## 2 System Dynamics Model of Vehicle-Curved Switch Rail
### Kik-Piotrowski Contact Theory and Archard Wear Model  
In the case of complex wheel-rail contact in the turnout area, the shape of the contact area between the wheels and rails being an ellipse cannot better reflect the actual situation. The normal stress distribution of the wheel and rail during rolling is assumed to be semi-elliptical by [PERSON] and [PERSON], and the contact area is confirmed by using the virtual penetration method [37; 38].  
The virtual penetration region between the wheel and rail is determined by the depth of penetration \(\delta_{0}\), which is taken as the contact area. The penetration function \(g(y)\) is expressed in Equation (1):  
\[g(y)=\begin{cases}\delta_{0}-f(y)&f(y)\leq\delta_{0}\\ 0&f(y)>\delta_{0}\end{cases} \tag{1}\]  
where \(f(y)\) is the distance in the rolling direction between the rail and wheel at the same coordinate.  
In the penetration area, the \(x\) coordinates of the front and rear ends are:  
\[x_{l}(y)=-x_{l}(y)\approx\sqrt{2 Rg(y)} \tag{2}\]  
where \(R\) is the wheel rolling circle radius.  
The distribution of normal contact stress \(p\) in the contact patch is shown in Equation (3):  
\[p(x,y)=\frac{p_{0}}{x_{l}(0)}\sqrt{x_{l}^{2}(y)-x^{2}} \tag{3}\]  
where \(p_{0}\) is the maximum normal contact stress in the contact patch.  
Integrating the above equation yields the total normal force of the contact patch \(N\).  
\[N=\frac{p_{0}}{x_{l}(0)}(\int_{y_{l}}^{y_{l}}\int_{-x_{l}}^{x_{l}}\sqrt{x_{l}^ {2}(y)-x^{2}dxdy}) \tag{4}\]  
Figure 5: The system dynamics model of the vehicle and curved switch rail.  
Using the Boussinesq function, the normal displacement at the point (0,0) is calculated using Equation (5):  
\[\frac{\delta_{0}}{2}=\omega_{0}=\frac{1-\sigma_{0}}{\pi E}\frac{p_{0}}{x_{l}(0)} (\int_{y_{r}}^{y_{l}}\int_{-x_{l}}^{x_{l}}\frac{\sqrt{x_{l}^{2}(y)-x^{2}}}{\sqrt {x^{2}+y^{2}}}dxdy) \tag{5}\]  
Through the above two equations, the calculation of normal force and maximum contact stress are obtained, as shown in Equations (6) and (7):  
\[N=\frac{\pi E\delta}{2(1-\sigma^{2})}(\int_{y_{r}}^{y_{l}}\int_{-x_{l}}^{x_{l} }\frac{\sqrt{x_{l}^{2}(y)-x^{2}}}{\sqrt{x^{2}+y^{2}}}dxdy)\overset{-1}{(\int_{ y_{r}}^{y_{l}}\int_{-x_{l}}^{x_{l}}\sqrt{x_{l}^{2}(y)-x^{2}}dxdy)} \tag{6}\]  
\[p_{0}=N\sqrt{2R\delta_{0}}\bigg{(}\int_{y_{r}}^{y_{l}}\int_{x_{r}}^{x_{l}} \sqrt{x_{l}^{2}(y)-x^{2}}dxdy\bigg{)}^{-1} \tag{7}\]  
The Archard model is one of the most important wear prediction models in the field of wheel-rail wear research. In the model, the wear volume is proportional to the normal force and sliding distance between the two contact bodies and inversely proportional to the material hardness [47], as shown in Equation (8):  
\[V=k\frac{NS}{H} \tag{8}\]  
where \(V\) is the wear volume, \(N\) is the normal force between the wheel and rail, \(S\) is the relative sliding distance, \(H\) is the material hardness, and \(k\) is the wear coefficient.  
The sliding distance between wheel and rail is calculated using Equation (9). The relative sliding speed of wheel and rail is calculated using Equation (10):  
\[S=|v_{s}|\frac{\Delta x}{v_{0}} \tag{9}\]  
\[v_{s}=v_{0}\Bigg{[}\begin{array}{c}\xi-x_{2}\phi-\frac{\partial u_{1}}{ \partial x_{1}}\\ \eta+x_{1}\phi-\frac{\partial u_{2}}{\partial x_{1}}\end{array}\Bigg{]} \tag{10}\]  
where \(v_{s}\) is the relative sliding speed of wheel and rail, \(v_{0}\) is the wheel rolling speed, and \(u_{1}\), \(u_{2}\) are the elastic displacements of the particles along the longitudinal and lateral directions, respectively. \(\xi\), \(\eta\), and \(\varphi\) are the longitudinal, lateral, and spin creepage values within the wheel-rail contact patch, respectively.  
[PERSON] [48] obtained the relationship between wear coefficient, contact stress, and sliding velocity by summarizing the results of wheel-rail wear tests, as shown in Figure 6.  
Figure 6: Wear coefficient distribution.