## 3 Simulation and Analysis Model of Train-Track-Station Coupled Vibration  
### Train Model and Track Irregularity Simulation  
When establishing the spatial vibration analysis model of the train, the basic assumptions about the train model in the work written by [PERSON] [22] are adopted: The train is regarded as a multi-rigid-body system composed of the car body, bogies, and wheelsets; the rigid bodies move at a constant speed in the direction along the bridge, and the longitudinal vibration of the rigid body system is neglected; it is assumed that each rigid body undergoes small vibrations; the springs in the primary and secondary suspensions are considered as linear elastic models with damping, and the influence of nonlinear creep is ignored; the wheelsets are always in close contact with the rails, and their vertical displacements are always the same. A rigid body has six degrees of freedom (DOFs): three translational DOFs and three rotational DOFs. In the model established in this paper, since the longitudinal vibration of the train is neglected, the degree of freedom in the direction of vehicle travel is not considered. Therefore, a car body has five DOFs, including sway, heave, roll, pitch, and yaw. Since it is assumed that the wheelsets are always in vertical contact with the rails, the vertical displacement and roll displacement of the wheelsets are determined by the rails. Thus, the heave and pitch degrees of freedom of the wheelsets are ignored, and each wheelset only has two DOFs, namely sway and yaw. In summary, in the train model studied in this paper, one train car has one car body, two bogies, and four wheelsets, with a total of 23 degrees of freedom. The CRH3 vehicle dynamics analysis model is shown in Figure 2.  
The specific details of the degrees of freedom are presented in Table 2, and the details of the remaining model parameters are shown in Table 3.  
Figure 2: Schematic Diagram of Train Dynamics Model. (**a**) elevation view; (**b**) side view; (**c**) plan view; (**d**) schematic diagram of the vehicle’s horizontal spring. The position of the train on the structure changes continuously as the train moves. First, it is necessary to calculate the total potential energy of a vehicle \(\Pi_{\text{v}}\), then employing the elastic system dynamics principle of total potential energy conservation \(\delta\Pi_{\text{v}}=0\) and the \"set-in-right-position\" rule that can form matrices, quickly proposed by [PERSON] [23; 24], the stiffness matrix, damping matrix, and mass matrix of the vehicle system, namely \([K_{\text{v}}]\), \([C_{\text{v}}]\), \([M_{\text{v}}]\) in the following formula, can be rapidly established. By combining these matrices with the external load vector \(\{P_{\text{v}}\}\), the equation of motion for a single-vehicle model can be derived:  
\[[M_{\text{v}}]\Big{\{}\ddot{\delta}_{\text{v}}\Big{\}}+[C_{\text{v}}]\Big{\{} \dot{\delta}_{\text{v}}\Big{\}}+[K_{\text{v}}]\big{\{}\delta_{\text{v}}\big{\}} =\{P_{\text{v}}\} \tag{1}\]  
During the construction and service life of rails, deviations between the actual and theoretical track profiles arise due to factors such as construction tolerances, material de-fects, wear from service, and subgrade displacements. Given the complexity of the track conditions and the stochastic nature of track irregularities, it is highly challenging to de-fine track irregularities with a specific mathematical relationship. Therefore, statistical parameters are commonly employed to characterize track irregularities, and the power spectral density (PSD) function is utilized to elucidate the relationship between the track spectral density and frequency. Track irregularities, primarily caused by construction errors and wear during operation, result in deformations of the rail geometry. These irregularities significantly increase train vibrations and wheel-rail interaction forces and are the main excitation sources for train-structure coupled vibrations.