# Out-of-Roundness Wheel Damage Identification in Railway Vehicles Using AutoEncoder Models
## 4 Characterization of the Simulated Data
### Railroad  
The numerical modeling of the track employed in this study is based on the research conducted by [PERSON] et al. (2020) [48] and is also implemented in the ANSYS\({}^{\circledR}\) software [31]. The model adopts a multilayer approach to represent the interaction between the track components, including ballast, sleepers, rails, and foundation [2; 45].  
Figure 6 illustrates the discretization of the model, where the rails and sleepers are modeled using beam elements with appropriate properties. The interfaces, such as ballast and pads/fasteners below and above the sleepers, are considered using spring and pressure point elements in all three directions. Finally, the foundation properties were considered by a simplified approach based on a Winkler model (presented in detail in [PERSON] et al. (2020) [49]). The 3D ballast track numerical model is validated with modal parameters as presented in [PERSON] et al. (2021) [50]. The material and geometric parameters considered in the simulation are detailed in Table 3.  
The geometric specifications and simulation parameters used for the tracks and trains analyzed in this study are also presented in detail by [PERSON] et al. (2020, 2021) [49; 51]. It is recommended to consult these works for additional information on the models adopted.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Symbol (Unit)** & **Value** \\ \hline Rail area & \(A_{r}\) (m\({}^{2}\)) & \(7.67\times 10^{-4}\) \\ Rail density & \(\rho_{r}\) (kg/m\({}^{3}\)) & 7850 \\ Rail second moment of area & \(I_{r}\) (m\({}^{4}\)) & \(30.38\times 10^{-6}\) \\ Rail Young’s modulus & \(E_{r}\) (N/m\({}^{2}\)) & \(210\times 10^{9}\) \\ Rail pad, longitudinal stiffness & \(k_{p,x}\) (N/m) & \(20\times 10^{6}\) \\ Rail pad, longitudinal damping & \(C_{p,x}\) (N \(\cdot\) s/m) & \(50\times 10^{3}\) \\ Rail pad, lateral stiffness & \(k_{p,y}\) (N/m) & \(20\times 10^{6}\) \\ Rail pad, lateral damping & \(C_{p,y}\) (N \(\cdot\) s/m) & \(50\times 10^{3}\) \\ Rail pad, vertical stiffness & \(k_{p,z}\) (N/m) & \(500\times 10^{6}\) \\ Rail pad, vertical damping & \(C_{p,z}\) (N \(\cdot\) s/m) & \(200\times 10^{3}\) \\ Sleeper density & \(\rho_{s}\) (kg/m\({}^{3}\)) & \(2590\) \\ Ballast, longitudinal stiffness & \(k_{x,x}\) (N/m) & \(900\times 10^{3}\) \\ Ballast, longitudinal damping & \(C_{b,x}\) (N \(\cdot\) s/m) & \(15\times 10^{3}\) \\ Ballast, lateral stiffness & \(k_{b,y}\) (N/m) & \(2250\times 10^{3}\) \\ Ballast, lateral damping & \(C_{b,y}\) (N \(\cdot\) s/m) & \(15\times 10^{3}\) \\ Ballast, vertical stiffness & \(k_{b,z}\) (N/m) & \(30\times 10^{6}\) \\ Ballast, vertical damping & \(C_{b,z}\) (N \(\cdot\) s/m) & \(15\times 10^{3}\) \\ Foundation, longitudinal stiffness & \(k_{f,x}\) (N/m) & \(20\times 10^{6}\) \\ Foundation, lateral stiffness & \(k_{f,y}\) (N/m) & \(20\times 10^{6}\) \\ Foundation, vertical stiffness & \(k_{f,z}\) (N/m) & \(20\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters for the numerical model of the railway track structure (Adapted from [PERSON] et al. (2021) [51].)  
Figure 6: Three-dimensional graphical representation of the numerical model of the track adopted in this study ([PERSON] et al. (2023)) [2].