Calculation of high-dimensional probability density functions of stochastically excited nonlinear mechanical systems

Abstract

Technical systems are subjected to a variety of excitations that cannot generally be described in deterministic ways. External disturbances like wind gusts or road roughness as well as uncertainties in system parameters can be described by random variables, with statistical parameters identified through measurements, for instance. For general systems the statistical characteristics such as the probability density function (pdf) may be difficult to calculate. In addition to numerical simulation methods (Monte Carlo Simulations, MCS) there are differential equations for the pdf that can be solved to obtain such characteristics, most prominently the Fokker–Planck equation (FPE). A variety of different approaches for solving FPEs for nonlinear systems have been investigated in the last decades. Most of these are limited to considerably low dimensions to avoid high numerical costs due to the “curse of dimension”. Problems of higher dimension, such as d=6, have been solved only rarely. In this paper we present results for stationary pdfs of nonlinear mechanical systems with dimensions up to d=10 using a Galerkin method, which expands approximative solutions (weighting functions) into orthogonal polynomials.