A Modified Implicit Euler Algorithm for Solving Vehicle Dynamic Equations

Abstract

Vehicle modelling is usually done by Multibody Systems. Very often the overall model consists of several subsystems, like the vehicle framework, the drive train and the steering system. Due to the tire forces and torques and due to small but essential compliances in the axle/wheel suspension systems the resulting differential equations are stiff. To improve the model quality dynamic models for some components like damper, and rubber elements are used. Again these models contain stiff parts. If the implicit Euler Algorithm is adopted to the specific problems in vehicle dynamics a very effective numerical solution can be achieved. Applied to vehicle dynamic equations the algorithm produces good and stable results even for integration step sizes in the magnitude of milliseconds. As it gets along with a minimum number of operations a very good run time performance is guaranteed. Hence, even with very sophisticated vehicle models real time applications are possible. Due to its robustness the presented algorithm is very well suited for co-simulations. The modifications in the implicit Euler Algorithm also make it possible to use a simple model for describing the dry friction in the damper and in the brake disks. A quarter car vehicle model with a longitudinal and a vertical compliancy in the wheel suspension and a dynamic damper model including dry friction is used to explain the algorithm and to show its benefits.