# Railway Vehicle Wheel Flat Detection with Multiple Records Using Spectral Kurtosis Analysis
## 3 Methodology for Flat Detection
### Track Response Obtained by the Strain Gauge Setup  
The passage of one vehicle of the Alfa Pendular and a freight wagon in a track stretch with the irregularity profile presented in Figure 3 is investigated. In all analyses where a wheel flat is considered, only the right wheel of the first wheelset is damaged with this type of defect. As shown in Figure 5, the three layout schemes considered in this study aim to optimize the number and position of the sensors. As illustrated in this figure, layout scheme 1 includes 12 virtual strain gauges installed on the rail from a track stretch with a length equivalent to a wheel perimeter. The envelope spectrum approach is used to verify the sensitivity of layout scheme 1, considering the unevenness profile presented in Figure 3. The speed is considered as 60 m/s for both the Alfa Pendular and the freight wagon. Figure 9 presents the power spectrum of the envelope signal for the defective and healthy wheels corresponding to the 12 shear responses obtained on the rail (position  
Figure 8: (**a**): Main signal evaluated from position 9 regarding layout scheme 1. (**b**): Spectral kurtosis for a signal corresponding to the Alfa Pendular train, \(\text{K}_{\text{max}}=60.25\) Hz, optimal window length = 16, center frequency = 0.1875 kHz, bandwidth frequency = 0.125 kHz. 1 to 12 shown in Figure 5), due to the passage of the Alfa Pendular and freight wagon, considering the unevenness profile of the track. In the following figures, the vertical dash lines indicate the wheel flat frequency as 22.21 Hz. By applying the envelop spectrum analysis, it is possible to observe a lag between the responses evaluated at the several SGs in the defective wheel, showing that a flat has been detected. On the other hand, as seen in Figure 9b,d, the responses in terms of the envelope spectrum in all SGs are very similar (no lag), indicating that there are no flats in both the Alfa Pendular and the freight wagon's wheels. Moreover, the amplitude variation of the envelope spectral signal can also be seen as an additional indicator to evaluate whether the wheel is healthy or not. As shown in this figure, the amplitude variation of the envelope spectral signal of a defective wheel is higher than a healthy one. The frequency content provides some interesting information. There are some high magnitude peaks corresponding to the bogies passing frequency (Figure 9b,d), which can be calculated as  
\[f_{b}=\frac{S}{a_{1}} \tag{12}\]  
where \(f_{b}\) is the bogie frequency and \(a_{1}\) is the regular spacing (see Figure 1), which for the Alfa Pendular corresponds to the spacing between bogies (\(a_{1}\) = 19 m) and for the freight wagon, since it has no bogies, corresponds to the spacing between axles (\(a_{1}\) = 6 m). Therefore, \(f_{b}\) was calculated as 3.15 and 6 Hz for the Alfa Pendular and the freight wagon, respectively. In other words, Figure 9b corresponds to the passage of the healthy wheel  
Figure 9: Envelope spectrum analysis for the 12 strain gauges (layout scheme 1) for the Alfa Pendular and the freight wagon, considering the unevenness profile of the track, (**a**) a defective wheel Alfa Pendular, (**b**) a healthy wheel Alfa Pendular, (**c**) a defective wheel freight wagon, and (**d**) a healthy wheel freight wagon. of the Alfa Pendular and presents the bogie passing frequency as 3.15 Hz and the wheel passing frequency as 22.17 Hz. However, in Figure 9a, corresponded to the passage of a defective wheel of the Alfa Pendular, only the frequency of the defective wheel (22.21 Hz) can be seen. Figure 10 shows the kurtosis values for healthy and defective wheels corresponding to 12 positions of the strain gauges (layout scheme 1) for the Alfa Pendular. As presented in this figure, a signal corresponding to a defective wheel has significantly greater impulsiveness, especially for strain gauge 9, which is located after the wheel flat impact location.