Response and Reliability of Poisson-Driven Systems by Path Integration

Abstract

The paper deals with the stochastic response and the reliability of a nonlinear and nonhysteretic single-degree-of-freedom (SDOF) oscillator subject to a stationary Poisson-driven train of impulses. The state vector made up of the displacement and the velocity components then becomes a Markov process. The applied solution method is based on path integration, which essentially implies that a mesh of discrete states of the Markov vector process is initially defined with a suitable distribution throughout the phase plane, next, the transition probability matrix related to the Markov chain originating from this discretization is calculated, assuming the transition time interval to be sufficiently small so that at most one impulse is likely to arrive during the interval. Obviously, this assumption is best fulfilled for processes with low pulse arrival rates. Consequently, the method is the most effective in such cases in contrast to all other approaches to the considered problems. The time-dependent joint probability density function (PDF) of the displacement and velocity is obtained by passing the system through a sequence of transient states. In the reliability problems, the probability mass is absorbed at the exit part of the boundary of the safe domain during transitions. The considered first passage time problem assumes time-invariant single or double barriers with deterministic or stochastic start in the safe domain. The method has been applied to a Duffing oscillator with linear viscous damping, and the computed results have been compared with those obtained from extensive Monte Carlo simulations.