# Investigations into the Recognisability of Gear Damage
## 3 Data Analysis
### Recognisability of Gear Damage in the Frequency Domain  
In this section, the previously segmented signals will be compared in the frequency domain in order to make statements about the recognisability of the damages via a further distinguishing feature. For the investigations in the frequency domain, the segmented signals are transformed into the frequency domain. For this, the discrete Fourier transformation \(X_{k}\) of a discrete signal \(x[n]\) is used. This was already implicitly used in the calculation of the STFT in Section 2.2. It results in  
\[X_{k}=\sum_{n=0}^{N-1}x[n]\mathrm{e}^{\frac{-2\pi\mathrm{km}}{N}}\text{ for }k=0,1,\ldots,(N-1). \tag{15}\]  
\(N\) describes the number of frequencies. A window function is used to avoid spectral leakage. The discrete Fourier transformation (DFT) of the windowed signal \(X_{k}^{\gamma}\) therefore results in  
\[X_{k}^{\gamma}=\sum_{n=0}^{N-1}x[n]\gamma[n]\mathrm{e}^{\frac{-2\pi\mathrm{km }}{N}}. \tag{16}\]  
\(\gamma[n]\) describes the window function. The FFT is used as an implementation of the DFT. To avoid unequal signal lengths in the frequency range, zero padding [23] is used and zeros are added to the vector after the windowing. The information content of the resulting frequency signal is not changed by zero padding, but the spectrum is smoothed by the increased frequency resolution. The new vector length is chosen based on the longest signal (at a rotational speed of 100 rpm) to the next higher power of two. After calculating all spectra \(X_{S_{i},k}^{\gamma}\) for each damage level and rotational speed, an averaged spectrum \(\overline{X}_{S_{i},k}^{\gamma}\) of the respective damage level is calculated. In the following, the averaged spectra for the measurement series of the gearbox with damage to the gear \(z_{1}\) and subsequently that for damage to the gear \(z_{3}\) are considered. The considered frequency range up to 22 kHz results from the linear range of the sensor up to this frequency. Figure 11 shows that all spectra have a more distinctive behaviour in the lower frequency range. Overall, the higher frequencies show lower amplitudes. In the range below 1 kHz and above 13.5 kHz, the spectra hardly show a distinguishable behaviour. The figure shows that the largest damage, \(S_{1}\), results in a particularly significant difference. The frequency range between 1 kHz and 9 kHz shows clearly more distinctive amplitudes. Further, smaller distinctive frequency bands occur at approximately 10 kHz, 11.5 kHz and 13 kHz. For the spectrum of the second largest damage \(S_{2}\), differences in the spectra are also noticeable. The frequency bands around 1.5 kHz and 7 kHz especially show higher amplitudes. The spectra of damages \(S_{3}\) and \(S_{4}\) show no recognisable differences to the reference signal at all. Figure 12 shows a significantly better resolution compared to the previous observation, due to the longer signal length resulting from the lower speed during the second gearbox stage. Furthermore, a decreasing signal amplitude can be recognised at higher frequencies. When comparing the spectra, a very similar spectra pattern can be observed for frequencies above 5 kHz, which do not show a clear differentiation. The greatest difference can once again be determined for damage level \(S_{1}\). From 0.5 kHz to 2.5 kHz, several distinctive frequency bands appear. Another distinct frequency band occurs at approximately 3.7 kHz. Likewise, more distinctive frequency bands also occur for damage level \(S_{2}\). These are mainly in ranges from 0.5 kHz to 1.5 kHz, 2.1 kHz to 2.5 kHz and at about 3.7 kHz. For damage levels \(S_{3}\) and \(S_{4}\), there are no noticeable differences to the reference signal. Overall, based on this investigation, damage levels \(S_{1}\) and \(S_{2}\) of both gearboxes show a recognisable behaviour in the frequency domain, as they stand out due to their distinctive frequencies. Damages \(S_{3}\) and \(S_{4}\) of this gearbox do not show any differences to the reference signal.