A Semi‐Analytical Method of Solving the Fokker‐Planck‐Equation for High‐Dimensional Nonlinear Mechanical Systems

Abstract

AbstractStochastic processes are a common way of describing systems that are subjected to random influences. Technical systems may be excited by road roughness or wind gusts, for example, as well as fluctuating system parameters, which can all be described by stochastic differential equations. In previous works by the author and others (see [1], for example) it has been demonstrated how a Galerkin‐method can be used to obtain global numerical solutions of the Fokker‐Planck‐Equation (FPE) for nonlinear random systems. Computational efforts are reduced by orthogonal polynomial expansion of approximate solutions so that probability density functions (pdfs) for comparably high‐dimensional problems have been computed successfully. Stationary mechanical systems with dimensions up to d = 10 have been investigated, including polynomial as well as non‐smooth nonlinearities. This article presents results for different energy‐harvester‐systems under stochastic excitation, a field of research that has become the subject of increasing attention in the last years. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)