## 1 Introduction
### Vibration Analysis  
Vibration analysis is an established method to monitor gear condition [22]. It is based on the acceleration of the gear teeth as they mesh. This process gives the Gear Mesh Frequency (GMF), which is very important for analysis, as well as its harmonics [3; 6; 23]. The GMF is always the same for two mating gears because both gears are mounted on different shafts and the shaft frequency is dependent on the number of teeth of a gear. The GMF and its harmonics are calculated as follows:  
\[\begin{array}{l}GMF_{i}=i\cdot f_{s}\cdot n\\ i:\text{order of GMF}\\ f_{s}:\text{shaft frequency}\\ n:\text{number of teeth}\end{array} \tag{1}\]  
As the gear teeth change due to heavy wear or damage, the vibration signals change. This change affects especially the amplitude of the \(GMF_{i}\)[4]. [PERSON] et al. suggested that harmonics should be considered at least up to the third order [23]. There are common metrics to detect a fault in gears based on this change in the vibration signal. These metrics are called Gear Condition Metrics (GCMs) and differ in their ability to detect severe faults, monitor faults over time and detect different types of wear mechanisms [24; 25].  
Two of these metrics have been chosen for this study. The GCMs are calculated as described in Table 1. Table 2 lists the components which are part of the regular, differential and residual vibration signal. More information about the metrics can be found in [24; 25].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**GCM** & **Formula** & & **Description** \\ \hline \multirow{4}{*}{Energy Ratio (ER)} & \multirow{4}{*}{\(ER=\frac{\sigma(d)}{\sigma(reg)}\)} & \multirow{4}{*}{(2)} & \(\sigma(d)\): standard deviation of the \\  & & & differential signal of a data sample \\ \cline{1-1}  & & & \(\sigma(reg)\): standard deviation of the regular \\ \cline{1-1}  & & & signal of a data sample \\ \cline{1-1}  & & & \(N\): number of data points in a \\ \cline{1-1}  & & & vibration signal \\ \cline{1-1}  & & & \(res\): residual vibration signal \\ \cline{1-1}  & & & \(res_{OK}\): residual vibration signal of a data \\ \cline{1-1}  & & & sample of the unworn gear \\ \hline \hline \end{tabular}
\end{table}
Table 1: GCMs used in this study.  
The effect of different wear patterns on the frequency spectrum was investigated by [PERSON] et al. [23] and [PERSON] et al. [26]. They found a change in the sidebands of the GMF as the wear progressed from pitting to abrasive. This change shows that the sidebands respond to abrasive wear, which is the main mechanism considered in this work. Cepstrum analysis uses the inverse Fourier transform to find the periodic components of a signal's spectrum, such as sidebands. Formula (4) is used to calculate the real cepstrum, which is used in this paper:  
\[\begin{array}{c}C_{r}=\mathcal{F}^{-1}\{\log(|\mathcal{F}\{f(t)\}|)\}\\ C_{r}:\text{ Cepstrum}\\ f(t):\text{ vibration signal}\end{array} \tag{4}\]  
[PERSON] et al. showed that it is possible to detect faults in POM gears using vibration analysis [8]. However, in most cases, it was not possible to distinguish between a healthy gear and pitting on a single tooth using vibration analysis. It should be noted that in some cases, even pitting spread over several teeth indicated an even healthier gear than pitting on a single tooth. The unclear results could be due to the greater damping of non-fibre-reinforced polymers compared to steel [5]. Better monitoring of gears reduces the uncertainty of wear, enabling use in a wider range of applications, the predictive maintenance of machinery and the reuse of gears.  
In conclusion, there are several vibration analysis methods for detecting wear with different objectives. It has already been shown that some of these methods can be applied to polymer gears, but the results are difficult to interpret.