## 3 Simulation of Low-Speed Steering Friction Force between the Tire and Pavement  
In this section, the finite element models of tire and pavement based on ABAQUS software are established to carry out the simulation and calculation of steering friction force under a low-speed steering condition. The law of steering friction force with running speed of tire is explored. The focus of this section is to study the friction between two surfaces, therefore, a few simplified treatments are made in tire modeling to reduce the difficulty of software calculation: The lateral tread pattern is ignored and only the longitudinal tread pattern is retained; the internal complex structure of tire is ignored and only the tread, carcass, cord layer and belt layer of the tire are retained; moreover, the rim would constrain the tire's deformation and should not be ignored, which is simplified into a flat-bottomed steel ring with a regular shape and assembled with the tire. Since the rubber material, which makes up the tire tread, has extremely complex characteristics, it is usually defined as the isotropic, incompressible and hyper-elastic material in the model. The Mooney-Rivlin material model is used in this section to describe the constitution of rubber and the material parameters of each component, as shown in Table 1[26; 27]. When modeling, the steel wires of the cord layer and belt layer are simulated by a rebar unit, which is firstly defined on the surface unit and then embedded into the corresponding entity unit of rubber. The tread and carcass of the tire are constrained by a tie. The rim, which is not deformed, is rigidly coupled with the central point of carcass' rotation axis, thus it can move with the tire carcass. The finite element models of the tire and pavement are shown in Figure 2. To clarify, the kingpin inclination and caster are usually not considered in 3D finite element models to simplify the models and facilitate calculation, and the tire just rolls and deflects around the central point which is coupled with the periphery of the rim. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Part**} & **Elasticity** & **Poisson’s** & **Density** & **Mooney Coefficients** \\  & **Modulus (MPa)** & **Ratio** & **(g/cm\({}^{3}\))** & **(MPa)** \\ \cline{3-5}  & & & \(C_{10}\) & \(C_{01}\) \\ \hline \hline \multirow{2}{*}{
\begin{tabular}{c} Tread \\ Carcass \\ Belt \\ Pavement \\ \end{tabular} } & & 1.14 & 2.0477 & 1.1859 \\ \cline{1-1} \cline{2-5}  & 794 & 0.45 & 1.39 & \multicolumn{1}{c}{\(\cline{1-1} \cline{2-5}  \\ \cline{1-1} \cline{2-5}  & 55,000 & 0.3 & 7.64 & \multicolumn{1}{c}{\(\cline{1-1} \cline{2-5}  \\ \cline{1-1} \cline{2-5}  \\ \cline{1-1} \cline{2-5}  & 1400 & 0.35 & 2.4 & \multicolumn{1}{c}{\(\cline{1-1} \cline{2-5}  \\ \hline \hline \end{tabular} } \\ \hline \end{tabular}
\end{table} 
Table 1: Material properties of the tire and pavement. In the simulation of low-speed steering friction, the tire pressure was set as 0.25 MPa, the vertical load on the tire was set as 3000 N and the friction coefficient of surface-to-surface contact was taken as 0.7, which is the initial value set in the software, indicating the contact property between two surfaces. The steered process of the tire was simulated by the tire's forward rolling and the pavement's horizontal tuning, respectively. The simulations were respectively performed in tire running speeds of 1 km/h, 2 km/h, 4 km/h, 6 km/h, 8 km/h and 10 km/h. The first 3 simulation results of the steering friction force are shown in Figure 3a. It can be seen that the friction force at each speed rose gradually and then kept a steady state. The steering friction force in the steady state was averaged in all simulation results, and then the average value was treated as the steering friction force at each corresponding speed as shown in Table 2. It can be seen that the steering friction force obviously decreased with the running speed of the tire.