## 4 Results
The axle box acceleration and vehicle speed in the time domain can be seen in Figure 9a. The axle box acceleration data are transformed into the frequency domain using a short-time Fourier transform (STFT), see Figure 9b. To compute the STFT the software Python (SciPy 1.10.1 signal package) is used, and the following parameters selected:  
* Moving window length: T = 0.5 s;
* Tukey window with shape parameter: p = 0.25;
* Moving window step size: dt = 0.01 s.  
Figure 6: Amplitude spectrum of axle box accelerations on the left side. Figure 7: Amplitude spectrum comparison left and right axle box accelerations. Figure 8: (**a**) Vehicle speed of three vehicles passing over the same track section. The section under investigation identified by the microphone measurements analysis is highlighted in yellow. Magnitude of the axle box acceleration in frequency domain (**b**) and in wavelength domain (**c**) of the three vehicle passages. Figure 4.0 (**a**) Axle box acceleration in time domain. The section under investigation identified by the microphone measurements analysis is highlighted in yellow; (**b**,**c**) Spectrogram in frequency domain. In Figure 9b,c, the increasing amplitude is represented with a colour change from dark blue to bright yellow. This representation allows a specific analysis of the excitations and the corresponding cause:  
* The vehicle and rail modes of vibration are constant excitations within a frequency range equal to the eigenfrequencies. In the spectrogram they can be seen as horizontal lines over many kilometres where the speed is high enough to excite the modes, see Figure 9b. The eigenfrequencies can also be estimated using the MBD simulation including the elastic model of the wheelsets. These values are shown in Figure 9c as horizontal pink lines. The estimated modes of vibration can guide the spectrogram analysis in the frequency domain to better identify the horizontal lines. However, it is important to notice that they are not necessary to apply the methodology. While searching for rail corrugation, any frequency-constant excitations can be discarded, whether they are identified as modes of vibration or not. * The bridges and switches shown in Figure 5 appear as impulsive vibrations that excite all frequencies in a very short time. Therefore, they can be seen as vertical lines in the spectrogram in Figure 9b and as highlighted vertical sky-blue lines in Figure 9c. Again, there is a good agreement between the construction positions and the resulting effects on the axle box accelerations. However, not all of them have a significant effect on the accelerations, especially where the vehicle is running at lower speed. * The wheel defects in the wavelength range of interest are mainly wheel corrugation (30-60 mm) and wheel out-of-roundness (OOR) within \(140-2\pi\)R mm, where R is the wheel radius [27]. These defects are constant in the wavelength domain, and, unlike rail corrugation, should be observed along the entire acceleration signal. Being constant in the wavelength domain means, for the frequency domain spectrogram, that they have the same shape as the vehicle speed. Some wheel defects can be observed in the spectrogram in Figure 9b and are highlighted with white dashed lines in (c). Once modes of vibration, construction and wheel defects have been excluded from the spectrogram, the remaining excitation is considered to be rail corrugation if it is constant in wavelength and lasts for a few hundred metres. Observing the section identified at the beginning of the analysis, some excitations can be seen in the 500-1000 Hz range. To better visualise the wavelength range of interest, the acceleration spectrogram is evaluated in the distance domain with respect to wavelength, see Figure 4.1 For the spectrogram in the wavelength domain a logarithmic scale is used for better visualisation. Figure 4.2 (**a**) Axle box acceleration in distance domain.