New partial differential equations governing the joint, response-excitation, probability distributions of nonlinear systems, under general stochastic excitation. II: numerical solution

摘要

In a companion paper (Athanassoulis and Sapsis 2006, this Conference) the problem of determining the probabilistic structure of the dynamical response of nonlinear systems subjected to general, external, stochastic excitation has been considered, and new partial differential equations have been derived, governing the joint, response-excitation, characteristic function and probability density function. These new equations are supplemented by a marginal compatibility condition (with respect to the known probability distribution of the forcing), which is of non-local character and, thus, difficult to implement. This is the price paid for discarding the assumption that the forcing is a process of independent increments, which implies that the response is now non-Markovian. In the present paper a method for the numerical solution of these new equations is introduced and illustrated through its application to a specific, simple, nonlinear problem. The solution method is based on the representation of the joint probability density function (or the joint characteris-tic function) by means of a convex superposition of kernel functions, which permits to satisfy a priori the non-local marginal compatibility condition. On the basis of this representation, the partial differential equation that governs the joint, response-excitation, probability density function (or the joint characteristic function) is even-tually transformed to a system of ordinary differential equations for the kernel parameters. Numerical results are presented for the case of a first order dynamical system, with cubic nonlinearity, under smooth stochastic excitations. An important feature of the proposed numerical solution method is its suitability to be implemented using parallel or distributed computing schemes.