## 3 Coupled Monorail Train-Bridge System
### Monorail Train-Bridge Interaction  
The vehicle system and the bridge system are coupled through the wheel-rail contact relationship. The force vector of the vehicles acting on the bridge can be expressed as:  
\[\mathbf{P_{b}}=\sum_{v=1}^{n_{v}}\sum_{m=1}^{3}\sum_{j=1}^{2}\sum_{i=1}^{2}\sum_{n=1 }^{2}\delta_{vmjinf}\mathbf{f_{vmjin}}, \tag{40}\]  
where \(\delta_{vmjin}\) is the distribution vector of the vehicle wheel loads delivered to each node of the bridge element; \(f_{vmjin}\) is the wheel load of the monorail trains acting on the bridge. \(n_{v}\) denotes the number of carariages in a monorail train; \(v\) denotes the number of carariages on the bridge; \(m=1\), \(2\), \(3\) indicating the traveling wheel, steering wheel, and the stabilizing wheel, respectively; \(j=1\), \(2\) indicating the front and rear bogies, respectively; \(i=1\), \(2\) indicating the front and rear bogies, respectively; \(n=1\), \(2\) indicating the left and right wheel, respectively. The expression of the wheel load fvmjin is as follows:  
\[\left\{ \begin{array}{l}f_{v1 jin}=k_{r}\cdot\left[\left(z_{bj}-z_{B}-r_{z} \right)+(-1)^{i}s_{1}\theta_{bj}+(-1)^{n}b_{2}\varphi_{bj}\right]+c_{r}\cdot \left[\left(\dot{z}_{bj}-\dot{z}_{B}-\dot{r}_{z}\right)+\right.\\ \left.\left(-1\right)^{i}s_{1}\dot{\theta}_{bj}+(-1)^{n}b_{2}\dot{\varphi}_{bj }\right],\ \ \text{traveling wheel},\\ f_{v2 jin}=k_{g}\cdot\left[\left(y_{bj}-y_{B}-r_{y}\right)-(-1)^{i}s_{2}\dot{ \varphi}_{bj}-h_{5}\varphi_{bj}\right]+c_{g}\cdot\left[\left(\dot{y}_{bj}-\dot {y}_{B}-\dot{r}_{y}\right)-\right.\\ \left.\left(-1\right)^{i}s_{2}\dot{\psi}_{bj}-h_{5}\dot{\varphi}_{bj}\right], \ \ \text{steering wheel},\\ f_{v3 jin}=k_{s}\cdot\left[\left(y_{bj}-y_{B}-r_{y}\right)-h_{4}\varphi_{bj} \right]+c_{s}\cdot\left[\left(\dot{y}_{bj}-\dot{y}_{B}-\dot{r}_{y}\right)-h_{4 }\dot{\varphi}_{bj}\right],\ \ \text{stabilizing wheel},\end{array} \right. \tag{41}\]  
According to the coordination condition of displacement, the surface of the bridge is always in contact with the wheels when the train passes the bridge. Meanwhile, the dynamic interaction between the bridge and the train is simulated based on the Darren Bell principle. In the process of the analysis for the train bridge coupled vibration, the force of a wheel acting on the bridge is equal to the force of the bridge acting on the wheel. Therefore, the governing equation of the train-bridge system can be determined as shown in Equation (42) by combining the motion equations of the monorail train and the bridge and the interaction force at the contact point [35]. \[\left[ \begin{array}{cc}\mathbf{M_{b}}&0\\ 0&\mathbf{M_{v}}\end{array} \right]\left[ \begin{array}{cc}\ddot{\mathbf{X}_{b}}\\ \dot{\mathbf{X}_{v}}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{C_{b}}&\mathbf{C_{ bv}}\\ \mathbf{C_{vb}}&\mathbf{C_{v}}\end{array} \right]\left[ \begin{array}{cc}\dot{\mathbf{X}_{b}}\\ \dot{\mathbf{X}_{v}}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{K_{b}}&\mathbf{K_{ bv}}\\ \mathbf{K_{vb}}&\mathbf{K_{v}}\end{array} \right]\left[ \begin{array}{cc}\mathbf{X_{b}}\\ \mathbf{X_{v}}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{P_{b}}\\ \mathbf{P_{v}}\end{array} \right], \tag{42}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) indicate the mass, damping, and stiffness matrices, respectively. \(\mathbf{P}\) is the force vector. Subscripts \(b\), \(v\), and \(bv\) (or \(vb\)) denote the bridge, vehicle, and vehicle-bridge interaction system, respectively. Figure 4: The track irregularity spectrum of the track beam: (**a**) under the traveling wheel; (**b**) under the steering wheel; (**c**) under the stabilizing wheel. The mass matrix of the bridge:  
\[\mathbf{M_{b}}=\left[ \begin{array}{cccc}\mathbf{M_{b1}^{e}}&0&0&0\\ 0&\mathbf{M_{b2}^{e}}&0&0\\ 0&0&\ddots&0\\ 0&0&0&\mathbf{M_{bn}^{e}}\end{array} \right],\ n=\text{Number of beam elements}, \tag{43}\]  
The mass matric of the train (a monorail train consists of 6 carriages):  
\[\mathbf{M_{v}}=\left[