## 2 Coupled Straddle-Type Monorail-Train System
### Bridge Model  
The superstructure of a monorail bridge consisted of two separate concrete track beams that have low stiffness. Simply supported beam bridges with a span of 22-26 m were generally used in straddle-type monorail transportation systems. The standard cross-sectional dimensions of the track beam, as shown in Figure 3, were 85 mm in width and 150 mm in height. Three middle crossbeams and two end crossbeams were built in each span. The center distance between the two separate track beams was 3.7 m. The piers were made of C30 concrete with cross-sectional dimensions of 1.7 m \(\times\) 1.7 m and \begin{table}
\begin{tabular}{c c c c c c} \hline
**Parameter** & **Notation** & **Value** & **Parameter** & **Notation** & **Value** \\ \hline Vehicle body length (mm) & \(L_{0}\) & 14,800 & Running wheel damping (N\(\times\)m) & \(C_{a}\) & 26100 \\ Middle vehicle body length (mm) & \(L_{1}\) & 13,900 & Guiding wheel stiffness (N/m) & \(K_{d}\) & 980,000 \\ Bogie centre distance (mm) & \(L\) & 9600 & Guiding wheel damping (N\(\times\)m) & \(C_{d}\) & 185,000 \\ Vehicle body width (mm) & \(D\) & 2900 & Stabilizing wheel stiffness (N/m) & \(K_{w}\) & 980,000 \\ Vehicle height (mm) & \(H_{c}\) & 5300 & Stabilizing wheel damping (N\(\times\)m) & \(C_{w}\) & 185,000 \\ Track beam width and height (mm) & \(B\&H\) & 850k150 & Lateral stiffness of the air spring (N/m) & \(K_{w}\) & 980,000 \\ Wheedbase of the running wheel (mm) & \(L_{e}\) & 1500 & Lateral damping of the air spring (N\(\times\)m) & \(C_{xy}\) & 10,000 \\ Wheedbase of the guiding wheel (mm) & \(l_{d}\) & 2500 & Vertical stiffness of the air spring (N/m) & \(K_{w}\) & 160,000 \\ Vehicle mass (kg) & \(M_{c}\) & 14,220 & Vertical damping of the air spring (N\(\times\)m) & \(C_{xx}\) & 45,400 \\ Bogie mass (kg) & \(M_{b}\) & 6200 & Shock absorber stiffness (N/m) & \(K_{y}\) & 9,810,000 \\ Running wheel mass (kg) & \(M_{w}\) & 54 & Longitudinal stiffness of the traction rubber & \(K_{\text{cp}}\) & 500,000 \\  & & & & & \\ Guiding wheel mass (kg) & \(M_{d}\) & 30 & Longitudinal damping of the traction rubber & \(C_{\text{cp}}\) & 50,000 \\  & & & & & \\ Stabilizing wheel mass (kg) & \(M_{w}\) & 30 & Lateral stiffness of the traction rubber & \(K_{\text{cp}}\) & 50,000 \\  & & & & & \\ Running wheel stiffness (N/m) & \(K_{w}\) & 1,180,000 & Lateral damping of the traction rubber & \(C_{\text{cp}}\) & 5000 \\  & & & & & \\ \end{tabular}
\end{table} 
Table 2: Main parameters for the monorail vehicle. Figure 2: Topology graph of a straddle-type monorail. a coping height of 1.5 m, and the height of the pier was at least 7 m in this paper, to satisfy the road boundary requirements that the track surface should be approximately 10 m from the ground. In this study, ANSYS was applied to establish a 3 m \(\times\) 22 m double-line simply supported beam bridge model for straddle-type monorail traffic. The dynamic response of a bridge was generally calculated by numerical integration or the mode superposition method. The equation of motion of a bridge model could be written as follows [36]:  
\[\mathbf{M}\ddot{u}+\mathbf{C}\dot{u}+\mathbf{K}u=\mathbf{F} \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass matrix, damping matrix, and stiffness matrix, respectively; \(\ddot{u}\), \(\dot{u}\), and \(u\) are the acceleration, velocity, and displacement of the bridge, respectively; and \(\mathbf{F}\) is the external force matrix of the bridge, which is generally the vehicle load matrix or the vehicle-bridge interaction force matrix for the bridge subsystem of a coupled vehicle-bridge system. Low-order modes are the main focus in vehicle-bridge coupling vibration problems; thus, the lumped mass matrix and the mode superposition method could be used to solve the dynamic response of the bridge.