# Railcar Dynamic Response during Braking Maneuvers Based on Frequency Analysis
## 3 Methodology
### Frequency Analisys before Braking  
In the frequency analysis, the methods described in Section 3 are applied to analyze the wheel's and the railcar's vibration signals in order to find a similarity between the wheel's and vehicle's dynamics. Time and frequency correlations indicate that tangential forces from the wheel are affecting the vehicle. With the oscillations that appear in the wheel's speed signal, it is possible to analyze the dynamics of the wheel throughout the entire vehicle's journey. In this study, the CWT is applied to the wheel encoder signal to determine which frequencies are present during the course. In this section, the results of a test that were conducted during normal operation (without applying the brakes) are presented. These results can be compared with the results of the braking tests, which are discussed in the following section. The frequencies depicted in the CWT spectrogram in Figure 11 correspond to the wheel's dynamics prior to braking on a clean rail. The spectrogram is drawn using the elements \(e_{i,j}\) that result from applying the CWT to _ws_. It can be observed that the predominant vibrations (brightest areas) oscillate between 120 and 200 Hz, but only for brief instants. Some of the frequencies found in this analysis relate to the wheel rolling (46 Hz) and the passage of sleepers (25 Hz). The frequencies of the system are summarized in Table 2.  
Figure 11 also depicts the frequency spectrum determined by FFT of the same signal in the same time interval. The peaks in this spectrum that are enclosed in black squares represent frequencies that are also present in the railcar accelerometer signals, as will be demonstrated in the following lines. Figure 12 depicts the FFT spectrum of the vehicle's body vibrations, which also exhibits very active dynamics; these results come from the same window of time as Figure 11. According to Figures 11 and 12, wheel and vehicle share a number of frequencies, including \begin{table}
\begin{tabular}{c c} \hline \hline
**Frequency (Hz)** & **Source** \\ \hline
25 & Sleeper passage \\
46 & Wheel rolling (at vehicle speed 1.2 m/s) \\
150 & Vehicle vertical \\
180 & Vehicle lateral \\
200 & Vehicle longitudinal \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The frequencies of the system. Figure 11: CWT and FFT results of the wheelset dynamics 0.4 s prior to braking on clean rails. The CWT results show some frequencies between 120 and 200 Hz, where the most prominent peak is 132 Hz by FFT. The black squares in the FFT spectrum highlight the common frequencies between wheel and vehicle dynamics. 66, 87, 132, 154, and 197 Hz. The spectrogram illustrates how these frequencies can also be visualized in time by using the results \(ax_{i,j}\), \(ay_{i,j}\), and \(az_{i,j}\). However, these frequency components do not manifest in all directions, exerting minimal influence only in the lateral and vertical directions, as can be seen in the correlation results. This correlation image was generated utilizing Equations (2) and (3), with \(u_{ax}\), \(u_{ay}\), \(u_{az}\), and \(u_{e}\) as equal to 1.5 times the mean value from \(Ax\), \(Ay\), \(Az\), and \(E\), respectively. This threshold is used for all correlations in this paper. Before braking, the wheel dynamics on sand-covered rails differ significantly from those on clean rails. As shown in Figure 13, the amplitudes are diminished and the frequencies appear to be more dispersed. Nonetheless, the detected frequencies are higher (around 168 Hz). This is partially due to the fact that the sand grains are evenly distributed over the rails, causing the wheels to be randomly excited. Figure 12: Frequency analysis results of the vehicle accelerations 0.4 s prior to braking on clean rails in the longitudinal (\(ax\)), lateral (\(ay\)), and vertical (\(az\)) directions. Only lateral and vertical accelerations exhibit a significant correlation between tangential forces and vehicle response. Figure 13: CWT and FFT results of the wheelset dynamics 0.4 s prior to braking on sand-scattered rails.