# Dynamic Analysis of CRTS II Ballastless Track-Vehicle System Based on Moving Unit Method
## 3 Establishment of the MEM Model  
The MEM principle is as follows. First, intercept a section of track and discrete it into a finite number of beam units, then place the point of the wheel-rail contact force in the middle of the whole track so that it meets a certain distance upstream and downstream, thus ignoring the boundary effect; at this time, the point on the coordinate axis \(x\)-axis and the corresponding coordinates are fixed. Then, take point a on the fixed coordinate axis \(x\)-axis, whose coordinates are \(x_{\rm a}\), and define the moving coordinate axis \(r\)-axis. The origin of the \(r\)-axis follows the train load movement, then point a corresponds to the moving coordinate \(r_{\rm a}\) on the \(r\)-axis as shown in Equation (5):  
\[r_{\rm a}=x_{\rm a}-Vt \tag{5}\]  
Substituting Equation (5) into Equations (2)-(4), applying the chain rule, and using a moving coordinate system instead of a fixed coordinate system, the discrete dynamic equations for the new three-layer Euler beam ballast track are derived as shown in Equations (6)-(8):  
\[ \begin{array}{l}E_{\rm r}I_{\rm r}\frac{\partial^{4}w}{\partial t^{2}}+m_{ \rm r}\Big{(}V^{2}\frac{\partial^{2}w}{\partial t^{2}}-2V\frac{\partial^{2} w}{\partial t\partial t}+\frac{\partial^{2}w}{\partial t^{2}}\Big{)}+\sum \limits_{i=1}^{n}c_{\rm r}\Big{(}\frac{\partial w}{\partial t}-V\frac{ \partial w}{\partial t}-\frac{\partial z}{\partial t}+\frac{\partial z}{ \partial t}\Big{)}\\ \times\delta(r+Vt-iL_{\rm s})+\sum\limits_{i=1}^{n}k_{\rm r}(w-z)\times\delta (r+Vt-iL_{\rm s})=\sum\limits_{j=1}^{m}F_{j}\times\delta\big{(}r-R_{j}\big{)} \end{array}  \tag{6}\]  
\[ \begin{array}{l}E_{\rm r}I_{\rm r}\frac{\partial^{4}z}{\partial t^{2}}+m_{ \rm t}\Big{(}V^{2}\frac{\partial^{2}z}{\partial t^{2}}-2V\frac{\partial^{2}z}{ \partial t\partial t}+\frac{\partial^{2}z}{\partial t^{2}}\Big{)}+\sum\limits _{i=1}^{n}c_{\rm r}\Big{(}\frac{\partial z}{\partial t}-V\frac{\partial z}{ \partial t}-\frac{\partial w}{\partial t}+\frac{\partial w}{\partial t}\Big{)} \\ \times\delta(r+Vt-iL_{\rm s})+\sum\limits_{i=1}^{n}k_{\rm r}(z-w)\times\delta (r+Vt-iL_{\rm s})+c_{\rm t}\Big{(}\frac{\partial z}{\partial t}-V\frac{\partial z }{\partial t}-\frac{\partial y}{\partial t}+\frac{\partial y}{\partial t}\Big{)} \\ +k_{\rm t}(z-y)=0\end{array}  \tag{7}\]\[ \begin{array}{l}E_{b}b_{b}\frac{\partial^{4}y}{\partial x^{2}}+m_{b}\left(V^{2} \frac{\partial^{2}y}{\partial x^{2}}-2V\frac{\partial^{2}y}{\partial x\partial t }+\frac{\partial^{2}y}{\partial t^{2}}\right)+c_{b}\left(\frac{\partial y}{ \partial t}-V\frac{\partial y}{\partial t}\right)\\ +c_{t}\left(\frac{\partial y}{\partial t}-V\frac{\partial y}{\partial t}-\frac {\partial z}{\partial t}+\frac{\partial z}{\partial t}\right)+k_{b}y-k_{t}(y- z)=0\end{array}  \tag{8}\]  
where \(L_{b}\) denotes the spacing between two adjacent track pads, \(R_{j}\) denotes the distance the \(j\)th wheel pair runs, \(F_{cj}\) denotes the contact force between the \(j\)th wheel pair and the track, and \(\delta(x)\) denotes the Dirac function. The C-ELE model diagram is shown in Figure 3, and a typical orbital unit of length l is taken for analysis. The derivation process of Lei [6] can be referred to be based on the moving coordinates and by introducing the form function \(N\). From the form function \(N\), the matrices \(N_{r}\), \(N_{t}\), and \(N_{b}\) can be obtained. The Euler beam model for the first layer of the rail is first calculated by multiplying Equation (6) by a weighting function and integrating over one unit length cell. [PERSON]'s method, the unit mass matrix \(\underline{M}_{r}\), the unit damping matrix \(\underline{C}_{r}\), and the unit stiffness matrix \(\underline{K}_{r}\) are obtained for the first layer of the C-ELE rail based on the track dynamics equations. Using the same method, the corresponding matrices for the second layer of the rail as well as the third layer of the rail can be found.