Moment equations and cumulant-neglect closure techniques for non-linear dynamic systems under renewal impulse process excitations

Abstract

The moment equations technique together with modified cumulant-neglect closure techniques is developed for a non-linear dynamic system subjected to a random train of impulses driven by an Erlang renewal counting process. The original non-Markov problem is converted into a Markov one by recasting the excitation process with the aid of an auxiliary, pure-jump stochastic process characterized by a Markov chain. Hence the conversion is carried out at the expense of augmentation of the state space of the dynamic system by Markov states of the auxiliary jump process. The problem is characterized by the set of joint probability density — discrete distribution function, which are the joint probabilities of the original state vector and of the Markov chain being in the particular jth state. Accordingly the statistical moments of the state variables are defined as integrals with respect to the mixed-type, probability density — discrete distribution function. The differential equations for moments are obtained with the aid of the forward integro-differential Chapman–Kolmogorov operator (Iwankiewicz, 2014). Two modified closure techniques are developed. The first one is the result of application of cumulant-neglect closure directly to unconditional moments. The second modified closure approximation technique is based on the representation of the joint probability density — discrete distribution function by conditioning it on two mutually exclusive and exhaustive events: that the Markov chain is in the jth state while the system is at rest (it is in the jth state for the first time) and that the Markov chain is in the jth state while the system is not at rest (it is in the jth state for any subsequent time). Thus the joint probability density function consists of a Dirac-delta spike and of the continuous part (cf Iwankiewicz et al. (1990)). The cumulant-neglect closure approximations are first formulated for the conditional moments resulting from the continuous part of the probability density function and next for the unconditional moments. As an example of a non-linear system the oscillator with cubic restoring force term is considered. The equations for moments up to the fourth-order are derived. Hence the modified cumulant-neglect closure approximations are derived for redundant fifth- and sixth order moments, both centralized and ordinary. The developed moment equations with modified cumulant-neglect closure techniques are verified against Monte Carlo simulations.