On non-linear stochastic dynamics of quarter car models

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.