# Cornering Stiffness Prediction Based on Geometric Method
## 2 Method Description
### Local Load Calculations
#### 2.1.1 The Tilting Method  
The author put forward the tilting method inspired by the geometric relationship between the tire and the road. The intersection of the undeformed tire and the road is used for indicating the tire's vertical deformation. Deformation without camber angle is shown in Figure 2a. By tilting the tire, Figure 2b shows the deformation change. In Figure 2b, the deformation volume boundaries of two tire parts are perpendicular to the road, because the load is perpendicular to the road. Figure 1: Diagrammatic view of the proposed calculation method. Figure 2: Cross-section diagrammatic drawing of deformation caused by the load. (**a**) Deformation at zero camber angle. \(w\) is the contact patch width of each part, \(d\) is the radial deformation of the tire, \(R_{L}\) is the loaded radius. (**b**) Deformation at non-zero camber angle \(\gamma\). \(d^{\prime}\) is the radial deformation of the cambered tire. \(R^{\prime}_{L}\) is the loaded radius. Points \(A\), \(B\) and \(C\) are the deformation boundary projections on Y-axis. Point \(O\) is the rim centre projection on Y-axis. The green dotted line \(L\) is located \(y\) away from point \(B\). (**c**) The lateral view of the plane where the green dotted line \(L\) is located in Figure 2b. \(R_{y}\) and \(d_{y}\) are the radius and vertical deformation of this plane, respectively. \(\theta_{y}\) is half of the central angle corresponding to the contact length of this plane. (**d**) The boundary conditions used in the calculations. \(w_{0}\) is half the width of the tire. \(d_{0}\) is the maximum thickness of the tread. In Figure 2, the deformation volume shown in red is proportional to the load on the left part, while that shown in yellow is proportional to the load on the right part. Taking the point \(O\) as the origin of the integral coordinate system in Figure 2b, we have:  
\[d_{0}=R_{0}-\sqrt{R_{0}^{2}-w_{0}^{2}} \tag{2}\]  
\[J=d^{\prime}-w_{0}\cdot\tan(\gamma), \tag{3}\]  
\[\overline{AB}=\left\{ \begin{matrix}\sqrt{R_{0}^{2}-\left(R_{L}^{\prime}\cdot \cos(\gamma)\right)^{2}}-R_{L}^{\prime}\cdot\sin(\gamma),&J-d_{0}\leq 0\\ \left(J-d_{0}\right)\cdot\sin(\gamma)+\frac{w_{0}}{\cos(\gamma)},&J-d_{0}>0 \end{matrix} ,\right. \tag{4}\]  
where \(R_{0}\) is the tire unloaded radius. \(J\) is used to determine the position of point \(A\) and choosing the formula of \(\overline{AB}\) (\(J-d_{0}\leq 0\), point \(A\) is on the tread. \(J-d_{0}>0\), and point \(A\) reaches its maximum, as is shown in Figure 2b). \[\overline{BO}=R_{L}^{\prime}\cdot\sin(\gamma), \tag{5}\]  
\[M=d^{\prime}+w_{0}\cdot\tan(\gamma), \tag{6}\]  
\[\overline{OC}=\left\{ \begin{matrix}\sqrt{R_{0}^{2}-\left(R_{L}^{\prime}\cdot \cos(\gamma)\right)^{2}}&,M-d_{0}\leq 0\\ \frac{w_{0}}{\cos(\gamma)}-\left(M-d_{0}\right)\cdot\sin(\gamma)-R_{L}^{\prime }\cdot\sin(\gamma),M-d_{0}>0\end{matrix} ,\right. \tag{7}\]  
where \(M\) is used to determine the position of \(C\) and choose the formula of \(\overline{OC}\) (\(M-d_{0}\leq 0\), point \(C\) is on the tread. \(M-d_{0}>0\), point \(C\) reaches its maximum, as is shown in Figure 2b). Here the point \(C\) is assumed to be to the right of \(O\), which requires \(\gamma<\arcsin(w_{0}/R_{0})\). \[d_{y}=\sqrt{R_{0}^{2}-\mathbf{y}^{2}}-R_{L}^{\prime}\cdot\cos(\gamma), \tag{8}\]  
\[R_{y}=R_{L}^{\prime}\cdot\cos(\gamma)+d_{y} \tag{9}\]  
\[\theta_{y}=arccos\left(\frac{R_{L}^{\prime}\cdot\cos(\gamma)}{\mathbf{R}_{y} }\right) \tag{10}\]  
\[V_{z1}=\int_{-(\overline{AB}+\overline{BO})}^{-\overline{BO}}R_{y}^{2}\cdot \left(\theta_{y}-\frac{sin(2\theta_{y})}{2}\right)d\mathbf{y}, \tag{11}\]  
\[V_{z2}=\int_{-\overline{BO}}^{\overline{OC}}R_{y}^{2}\cdot\left(\theta_{y}- \frac{sin(2\theta_{y})}{2}\right)d\mathbf{y}, \tag{12}\]  
\[F_{z1}=F_{z}\cdot\frac{V_{z1}}{V_{z1}+V_{z2}} \tag{13}\]  
\[F_{z2}=F_{z}\cdot\frac{V_{z2}}{V_{z1}+V_{z2}} \tag{14}\]  
\(V_{z1}\) and \(V_{z2}\) are the deformation volumes of the left and right parts of the tire. \(F_{z}\) is the total load on the tire.