## 3 Simulation and Analysis Model of Train-Track-Station Coupled Vibration
### Establishment and Solution of the Coupled Vibration Equations for the Train-Track-Station System  
This paper employs the method proposed by [PERSON] in Section 3.2 to establish the coupled vibration equations of the vehicle-track-station system. The process of establishing the equations is as follows:  
Figure 4: Finite element model of Station Type B (**a**) axonometric view; (**b**) front elevation. Figure 3: Finite element model of Station Type A. (**a**) axonometric view; (**b**) front elevation. \begin{table}
\begin{tabular}{c c c} \hline Station Structure Parameters & Station Type A & Station Type B \\ \hline Material Strength of Beams, & C45 & C45 \\ Columns, and Slabs & 1089 & 989 \\ Number of beam elements & 1452 & 1342 \\ Number of shell elements & 768 & 768 \\ \hline \end{tabular}
\end{table} 
Table 5: Station Structure Parameters. 1. The total potential energy of the spatial vibration of the vehicle-track-station system is obtained by calculating and summing the spatial vibration potential energies of the track-station and the train at a certain moment. The calculation formula is shown in Equation (6): \[\Pi_{d}=U_{i}+V_{m}+V_{c}+V_{F}+V_{P}+V_{g}\] (6) where, \(U_{i}\), \(V_{m}\), \(V_{c}\), \(V_{F}\), \(V_{P}\), \(V_{g}\), respectively, represent the elastic strain energy, inertial force potential energy, damping force potential energy, frictional force potential energy, disturbing force potential energy, and gravitational potential energy of the system. (2) By applying the principle of constant total potential energy of elastic system dynamics, the variation of the total potential energy is set to zero, as shown in Equation (7): \[\delta_{e}\Pi_{d}=\delta_{e}\big{(}U_{i}+V_{m}+V_{c}+V_{F}+V_{P}+V_{g}\big{)}=0\] (7) 3. The track irregularity is regarded as a self-excitation source. Based on the total potential energy variation equation and using the \"position-matching\" rule for forming matrices, the mass matrix \([M(t)]\), damping matrix \([C(t)]\), stiffness matrix \([K(t)]\), and load vector \(\{P(t)\}\) of the system at any time \(t\) are calculated. The vibration equation of the coupled vehicle-track-station system is established, as shown in Equation (8): \[[M(t)]\Big{\{}\tilde{\delta}(t)\Big{\}}+[C(t)]\Big{\{}\dot{\delta}(t)\Big{\}}+ [K(t)]\big{\{}\delta(t)\big{\}}=\{P(t)\}\] (8) where, \(\Big{\{}\tilde{\delta}(t)\Big{\}}\), \(\Big{\{}\dot{\delta}(t)\Big{\}}\), \(\{\delta(t)\}\), respectively, represent the acceleration, velocity, and displacement vectors of the vehicle-track-station system at time \(t\). For the solution of linear structural vibration equations, the modal superposition method is generally adopted. On the one hand, the calculation process is relatively simple, and on the other hand, it allows for selective computation of any order of modes as needed. The train-track-station system investigated in this paper is a highly complex time-variant system. Although the vibration equations formed by the system are linear, the linear vibration equations formed are variable-coefficient equations due to the changes in spring and damping characteristics throughout the process. Therefore, the modal superposition method cannot be used. A more suitable approach is the step-by-step integration method. This method discretizes the entire time span during which the train passes through the system into numerous small time increments \(\Delta t\), assuming that the system properties remain constant within each time increment. The dynamic response of the system at the end of one time increment \(\Delta t\) is then used as the initial condition for solving the response in the subsequent time increment. This process is iterated until the dynamic response of the system over the entire time history is obtained.