# Comparison of Crack Detection Performance of Low and High Contact Ratio Spur Gears
## 5 Comparison of Statistical Indicators for LCR and HCR Gears  
In this study, changes in dynamic force signals were statistically examined to identify tooth root cracks for LCR and HCR gears. Statistical indicators were calculated for each case examined in the study. Indicators for healthy status have been determined as reference values. The % change of the indicators was analyzed according to the crack length. In this way, the stage of the crack was determined, and whether it could be easily detected from case to case. Kurtosis, RMS, crest factor, mean, standard deviation, and variance statistical indicators were used to detect tooth root cracks and levels for LCR and HCR gears. Since the results of this study are obtained directly from a dynamic model, fundamental statistical indicators will be sufficient for the solution since there is no noise or signal-disturbing factor. When the literature is examined, it is seen that in numerical studies, examining the change of one or two indicators is generally sufficient for error detection [24]. RMS is the root mean square value of the data. It is defined as a measure of the power content of the vibration and audio signal. \[\text{RMS}=\sqrt{\frac{1}{n}\overset{\text{n}}{\underset{i=1}{\sum}}|\text{x} _{i}|^{2}}. \tag{46}\]  
Kurtosis is defined as the fourth-order normalized moment of a signal. It reveals the peak characteristic of the signal. Increases in gear vibrations due to damage will also cause the kurtosis value to increase. In healthy systems, this value is around 3. Figure 10: Dynamic analysis results for LCR (Case I, **(A–D)**) and HCR (Case V, **(E–H)**) gears: (**A–E**) healthy, (**B–F**) 1.5 mm, (**C–G**) 3 mm, (**D–H**) 4.5 mm crack length. \[\text{Kurtosis}=\frac{1}{n}{\sum_{i=1}^{n}}(x_{i}-\overline{x})^{2}/\left(\frac{1}{n }{\sum_{i=1}^{n}}(x_{i}-\overline{x})^{2}\right)^{2}. \tag{47}\]  
The crest factor is defined as the ratio of the maximum positive peak value in the signal to the RMS value. In healthy gears, it is expected to be between 2 and 6. For a sine function, the relationship between the crest factor, RMS, and the maximum point of the signal is given in Figure 11.  
\[\text{CF}=\frac{\text{max}|x_{i}|}{\sqrt{\frac{1}{n}{\sum_{i=1}^{n}}{|x_{i}|}^ {2}}}. \tag{48}\]  
Mean is defined as the average value of the signal obtained by dividing all values in the mean signal by the number of points in the signal. The mean value for any signal can be calculated with the help of Equation (39). \[\mu=\frac{1}{N}{\sum_{i=1}^{N}}\text{A}_{i}. \tag{49}\]  
Standard deviation is a measure of how dispersed the data are in relation to the mean. To calculate the standard deviation, the average value of the system must also be calculated. \[\text{Sd}=\sqrt{\frac{1}{N-1}{\sum_{i=1}^{N}}|A_{I}-\mu|^{2}}. \tag{50}\]  
where \(\mu\) is the mean. Standard deviation gives information about how close the data in a signal are to the average value. If the standard deviation value is small, the data in the signal are distributed close to the average value. The data are homogeneous and the discrimination of the data is low. If the standard deviation value is large, it follows that \"the data are scattered at points far from the average value\". There is a heterogeneous dataset. Therefore, if all values in a signal are equal, the standard deviation value will be zero. Therefore, it is expected that the standard deviation will increase as the crack length increases. Variance can be expressed as the sum of squares of the deviations of the data in a signal from the average values. In short, it is the square of the standard deviation. Since there is a root-taking process in calculating the standard deviation, the variance value was  
Figure 11: RMS, peak, and crest factor relationship for the sine function. also examined as an error indicator in this study in order to see the change in the data more clearly.