# Railway Vehicle Wheel Flat Detection with Multiple Records Using Spectral Kurtosis Analysis
## 2 Train-Track Dynamic Interaction Modeling  
### Numerical Modeling for Alfa Pendular Vehicle and Freight Wagon  
In order to enhance the parametric study carried out in the present paper, two different train models were developed, namely a passenger (Alfa Pendular) train and a container freight (Laagrss wagon). The numerical models of both trains were performed in the Finite Element Method (FEM) package ANSYS\({}^{\text{\textregistered}}\)[38] using spring-damper elements COMBIN14 to simulate the suspensions that connect the various components of the trains, including car body, bogies, and wheelsets. The mass and the rotary inertia of the mentioned components are taken into account through mass point elements MASS21. The dynamic models adopted for each train are depicted in Figure 1. As presented in this figure, \(k\), \(c\), \(m\), and \(I\) denote stiffness, damping, concentrated mass, and rotary inertia. Moreover, the subscripts \(c_{b}\), \(b\), and \(w\) refer to the car body, bogies, and wheelsets. \(r_{w}\) is the wheel radius, and gauge is indicated by s. Finally, the characters \(a\), \(b\), and \(h\) denote the longitudinal, transversal, and vertical distances. The geometrical and mechanical properties of both trains are presented in Tables 1 and 2. Note that in all analyses, only one car was taken into account to simplify the numerical computations. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Car body mass, \(m_{cb}\) & 35,640 kg \\ Car body roll moment of inertia, \(I_{cb,x}\) & 55,120 kg\(\cdot\)m\({}^{2}\) \\ Car body pitch moment of inertia, \(I_{cb,y}\) & 1,475,000 kg\(\cdot\)m\({}^{2}\) \\ Car body yaw moment of inertia, \(I_{cb,z}\) & 1,477,000 kg\(\cdot\)m\({}^{2}\) \\ Bogie mass, \(m_{b}\) & 2829 kg \\ Bogie roll moment of inertia, \(I_{b,x}\) & 2700 kg\(\cdot\)m\({}^{2}\) \\ Bogie pitch moment of inertia, \(I_{b,y}\) & 1931.49 kg\(\cdot\)m\({}^{2}\) \\ Bogie yaw moment of inertia, \(I_{b,z}\) & 3878.76 kg\(\cdot\)m\({}^{2}\) \\ Wheelset mass, \(m_{w}\) & 1711 kg \\ Wheelset roll moment of inertia, \(I_{w,x}\) & 733.430 kg\(\cdot\)m\({}^{2}\) \\ Wheelset yaw moment of inertia, \(I_{w,z}\) & 733.4303 kg\(\cdot\)m\({}^{2}\) \\ Stiffness of the primary longitudinal suspension, \(k_{1,x}\) & 44,981,000 N/m \\ Stiffness of the primary transversal suspension, \(k_{1,y}\) & 30,948,200 N/m \\ Stiffness of the primary vertical suspension, \(k_{1,z}\) & 1,652,820 N/m \\ Damping of the primary vertical suspension, \(c_{1,z}\) & 16,739 N\(\cdot\)s/m \\ Stiffness of the secondary longitudinal suspension, \(k_{2,x}\) & 4,905,000 N/m \\ Stiffness of the secondary transversal suspension, \(k_{2,y}\) & 2,500,000 N/m \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters for the Alfa Pendular vehicle (adapted from [PERSON] et al. (2020) [39]). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Car body mass, \(m_{cb}\) & 41,100 kg \\ Car body roll moment of inertia, \(l_{cb,x}\) & 48,997.023 kg\(\cdot\)m\({}^{2}\) \\ Car body pitch moment of inertia, \(l_{cb,y}\) & 673,322.463 kg\(\cdot\)m\({}^{2}\) \\ Car body yaw moment of inertia, \(l_{cb,z}\) & 665,107.6 kg\(\cdot\)m\({}^{2}\) \\ Wheelest mass, \(m_{w}\) & 1246.52 kg \\ Wheelest roll moment of inertia, \(l_{w,x}\) & 311.839 kg\(\cdot\)m\({}^{2}\) \\ Wheelest yaw moment of inertia, \(l_{w,z}\) & 311.839 kg\(\cdot\)m\({}^{2}\) \\ Stiffness of the longitudinal suspension, \(k_{1,x}\) & 44,981,000 N/m \\ Stiffness of the transversal suspension, \(k_{1,y}\) & 30,948,200 N/m \\ Stiffness of the vertical suspension, \(k_{1,z}\) & 1,860,000 N/m \\ Damping of the vertical suspension, \(c_{1,z}\) & 16,739 N\(\cdot\)s/m \\ The static load transmitted by each wheel & 107,000 N \\ Longitudinal distance between wheelsets, \(a_{1}\) & 6 m \\ Transversal distance between vertical suspensions, \(b_{1}\) & 2.17 m \\ Vertical distance between car body center and suspension, \(h_{1}\) & 1.867 m \\ Nominal rolling radius, \(r_{w}\) & 0.43 m \\ Gauge, \(s\) & 1.67 m \\ \hline \hline \end{tabular}
\end{table}