## 3 Tire-Road Contact Force Equations through Numerical Analysis
### Tire Contact Force Equations
The results indicate that the contact forces estimated by the equation based on the oval assumption have a large difference from the simulation results. On the contrary, the equation based on the modified footprint achieved excellent results in the calculation of tire force. Compared to passenger car tires, truck tires always have higher inflation pressure and load. Meanwhile, the deformation process of truck tires is also different from that of passenger car tires. It is noted that based on the statistical analysis of the measured data, the actual inflation pressure of the truck generally exceeds the standard inflation pressure [41]. Thus, the inflation pressure of the truck tire was set in the range of 0.7 MPa to 1 MPa, and the vertical deflection was in the range of 10 mm to 20 mm. Figure 13 shows the footprint results of a truck tire under different vertical deflections. Figure 12: Estimation error of passenger car tire contact force: (**a**) different vertical deflection; (**b**) different inflation pressure. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Tire Type** & **Inflation** & **Vertical** & **Simulation** & **Oval Equation (N)** & **Modified** \\  & **Pressure (MPa)** & **Deflection (mm)** & **Force (N)** & **Oval Equation (N)** & **Equation (N)** \\ \hline
205/50R16 & 0.25 & 3 & 1549.5 & 2547.9 & 1622.0 \\  & 0.25 & 10 & 4235.1 & 8428.5 & 4471.2 \\  & 0.25 & 15 & 5784.6 & 12,573.6 & 5453.3 \\  & 0.25 & 20 & 6762.7 & 16,672.6 & 6270.5 \\  & 0.2 & 10 & 3322.3 & 6742.8 & 3576.9 \\  & 0.225 & 10 & 3974.1 & 7585.6 & 4024.1 \\  & 0.275 & 10 & 4552.0 & 9271.3 & 4918.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Contact forces of passenger car tire. As for the truck tire with heavy load, the tire contact length is usually larger than the contact width. In Figure 13a-c, as the deformation of the tire is small, the contact width is smaller than the tread width and the shape of the footprint looks like an oval. Referring to the assumption in Section 2, the contact width \(b\) of the truck tire is assumed to be 0.6\(a\). Similarly, when the contact width is equal to the tread width as shown in Figure 13d, the footprint is likely to be a combination of a rectangle with two half circles on each side, which is the same as the assumption in Section 2. Accordingly, the improved contact force equation of truck tires can be expressed as:  
\[S_{Tuck}=\left\{ \begin{array}{ll}0.6\pi\delta(D+2H-\delta)&,\ b<B^{\prime}\\ B^{\prime}[2\sqrt{(D+2H)\delta-\delta^{2}}-(1-0.25\pi)B^{\prime}]&,\ b=B^{ \prime}\end{array} \right. \tag{11}\]  
By substituting Equation (11) into Equation (1), the tire-road contact force of the passenger car tire can be expressed as  
\[F_{Tuck}=\left\{ \begin{array}{ll}0.6\pi P\delta(D+2H-\delta)&,\ b<B^{\prime} \\ PB^{\prime}[2\sqrt{(D+2H)\delta-\delta^{2}}-(1-0.25\pi)B^{\prime}]&,\ b=B^{ \prime}\end{array} \right. \tag{12}\]  
The calculated results for the tire-contact force under different footprint assumptions are shown in Table 4.  
The relative errors are shown in Figure 14. It should be noted that the values calculated by the rectangular equation (i.e., Equation (6)) in Table 4 are much larger than the simulation value. Thus, the error of the rectangular equation is not analyzed in the figure. From Figure 14a,b, we can see that the change trend in the Huang equation (i.e., Equation (7)) and the modified equation (i.e., Equation (12)) is similar. Meanwhile, the force estimation error is smaller compared with that of the passenger car. The error of the Huang equation is in the range of 11.67\(\sim\)20.21% while the error of the modified equation is in the range of 0.68\(\sim\)8.37%. This means that the footprint-modified equation significantly improves the results.