## 3 Results and Discussion  
### Comparison of the Feature Extraction Methods  
The performance of the VMD and empirical mode decomposition (EMD) in feature extraction is first compared. As shown in Figure 8a, one raw vibration signal was decomposed into eight IMFs by EMD. The mode waveforms and central frequencies were mostly concentrated in the first three IMFs, while the last five IMFs almost contained a few signal features. The amplitude of the IMFs based on the EMD method varied greatly, and most of the signal energy was concentrated in IMF1. The frequency variation based on the EMD method was also large, which led to the result that most of the signal energy was concentrated in IMF1 and IMF2. However, as shown in Figure 8b, the same raw vibration signal was decomposed into eight IMFs by VMD, and these eight IMFs contained effective waveforms and central frequencies features. The frequency of the IMFs based on VMD showed a steady gradient decrease, while the amplitude of each IMF was consistent on the whole, which ensured that every IMF contained the effective features of the raw vibration signal.  
Figure 7: Flow process diagram of the fault diagnosis experiment and analysis.  
Discrete Fourier transform was performed on the eight IMFs decomposed by EMD and VMD. According to Equation (4), the energies of each IMF were calculated, as shown in Figure 9. Compared with EMD, the signal energy based on VMD had more intuitive differentiation. Therefore, the VMD method was superior to the EMD method in feature extraction.  
The signal energy of eight IMFs corresponding to 10 types of bearing fault signals was calculated according to Equation (4) to further compare the feature extraction ability of EMD and VMD. As shown in Figure 10, the regularity of the energy feature extracted by the EMD method was not as obvious as that of the VMD method. Compared with EMD, the extracted energy feature based on the VMD method changed more regularly with the IMFs as a whole, which is conducive to finding the internal principle of the bearing fault in the training process of the ANN. In addition, for the extracted energy feature based on EMD, the energy of the first three IMFs was obviously larger than that of the last five IMFs based on EMD, which may count against the subsequent ANN training process.  
Figure 8: Comparison of the decomposition performance of EMD and VMD on the time domain: (**a**) 8 IMFs decomposed by EMD in the time domain; (**b**) 8 IMFs decomposed by VMD in time domain.  
Figure 9: The discrete Fourier transform of the IMFs decomposed by EMD and VMD: (**a**) discrete Fourier transform of the IMFs decomposed by EMD; (**b**) discrete Fourier transform of the IMFs decomposed by VMD.  
Furthermore, with the features extracted through the raw vibration signal, EMD and VMD as the input for the ANN, the results of the prediction accuracy and computation time of bearing fault diagnosis were obtained to demonstrate the superiority of the VMD. The architecture of the ANN adopted the parameters shown in Table 2. As shown in Figure 11, the ANN based on the VMD method had the highest prediction accuracy of 99.3%, with a computation time of 2.4 s, and the prediction accuracy based on the EMD method was 94.2%, with a computation time of 2.3 s. In terms of the computation time, there was little difference between the VMD method and the EMD method. The ANN based on the raw vibration signal of the rolling bearings not only had the lowest prediction accuracy of 81.2% but also had the longest computation time of 5.6 s. Therefore, feature extraction based on VMD plays a key role in improving the performance of ANNs.