# Investigations into the Recognisability of Gear Damage
## 3 Data Analysis
### Recognisability of Gear Damage in the Time Domain  
In order to examine the experimentally possible recognisability of the synthetic damages, the segmented signals are examined more closely in the time domain. For this purpose, the cross-correlation between the individual signals determined beforehand is calculated in order to use the correlation coefficients to make statements about the damage levels and the recognisability of damage. The cross-correlation function (CCF) is a modification of an \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \multicolumn{3}{c}{**Rotational Speed [rpm]**} \\ \cline{2-5}  & **100** & **200** & **300** & **400** \\ \hline \(n_{\text{S},x_{1}}\) & 13 & 29 & 41 & 50 \\ \(n_{\text{S},x_{3}}\) & 14 & 34 & 48 & 64 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Number of signals \(n_{\text{S},x_{1}}\) for each of the four damages on gear \(z_{1}\) and \(n_{\text{S},x_{3}}\) for each of the four damages on z\({}_{3}\) for the respective rotational speeds. autocorrelation function (ACF). A discrete autocorrelation \(R_{x_{1}x_{1},m}\) describes the correlation of a signal \(x_{1}[n]\) with itself at another time \(x_{1}[n+m]\) over a considered interval \([-M,M]\) and is described by  
\[R_{x_{1}x_{1},m}=\lim_{M\to\infty}\frac{1}{2M+1}\sum_{n=-M}^{M}x_{1}[n]\cdot x _{1}[n+m]. \tag{11}\]  
The autocorrelation function thus describes the similarity of a signal part to previous times in the signal [20]. The cross correlation is an extension of the ACF in which a signal \(x_{1}[n]\) is compared with a second different signal \(x_{2}[n]\)[20]. For the discrete case, the cross correlation \(R_{x_{1}x_{2},m}\) between two signals is calculated according to  
\[R_{x_{1}x_{2},m}=\lim_{M\to\infty}\frac{1}{2M+1}\sum_{n=-M}^{M}x_{1}[n]\cdot x _{2}[n+m]. \tag{12}\]  
The cross correlation describes the similarity of two signals \(x_{1}[n]\) and \(x_{2}[n]\). The cross correlation is normalised over the values of the ACF without shift \(R_{x_{1}x_{1},0}\) and \(R_{x_{2}x_{2},0}\) of both signals to  
\[R_{x_{1}x_{2},\text{coeff},m}=\frac{1}{\sqrt{R_{x_{1}x_{1},0}R_{x_{2}x_{2},0}} }R_{x_{1}x_{2},m}. \tag{13}\]  
This results in values in the range \([-1,1]\) for the normalised CCF, which are thus comparable to the values of the correlation coefficients of the Pearson correlation [21]. Since it cannot be assumed that all signals recorded are ideally aligned, the maximum value of the CCF is calculated. Signals can be aligned via the associated temporal offset of the maximum value [22]. The maximum value of the cross correlation function \(r_{\text{max}}\) thus describes the similarity of two signals aligned to each other and results in  
\[r_{\text{max}}=\text{max}\left(R_{x_{1}x_{2},\text{coeff},m}\right). \tag{14}\]  
In the following, a correlation matrix of the recorded vibration signals of the gearbox with damage to gear \(z_{1}\) at a rotational speed of 200 rpm is first considered as an example. For this purpose, 34 signals are considered for each damage level \(S_{1}\), \(S_{2}\), \(S_{3}\) and \(S_{4}\), as well as for the reference tooth mesh of the intact teeth \(S_{0}\). Figure 9 shows the correlation matrix for all 170 signals considered. This allows for the different vibration behaviour of the individual damage levels (\(S_{1}\)-\(S_{4}\)) of the gear to be compared with the intact tooth mesh \(S_{0}\). Based on the periodic behaviour of the tooth meshes, a statistically representative set of 34 meshes of each damage level as well as the intact mesh were recorded.The individual damage levels are summarised as blocks and labelled on the y- and x-axis in the figure. The main diagonal of the matrix shows the correlation of the respective signal with itself. Accordingly, the main diagonal shows correlation coefficients of 1. Furthermore, the blocks on the main diagonal also show increased values, due to the similarity of the individual signals of a damage level. The other blocks thus show the similarity between the individual damage levels.