# Gear Fault Diagnosis Based on Kurtosis Criterion VMD and SOM Neural Network
## 3 Based on the Kurtosis Criterion VMD Gear Fault Feature Extraction Method  
### VMD Signal Decomposition  
The VMD algorithm was proposed by [PERSON] and [PERSON] in 2014. The VMD algorithm decomposes the collected gear vibration signal \(x(t)\) by constructing a variational model, and adaptively decomposes the gear vibration signal \(x(t)\) by searching for the constrained variational optimal solution. The signal is adaptively decomposed into k IMFs \(x_{k}(t)\)[23]:  
Finally, the decomposed IMFs \(x_{k}(t)\) are used to construct the squared \(L2\) norm of the VMD algorithm is expressed as:  
\[ \begin{array}{l}\min\limits_{\{u_{k}\},|\omega_{k}|\}\left\{\sum\limits_{k} \left\|\partial_{t}\!\left[\left(\delta(t)+\frac{j}{\pi t}\right)\times x_{k} (t)\right]\!e^{-j\omega_{k}t}\right\|_{2}^{2}\right\}\\ s.t.\sum\limits_{k=1}u_{k}=x(t)\end{array}  \tag{1}\]  
where: \(\partial_{t}\) is the partial derivative of \(t\), \(x(t)\) is the original signal, \(\omega_{k}\) is the bandwidth center frequency, and \(\delta_{t}\) is the pulse signal. In this paper, \(k=4\) is derived based on the magnitude of the instantaneous frequency value. Figure 3: Fault diagnosis flow chart based on kurtosis criterion variational mode decomposition (VMD) and self-organizing map (SOM) neural network. The time-domain and frequency-domain diagrams of vibration signals collected under four working conditions of normal gear, gear with tooth wear, gear with tooth crack, and gear with tooth break are shown in Figure 4.  
It can be seen from Figure 4 that the four types of gear failure signals are obtained after Fourier transform, and the four types of signals have the same frequency band peaks in the frequency spectrum. The noise contained in the signal and the acquisition time is also different. After performing the Fourier transform, the characteristic frequency of the signal may be masked. Therefore, a fault diagnosis algorithm is needed to intelligently diagnose gear faults. In the VMD decomposition of the signal, the decomposition number \(k\) and the penalty factor \(\alpha\) are selected empirically. According to experience, the number of values of \(k\) is often decomposed inaccurately, resulting in the value of \(k\) is often too large. After VMD decomposition, some intermittent and useless frequency bands will appear. Therefore, the choice of \(k\) value is also very important in VMD decomposition. In this paper, the instantaneous frequency mean is selected as the evaluation index, and the change of instantaneous frequency mean when the number of decomposition is 1-9 is selected as the basis for selecting the value of decomposition number \(k\)[24]. Figure 5 respectively shows the instantaneous frequency mean size of the fault signal under the four working conditions of normal gear, gear with tooth crack, and gear with tooth break at the decomposition number of 1-9:  
According to the analysis of Figure 5, after the \(k\) value is 5, the high frequency component is somewhat bent and intermittent. When the number of decomposition is greater than 4, some useless components may be decomposed. So \(k=4\). Figure 4: Time-domain and frequency-domain diagram of gear vibration signal: (**a**) Normal gear vibration signal, (**b**) gear with tooth wear vibration signal, (**c**) gear with tooth crack vibration signal, and (**d**) gear with tooth break vibration signal. The constrained variational problem in the formula is further solved, and the quadratic penalty factor \(\alpha\) and _Lagrange_ multiplication operator \(\lambda(t)\) are introduced in consideration of the fact that the constraint becomes non-constraint. The quadratic penalty factor \(\alpha\) guarantees the signal reconstruction accuracy in the noise environment, and the _Lagrange_ multiplication operator \(\lambda(t)\) keeps the constraint condition strict.