Moment Equations for Non-Linear Systems Under Renewal-Driven Random Impulses with Gamma-Distributed Interarrival Times

Abstract

The moment equations technique is devised for non-linear dynamic systems subjected to random trains of impulses driven by an ordinary renewal point process with gamma-distributed integer parameter interarrival times (Erlang process). Since the renewal point process has not independent increments the state vector of the system, consisting of the generalized displacements and velocities, is not a Markov process. Based on the fact that for this class of renewal processes the renewal events are every kth Poisson events (k - being the integer parameter of the gamma distribution) the renewal impulse process is recast in such a way as to express it in terms of the stationary Poisson counting process. This results in the introduction of additional state variables, for which the stochastic equations are also formulated. The resulting state vector augmented by the additional variables is now a Markov vector process.

Next, the equations for the joint central moments of the state variables are obtained based on the generalized Itô’s differential rule valid for Poisson driven processes. As the example problem the Duffing oscillator is considered subjected to the renewal impulse processes with k = 2, k = 3 and k = 4. The cumulant neglect closure is used to truncate the equations for moments at fourth order moments. The computed response mean values and variances are verified against the results of Monte Carlo simulations.