# Evaluation of a Novel Condition Indicator with Comparative Application to Hypoid Gear Failures
## 3 Condition Monitoring
### RSD Indicator  
The frequency spectrum of the signal is divided into \(n\) equidistantly spread bands, up to the highest frequency monitored, \(f_{max}\), resulting in  
\[S(i)=\max_{f\in[i\star\Delta f\pm\frac{1}{2}\Delta f]}(S(f)) \tag{1}\]  
where \(i\) is the number of the band. The frequency range \(\Delta f=\frac{f_{max}}{n}\) for each band has to be smaller than the smallest difference between any frequencies of interest. The resulting vector \(S(i)\) contains the maximum value of \(S(f)\) for each of the bands, \(i\). The very same FFT settings are used in the learning phase and the evaluation phase. In the learning phase, the sample is considered to be intact. The learning phase has the duration of \(m\) frequency spectra where \(\mathbb{M}=\{\)spectra where the sample was considered to be in a good state\(\}\). During the learning phase, the average and the standard deviation of the spectrum are calculated for each band in the frequency domain; \(S_{avg}(i)\), \(S_{std}(i)\). \(S_{avg}(i)\) is the vector displaying the average reference spectrum, while \(S_{std}(i)\) is a vector displaying the standard deviation for each value of \(S(i)\) for the learning phase.  
\[S_{avg}(i)=\frac{1}{m}\sum_{j=1}^{m}S_{j}(i);j\in\mathbb{M} \tag{2}\]  
\[S_{std}(i)=\sqrt{\frac{\sum_{j=1}^{m}(S_{j}(i)-S_{avg}(i))^{2}}{n}};j\in \mathbb{M} \tag{3}\]  
With these spectra, the so-called upper and lower reference spectra are calculated, \(S_{ref}^{upper}(i)\), \(S_{ref}^{lower}(i)\),  
\[S_{ref}^{upper}(i)=S_{avg}(i)+S_{std}(i)\cdot k \tag{4}\]\[S_{ref}^{lower}(i)=S_{avg}(i)-S_{std}(i)\cdot k \tag{5}\]  
where \(k\) is the factor, which defines the distance between the upper and lower references from the \(S_{avg}\). As a first approximation, it was supposed that the frequency bands have a Gaussian distribution in time and that \(k=2\). It must be noted that the standard deviation, more specifically the sample standard deviation, varies with the sampling time. The sample standard deviation should approximate the population standard deviation more precisely with increasing learning time. In this respect, it is important to define the learning phase correctly and, thus, determine the sample standard deviation estimating the population standard deviation well. Similar considerations are also valid for the calculation of the average.  
After the learning phase, the evaluation phase begins and the relative difference \(S_{diff}^{rel}(i)\) between the actual spectrum \(S(i)\) and the upper and lower reference spectra \(S_{ref}^{upper}(i)\), \(S_{ref}^{lower}(i)\) for each band can be calculated continuously, where \(i\) denotes the band.  
\[S_{diff}^{rel}(i)=\begin{cases}\frac{S(i)-S_{ref}^{upper}(i)}{S_{ref}^{upper}( i)}&\text{ if }S(i)>S_{ref}^{upper}(i)\\ 0&\text{ if }S_{ref}^{lower}(i)\leq S(i)\geq S_{ref}^{upper}(i)\\ \frac{S_{ref}^{lower}(i)-S(i)}{S_{ref}^{lower}(i)}&\text{ if }S(i)<S_{ref}^{lower}(i) \end{cases} \tag{6}\]  
Additionally, the maximum and minimum values over all \(m\) spectra can be learned. These references can be used to set thresholds for the automated shutdown of the test bed, but have not been used for the validation of the \(RSD\) indicator.  
Based on the relative difference for each band \(S_{diff}^{rel}\), the \(RSD\) can be calculated.  
\[RSD_{abs}=\sum_{i=1}^{n}S_{diff}^{rel}(i) \tag{7}\]  
Alternatively, the indicator can be calculated relative to the number of bands used:  
\[RSD_{rel}=\frac{1}{n}\sum_{i=1}^{n}S_{diff}^{rel}(i) \tag{8}\]