New Resonances and Velocity Jumps in Nonlinear Road-vehicle Dynamics

Abstract

When vehicles are riding on uneven roads, they are excited to vertical vibrations described by linear equations of motion. The vertical vibrations of the vehicle determine the characteristics of the stationary driving force needed to control and maintain a constant velocity. More realistic is the inverse problem of a constant driving force meanwhile the velocity process of the vehicle is stationary, fluctuating around a mean value as the consequence of the random up and down of road surfaces. First and second order road profiles are modeled by linear filter equations under white noise which allow for- and backward drives. The contact between road and vehicle leads to a nonlinear resistance force determined by the damper and spring force of the vehicle multiplied by the road process which represents the vertical road velocity relative to the moving vehicle.The paper investigates new resonances of linear half-car models on road. For multi-body vehicle systems with n wheels on road, the classical covariance matrix equation is extended by double sums of exponential matrix functions in dependence on time differences effected by each wheel pair. Applications are given for half-car models with two wheels and four degrees of freedom. In case of nonlinear quarter car models with one and a half degree of freedom, calculated velocity fluctuations show mean velocity jumps and new bifurcations to bimodal probability densities when the driving force approaches the critical value where the vertical car vibrations become resonant. The original problem of maintaining of constant car velocities, however, leads to largely distributed force distributions of Gaussian product processes which are physically difficult to realize.