# Tire Model with Temperature Effects for Formula SAE Vehicle
## 4 Modified Magic Formula Incorporating Temperature Effect
### Proposed Modification for Temperature Effect  
To change the peak friction and the stiffness of the force curve according to temperature variation, the magic formula was modified to affect the tire forces and moments. The proposed modifications are only demonstrated for the basic magic formula; meanwhile, they were integrated with the extended version of the magic formula. To represent temperature change, a delta temperature \(\Delta T\) was defined:  
\[\Delta T=\frac{T-T_{ref}}{T_{ref}} \tag{23}\]  
where \(T_{ref}\) is reference temperature (the performance is known from measurements). To evaluate the performance of the proposed modification, the normalized friction values from measurement data and simulation results were compared. The measured temperature was used as an input to the modified magic formula. Longitudinal Force. Four temperature coefficients were introduced to capture the temperature effect on the longitudinal force. The coefficients related to temperature effect are: \begin{tabular}{l l} \(T_{X1}\) & Linear temperature effect on longitudinal slip stiffness \\ \(T_{X2}\) & Quadratic temperature effect on longitudinal slip \\  & stiffness \\ \(T_{X3}\) & Linear temperature effect on longitudinal friction \\ \(T_{X4}\) & Quadratic temperature effect on longitudinal friction \\ \end{tabular}   
To affect the longitudinal slip stiffness and longitudinal friction, the above-mentioned coefficients were used. The modified peak factor \(D_{x}\) and longitudinal slip stiffness \(K_{x}\) are:  
\[ \begin{split} D_{x}&=\Big{(}1+T_{X3}\Delta T+T_{X4} \Delta T^{2}\Big{)}D_{xbase}\\ K_{x}&=\Big{(}1+T_{X1}\Delta T+T_{X2}\Delta T^{2} \Big{)}K_{xbase}\end{split}  \tag{24}\]  
where \(D_{xbase}\) is the peak factor of pure longitudinal force \(D\) and \(K_{xbase}\) is the longitudinal slip stiffness \(K_{x}\) according to the basic magic formula. As can be observed in Figure 9, the measurement data and simulation results of normalized longitudinal friction are close to each other. The quantitative comparison between simulation and experimental results for longitudinal force is summarized in Table 3. It also can be noted that temperature changes longitudinal stiffness and the longitudinal friction similar to experimental data. The main difference is related to the measurement noise and fluctuations due to road surface irregularities. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Temperature Range** & **Longitudinal Force \(F_{x}\)** & **Lateral Force \(F_{y}\)** \\ \hline \(T<50\)\({}^{\circ}\)C & - & 18.35\% \\ \(50\)\({}^{\circ}\)C \(\leq T<60\)\({}^{\circ}\)C & 14.35\% & 11.53\% \\ \(60\)\({}^{\circ}\)C \(\leq T<70\)\({}^{\circ}\)C & 12.9\% & 4.9\% \\ \(70\)\({}^{\circ}\)C \(\leq T<80\)\({}^{\circ}\)C & 12.5\% & - \\ \(
Lateral Force and Self-Aligning Moment. Four temperature coefficients were introduced to incorporate the temperature effect in lateral force variation:  
\(T_{Y1}\) Temperature effect on cornering stiffness magnitude  
\(T_{Y2}\) Temperature effect on location of cornering stiffness peak  
\(T_{Y3}\) Linear temperature effect on lateral friction  
\(T_{Y4}\) Quadratic temperature effect on lateral friction  
To affect the cornering stiffness and lateral friction respectively, the above-mentioned coefficients were used. The modified peak factor \(D_{y}\) and the cornering stiffness \(K_{y}\) are described:  
\[ \begin{split}& D_{y}=\left(1+T_{Y3}\Delta T+T_{Y4}\Delta T^{2} \right)D_{ybase}\\ & K_{y}=(1+T_{Y1}\Delta T)P_{KY1}F_{z0}\sin(\arctan(\frac{F_{z}} {P_{KY2}F_{z0}(1+T_{Y2}\Delta T)}))\end{split}  \tag{25}\]  
where \(D_{ybase}\) is the peak factor of pure lateral force \(D\) in the basic magic formula; \(F_{z0}\) is the nominal normal load; \(P_{KY1}\) is the maximum value of stiffness \(K_{f\!y}/F_{z0}\); \(P_{KY2}\) is the load at which \(K_{f\!y}\) reaches maximum value. The results of normalized lateral force from measurement data and simulation results are shown in Figure 10 and demonstrate a similar behavior while temperature changes.