## 2 Coupled Straddle-Type Monorail-Train System
### Straddle-Type Monorail Model  
A straddle-type monorail train consists of two bogies and a vehicle body. Each bogie is composed of a bogie frame, rubber wheels, a suspension system, and a braking system. There are three types of rubber wheels--walking wheels, which provide movement and bear most of the vertical force; stabilizing wheels, which are installed on each side of the bogie to maintain vehicle balance and prevent vehicle roll; and guiding wheels, which play an important role in guiding the direction of the vehicle, especially when passing curves. The suspension system acts as a force transmission and vibration attenuation device and consists of two air springs, two shock absorbers, and a traction rubber pile. The dynamics analysis model of a straddle-type monorail vehicle is shown in Figure 1.  
Rigid bodies are used to simulate the vehicle body, bogies, and wheels in SIMPACK, the mechanical characteristics of the suspension system are simulated by force elements, and the elastic characteristics of the rubber wheels are simulated by the tire elements. The DOFs of the vehicle model are defined by hinges and constraints. In this study, six rigid body DOFs, which are described as longitudinal, vertical and lateral translation, rolling, pitching, and yawing, were considered for the vehicle body and the bogies, and one rotation freedom was considered for each rubber tire. Thus, the monorail vehicle model had 38 DOFs, as shown in Table 1.  
A topology graph that clearly describes the relationship between the various components of the monorail vehicle was introduced as the modeling basis, which was convenient for model troubleshooting. The topology graph of a straddle-type monorail is presented in Figure 2. The tire was adopted to simulate the tire force in the vehicle model; it was a connecting element in a multi-body system composed of springs and dampers connected in parallel. The main parameters for vehicle modeling are listed in Table 2[32; 35].  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Components**} & \multirow{2}{*}{\begin{tabular}{c} **Longitudinal** \\ **Translation** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Vertical** \\ **Translation** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Lateral** \\ **Translation** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Rolling** \\ \(\alpha\) \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Pitching** \\ \(\beta\) \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Yawing** \\ \(\gamma\) \\ \end{tabular} } \\  & \(x\) & \(y\) & \(z\) & & & \\ \hline Vehicle body and bogies & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\ Walking wheels & - & - & - & - & \(\surd\) & - \\ Guiding wheels & - & - & - & - & - & \(\surd\) \\ Stabilizing wheels & - & - & - & - & - & \(\surd\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Degrees of Freedom (DOFs) of a straddle-type monorail vehicle.  
Figure 1: Dynamics analysis model of a straddle-type monorail.  
To simplify the dynamic analysis process and improve the efficiency of the solution, the four following basic assumptions were proposed for the vehicle model:  
1. The vehicle moved at a constant speed, and the walking wheels always clung to the upper surface of the track beam while driving.
2. The vehicle model was symmetrical about the X and Y axes.
3. The braking and power systems were ignored during modeling; however, their mass and inertia were superimposed on the bogie frames.
4. The body, bogies, and wheels vibrated with minute displacements at their equilibrium positions.