# 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 extended Lagrange is defined as Equation (2):  
\[L(\{x_{k}\},\{\omega_{k}\},\{\lambda\})=\] \[\alpha\sum_{k}\left\|\partial_{t}\big{[}(\delta(t)+\frac{j}{ \pi t})\times x_{k}(t)\big{]}e^{-j\omega_{k}t}\right\|_{2}^{2}+\left\|f(t)- \sum_{k}x_{k}(t)\right\|_{2}^{2}+\left\langle\lambda(t),f(t)-\sum_{k}x_{k}(t)\right\rangle \tag{2}\]  
To solve the optimal solution of Equation (5), the alternating direction multiplication operator algorithm is used. The specific implementation steps of the algorithm are as follows:  
Firstly, initialize the parameters, including: \(\left\{\hat{x}_{k}^{1}\right\}\), \(\left\{\hat{\alpha}_{k}^{1}\right\}\), \(\left\{\hat{\lambda}_{k}^{1}\right\}\), mode function is 4, error \(\varepsilon\), output \(\{x_{k}\},\{\omega_{k}\}\), \(\lambda\). Step 1. Execution loop \(n=n+1\);  
Step 2. Update \(\{\hat{x}_{k}\}\) for all \(\omega\geq 0\);  
\[\hat{x}_{k}^{n+1}(\omega)=\frac{\hat{f}(\omega)-\sum\limits_{i\
eq k}\hat{x}_{ i}(\omega)+\lambda_{i}(\omega)/2}{1+2\alpha(\omega-\omega_{k})^{2}} \tag{3}\]  
\(k\in[1,4]\)  
Figure 5: The instantaneous frequency mean of gear vibration signal at the decomposition number of 1–9: (**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. Step 3. Update the modal center frequency \(\{\hat{\omega}_{k}\}\);  
\[\hat{\omega}_{k}{}^{n+1}=\frac{\int_{0}^{\infty}\omega\big{|}u(\omega)\big{|}^{2 }d\omega}{\int_{0}^{\infty}\big{|}u_{k}(\omega)\big{|}^{2}d\omega},k\in[1,4] \tag{4}\]  
Step 4. Update \(\lambda\);  
\[\hat{\lambda}^{n+1}(\omega)=\hat{\lambda}^{n}(\omega)+\tau(f(\omega)-\sum_{k} x_{k}{}^{n+1}(\omega)) \tag{5}\]  
Step 5. Repeat Steps 1-4 until the iteration stop condition is satisfied;  
\[\sum_{k}\Big{\|}\hat{x}_{k}{}^{n+1}-\hat{x}_{k}{}^{n}\Big{\|}_{2}^{2}\Big{/} \Big{\|}\hat{x}_{k}{}^{n}\Big{\|}_{2}^{2}<\varepsilon \tag{6}\]  
Step 6. At the end of iteration, four IMFs are obtained;  
Where: \(f(\omega),\hat{x}_{i}(\omega),\hat{\lambda}(\omega)\) represents the Fourier transform of \(f(\omega),x_{i}(\omega),\lambda(\omega)\), respectively, and \(\varepsilon\) represents the discriminant accuracy. After VMD decomposition of the signal, the penalty factor a was selected \(\alpha\) is 2000 according to experience, and the discriminant accuracy \(\varepsilon\) was. \(10^{-7}\) After repeated experimental analysis, the \(k\) value was 4. In this paper, VMD decomposition time-domain waveform and frequency spectrum of vibration signals of normal gear, gear with tooth wear, gear with tooth crack, and gear with tooth break under four working conditions is shown in Figure 4.0  
Figure 4.1 Gear vibration signal VMD decomposition time-domain waveform and spectrum: (**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. As can be seen from the above figure, the VMD concentrates the modal components near the center of each frequency, effectively improving the modernixing, thus verifying the correctness of the \(k\) selection. After parameter optimization, the VMD algorithm is compared with EMD algorithm. In order to verify the effectiveness of the proposed method, a simulation signal is constructed to decompose EMD and VMD respectively, and the simulation signal [9] is as follows:  
\[\left\{\begin{array}{c}x_{1}(t)=\cos(2\pi\omega_{1}t)\\ x_{2}(t)=\frac{\cos(2\pi\omega_{2}t)}{4}\\ x_{3}(t)=\frac{\cos(2\pi\omega_{3}t)}{16}\\ n(  
It can be seen from Figure 8 that the frequency of the main fault signal can be obtained after EMD decomposition of the simulation signal. However, it can be seen from the frequency-domain diagram of IMF4 that it has mode mixing with IMF3 and IMF2, indicating that the component signal with characteristic frequency \(\omega_{2}=25\) also exists in the component signals of IMF3 and IMF2.