Dynamic response of non-linear systems to renewal-driven random pulse trains

Abstract

The moment equations technique is developed for the non-linear dynamical systems under random pulse trains driven by a class of renewal processes. Since the increments of the considered point process are not statistically independent, the direct application of the generalized Itô's differential rule does not yield the explicit equations for moments. Hence, the approach is suitably modified. First, the excitation term is recast, for an ordinary renewal process with gamma-distributed, with k = 2, interarrivai times, as a transformation of a Poisson counting process, which allows to perform the averaging of the differential rule. Next, for the additional unknown expectations which consequently appear in the equations for moments, the differential equations in the form of the correlation splitting formulae are derived. The technique developed is applied to a linear oscillator and to a Duffing oscillator. In the latter case, suitable closure approximations are used in order to truncate the hierarchy of moment equations. The analytical results (transient response moments up to fourth order) are verified against the results of Monte Carlo simulations.