## 2 Dynamic Coupled Equation
### The Equation of Motion for Bridge  
The bridge is divided into a series of uniformly spaced beam elements, the length of which is denoted by \(l_{b}\). The longitudinal displacement, vertical displacement, lateral displacement, torsion, angular around Z axes, and angular around Y axes of each node are denoted by \(u_{b}\), \(v_{b}\), \(w_{b}\), \(\theta_{bx}\), \(\theta_{by}\), and \(\theta_{bz}\) respectively. There are \(m\) springs and dashpots in each element to connect the bridge with the track. The symbols \(k^{v}_{rb}\) and \(c^{v}_{rb}\) denote the vertical stiffness and damping coefficient, and the symbols \(k^{w}_{rb}\) and \(c^{v}_{rb}\) denote the lateral stiffness and damping coefficient at each join point respectively. The superscripts \"\(\cdot\)\" and \"\(\cdot\)\" denote the first and second derivatives with respect to time, and the superscripts \"\(\cdot\)\" and \"\(\cdot\)\" denote the first and second derivatives with respect to coordinates.  
The stress of the bridge in the equilibrium position is divided into inertial force, its elastic resilience and damping force, and the elastic and damping forces of the bridge-rail connection. The equation of motion for the element of bridge based on the virtual work principle can be expressed by:  
\[\begin{split}&\int_{0}^{l_{b}}m_{b}(\ddot{u}_{b}\delta u_{b}+ \ddot{v}_{b}\delta v_{b}+\ddot{w}_{b}\delta w_{b})+\rho_{b}J_{b}\ddot{\theta}_ {b}\delta\theta_{b}\text{d}x+\int_{0}^{l_{b}}E_{b}(A_{b}u^{\prime}_{b}\delta u ^{\prime}_{b}+I_{by}v^{\prime\prime}_{b}\delta v^{\prime\prime}_{b}+I_{bz}w^{ \prime\prime}_{b}\delta w^{\prime\prime}_{b})+G_{b}J_{b}\theta^{\prime}_{b} \delta\theta^{\prime}_{b}\text{d}x\\ &+\int_{0}^{l}c_{b}\Big{(}u_{b}\delta u_{b}+\dot{v}_{b}\delta v _{b}+\ddot{w}_{b}\delta w_{b}+\dot{\theta}_{b}\delta\theta_{b}\Big{)}\text{d}x +\sum_{i=1}^{m}k^{v}_{rb}(v_{bi}+f\theta_{bi}-v_{rLi})\delta(v_{bi}+f\theta_{ bi})\\ &+\sum_{i=1}^{m}c^{v}_{rb}\Big{(}\dot{v}_{bi}+f\dot{\theta}_{bi}- \dot{v}_{rLi}\Big{)}\delta(v_{bi}+f\theta_{bi})+\sum_{i=1}^{m}k^{w}_{rb}(w_{bi} +h_{b}\theta_{bi}-w_{rLi})\delta(w_{bi}+h_{b}\theta_{bi})\\ &+\sum_{i=1}^{m}c^{w}_{rb}\Big{(}\dot{w}_{bi}+h_{b}\dot{\theta}_{ bi}-\dot{w}_{rLi}\Big{)}\delta(w_{bi}+h_{b}\theta_{bi})+\sum_{i=1}^{m}k^{v}_{rb}(v_{bi}-f \theta_{bi}-v_{rRi})\delta(v_{bi}-f\theta_{bi})\\ &+\sum_{i=1}^{m}c^{v}_{rb}\Big{(}\dot{v}_{bi}-f\dot{\theta}_{bi}- \dot{v}_{rRi}\Big{)}\delta(v_{bi}-f\theta_{bi})+\sum_{i=1}^{m}k^{w}_{rb}(w_{bi} -h_{b}\theta_{bi}-w_{rRi})\delta(w_{bi}-h_{b}\theta_{bi})\\ &+\sum_{i=1}^{m}c^{w}_{rb}\Big{(}\dot{w}_{bi}-h_{b}\dot{\theta}_{ bi}-\dot{w}_{rRi}\Big{)}\delta(w_{bi}-h_{b}\theta_{bi})=\langle\delta d_{b} \rangle\{f_{b}\}\\ &\text{where $f$ denotes the eccentric distance of the left-side rail and the right-side rail; $h_{b}$ is the vertical distance between the bridge deck and the torsional center of the bridge.}\\ &\text{The node displacement components at both ends of each bridge element can be expressed by:  
\[\{d_{b}\}=\Big{\langle}u_{b1}\;\;v_{b1}\;\;w_{b1}\;\;\theta_{bx1}\;\;\theta_{ by1}\;\;\theta_{bz1}\;\;u_{b2}\;\;v_{b2}\;\;w_{bz2}\;\;\theta_{bx2}\;\;\theta_{by2}\;\; \theta_{bz2}\Big{\rangle}^{T}. \tag{2}\]The longitudinal displacement, vertical displacement, lateral displacement, and angular of each bridge element at the coordinates \(x\) using the Hermite interpolation function can be expressed by:  
\[\begin{array}{l}u_{b}(x)=\langle N_{u}\rangle_{b}\{d_{b}\}\\ v_{b}(x)=\langle N_{v}\rangle_{b}\{d_{b}\}\\ w_{b}(x)=\langle N_{w}\rangle_{b}\{d_{b}\}\\ \theta_{b}(x)=\langle N_{\theta}\rangle_{b}\{d_{b}\}\\ \end{array} \tag{3}\]