## 2 Train-Track-Bridge Coupled Vibration Model
### Track and Bridge Model  
This paper takes CRTS II slab ballastless track as an example to establish the track model. The track structure is mainly composed of a base, CA mortar layer, track board, elastic fasteners, rails and other components [30; 31]. The steel rail is simulated as a beam element, and the track plate and base are both simulated as plate elements. The base is connected to the bridge, the track plate is connected to the base, and the steel rail is connected to the track plate as a linear spring damper, as shown in Figure 2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Vertical** & **Longitudinal** & **Lateral** & **Yaw** & **Roll** & **Pitch** \\ \hline Car body & \(z_{c}\) & \(x_{c}\) & \(y_{c}\) & \(\theta_{zc}\) & \(\theta_{xc}\) & \(\theta_{yc}\) \\ Front bogie & \(z_{tf}\) & \(x_{tf}\) & \(y_{tf}\) & \(\theta_{ztf}\) & \(\theta_{xff}\) & \(\theta_{uff}\) \\ Rear bogie & \(z_{tb}\) & \(x_{tb}\) & \(y_{tb}\) & \(\theta_{ztb}\) & \(\theta_{xtb}\) & \(\theta_{ytb}\) \\ First wheel set & \(z_{w1}\) & \(x_{w1}\) & \(y_{w1}\) & \(\theta_{zw1}\) & \(\theta_{xw1}\) & - \\ Second wheel set & \(z_{w2}\) & \(x_{w2}\) & \(y_{w2}\) & \(\theta_{zw2}\) & \(\theta_{xw2}\) & - \\ Third wheel set & \(z_{w3}\) & \(x_{w3}\) & \(y_{w3}\) & \(\theta_{zw3}\) & \(\theta_{xw3}\) & - \\ Fourth wheel-set & \(z_{w4}\) & \(x_{w4}\) & \(y_{w4}\) & \(\theta_{zw4}\) & \(\theta_{xw4}\) & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Distribution of 38 degrees of freedom for a single train section.  
Figure 1: Single train model.  
The bridge adopts a simply supported concrete box girder at both ends and a concrete pier, as shown in Figure 3. The connection between the steel beam and the steel rail is also considered as a linear spring damper connection. A bridge model is established based on the finite element method, and the pier and beam are simulated as Euler-Bernoulli beam elements. Each element has two nodes, and each node contains six degrees of freedom.  
The mass matrix \(\mathbf{M}_{rr}\), \(\mathbf{M}_{bb}\), stiffness matrix \(\mathbf{K}_{rr}\), \(\mathbf{K}_{bb}\) and damping matrix \(\mathbf{C}_{rr}\), \(\mathbf{C}_{bb}\) of the bridge and track can be easily calculated via the finite element method [32; 33]. The damping matrix \(\mathbf{C}_{bb}\) of the bridge adopts Rayleigh damping, which can be composed of a mass matrix \(\mathbf{M}_{bb}\) and stiffness matrix \(\mathbf{K}_{bb}\) in a linear way.  
\[\mathbf{C}_{bb}=\frac{2\omega_{1}\omega_{2}\zeta}{\omega_{1}+\omega_{2}} \mathbf{M}_{bb}+\frac{2\zeta}{\omega_{1}+\omega_{2}}\mathbf{K}_{bb} \tag{1}\]  
where \(\omega_{1},\ \omega_{2}\) denote first-order and second-order natural frequencies (rad) of the bridge structure, respectively, and \(\zeta\) denotes the bridge damping ratio. Here, it is assumed that the first-order and second-order corresponding bridge damping ratios are equal. Moreover,  
Figure 3: Cross section view of bridge model.  
Figure 2: Cross section view of track model.  
the stiffness and damping matrices (\(\mathbf{K}_{rb},\ \mathbf{K}_{br},\ \mathbf{C}_{rb},\ \mathbf{C}_{br}\)) of the interaction between the track and the bridge are also easily obtained, with the following relationships:  
\[\mathbf{K}_{br}=\mathbf{K}_{rb}^{\mathrm{T}},\mathbf{C}_{br}=\mathbf{C}_{rb}^{ \mathrm{T}} \tag{2}\]