Moment equations and quasi-moment neglect closure approximations for non-linear dynamic systems under Erlang renewal impulse process excitations

Abstract

In the present paper the moment equations technique together with two closure approximations is developed for a non-linear dynamic system subjected to an impulse process excitation driven by an Erlang renewal counting process. The original non-Markov problem is converted into a Markov one at the expense of augmentation of the state space of the dynamic system by the Markov states of the introduced auxiliary jump process. Differential equations for moments are derived from the explicit integro-differential equations governing the joint probability density — discrete distribution function (Iwankiewicz,2008), which is a different approach from that of a generating equation for moments (Iwankiewicz,2014). For non-linear dynamic systems with polynomial-type non-linearities two closure approximation techniques, based on neglecting of quasi-moments above certain order, are developed. The first closure technique is the result of application of quasi-moment closure directly to the usual, unconditional moments. The second one, a modified closure approximation technique, is based on the representation of the joint probability density — discrete distribution function in the form consisting of a spike and of the continuous part (cf. (Iwankiewicz,1990). 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 where the usual and the modified quasi-moment neglect closure approximations are used for redundant fifth- and sixth order moments. Mean value and variance of the response have been determined with the aid of the developed approximate techniques and verified against Monte Carlo simulations.