# Tire Model with Temperature Effects for Formula SAE Vehicle
## 3 Thermal Model  
### Proposed Thermal Model  
A lumped parameter approach is used according to [18] and the thermal model consists of three bodies such as the tread with temperature \(T_{trad}\), the carcass with temperature \(T_{across}\), and the inflation gas with temperature \(T_{gas}\). The road surface with ambient air provides the boundary conditions with fixed temperatures \(T_{trad}\) and \(T_{amb}\). In contrast to the study [19], a single value of tread temperature was calculated without the need to define a complex temperature distribution across the contact patch area. The temperature change of tread, carcass, and gas is described as:  
\[ \begin{array}{l}\dot{T}_{trad}=\frac{Q_{sliding}-Q_{trand-rand}+Q_{across- normal}-Q_{trand-amb}}{S_{trand}M_{trad}}\\ \dot{T}_{across}=\frac{Q_{damp}-Q_{across- normal}-Q_{across- normal}-Q_{across-gas}}{S_{across}M_{across}}\\ \dot{T}_{gas}=\frac{Q_{across-gas}}{S_{gas}M_{gas}}\end{array}  \tag{1}\]  
where \(Q_{sliding}\) is the heat flow due to sliding; \(Q_{trad\to rad}\) is the heat flow between tread and road; \(Q_{across-trad}\) is the heat flow between carcass and tread; \(Q_{trand-amb}\) is the heat flow between tread and ambient air; \(Q_{damp}\) is the heat flow due to carcass deflection; \(Q_{across-amb}\) is the heat flow between carcass and ambient air; \(Q_{across-gas}\) is the heat flow between carcass and inflation gas; \(S_{trad}\) is the specific heat capacity of carcass; \(S_{across}\) is the specific heat capacity of carcass; \(S_{gas}\) is the specific heat capacity of inflation gas; \(M_{trad}\) is the tread mass; \(M_{across}\) is the carcass mass; \(M_{gas}\) is the inflation gas mass. According to [18], two heat generation processes should be considered:  
* Due to carcass deflection  
\[Q_{damp}=\Big{(}E_{x}|F_{x}|+E_{y}\big{|}F_{y}\big{|}+E_{z}|F_{z}|\Big{)}V_{x} \tag{2}\]  
where \(E_{x}\), \(E_{y}\), and \(E_{z}\) are the carcass longitudinal, lateral, and vertical force efficiency factors; \(F_{x}\), \(F_{y}\), and \(F_{z}\) are the longitudinal, lateral, and normal forces; \(V_{x}\) is the longitudinal velocity. * Due to sliding friction in the contact patch  
\[Q_{sliding}=\mu_{d}F_{z}v_{s} \tag{3}\]  
where \(\mu_{d}\) is the dynamic friction coefficient; \(v_{s}\) is the sliding velocity. To define the shift along the \(\mu\) axis with compound temperature, the dynamic friction model [18] was modified. Based on the friction model [20], Equation (4) is proposed incorporating the shift along  
Figure 5: Temperature effects on tire forces for Apollo tire (\(F_{z}\) = 1000 N): **(a)** normalized longitudinal friction \(\mu\) vs. wheel slip \(\kappa\); **(b)** normalized lateral friction \(\mu\) vs. slip angle \(\alpha\). the \(\mu\) axis due to temperature using the parameters \(\mu\) and \(h\) as temperature dependent. The idea was taken from research [21] where the parameters of the friction models from [20] and [22] were made temperature-dependent. The final equation for the dynamic friction model is described:  
\[\mu_{d}(T)=\mu_{base}+\left[\mu_{peak}(T)-\mu_{base}\right]e^{-(h(T)(\log_{10} \left(v_{\max}^{\frac{v_{3}}{\max}}\right)-K_{shift}(T_{trand}-T_{ref})))^{2}} \tag{4}\]  
The temperature-dependent peak friction coefficient \(\mu_{peak}\) is defined:  
\[\mu_{peak}(T)=a_{1}T^{2}+a_{2}T+a_{3} \tag{5}\]  
where \(a_{1}\), \(a_{2}\), and \(a_{3}\) are the tuning parameters for a second order polynomial function. Dimensionless parameter \(h\) is related to the width of the speed range in which the friction coefficient varies significantly [23]. The following temperature-dependent adjustment is made similar to the adaptation of the compound shear modulus in [18]:  
\[h(T)=b_{1}\frac{e^{b_{2}T_{trand}}}{e^{b_{2}T_{ref}}} \tag{6}\]  
where \(b_{1}\) and \(b_{2}\) are the tuning parameters. The proposed modifications cover both sliding and non-sliding friction components. Temperature-dependent peak friction coefficient \(\mu_{peak}\) corresponds to the peak friction properties of the tire compound.