FAST CELL-TO-CELL MAPPING (PATH INTEGRATION) FOR NONLINEAR WHITE-NOISE AND POISSON DRIVEN SYSTEMS

The stochastic response of nonlinear nonhysteretic single-degree-of-fr eedom oscillators subject to random excitations with independent incre ments is studied, where the state vector made up of the displacement a nd the velocity components becomes a Markov process. Random stationary white noise excitations and homogeneous Poisson driven impulses are c onsidered as common examples of random excitations with independent in crements. 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 distri bution throughout the phase plane and the transition probability matri x 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 quan tities in combination with a Gaussian closure scheme, For Poisson driv en systems, the transition time interval is taken sufficiently small s o that at most one impulse is likely to arrive during the interval. Th e 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 sy stem through a sequence of transient states. The formulation allows fo r 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 sta tionary jpdf and extreme values have been compared to analytically ava ilable results for white noise driven systems acid to those obtained f rom extensive Monte Carlo simulations for Poisson driven systems.