The problem of estimating the probability distribution of the response amplitude of a hardening. Duffing oscillator subjected to narrow band excitation is ...
Computer Methods in Applied Mechanics and Engineering 105 (1993) 143-150 North-Holland CMA 353
Response amplitude probability functions of a hardening Duffing oscillator subjected to filtered white noise P.K. Koliopulos and S.R. Bishop Department of Civil Engineering, University College London, Gower Street, London WC1 E 6BT, UK
G.D. Stefanou Department of Civil Engineering, University of Patras, GR-26110 Patras, Greece Received 1 July 1992 The problem of estimating the probability distribution of the response amplitude of a hardening Duffing oscillator subjected to narrow band excitation is addressed. A method is presented which is based on a quasi-static approximation of system behaviour and is capable of reproducing the resulting concave shape of probability functions when a jump phenomenon occurs. Predictions ~re compared with those obtained via stochastic averaging techniques and with digital simulations.
1. Introduction
Many applications in engineering dynamics involve systems that exhibit strong nonlinearities under environmental forces that are random in nature. The response of nonlinear oscillators in a randomly fluctuating environment has been traditionally studied under the assumption that the force can be modelled as a white noise process. This hypothesis enables the analyst to implement the well known Fokker-Planck-Koimogorov differential equation which governs the diffusion of the probability of a Markov process. It has been realised, however, that realistic excitations such as winds or sea waves, often violate the white noise assumption, exhibiting a narrow band power spectrum. Simulation studies have indicated that the variation of input bandwidth may have a significant effect on the shape of the probability distribution of the response amplitude [1]. Furthermore if the bandwidth is sufficiently small, the system may realise multiple response states [2-4]. Certain types of failure (e.g. fatigue damage accumulation) depend on the statistical behaviour of the peaks of the response process. Thus the effect of multiple response levels on the probability function of response amplitude is of major importance. In recent years, a number of techniques have been suggested to deal with the problem of estimating the response probability of nonlinear systems Correspondence to: Professor G.D. Stefanou, University of Patras, Faculty of Engineering, Department of Civil Engineering, University Campus, GR-26110 Patras, Greece. 0045-7825/93 / $06.00 (~) 1993 Elsevier Science Publishers B.V. All rights reserved
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P.K. Koliopulos et ai., Response amplitude probability functions
subjected tO coloured noise. One of the most successful appears to be the stochastic averaging method which models the total energy of the system as a one-dimensional Markov process, taking into account the shape of the input power spectrum [1]. This technique can be modified in order to be able to account for the multi-valued behaviour of the response with random jumps between pseudo-stable states [5]. In this study an alternative method is presented which is based on a quasi-static assumption of the response process of the nonlinear oscillator. A criterion is proposed for the assessment of when multi-valued response statistics are likely to occur and in such a case the effect of these jumps on the statistical distribution of the peaks of the response is evaluated via a suggested mixture technique. A comparison of the performance of the proposed methodology with the stochastic averaging technique and with simulation, concludes this study.
2. The model
In this work the class of problems considered involve a hardening Duffing oscillator subjected to white noise random excitation which is filtered through a second order shaping filter. The resulting set of stochastic differential equations is of the form Ji + 2 a ~ + tO2o(X + yx 3) = F(t),
(1)
P + 2~ttotP + to2tF = W(t).
(2)
where
The excitation model of (2) has been implemented in the literature to represent a variety of environmental forces such as hydrodynamic forces due to sea-wave kinematics [6] and stationary ground accelerations filtered through soil layers [7].
3. Quasi-static formulation
The fundamental assumption of the method is that both excitation and response are narrow band processes so that both can be represented as sinusoids with slowly varying amplitude and phase. In addition, the excitation process is considered to be narrow band with respect to the system's bandwidth, so that the forcing amplitude and phase are approximated as constant random variables within any particular cycle of oscillation. Under these assumptions the input force F(t) and the resulting response x(t) may be represented as
x(t) = Xo(t ) cos(to:t + $(t)),
F(t) = Fo(t)cos(,o:t + q,(t)),
(3)
where Xo(t), Fo(t ) and ~(t), ~,(t) all have slowly varying amplitudes and phase. It can be shown (e.g. [8]) that in the limit of zero time derivatives of the random variables involved, the stochastic averaging method yields an algebraic relation between the amplitude of the response and that of the excitation which is identical to that obtained via classical harmonic-
P.K. Koliopulos et ai., Response amplitude probability functions
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balance deterministic analysis. Setting for convenience, v = tof/too,
y ( t ) = ~/Xo(t)2 ,
$ = a/to o ,
O(t) = F o ( t ) 2 y / 2 t o 4 ,
(4)
this relation reads y 3 + g8( l _
v2 ) y 2 + _~_ [ ( l _
v2)2 +482~,21y=-~-0. 32
(5)
The above expression is a memoryless nonlinear transformation of the input random variable 0 and hence can be used to express various statistical characteristics of the response variable y in terms of those of 0. Since the input process F ( t ) models the output of r~ linear shaping filter driven by white noise, it follows the Gaussian distribution, and hence its amplitude F 0 is Rayleigh distributed: p(F0) = F--~ e x p [ - ~1 (F0/trr)2]
= 2tr 2 .
(6)
O" F
The transformation rule for the probability density of functions of random variables reads
p o[Fo(O)] po(o)= aolaFo "
(7)
Substituting this result into the expression of 0 in (4) and introducing a new variable , / = (0), we obtain 2
p o ( O ) = _1 exp(-O/r/) ,
r/
r/ = ( 0 ) = ---7"TtrF• tOo
(8)
A subsequent implementation of the transformation rule in order to obtain the probability density of y is prevented, however, due to the fact that (5) exhibits multiple solutions. For each pair of parameters (8, v), a computer search of the real and positive roots of (5) can evaluate the pair of corresponding threshold values (0min, emax) which define the region of acceptable multiple solutions (corresponding to jumps). When the excitation variable 0 is outside this region, the implementation of (7) results in the computation of the response probability density. The problem arises for the values 0mi. ~
