## 2 Materials and Methods
### Modelling Method of a Vehicle-Slab Track Dynamic System
#### 2.1.3 Wheel-Rail Contact  
The well-known Hertz non-linear elastic contact theory expressed by Equation (3) was applied to calculate the wheel-rail normal force [44].  
\[p(t)=\left[\frac{1}{G}\delta z(t)\right]^{3/2} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parts** & **Parameters** & **Units** & **Values** \\ \hline \multirow{3}{*}{Rail} & Density & kg/m\({}^{3}\) & 7830 \\  & Elastic modulus & GPa & 205 \\  & Poisson’s ratio & / & 0.30 \\  & Size & m & \(5.60\times 2.50\times 0.21\) \\ \multirow{3}{*}{Slab} & Density & kg/m\({}^{3}\) & 2500 \\  & Elastic modulus & GPa & 36.50 \\  & Poisson’s ratio & / & 0.20 \\  & Depth & m & 0.10 \\ \multirow{3}{*}{Self-compacting concrete layer} & Density & kg/m\({}^{3}\) & 2500 \\  & Elastic modulus & GPa & 32.50 \\ \multirow{3}{*}{Self-compacting concrete layer} & Poisson’s ratio & / & 0.20 \\  & Width \(\times\) Depth & m & \(3.10\times 0.10\) \\ \multirow{3}{*}{Foundation plate} & Density & kg/m\({}_{3}\) & 2500 \\  & Elastic modulus & GPa & 32.5 \\ \multirow{3}{*}{Concave groove} & Poisson’s ratio & / & 0.20 \\ \multirow{3}{*}{Cushion} & Size & m & \(0.60\times 0.40\times 0.10\) \\  & Stiffness & kN/mm & 250 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Structure parameters of CRTS III slab track.  
Figure 3: Structural components of CRTS III slab track.  
In Equation (3), G refers to the wheel-rail contact constant (m/N\({}^{2/3}\)); \(\delta z(t)\) represents the elastic compression between the wheel and rail (m); \(t\) refers to time (s).  
The worn profile tread wheel was applied in this study; thus, the wheel-rail contact constant G was calculated by  
\[\text{G}=3.86R^{-0.115}\times 10^{-8} \tag{4}\]  
In Equation (4), \(R\) represents the wheel radius.  
The wheel-rail tangential force is calculated by [PERSON] linear creep theory. Creep force in three directions \(T_{x}\), \(T_{y}\), \(M_{z}\) can be calculated by Equation (5):  
\[\left\{\begin{array}{c}T_{x}=-f_{11}\xi_{x}\\ T_{y}=-f_{22}\xi_{y}-f_{23}\xi_{sp}\\ M_{z}=f_{32}\xi_{y}-f_{33}\xi_{sp}\end{array}\right. \tag{5}\]  
In Equation (5), \(\xi_{x}\), \(\xi_{x}\), \(\xi_{sp}\) refer to creepage in three directions; \(f_{11}\), \(f_{22}\) refer to longitudinal and lateral creep coefficients, respectively; \(f_{23}\), \(f_{32}\) refer to lateral/spin creep coefficients, \(f_{23}\) =\(f_{32}\);\(f_{33}\) refers to spin creep coefficient.  
\(\xi_{x}\), \(\xi_{x}\), \(\xi_{sp}\) could be calculated by Equation (6):  
\[\left\{\begin{array}{c}\xi_{x}=\frac{V_{w1}-V_{r1}}{V}\\ \xi_{x}=\frac{V_{w2}-V_{r2}}{V}\\ \xi_{sp}=\frac{\Omega_{w3}-\Omega_{\beta}}{V}\end{array}\right. \tag{6}\]  
In Equation (6), \(V_{w1}\), \(V_{w2}\), \(\Omega_{w3}\) refer to longitudinal linear speed, lateral linear speed and spin angular speed of the wheel at the contact point, respectively; \(V_{r1}\), \(V_{r2}\), \(\Omega_{r3}\) refer to longitudinal linear speed, lateral linear speed and spin angular speed of the rail at the contact point, respectively; \(V\) refers to nominal speed of the wheel.  
\(f_{11}\), \(f_{22}\), \(f_{23}\), \(f_{33}\) can be determined by Equation (7):  
\[\left\{\begin{array}{c}f_{11}=EabC_{11}\\ f_{22}=EabC_{22}\\ f_{23}=E(ab)^{2/3}C_{23}\\ f_{33}=E(ab)^{2}C_{23}\end{array}\right. \tag{7}\]  
In Equation (7), \(a\) and \(b\) refer to the major and minor axes of the wheel-rail contact ellipse area; \(E\) is the elastic modulus of wheel set and rail material; \(C_{ij}\) refers to the Kalker coefficient, which can be obtained according to table lookup [45].  
In order to avoid non-convergence in the iterative process caused by the non-positive definite coefficient matrices, the penalty function method is used to solve the wheel-rail contact behavior by introducing the normal and tangential contact conditions into the functional equation [38].  
The random track irregularity recommended by the Chinese Track Irregularity Spectrum of High-Speed Railway Ballastless Track was used for the excitation of the wheel-rail system [24], as shown in Figure 4.