Stochastic Car Vibrations With Strong Nonlinearities

High dimensional probability density functions of nonlinear dynamical systems are calculated by solutions of Fokker-Planck equations. First approximations are derived via the solutions of the associated linear system and the analytical results of the expected values. These first approximations are utilized as weighting functions for the construction of generalized orthogonal polynomials. The Fokker-Planck equation is expanded into these polynomials and solved by a Galerkin method. As an example, a simple model of a quarter car with nonlinear damping subjected to white or coloured noise excitation is considered. The damping characteristic is piecewisely linear and highly non-symmetrical. The excitation is generated by the roughness of the road surface on which the car is driving with constant velocity. The main result is a non-vanishing mean value of the vertical car vibrations. Monte-Carlo simulations and analytical results are applied for comparison and tests.

A new proof of a theorem about generalized orthogonal polynomials

In this note it is shown that a fairly recent result on the asymptotic distribution of the zeros of generalized polynomials can be deduced from an old theorem ofG. Polya, using a minimum of orthogonal polynomial theory.