## 3 Simulation and Analysis Model of Train-Track-Station Coupled Vibration
### Train Model and Track Irregularity Simulation
Given their substantial impact on simulating train operations, effective simulation of track irregularities is essential. Track irregularities exhibit high uncertainty and are difficult to represent with specific formulas. However, their characteristics can be effectively described using statistical methods. Currently, the power spectral density function is widely used to describe the wavelength characteristics of track irregularities. Various countries have proposed different track irregularity spectra based on field measurements, such as the German low-disturbance track spectrum, the U.S. six-level spectrum, the Japanese spectrum, and the fitted spectrum for existing railway mainlines in China \begin{table}
\begin{tabular}{c c c c c c} \hline Vehicle Budy & \(Y_{g0}\) & \(Z_{g0}\) & \(\phi_{g0}\) & \(J_{g0}\) & \(\theta_{g0}\) \\ Front Bogie Frame & \(Y_{c_{1}}\) & \(Z_{c_{1}}\) & \(\phi_{c_{1}}\) & \(J_{c_{1}}\) & \(\theta_{c_{1}}\) \\ Rear Bogie Frame & \(Y_{c_{2}}\) & \(Z_{c_{2}}\) & \(\phi_{c_{2}}\) & \(J_{c_{2}}\) & \(\theta_{c_{2}}\) \\ First Wheelset & \(Y_{v_{1}}\) & / & \(\psi_{v_{1}}\) & / & / \\ Second Wheelset & \(Y_{v_{1}}\) & / & \(\phi_{v_{2}}\) & / & / \\ Third Wheelset & \(Y_{v_{3}}\) & / & \(\phi_{v_{3}}\) & / & / \\ Fourth Wheelset & \(Y_{v_{4}}\) & / & \(\phi_{v_{4}}\) & / & / \\ \hline \end{tabular}
\end{table} 
Table 2: Summary of Degrees of Freedom (DOF) for Each Vehicle Model. \begin{table}
\begin{tabular}{c c} \hline Parameters & **Notation** \\ \hline Vehicle body mass, bogie mass, wheelset mass & \(M_{c}\), \(M_{g}\), \(M_{\text{s}}\) \\ Moment of Inertia of the vehicle body and Bogie & \(J_{g0r}\), \(J_{g0}\) \\ Stiffness of Primary Suspension System & \(K_{x}^{d}\), \(K_{y}^{d}\), \(K_{z}^{d}\) \\ Stiffness of Secondary Suspension System & \(K_{x}^{u}\), \(K_{y}^{d}\), \(K_{z}^{u}\) \\ Damping Coefficient of Primary Suspension System & \(C_{x}^{d}\), \(C_{y}^{d}\), \(C_{z}^{d}\) \\ Damping Coefficient of Secondary Suspension System & \(C_{x}^{u}\), \(C_{y}^{d}\), \(C_{z}^{u}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Vehicle Model Parameters. This study investigates railway bridge-type stations with train speeds exceeding 160 km/h. It uses the German low-interference spectrum, which is suitable for high-speed applications, to simulate track irregularities. The randomly generated track irregularities based on the German low-interference spectrum are used as excitation for the track system. The power spectral density (PSD) functions for track direction, vertical alignment, horizontal alignment, and gauge irregularities are shown as follows, respectively:  
\[S_{a}(\Omega)=\frac{A_{a}\Omega_{c}^{2}}{(\Omega_{x}^{2}+\Omega^{2})(\Omega_{ c}^{2}+\Omega^{2})} \tag{2}\]  
\[S_{v}(\Omega)=\frac{A_{v}\Omega_{c}^{2}}{(\Omega_{x}^{2}+\Omega^{2})(\Omega_{ c}^{2}+\Omega^{2})} \tag{3}\]  
\[S_{c}(\Omega)=\frac{A_{v}b^{-2}\Omega_{c}^{2}\Omega^{2}}{(\Omega^{2}+\Omega_{ x}^{2})(\Omega^{2}+\Omega_{c}^{2})(\Omega^{2}+\Omega_{s}^{2})} \tag{4}\]  
\[S_{g}(\Omega)=\frac{A_{g}\Omega_{c}^{2}\Omega^{2}}{(\Omega^{2}+\Omega_{x}^{2}) (\Omega^{2}+\Omega_{c}^{2})(\Omega^{2}+\Omega_{s}^{2})} \tag{5}\]  
where, \(S_{a}(\Omega)\), \(S_{v}(\Omega)\), \(S_{g}(\Omega)\), respectively, represent the power spectral densities (PSDs) corresponding to track alignment irregularity, vertical profile irregularity, and gauge irregularity, all units are \((m^{2}/(rad/m))\); \(S_{c}(\Omega)\) represent the power spectral density of horizontal irregularities, and because of the use of an inclination angle measure, its unit is \((1/(rad/m))\); \(A_{v}\), \(A_{a}\), \(A_{g}\), respectively, represent roughness constant, its unit is \((m^{2}\times(rad/m))\); \(\Omega_{c}\), \(\Omega_{s}\), \(\Omega_{r}\) represent cutoff frequency (rad/m); b represent half of the distance between the left and right rolling circles. The roughness constants and cutoff frequencies for simulating track irregularities using the German low-interference spectrum are shown in Table 4.