# Vibration and Stress Response of High-Speed Train Gearboxes under Different Excitations
## 2 Modeling of High-Speed Train with Gear Transmission  
### Multibody System Dynamics Model  
The typical high-speed train model of China was established in this study, as shown in Figure 3, which is composed of a car body, two bogies, and four wheelsets. Each wheelset is mounted with a traction motor and a gear transmission system. The detailed model has been illustrated in Reference [20], and the total degrees of freedom of the train model are listed in Table 1. The primary suspension parameters of the train are given in the literature [17]. The dynamic model of the gear transmission system has been illustrated in Reference [26]. The main design parameters of the gear transmission are listed in Table 2. The multi-body dynamic model is established by [PERSON]. The formula of motion of the multi-body dynamic model is as follows:  
\[M\ddot{x}+\dot{C}\dot{x}+Kx=F \tag{1}\]  
where \(M\), \(C\), \(K\) are the mass matrix, the damping matrix, and the stiffness matrix, respectively. \(X\) is the displacement vector and \(F\) is the vector of loads.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Vehicle Model**} & \multicolumn{8}{c}{**Type of Motion**} \\ \cline{2-9}  & **Longitudinal** & **Horizontal** & **Vertical** & **Roll** & **Yaw** & **Pitch** & \\ \hline Carbody & \(X_{c}\) & \(Y_{c}\) & \(Z_{c}\) & \(\phi_{c}\) & \(\psi_{c}\) & \(\beta_{c}\) & \\ Bogie frame & \(X_{hi}\) & \(Y_{hi}\) & \(Z_{hi}\) & \(\phi_{hi}\) & \(\psi_{hi}\) & \(\beta_{hi}\) & \(i=1\)\(\sim\)2 \\ Wheelset & \(X_{ui}\) & \(Y_{ui}\) & \(Z_{ui}\) & \(\phi_{ui}\) & \(\psi_{ui}\) & \(\beta_{ui}\) & \(i=1\)\(\sim\)4 \\ Stator & \(X_{mi}\) & \(Y_{mi}\) & \(Z_{mi}\) & \(\phi_{mi}\) & \(\psi_{mi}\) & \(\beta_{mi}\) & \(i=1\)\(\sim\)4 \\ Rotor & & & & & & \(\beta_{ri}\) & \(i=1\)\(\sim\)4 \\ Gear wheel & & & & & & \(\beta_{pi}\) & \(i=1\)\(\sim\)4 \\ Pinion wheel & & & & & & \(\beta_{gi}\) & \(i=1\)\(\sim\)4 \\ Gearbox housing & & & & & & \(\beta_{gi}\) & \(i=1\)\(\sim\)4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Degrees of freedom for the railway vehicle.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Specification**} & \multicolumn{2}{c}{**Value**} \\ \cline{2-4}  & **Pinion** & **Gear** \\ \hline Tooth shape & Involute & Involute \\ Material & Steel & Steel \\ Normal module & 6 mm & 6 mm \\ Normal pressure angle & 20\({}^{\circ}\) & 20\({}^{\circ}\) \\ Dedendum diameter & 222 mm & 522 mm \\ Addendum diameter & 195 mm & 495 mm \\ Flank width & 65 mm & 65 mm \\ Number of teeth & 35 & 85 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Main design parameters of the gear transmissions.  
The wheel-rail contact force is evaluated by the Hertzian nonlinear contact theory, which can be expressed as follows:  
\[f_{i}(t)=\left\{\begin{pmatrix}\frac{1}{C}\delta Z_{i}(t)\end{pmatrix}^{\frac{3} {2}}&\text{if }\delta Z_{i}(t)>0\\ 0&\text{if }\delta Z_{i}(t)\leq 0\end{pmatrix}\right. \tag{2}\]  
where \(G\) is the wheel-rail contact constant and \(\delta Z_{i}(t)\) is the elastic compression deformation of the \(i\)th wheelset and rail at the contact point in the normal direction, as follows [17]:  
\[\delta Z_{i}(t)=Z_{\omega i}(t)-Z_{r}(x_{i},t)-Z_{0i}(t)(\,i=1\,\sim 4) \tag{3}\]  
where \(Z_{\omega i}(t)\) is the displacement of the \(i\)th wheelset, \(Z_{r}(x_{i},t)\) is the rail displacement under the \(i\)th wheelset, and \(Z_{0i}(t)\) is the track irregularity under the \(i\)th wheelset.