## 2 Theoretical Background
### Vertical Coupling Vibration Model of Train-Rail-Bridge
#### 2.2.2 Bridge Element Equation  
The dynamic equation of the bridge element is:  
\[\mathbf{M}_{\text{b}}^{\text{e}}\mathbf{z}_{\text{b}}^{\text{e}}+\mathbf{C}_{ \text{b}}^{\text{e}}\mathbf{z}_{\text{b}}^{\text{e}}+\mathbf{K}_{\text{b}}^{\text {e}}\mathbf{z}_{\text{b}}^{\text{e}}=\mathbf{P}_{\text{b}}^{\text{e}}-\int_{0}^{ l_{b}}\mathbf{N}_{\text{b}}^{\text{T}}q_{\text{rb}}(x,t)\text{d}x \tag{7}\]  
Figure 3: Coupling elements of an arbitrary length combination.  
where \(\mathbf{M}_{\text{b}}^{\text{e}}\), \(\mathbf{C}_{\text{b}}^{\text{e}}\), and \(\mathbf{K}_{\text{b}}^{\text{e}}\) are bridge element mass matrix, damping matrix, and stiffness matrix, respectively; \(\mathbf{z}_{\text{b}}^{\text{e}}\) is the node displacement vector of bridge element; \(\mathbf{P}_{\text{b}}^{\text{e}}\) is the load vector of the element node, and \(l_{\text{b}}\) is the length of a bridge element.  
The distributed force \(q_{\text{rb}}\) transmitted from the rail to the bridge is:  
\[q_{\text{rb}}(x,t)=-k_{\text{rb}}\big{(}z_{\text{pr}}-z_{\text{b}}\big{)}-c_{ \text{rb}}\big{(}\dot{z}_{\text{pr}}-\dot{z}_{\text{b}}\big{)} \tag{8}\]  
\[z_{\text{pr}}=\mathbf{N}_{\text{pr}}\mathbf{z}_{\text{pr}}^{\text{e}},\quad \dot{z}_{\text{pr}}=\mathbf{N}_{\text{pr}}\dot{z}_{\text{pr}}^{\text{e}}\]  
where \(k_{\text{rb}}\) and \(c_{\text{rb}}\) are the equivalent stiffness and damping between the bridge rails, \(\mathbf{z}_{\text{pr}}^{\text{e}}\) is the displacement vector of the \(p\)th rail element corresponding to the wheel acting on the \(k\)th bridge element, and \(z_{\text{pr}}\) is the rail displacement value of the wheel action point.  
Considering that there are \(m\) rail elements on a bridge element, and substituting Equation (8) into Equation (10), the following can be obtained:  
\[\mathbf{M}_{\text{b}}^{\text{e}}\ddot{\mathbf{z}}_{\text{b}}^{\text{e}}+\left( \mathbf{C}_{\text{b}}^{\text{e}}+\mathbf{C}_{\text{kb}}^{\text{e}}\right)\dot {\mathbf{z}}_{\text{b}}^{\text{e}}+\left(\mathbf{K}_{\text{b}}^{\text{e}}+ \mathbf{K}_{\text{kb}}^{\text{e}}\right)\dot{\mathbf{z}}_{\text{b}}^{\text{e} }-\sum_{p=1}^{m}\left(\mathbf{C}_{\text{prb}}^{\text{e}}\dot{z}_{\text{pr}}^{ \text{e}}+\mathbf{K}_{\text{prb}}^{\text{e}}\mathbf{z}_{\text{pr}}^{\text{e}} \right)=0 \tag{9}\]  
\[\mathbf{C}_{\text{pb}}^{\text{e}}=c_{\text{rb}}\int_{(p-1)l_{r}}^{p\text{r}_{ r}}\mathbf{N}_{\text{b}}^{\text{T}}\mathbf{N}_{\text{b}}\text{d}x;p=1,2,\cdots m\]  
\[\mathbf{C}_{\text{kb}}^{\text{e}}=\sum_{p=1}^{m}\mathbf{C}_{\text{pb}}^{\text{ e}}=c_{\text{rb}}\int_{0}^{l_{b}}\mathbf{N}_{\text{b}}^{\text{T}}\mathbf{N}_{ \text{b}}\text{d}x \tag{10}\]  
where \(\mathbf{K}_{\text{rb}}^{\text{e}}\) and \(\mathbf{C}_{\text{kb}}^{\text{e}}\) are the additional damping matrix and stiffness matrix of the \(k\)th bridge element due to elastic support, respectively. \(\mathbf{C}_{\text{prb}}^{\text{e}}\) and \(\mathbf{K}_{\text{prb}}^{\text{e}}\) are the coupling damping and stiffness matrix of the \(p\)th rail element of the \(k\)th bridge element due to elastic support, respectively. \(l_{\text{r}}\) and \(l_{\text{b}}\) are the length of the rail and bridge element, respectively.  
The coupling damping matrix \(\mathbf{C}_{\text{prb}}^{\text{e}}\) can be expressed as:  
\[\mathbf{C}_{\text{prb}}^{\text{e}}=c_{\text{rb}}\int_{(p-1)l_{r}}^{p\text{r}_ {r}}\mathbf{N}_{\text{b}}^{\text{T}}\mathbf{N}_{\text{pr}}\text{d}x;\quad p=1,2,\cdots m \tag{11}\]