# Speed Limit of Linear Induction Motor Subway Trains Running through 65 m Radius Curves on Yard Line
## 2 Train-Track Coupling Dynamic Model
### Track System  
#### 2.2.1 Rail System  
The rail is considered as a Timoshenko beam with \"simply supported-simply supported\" boundary conditions. The equations of motion in the transverse, vertical, and torsional directions are shown.  
Lateral motion:  
\[\left\{\begin{array}{ll}m\frac{\partial^{2}y(x,t)}{\partial t^{2}}+\kappa_{y }GA\left(\frac{\partial\phi_{y}(x,t)}{\partial x}-\frac{\partial^{2}y(x,t)}{ \partial x^{2}}\right)=-\sum\limits_{i=1}^{N_{\rm{s}}}R_{yi}(t)\delta(x-x_{i \rm{s}i})+\sum\limits_{j=1}^{N_{\rm{w}}}F_{\rm{wryi}}(t)\delta\big{(}x-x_{\rm{ wj}}\big{)}\\ \rho I_{z}\frac{\partial^{2}\phi_{y}(x,t)}{\partial t^{2}}+\kappa_{y}GA\left( \psi_{y}(x,t)-\frac{\partial y(x,t)}{\partial x}\right)-EI_{z}\frac{\partial^ {2}\phi_{y}(x,t)}{\partial x^{2}}=0\end{array}\right. \tag{22}\]  
Vertical motion:  
\[\left\{\begin{array}{ll}m\frac{\partial^{2}z(x,t)}{\partial t^{2}}+\kappa_{z }GA\left(\frac{\partial\phi_{z}(x,t)}{\partial x}-\frac{\partial^{2}z(x,t)}{ \partial x^{2}}\right)=-\sum\limits_{i=1}^{N_{\rm{s}}}R_{zi}(t)\delta(x-x_{i\rm {s}i})+\sum\limits_{j=1}^{N_{\rm{w}}}F_{\rm{wrzi}}(t)\delta\big{(}x-x_{\rm{wj}} \big{)}\\ \rho I_{y}\frac{\partial^{2}\phi_{z}(x,t)}{\partial t^{2}}+\kappa_{z}GA\left( \psi_{z}(x,t)-\frac{\partial z(x,t)}{\partial x}\right)-EI_{y}\frac{\partial^ {2}\phi_{y}(x,t)}{\partial x^{2}}=0\end{array}\right. \tag{23}\]  
Twisting motion:  
\[\rho I_{0}\frac{\partial^{2}\phi(x,t)}{\partial t^{2}}-GK\frac{\partial^{2} \phi(x,t)}{\partial x^{2}}=-\sum\limits_{i=1}^{N_{\rm{s}}}M_{\rm{s}i}(t)\delta (x-x_{\rm{s}i})+\sum\limits_{j=1}^{N_{\rm{w}}}M_{\rm{Gj}}(t)\delta\big{(}x-x_{ \rm{wj}}\big{)} \tag{24}\]  
Figure 7: Coupler draft gear device force element.  
where \(y\), \(z\) and \(\phi\) are the transverse, vertical and torsional displacements of the rail. \(\psi_{z}\) and \(\psi_{y}\) are the angular deformation of the rail cross section on \(z\) and \(y\) axles, respectively. \(R_{yi}\) and \(R_{zi}\) are the lateral and vertical reaction forces of the \(i\)-th fulcrum. \(F_{wwij}\) and \(F_{wwij}\) are the transverse and vertical loads of the \(j\)th wheel on the rail, respectively. \(M_{yi}\) and \(M_{Gj}\) are the moment of reaction force of the of the \(i\)th rail and the jth wheel on the rail. \(I_{y}\) and \(I_{z}\) are the inertia moment of rail cross section on \(y\) and \(z\) axles. \(G\) is the rail shear modulus. \(A\) is the cross-sectional area of the rail. \(N_{s}\) is the number of fasteners within the calculation area. \(N_{\mathrm{w}}\) is the number of wheelsets. While \(x_{y}\) and \(x_{z}\) are shear factors of rail transverse and vertical cross section, the values are 0.4507 and 0.532.  
The vibration response of the rail is analyzed by separation variable method. Regular mode shape function and regular coordinates represent the transverse, vertical, and torsional displacement. First, the rail vibration partial differential equation is simplified into a second-order ordinary differential equation about the regular mode coordinates, by using the orthogonality of the regular mode and the properties of the Dirichlet function. Then, find the canonical coordinates by the rail boundary. Finally, the transverse, vertical, and torsional displacements of the rail are obtained by multiplying and superimposing with the regular mode function and the canonical coordinates.