On non-linear stochastic dynamics of quarter car models

Abstract

In this paper, high-dimensional probability density functions of non-linear dynamical systems are calculated solving the corresponding Fokker–Planck equations. Zeroth approximations are derived from solutions of corresponding linear systems and analytical results for first- and second-order expected values. The zeroth approximations are used as weighting functions for the construction of generalized Hermite polynomials. The Fokker–Planck equation is expanded in terms of these polynomials and subsequently solved by a Galerkin method. As an example, models of a quarter car with non-linear damping subjected to white or colored noise excitation are considered. The damping is piecewise linear and asymmetric leading to a non-vanishing expected value of the displacement of the car. The excitation is realized by the roughness of the road and the car moves with constant velocity. Monte-Carlo simulations and analytical results are used for comparison.