## 2 Theoretical Background
### Vertical Coupling Vibration Model of Train-Rail-Bridge
#### 2.2.5. Train-Rail-Bridge Coupling Time-Varying Equation  
Accordingly, the coupled element dynamic Equation (17) of an arbitrary length combination, as shown in Figure 2, can be obtained by simultaneous solution of Equations (12)-(16).  
\[\left(\begin{array}{cccc}\mathbf{M}_{\mathrm{b}}^{e}&&&\text{ sym.}\\ 0&\mathbf{M}_{\mathrm{lr}}^{e}&&&\\ \vdots&\vdots&\ddots&\\ 0&0&\dots&\mathbf{M}_{\mathrm{mr}}^{e}&\\ 0&0&\dots&0&m_{\mathrm{w}}\end{array}\right)\left(\begin{array}{cccc}\dot{ \mathbf{z}}_{\mathrm{b}}^{e}\\ \dot{\mathbf{z}}_{\mathrm{lr}}^{e}\\ \dot{\mathbf{z}}_{\mathrm{mr}}^{e}\\ \dot{y}_{\mathrm{w}}\end{array}\right)+\left(\begin{array}{cccc}\mathbf{C}_{ \mathrm{b}}^{e}+\mathbf{C}_{\mathrm{b}}^{e}\\ -\mathbf{C}_{\mathrm{lbr}}^{e}&\mathbf{C}_{\mathrm{lr}}^{e}+\mathbf{C}_{ \mathrm{lrr}}^{e}\\ \vdots&\vdots&\ddots\\ 0&-\mathbf{C}_{\mathrm{mbr}}^{e}&0&\dots&\mathbf{C}_{\mathrm{mr}}^{e}+\mathbf{ C}_{\mathrm{mr}}^{e}\\ 0&0&\dots&0&0\end{array}\right)\left(\begin{array}{c}\dot{\mathbf{z}}_{ \mathrm{b}}^{e}\\ \dot{\mathbf{z}}_{\mathrm{lr}}^{e}\\ \dot{\mathbf{z}}_{\mathrm{mr}}^{e}\\ \dot{y}_{\mathrm{w}}\end{array}\right)+ \tag{17}\]  
As previously stated, the train system is based on a theoretical model of a 10 DOFs multi-rigid-body system, which is illustrated in reference [39], along with its system equation, stiffness, and damping matrix.  
According to the principle that the wheel-rail interaction unit and the total potential energy of the power system are fixed, the finite element motion equation of the train-rail-bridge interaction system is as follows [40]:  
\[\left[\begin{array}{cccc}\mathbf{M}_{\mathrm{bb}}&&&\text{ sym.}\\ 0&\mathbf{M}_{\mathrm{rr}}&&&\\ 0&0&&\mathbf{M}_{\mathrm{vv}}\end{array}\right]\left\{\begin{array}{c}\ddot{ \mathbf{z}}_{\mathrm{b}}\\ \dot{\mathbf{z}}_{\mathrm{r}}\\ \dot{\mathbf{z}}_{\mathrm{v}}\end{array}\right\}+\left[\begin{array}{cccc} \mathbf{C}_{\mathrm{bb}}&&&\text{ sym.}\\ \dot{\mathbf{z}}_{\mathrm{r}}&\\ 0&0&&\mathbf{C}_{\mathrm{vv}}\end{array}\right]\left\{\begin{array}{c}\dot{ \mathbf{z}}_{\mathrm{b}}\\ \dot{\mathbf{z}}_{\mathrm{r}}\\ \dot{\mathbf{z}}_{\mathrm{v}}\end{array}\right\}+\left[\begin{array}{cccc} \mathbf{K}_{\mathrm{bb}}&&&\text{ sym.}\\ \dot{\mathbf{z}}_{\mathrm{r}}&\\ 0&0&&\mathbf{C}_{\mathrm{vv}}\end{array}\right]\left\{\begin{array}{c}\dot{ \mathbf{z}}_{\mathrm{b}}\\ \dot{\mathbf{z}}_{\mathrm{r}}\\ \dot{\mathbf{z}}_{\mathrm{v}}\end{array}\right\}+\left[\begin{array}{cccc} \mathbf{K}_{\mathrm{bb}}&&&\text{ sym.}\\ \mathbf{K}_{\mathrm{br}}&\mathbf{K}_{\mathrm{rr}}+\mathbf{K}_{\mathrm{cr}}&\\ 0&\mathbf{K}_{\mathrm{rv}}&\mathbf{K}_{\mathrm{vv}}\end{array}\right]\left\{ \begin{array}{c}\mathbf{z}_{\mathrm{b}}\\ \mathbf{z}_{\mathrm{r}}\\ \mathbf{z}_{\mathrm{v}}\end{array}\right\}=\left\{\begin{array}{c}\mathbf{F} _{\mathrm{b}}\\ \mathbf{F}_{\mathrm{r}}+\mathbf{F}_{\mathrm{cr}}\\ \mathbf{F}_{\mathrm{v}}+\mathbf{F}_{\mathrm{cv}}\end{array}\right\} \tag{18}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) represent mass, damping, and stiffness submatrix, respectively. \(\mathbf{z}\) and \(\mathbf{F}\) represent displacement and force subvector, respectively. In addition, subscripts b, r, v, and c represent the interaction effects caused by the bridge, rail, train, and wheel-rail contact.