# Speed Limit of Linear Induction Motor Subway Trains Running through 65 m Radius Curves on Yard Line
## 2 Train-Track Coupling Dynamic Model
### Train System  
#### 2.1.1 Vehicle System  
The vehicle system was simplified into one car body, two bogie frames, and four wheelsets, each part of which was regarded as rigid body. The LIM was simplified into Euler beam. All parts are connected through the vehicle suspension system and simulated by a spring-damping connection unit. The degree of freedom contained in the LIM vehicle system is shown in Table 1.  
Force analysis is carried out on each vehicle component, as shown in Figures 3-5. Based on the force analysis, motion differential equations of each part are established. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Degree of Freedom** & **Longitudinal** & **Lateral** & **Vertical** & **Swing** & **Nodding** & **Yaw** \\ \hline Car body & \(X_{c}\) & \(Y_{c}\) & \(Z_{c}\) & \(\Phi_{c}\) & \(\beta_{c}\) & \(\Psi_{c}\) \\ \hline Bogie frames (\(i\) = 1–2) & \(X_{ii}\) & \(Y_{ii}\) & \(Z_{ii}\) & \(\Phi_{ii}\) & \(\beta_{ii}\) & \(\Psi_{ii}\) \\ Wheelsets (\(i\) = 1–4) & \(X_{uzi}\) & \(Y_{uzi}\) & \(Z_{wi}\) & \(\Phi_{wi}\) & \(\beta_{wi}\) & \(\Psi_{wi}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Degree of freedom contained in the LIM vehicle system. Figure 2: LIM subway vehicle-track coupling dynamic six-group model. Equation of car body motion:  
Longitudinal motion:  
\[M_{\rm c}\ddot{\rm X}_{\rm c}=-F_{\rm xtL1}-F_{\rm xtR1}-F_{\rm xtL2}-F_{\rm xtR2 }-F_{\rm xcf}-F_{\rm xcb} \tag{1}\]  
Lateral motion:  
\[ \begin{array}{rcl}M_{\rm c}\Big{[}\ddot{\rm Y}_{\rm c}+\frac{y^{2}}{R_{\rm c}}+ \big{(}r_{0}+H_{\rm bw}+H_{\rm Bt}+H_{\rm CB}\big{)}\ddot{\phi}_{\rm sec}\Big{]}& =F_{\rm ytL1}+F_{\rm ytR1}+F_{\rm ytL2}+F_{\rm ytR2}\\ &&-F_{\rm xcf}-F_{\rm xcb}+M_{\rm c}\delta y_{\rm sec}\end{array}  \tag{2}\]  
Figure 4: Force analysis of car bogie frame. Figure 5: Force analysis of wheelset. Figure 3: Force analysis of car body. Vertical motion:  
\[M_{\rm c}\bigg{[}\ddot{Z}_{\rm c}-a_{0}\ddot{\phi}_{\rm sec}-\frac{v^{2}}{R_{\rm c }}\phi_{\rm sec}\bigg{]}=-F_{\rm ztL1}-F_{\rm ztR1}-F_{\rm ztR2}-F_{\rm ztR2}-F_{ \rm zcf}-F_{\rm zcb}+M_{\rm c}g \tag{3}\]  
Swing motion:  
\[\begin{array}{ll}I_{\rm cr}\left[\ddot{\phi}_{\rm c}+\ddot{\phi}_{\rm sec} \right]&=-\big{(}F_{\rm ytL1}+F_{\rm ytR1}+F_{\rm ytL2}+F_{\rm ytR2}\big{)}H_{ \rm cB}+(F_{\rm zcf}+F_{\rm zcb})H_{\rm cc}\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+(F_ {\rm ztR1}-F_{\rm ztR1}+F_{\rm ztL2}-F_{\rm ztR2})d_{\rm s}  
\[ \begin{array}{l}\text{ Lateral motion:}\\ \\ M_{\text{w}}\bigg{[}\dot{Y}_{\text{w}i}+\frac{v^{2}}{R_{\text{w}i}}+r_{0}\ddot{\phi} _{\text{sew}i}\bigg{]}=-F_{\text{yftLi}}-F_{\text{yftRi}}+F_{\text{L}yi}+F_{ \text{R}yi}+N_{\text{L}yi}+N_{\text{R}yi}\\ \\ \text{ Vertical motion:}\\ \\ M_{\text{w}}\bigg{[}\ddot{Z}_{wi}-a_{0}\ddot{\phi}_{\text{sew}i}-\frac{v^{2}}{ R_{\text{w}i}}\phi_{\text{sew}i}\bigg{]}=F_{\text{zftLi}}+F_{\text{zfRi}}-F_{\text{L}zi}-F_{ \text{R}zi}-N_{\text{L}zi}-N_{\text{R}zi}+F_{\text{zlim}i}+M_{\text{w}}g\\ \\ \text{ Swing motion:}\\ \\ I_{\text{w}x}\bigg{(}\ddot{\phi}_{\text{sew}i}+\ddot{\phi}_{\text{w}i}\bigg{)}- I_{\text{w}y}\Big{(}\dot{\beta}_{\text{w}i}-\Omega\Big{)}\Big{(}\dot{\psi}_{ \text{w}i}+\frac{v}{R_{\text{w}i}}\Big{)}\\ \\ \hskip 14.226378 pt-r_{\text{L}i}\big{(}F_{\text{L}yi}+N_{\text{L}yi}\big{)}-r_{ \text{R}i}\big{(}F_{\text{R}yi}+N_{\text{R}yi}\big{)}\\ \hskip 14.226378 pt+d_{\text{w}}\big{(}F_{\text{zftRi}}-F_{\text{zftLi}}\big{)} \end{array}  \tag{16}\]  
\[ \begin{array}{l}\\ I_{\text{w}y}\ddot{\beta}_{\text{w}i}=r_{\text{R}i}F_{\text{R}xi}+r_{\text{L}i}F_ {\text{L}xi}+r_{\text{R}i}\psi_{\text{w}i}\big{(}F_{\text{R}yi}+N_{\text{R}yi }\big{)}+r_{\text{L}i}\psi_{\text{w}i}\big{(}F_{\text{L}yi}+N_{\text{L}yi} \big{)}\\ \\ \hskip 14.226378 pt+M_{\text{L}yi}+M_{\text{R}yi}+N_{\text{L}xi}r_{\text{L}i}+N _{\text{R}xi}r_{\text{R}i}\end{array}  \tag{17}\]  
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