## 2 Numerical Simulation
### Vehicle-Track-Subgrade Coupled Model
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Structural Layer**} & \multicolumn{3}{c}{**Model Parameters**} & \multicolumn{3}{c}{**Rayleigh Damping Coefficients**} \\ \cline{2-5}  & **Parameter** & **Unit** & **Value** & **Parameter (Unit s\({}^{-1}\))** & **Value** \\ \hline \multirow{4}{*}{Rail} & Elastic modulus & GPa & 210 & \multirow{4}{*}{\(\alpha\)} & 0.0328 \\  & Poisson’s ratio & & 0.3 & & 0.0328 \\  & Cross-sectional area & cm\({}^{3}\) & 76.45 & & \\  & Density & kg\(\cdot\)m\({}^{-3}\) & 7800 & \(\beta\) & 0.0031 \\ \hline \multirow{4}{*}{Track slab} & Elastic modulus & GPa & 35.5 & \multirow{4}{*}{\(\alpha\)} & 0.0983 \\  & Poisson’s ratio & & 0.2 & & \\ \cline{1-1}  & Length \(\times\) Width \(\times\) & \multirow{2}{*}{\(\text{m}\times\text{m}\times\text{m}\)} & \multirow{2}{*}{\(100\times 2.55\times 0.2\)} & \multirow{2}{*}{\(\beta\)} & 0.0092 \\  & Thickness & & & & \\ \cline{1-1}  & Density & kg\(\cdot\)m\({}^{-3}\) & 2500 & & \\ \hline \multirow{4}{*}{CA mortar} & Elastic modulus & MPa & 7000 & \multirow{4}{*}{\(\alpha\)} & 0.0983 \\  & Poisson’s ratio & & 0.2 & & \\ \cline{1-1}  & Thickness & M & 0.03 & & \\ \cline{1-1}  & Density & kg\(\cdot\)m\({}^{-3}\) & 1800 & \(\beta\) & 0.0092 \\ \hline \multirow{4}{*}{Supporting layer} & Elastic modulus & GPa & 22 & \multirow{4}{*}{\(\alpha\)} & 0.0983 \\  & Poisson’s ratio & & 0.2 & & \\ \cline{1-1}  & Span \(\times\) Thickness & m \(\times\) m & \((2.95/3.25)\times 0.3\) & \multirow{2}{*}{\(\beta\)} & 0.0092 \\  & Density & kg\(\cdot\)m\({}^{-3}\) & 2500 & & \\ \hline \multirow{4}{*}{Subgrade surface layer} & Elastic modulus & GPa & 220 & \multirow{4}{*}{\(\alpha\)} & 0.2620 \\  & Poisson’s ratio & & 0.25 & & \\ \cline{1-1}  & Density & kg\(\cdot\)m\({}^{-3}\) & 2100 & & \\ \cline{1-1}  & Cohesion & KPa & 35 & & \\ \cline{1-1}  & Internal friction angle & \({}^{\circ}\) & 43 & \(\beta\) & 0.0244 \\ \cline{1-1}  & Length \(\times\) Thickness & m \(\times\) m & \(100\times 0.4\) & & \\ \hline \multirow{4}{*}{Subgrade bottom layer} & Elastic modulus & GPa & 130 & \multirow{4}{*}{\(\alpha\)} & 0.2293 \\  & Poisson’s ratio & & 0.3 & & \\ \cline{1-1}  & Density & kg\(\cdot\)m\({}^{-3}\) & 1900 & & \\ \cline{1-1}  & Decision & KPa & 13 & & \\ \cline{1-1}  & Internal friction angle & \({}^{\circ}\) & 28 & \(\beta\) & 0.0214 \\ \cline{1-1}  & Length \(\times\) Thickness & m \(\times\) m & \(100\times 2.3\) & & \\ \hline \multirow{4}{*}{Subgrade body} & Elastic modulus & GPa & 50 & \multirow{4}{*}{\(\alpha\)} & 0.2750 \\  & Poisson’s ratio & & 0.2 & & \\ \cline{1-1}  & Density & kg\(\cdot\)m\({}^{-3}\) & 1800 & & \\ \cline{1-1}  & Cohesion & KPa & 30 & & \\ \cline{1-1}  & Internal friction angle & \({}^{\circ}\) & 25 & \(\beta\) & 0.0305 \\ \cline{1-1}  & Length \(\times\) Thickness & m \(\times\) m & \(100\times 1.3\) & & \\ \hline \multirow{4}{*}{Surface soil layer of the foundation} & Elastic modulus & GPa & 50 & \multirow{4}{*}{\(\alpha\)} & 0.2454 \\  & Poisson’s ratio & & 0.2 & & \\ \cline{1-1}  & Density & kg\(\cdot\)m\({}^{-3}\) & 1800 & & \\ \cline{1-1}  & Cohesion & KPa & 30 & & \\ \cline{1-1}  & Internal friction angle & \({}^{\circ}\) & 25 & \(\beta\) & 0.0016 \\ \cline{1-1}  & Length \(\times\) Width \(\times\) & \multirow{2}{*}{\(\text{m}\times\text{m}\times\text{m}\)} & \multirow{2}{*}{\(100\times 80\times 3\)} & & \\ \cline{1-1}  & Thickness & & & & \\ \hline \multirow{4}{*}{Bedrock} & Elastic modulus & GPa & 190 & \multirow{4}{*}{\(\alpha\)} & 0.2454 \\ \cline{1-1}  & Poisson’s ratio & & 0.25 & & \\ \cline{1-1}  & Density & kg\(\cdot\)m\({}^{-3}\) & 2650 & & \\ \cline{1-1}  & Cohesion & KPa & 20 & & \\ \cline{1-1}  & Internal friction angle & \({}^{\circ}\) & 44.3 & \(\beta\) & 0.0016 \\ \cline{1-1}  & Length \(\times\) Width \(\times\) & \multirow{2}{*}{\(\text{m}\times\text{m}\times\text{m}\)} & \multirow{2}{*}{\(100\times 80\times 27\)} & & \\ \cline{1-1}  & Thickness & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the ballastless track-subgrade model.