# Investigations into the Recognisability of Gear Damage
## 3 Data Analysis
### Data Segmentation
An analogous consideration for the gearbox with artificially damaged gear \(\mathrm{z}_{3}\) is shown in Figure 7.  
Figure 6: Vibration signal and spectrogram of the artificially damaged gear \(\mathrm{z}_{1}\). Figure 7: Vibration signal and spectrogram of the artificially damaged gear \(\mathrm{z}_{3}\). Here again, one gear revolution is considered. Compared to the previous observation, this results in a length of about 900 ms due to the gear ratio \(i_{1}=3.57\). In this observation, no concrete deviation from the rest of the signal can be recognised in the time domain even in the time window \(S_{1}\) of the greatest damage. Only when looking at the time-frequency domain in the bottom illustration, can the differences be recognised. Above all, \(S_{1}\) is characterised by strongly distinctive frequencies in the range of about 1 kHz. This distinctive frequency range also occurs for \(S_{2}\). \(S_{3}\) and \(S_{4}\) show no differences to the intact tooth meshes. For a statistical evaluation of the tooth meshing of the synthetically damaged teeth, the individual measurements are first segmented. Based on the previous observation, it could be shown that for the two examined gears, the most severe damage \(S_{1}\) is recognisable in each case by a more distinctive frequency behaviour in the spectrogram. In order to first determine the times of these damaged tooth meshes, the mean signal power of the spectrogram \(\bar{s}_{\text{xM,m,dB}}\) in the dominant frequency ranges \([f_{1},f_{2}]\), which were previously derived, is calculated according to  
\[s_{\text{xM,m,dB}}=20\cdot\log_{10}\left(\frac{\sum\limits_{k=f_{1}}^{f_{2}} \mathcal{S}_{\text{x,m,k}}}{(f_{2}-f_{1})}\right). \tag{5}\]  
As an example for the segmentation of the tooth meshes of the damage \(S_{1}\), a spectrogram as well as the mean signal power \(\bar{s}_{\text{xM,m,dB}}\) in the range of 1 kHz to 7 kHz is shown in Figure 8. The peaks in the bottom plot correspond to the damaged tooth meshes. In addition, it can be seen that there are no anomalies in the signals in the areas after the damage, and thus the intact meshes are not influenced by the damaged ones before them. The vibration signal used for this spectrogram was recorded at the gearbox with damage to gear z\({}_{1}\). The measured speed is initially constant at 207 rpm and increases to 322 rpm after about one second. In the spectrogram, recurring significant distinct frequencies can be recognised. Furthermore, an increase in the total signal power can be seen from about one second onwards, due to the increase in rotational speed at this point. The average calculated signal power  
Figure 8: Spectrogram (**top**) and mean signal power (**bottom**) of the vibration signal of the gearbox with synthetic damage at gear z\({}_{1}\) in the range of 1–7 kHz. in the range of 1-7 kHz also shows the signal peaks. The total average power increase due to the increased speed can also be seen in this diagram. Furthermore, it can be seen that the period duration decreases due to the increasing speed. The average period \(\bar{T}_{x_{1}}\) between the determined signal peaks in the range of the measured speed of \(\omega[t]=207\) rpm is \(\bar{T}_{x_{1}}=0.2915\) s. The theoretical period duration \(\bar{T}_{x_{1},\text{theo}}\) for the occurrence of the damage at z\({}_{1}\) is given by  
\[\bar{T}_{x_{1},\text{theo}}=\frac{1}{\omega[t]/60}=\frac{1}{207\text{ rpm}/60}=0.2899\,\text{s} \tag{6}\]  
and thus coincides within the scope of the measurement accuracy with the period durations determined via the average signal power. The mean period between the signal peaks in the speed range of 322 rpm is 0.1849 s. This again coincides with the theoretical period duration of 0.1863 s for this speed. As a result, the signal peaks can be assigned to the damage \(S_{1}\). The signal peaks are defined as the times \(t_{i}(S_{1})\) of the tooth meshes \(S_{1}\).