## 3 Model and Calculation Method
### Wheel-Rail Interaction  
The wheel-rail interaction is based on the compression between a wheel and rail. The nonlinear Hertzian elastic contact theory [31; 35; 36] is used to calculate the wheel-rail force. Therefore, the wheel-rail force is simulated by Equations (4) and (5):  
\[\left\{\begin{array}{ll}[\frac{1}{G_{Hertz}}\Delta Z(t)]^{3/2},&\Delta Z(t)>0 \\ 0,&\Delta Z(t)\leq 0\end{array}\right. \tag{4}\]  
Figure 4: (**a**) Type-III pre-stressed concrete sleepers. (**b**) Composite sleepers.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Parameters** & **Unit** & **Value** \\ \hline \multirow{4}{*}{Rail} & Elastic modulus & N/m\({}^{2}\) & 2.06 \(\times\) 10\({}^{11}\) \\  & Moment of inertia & m\({}^{4}\) & 3.217 \(\times\) 10\({}^{-5}\) \\  & Mass per unit length & kg/m & 6.064 \(\times\) 10 \\  & Poisson ratio & - & 3.0 \(\times\) 10\({}^{-1}\) \\ \hline \multirow{2}{*}{Fastener} & Vertical stiffness & N/m & 6.0 \(\times\) 10\({}^{7}\) \\  & Vertical damping & N-s/m & 7.5 \(\times\) 10\({}^{4}\) \\ \hline \multirow{2}{*}{Composite sleeper} & Density & kg/m\({}^{3}\) & 1.097 \(\times\) 10\({}^{3}\) \\  & Length/width/height & m & 2.7/0.22/0.18 \\ \hline \multirow{4}{*}{Type-III sleeper} & Elastic modulus & N/m\({}^{2}\) & 3.6 \(\times\) 10\({}^{10}\) \\  & Density & kg/m\({}^{3}\) & 2.5 \(\times\) 10\({}^{3}\) \\ \cline{1-1}  & Poisson ratio & - & 2.0 \(\times\) 10\({}^{-1}\) \\ \cline{1-1}  & Loss factor & - & 1.0 \(\times\) 10\({}^{-2}\) \\ \cline{1-1}  & Length/width/height & m & 2.6/0.3/0.23 \\ \hline \multirow{4}{*}{Ballast bed} & Mass & kg & 6.5 \(\times\) 10\({}^{2}\) \\  & Vertical stiffness & N/m & 2.4 \(\times\) 10\({}^{8}\) \\ \cline{1-1}  & Vertical damping & N-s/m & 6.0 \(\times\) 10\({}^{4}\) \\ \cline{1-1}  & Shear stiffness & N/m & 7.8 \(\times\) 10\({}^{7}\) \\ \cline{1-1}  & Shear damping & N-s/m & 8.0 \(\times\) 10\({}^{4}\) \\ \hline \multirow{2}{*}{Subgrade} & Vertical stiffness & N/m & 2.4 \(\times\) 10\({}^{8}\) \\ \cline{1-1}  & Vertical damping & N-s/m & 6.0 \(\times\) 10\({}^{4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The parameters of the track.  
\[\Delta Z(t)=Z_{w}(t)-Z_{r}(t)-\eta(t) \tag{5}\]  
where \(G_{Hertz}\) is the contact constant of the wheel and rail (m/N\({}^{2/3}\)), \(\Delta Z(t)\) is the normal compression at the wheel-rail contact point, \(Z_{w}(t)\) is the vertical displacement of the wheel, \(Z_{r}(t)\) is the vertical displacement of the rail at the relevant contact point, and \(\eta(t)\) is the track vertical irregularity.  
As we know, random track irregularities exist in railway lines everywhere and influence the dynamic performance of the vehicle-track system. Therefore, random track irregularities should be considered in the vehicle-track coupled model. The German High-speed Railway low irregularity is adopted. The power spectrum density (PSD) function of the German HSR low irregularity is expressed as follows [30]:  
\[S_{v}(\Omega)=\frac{A_{v}\Omega_{c}^{2}}{(\Omega^{2}+\Omega_{r}^{2})\left( \Omega^{2}+\Omega_{c}^{2}\right)} \tag{6}\]  
where \(\Omega_{c}\), \(\Omega_{r}\), and \(\Omega\) denote the cut-off frequencies of track irregularities, while Av denotes the roughness constant of track elevation irregularities. In Equation (6), \(\Omega_{c}~{}=~{}0.8246~{}(\text{rad}/\text{m})\), \(\Omega_{r}=0.0206~{}(\text{rad}/\text{m})\), and \(A_{v}=4.032\times 10^{-7}~{}(\text{m}^{2}\). Rad\(/\text{m})\). The PSD function would be transformed into vertical irregularities along the longitudinal distance of the track by means of a time-frequency transformation technique, as shown in Figure 5.