# Dynamic Analysis of CRTS II Ballastless Track-Vehicle System Based on Moving Unit Method
## 2 Vehicle and Ballastless Track Models
### Ballastless Track Modelling  
The ballastless track model based on the CRTS II slab ballast track consists of track, elastic fasteners (discrete trackpads), precast track slabs, CA mortar layers, and concrete foundations. The discrete steel rails, track slabs, and concrete foundations are discrete and are considered as Euler beams; a three-level Euler beam ballast track model is established out of those, with the steel rails and track slabs being connected by discrete viscoelastic track pads, respectively, and the CA mortar layers between the track slabs and concrete foundations, as well as the roadbed under the concrete foundations, being considered as viscoelastic supports, as shown in Figure 1. The established model is shown in Figure 2.  
In Figure 1, \(E_{\rm r}\), \(E_{\rm t}\), and \(E_{\rm b}\) denote the modulus of the elasticity of the rail, the rail slab, and the concrete foundation, respectively, \(I_{\rm r}\), \(I_{\rm t}\), and \(I_{\rm b}\) denote the moment of inertia of the rail, rail slab, and concrete foundation, respectively, and \(m_{\rm r}\), \(m_{\rm t}\), and \(m_{\rm b}\) denote the unit mass of the rail, the rail slab, and concrete foundation, respectively. \(k_{\rm r}\) and \(c_{\rm r}\) denote the stiffness coefficient and damping coefficient, respectively, of the equidistant bulk track pads supporting the track, \(k_{\rm t}\) and \(c_{\rm t}\) denote the CA mortar stiffness factor and damping factor for supporting the track slab, respectively, \(k_{\rm b}\) and \(c_{\rm b}\) denote the roadbed stiffness factor and the damping factor for supporting the concrete foundation, respectively, \(k_{\rm ss}\) and \(c_{\rm ss}\) denote the stiffness factor and damping factor for shear springs and dampers at the expansion joint connection between adjacent track slabs, respectively, and \(w\), \(z\), and \(y\) denote the vertical displacement of the track, track slab, and concrete foundation, respectively. Figure 1: Three-layer Euler beam ballastless track model. Figure 2: CRTS II type slab track model. In building the ballastless track model, the particular ballastless track element (P-ELE) and the common ballastless track element (C-ELE) in the moving coordinate system are established by referring to Xu [12]. The P-ELE is a three-layer Euler beam ballast track element containing the connecting part of the track slab, as shown in the dashed box in Figure 1, while the rest of the track model can be decomposed into C-ELE [12]. In the ballastless track model, the trackpad is simplified to discrete viscoelastic support, and in deriving the dynamic equations, the influence of the discrete trackpad is also realized by considering the constant repetitive cycle change in the discrete support, according to Euler beam theory, by tracking the cycle change in the support position and calculating the stiffness and damping matrix of the moving unit in one cycle and storing it. The dynamic equations of the three-layer beam model considering the effect of the discrete support of the trackpad can be established by using the displacements \(w\), \(z\), and \(y\) of the three-layer beam model as shown in Equations (2)-(4):  
\[ \begin{array}{l}E_{\rm r}I_{\rm r}\frac{\partial^{4}w}{\partial x^{4}}+m_{ \rm r}\frac{\partial^{2}w}{\partial t^{2}}+\sum\limits_{i=1}^{n}c_{\rm r}\Big{(} \frac{\partial w}{\partial t}-\frac{\partial z}{\partial t}\Big{)}\times\delta (x-iL_{\rm s})\\ +\sum\limits_{i=1}^{n}k_{\rm r}(w-z)\times\delta(x-iL_{\rm s})=\sum\limits_{j= 1}^{m}F_{j}\times\delta\big{(}x-x_{j}(t)\big{)}\end{array}  \tag{2}\]  
\[ \begin{array}{l}E_{\rm r}t_{\rm r}\frac{\partial^{4}z}{\partial x^{4}}+m_{ \rm t}\frac{\partial^{2}z}{\partial t^{2}}+\sum\limits_{i=1}^{n}c_{\rm r}\Big{(} \frac{\partial z}{\partial t}-\frac{\partial w}{\partial t}\Big{)}\times \delta(x-iL_{\rm s})\\ +\sum\limits_{i=1}^{n}k_{\rm r}(z-w)\times\delta(x-iL_{\rm s})+c_{\rm t}\Big{(} \frac{\partial z}{\partial t}-\frac{\partial y}{\partial t}\Big{)}+k_{\rm r}( z-y)=0\end{array}  \tag{3}\]  
\[