# Measurement and Modeling of a Cargo Bicycle Tire for Vehicle Dynamics Simulation
## 3 Results
### Impact of Tire Pressure and Normal Force  
To investigate the impact of tire pressure on the measurements, Figure 12 shows the model curves of \(F_{t,x}(\kappa)\), \(F_{t,y}(\alpha)\) and \(T_{t,z}(\alpha)\) at a normal force of 625 N and tire pressures of 3.0, 3.5 and 4.0 bar. As can be seen from (a), (b) and (c) in Figure 12, a change from 1 bar to the tire pressure only had a minor effect on the maximum transmittable longitudinal and lateral forces, as well as the aligning torque of the bicycle tire tested when \(F_{N,z}=625\) N. In addition, the slope of the models in the linear range remains almost the same. The impact of changing the tire pressure from 3 bar to 4 bar in steps of 0.5 bar on the longitudinal and lateral forces \(F_{t,x}(\kappa)\) and \(F_{t,y}(\alpha)\) as well as the aligning torque \(T_{t,z}(\alpha)\) at \(F_{N,z}=765\) N is shown in Figure 13. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Test No.** & \(B_{\mathbf{z}}\) & \(C_{\mathbf{z}}\) & \(D_{\mathbf{z}}\) & \(E_{\mathbf{z}}\) & \(S_{H,z}\) & \(S_{V,z}\) & \(\mathbf{R^{2}}\) & **NRMSE** \\ \hline
1.1 & 0.185 & 7.715 & 3.600 & 1.340 & 1.150 & 0.000 & 0.941 & 0.061 \\
1.2 & 0.082 & 9.000 & 4.397 & 1.717 & 2.200 & 0.000 & 0.969 & 0.047 \\
2.1 & 0.189 & 7.442 & 3.500 & 1.339 & 1.160 & 0.000 & 0.912 & 0.065 \\
2.2 & 0.101 & 8.381 & 4.000 & 1.607 & 1.910 & 0.000 & 0.937 & 0.058 \\
3.1 & 0.126 & 8.611 & 3.700 & 1.627 & 1.490 & 0.000 & 0.902 & 0.074 \\
3.2 & 0.128 & 7.572 & 4.200 & 1.441 & 1.680 & 0.000 & 0.905 & 0.072 \\ \hline \end{tabular}
\end{table} 
Table 6: Magic formula coefficients and model fit for \(T_{t,z}(\kappa)\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Test No.** & \(B_{\mathbf{x}}\) & \(C_{\mathbf{x}}\) & \(D_{\mathbf{x}}\) & \(E_{\mathbf{x}}\) & \(S_{H,x}\) & \(S_{V,x}\) & \(\mathbf{R^{2}}\) & **NRMSE** \\ \hline
1.1 & 0.133 & 1.364 & 678.7 & 0.000 & 0.000 & \(-\)36.080 & 0.996 & 0.026 \\
1.2 & 0.094 & 1.700 & 892.4 & 0.665 & 0.000 & \(-\)17.260 & 0.998 & 0.018 \\
2.1 & 0.108 & 1.640 & 663.4 & 0.680 & 0.000 & \(-\)42.600 & 0.997 & 0.023 \\
2.2 & 0.105 & 1.436 & 826.9 & 0.122 & 0.000 & \(-\)57.670 & 0.997 & 0.023 \\
3.1 & 0.121 & 1.611 & 675.2 & 0.713 & 0.000 & \(-\)17.170 & 0.998 & 0.018 \\
3.2 & 0.067 & 2.146 & 831.1 & 0.982 & 0.000 & \(-\)12.100 & 0.999 & 0.015 \\ \hline \end{tabular}
\end{table} 
Table 4: Magic formula coefficients and model fit for \(F_{t,x}(\kappa)\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Test No.** & \(B_{y}\) & \(C_{y}\) & \(D_{y}\) & \(E_{y}\) & \(S_{H,y}\) & \(S_{V,y}\) & \(\mathbf{R^{2}}\) & **NRMSE** \\ \hline
1.1 & 0.210 & 1.412 & 774.0 & 0.633 & 0.000 & 0.000 & 0.995 & 0.017 \\
1.2 & 0.127 & 1.373 & 1045.0 & 0.409 & 0.000 & 0.000 & 0.999 & 0.009 \\
2.1 & 0.250 & 1.119 & 789.7 & \(-\)0.461 & 0.000 & 0.000 & 0.994 & 0.019 \\
2.2 & 0.123 & 1.542 & 922.1 & 0.388 & 0.000 & 0.000 & 0.998 & 0.011 \\
3.1 & 0.174 & 1.561 & 788.1 & 0.618 & 0.000 & 0.000 & 0.993 & 0.020 \\
3.2 & 0.161 & 1.216 & 912.0 & 0.397 & 0.000 & 0.000 & 0.997 & 0.015 \\ \hline \end{tabular}
\end{table} 
Table 5: Magic formula coefficients and model fit for \(F_{t,y}(\alpha)\). Compared with the measurements at \(F_{N,z}=625\) N, a contact force of \(F_{N,z}=765\) N results in a significantly larger spread of the force curves \(F_{t,x}\) and \(F_{t,y}\) in the range of \(\kappa>10\%\) and \(\alpha>4^{\circ}\) at different tire pressures. Furthermore, the three-bar model curve can reach the highest force values in both the longitudinal and lateral direction. However, the four-bar curve produces the lowest force values in both directions. By comparing the curves of \(F_{N,z}=625\) N and \(F_{N,z}=765\) N, it is clear that the effects of different tire pressures are more significant for a higher value of \(F_{N,z}\).