Equations for probability density of response of dynamic systems to a class of non-Poisson random impulse process excitations

The excitation considered in the present paper is a random train of impulses driven by two classes of non-Poisson counting processes. The impulse processes are obtained by selecting impulses from a Poisson and from an Erlang-driven trains of impulses with the aid of an additional, purely jump stochastic process, assumed as an auxiliary state variable. The variable introduced for the first class of non-Poisson processes is governed by the stochastic differential equation driven by two independent Poisson processes, with different parameters, and is tantamount to a two-state Markov chain. The variable introduced for the second class of non-Poisson processes is governed by the stochastic differential equation driven by two independent Erlang processes, with different parameters. As each Erlang process is tantamount to a number of Markov states, the Markov chain for the whole problem is constructed. The equations governing the joint probability density-distribution function of the state vector of the dynamic system and of the Markov states are derived from the general integro-differential forward Chapman–Kolmogorov equation. The necessary jump probability intensity functions are evaluated for both classes of impulse processes and for purely external as well as parametric excitations. Parametric excitation multiplicative to the displacement and to the velocity state variable is considered. The resulting set of coupled integro-partial differential equations is obtained.