# Railway Vehicle Wheel Flat Detection with Multiple Records Using Spectral Kurtosis Analysis
## 2 Train-Track Dynamic Interaction Modeling
### Train-Track Interaction  
The train-track dynamic interaction coupling system is solved using the numerical tool \"VSI-Vehicle-Structure Interaction Dynamic Analysis.\" This tool is based on the Lagrange multipliers method, in which the constraint equations that relate the train displacements with the nodal displacements of the track, including the track irregularities, are added to the equilibrium equations, forming a single system that defines the coupling between the  
Figure 3: Track irregularity sample: (**a**) vertical and (**b**) lateral direction. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Parameter**} & \multicolumn{2}{c}{**Value**} \\ \hline \multirow{4}{*}{Rail} & \(A_{r}\) (m\({}^{2}\)) & 7.67 \(\times\) 10\({}^{-4}\) & [41] \\  & \(\rho_{r}\) (kg/m\({}^{3}\)) & 7850 & [41] \\  & \(l_{r}\) (m\({}^{4}\)) & 30.38 \(\times\) 10\({}^{-6}\) & [41] \\  & \(v_{r}\) & 0.28 & [41] \\  & \(E_{r}\) (N/m\({}^{2}\)) & 210 \(\times\) 10\({}^{9}\) & [41] \\ \hline \multirow{2}{*}{Rail pad, longitudinal} & \(K_{p}\) (N/m) & 20 \(\times\) 10\({}^{6}\) & [42] \\  & \(C_{p}\) (N-s/m) & 50 \(\times\) 10\({}^{3}\) & [42] \\ \hline \multirow{2}{*}{Rail pad, lateral} & \(K_{p}\) (N/m) & 20 \(\times\) 10\({}^{6}\) & [42] \\  & \(C_{p}\) (N-s/m) & 50 \(\times\) 10\({}^{3}\) & [42] \\ \hline \multirow{2}{*}{Rail pad, vertical} & \(K_{p}\) (N/m) & 500 \(\times\) 10\({}^{6}\) & [43] \\  & \(C_{p}\) (N-s/m) & 200 \(\times\) 10\({}^{3}\) & [43] \\ \hline \multirow{2}{*}{Sleeper} & \(\rho_{s}\) (kg/m\({}^{3}\)) & 2590 & \\  & \(\
u_{s}\) & 0.2 & \\  & \(E_{s}\) (N/m\({}^{2}\)) & 40.9 \(\times\) 10\({}^{9}\) & \\ \hline \multirow{2}{*}{Ballast, longitudinal} & \(K_{b,x}\) (N/m) & 9000 \(\times\) 10\({}^{3}\) & [44] \\  & \(\text{C}_{b,x}\) (N-s/m) & 15 \(\times\) 10\({}^{3}\) & [45] \\ \hline \multirow{2}{*}{Ballast, lateral} & \(K_{b,y}\) (N/m) & 2250 \(\times\) 10\({}^{3}\) & [46] \\  & \(\text{C}_{by}\) (N-s/m) & 15 \(\times\) 10\({}^{3}\) & [45] \\ \hline \multirow{2}{*}{Ballast, vertical} & \(K_{b,z}\) (N/m) & 30 \(\times\) 10\({}^{6}\) & [46] \\  & \(\text{C}_{b,z}\) (N-s/m) & 15 \(\times\) 10\({}^{3}\) & [45] \\ \hline \multirow{2}{*}{Foundation, longitudinal} & \(K_{f,x}\) (N/m) & 20 \(\times\) 10\({}^{6}\) & [47] \\  & \(C_{f,x}\) (N-s/m) & 5.01 \(\times\) 10\({}^{2}\) & [47] \\ \hline \multirow{2}{*}{Foundation, lateral} & \(K_{f,y}\) (N/m) & 20 \(\times\) 10\({}^{6}\) & [47] \\  & \(C_{f,y}\) (N-s/m) & 5.01 \(\times\) 10\({}^{2}\) & [47] \\ \hline \multirow{2}{*}{Foundation, vertical} & \(K_{f,z}\) (N/m) & 20 \(\times\) 10\({}^{6}\) & [47] \\  & \(C_{f,z}\) (N-s/m) & 5.01 \(\times\) 10\({}^{2}\) & [47] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mechanical properties of the track. two sub-structures. Note that when including the track irregularities in the constraint equations, it is not necessary to include them explicitly in the FEM model, which considerably simplifies the analysis. Moreover, as will be mentioned later, it is also possible to include the wheel flats as a periodic rail irregularity. To solve the train structure detachment, [PERSON] et al. [48] and [PERSON] et al. [49] extended the original formulation developed by [PERSON] et al. [50]. Subsequently, [PERSON] et al. [51] reformulated the method to consider the lateral interaction by taking into account the wheel and rail geometry and the contact model between them. In the last upgrade, the vertical and lateral contact forces were computed and made the software possible to deal with different scenarios. The wheel-rail contact model adopted in this numerical tool has been validated with numerical examples and experimental data (see [48; 51]) and is based on a specially developed finite contact element that considers the geometry of the contact interface by parameterizing the wheel and rail profiles with cubic splines. The element uses the Lagrange multipliers method to guarantee the coupling between the vehicle and the track structure when contact occurs, but can also lead to wheel-rail detachment.