## 2 Wheel-Track Coupling Model
### Numerical Model
#### 2.2.4 Numerical Model Building  
After building all submodels, these are connected by spring-damper systems, and a contact element based on the Hertz contact theory is generated between the wheel and rail to simulate the wheel drop impact. Because the wheelset drop test can be regarded as an impact process, no excitation or rail roughness is considered in the model. By coupling the Equation (14) and Equation (17) together while adopting the superscript \(M\) and \(F\), which  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Rail elastic modulus (N\(\cdot\)m\({}^{-2}\)) & 2.059 \(\times\) 10\({}^{11}\) \\ Moment of inertia of rail section (m\({}^{4}\)) & 3.217 \(\times\) 10\({}^{-5}\) \\ Rail mass per unit length (kg\(\cdot\)m\({}^{-1}\)) & 60.64 \\ Sleeper mass (kg) & 125.5 \\ Vertical stiffness of fastening system (N\(\cdot\)m\({}^{-1}\)) & 6.5 \(\times\) 10\({}^{7}\) \\ Vertical damping of fastening system (N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) & 7.5 \(\times\) 10\({}^{4}\) \\ Lateral stiffness of fastening system (N\(\cdot\)m\({}^{-1}\)) & 5.5 \(\times\) 10\({}^{6}\) \\ Lateral damping of fastening system (N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) & 4.5 \(\times\) 10\({}^{3}\) \\ Sleeper spacing distance (m) & 0.6 \\ Ballast density (kg\(\cdot\)m\({}^{-3}\)) & 1.8 \(\times\) 10\({}^{3}\) \\ Ballast elastic modulus (N\(\cdot\)m\({}^{-2}\)) & 1.1 \(\times\) 10\({}^{8}\) \\ Ballast damping (N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) & 5.88 \(\times\) 10\({}^{4}\) \\ Shear stiffness of ballast (N\(\cdot\)m\({}^{-1}\)) & 7.84 \(\times\) 10\({}^{7}\) \\ Shear damping of ballast (N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) & 8.0 \(\times\) 10\({}^{4}\) \\ Ballast height (m) & 0.45 \\ Stiffness of ballast mat (N\(\cdot\)mm\({}^{-3}\)) & 0.02 \\ Damping of ballast mat (N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) & 1.36 \(\times\) 10\({}^{6}\) \\ Subgrade elastic modulus (N\(\cdot\)m\({}^{-2}\)) & 0.9 \(\times\) 10\({}^{8}\) \\ Stiffness of subgrade (N\(\cdot\)m\({}^{-1}\)) & 7.75 \(\times\)10\({}^{7}\) \\ Damping of subgrade (N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) & 3.115 \(\times\)10\({}^{4}\) \\ Wheel mass (kg) & 1.2 \(\times\)10\({}^{3}\) \\ Wheel drop height (mm) & 25 \\ \hline \hline \end{tabular}
\end{table}
Table 1: System parameters.  
denote the dynamic equations based on the MBS theory and FEM theory, respectively, the system equations can be obtained as follows:  
\[\left[\begin{array}{cc}M_{w}^{M}&\\ M_{T}^{F}\end{array}\right]\left[\begin{array}{c}\ddot{\delta}_{w}^{M}\\ \ddot{\delta}_{T}^{F}\end{array}\right]+\left[\begin{array}{cc}0&\\ 0&\\ C_{T}^{F}\end{array}\right]\left[\begin{array}{c}\dot{\delta}_{w}^{M}\\ \ddot{\delta}_{T}^{F}\end{array}\right]+\left[\begin{array}{cc}K_{w}^{M}&\\ \ddot{\delta}_{w}^{F}\end{array}\right]\left[\begin{array}{c}\delta_{w}^{M}\\ \ddot{\delta}_{T}^{F}\end{array}\right]=\left[\begin{array}{c}F_{w}^{M}\\ F_{T}^{F}\end{array}\right] \tag{18}\]  
In the above equation, the subscript \(T\) denotes for the track structure, and the other notations are as defined earlier. The wheel-rail impact force acts as an internal force on the system and is exerted on both the wheel and the track when the contact is closed. The motions of the wheel and track structure are coupled and updated in each time step.  
The entire model established in Multibody System (MBS) software is shown in Figure 4. In the model, the blue cubes represent the rail elements, which are modelled with Timoshenko beam formulation. The displacement at arbitrary location between two adjacent nodes is obtained through Hermite interpolation. The yellow cuboids represent the sleeper, and the green cubes represent the ballast mass. Spring--damper systems are used to simulate the connection between the rail, sleeper and ballast. The red bars represent the shear spring-damper systems, which connect the two adjacent ballast mass as shown in Figure 4.