# Out-of-Roundness Wheel Damage Identification in Railway Vehicles Using AutoEncoder Models
## 4 Characterization of the Simulated Data
### Wheel Polygonization Profiles  
The wheels of railway vehicles, although designed to be circular, often present irregularities that significantly affect the dynamic performance of the system. One such imperfection, known as polygonization, consists of radial undulations along the tread. The severity of these undulations, when greater than 0.2 mm, can induce considerable variations in the wheel-rail contact forces, generating vibrations and noise. The geometric characterization of polygonization involves the calculation of the wavelength (\(\lambda\)) of the irregularities, which is a function of the wheel radius (\(R_{w}\)) and the harmonic order (\(\theta\)) of the undulation, as expressed by Equation (8), where \(\theta\) takes positive integer values. \[\lambda=\frac{2\pi R_{w}}{\theta},\quad\theta=1,2,3,...,n \tag{8}\]  
To cover a wide range of operating conditions, two sets of wheel profiles were generated for the numerical simulations. These profiles were constructed from real wheel data, both under initial polygonization damage conditions (Damage 1) and under more severe polygonization damage conditions (Damage 2), with their respective irregularity spectra presented in Figure 9.  
The measurement data used to characterize the wheel profiles were obtained from two main sources: (i) [PERSON] et al. (2005) [PERSON] et al. (2005) provided data for intact polygonal wheels (Damage 1), with a wavelength spectrum extending across 20 harmonics, ranging from 0.135 to 2.7 m (Figure 9a); (ii) [PERSON] et al. (2019) [PERSON] et al. (2019) provided data for polygonal wheels with damage (Damage 2), where the first 30 harmonics were considered. For these latter data, the sixth to eighth harmonic orders were predominant, indicating a specific wear pattern. A variety of wheel profiles with different levels of irregularities were numerically synthesized. The methodology employed involved the superposition of multiple sinusoidal functions (H = 30), as described in Equation (9). \[w(x_{w})=\sum_{\theta=1}^{H}A_{\theta}\sin\left(\frac{2\pi}{\lambda}x_{w}+ \psi_{\theta}\right) \tag{9}\]  
The amplitude of each sinusoidal component, \(A_{\theta}\), was determined using Equation (10), where \(\omega_{\text{ref}}\) represents a reference wavelength. \[A_{\theta}=\sqrt{2}\cdot 10^{\frac{L_{w}}{2\theta}}\cdot w_{ref} \tag{10}\] \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Scenario** & **Number of** & **Number of** & **Points per Sample** & **Points per Sample** \\  & **Samples (Before)** & **Samples (After)** & **(Before)** & **(After)** \\ \hline No Damage (reference) & 10 & 100 & 39,169 & 3916 \\ Damage 1 (left rear wheel, 3 rd wagon) & 10 & 100 & 39,169 & 3916 \\ Damage 2 (right rear wheel, 5 th wagon) & 10 & 100 & 39,169 & 3916 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of data before and after segmentation for each scenario. Figure 9 illustrates the irregularity spectra used to calibrate the numerical models, both for the initial polygonization condition (Figure 9a) and for the severe polygonization condition (Figure 9b). For each spectrum, a set of profiles was generated by randomly varying the initial phases \(\psi_{\theta}\) of the sinusoidal functions within the range of 0 to \(2\pi\). This approach enabled the simulation of the inherent randomness associated with wheel manufacturing and wear processes. The high similarity between the acceleration signals obtained for healthy wheels and those with different levels of polygonization can be verified in Figure 10. This figure illustrates the complexity of the task of classifying defects in railway wheels using Machine Learning techniques. The small variation in signal amplitude and the similarity between them make the distinction between the classes a complex task, requiring the extraction of high-order features for accurate classification. Figure 10: Original signals from accelerometer 1 for the conditions: (**a**) No damage; (**b**) Damage 1; (**c**) Damage 2.