## 3 Tire-Road Contact Force Equations through Numerical Analysis
### Tire Contact Force Equations  
The verified tire FE models (205/50R16 passenger car tire and 11.00R20 truck tire) were analyzed under different conditions. For the passenger car tire, the following cases were analyzed: tire inflation pressure of 0.2, 0.225, 0.25, and 0.275 MPa, and vertical deflection of 3, 10, 15, and 20 mm. \begin{table}
\begin{tabular}{c c c c} \hline  & **Contact Length** & **Contact Width** & **Contact Area** \\ \hline Simulation results & 133 mm & 153 mm & 14,790 mm \\ Measured results & 126 mm & 154 mm & 14,316 mm \\ Absolute error (AE) & 5.2\% & 0.6\% & 3.3\% \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of the tire footprint. Figure 8: Test for obtaining the tire footprint. Figure 9: Footprint of the tire: (**a**) measured result; (**b**) simulation result. The tire footprints under different vertical deflections were extracted as shown in Figure 10. It can be seen that under constant inflation pressure, the area of the tire footprint increases with the increase in tire deflection. When the vertical deflection is small and the contact width is smaller than the tread width, the footprint edge looks like a rhombus, as shown in Figure 10a. As the vertical deflection increases, the contact width reaches the tread width, and thereafter the footprint keeps increasing along the longitudinal direction. The footprint edge becomes a hexagon, as shown in Figures 10b and 11d. In addition, under the constant vertical deflection, the footprint under different inflation pressure is almost the same. This implies that the footprint shape is mainly affected by the vertical deflection. In terms of the above simulation results, the footprint of passenger car tire is not always an oval shape as assumed in Equation (5). It is actually a rhombus when the contact width is smaller than the tread width, and a hexagon when the contact width is larger than the tread width. Based on the analysis of multiple sets of simulation results, the length of the hexagon edge is about half the contact length, thus it is assumed to be \(0.5a\), as shown in Figure 11.  
Accordingly, Equation (5) can be modified as  
\[S_{Car}=\left\{ \begin{array}{ll}2\delta\sqrt{D+2H-\delta}\sqrt{h+0.25\frac{ B^{\prime 2}}{h}-\delta}&,\ b<B^{\prime}\\ 1.5B^{\prime}\sqrt{(D+2H)\delta-\delta^{2}}&,\ b=B^{\prime}\end{array} \right. \tag{8}\]  
By substituting Equation (8) into Equation (1), the tire-road contact force of the passenger car tire can be expressed as  
\[F_{Car}=\left\{ \begin{array}{ll}2P\delta\sqrt{D+2H-\delta}\sqrt{h+0.25\frac{ B^{\prime 2}}{h}-\delta}&,\ b<B^{\prime}\\ 1.5 PB^{\prime}\sqrt{(D+2H)\delta-\delta^{2}}&,\ b=B^{\prime}\end{array} \right. \tag{9}\]  
The calculated results of the tire contact force under different area assumptions are shown in Table 3.  
Figure 11: Calculation of passenger car tire footprint. The relative error (RE) is used to evaluate the accuracy of the contact force estimation as  
\[e_{RE}=\left|\frac{F_{E_{\theta}}-F_{FEA}}{F_{FEA}}\right|\times 100\% \tag{10}\]  
where \(F_{E_{\theta}}\) is the \"calculated force\"; and \(F_{FEA}\) is the \"simulation force\". The comparison of the error results is shown in Figure 12.  
Figure 12a represents the relative error of the contact force under different vertical deflections with an inflation pressure of 0.25 MPa. It is seen that the contact force estimated by the oval equation (i.e., Equation (5)) is much larger than the simulation results, and the error increases with the increase in tire vertical deflection, with a maximum error of 146.54%. Although the error of the modified equation (i.e., Equation (9)) also increases during the vertical deflection from 3 mm to 20 mm, it is in the range of 4.68-7.28%. Figure 12b represents the relative error under different inflation pressure with constant vertical deflection. The error of the oval equation is about 100%. In contrast, the modified equation yields satisfactory results and the maximum error is 8.05%.