# Gear Fault Diagnosis Based on Kurtosis Criterion VMD and SOM Neural Network
## 3 Based on the Kurtosis Criterion VMD Gear Fault Feature Extraction Method
### Feature extraction Based on Kurtosis Criterion VMD  
Kurtosis is a dimensionless parameter, which reflects the numerical statistics of the random variable distribution characteristics of the signal, and also represents the normalized fourth order central moment of the signal. With the emergence and development of gear faults, the kurtosis of vibration signals increases. The higher the kurtosis value of the vibration signal, the more serious the gear fault and the easier of the gear faults information is to be extracted. Its mathematical formula is shown in Equation (8):  
\[K=\frac{E(x-\mu)^{4}}{\sigma^{4}} \tag{8}\]  
where, \(\mu\) and \(\sigma\) are the mean and standard deviation of the vibration signal, respectively.  
Taking the instantaneous frequency mean as the VMD evaluation index, the number of VMD decomposition was determined after K=4, and the original vibration signal is decomposed into four components by VMD. Since the original vibration signal contains some characteristics of the original information after VMD decomposition, this paper extracts the kurtosis value of each mode to form feature vectors, and then normalizes the kurtosis value to form new feature vectors, which are input into the SOM neural network for gear fault mode classification and identification.  
The 16 groups of data collected under 4 states were decomposed into VMD, the 4 IMF components decomposed by VMD were selected to extract kurtosis value, and then the feature vectors were formed. K1 represents the kurtosis of IMF1, K2 represents the kurtosis of IMF2, K3 represents the kurtosis of IMF3, and K4 represents the kurtosis of IMF4. The kurtosis value matrix of each component is shown in Table 1. After the kurtosis value of each component is extracted, in order to eliminate the influence of dimension, the kurtosis value matrix row vector normalization is adopted. The processing results are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Gear Fault Type** & **Sample Number** & **K1** & **K2** & **K3** & **K4** \\ \hline \multirow{4}{*}{Normal gear} & 1 & 5.8307 & 1.3609 & 3.4593 & 2.6815 \\  & 2 & 5.8335 & 1.3690 & 3.4578 & 2.6883 \\  & 3 & 5.8331 & 1.3675 & 3.4592 & 2.6875 \\  & 4 & 5.8341 & 1.3603 & 3.4564 & 2.6883 \\ \hline \multirow{4}{*}{Gear with tooth wear} & 1 & 6.7332 & 2.8831 & 4.9074 & 3.4868 \\  & 2 & 6.7742 & 2.8387 & 4.9679 & 3.4013 \\  & 3 & 6.7018 & 2.8033 & 4.9781 & 3.4829 \\  & 4 & 6.7793 & 2.8965 & 4.9508 & 3.4726 \\ \hline \multirow{4}{*}{Gear with tooth crack} & 1 & 4.2817 & 1.9227 & 3.8432 & 2.7852 \\  & 2 & 4.2891 & 1.9228 & 3.8209 & 2.7972 \\  & 3 & 4.2999 & 1.9215 & 3.8397 & 2.7717 \\  & 4 & 4.2886 & 1.9225 & 3.8470 & 2.7886 \\ \hline \multirow{4}{*}{Gear with tooth break} & 1 & 8.9575 & 7.7928 & 9.8567 & 4.6750 \\  & 2 & 8.9281 & 7.7706 & 9.8311 & 4.6933 \\ \cline{1-1}  & 3 & 8.9978 & 7.7454 & 9.8819 & 4.6157 \\ \cline{1-1}  & 4 & 8.9271 & 7.7580 & 9.8879 & 4.6864 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Kurtosis values of each intrinsic mode functions (IMFs).