## 2 Numerical Analysis Model
### Vehicle-Bridge Coupled Model
#### 2.2.2 Bridge Spatial Vibration Model  
The spatial vibration of the box girder bridge is studied by the FE method of beam segment element [23]. The main assumptions in the calculation are as follows:  
1. The tapered edge sections and reinforcement are not considered in the limit element model.
2. The deflection on both sides at each corner of the box section is assumed to be equal to \(\gamma/2\).
3. The dynamic modulus of elasticity of concrete is set at 115-120% the static modulus with elasticity.  
The box girder section has a torsion center k and shape center c, and the coordinate origin is set at the torsion center as depicted in Figure 2. There are four DOFs in the space of the box girder section, i.e., transverse displacement \(U\), vertical displacement \(V\), torsion angle \(\varphi\), distortion angle \(\gamma\)[24]. The displacement at any point in the cross-section of the beam can be separated into the transverse displacement \(u\) (\(z\), \(t\)), the vertical displacement \(v\) (\(z\), \(t\)), the torsion angle \(\varphi\) (\(z\), \(t\)) and the distortion angle \(\gamma\) (\(z\), \(t\)) of the cross-section.  
Calculating the damping matrix of a bridge accurately is very difficult. According to the empirical data of bridge damping, this paper determines the damping matrix's ap-proximation by the bridge structure's overall energy consumption during the vibration process [19].  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Item** & **Notation** \\ \hline Mass of car body, bogie, wheelset & \(M_{c},M_{g},M_{s}\) \\ One half of the longitudinal distance between the front & \(L\) \\ and rear bogie centers of gravity & \(L_{1}\) \\ Half the length of the wheelbase & \(h_{1}\) \\ Distance from the center of gravity of the vehicle to the central transverse spring & \(h_{2}\) \\ Distance from the center of gravity of the frame to the central transverse spring & \(h_{3}\) \\ Distance from the center of gravity of the frame to the transverse spring of the axle box & \(h_{4}\) \\ Stiffness of the suspension systems between the car & \(K_{ax},K_{ay},K_{uz}\) \\ body and the bogie & \(K_{ax},K_{ay},K_{uz}\) \\ Stiffness of the suspension systems between the bogie & \(K_{dx},K_{dy},K_{dz}\) \\ and wheelset & \(K_{dx},K_{dy},K_{dz}\) \\ Damping of the suspension systems between the car & \(C_{ax},C_{ay},C_{uz}\) \\ body and the bogie & \(C_{dx},C_{dy},C_{dz}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters of the train model.  
The beam stiffness matrix [K] can be derived from the strain energy of each element, diaphragm plate and partition wall. A linear combination of the mass matrix [M] and the stiffness matrix [K] is used to represent the damping matrix of the beam structure.  
\[[C]=\alpha[M]+\beta[K] \tag{4}\]  
where \(\alpha\) and \(\beta\) are parameters with the following expressions:  
\[\left\{\begin{array}{c}\alpha=\frac{2\left(\xi_{i}\omega_{j}-\xi_{i}\omega_{ j}\right)}{\omega_{j}^{2}-\omega_{i}^{2}}\alpha_{i}\xi\omega_{j}\\ \beta=\frac{2\left(\xi_{i}\omega_{j}-\xi_{i}\omega_{i}\right)}{\omega_{j}^{2} -\omega_{i}^{2}}\end{array}\right. \tag{5}\]  
where \(\omega_{i}\) and \(\xi_{i}\) denote the \(i\)-order natural frequency and the \(i\)-order damping ratio, respectively. \(\omega_{j}\) and \(\xi_{j}\) denote the \(j\)-order natural frequency and the \(j\)-order damping ratio, respectively.