An analytical solution to the discontinuity of the brush tire model: using an asymmetric vertical pressure distribution

In this paper, a continuous version of the brush tire model is introduced to facilitate its application in vehicle dynamics analysis and controller design, especially when considering transient behaviour of the tires. An asymmetric vertical pressure distribution satisfying the boundary conditions is proposed, and the equations for calculating the tire forces and moment under the pure translational slip conditions are derived. When the bias factor of the vertical pressure distribution satisfies the given condition, the tire slip characteristics can be described using continuous functions for all slip conditions. The model is further extended to the analysis of transient tire dynamics. Compared with the classical model, the advantages of the continuous model in numerical simulation and analysis are further illustrated. The dynamic friction characteristics is considered to improve the express accuracy of the model. The accuracy of the model is verified by comparing the proposed model with the classical brush tire model and experimental data. The results show that the expression accuracy of the proposed simple model is close to that of the classical brush tire model, and both of them can reproduce the experimental data under a wide range of operating conditions.