Dynamic Response of Non-Linear Systems to Random Trains of Non-Overlapping Pulses

The stochastic excitation considered is a random train of rectangular, non-overlapping pulses, with random durations completed at latest at the next pulse arrival. For Erlang distributed interarrival times and for the actual distributions of pulse durations determined from the primitive Erlang distribution, the formulation of the problem in terms of a Markov chain allows to evaluate the mean value, the autocorrelation function and the characteristic function of the excitation process. However, the state vector of the dynamical system is a non-Markov process. The train of non-overlapping pulses with parameters ν, 1 and μ, 1 is then demonstrated to be a process governed by a stochastic equation driven by two independent Poisson processes, with parameters ν and μ, respectively. Hence, the state vector of the dynamical system augmented by this additional variable becomes a Markov process. The generalized Itô’s differential rule is then used to derive the equations for the characteristic function and for moments of the response of a non-linear oscillator.