# Investigations into the Recognisability of Gear Damage
## 3 Data Analysis
### Data Segmentation  
In the following, the vibration signal of one gear revolution at a rotational speed approximately of 200 rpm is considered as an example for each gear. For this purpose, the recorded vibration signal is transformed into the time-frequency domain as a spectrogram. Spectrograms of vibration signals are determined in digital signal processing via the discrete short-time Fourier transform (STFT). The discrete STFT \(\mathcal{F}_{\chi,m,k}^{\gamma}\)[19] is given by  
\[\mathcal{F}_{\chi,m,k}^{\gamma}=\sum_{n=0}^{N-1}x[n]\cdot\gamma^{*}[n-m\Delta M ]\cdot\mathrm{e}^{\frac{-2\pi\mathrm{thr}}{N}}. \tag{3}\]  
Here, \(x[n]\) describes a discrete-time signal and \(\gamma^{*}[n-m\Delta M]\cdot\mathrm{e}^{\frac{-2\pi\mathrm{thr}}{N}}\) a time- and frequency-shifted window function in the considered interval \([0,N-1]\). Since only discrete frequencies and time points are considered, \(m=0,1,\ldots,M-1\) applies. \(\Delta M\) describes the considered window size. The complex-valued short-time Fourier transform is converted into real numbers via the magnitude square for the pictorial representation in a spectrogram \(\mathcal{S}_{\chi,m,k}\):  
\[\mathcal{S}_{\chi,m,k}=\left|\mathcal{F}_{\chi,m,k}^{\gamma}\right|^{2}=\left| \sum_{n=0}^{N-1}x[n]\cdot\gamma^{*}[n-m\Delta M]\cdot\mathrm{e}^{\frac{-2\pi \mathrm{thr}}{N}}\right|^{2} \tag{4}\]  
The spectrogram in decibels \(\mathcal{S}_{\chi,m,k,\mathrm{dB}}\) results in \(\mathcal{S}_{\chi,m,k,\mathrm{dB}}=20\cdot\log_{10}(\mathcal{S}_{\chi,m,k})\). First, in Figure 6, the vibration signal is considered above and the corresponding spectrogram for the measurement of the gearbox with synthetic damage to gear \(\mathrm{z}_{1}\) is considered below. The vibration signal shown was sampled at 51.2 kHz. A resampling to 44 kHz was applied. For the calculation of the following spectrograms, an FFT length of 512 values, a Blackman window function with a length of 512 values and an overlap of 511 values were chosen. Due to the short FFT length and the large overlap, a good temporal resolution can be achieved, which is necessary for the assignment of the individual teeth. Furthermore, the black lines show the parts of the signal in which the tooth meshes of the synthetically damaged teeth occur. These time windows are labelled according to the damaged teeth \(S_{1}\), \(S_{2}\), \(S_{3}\), \(S_{4}\) from Section 2.3. The highlighted signal parts were determined based on the spectrogram, the rotational speed and the known geometry of the synthetic damage. The highlighted areas correspond to the proportionate duration of the respective teeth meshes. The largest damage (\(S_{1}\)) was determined in the spectrogram via its period duration and subsequently the positions of \(S_{2}\), \(S_{3}\), \(S_{4}\) were calculated. The procedure for selecting the time windows will be discussed in more detail later in this section. The signal in the time domain shows the vibrations over a time interval of about 200 ms. The vibration signal is composed of many individual transients, which are caused by the metallic contacts of the tooth meshes and roller bearings. In \(S_{1}\), a more distinctive transient can be seen compared to the rest of the signal. In terms of amplitude, however, this can hardly be distinguished from the other transients. For \(S_{2}\), \(S_{3}\) and \(S_{4}\), no differences to the undamaged tooth meshes can be seen in the time domain. In the spectrogram of the vibration signal, the tooth mesh of the synthetic damaged tooth in \(S_{1}\) can be clearly recognised by the more distinctive frequency components in a range from 1 kHz to 7 kHz. This transient can be clearly separated from the entire signal shown. \(S_{2}\) shows smaller distinct frequency ranges which cannot be clearly assigned to the damage due to its comparable occurrence in the intact tooth meshes. For \(S_{3}\) and \(S_{4}\), no noticeable distinctive frequencies occur.