# Tire Model with Temperature Effects for Formula SAE Vehicle
## 3 Thermal Model
### Comparison with the Established Models  
The Sorniotti model [12] describes an empirical model to estimate tire temperature as the function of the actual working conditions of the component. To evaluate the temperature effect on tire forces, a combination of the estimated temperature with a tire brush model [24] was used. The model was empirically tuned using experimental data to demonstrate the variation of tire performance as temperature function. The thermal model considers distinct thermal capacities related to the tread and carcass. The tread thermal capacity is related to the heat flux caused by the tire-road forces and carcass thermal capacity is affected by rolling resistance and exchanges heat with the external ambient. Other heat fluxes corresponded to the ambient and exchange between the two capacities. The original model [12] did not consider the heat flow between the road and tread. Therefore, the heat flow term \(P_{trad,road}\) was introduced to the original model to improve the accuracy according to [25]. The model is described as:  
\[ \begin{array}{l}C_{eq\_carcass}\frac{dT_{crosses}}{dt}=P_{rolling\_resi\tan\alpha}+P_{ conduction}+P_{ambient\_carcass}\\ C_{eq\_trad}\frac{dT_{total}}{dt}=P_{Fx,tire}+P_{Fy,tire}-P_{ conduction}+P_{ambient\_trad}+P_{trad\_road}\end{array}  \tag{14}\]Power fluxes corresponding to the cooling flux due to the temperature difference between carcass and ambient \(P_{ambient,across}\), and tire tread and ambient \(P_{ambient,trend}\) are defined as:  
\[ \begin{array}{l}P_{ambient,across}=h_{across}(T_{ambient}-T_{across})\\ P_{ambient,trend}=h_{trend}(T_{ambient}-T_{trend})\end{array}  \tag{15}\]  
Power fluxes related to conduction between tread and carcass \(P_{conduction}\) and between tread and road \(P_{trend,read}\) are calculated as:  
\[ \begin{array}{l}P_{conduction}=h_{conduction}(T_{trend}-T_{across})\\ P_{trend,road}=H_{trend-road}(T_{trend}-T_{road})\end{array}  \tag{16}\]  
Compared to the Sorniotti model, the proposed thermal model due to a higher differential order should capture more dynamics related to heat flow and potentially produce better results. The Kelly and [PERSON] model [18] states that in racing applications, the temperature of the tread significantly affects both the tire stiffness and the contact patch friction. It should be noted that the effect on the friction is higher. Since the rubber viscoelastic properties depend on temperature, the maximum performance on the racetrack is only available in a specific temperature range. The tire model is also based on the brush model. Using the bristle stiffness \(c_{p}\) the adhesion part can be presented:  
\[c_{p}=\frac{w_{cp}G_{trend}}{h_{trend}} \tag{17}\]  
where \(w_{cp}\) is the contact patch width; \(h_{trend}\) is the tread height. The shear modulus of the tread \(G_{trend}\) is defined as:  
\[G_{trend}=\frac{G_{TA}-G_{\text{limit}}}{e^{-K_{G}T_{GA}}}e^{-K_{G}T_{trend}}+ G_{\text{limit}},\text{ where }K_{G}=\frac{\log(G_{TA}-G_{\text{limit}})-\log(G_{TB}-G_{\text{limit}})}{T_{ GB}-T_{GA}} \tag{18}\]  
where \(G_{TA}\) is the reference shear modulus at temperature A; \(G_{TB}\) is the reference shear modulus at temperature B; \(G_{limit}\) is the limit shear modulus value at a high temperature; \(T_{GA}\) is the reference temperature A; \(T_{GB}\) is the reference temperature B. The sliding part of the contact patch is described using the dynamic friction coefficient.