# Tire Model with Temperature Effects for Formula SAE Vehicle
## 5 Integration into Full FSAE Vehicle Model
### Validation  
To validate the overall model, two test scenarios were considered. The first test scenario was a skidpad maneuver to evaluate a steady-state lateral behavior. The second test scenario was a full lap maneuver performed for a longer time. Skidpad Maneuver. The skidpad course consists of two pairs of concentric circles in a figure of eight pattern. The centers of these circles are 18.25 m apart. The inner circles are 15.25 m in diameter and the outer circles are 21.25 m in diameter. The driving path is the 3 m wide path between the inner and outer circles [31]. To increase the signal-to-noise ratio without significant signal distortion, the raw sensor data was filtered using a Savitzky-Golay digital filter [32]. The filtered data was used for comparison with simulation results. The longitudinal and lateral accelerations, and yaw rate are shown in Figure 13.  
The simulation shows more oversteered behavior compared to the measured data. The mismatch can be caused by the modeling of chassis roll stiffness. A real chassis has a limited roll stiffness; however, as mentioned, the chassis was assumed as a rigid body. Hence, the chassis was infinitely stiff in the simulation model. It could result in a higher yaw rate and increase the root mean square error (RMSE) of the yaw rate. The longitudinal and lateral accelerations, and yaw rate were better aligned with the measurement data in the left turn circles. This observation can be explained by the fact that the elevation change in the skidpad circuit affected the measurements from the accelerometer sensor. Figure 12: Model structure. Figure 14 shows the evolution of tire temperature during skidpad maneuver for four wheels: front left (FL), front right (FR), rear left (RL), and rear right (RR). As can be observed, due to higher normal loads, the outer tires (initially left tires for the right turn) start heating up first. When the vehicle steered in another direction (at 15 s), the right tires start heating up. Full Lap. To validate transient behavior of the simulation model, a full lap maneuver was investigated. The measurement data correspond to one of the endurance runs with a duration of \(-22\) km driving (18-20 laps). The simulation was conducted for one lap of the circuit performed approx. in 50 s.  
In a similar way as was done for the skidpad scenario, the raw measurement data of longitudinal and lateral accelerations, and yaw rate, were filtered and the RMSE calculated. The RMSE for the longitudinal and lateral accelerations, as well as yaw rate are summarized in Table 4, including also the simulation results performed without the proposed model. The comparison between simulation results with a temperature effect and measurement data is shown in Figure 15. It can be noted that the yaw rate from the simulation results was aligned with the measured data; however, an offset can be observed which might relate to the above-mentioned difference in the chassis roll stiffness. The RMSE for longitudinal acceleration is slightly higher compared to the skidpad scenario; however, the longitudinal acceleration in the skidpad maneuver was much lower compared to a full lap scenario. Thus, the simulation results of longitudinal dynamics using wheel torques as an input matched the longitudinal acceleration as sufficiently accurate. The highest difference was found in lateral dynamics by the RMSE of lateral acceleration of 3.19 m/s\({}^{2}\). The potential reasons can be related to: (i) combined slip conditions; (ii) camber effect on lateral \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Skidpad Maneuver**} & \multicolumn{2}{c}{**Full Lap Maneuver**} \\ \hline Temperature effect & without & with & without & with \\ \hline Longitudinal acceleration, m/s\({}^{2}\) & 0.62 & 0.65 & 1.50 & 1.42 \\ Lateral acceleration, m/s\({}^{2}\) & 1.86 & 1.26 & 4.77 & 3.19 \\ Yaw rate, rad/s & 0.27 & 0.16 & 0.37 & 0.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The root mean square error (RMSE) for both simulation maneuvers.