
(15a)
and in the case of general pulses (of. eqn (6c)) #l
=
#2
g2 =
-atO2o#) - 2(O9o#: - (1 - 0t)cog#3 + /~5
#3 = E[g(X,, X3)]
(15b)
#4 = E[h(X2, X3)] Us
=
#6
P6 = - D ~ m - 2zD.op6 + v(t)E[P] where
It may be shown that the probability distribution of the response of a linear system to a Poisson-distributed train of non-zero-mean impulses (E[P] # 0) is not symmetric regardless of the type of probability distribution of random magnitudes P of impulses. This is so because, as is well known, the third-order cumulant (and hence the skewness coefficient) of the response depends on the third-order ordinary moment of the random variable p.21,37 In the case of zero-mean impulses the response probability distribution is not symmetric if the probability density function of the random variable P is not symmetric, i.e. iffp(rl) ¢ fp(-r/). The probability distribution of the response of a system with symmetrical non-linearity, e.g. of a hysteretic system specified by eqn (2), should be predicted, in the cases described above, to be not symmetrical. This means that the steady-state response of such a system to zero-mean random impulses (i.e. ifE[P] = 0) has to be predicted to be non-zero-mean if the probability density function of the random variable P is not symmetrical. This is similar to the non-zeromean response problem in the case of asymmetrical nonlinearities. However, if the governing equations were substituted by equivalent linear equations, the response of the equivalent system to the zero-mean excitation (zero-mean impulses) would be zero-mean. Hence the formulation of the equivalent linearization used in the present paper is similar to the one applied to the problems with asymmetrical non-linearities 23 or to non-zero-mean problems. 24 First the equations for mean responses are formulated and next the equations for zero-mean responses (centralized processes) are substituted by equivalent linear equations. Equations for mean responses /~,. = E[X~] are obtained, by taking the expectations of both sides of eqn
Elg(X~, X , ) ]
=
a~
-
,aE,
-
~,E,
(15c)
E, = E[IX, IX~IX, I"-']
(lSd)
E2 =
(lSe)
E[X21X3["]
E[h(X2, X3)]
~---
E[X2X3]
(15f)
Equations for zero-mean responses =
x,-u,
obtained by subtracting (14) from (10), become dY =
{a°(Y + /~) - ev(t)E[P]}dt
+ e ~ qM(dt, dr/)
(16a)
where a°(Y + /~) = a ( Y + p ) -
E[a(Y + /0]
(16b)
where/A denotes the column matrix of mean values #i. Equivalent linearization consists of considering a linear system in which a°(Y + p) is assumed as the following linear form a°~q(Y + /~) = ~ B~Ym
(17)
/rl
In order to evaluate the equivalent coefficients B~, of the linear system it is required that the mean square of the error ei(Y + p) =
a,°.fY + P) - a,°..~q(Y + P)
(18)
be minimized. The minimization conditions are E[sT(Y + p)8(Y + /~)1 = 0 where ( )x denotes transpose of a matrix.
(19)
140
R. Iwankiewicz, S.R.K. Nielsen
Consequently the equivalent coefficients satisfy the following set of linear algebraic equations U,,~B~ = E[Yja,.°(Y + p)]
I +
Z i+1=3
+ ~.. i + ] = 3
where #~j = E[YmYj]
(20b)
From eqn (16b) it follows that E[Y~a°(Y + n)l = E[Y~a,(Y + P)I
(21a)
where a3(Y + ~)
=
g ( Y + p)
-- ?(Y~ + #2)1Y3 + /z31~
(21b)
= h(Y + p) = (Y2 + #2)(Ya + U3) (22)
For example in the case of eqn (16a) the term a°(¥ + p) = g(Y + p ) -
E[g(Y + p ) ]
(23)
(24)
and the equivalent coefficients b~ and b2 are then obtained from the following linear algebraic equation
b2
kE[g3g( Y + P)]
In eqn (25) a different notation for moments is used than in eqn (20b), i.e. /~u = E[Y~Y{]
(26)
In view of the non-algebraic form of the non-linear function g(Y + p), the expectations E[g(¥ + p)] in eqn (15) and E[Yjg(Y + /t)] in eqn (25) can only be performed as the pertinent integrals with respect to the joint probability densityfr~ r~( Y2, Y3) of the response processes I12 and Y3. Such a density function is usually assumed as the probability density of the response of a linearized system. In the case of a Gaussian excitation the response of a linearized system is also a Gaussian-distributed process hence fY:Y~(Y2, Y3) is assumed as a bivariate Gaussian density function. However the exact form of the joint probability density of the response processes I"2 and Y3 of a linearized system to Poisson-distributed impulses is not known. Therefore an approximate joint probability density function is assumed in the form of a truncated bivariate Gram-Charlier expansion (cf. for example Refs 37-39)
fr, r3(Y2, Y3) =
1 271:0-20.3 ~/1 -- p2
x e x p [ - - ~ 2 --~ 22~2~3P 1 2--( 1p2)+
i ! j ! k ! l ! H~+k,j+~(~2, ~3, P) + . . .
where ~2 =
Y2/0"2,
~3 =
Y3/0"3
0-2 =
~20,
0.3 =
V/~2,
//H p
=
(28)
0.2if3
2o =
0. 0-i
where the joint moments #ij are defined by eqn (26), x o is the joint i + j t h order cumulant of the processes Y:, Y3 and H0(~2, ~3, P) is the family of bivariate, standardized Hermite polynomials of order (i, j ) which satisfy the relationships 1. 2 H0(¢2, ~3, P) exp [-2(~2
-
-
2~2¢3P + ~)/(1 -- p2)]
0i+2
is substituted by the linear form a°~q(Y + p) = b, Y2 + b2 Y3
k+l=3
tcq
-- fllY2 + #2[(Y3 + u3)lr3 + #31"-'
a,(¥ +
•
(27)
a(r2 + la2)
=
¢,,p) •
(20a)
m
]~11~02
x
= (-1) i"
i
i
0~20~3
xexp [ - ½ ( ~ - 2~2~3p + ~)/(1 - p2)] (29) Expansion (27) can be truncated at any level required depending on the number of terms which are to be included in the approximate expression. In the classical equivalent linearization technique the equations for second-order moments are then derived. However in the present approach the cumulants which appear in the Gram-Charlier expansion have to be calibrated from the moments, hence equations for moments up to the nth order have to be derived if the terms including the cumulants up to the nth order are to be retained in the expansion (27). The Gram-Charlier expansion will be truncated at fourth-order cumulants, i.e. at N = 4. The equations for moments are derived from the linearized system obtained by combining eqns (16a), (16b) and (17), with the help of a general form of eqn (13). The necessary equations for the moments up to the fourth order become
IJU = B~mP,,j + BjmPm, + v(t)E[p2]eiej
ftqk = Bim#myk + ByrnPmik + Bkm#mq + v(t)E[p3]eiejek
(30a) (30b)
#okl = B~,,#mjk~ + Bjml~m~k~+ BkmlZmVt + B~,l#.Ok + v(OE[p2](e, ej#kl + eieklgit + e~ed~jk + ejeklgit + ejetlgik + ekedgij) + v(OE[P4]eiejeket
(30c)
where the following notation of moments is assumed
].lil...in
=
E [ Y q . . . Yi.]
(31)
and the summation convention is applied over the dummy indices.
Dynamic Response of Hysteretic Systems to Poisson-distributed Pulse Trains When the expectations E[g(Y2 + #2, I"3 + #3)], E[Y2g(Y2 + //2, Y3 + #3)1, E[Y3g(Y2 + #2, Y3 + #3)1 are performed with the help of a Gram-Charlier joint density function (27) they become expressed in terms of the moments and cumulants of the linearized system response. The explicit expressions for these expectations evaluated with the help of the Gram-Charlier expansion confined to the one-fold sum in (27) and truncated at the order N = 4 and for n = 1 are given in the Appendix. The equations for moments (30a)-(30c) form a closed set, but they are coupled with eqns (14) for the mean responses, because the higher-order moments enter these equations through the expectations E[g(Y + /4)] and E[h(Y + /0]. Due to the fact that the equivalent coefficients B~,, as given by eqns (20a) or (25) depend on the moments, the equations for moments become non-linear. It is interesting to investigate the steady-state solution for the mean responses governed by eqn (15a). The derivatives on the left-hand side of eqn (15a) are set equal to zero, thus 0 = //2
0 = -- Ctt.O02//l-- 2(tO0//2 0 = a//2 - t e l - ?E2
(32)
Hence //~ = 0 and if the impulses are zero-mean, i.e. E[P] = 0, then 0 =
-~tog//~ - (1 - ~t)to02//3
(33a)
0 =
- tEl
(33b)
-
)~E2
It may be shown that//3 = 0 does not satisfy eqn (33b) if it is assumed that the response probability distribution is not symmetric, i.e. if all third-order joint central moments do not vanish. If//3 = 0 is substituted into eqn (33b) its right-hand side becomes (see Appendix) _ fiE l
_
yE2
=
fl (//222//23
0"2 ~
+ ~
~, 3[/22
'7
(//333//23 \ 3//33
the non-linearities are not severe. The idea of this technique is to consider an equivalent non-linear system, in which a°(V+t0
=
//223
)
//233
)
(34)
In general expression (34) is not equal to zero. Hence the solution must be for some //3 # 0, and from (33a) it follows that//t ~ 0 as well. Hence the stationary value of the mean displacement response and of the mean hysteretic restoring force are non-zero even if the excitation is zero-mean (zero-mean impulses). Moreover from (33a) it follows that the stationary values of E[X] = #j and E[Z] = //3 have opposite signs. 3.3. Equivalent cubic expansion technique
a ~ ( Y + /~) =
A , + B~Ym + C,~Y,,Y~
+ D ~ , Yr. Y, Yp (35) The equivalent coefficients are evaluated from the minimization conditions of the mean square of the error 5,(v + p) =
¢ ( Y + /~) - a°~q(Y + P)
d ~A~ E[ds]
0,
i.e. =
O E[srs] = 0C~
0 0B~ E[sXs] =
0
0 0,
(36)
aD~p
E[ds]
= 0
The following linear algebraic equations for the equivalent coefficients are obtained from the minimization conditions (36) (cf. Refs 13, 14). B~,//mj = C~#~.jk
=
- C~#m. - Dim.p#..v E[Yja °] - C ~ / / . w -- D~//m,~j E[YjYka~°] - A,#jk -- B~.#m~k- Dimnp//~mpjk
D,.~pp.~pjkt =
E[Yj Yk Yt a°]
-
Ai//jkl
--
Bira//mjk,
- C~//mm
(37)
Again the expectations E[Yja°], E[YjY~a°] and E[Yj Yk Yla °] may only be performed with the help of an approximate joint probability density function. This function will be assumed to be in the form of a GramCharlier expansion (27) truncated at fourth-order cumulants and accordingly the equations for moments up to the fourth order will be derived (cf. Refs 13, 14). Then the unavailable fifth- and sixth-order moments which appear in eqns (37) have to be evaluated with the help of a Gram-Charlier expansion. Strictly speaking all the expectations, including the moments, which appear in eqns (37) have to be evaluated with respect to the same probability measure - - the bivariate joint density function assumed here as the truncated Gram-Charlier expansion. However all the moments of the order which is not beyond the truncation level, evaluated with the help of a Gram-Charlier expansion, are exactly the same as those obtained from the equations for moments. In the case of the third equation of (15a) or (15b) the term a~(Y + /0 given by eqn (23) is substituted by the following cubic form ~,~q(Y + /4) =
The equivalent polynomial expansion technique has been applied with some success to the problem of hysteretic systems subject to a Gaussian white noise excitation, if
a(¥+~)-E[a(V+t0]
is substituted by the following cubic form in the state variables y, m4
Ai =
-- (1 -- ~t)to~//3 + v(t)E[P]
141
al + blY2 + b2Ya + cly2 + c2Y2Y3 + c3Y 2 + d , Y 3 + d2y2y3 + d3Y2Y~ + d4Y 3
(38)
R. Iwankiewicz, S.R.K. Nielsen
142
The expansion coefficients are then evaluated from the following set of linear algebraic equations
18 16
-
"--
14
I
0
0
#20
~'~11 #02
#30
#21
#l 2
#o3" - a, ]
#20
#ll
#30
#21
#40
#31
#22
#t3
hi [
#o3 #3~ #22 #~3 /q4
b2 I
6
#12
#0~ #:~ #n #4o
12
8
#3~
#22
#50
#4~
#32
#23
C,I
4 2
#22
#13
#41
#32
#23
#~4
c~ I
0
#04
#32
/'/23 /['/14 #o5
c~ I
#60
#51
#42
#33
dl I
#42
#33
#2~
32 I
/,/24
#~
Symmetry
!
d~ I
t/TO
Fig. 1. Mean displacement response for non-zero-mean impulses. obtained by combining eqns (16a), (16b) and (35). Making use of the general formula (13) leads to the following equations for moments up to the fourth order
~i: = Bin#m: + gjm#mi + C~n#rnnj + Cjmn#mni
0
+ D~,~p#m~pj + Dj,~p#m~.~ + v(t)E[P2]e~ej
E[Y2g]
(42)
E[Y3g]
/iij,~ = A.ujk + Aj#ik + Ak#u + B.,,#,,,jk + Bj,.,u.,~k
E[y2g °]
+ Bk,.#,.u + C~.,n#,,~jk + Cj,,,.#,,~k
E[Y2 Y3g °]
+ C,,,,,,,Um,,~ + D~.p#,,~pjk + Dj,,,~p#,.,,p~k
E[ Y32g ° ]
+ Dk.~npg.~pU+ v(t)E[p3]eiejek
E[y3g °]
I~jk~ = A~#jkt + Ad4kt + Ak#ut + At#u~
E[Y2 r3g °]
q- Bim#mjkI + Bjm#mikt -t- Bk.,#mul
E[Y: rE gO]
+ BtmgmOk + Ci.,~#.~jkt + Cj'~#.~k~
E[ Y3~g° ] (39)
+ Ckmn#m.ijt + Cl,,~#.~jk + D~p#,,~pj~t
Dt.ranp#mnpifl-}- Dtmnp#mnpijk
"3I- Ojmnp#ranpikl "~
where the notation for moments is given by eqn (26) and gO = g ( y + /a) -- E[g(V + /0]
=
(Y2 ~ #2)(Y3 "~- #3) -- E[(Y2 ~- #2)(Y3 -]- #3)] -E[Y2Y3]
+
# 3 Y 2 q- # 2 Y 3 -I-
Y2Y3
(40)
hence in this case one has directly a, =
-E[Y2Y3]
b~ = #2,
c2 =
=
-#,,, 1
+ v(t)E[P2](eiej#kt + eiek#jt + eiel#jk + ejekpit + ejet#ik + eked.lij)
The fourth non-linear equation in (15a) or (15b) becomes ]74 =
(43)
b, =
[-/3
+ v(t)E[P4]eie/eket
where the notation for moments is given by eqn (31). The equations for moments form an infinite hierarchy. In particular in the above equations unknown fifth-order 0.8
(41)
Since the equations for moments up to the fourth order will be derived, the moments #u, for i + j = 5 and 6 which appear at the left-hand side of eqn (30) remain unknown. They have to be evaluated as the expectations with respect to the same probability measure as the expectations on the fight-hand side of eqn (39), i.e. with respect to the Gram-Charlier expansion (27) including only one-fold sum and truncated at N = 4. Equations for moments are derived now from the equations governing the equivalent non-linear system
(44)
-
0.7-
0.6 0,5 ~0.4 ~a 0,3 0,2
O.i 0.0
• N~
~.
....
t/TO
Fig. 2. Mean hysteretic restoring force for non-zero-mean impulses.
Dynamic Response of Hysteretic Systems to Poisson-distributed Pulse Trains 30-
0.40_
25-
0.35 0.300.250.202 0,150.i0
20-
ts. 105.
4
6 t/TO
8
I0
(45)
f'O~,m. = 15{#o~,~}s + lO{uo~,,,..}s - 2.15{#0#u#,~} ~
(46)
where { . . . } , denotes the symmetrizing operation.
4 NUMERICAL STUDIES 4.1 S D O F system subjected to a random train of impulses
To illustrate the applications of the techniques devdoped, consider as the first example a nearly elasto-plastic system (n = 1) subjected to a random train o f impulses, cf. eqns (6a), (6b). The following parameters of the system are considered: a = 1,// = ~, = 0-5, ct = 0.05, = 0"01, coo = 1. The Poisson process is assumed to be homogeneous, with constant expected arrival rate of events v. Two cases o f random impulses of magnitudes P are considered:
30t
(i) non-zero-mean, Rayleigh distributed random variable P, and (ii) zero-mean variable P = R - E[R], where R is a Rayleigh distributed random variable. It is known that the Poisson-distributed train of impulses, which is also a delta-correlated process tends to a Gaussian white noise when v ~ oo in such a way that vE[P 2] is kept constantJ ° Because the effect of substitution of the random train of impulses by a Gaussian white noise will be investigated, the values of the excitation parameters, i.e. v and the Rayleigh distribution parameter th, are assumed in such a way as to obtain the same excitation mean-square level, as in Ref. 24, i.e. corresponding to the power spectrum level Go = 0.1. In the case of the random train of impulses the product vE[P ~] is decisive for the mean square level, hence it is assumed as vE[P 2] = 2riG0 = 0.2n. Moreover in case (i) the mean excitation level is assumed to be rE[P] = 0.8 (cf. #r in Ref. 24). Consequently the following data are assumed: (i) v = 1-297co0, (ii) v = 1"297co0,
a e = 0.49216m/s ae = 1.0624m/s
A fourth-order R u n g e - K u t t a method perform the numerical integration of moments. A time step was chosen as To = 2n/co0 is the natural period of the linear system.
7060 ~
~
was used to equations for T0/40, where corresponding
lo-
~
"
J
~4o~j
~ - -
:. •,,,,,~ _ ...._.
20_ f . ~
4
i0 0 0
50 2
~
50
>~ 15"I..","
0
Fig. 5. Hysteretic restoring force variance for non-zero-mean impulses.
80 ¸
25
20
; t/TO
central moments ##,.~ and sixth-order central moments #0kt~ appear. This hierarchy will be truncated with the help o f the cumulant-neglect closure technique. In the present case the cumulants above the fourth order are neglected which yields the following relationships for fifth- and sixth-order joint central moments
= lO{~,j~,,.}s
~
o
12
Fig. 3. Mean cumulative damage for non-zero-mean impulses.
~U*tm
.°.....
0.05 L' 0.00 2
143
6 t/TO
8
i0
12
Fig. 4. Displacement response variance for non-zero-mean impulses.
,~,~.~ ~..
~"~ ~'.,.L~L . ~ a~m.L~.~.ji~ 2
4
6 t/TO
8
I0
12
Fig. 6. Variance of the cumulative damage for non-zero-mean impulses.
144
R. lwankiewicz, S.R.K. Nielsen 22.5
f
0.6
f "
20.0
/
0.5.
17.5 15.0
0.4-
12.5 0.3-
i0.0
v
t~
0.L
7.5 5.0
0.1-
2.5 0.0 0
2
4
6
8
I0
O. O}'
12
2
t/TO
4
6
8
10
12
t/TO
Fig. 7. Mean displacement response for zero-mean impulses.
Fig. 9. Mean cumulative damage for zero-mean impulses.
To verify the validity of the approximate analytical techniques the response moments were also obtained from Monte Carlo simulation based on averaging over 50 000 independent sample curves, each obtained by the numerical integration of the governing equations (6a) driven by the excitation sample function, i.e. the trains of impulses with interarrival times generated from the negative exponential distribution and with strengths generated from the Rayleigh distribution. The calculation time used for simulations was 3.07 h in the case of non-zero-mean impulses and 3.12 h in the case of zero-mean impulses. The calculation time used for the equivalent linearization and equivalent cubic expansion amounted to 0.88% and 6.0%, respectively, of the time used for the simulations. In Figs 1-12 the mean values and variances of the response variables X, Z and D are plotted against the non-dimensional time t/To. Solid lines represent the simulation results, the broken-and-dotted line denotes the solution obtained from the equivalent linearization technique developed in the present paper, while the broken line corresponds to the solution obtained with the help of the equivalent cubic expansion. The results obtained in the case of an equivalent white noise excitation and with the help of the classical equivalent linearization technique are shown by the dotted line. It is seen in Figs 1-3 that in the non-zero-mean case the equivalent linearization technique developed herein gives very good predictions of the response mean values. However the variances are not well predicted, as shown
in Figs 4-6. Only in the case ofvar (X) is the value of the transient maximum in quite good agreement with the simulated curve. However in the case of var (Z) the prediction is merely qualitative and the prediction of var (D) is completely incorrect. The variance estimates obtained from the equivalent cubic expansion technique are in much better agreement with those obtained from simulations. The relatively good agreement for the variance of the damage indicator, var (D), is mainly due to the fact that the damage rate is a quadratic form in and Z, and hence it is exactly represented by a cubic expansion, cf. Ref. 40. Substitution of the random train of impulses by an equivalent white noise results in good estimates for mean values (mean levels of both excitations are the same) but the variance estimates are of the same quality as in the case of the equivalent linearization technique developed herein. Certainly the estimate of var (X) obtained with the help of a white noise approximation is very inaccurate. It is worthwhile to note that mean responses E[X], E[Z] and variances var (X) and var (Z), obtained by substituting the random train of impulses by a Gaussian white noise, and shown in Figs 1, 2, 4 and 5, respectively, are exactly the same as those given in the paper by Baber. 24
In the case of zero-mean impulses (E[P] = 0) the mean values E[X] and E[Z] approach the non-zero values in the stationary state, as has been predicted (see Figs 7 and 12
0.00 -0.
,°°.°
.°°°
°. °°.°.°°°,°°°
f
i0.
°°.°°..°°.
005
f ,
O. 0 2 5
-
O. 0 3 0 -
"-
,°. °.,,..
° o°. °. ° ° °,°
4.
N
2-
%,
O.
o
- 0.035 2
°°°o.,.
6
\
~-0.020 e<
%
I
/,/
8.
-0.010 -0.015
I
4
6 t/TO
8
I0
12
Fig. 8. Mean hysteretic restoring force for zero-mean impulses.
2'
}a
8
10
12
t/TO Fig. 10. Displacement response variance for zero-mean impulses.
Dynamic Response of Hysteretic Systems to Poisson-distributedPulse Trains
145
0.40 0.35
°°°°oo
°°°°°°°°°°°°°
0.300.25-
~10
0,20. LU
6
0.15 0.i0 0.05
2
0.00
&
6
8
10
12
0
t/T o
t/TO
Fig. 11. Hysteretic restoring force variance for zero-mean impulses.
Fig. 13. Mean displacement response for general pulses. parameter
8). The estimates of mean responses, shown in Figs 7-9 are at least qualitatively correct. However, approximation of the excitation by a white noise and application of the classical equivalent linearization technique gives completely incorrect predictions of mean values: both E[X] and E[Z] are zero. Theoretical predictions of variances var (X) and var (Z), shown in Figs 10 and 11, respectively, in the case of the equivalent linearization technique are only qualitative, the prediction of var (D) is again absolutely incorrect; the curve is not shown because it coincides with the time axis in Fig. 12. Response variances evaluated in the case of the substitution of the excitation by Gaussian white noise are also very inaccurate, especially the estimate for var (D), which is equal to zero. Application of the equivalent cubic expansion technique improves the theoretical predictions of responses mean values (Figs 7-9) and varances (Figs 10-12).
4.2 SDOF system subjected to a random train of general pulses As a second example consider the same SDOF nearly elasto-plastic system subjected to a train of non-zeromean general pulses, cf. eqns (6a) and (6c). As before the Poisson process is assumed to be homogeneous and the impulse magnitudes P are Rayleigh distributed with
60
COo = 1.0,( = 0.01,0¢ = 0.05, a = 1.0, fl = 0.5, 7 =
0.5, n =
1, D.0 =
v =
1.0,~p =
3-70834.
2-40192, Z =
0.95,
Also in this case the approximate analytical results were verified against the Monte Carlo simulations. The response moments were obtained from a simulation based on averaging over 50000 independent sample curves. Likewise, the calculation times used for the equivalent linearization and cubic expansion techniques were 0-88% and 6.0%, respectively, of the calculation time used for the simulation. In Figs 13-18 the time histories of the means and variances of X, Z and D are shown, versus the nondimensional time t/To. The solid line corresponds to the simulation, the broken-and-dotted line to equivalent linearization and the broken line to equivalent cubic expansion. As seen (Figs 13-15) the equivalent linearization can predict the mean values reasonably well. The variances are less accurately predicted (Figs 16-18). The variance estimates in the case of equivalent cubic expansion are in much better agreement with those obtained from numerical simulation. The results show that the proposed approximate analytical technique is equally effective in a more complex case of general pulses as in the simplest case of Dirac delta impulses.
j
50
0.8
40
0.6
~3o
E
t~J
2O
0.2
~'~
10
0.0 0
.
2
.
.
4
.
6
.
8
i
.
I0
12
0
2
&
i
6
8
10
12
t/TO
t/T o
Fig. 12. Variance of the cumulative damage for zerb-mean impulses.
Fig. 14. Mean hystereti¢ restoring force for general pulses.
146
R. lwankiewicz, S.R.K. Nielsen 0.6
40
0.5
RO.4 >~0.3
~20
7 &1-t~--~----
. . . . . . .
0.2
10
0.1 i
2
4
i
i
6
8
i
10 12
2
4
t/T o
i
i
i
i
6
8
10
12
t/T O
Fig. ]5. M e a n cumulative damage f o r general pulses.
Fig. 17. Hysteretic restoring force variance for general pulses.
5 CONCLUSIONS
For the hysteretic system subjected to non-Gaussian stochastic excitations in form of Poisson-distributed trains of random pulses a formulation within the framework of Poisson-driven Markov processes is given. Based on the governing stochastic integro-differential equations, two approximate analytical techniques are developed in order to evaluate the response moments. The first one is a version of the equivalent linearization technique which differs from the usual one in that the excitation is not a Gaussian process; hence the response of a linearized system cannot be assumed to be Gaussian distributed. Hence the joint probability density of the pertinent response processes, the exact form of which is not known in the present problem, is assumed in the approximate form of a truncated, generalized GramCharlier expansion. The second technique consists of substituting the non-analytical non-linearities in the original system by the equivalent cubic form of the pertinent state variables. The comparison of the approximate analytical results with those obtained from Monte Carlo simulations shows that the equivalent linearization technique developed in the present paper allows prediction of the mean responses reasonably well, but the predictions of variances are poor, perhaps only of qualitative nature in some cases. The equivalent cubic expansion technique allows us to obtain very good predictions of both mean values and variances.
30 20
30 5 0
2
4
6
8
10
12
t/T o
Fig. 16. Displacement response variance for general pulses.
120" 100
~to 20
o
2
4
6
8 10 12 t/T O
Fig. 18. Variance of the cumulative damage for general pulses.
Comparison of the calculation times shows that the analytical technique developed is effective and competitive, compared with simulations. It has been also shown that excitation in the form of a random train of impulses cannot be, in this kind of problem, substituted by an equivalent Gaussian white noise.
ACKNOWLEDGEMENTS
The first author was supported by the Polish Academy of Sciences through the Research Programme C.P.B.P.02.02/ 5.3. This support is gratefully acknowledged.
REFERENCES 1. Caughey, T.K., Random excitation of a system with bilinear hysteresis. Trans. ASME, J. Applied Mechanics, 27 (1963) 649-52. 2. Kaul, M.K. & Penzien, J., Stochastic seismic analysis of yielding offshore towers. Proc. ASCE, 100 (EM5) (1974) 1025-38. 3. Wen, Y.-K., Method for random vibration of hysteretic systems. Proc. ASCE, J. Engng Mechanics Division, 102 (1976) 249-63. 4. Roberts, J.B., The response of an oscillator with bilinear hysteresis to stationary random excitation. Trans. ASME, J. Applied Mechanics, 45 (1978) 922-8. 5. Roberts, J.B., The yielding behaviour of a randomly excited elastoplastic structure. J. Sound and Vibration, 72 (1980) 71-85.
Dynamic Response of Hysteretic Systems to Poisson-distributed Pulse Trains 6. Spanos, P.T.D., Hysteretical structural vibrations under random load. J. Acoustical Society of America, 65(2) (1979) 404-10. 7. Iyengar, R.N. & Dash, P.K., Study of the random vibration of nonlinear systems by the Gaussian closure technique. Trans. ASME, J. Applied Mechanics, 45 (1978) 393-9. 8. Wen, Y.-K., Equivalent linearization for hysteretic systems under random excitation. Trans. ASME, J. Applied Mechanics, 47 (1980) 150-4. 9. Baber, T.T. & Wen, Y-K., Stochastic response of multistorey yielding frames. Earthq. Engng and Struct. Dynamics, 10 (1982) 403-16. 10. Casciati, F. & Faravelli, L., Reliability assessment for nonlinear random frames. Proc. of Weibull IUTAM Symposium, Stockholm, Sweden, 1984, Springer Verlag, Berlin, 1985, pp. 469-78. 11. Casciati, F. & Faravelli, L., Stochastic equivalent linearization in 3-D hysteretic frames. Struct. Mechanics in Reactor Technology, M (1987) 453-8. 12. Baber, T.T., Modal analysis for random vibration of hysteretic frames. Earthq. Engng and Struct. Dynamics, 14 (1986) 841-59. 13. Nielsen, S.R.K., Merk, K.J. & Thoft-Christensen, P., Stochastic response of hysteretic systems. Structural Safety, 9 (1990) 59-71. 14. Nielsen, S.R.K., Mork, K.J. & Thoft-Christensen, P., Response analysis of hysteretic multi-storey frames under earthquake excitation. Earthq. Engng and Struct. Dynamics, 5 (1989) 655-66. 15. Borgman, L.E., Random hydrodynamic forces on objects. Annals of Math. Statistics, 38 (1967) 37-51. 16. Tung, C.C., Random response of highway bridges to vehicle loads. Proc. ASCE, J. Engng Mechanics Division, 93 (1967) 79-94. 17. Liepmann, H.W., On the application of statistical concepts to the buffeting problem. J. Aeronautical Sciences, 19 (1952) 793-800, 822. 18. Cornell, C.A., Stochastic process models in structural engineering. Stanford University, Technical Report 34, 1964. 19. Merchant, D.H., A stochastic model of wind gusts. Stanford University, Technical Report 48, 1964. 20. Lin, Y.K., Application of nonstationary shot noise in the study of system response to a class of nonstationary excitations. Trans. ASME, J. Applied Mechanics, 30 (1963) 555-8. 21. Roberts, J.B., On the response of a simple oscillator to random impulses. J. Sound and Vibration, 4 (1966) 51-61. 22. Madsen, H.O., Engineering Academy of Denmark, Lyngby, 1988, pers. comm. 23. Spanos, P.-T.D., Formulation of stochastic linearization for symmetric or asymmetric MDOF nonlinear systems. Trans. ASME, J. Applied Mechanics, 47 (1980) 209-11. 24. Baber, T.T., Nonzero mean random vibration of hysteretic systems. Proc. ASCE, J. Engng Mechanics Division, 110 (1984) 1036--49. 25. Snyder, D.L., Random Point Processes. John Wiley, New York, 1975. 26. Iwankiewicz, R. & Nielsen, S.R.K., Dynamic response of non-linear systems to Poisson-distributed random impulses. Inst. of Materials Science and Appl. Mech., Report No. 11, Technical University of Wrociaw, Poland, 1989. 27. Gikhman, I.I. & Dorogovstsev, A.J., On stability of solutions of stochastic differential equations. Ukrainian Mathematical Journal, 17 (1965) (in Russian). 28. Renger, A., Equation for probability density of vibratory
29. 30. 31. 32. 33. 34.
35.
36. 37. 38. 39. 40.
147
systems subjected to continuous and discrete stochastic excitation. Zeitschrift J~r Angew. Math. und Mechanik, 59 (1979) 1-13 (in German). Tylikowski, A., Nonstationary response of finear discrete systems to Poissonian impulse sequences. Facta Universitatis (Yugoslavia) (in press). Tylikowski, A. & Marowski, W., Vibration of a non-linear single-degree-of-freedom system due to Poissonian impulse excitation. Int. J. Non-linear Mechanics, 21 (1986) 229-38. Gikhman, I.I. & Skorokhod, A.V., Stochastic Differential Equations. Springer Verlag, Berlin, 1972. Assaf, S.A. & Zirkle, L.D., Approximate analysis of nonlinear stochastic systems. Int. J. Control, 23 (1976) 477-92. Crandall, S.H., Non-Gaussian closure for random vibration of non-linear oscillators. Int. J. Non-Linear Mechanics, 15 (1980) 303-13. Beaman, J.J. & Hedrick, J.K., Improved statistical linearization for analysis and control of non-linear stochastic systems: Part I: An extended statistical linearization technique. J. Dynamic Systems Measurement and Control (ASME), 103 (1981) 14--21. Minai, R. & Suzuki, Y., Seismic reliability analysis of building structures, Proc. of ROC-Japan Joint Seminar on Multiple Hazards Mitigation, National Taiwan University, ROC, 1985. pp. 193-208. Suzuki, Y. & Minai, R., Seismic reliability analysis of hysteretic structures based on stochastic differential equations. Proc. oflCOSSAR-4, 1985. Vol. 2, pp. 177-86. Roberts, J.B., System response to random impulses. J. Sound and Vibration, 24 (1972) 23-34. Longuet-Higgins, M.S., Modified Gaussian distributions for slightly non-linear variables. Radio Science J. of Research, 68D (1964) 1049-62. Iwankiewicz, R. & W6jcicki, Z., Threshold crossings in a linear oscillator due to a Poissonian train of general pulses. Archives of Mechanics, 39 (1987) 527-39. Roberts, J.B., The response of linear vibratory systems to random impulses. J. Sound and Vibration, 2 (1965) 375-90.
APPENDIX: EXPECTATIONS OF THE NON-ALGEBRAIC F U N C T I O N S O F T H E RESPONSE PROCESSES, FOR n = 1
Notations: g = g(Y2 + #2, I"3 + #3) =
ar2
-
-
fl[Y2 +
/~21(r3 + #3)
~'(Y2 + ,2)1Y3 +/~3(
#qkl = 0"2 =
N~22 ,
0"3 =
~33
/~2 =
~2/a2,
133 =
~3/'~3
P =
-, ~2 ~3
/'/23
~'iyu =
Kijkl a i ayo"ko"/
rOkl - - fourth-order cumulant of the processes Y . Yj, Yk and Yt 1
erf (x) = x ~
f~-~ exp (--z2/2)dz
E[g(Y2 + /z2, r3 + #3)1 =
a#2 - flE,(Y2, Y3)
148
R. Iwankiewicz, S.R.K. Nielsen -
El(Y2, Y3) =
~,E~(r:,
r~)
~2.20"3
+ --ST-[&n~p(~ -/~ - 2)
E[IY2 + /2zl(Y~ + m)]
E2(Y2, Y3) = E[(Y: + /2~)lr~ +/2sl] = E,(Y3, Y2) E[Y2g] = a/222 - flE3(r2, Y3) -- )'E,(Y2, I:3)
+ 6,hn~(/~+ 2)]1
E[Y3g]
+ (/22#23 + #223)[1 - 2 erf (-f12)]
=
a#23 - flE4(Y3, Yz) - yE3(Y3, Y2)
Es(Y2, Ys) =
#3LI(Y2) + L2(72, Y3)
L3 ( Y2, }'3) = E[YEIY3 + #31]
E4(Y2, Y3) = /22L3(Y2, Y3) + L4(Y2, Y3) = : E,(y2, y3) = ~/!~exp(_½fl2){/h(0": 0"20"3 + ~ [&22p~ -
I) + 3,h2d
24
/220"3,~,2222p(~
-
3)
-
4/220"3,~,2223] }
+ (#23 + /22/23)[1 -- 2 erf(-fl2)] Lt(Y2) =
E[Y2IY2 + /221]
= ~2- exp (-½,~) { - ~ /220222222
24
[~.333P(fl]- l)-.+-.32233] [422333 "1" ,~3333P(fl2 -- 3)]}
+ #2311 - 2 erf(-fl3)] L4 ( Y2 , Y3 ) = E[Y221Y3+ #31]
= :exp(--½fl])(0.3#22(1
+ p 2)
(,~ + 2)
(#~ + 1) }
+ /222[1 - 2 erf ( - f12)] L2(Y2, Y3) = E[Y2Y31Y2 +/2~1] = :
X {'~ #30'2
l
+ ~ ~ 0":,~n~(/~ - I) -
exp (-½fl~)
/2222226)
/23~22 [)~333p2(fl~ -- 3)
6
+ 6,~233P -I- "~333] "t- 0.3fl2___~2
24
× [ & . 3 ~ - 1 + p 2 ~ - 6fl~ + 3))
exp(- ½fl~) {20.2/233 + 8,~2333P(fl] -- 1) + 12,~2233] } /220"20"3 [~'222P(fl~ + 1) + 32223]
6
+ (#3#22 + /2223)[1 - 2 erf (-f13)]
Dynamic response of hysteretic systems to Poisson-distributed pulse trains Radoslaw lwankiewicz Institute of Materials Science and Applied Mechanics, Technical University of Wroclaw, Wybrze~e Wyspimlskiego 27, PL-50-370 Wroctaw, Poland
& Soren R. K. Nielsen Institute of Building Technology and Structural Engineering, University of Aalborg, Sohngaardsholmswej 57, DK-9000 Aalborg, Denmark (Received January 1990; accepted August 1991) A single-degree-of-freedom hysteretic system subjected to a specific non-Gaussian random excitation in the form of a Poisson-distributed train of random pulses is considered. The hysteretic behaviour is described by the Bouc-Wen model of smooth hysteresis. The total hysteretic energy dissipation is assumed as a cumulative damage indicator. The state variables of the system together with the damage indicator form in that case a Poisson-driven, non-diffusive Markov vector process. Two equivalent systems are introduced by substituting the original non-analytical, non-algebraic non-linearities by equivalent linear and cubic forms in the pertinent state variables. Equations for mean responses are obtained by direct averaging of the governing equations, whereas the equations for second- and higher-order joint central moments are derived from the equivalent systems with the help of a generalized Ito's differential rule. Appearing in the equations for mean responses and for equivalent coefficients, expectations of non-algebraic functions of state variables are performed with respect to a non-Gaussian joint probability density function assumed in the form of a generalized, bivariate Gram-Charlier expansion. In the case of equivalent cubic non-linearities, the equations for moments form an infinite hierarchy which is truncated with the help of a cumulant-neglect closure technique. Mean values and variances of the response variables are evaluated by the numerical integration of equations for moments for two equivalent systems. For the sake of comparison the excitation process is also substituted by a Gaussian white noise and the usual equivalent linearization technique, combined with the Gaussian closure, is applied. The cases of non-zero-mean as well as zero-mean excitation processes are included. The case of general pulses is dealt with by a suitable augmentation of the state vector. The accuracy of the analytical techniques developed is verified against Monte Carlo simulations.
1 INTRODUCTION
to analytical treatment, has fast progress of the studies and development of the methods been accomplished. Different methods were proposed for the analysis of hysteretic systems under r a n d o m excitations, for example methods based on the F o k k e r - P l a n c k - K o l m o g o r o v equation, such as a direct linearization of this equation for the system with bilinear hysteresis 2 or a generalized Galerkin solution to the F o k k e r - P l a n c k - K o l m o g o r o v equation for the system with smooth hysteresis. 3 Also the stochastic averaging method ~ and a Gaussian closure of
Considerable attention has been given in the literature over the last two decades to the dynamic response of inelastic, hysteretic systems to r a n d o m excitations. In the earliest studies a bilinear hysteresis model was used (e.g. Refs 1, 2). However only since Wen introduced the Bouc model of smooth hysteresis, 3 which lends itself very well
Probabilistic Engineering Mechanics 0266-8920/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 135
136
R. lwankiewicz, S.R.K. Nielsen
moment equations 7 have been used to study the response of hysteretic systems. The equivalent linearization technique was applied for the first time to the analysis of a hysteretic system by Wen, 8 and ever since it has perhaps become the most widely used technique for this kind of problem. It has been used not only for single-degree-of-freedom systems but also for the analysis of hysteretic structures, e.g. frames. 9-~2 Recently the technique of the equivalent polynomial expansion has been developed for hysteretic systems. 13't4 This technique consists of considering an equivalent nonlinear system obtained by substituting the original nonanalytical, non-algebraic non-linear functions of the state variables by the polynomials in these variables. This technique has been shown to improve the results over those obtained from the equivalent linearization technique. A crucial assumption which is made in all the methods mentioned above is that the excitation is characterized by a Gaussian random process (a Gaussian white noise or a filtered white noise). However, in structural dynamics there are important excitation processes for which a Gaussian assumption is not justified. Some of them, e.g. wave-induced forces acting on structures, can be represented as non-linear functions of a Gaussian process.~5 Another class of non-Gaussian excitations may be idealized as a random train of pulses. Such an idealization is especially suitable for traffic loads on a bridge. ~6 However it also finds applications in other problems such as, for example, a random 'down-wash' exciting intermittently the tail of an airplane, ~7 randomly occurring wind gusts associated with eddies, 18't9 or in some cases ground motion acceleration due to earthquakes. 2° Another example is the behaviour of the vehicle travelling over rough ground, where the vehicle structure is subjected to shocks produced by sudden bumps in the ground surface) ~ Finally, the dynamic loading due to wave slamming can also be idealized as a random pulse train. Wave slamming loading has been known for a long time to act upon ship hull structures, and recently it has been observed to be of relevance to off-shore structures, in particular to horizontal members of trussed towers, z2 Inelastic effects are known to occur in these structures. Under the assumption that the pulses are independent, i.e. that they obey a Poisson distribution, a random pulse train is a filtered Poisson process, or a Poisson-driven shot noise. The subject of the present paper is the dynamic response of a single-degree-of-freedom system to Poissondistributed trains of random pulses. The main objectives of the study are to develop and verify the appropriate approximate analytical techniques and to gain an insight into the dynamic behaviour ofhysteretic systems subjected to this particular type of non-Gaussian excitations.
Pulses of two types are taken into account: Dirac delta impulses and general pulses. Two analytical approaches are developed; a version of the equivalent linearization technique and the technique of an equivalent system with cubic non-linearities. The version of the equivalent linearization technique developed herein has two specific properties. The first is that prerequisite equations for mean responses are formulated by direct averaging of the governing stochastic equations, and next the equations for zero-mean responses are substituted by linear ones. Such a procedure is used in the case of asymmetrical non-linearities, ~3 since it allows preservation of the property of a non-zero-mean stationary response to zero-mean excitation, in the case of the equivalent linearization technique. Such a procedure is also used for non-zero-mean problems. 24 In the present case the non-linearity is symmetrical but the similar effect should be predicted because of the asymmetrical distribution of the response probability density. The second property of the version of the equivalent linearization technique used in the present paper is that the exact form of the joint probability density function of the response of the equivalent linear system is not known, hence in order to perform the pertinent expectations of the non-algebraic functions of the response processes, this joint density function is assumed in an approximate form of a truncated bivariate Gram-Charlier series. The expansion coefficients of this series (joint cumulants of the response processes) are calibrated from the moments and hence the equations for moments of order higher than two have to be derived. The required number of the equations for moments is determined by the truncation level of the Gram-Charlier expansion. In the case of the equivalent non-linear system the equations for moments form an infinite hierarchy, and the cumulant-neglect closure technique is used to truncate it. Also in this case the bivariate GramCharlier expansion is assumed to perform the pertinent expectations of non-algebraic functions of response processes. The equations for moments are integrated numerically, which gives transient response moments. The computed time histories of the response mean values and variances are compared with those obtained from the Monte Carlo simulations.
2 STATEMENT OF THE PROBLEM: STOCHASTIC EQUATIONS GOVERNING THE POISSON-DRIVEN MARKOV VECTOR RESPONSE PROCESS Consider a single-degree-of-freedom hysteretic system under a Poisson-distributed train of random pulses. The response is governed by the equation
D y n a m i c Response o f H y s t e r e t i c S y s t e m s to Poisson-distributed Pulse Trains
,~ + 2¢coo,¢ + ~O~o~X+ (1 - ~)~oo~Z N(O
=
(1)
E e i w ( t -- ti) i=I
where w(t - t~) is the pulse shape function, usually defined in the interval t E (t;,t~ + T) and equal to zero outside this interval, T being the pulse duration. The hystereticcomponent Z of the restoring force is given by the Bouc-Wen model) which can be represented in the form 2
=
a:(-
[3l:?lZlZI "-I -
~:?lZl n =
Governing equations can be converted into the following set of first-order differential equations d Nft) d-~X(t) = a(X(t)) + e ~ P~6(t - t;) (6a) i=l
where in the case of Dirac delta impulses
x(t) X(t) X(t) =
D(t)
[
(2)
=
a(X) =
g(.¢, z) h(:?, z )
eIi
(6b)
and in the case of general pulses
-x(t) f((t) z(t)
X(t) =
D(t) U(t)
~, P : ( t -- ti) i=l
is, by virtue of the linear superposition principle, the response of the auxiliary filter to the train of Dirac delta impulses, governed by the equation
_ (J(t)
~'(t)
0 + 2Zl'~00 + D.~U
=
"
-~co0~x- 2~co02- (1 - ~)~0Z
:¢(0
U(t)
,
z(t)
g(:?, z )
In eqn (2) ¢ denotes the damping ratio, COois the natural frequency of the corresponding linear system, a is the fraction of the restoring force made up of the linear elastic component, and a, t, 7 and n are constant parameters which describe the hysteresis loop and which must be calibrated to empirical data. If the excitation is a train of Dirac delta impulses, then w(t - t~) = 6(t - t~). In the case of general pulses let us consider the situation when the pulse shape has the form of, or may be idealized as, the impulse response function s(t - t~) of an auxiliary linear filter with properly chosen eigenfrequency and damping ratio. Indeed, if the filter damping ratio is high, close to 1, then the impulse response function becomes the single pulse with duration equal to half of the filter natural period and with a rapidly decaying with time, insignificant tail. Then the train of general pulses
137
E P~c~(t - t,)
-~co02X - 2~co0~: - (1 - a)co~Z + U
(3)
g(:/', z) h(:?, z)
i=1
where ~ and ;f are the filter eigenfrequency and damping ratio, respectively. Evolution of damage of the system due to the plastic deformations can be described by an extra equation. If the damage is modelled as the hysteretic energy, this equation has the following form /~ = )?Z = h(~', Z) (4) In eqn (1), N ( t ) is the Poisson counting process, which gives the number of impulse arrivals in the time interval [to, t). The expected arrival rate of events (or the expected number of events per unit time) is v(t), i.e. E[N(dt)] = v(0dt + o(dt) (5) where N(dt) is the random number of impulses in the infinitesimal time interval [t, t + dt). Random magnitudes P~ of the impulses are assumed to be mutually independent random variables, which are also independent of the counting process N ( O , and are identically distributed as a random variable P.
a(X) =
0
-o~v- 2z~0 -0 0 0 e
=
(6c)
0 0 _I with the general initial conditions X(t0) = X0.
(6d)
As is seen the case of general pulses is dealt with by the suitable augmentation of the state vector.
R. lwankiewicz, S.R.K. Nielsen
138
Stochastic integral equations are obtained from the governing equations (6a) by the integration over t and substitution of the compound Poisson process by the counting integral (cf. Refs 25, 26)
fying eqn (6a), the following differential rule is valid25'31 dV(X(/), t) = + ~, a,(X(t))
N(O
X P/ -- f:0 fa, tiM(dz, dti)
where M(dz, dr/) is the Poisson random measure, 25'27 which specifies the increment of the compound Poisson process in the infinitesimal time interval IT, z + dz) and denotes the whole sample space of the random variable P. The Poisson random measure has the following properties N(dt)
t i> to (10)
where dX(t) = X(t + dt) - X(t) denotes the infinitesimal increment in X(t) during the time interval [t, t + dO. In what follows, zero initial conditions are assumed. The process X(t) which satisfies eqn (10) is the Poissondriven Markov vector process25 whose joint probability density fx(x, t) is governed by the following integrodifferential partial equation, an analogue of the FokkerPlanck-Kolmogorov equation for diffusion processes (cf. Ref. 28) ~fx(X, t) + ~ ~Xi [fx(X, t)a,(x)] + v(tff'x(X, t) -
where V(X(t), t) must be bounded for all t and (X~. . . . . X,) finite and must be once continuously differentiable with respect to all its arguments. Performing averaging of both sides of eqn (12) yields the equations for moments which have the general form d E[V(X, t)] = E [dV(X, t)] d-t [ ~-t
(9)
where fp(ti) is the probability density of the random variable P. The resulting stochastic integral equations can be represented in an equivalent differential form
- v(t) f, fx(x
~ V(X(t), t) OX/ dt
+ f~, [V(X(t) + tle, t) - VX(t), t)]M(dt, dr/) (12)
(8)
E[n(dt, dr/)] = v(t)fe(ti)dtdti
dX(t) = a(X(t))dt + e f tiM(dt, dr/),
Ot
(7)
i~l
f~ M(dt, dti) -
aV(X(t), t) dt
tie, t)fp(ti)dti = 0
(11)
Direct solution to this equation is, unfortunately, unknown. Instead the application of a Fourier transform to eqn (11) yields the first-order partial differential equations for the characteristic function. 2s-3° In the case of a linear system, the equation for the characteristic function can be solved to give an explicit expression for the response characteristic function. 29'3° An alternative approach is to derive the equations for moments of the Poisson-driven Markov vector process x(t).
3 EQUATIONS FOR RESPONSE MOMENTS 3.1 General form of the equations for moments For the Poisson-driven Markov vector process X(t) satis-
+ ~.E . [ai(X)c3V(X'~3ji~/t)] + v(t) f, EtV(X + tie, t) - V(X, t)lfp(ti)dti (13) Equations for the response nth order moments may be obtained from eqn (13) by substituting V(X, t ) = X~,X ~ . . . X~,,where ir = 1. . . . . 4 or 1 , . . . , 6 for each
i,. Due to the non-linearity of the drift terms a3(X) = g()/', Z) and a4(X) = h(8, Z), the typical problem for non-linear systems arises, i.e. to close the set of equations for moments by evaluating the expectations of non-linear functions of the state variables. Since the function g(.Y, Z) is non-algebraic the pertinent expectations only can be performed as mult-fold integrals with respect to the joint probability density functions of the relevant processes. Such operations are often performed in non-linear problems. It is obvious that the quality (or accuracy) of the results obtained depends on how close the assumed, or tentative, form of this probability density function is to the true one. Often the tentative joint probability density function which must reveal the non-Gaussian behaviour is assumed in the form of the truncated GramCharlier expansion, e.g. Refs 32-34. A similar approach was used by Minai & Suzuki 35'36 who considered the response of an elasto-plastic structure to a Gaussian excitation and used a modified Gram-Charlier form of the tentative probability density function. The modification consisted of including a couple of Dirac delta spikes, which correspond to the fact that the sample space of the hysteretic component is in that case the finite interval between the two yield points, which are related with finite probabilities. In the present case the joint multidimensional probability density functions of all the four or six state variables X,. would be required to perform the expectations. Because the evaluation of integrals with respect to such density functions may be cumbersome, and for a largedimensional state vector it may even become prohibitive,
Dynamic Response of Hysteretic Systems to Poisson-distributed Pulse Trains another approach is developed. Instead of dealing with the original system, another physical system is considered in which the non-linear function is substituted by the equivalent polynomial form (in particular a linear form and the cubic form) in the pertinent state variables. It should be emphasized that since the original non-linear function g(,~', Z) is non-analytical, the polynomial form cannot be obtained as a truncated Taylor expansion (which does not exist). The coefficients of the equivalent system are determined by the least squares criterion and updated in each time point of the assumed discrete set (in each numerical integration step). Then it can be proven ~3 that if the polynomial expansion of order M in the centralized state variables is considered, the mean values and joint central moments up to and including the order M + 1 obtained from the equivalent polynomial system are identical with those obtained from the original system. The main advantage is that only joint bivariate probability density function of the state variables ,~" and Z is then required.
3.2 Equivalent Hnearization technique
139
(10), as Ela,(X)] + v(t)E[P]e,
/~i =
(14)
The specific form of the equations for means in the case of impulses is (cf. eqn (6b)) ti,
=
/~2
/J2 =
-otc°2PJ - 2(c°0;h - (1 - ~t)t02#3 + v(t)E[P]
/i~
E[g(X~, X~)]
=
/~, = E[h(X2 ' X3)]
(15a)
and in the case of general pulses (of. eqn (6c)) #l
=
#2
g2 =
-atO2o#) - 2(O9o#: - (1 - 0t)cog#3 + /~5
#3 = E[g(X,, X3)]
(15b)
#4 = E[h(X2, X3)] Us
=
#6
P6 = - D ~ m - 2zD.op6 + v(t)E[P] where
It may be shown that the probability distribution of the response of a linear system to a Poisson-distributed train of non-zero-mean impulses (E[P] # 0) is not symmetric regardless of the type of probability distribution of random magnitudes P of impulses. This is so because, as is well known, the third-order cumulant (and hence the skewness coefficient) of the response depends on the third-order ordinary moment of the random variable p.21,37 In the case of zero-mean impulses the response probability distribution is not symmetric if the probability density function of the random variable P is not symmetric, i.e. iffp(rl) ¢ fp(-r/). The probability distribution of the response of a system with symmetrical non-linearity, e.g. of a hysteretic system specified by eqn (2), should be predicted, in the cases described above, to be not symmetrical. This means that the steady-state response of such a system to zero-mean random impulses (i.e. ifE[P] = 0) has to be predicted to be non-zero-mean if the probability density function of the random variable P is not symmetrical. This is similar to the non-zeromean response problem in the case of asymmetrical nonlinearities. However, if the governing equations were substituted by equivalent linear equations, the response of the equivalent system to the zero-mean excitation (zero-mean impulses) would be zero-mean. Hence the formulation of the equivalent linearization used in the present paper is similar to the one applied to the problems with asymmetrical non-linearities 23 or to non-zero-mean problems. 24 First the equations for mean responses are formulated and next the equations for zero-mean responses (centralized processes) are substituted by equivalent linear equations. Equations for mean responses /~,. = E[X~] are obtained, by taking the expectations of both sides of eqn
Elg(X~, X , ) ]
=
a~
-
,aE,
-
~,E,
(15c)
E, = E[IX, IX~IX, I"-']
(lSd)
E2 =
(lSe)
E[X21X3["]
E[h(X2, X3)]
~---
E[X2X3]
(15f)
Equations for zero-mean responses =
x,-u,
obtained by subtracting (14) from (10), become dY =
{a°(Y + /~) - ev(t)E[P]}dt
+ e ~ qM(dt, dr/)
(16a)
where a°(Y + /~) = a ( Y + p ) -
E[a(Y + /0]
(16b)
where/A denotes the column matrix of mean values #i. Equivalent linearization consists of considering a linear system in which a°(Y + p) is assumed as the following linear form a°~q(Y + /~) = ~ B~Ym
(17)
/rl
In order to evaluate the equivalent coefficients B~, of the linear system it is required that the mean square of the error ei(Y + p) =
a,°.fY + P) - a,°..~q(Y + P)
(18)
be minimized. The minimization conditions are E[sT(Y + p)8(Y + /~)1 = 0 where ( )x denotes transpose of a matrix.
(19)
140
R. Iwankiewicz, S.R.K. Nielsen
Consequently the equivalent coefficients satisfy the following set of linear algebraic equations U,,~B~ = E[Yja,.°(Y + p)]
I +
Z i+1=3
+ ~.. i + ] = 3
where #~j = E[YmYj]
(20b)
From eqn (16b) it follows that E[Y~a°(Y + n)l = E[Y~a,(Y + P)I
(21a)
where a3(Y + ~)
=
g ( Y + p)
-- ?(Y~ + #2)1Y3 + /z31~
(21b)
= h(Y + p) = (Y2 + #2)(Ya + U3) (22)
For example in the case of eqn (16a) the term a°(¥ + p) = g(Y + p ) -
E[g(Y + p ) ]
(23)
(24)
and the equivalent coefficients b~ and b2 are then obtained from the following linear algebraic equation
b2
kE[g3g( Y + P)]
In eqn (25) a different notation for moments is used than in eqn (20b), i.e. /~u = E[Y~Y{]
(26)
In view of the non-algebraic form of the non-linear function g(Y + p), the expectations E[g(¥ + p)] in eqn (15) and E[Yjg(Y + /t)] in eqn (25) can only be performed as the pertinent integrals with respect to the joint probability densityfr~ r~( Y2, Y3) of the response processes I12 and Y3. Such a density function is usually assumed as the probability density of the response of a linearized system. In the case of a Gaussian excitation the response of a linearized system is also a Gaussian-distributed process hence fY:Y~(Y2, Y3) is assumed as a bivariate Gaussian density function. However the exact form of the joint probability density of the response processes I"2 and Y3 of a linearized system to Poisson-distributed impulses is not known. Therefore an approximate joint probability density function is assumed in the form of a truncated bivariate Gram-Charlier expansion (cf. for example Refs 37-39)
fr, r3(Y2, Y3) =
1 271:0-20.3 ~/1 -- p2
x e x p [ - - ~ 2 --~ 22~2~3P 1 2--( 1p2)+
i ! j ! k ! l ! H~+k,j+~(~2, ~3, P) + . . .
where ~2 =
Y2/0"2,
~3 =
Y3/0"3
0-2 =
~20,
0.3 =
V/~2,
//H p
=
(28)
0.2if3
2o =
0. 0-i
where the joint moments #ij are defined by eqn (26), x o is the joint i + j t h order cumulant of the processes Y:, Y3 and H0(~2, ~3, P) is the family of bivariate, standardized Hermite polynomials of order (i, j ) which satisfy the relationships 1. 2 H0(¢2, ~3, P) exp [-2(~2
-
-
2~2¢3P + ~)/(1 -- p2)]
0i+2
is substituted by the linear form a°~q(Y + p) = b, Y2 + b2 Y3
k+l=3
tcq
-- fllY2 + #2[(Y3 + u3)lr3 + #31"-'
a,(¥ +
•
(27)
a(r2 + la2)
=
¢,,p) •
(20a)
m
]~11~02
x
= (-1) i"
i
i
0~20~3
xexp [ - ½ ( ~ - 2~2~3p + ~)/(1 - p2)] (29) Expansion (27) can be truncated at any level required depending on the number of terms which are to be included in the approximate expression. In the classical equivalent linearization technique the equations for second-order moments are then derived. However in the present approach the cumulants which appear in the Gram-Charlier expansion have to be calibrated from the moments, hence equations for moments up to the nth order have to be derived if the terms including the cumulants up to the nth order are to be retained in the expansion (27). The Gram-Charlier expansion will be truncated at fourth-order cumulants, i.e. at N = 4. The equations for moments are derived from the linearized system obtained by combining eqns (16a), (16b) and (17), with the help of a general form of eqn (13). The necessary equations for the moments up to the fourth order become
IJU = B~mP,,j + BjmPm, + v(t)E[p2]eiej
ftqk = Bim#myk + ByrnPmik + Bkm#mq + v(t)E[p3]eiejek
(30a) (30b)
#okl = B~,,#mjk~ + Bjml~m~k~+ BkmlZmVt + B~,l#.Ok + v(OE[p2](e, ej#kl + eieklgit + e~ed~jk + ejeklgit + ejetlgik + ekedgij) + v(OE[P4]eiejeket
(30c)
where the following notation of moments is assumed
].lil...in
=
E [ Y q . . . Yi.]
(31)
and the summation convention is applied over the dummy indices.
Dynamic Response of Hysteretic Systems to Poisson-distributed Pulse Trains When the expectations E[g(Y2 + #2, I"3 + #3)], E[Y2g(Y2 + //2, Y3 + #3)1, E[Y3g(Y2 + #2, Y3 + #3)1 are performed with the help of a Gram-Charlier joint density function (27) they become expressed in terms of the moments and cumulants of the linearized system response. The explicit expressions for these expectations evaluated with the help of the Gram-Charlier expansion confined to the one-fold sum in (27) and truncated at the order N = 4 and for n = 1 are given in the Appendix. The equations for moments (30a)-(30c) form a closed set, but they are coupled with eqns (14) for the mean responses, because the higher-order moments enter these equations through the expectations E[g(Y + /4)] and E[h(Y + /0]. Due to the fact that the equivalent coefficients B~,, as given by eqns (20a) or (25) depend on the moments, the equations for moments become non-linear. It is interesting to investigate the steady-state solution for the mean responses governed by eqn (15a). The derivatives on the left-hand side of eqn (15a) are set equal to zero, thus 0 = //2
0 = -- Ctt.O02//l-- 2(tO0//2 0 = a//2 - t e l - ?E2
(32)
Hence //~ = 0 and if the impulses are zero-mean, i.e. E[P] = 0, then 0 =
-~tog//~ - (1 - ~t)to02//3
(33a)
0 =
- tEl
(33b)
-
)~E2
It may be shown that//3 = 0 does not satisfy eqn (33b) if it is assumed that the response probability distribution is not symmetric, i.e. if all third-order joint central moments do not vanish. If//3 = 0 is substituted into eqn (33b) its right-hand side becomes (see Appendix) _ fiE l
_
yE2
=
fl (//222//23
0"2 ~
+ ~
~, 3[/22
'7
(//333//23 \ 3//33
the non-linearities are not severe. The idea of this technique is to consider an equivalent non-linear system, in which a°(V+t0
=
//223
)
//233
)
(34)
In general expression (34) is not equal to zero. Hence the solution must be for some //3 # 0, and from (33a) it follows that//t ~ 0 as well. Hence the stationary value of the mean displacement response and of the mean hysteretic restoring force are non-zero even if the excitation is zero-mean (zero-mean impulses). Moreover from (33a) it follows that the stationary values of E[X] = #j and E[Z] = //3 have opposite signs. 3.3. Equivalent cubic expansion technique
a ~ ( Y + /~) =
A , + B~Ym + C,~Y,,Y~
+ D ~ , Yr. Y, Yp (35) The equivalent coefficients are evaluated from the minimization conditions of the mean square of the error 5,(v + p) =
¢ ( Y + /~) - a°~q(Y + P)
d ~A~ E[ds]
0,
i.e. =
O E[srs] = 0C~
0 0B~ E[sXs] =
0
0 0,
(36)
aD~p
E[ds]
= 0
The following linear algebraic equations for the equivalent coefficients are obtained from the minimization conditions (36) (cf. Refs 13, 14). B~,//mj = C~#~.jk
=
- C~#m. - Dim.p#..v E[Yja °] - C ~ / / . w -- D~//m,~j E[YjYka~°] - A,#jk -- B~.#m~k- Dimnp//~mpjk
D,.~pp.~pjkt =
E[Yj Yk Yt a°]
-
Ai//jkl
--
Bira//mjk,
- C~//mm
(37)
Again the expectations E[Yja°], E[YjY~a°] and E[Yj Yk Yla °] may only be performed with the help of an approximate joint probability density function. This function will be assumed to be in the form of a GramCharlier expansion (27) truncated at fourth-order cumulants and accordingly the equations for moments up to the fourth order will be derived (cf. Refs 13, 14). Then the unavailable fifth- and sixth-order moments which appear in eqns (37) have to be evaluated with the help of a Gram-Charlier expansion. Strictly speaking all the expectations, including the moments, which appear in eqns (37) have to be evaluated with respect to the same probability measure - - the bivariate joint density function assumed here as the truncated Gram-Charlier expansion. However all the moments of the order which is not beyond the truncation level, evaluated with the help of a Gram-Charlier expansion, are exactly the same as those obtained from the equations for moments. In the case of the third equation of (15a) or (15b) the term a~(Y + /0 given by eqn (23) is substituted by the following cubic form ~,~q(Y + /4) =
The equivalent polynomial expansion technique has been applied with some success to the problem of hysteretic systems subject to a Gaussian white noise excitation, if
a(¥+~)-E[a(V+t0]
is substituted by the following cubic form in the state variables y, m4
Ai =
-- (1 -- ~t)to~//3 + v(t)E[P]
141
al + blY2 + b2Ya + cly2 + c2Y2Y3 + c3Y 2 + d , Y 3 + d2y2y3 + d3Y2Y~ + d4Y 3
(38)
R. Iwankiewicz, S.R.K. Nielsen
142
The expansion coefficients are then evaluated from the following set of linear algebraic equations
18 16
-
"--
14
I
0
0
#20
~'~11 #02
#30
#21
#l 2
#o3" - a, ]
#20
#ll
#30
#21
#40
#31
#22
#t3
hi [
#o3 #3~ #22 #~3 /q4
b2 I
6
#12
#0~ #:~ #n #4o
12
8
#3~
#22
#50
#4~
#32
#23
C,I
4 2
#22
#13
#41
#32
#23
#~4
c~ I
0
#04
#32
/'/23 /['/14 #o5
c~ I
#60
#51
#42
#33
dl I
#42
#33
#2~
32 I
/,/24
#~
Symmetry
!
d~ I
t/TO
Fig. 1. Mean displacement response for non-zero-mean impulses. obtained by combining eqns (16a), (16b) and (35). Making use of the general formula (13) leads to the following equations for moments up to the fourth order
~i: = Bin#m: + gjm#mi + C~n#rnnj + Cjmn#mni
0
+ D~,~p#m~pj + Dj,~p#m~.~ + v(t)E[P2]e~ej
E[Y2g]
(42)
E[Y3g]
/iij,~ = A.ujk + Aj#ik + Ak#u + B.,,#,,,jk + Bj,.,u.,~k
E[y2g °]
+ Bk,.#,.u + C~.,n#,,~jk + Cj,,,.#,,~k
E[Y2 Y3g °]
+ C,,,,,,,Um,,~ + D~.p#,,~pjk + Dj,,,~p#,.,,p~k
E[ Y32g ° ]
+ Dk.~npg.~pU+ v(t)E[p3]eiejek
E[y3g °]
I~jk~ = A~#jkt + Ad4kt + Ak#ut + At#u~
E[Y2 r3g °]
q- Bim#mjkI + Bjm#mikt -t- Bk.,#mul
E[Y: rE gO]
+ BtmgmOk + Ci.,~#.~jkt + Cj'~#.~k~
E[ Y3~g° ] (39)
+ Ckmn#m.ijt + Cl,,~#.~jk + D~p#,,~pj~t
Dt.ranp#mnpifl-}- Dtmnp#mnpijk
"3I- Ojmnp#ranpikl "~
where the notation for moments is given by eqn (26) and gO = g ( y + /a) -- E[g(V + /0]
=
(Y2 ~ #2)(Y3 "~- #3) -- E[(Y2 ~- #2)(Y3 -]- #3)] -E[Y2Y3]
+
# 3 Y 2 q- # 2 Y 3 -I-
Y2Y3
(40)
hence in this case one has directly a, =
-E[Y2Y3]
b~ = #2,
c2 =
=
-#,,, 1
+ v(t)E[P2](eiej#kt + eiek#jt + eiel#jk + ejekpit + ejet#ik + eked.lij)
The fourth non-linear equation in (15a) or (15b) becomes ]74 =
(43)
b, =
[-/3
+ v(t)E[P4]eie/eket
where the notation for moments is given by eqn (31). The equations for moments form an infinite hierarchy. In particular in the above equations unknown fifth-order 0.8
(41)
Since the equations for moments up to the fourth order will be derived, the moments #u, for i + j = 5 and 6 which appear at the left-hand side of eqn (30) remain unknown. They have to be evaluated as the expectations with respect to the same probability measure as the expectations on the fight-hand side of eqn (39), i.e. with respect to the Gram-Charlier expansion (27) including only one-fold sum and truncated at N = 4. Equations for moments are derived now from the equations governing the equivalent non-linear system
(44)
-
0.7-
0.6 0,5 ~0.4 ~a 0,3 0,2
O.i 0.0
• N~
~.
....
t/TO
Fig. 2. Mean hysteretic restoring force for non-zero-mean impulses.
Dynamic Response of Hysteretic Systems to Poisson-distributed Pulse Trains 30-
0.40_
25-
0.35 0.300.250.202 0,150.i0
20-
ts. 105.
4
6 t/TO
8
I0
(45)
f'O~,m. = 15{#o~,~}s + lO{uo~,,,..}s - 2.15{#0#u#,~} ~
(46)
where { . . . } , denotes the symmetrizing operation.
4 NUMERICAL STUDIES 4.1 S D O F system subjected to a random train of impulses
To illustrate the applications of the techniques devdoped, consider as the first example a nearly elasto-plastic system (n = 1) subjected to a random train o f impulses, cf. eqns (6a), (6b). The following parameters of the system are considered: a = 1,// = ~, = 0-5, ct = 0.05, = 0"01, coo = 1. The Poisson process is assumed to be homogeneous, with constant expected arrival rate of events v. Two cases o f random impulses of magnitudes P are considered:
30t
(i) non-zero-mean, Rayleigh distributed random variable P, and (ii) zero-mean variable P = R - E[R], where R is a Rayleigh distributed random variable. It is known that the Poisson-distributed train of impulses, which is also a delta-correlated process tends to a Gaussian white noise when v ~ oo in such a way that vE[P 2] is kept constantJ ° Because the effect of substitution of the random train of impulses by a Gaussian white noise will be investigated, the values of the excitation parameters, i.e. v and the Rayleigh distribution parameter th, are assumed in such a way as to obtain the same excitation mean-square level, as in Ref. 24, i.e. corresponding to the power spectrum level Go = 0.1. In the case of the random train of impulses the product vE[P ~] is decisive for the mean square level, hence it is assumed as vE[P 2] = 2riG0 = 0.2n. Moreover in case (i) the mean excitation level is assumed to be rE[P] = 0.8 (cf. #r in Ref. 24). Consequently the following data are assumed: (i) v = 1-297co0, (ii) v = 1"297co0,
a e = 0.49216m/s ae = 1.0624m/s
A fourth-order R u n g e - K u t t a method perform the numerical integration of moments. A time step was chosen as To = 2n/co0 is the natural period of the linear system.
7060 ~
~
was used to equations for T0/40, where corresponding
lo-
~
"
J
~4o~j
~ - -
:. •,,,,,~ _ ...._.
20_ f . ~
4
i0 0 0
50 2
~
50
>~ 15"I..","
0
Fig. 5. Hysteretic restoring force variance for non-zero-mean impulses.
80 ¸
25
20
; t/TO
central moments ##,.~ and sixth-order central moments #0kt~ appear. This hierarchy will be truncated with the help o f the cumulant-neglect closure technique. In the present case the cumulants above the fourth order are neglected which yields the following relationships for fifth- and sixth-order joint central moments
= lO{~,j~,,.}s
~
o
12
Fig. 3. Mean cumulative damage for non-zero-mean impulses.
~U*tm
.°.....
0.05 L' 0.00 2
143
6 t/TO
8
i0
12
Fig. 4. Displacement response variance for non-zero-mean impulses.
,~,~.~ ~..
~"~ ~'.,.L~L . ~ a~m.L~.~.ji~ 2
4
6 t/TO
8
I0
12
Fig. 6. Variance of the cumulative damage for non-zero-mean impulses.
144
R. lwankiewicz, S.R.K. Nielsen 22.5
f
0.6
f "
20.0
/
0.5.
17.5 15.0
0.4-
12.5 0.3-
i0.0
v
t~
0.L
7.5 5.0
0.1-
2.5 0.0 0
2
4
6
8
I0
O. O}'
12
2
t/TO
4
6
8
10
12
t/TO
Fig. 7. Mean displacement response for zero-mean impulses.
Fig. 9. Mean cumulative damage for zero-mean impulses.
To verify the validity of the approximate analytical techniques the response moments were also obtained from Monte Carlo simulation based on averaging over 50 000 independent sample curves, each obtained by the numerical integration of the governing equations (6a) driven by the excitation sample function, i.e. the trains of impulses with interarrival times generated from the negative exponential distribution and with strengths generated from the Rayleigh distribution. The calculation time used for simulations was 3.07 h in the case of non-zero-mean impulses and 3.12 h in the case of zero-mean impulses. The calculation time used for the equivalent linearization and equivalent cubic expansion amounted to 0.88% and 6.0%, respectively, of the time used for the simulations. In Figs 1-12 the mean values and variances of the response variables X, Z and D are plotted against the non-dimensional time t/To. Solid lines represent the simulation results, the broken-and-dotted line denotes the solution obtained from the equivalent linearization technique developed in the present paper, while the broken line corresponds to the solution obtained with the help of the equivalent cubic expansion. The results obtained in the case of an equivalent white noise excitation and with the help of the classical equivalent linearization technique are shown by the dotted line. It is seen in Figs 1-3 that in the non-zero-mean case the equivalent linearization technique developed herein gives very good predictions of the response mean values. However the variances are not well predicted, as shown
in Figs 4-6. Only in the case ofvar (X) is the value of the transient maximum in quite good agreement with the simulated curve. However in the case of var (Z) the prediction is merely qualitative and the prediction of var (D) is completely incorrect. The variance estimates obtained from the equivalent cubic expansion technique are in much better agreement with those obtained from simulations. The relatively good agreement for the variance of the damage indicator, var (D), is mainly due to the fact that the damage rate is a quadratic form in and Z, and hence it is exactly represented by a cubic expansion, cf. Ref. 40. Substitution of the random train of impulses by an equivalent white noise results in good estimates for mean values (mean levels of both excitations are the same) but the variance estimates are of the same quality as in the case of the equivalent linearization technique developed herein. Certainly the estimate of var (X) obtained with the help of a white noise approximation is very inaccurate. It is worthwhile to note that mean responses E[X], E[Z] and variances var (X) and var (Z), obtained by substituting the random train of impulses by a Gaussian white noise, and shown in Figs 1, 2, 4 and 5, respectively, are exactly the same as those given in the paper by Baber. 24
In the case of zero-mean impulses (E[P] = 0) the mean values E[X] and E[Z] approach the non-zero values in the stationary state, as has been predicted (see Figs 7 and 12
0.00 -0.
,°°.°
.°°°
°. °°.°.°°°,°°°
f
i0.
°°.°°..°°.
005
f ,
O. 0 2 5
-
O. 0 3 0 -
"-
,°. °.,,..
° o°. °. ° ° °,°
4.
N
2-
%,
O.
o
- 0.035 2
°°°o.,.
6
\
~-0.020 e<
%
I
/,/
8.
-0.010 -0.015
I
4
6 t/TO
8
I0
12
Fig. 8. Mean hysteretic restoring force for zero-mean impulses.
2'
}a
8
10
12
t/TO Fig. 10. Displacement response variance for zero-mean impulses.
Dynamic Response of Hysteretic Systems to Poisson-distributedPulse Trains
145
0.40 0.35
°°°°oo
°°°°°°°°°°°°°
0.300.25-
~10
0,20. LU
6
0.15 0.i0 0.05
2
0.00
&
6
8
10
12
0
t/T o
t/TO
Fig. 11. Hysteretic restoring force variance for zero-mean impulses.
Fig. 13. Mean displacement response for general pulses. parameter
8). The estimates of mean responses, shown in Figs 7-9 are at least qualitatively correct. However, approximation of the excitation by a white noise and application of the classical equivalent linearization technique gives completely incorrect predictions of mean values: both E[X] and E[Z] are zero. Theoretical predictions of variances var (X) and var (Z), shown in Figs 10 and 11, respectively, in the case of the equivalent linearization technique are only qualitative, the prediction of var (D) is again absolutely incorrect; the curve is not shown because it coincides with the time axis in Fig. 12. Response variances evaluated in the case of the substitution of the excitation by Gaussian white noise are also very inaccurate, especially the estimate for var (D), which is equal to zero. Application of the equivalent cubic expansion technique improves the theoretical predictions of responses mean values (Figs 7-9) and varances (Figs 10-12).
4.2 SDOF system subjected to a random train of general pulses As a second example consider the same SDOF nearly elasto-plastic system subjected to a train of non-zeromean general pulses, cf. eqns (6a) and (6c). As before the Poisson process is assumed to be homogeneous and the impulse magnitudes P are Rayleigh distributed with
60
COo = 1.0,( = 0.01,0¢ = 0.05, a = 1.0, fl = 0.5, 7 =
0.5, n =
1, D.0 =
v =
1.0,~p =
3-70834.
2-40192, Z =
0.95,
Also in this case the approximate analytical results were verified against the Monte Carlo simulations. The response moments were obtained from a simulation based on averaging over 50000 independent sample curves. Likewise, the calculation times used for the equivalent linearization and cubic expansion techniques were 0-88% and 6.0%, respectively, of the calculation time used for the simulation. In Figs 13-18 the time histories of the means and variances of X, Z and D are shown, versus the nondimensional time t/To. The solid line corresponds to the simulation, the broken-and-dotted line to equivalent linearization and the broken line to equivalent cubic expansion. As seen (Figs 13-15) the equivalent linearization can predict the mean values reasonably well. The variances are less accurately predicted (Figs 16-18). The variance estimates in the case of equivalent cubic expansion are in much better agreement with those obtained from numerical simulation. The results show that the proposed approximate analytical technique is equally effective in a more complex case of general pulses as in the simplest case of Dirac delta impulses.
j
50
0.8
40
0.6
~3o
E
t~J
2O
0.2
~'~
10
0.0 0
.
2
.
.
4
.
6
.
8
i
.
I0
12
0
2
&
i
6
8
10
12
t/TO
t/T o
Fig. 12. Variance of the cumulative damage for zerb-mean impulses.
Fig. 14. Mean hystereti¢ restoring force for general pulses.
146
R. lwankiewicz, S.R.K. Nielsen 0.6
40
0.5
RO.4 >~0.3
~20
7 &1-t~--~----
. . . . . . .
0.2
10
0.1 i
2
4
i
i
6
8
i
10 12
2
4
t/T o
i
i
i
i
6
8
10
12
t/T O
Fig. ]5. M e a n cumulative damage f o r general pulses.
Fig. 17. Hysteretic restoring force variance for general pulses.
5 CONCLUSIONS
For the hysteretic system subjected to non-Gaussian stochastic excitations in form of Poisson-distributed trains of random pulses a formulation within the framework of Poisson-driven Markov processes is given. Based on the governing stochastic integro-differential equations, two approximate analytical techniques are developed in order to evaluate the response moments. The first one is a version of the equivalent linearization technique which differs from the usual one in that the excitation is not a Gaussian process; hence the response of a linearized system cannot be assumed to be Gaussian distributed. Hence the joint probability density of the pertinent response processes, the exact form of which is not known in the present problem, is assumed in the approximate form of a truncated, generalized GramCharlier expansion. The second technique consists of substituting the non-analytical non-linearities in the original system by the equivalent cubic form of the pertinent state variables. The comparison of the approximate analytical results with those obtained from Monte Carlo simulations shows that the equivalent linearization technique developed in the present paper allows prediction of the mean responses reasonably well, but the predictions of variances are poor, perhaps only of qualitative nature in some cases. The equivalent cubic expansion technique allows us to obtain very good predictions of both mean values and variances.
30 20
30 5 0
2
4
6
8
10
12
t/T o
Fig. 16. Displacement response variance for general pulses.
120" 100
~to 20
o
2
4
6
8 10 12 t/T O
Fig. 18. Variance of the cumulative damage for general pulses.
Comparison of the calculation times shows that the analytical technique developed is effective and competitive, compared with simulations. It has been also shown that excitation in the form of a random train of impulses cannot be, in this kind of problem, substituted by an equivalent Gaussian white noise.
ACKNOWLEDGEMENTS
The first author was supported by the Polish Academy of Sciences through the Research Programme C.P.B.P.02.02/ 5.3. This support is gratefully acknowledged.
REFERENCES 1. Caughey, T.K., Random excitation of a system with bilinear hysteresis. Trans. ASME, J. Applied Mechanics, 27 (1963) 649-52. 2. Kaul, M.K. & Penzien, J., Stochastic seismic analysis of yielding offshore towers. Proc. ASCE, 100 (EM5) (1974) 1025-38. 3. Wen, Y.-K., Method for random vibration of hysteretic systems. Proc. ASCE, J. Engng Mechanics Division, 102 (1976) 249-63. 4. Roberts, J.B., The response of an oscillator with bilinear hysteresis to stationary random excitation. Trans. ASME, J. Applied Mechanics, 45 (1978) 922-8. 5. Roberts, J.B., The yielding behaviour of a randomly excited elastoplastic structure. J. Sound and Vibration, 72 (1980) 71-85.
Dynamic Response of Hysteretic Systems to Poisson-distributed Pulse Trains 6. Spanos, P.T.D., Hysteretical structural vibrations under random load. J. Acoustical Society of America, 65(2) (1979) 404-10. 7. Iyengar, R.N. & Dash, P.K., Study of the random vibration of nonlinear systems by the Gaussian closure technique. Trans. ASME, J. Applied Mechanics, 45 (1978) 393-9. 8. Wen, Y.-K., Equivalent linearization for hysteretic systems under random excitation. Trans. ASME, J. Applied Mechanics, 47 (1980) 150-4. 9. Baber, T.T. & Wen, Y-K., Stochastic response of multistorey yielding frames. Earthq. Engng and Struct. Dynamics, 10 (1982) 403-16. 10. Casciati, F. & Faravelli, L., Reliability assessment for nonlinear random frames. Proc. of Weibull IUTAM Symposium, Stockholm, Sweden, 1984, Springer Verlag, Berlin, 1985, pp. 469-78. 11. Casciati, F. & Faravelli, L., Stochastic equivalent linearization in 3-D hysteretic frames. Struct. Mechanics in Reactor Technology, M (1987) 453-8. 12. Baber, T.T., Modal analysis for random vibration of hysteretic frames. Earthq. Engng and Struct. Dynamics, 14 (1986) 841-59. 13. Nielsen, S.R.K., Merk, K.J. & Thoft-Christensen, P., Stochastic response of hysteretic systems. Structural Safety, 9 (1990) 59-71. 14. Nielsen, S.R.K., Mork, K.J. & Thoft-Christensen, P., Response analysis of hysteretic multi-storey frames under earthquake excitation. Earthq. Engng and Struct. Dynamics, 5 (1989) 655-66. 15. Borgman, L.E., Random hydrodynamic forces on objects. Annals of Math. Statistics, 38 (1967) 37-51. 16. Tung, C.C., Random response of highway bridges to vehicle loads. Proc. ASCE, J. Engng Mechanics Division, 93 (1967) 79-94. 17. Liepmann, H.W., On the application of statistical concepts to the buffeting problem. J. Aeronautical Sciences, 19 (1952) 793-800, 822. 18. Cornell, C.A., Stochastic process models in structural engineering. Stanford University, Technical Report 34, 1964. 19. Merchant, D.H., A stochastic model of wind gusts. Stanford University, Technical Report 48, 1964. 20. Lin, Y.K., Application of nonstationary shot noise in the study of system response to a class of nonstationary excitations. Trans. ASME, J. Applied Mechanics, 30 (1963) 555-8. 21. Roberts, J.B., On the response of a simple oscillator to random impulses. J. Sound and Vibration, 4 (1966) 51-61. 22. Madsen, H.O., Engineering Academy of Denmark, Lyngby, 1988, pers. comm. 23. Spanos, P.-T.D., Formulation of stochastic linearization for symmetric or asymmetric MDOF nonlinear systems. Trans. ASME, J. Applied Mechanics, 47 (1980) 209-11. 24. Baber, T.T., Nonzero mean random vibration of hysteretic systems. Proc. ASCE, J. Engng Mechanics Division, 110 (1984) 1036--49. 25. Snyder, D.L., Random Point Processes. John Wiley, New York, 1975. 26. Iwankiewicz, R. & Nielsen, S.R.K., Dynamic response of non-linear systems to Poisson-distributed random impulses. Inst. of Materials Science and Appl. Mech., Report No. 11, Technical University of Wrociaw, Poland, 1989. 27. Gikhman, I.I. & Dorogovstsev, A.J., On stability of solutions of stochastic differential equations. Ukrainian Mathematical Journal, 17 (1965) (in Russian). 28. Renger, A., Equation for probability density of vibratory
29. 30. 31. 32. 33. 34.
35.
36. 37. 38. 39. 40.
147
systems subjected to continuous and discrete stochastic excitation. Zeitschrift J~r Angew. Math. und Mechanik, 59 (1979) 1-13 (in German). Tylikowski, A., Nonstationary response of finear discrete systems to Poissonian impulse sequences. Facta Universitatis (Yugoslavia) (in press). Tylikowski, A. & Marowski, W., Vibration of a non-linear single-degree-of-freedom system due to Poissonian impulse excitation. Int. J. Non-linear Mechanics, 21 (1986) 229-38. Gikhman, I.I. & Skorokhod, A.V., Stochastic Differential Equations. Springer Verlag, Berlin, 1972. Assaf, S.A. & Zirkle, L.D., Approximate analysis of nonlinear stochastic systems. Int. J. Control, 23 (1976) 477-92. Crandall, S.H., Non-Gaussian closure for random vibration of non-linear oscillators. Int. J. Non-Linear Mechanics, 15 (1980) 303-13. Beaman, J.J. & Hedrick, J.K., Improved statistical linearization for analysis and control of non-linear stochastic systems: Part I: An extended statistical linearization technique. J. Dynamic Systems Measurement and Control (ASME), 103 (1981) 14--21. Minai, R. & Suzuki, Y., Seismic reliability analysis of building structures, Proc. of ROC-Japan Joint Seminar on Multiple Hazards Mitigation, National Taiwan University, ROC, 1985. pp. 193-208. Suzuki, Y. & Minai, R., Seismic reliability analysis of hysteretic structures based on stochastic differential equations. Proc. oflCOSSAR-4, 1985. Vol. 2, pp. 177-86. Roberts, J.B., System response to random impulses. J. Sound and Vibration, 24 (1972) 23-34. Longuet-Higgins, M.S., Modified Gaussian distributions for slightly non-linear variables. Radio Science J. of Research, 68D (1964) 1049-62. Iwankiewicz, R. & W6jcicki, Z., Threshold crossings in a linear oscillator due to a Poissonian train of general pulses. Archives of Mechanics, 39 (1987) 527-39. Roberts, J.B., The response of linear vibratory systems to random impulses. J. Sound and Vibration, 2 (1965) 375-90.
APPENDIX: EXPECTATIONS OF THE NON-ALGEBRAIC F U N C T I O N S O F T H E RESPONSE PROCESSES, FOR n = 1
Notations: g = g(Y2 + #2, I"3 + #3) =
ar2
-
-
fl[Y2 +
/~21(r3 + #3)
~'(Y2 + ,2)1Y3 +/~3(
#qkl = 0"2 =
N~22 ,
0"3 =
~33
/~2 =
~2/a2,
133 =
~3/'~3
P =
-, ~2 ~3
/'/23
~'iyu =
Kijkl a i ayo"ko"/
rOkl - - fourth-order cumulant of the processes Y . Yj, Yk and Yt 1
erf (x) = x ~
f~-~ exp (--z2/2)dz
E[g(Y2 + /z2, r3 + #3)1 =
a#2 - flE,(Y2, Y3)
148
R. Iwankiewicz, S.R.K. Nielsen -
El(Y2, Y3) =
~,E~(r:,
r~)
~2.20"3
+ --ST-[&n~p(~ -/~ - 2)
E[IY2 + /2zl(Y~ + m)]
E2(Y2, Y3) = E[(Y: + /2~)lr~ +/2sl] = E,(Y3, Y2) E[Y2g] = a/222 - flE3(r2, Y3) -- )'E,(Y2, I:3)
+ 6,hn~(/~+ 2)]1
E[Y3g]
+ (/22#23 + #223)[1 - 2 erf (-f12)]
=
a#23 - flE4(Y3, Yz) - yE3(Y3, Y2)
Es(Y2, Ys) =
#3LI(Y2) + L2(72, Y3)
L3 ( Y2, }'3) = E[YEIY3 + #31]
E4(Y2, Y3) = /22L3(Y2, Y3) + L4(Y2, Y3) = : E,(y2, y3) = ~/!~exp(_½fl2){/h(0": 0"20"3 + ~ [&22p~ -
I) + 3,h2d
24
/220"3,~,2222p(~
-
3)
-
4/220"3,~,2223] }
+ (#23 + /22/23)[1 -- 2 erf(-fl2)] Lt(Y2) =
E[Y2IY2 + /221]
= ~2- exp (-½,~) { - ~ /220222222
24
[~.333P(fl]- l)-.+-.32233] [422333 "1" ,~3333P(fl2 -- 3)]}
+ #2311 - 2 erf(-fl3)] L4 ( Y2 , Y3 ) = E[Y221Y3+ #31]
= :exp(--½fl])(0.3#22(1
+ p 2)
(,~ + 2)
(#~ + 1) }
+ /222[1 - 2 erf ( - f12)] L2(Y2, Y3) = E[Y2Y31Y2 +/2~1] = :
X {'~ #30'2
l
+ ~ ~ 0":,~n~(/~ - I) -
exp (-½fl~)
/2222226)
/23~22 [)~333p2(fl~ -- 3)
6
+ 6,~233P -I- "~333] "t- 0.3fl2___~2
24
× [ & . 3 ~ - 1 + p 2 ~ - 6fl~ + 3))
exp(- ½fl~) {20.2/233 + 8,~2333P(fl] -- 1) + 12,~2233] } /220"20"3 [~'222P(fl~ + 1) + 32223]
6
+ (#3#22 + /2223)[1 - 2 erf (-f13)]