## 3 Coupled Monorail Train-Bridge System
### Monozial Train
\begin{array}{l}T=[\frac{1}{2}m_{c}z_{c}^{2}+\frac{1}{2}m_{c}\dot{y}_{c}^{2}+ \frac{1}{2}I_{c\varphi}\dot{\varphi}_{c}^{2}+\frac{1}{2}I_{c\varphi}\dot{ \psi}_{c}^{2}+\frac{1}{2}I_{c\theta}\dot{\theta}_{c}^{2}+\sum_{j=1}^{2}[\frac {1}{2}m_{b}\dot{z}_{bj}^{2}+\frac{1}{2}m_{b}\dot{y}_{bj}^{2}+\\ \frac{1}{2}I_{b\varphi}\dot{\varphi}_{bj}^{2}+\frac{1}{2}I_{b\psi}\dot{\psi}_ {bj}^{2}+\frac{1}{2}I_{b\theta}\dot{\theta}_{bj}^{2}],\\ \\ U=\sum_{j=1}^{2}\{\frac{1}{2}k_{2x}\bigg{[}b^{2}\Big{(}\psi_{c}-\psi_{bj} \Big{)}^{2}+\Big{(}h_{1}\theta_{c}+h_{2}\theta_{bj}\Big{)}^{2}\bigg{]}+\frac{ 1}{2}k_{2y}\Big{(}y_{c}-y_{bj}-h_{1}\varphi_{c}-h_{2}\varphi_{bj}+\\ \\ \eta_{j}s\psi_{c}\Big{)}^{2}+\frac{1}{2}k_{2z}[\Big{(}z_{c}-z_{bj}+\eta_{j}s \theta_{c}\Big{)}^{2}+b^{2}\Big{(}\varphi_{c}-\varphi_{bj}\Big{)}^{2}]\},\\ \\ Q=\sum_{j=1}^{2}\{\frac{1}{2}c_{2x}\bigg{[}b^{2}\Big{(}\dot{\psi}_{c}-\dot{ \psi}_{bj}\Big{)}^{2}+\Big{(}h_{1}\dot{\theta}_{c}+h_{2}\dot{\theta}_{bj} \Big{)}^{2}\bigg{]}+\frac{1}{2}c_{2y}(\dot{y}_{c}-\dot{\psi}_{bj}-h_{1}\dot{ \varphi}_{c}-h_{2}\dot{\varphi}_{bj}+\\ \\ \eta_{j}\dot{s}\dot{\psi}_{c}\Big{)}^{2}+\frac{1}{2}c_{2z}[\big{(}\dot{z}_{c}- \dot{z}_{bj}+\eta_{j}\dot{s}\dot{\theta}_{c}\big{)}^{2}+b^{2}\big{(}\dot{ \varphi}_{c}-\dot{\varphi}_{bj}\big{)}^{2}]\},\end{array}  \tag{11}\]  
where \(j=1\), when \(\eta=-1\), or \(j=2\), when \(\eta=1\). The displacement equation of the monorail train can be derived by substituting Equations (10)-(12) into Equation (9). Yawing of the car body:  
\[m_{c}\ddot{y}_{c}+\sum_{j=1}^{2}k_{2y}\Big{(}y_{c}-y_{bj}-h_{1}\varphi_{c}-h_{ 2}\varphi_{bj}\Big{)}+\sum_{j=1}^{2}c_{2y}\Big{(}\dot{y}_{c}-\dot{y}_{bj}-h_{1} \dot{\varphi}_{c}-h_{2}\dot{\varphi}_{bj}\Big{)}=0, \tag{13}\]  
Vertical settlement of the car body:  
\[m_{c}\ddot{z}_{c}+\sum_{j=1}^{2}k_{2z}\Big{(}z_{c}-z_{bj}\Big{)}+\sum_{j=1}^{2 }c_{2z}(\dot{z}_{c}-\dot{z}_{bj})=0, \tag{14}\]  
Side-rolling of the car body:  
\[ \begin{array}{l}I_{c\varphi}\ddot{\varphi}_{c}-\sum_{j=1}^{2}h_{1}k_{2y} \Big{(}y_{c}-y_{bj}-h_{1}\varphi_{c}-h_{2}\varphi_{bj}\Big{)}+\sum_{j=1}^{2} b^{2}k_{2z}\Big{(}\varphi_{c}-\varphi_{bj}\Big{)}-\\ \sum_{j=1}^{2}h_{1}c_{2y}\Big{(}\dot{y}_{c}-\dot{y}_{bj}-h_{1}\dot{\varphi}_{c} -h_{2}\dot{\varphi}_{bj}\Big{)}+\sum_{j=1}^{2}b^{2}c_{2z}\Big{(}\dot{\varphi}_{ c}-\dot{\varphi}_{bj}\Big{)}=0,\end{array}  \tag{15}\]  
Head shaking of the car body:  
\[ \begin{array}{l}I_{c\varphi}\ddot{\psi}_{c}+b^{2}k_{2x}(2\psi_{c}-\psi_{b1}- \psi_{b2})+sk_{2y}[(y_{b1}-y_{b2})+h_{2}(\varphi_{b1}-\varphi_{b2})+2s\psi_{c}] +\\ b^{2}c_{2x}(2\dot{\psi}_{c}-\dot{\psi}_{b1}-\dot{\psi}_{b2})+sc_{2y}\big{[}( \dot{y}_{b1}-\dot{y}_{b2})+h_{2}\big{(}\dot{\varphi}_{b1}-\dot{\varphi}_{b2} \big{)}+2s\dot{\psi}_{c}]=0,\end{array}  \tag{16}\]  
Nodding of the car body:  
\[ \begin{array}{l}I_{c\theta}\ddot{\theta}_{c}+h_{1}k_{2x}[2h_{1}\theta_{c}+h_{ 2}(\theta_{b1}+\theta_{b2})]+sk_{2z}[(z_{b1}-z_{b2})+2s\theta_{c}]+h_{1}c_{2x}[2 h_{1}\dot{\theta}_{c}+\\ h_{2}(\dot{\theta}_{b1}+\theta_{b2})]+sc_{2z}\big{[}(\dot{z}_{b1}-\dot{z}_{b2})+2s \dot{\theta}_{c}]=0,\end{array}  \tag{17}\]
Yawing of the bogie:  
\[ \begin{array}{l}m_{b}\ddot{y}_{bj}-k_{2y}\Big{(}y_{c}-y_{bj}-h_{1}\varphi_{c}-h _{2}\varphi_{bj}+\eta_{j}s\dot{\psi}_{c}\Big{)}-c_{2y}\Big{(}\dot{y}_{c}-\dot{y} _{bj}-h_{1}\dot{\varphi}_{c}-h_{2}\dot{\varphi}_{bj}+\\ \eta_{j}s\dot{\psi}_{c}\Big{)}=-4k_{g}\Big{(}y_{bj}-y_{B}-r_{y}-h_{5}\varphi_{bj }\Big{)}-4c_{g}\Big{(}\dot{y}_{bj}-\dot{y}_{B}-\dot{r}_{y}-h_{5}\dot{\varphi}_{bj }\Big{)}-\\ 2k_{s}\Big{(}y_{bj}-y_{B}-r_{y}-h_{4}\varphi_{bj}\Big{)}-2c_{s}\Big{(}\dot{y}_{bj }-\dot{y}_{B}-\dot{r}_{y}-h_{4}\dot{\varphi}_{bj}\Big{)},\end{array}  \tag{18}\]  
Vertical settlement of the bogie:  
\[ \begin{array}{l}m_{b}\ddot{z}_{bj}-k_{2z}\Big{(}z_{c}-z_{bj}+\eta_{j}s\theta _{c}\Big{)}-c_{2z}(\dot{z}_{c}-\dot{z}_{bj}+\eta_{j}\dot{s}\dot{\theta}_{c})=- 4k_{r}\Big{(}z_{bj}-z_{B}-r_{z}\Big{)}-\\ 4c_{z}\Big{(}\dot{z}_{bj}-\dot{z}_{B}-\dot{r}_{z}\Big{)},\end{array}  \tag{19}\]  
Side-rolling of the bogie:  
\[