# Investigations into the Recognisability of Gear Damage
## 5 Calculation of Digital Filter Limits Using Difference Spectra  
To determine appropriate filter limits, the linear prediction spectra \(X_{\text{LPS}_{1}}\) and \(X_{\text{LPS}_{2}}\) for damage levels \(S_{1}\) and \(S_{2}\) are calculated. Furthermore, \(X_{\text{LPS}_{0}}\) describes the reference spectrum of the time windows \(S_{0}\). First, difference spectra \(\widetilde{X}_{\Delta S_{1}}=X_{\text{LPS}_{1}}-X_{\text{LPS}_{0}}\) and \(\widetilde{X}_{\Delta S_{2}}=X_{\text{LPS}_{2}}-X_{\text{LPS}_{0}}\) are defined to determine the differing frequency ranges. Of particular interest here are the ranges where the difference is greater than 0 dB, since a more distinctive behaviour of the vibration signal due to the damage is recognisable. The difference spectra are shown in Figure 14 above for the gearbox with damage to gear \(z_{1}\) and below for \(z_{3}\). Figure 13: Comparison of the spectra via prediction coefficients of different orders \(p\) with an FFT spectrum in the frequency range up to 10 kHz. It can be seen that in both cases the largest damage \(S_{1}\) shows the most distinctive behaviour. Overall, both difference spectra are strongly correlated. For the measurement series with damage to gear \(z_{1}\) in the upper curve, it can be seen that the distinctive areas occur mainly in the lower frequency range, while damage \(S_{2}\) shows a clearly narrower behaviour than \(S_{1}\) in the frequency domain. For the measurement series with damage to \(z_{3}\), a clearly more distinctive behaviour occurs in the lower frequency range. The differences between the frequency ranges are mainly associated with the different geometry of the gears and the different transmission paths of the vibration to the sensor. In order to determine appropriate digital filter limits for each gearbox stage, the frequency ranges with more distinctive behaviour are searched for in the following. For this purpose, the function \(f^{+}(x)\) is defined, which limits the difference spectra to the positive range according to  
\[f^{+}(x)=\left\{ \begin{array}{rl}1,&\text{for }x>0\\ 0,&\text{for }x\leq 0\end{array} \right. \tag{25}\]  
This replaces negative function values to zero and positive to 1. The overlap regions \(\widetilde{X}_{\text{overlap}}\) of the difference spectra are subsequently defined as  
\[\widetilde{X}_{\text{overlap}}=f^{+}(\widetilde{X}_{\Delta S_{1}})\cdot f^{ +}(\widetilde{X}_{\Delta S_{2}}). \tag{26}\]  
\(\widetilde{X}_{\text{overlap}}\) thus describes equal distinct frequencies for both damage levels. Figure 15 shows the overlap of the difference spectra for the measurement series with damage at gear \(z_{1}\) (blue) and gear \(z_{3}\) (red). Frequencies greater than \(11\,\text{kHz}\) are not shown because no overlap occurs. The figure shows the frequency ranges of the distinct frequencies which occur in the difference spectrum of damage \(S_{1}\) as well as damage \(S_{2}\) for both examined gearbox stages. For the first gearbox with damage to gear \(\mathrm{z}_{1}\), frequencies from 687 Hz to 1697 Hz, 5693 Hz to 6445 Hz and 8013 Hz to 10,527 Hz result. For the gearbox with damage to gear \(\mathrm{z}_{3}\), frequencies from 0 Hz to 1611 Hz, 2129 Hz to 2669 Hz, 3574 Hz to 3947 Hz and 4887 Hz to 5392 Hz result. In the following, the determined frequency bands are examined exemplarily as filter limits for the examined measurements. For this purpose, a multiband filter with finite impulse response (FIR) is used. The results of the digital filtering are shown for the two investigated gearboxes in Figure 16. Both measurements were recorded at a rotational speed of 200 rpm and a torque of 30 Nm on the input shaft. Furthermore, the areas of the damaged tooth meshings \(S_{1}\) (light gray) and \(S_{2}\) (dark gray) are highlighted. When looking at the unfiltered vibration signal, damage level \(S_{1}\) can already be recognised in curve (a). Further damage cannot be seen directly in both (a) and (b).