## 2 Fault Diagnosis of Rolling Bearing Based on VMD and ANN
### Feature Extraction Based on VMD  
The variational mode decomposition (VMD) decomposes the raw vibration signal of a bearing \(x(t)\) into several numbers, \(N\), of intrinsic mode functions (IMFs), \(u_{k}(t)\)[21]:  
\[x(t)=\sum_{k=1}^{K}u_{k}(t)=A_{k}(t)\cos(\phi_{k}(t)) \tag{1}\]  
Figure 1: The data partition method for the bearing vibration data of the different fault types.  
Figure 2: The resampling method for obtaining different sampling periods.  
where \(k=1,\ 2,\ \dots,\ 8\) is the number of the decomposed IMFs, \(\phi_{k}(t)\) is mode phase and \(A_{k}(t)\) is the envelope. To obtain the IMFs, \(u_{k}(t)\), and central frequency, \(f_{k}\), the optimum of the augmented Lagrangian was performed as follows:  
\[\begin{split} L(u_{k}(t),\ f_{k},\ \lambda_{t})=\alpha\sum \limits_{k=1}^{K}\|\frac{d}{dt}\Big{[}\Big{(}\delta(t)+\frac{j}{m}\Big{)} \times u_{k}(t)\Big{]}e^{-j2\pi\hat{f}_{k}t}\|_{2}^{2}\\ +\|x(t)-\sum\limits_{k=1}^{K}u_{k}(t)\|_{2}^{2}+\lambda_{t},\ x_ {t}-\sum\limits_{k=1}^{K}u_{k}(t)\end{split} \tag{2}\]  
where \(\lambda_{t}\) is the Lagrangian coefficient, \(\alpha\) is the scale factor, \(\delta\) is the Fermi-Dirac distribution, \(\|\cdot\|_{2}^{2}\) is the 2-norm and \(\langle\cdot\rangle\) is the inner product. The VMD method captures the mode waveforms and central frequencies of the raw rolling bearing vibration signal in the frequency domain.  
The VMD method extends the signal by mirroring half of its length on both sides. As shown in Figure 3, the vibration signal with 1024 sampling points was decomposed to obtain eight IMF components. The waveforms and central frequencies of the eight IMFs clearly differed from each other. As shown by the study, defining an appropriate sampling period could not only guarantee the fault classification result with high accuracy but also reduce the computation time of the algorithm. As shown in Figure 2, the optimal sampling period in this paper was 1/6 ms, that is, 128 sampling points. Therefore, after resampling and VMD processing, 1024 sampling points of each raw vibration signal became 128 points.  
In order to capture the quantization features, discrete Fourier transform (DFT) was performed on the eight IMFs to obtain the amplitude frequency of each IMF, as follows:  
\[X_{k}(f)=\frac{1}{N}\sum\limits_{n=0}^{N-1}u_{k}(n)e^{-j\pi nf/N} \tag{3}\]  
where \(X_{k}(f)\) is the amplitude frequency of each IMF, \(n\) is the length of each IMF and \(N\) = 128. According to Equation (3), the discrete Fourier transform results of the eight IMFs are shown in Figure 4.  
Furthermore, inspired by Shannon entropy, a newly defined energy was used to characterize the central frequency and amplitude frequency features of each IMF. The new defined energy, \(E\), of each IMF's amplitude frequency was calculated using the equation below:  
\[E_{k}=-\sum\limits_{n=0}^{N}f_{n}\times\log_{10}(X_{k}(f_{n})) \tag{4}\]  
Figure 3: The 8 IMFs of the vibration signal with the 1024 samples based on VMD.  
where \(k\) = 1, 2,..., 8 is the index of the eight IMF components, \(n\) = 1, 2,..., 128 is the signal length of each IMF component and \(X_{k}(f)\) is the amplitude frequency of the IMFs corresponding to frequency \(f\). Then, the total energy was obtained using the following equation:  
\[E=\sum_{k=1}^{8}E_{k} \tag{5}\]  
The normalized energy eigenvector of each IMF's amplitude frequency can be obtained using:  
\[\mathbf{X}=\left[\frac{E_{1}}{E},\;\frac{E_{2}}{E},\;\ldots,\frac{E_{8}}{E}\;\right] \tag{6}\]  
The normalized energy eigenvector \(\mathbf{X}\) contains the signal characteristics of the raw vibration signal at different time scales. In addition, taking the normalized energy eigenvector of each IMF amplitude frequency as the input for the ANN can not only improve the prediction accuracy of the bearing fault type but also reduce the data size and, thus, improve the computational efficiency.