## 3 Coupled Monorail Train-Bridge System
### Monozial Train
\begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{1}{c}{**Parameters**} & **Value** & **Unit** \\ \hline Moment of inertia of the car body on the X-axis (\(l_{cp}\)) & 53,900 & kg m\({}^{2}\) \\ Moment of inertia of the car body on the Y-axis (\(l_{cp}\)) & 530,000 & kg m\({}^{2}\) \\ Stiffness of the traveling wheel (\(k_{x}\)) & 1180 & kN/m \\ Stiffness of the steering wheel (\(k_{y}\)) & 980 & kN/m \\ Stiffness of the stabilizing wheel (\(k_{x}\)) & 980 & kN/m \\ Damping of the traveling wheel (\(c_{x}\)) & 26.1 & kN s/m \\ Damping of the steering wheel (\(c_{x}\)) & 186 & kN s/m \\ Damping of the stabilizing wheel (\(c_{x}\)) & 186 & kN s/m \\ Longitudinal stiffness of the secondary spring (\(k_{2x}\)) & 130 & kN/m \\ Transverse stiffness of the secondary spring (\(k_{2y}\)) & 130 & kN/m \\ Vertical stiffness of the secondary spring (\(k_{2z}\)) & 160 & kN/m \\ Longitudinal damping of the secondary suspension system (\(c_{2z}\)) & 333.6 & kN s/m \\ Transverse damping of the secondary suspension system (\(c_{2y}\)) & 333.6 & kN s/m \\ Vertical damping of the secondary suspension system (\(c_{2z}\)) & 22.8 & kN s/m \\ Vertical distance between the center of the car body and the center of the secondary spring (\(h_{1}\)) & 0.529 & m \\ Vertical distance between the center of the bogie and the center of the secondary spring (\(h_{2}\)) & 0.352 & m \\ Transverse distance between the center of the car body and the upper endpoint of the secondary spring (\(b\)) & 1.025 & m \\ Longitudinal distance between the center of the car body and the upper endpoint of the secondary spring (\(s\)) & 4.8 & m \\ Height between the centers of the bogie and the traveling wheel (\(l_{3}\)) & \(-\)0.221 & m \\ Transverse distance between the centers of the bogie and the traveling wheel (\(b_{2}\)) & 0.2 & m \\ Longitudinal distance between the centers of the bogie and the traveling wheel (\(s_{1}\)) & 0.75 & m \\ Height between the centers of the bogie and the steering wheel (\(b_{1}\)) & \(-\)0.061 & m \\ Transverse distance between the centers of the bogie and the steering wheel (\(b_{1}\)) & 0.782 & m \\ Longitudinal distance between the centers of the bogie and the steering wheel (\(s_{2}\)) & 1.2 & m \\ Height between the centers of the bogie and the stabilizing wheel (\(h_{4}\)) & 1.025 & m \\ Transverse distance between the centers of the bogie and the stabilizing wheel (\(b_{1}\)) & 0.782 & m \\ Half-width of the track beam (\(b_{3}\)) & 0.345 & m \\ Height between the centers of the bogie and the track beam (\(h_{4}\)) & 0.725 & m \\ Length of the vehicle (\(L\)) & 15.5 & m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._bogies, respectively. The governing equations of the train are derived using an energy method based on the Lagrange equation, as shown in Equation (9) [28]. \[\frac{d}{dt}\bigg{(}\frac{\partial T}{\partial\dot{a}_{i}}\bigg{)}-\frac{\partial T }{\partial a_{i}}+\frac{\partial U}{\partial a_{i}}+\frac{\partial Q}{\partial \dot{a}_{i}}=0, \tag{9}\]  
where \(T\) is the kinetic energy, \(U\) is the potential energy, \(Q\) is the dissipation energy of the system, and \(a_{i}\) is a generalized coordinate. The kinetic energy, potential energy, and dissipation energy of a train on a bridge are developed by modifying the energy equation for a vehicle on a highway bridge [32]. These energy equations are expressed in a set of generalized coordinates as follows:  
\[