## 2 Dynamic Coupled Equation
### The Equation of Motion for Track  
The single-line track is assembled by the left-side rail and the right-side rail, which are symmetric with respect to the central line of the bridge. The track system is composed of a single-layer rail ignoring the elastic contact between the track slab and the bridge deck. The elastic support under the track is provided by rubber pad. The left-side and right-side rails are divided into a series of uniformly spaced beam elements, the length of which are respectively denoted by \(l_{rl}\) and \(l_{rR}\). The mathematical model avoids the separate modeling of wheel-rail vertical contact and wheel-rail vertical separation. Based on assumption (2), the linear stiffness coefficient of the vertical Hertz contact is uniformly expressed by \(ak_{wr}^{v}\). The contact coefficient \(a\) should satisfy the following piecewise function considering no tension in the vertical wheel-rail connection. \[ \begin{cases}a=1&y_{k}^{s}>0\\ a=0&y_{k}^{s}<0\end{cases} . \tag{4}\]  
The vertical geometric compatibility condition at the \(k\)th wheelset can be expressed by:  
\[y_{k}^{s}=v_{r}(x_{k},t)-Y_{wk}+{(-1)}^{n}e\theta_{wk}-IR_{y}(x_{k}), \tag{5}\]  
where \(Y_{w}\) and \(\theta_{w}\) denote the vertical and rolling motion of the wheelset; \(n=1\), \(2\) denotes the left-side and right-side wheel respectively; \(v_{r}\) denotes the vertical displacement of the rail at the contact coordinates \(x_{k}\) and the moment \(t\); \(IR_{y}\) denotes the vertical irregularity of the rail. The stress of the rail in the equilibrium position is divided into inertial force, its elastic resilience and damping force, the elastic and damping forces of the bridge-rail connection, and the elastic force of the wheel-rail connection. The equation of motion for the element of left-side rail based on the virtual work principle can be expressed by:  
\[ \begin{array}{l}\int_{0}^{l_{rl}}m_{r}(\bar{u}_{rl}\delta u_{rl}+\tilde{v}_ {rl}\delta v_{rl}+\bar{w}_{rl}\delta w_{rl})\text{d}x+\int_{0}^{l_{rl}}E_{r}( A_{rl}u_{rl}^{\prime}\delta u_{rl}^{\prime}+I_{ry}v_{rl}^{\prime\prime} \delta v_{rl}^{\prime\prime}+I_{rz}w_{rl}^{\prime\prime}\delta w_{rl}^{\prime \prime})\text{d}x\\ +\int_{0}^{l_{rl}}c_{r}\big{(}\dot{u}_{rl}\delta u_{rl}+\dot{v}_{rl}\delta v_{rl }+\dot{w}_{rl}\delta w_{rl})\text{d}x+\sum\limits_{i=1}^{m}k_{rb}^{v}(v_{rl}- v_{bi}-f\theta_{bi})\delta v_{rl}\\ +\sum\limits_{i=1}^{m}c_{rb}^{v}\big{(}\dot{v}_{rl}-\dot{v}_{bi}-f\dot{\theta} _{bi}\big{)}\delta v_{rl}+\sum\limits_{i=1}^{m}k_{rb}^{w}(w_{rl}-w_{bi}-h_{b} \theta_{bi})\delta w_{rl}\\ +\sum\limits_{i=1}^{m}c_{rb}^{w}\big{(}\dot{w}_{rl}-\dot{w}_{bi}-h_{b}\dot{ \theta}_{bi}\big{)}\delta w_{rl}+ak_{wr}^{v}\big{(}v_{rl}-Y_{w}-e\theta_{w}- IR_{yl}\big{)}\delta v_{rl}\\ +k_{wr}^{w}(w_{rl}-Z_{w}-R\theta_{w}-IR_{xL})\delta w_{rl}=\langle\delta d_{rl} \rangle\big{\{}f_{rl}\big{\}}\\ \end{array}  \tag{6}\]  
where \(k_{wr}^{w}\) denotes lateral wheel-rail contact stiffness; \(IR_{yL}\) and \(IR_{zL}\) denote the left-side vertical and lateral rail irregularity respectively; \(e\) is the half of lateral distance between contact points of wheelset and rail. The node displacement components at both ends of the left-side rail element can be expressed by:  
\[\{d_{rl}\}=\big{\langle}u_{rl1}&v_{rl1}&w_{rl1}&\theta_{rlx1}&\theta_{rly1}& \theta_{rlz1}&u_{rl2}&v_{rl2}&w_{rl2}&\theta_{rlx2}&\theta_{rly2}&\theta_{rlz2} &\theta_{rlz2}\big{\rangle}^{T}. \tag{7}\]The longitudinal displacement, vertical displacement, lateral displacement, and angular of the left-side rail at the coordinates \(x\) using the Hermite interpolation function can be expressed by:  
\[ \begin{array}{l}u_{rL}(x)=\left\langle N_{u}\right\rangle_{rL}\{d_{rL}\}\\ v_{rL}(x)=\left\langle N_{v}\right\rangle_{rL}\{d_{rL}\}\\ w_{rL}(x)=\left\langle N_{w}\right\rangle_{rL}\{d_{rL}\}\\ \theta_{rL}(x)=\left\langle N_{\theta}\right\rangle_{rL}\{d_{rL}\}\\ \end{array}  \tag{8}\]  
The equation of motion for the element of the right-side rail based on the virtual work principle can be expressed by:  
\[