## 3 Coupled Monorail Train-Bridge System
### Monozial Train
\begin{array}{l}l_{bp}\ddot{\varphi}_{bj}-h_{2}k_{2y}\Big{(}y_{c}-y_{bj}-h_{ 1}\varphi_{c}-h_{2}\varphi_{bj}+\eta_{j}s\psi_{c}\Big{)}-b^{2}k_{2z}\Big{(} \varphi_{c}-\varphi_{bj}\Big{)}-\\ h_{2}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{)}-b^{2}c_{2z}\Big{(}\dot{\varphi}_{ c}-\dot{\varphi}_{bj}\Big{)}=-4b_{2}^{2}k_{r}\varphi_{bj}-\\ 4b_{2}^{2}\dot{c}_{r}\dot{\varphi}_{bj}+4k_{g}h_{5}\Big{(}y_{bj}-y_{B}-r_{y}-h_ {5}\varphi_{bj}\Big{)}+4c_{g}h_{5}\Big{(}\dot{y}_{bj}-\dot{y}_{B}-\dot{r}_{y}- h_{5}\dot{\varphi}_{bj}\Big{)}+\\ 2k_{s}h_{4}\Big{(}y_{bj}-y_{B}-r_{y}-h_{4}\varphi_{bj}\Big{)}+2c_{s}h_{4}\Big{(} \dot{y}_{bj}-\dot{y}_{B}-\dot{r}_{y}-h_{4}\dot{\varphi}_{bj}\Big{)},\end{array}  \tag{20}\]  
Head shaking of the bogie:  
\[l_{bp}\ddot{\psi}_{bj}-b^{2}k_{2x}(\psi_{c}-\psi_{bj})-b^{2}c_{2x}(\dot{\psi}_{ c}-\dot{\psi}_{bj})=-4k_{g}s_{2}^{2}\psi_{bj}-4c_{g}s_{2}^{2}\dot{\psi}_{bj}, \tag{21}\]  
Nodding of the bogie:  
\[l_{be}\ddot{\theta}_{bj}+h_{2}k_{2x}\Big{(}h_{1}\theta_{c}+h_{2}\theta_{bj} \Big{)}+h_{2}c_{2x}(h_{1}\dot{\theta}_{c}+h_{2}\dot{\theta}_{bj})=-4k_{r}s_{1}^ {2}\theta_{bj}-4c_{r}s_{1}^{2}\dot{\theta}_{bj}, \tag{22}\]  
where \(y_{B}\) = the lateral displacement of bridge; \(z_{B}\) = the vertical displacement of bridge; \(r_{y}\) = the track lateral irregularity; and \(r_{z}\) = the track vertical irregularity. According to the dynamic-static method, the dynamic balance equation of the monorail train can be expressed as follows:  
\[[\mathbf{M}_{\varphi}]\Big{\{}\bar{\mathbf{X}}_{\varphi}\Big{\}}+[\mathbf{C}_{\varphi}] \Big{\{}\dot{\mathbf{X}}_{\varphi}\Big{\}}+[\mathbf{K}_{\varphi}]\{\mathbf{X}_{\varphi}\}= \{\mathbf{P}_{g}\}+\{\mathbf{P}_{vb}\}, \tag{23}\]  
where \(v\) denotes the monorail vehicle; \(\{\mathbf{P}_{g}\}\) denotes the force vector of the gravity of the vehicle; \(\{\mathbf{P}_{vb}\}\) denotes the external force applied to the vehicle through the bridge. The specific expressions of the components in Equation (23) are as follows. The displacement vector of the vehicle:  
\[\mathbf{X}_{\varphi}=[\mathbf{y}_{c}\ \mathbf{z}_{c}\ \mathbf{\varphi}_{c}\ \mathbf{\psi}_{c}\ \mathbf{\theta}_{c}\ \mathbf{y}_{b1}\ \mathbf{z}_{b1}\ \mathbf{\varphi}_{b1}\ \mathbf{\psi}_{b1}\ \mathbf{\theta}_{b1}\ \mathbf{y}_{b2}\ \mathbf{z}_{b2}\ \mathbf{\varphi}_{b2}\ \mathbf{\psi}_{b2}\ \mathbf{\theta}_{b2}]^{T}, \tag{24}\]  
The mass matrix of the vehicle:  
\[\mathbf{M}_{\varphi}=\left[ \begin{array}{ccc}\mathbf{M}_{c}&0&0\\ 0&\mathbf{M}_{b1}&0\\ 0&0&\mathbf{M}_{b2}\end{array} \right], \tag{25}\]  
in which \(\mathbf{M}_{c}=diag\left[\mathbf{m}_{c}\ \mathbf{m}_{c}\ \mathbf{L}_{cp}\ \mathbf{I}_{c\mathbf{\phi}}\ \mathbf{I}_{c\mathbf{\phi}}\right]\), \(\mathbf{M}_{bj}=diag\left[\mathbf{m}_{bj}\ \mathbf{m}_{bj}\ \mathbf{I}_{b\mathbf{\varphi}}\ \mathbf{I}_{b\mathbf{ \phi}}\right]\). The stiffness matrix of the vehicle:  
\[K_{\rm p}=\left[\begin{array}{cccc} \begin{array}{cccc}\begin{array}{c} \mathbf{K_{c}}&\mathbf{K_{cb_{1}}}&\mathbf{K_{cb_{2}}}\\ \mathbf{K_{b_{1}c}}&\mathbf{K_{b_{1}}}&0\\ \mathbf{K_{b_{2}c}}&0&\mathbf{K_{b_{2}}}\end{array}\end{array} \right], \tag{26}\]  
\[K_{\rm c}=\left[\begin{array}{cccc} \begin{array}{cccc}\begin{array}{c}4k_{ 2y}&0&-4h_{1}k_{2y}&0&0\\ 0&4k_{2z}&0&0&0\\ -4h_{1}k_{2y}&0&4h_{1}^{2}k_{2y}+4b^{2}k_{2z}&0&0\\ 0&0&0&4s^{2}k_{2z}&0\\ 0&0&0&0&4s^{2}k_{2y}\end{array}\end{array} \right], \tag{27}\]  
\[K_{\rm b_{1}}=K_{\rm b_{2}}=\left[\begin{array}{cccc} \begin{array}{cccc} \begin{array}{c}2k_{2y}+4k_{x}+2k_{x}&0&2h_{2}k_{2y}-4h_{y}k_{x}-2h_{x}k_{x} &0&0\\ 0&2k_{2z}+4k_{x}&0&0\\ 2h_{2}k_{2y}-4h_{y}k_{x}-2h_{x}k_{x}&0&2h_{2}^{2}k_{2y}+2h^{2}k_{2z}+4h_{x}^{2} k_{x}+4h_{x}^{2}k_{x}+2h_{x}^{2}k_{x}&0&0\\ 0&0&0&4s^{2}_{2}k_{x}&0\\ 0&0&0&0&4s^{2}_{2}k_{x}\end{array}\end{array} \right]. \tag{28}\]  
\[K_{\rm cb_{1}}=K_{\rm b_{1}e}^{T}\left[ \begin{array}{cccc}-2k_{2y}&0&2h_{1}k _{2y}&0&-2 sk_{2y}\\ 0&-2k_{2z}&0&-2 sk_{2z}&0\\ -2h_{2}k_{2y}&0&2h_{1}h_{2}k_{2y}-2h^{2}k_{2z}&0&-2 sl_{2}k_{2y}\\ 0&0&0&0&0\\ 0&0&0&0&0\end{array}