# Dynamic Analysis of CRTS II Ballastless Track-Vehicle System Based on Moving Unit Method
## 3 Establishment of the MEM Model
Combining the unit matrices of the three-layer Euler beam model, the unit mass matrix \(\underline{M}_{e}\), unit damping matrix \(\underline{C}_{e}\), and unit stiffness matrix \(\underline{K}_{e}\) of the C-ELE can be obtained. The P-ELE is distributed at the joint of the two adjacent track slabs of the track slab Euler beam with the expansion joint located inside the cell; the P-ELE model is shown in Figure 4.  
In the study of the ballastless track model, the ballastless track is cut into one track slab ignoring the expansion joints between the slabs, while assuming that the cross sections of  
Figure 4: P-ELE model. Figure 3: C-ELE model. the track slabs are identical in terms of the mechanical properties and structural shape. This shows that the physical properties of the P-ELE and C-ELE can be generalized, except for the difference in the continuity and degrees of freedom, so the transformation method for degenerating the P-ELE to the C-ELE, reversing the transformation, can be used to obtain the corresponding matrix of the P-ELE by deriving the cell mass, damping, and stiffness matrices of the C-ELE. The unit mass, damping, and stiffness matrices for the second layer of the rail after truncation of the \(I_{\rm w}\)-long track plate, adding the effects of the shear spring \(\underline{K}_{\rm ss}\) and the damper \(\underline{C}_{\rm ss}\), provides the cell damping and stiffness matrices of the P-ELE as shown in Equations (9)-(11). \[\underline{M}_{\rm t}^{p}=\underline{M}_{\rm t}-m_{\rm t}\int_{0}^{l_{\rm w}} \underline{N}_{\rm t}^{\rm T}\underline{N}_{\rm t}{\rm d}r \tag{9}\]  
\[\underline{C}_{\rm t}^{p}=\underline{C}_{\rm t}-c_{\rm t}\int_{0}^{l_{\rm w}} \underline{N}_{\rm t}^{\rm T}\underline{N}_{\rm t}{\rm d}r+\underline{C}_{\rm ss} \tag{10}\]  
\[\underline{K}_{\rm t}^{p}=\underline{K}_{\rm t}-E_{\rm t}{\rm I}_{\rm t}\int_{ 0}^{l_{\rm w}}\underline{N}_{\rm t,rr}^{\rm T}\underline{N}_{\rm t,rr}{\rm d }r+\underline{K}_{\rm ss} \tag{11}\]  
Additionally, the shear spring \(\underline{K}_{\rm ss}\) and the damper \(\underline{C}_{\rm ss}\) can be obtained by means of Equations (12) and (13). Referring to the model of Xu [12], \(\underline{N}_{\rm ss}\) = [1; 2; 3; 4]. \[\underline{K}_{\rm ss}=k_{\rm ss}\underline{N}_{\rm ss}^{\rm T}\underline{N}_{ \rm ss} \tag{12}\]  
\[\underline{C}_{\rm ss}=c_{\rm ss}\underline{N}_{\rm ss}^{\rm T}\underline{N}_{ \rm ss} \tag{13}\]  
Additionally, because the discontinuity of the track plate affects only the second layer of the track plate in the dynamic equations, the unit mass, damping, and stiffness matrices of the first and third layers remain unchanged. Through the addition of the cell mass matrix, the cell damping matrix, and the cell stiffness matrix of each layer of the P-ELE, we obtain the P-ELE cell mass matrix \(M_{\rm c}^{p}\), cell damping matrix \(C_{\rm c}^{p}\), and cell stiffness matrix \(K_{\rm c}^{p}\). The ballastless track is laid by numerous prefabricated track slabs. The unit mass matrix, unit damping matrix, and unit stiffness matrix of each part are combined to obtain the total mass matrix \(\mathbf{M}_{\rm D}\), total damping matrix \(\mathbf{C}_{\rm D}\), and total stiffness matrix \(\mathbf{K}_{\rm D}\) of the ballastless track model to establish the dynamic equations of the ballastless track model as shown in Equation (14):  
\[\mathbf{M}_{\rm D}\ddot{\mathbf{Z}}_{\rm D}+\mathbf{C}_{\rm D}\dot{\mathbf{Z} }_{\rm D}+\mathbf{K}_{\rm D}\mathbf{Z}_{\rm D}=\mathbf{F}_{\rm D} \tag{14}\]  
where \(\ddot{\mathbf{Z}}_{\rm D}\), \(\dot{\mathbf{Z}}_{\rm D}\) and \(\mathbf{Z}_{\rm D}\) denote the acceleration vector, velocity vector, and displacement vector of the ballast track model, respectively. \(\mathbf{F}_{\rm D}\) denotes the force vector of the ballastless track model.