Fast cell-to-cell mapping (path integration) with probability tails for the stochastic response of non-linear white noise and Poisson driven systems

摘要

The stochastic response of non-linear non-hysteretic single-degree-of-freedom oscillators subject to random excitations with independent increments is studied, where the state vector made up of the displacement and the velocity components becomes a Markov process. Random stationary white noise excitations and homogeneous Poisson driven impulses are considered as common examples of random excitations with independent increments. The applied method for the solution of the joint probability density function (jpdf) of the response is based on the cell-to-cell mapping (path integration) method, in which a mesh of discrete states of the Markov vector process is initially defined by a suitable distribution throughout the phase plane and the transition probability matrix related to the Markov chain originating from this discretization is approximately calculated. For white noise driven systems, transitions are assumed to be locally Gaussian and the necessary conditional mean values and covariances for only the first time step are obtained from the numerical integration of the differential equations for these quantities in combination with a Gaussian closure scheme. For Poisson driven systems, the transition time interval is taken sufficiently small so that at most one impulse is likely to arrive during the interval. The conditional transitional jpdf for exactly one impulse occurrence in the transition time interval is obtained by a new technique in which a convection expansion in terms of pulse intensities is employed. Next, the time dependent jpdf of the response is obtained by passing the system through a sequence of transient states. The formulation allows for a very fast and very accurate calculation of the stationary jpdf of the displacement and velocity by solving an eigenvector problem of the transition probability matrix with eigenvalue equal to 1. The method has been applied to the Duffing oscillator, and the results for the stationary jpdf and extreme-values have been compared to analytically available results for white noise driven systems and to those obtained from extensive Monte Carlo simulations for Poisson driven systems.