## 2 Model and Method
### Model and Method for the Rail Mobility/_Receptance_
#### 2.1.3 Spectral Element Method  
The SEM is used to solve the Timoshenko-beam track model. As the spectral element size has no influence on the calculation precision of the SEM, the rail between two adjacent fasteners can be modeled as only one spectral Timoshenko beam element. To simulate the non-reflecting boundary conditions, two Timoshenko beam throw-off elements are modeled at two ends of the long rail. The vertical extension spring is modeled with a two-node spectral spring element. The element division is shown in Figure 3 where the black dots represent the nodes. The two-node spectral Timoshenko beam element is shown in Figure 4.  
\(E\) is the Young's modulus, \(I\) is the area moment of inertia, \(\rho\) is the mass density, \(A\) is the cross-sectional area, \(G\) is the shear modulus, \(L\) is the length of the spectral element, \(\kappa\) is the shear correction factor and \(\eta\) is the damping loss factor. \(U_{1}\), \(\Theta_{1}\), \(F_{1}\) and \(M_{1}\) are the spectral vertical displacement, spectral bending rotation, spectral vertical force and spectral bending moment at the left node, respectively, while \(U_{2}\), \(\Theta_{2}\), \(F_{2}\) and \(M_{2}\) are the corresponding items at the right node. The relation between the spectral nodal displacements and the spectral nodal forces in a spectral Timoshenko beam element can be expressed as:  
\[\mathbf{F_{t}}=\mathbf{S_{t}}(\omega)\mathbf{U_{t}} \tag{2}\]  
where \(\mathbf{F_{t}}=(F_{1},M_{1},F_{2},M_{2})^{\rm T}\) is the spectral nodal force vector, \(\mathbf{U_{t}}\) = (\(U_{1}\), \(\Theta_{1}\), \(U_{2}\), \(\Theta_{2}\))\({}^{\rm T}\) is the spectral nodal displacement vector, \(\mathbf{S_{t}}(\omega)\) is the frequency-dependent spectral element stiffness matrix, and \(\omega\) is the angular frequency. The derivation of Equation (2) is based on the free vibration of a uniform Timoshenko beam and the spectral analysis. The spectral element stiffness matrix \(\mathbf{S_{t}}(\omega)\) is obtained by substituting the wave solution into the governing differential equations. The expression of \(\mathbf{S_{t}}(\omega)\) can be given by:  
\[\mathbf{S_{t}}(\omega)=\left\{ \begin{array}{c}E\mathbf{I}\mathbf{N}^{\
u}(0,\omega)+\rho\omega^{2}\mathbf{N}(0,\omega)\\ -E\mathbf{I}\mathbf{N}^{\
u}(0,\omega)\\ -E\mathbf{I}\mathbf{N}^{\
u}(L,\omega)\\ E\mathbf{I}\mathbf{N}^{\
u}(L,\omega)\end{array} \right\}\left(\begin{array}[] {cccc}(r_{1}+r_{2}\epsilon_{2})&-R_{2}(r_{1}-r_{2}\epsilon_{2})&-(r_{1} \epsilon_{2}+r_{2})&R_{2}(r_{1}\epsilon_{2}-r_{2})\\ -(r_{1}+r_{2}\epsilon_{1})&R_{1}(r_{1}-r_{2}\epsilon_{1})&(r_{1}\epsilon_{1}+ r_{2})&-R_{1}(r_{1}\epsilon_{1}-r_{2})\\ (r_{1}\epsilon_{2}+r_{2})&R_\[k_{1}=\sqrt{\frac{a_{1}}{2}+\sqrt{\frac{a_{1}^{2}}{4}-a_{2}}},\ k_{2}=\sqrt{\frac{a _{1}}{2}-\sqrt{\frac{a_{1}^{2}}{4}-a_{2}}} \tag{5}\]  
\[a_{1}=\frac{GA\kappa pla^{2}+EI\rho A\omega^{2}}{GA\kappa EI},\ a_{2}=\frac{\rho A \omega^{2}(\rho l\omega^{2}-GA\kappa)}{GA\kappa EI} \tag{6}\]  
\[R_{m}=\frac{\mathrm{i}GA\kappa k_{m}}{GA\kappa k_{m}^{2}-\rho A\omega^{2}}\ (m=1,\ 2) \tag{7}\]  
\[r_{1}=(R_{1}-R_{2})(1-e_{1}e_{2}),\ r_{2}=(R_{1}+R_{2})(e_{1}-e_{2}),\ e_{1}= \mathrm{e}^{-\mathrm{i}k_{1}L},\ e_{2}=\mathrm{e}^{-\mathrm{i}k_{2}L},\ \Delta=r_{1}^{2}-r_{2}^{2} \tag{8}\]  
where \(k_{1}\) and \(k_{2}\) are the wavenumbers. The derivation of \(\mathbf{S_{t}}(\omega)\) has been demonstrated in the previous work [16], which is omitted in this paper. \(\mathbf{F_{t}}\) is the input determined by the external loads. In order to simulate the non-reflecting boundary conditions, the Timoshenko beam throw-off elements are modeled at both ends of the rail. The reflected waves can be prohibited by removing the reflection items in the wave solutions of the Timoshenko beam.