# Mechanical Incipient Fault Detection and Performance Analysis Using Adaptive Teager-VMD Method
## 4 Fault Experiment Analysis
The experimental results show that using CTEO to preprocess raw faulty signals can effectively enhance the impact component. Furthermore, comparing the envelope spectra  
Figure 14: Envelope spectrum of BIRF signal. **(a)** CEEMD. **(b)** LMD. **(c)** ITD. **(d)** VMD. obtained with the classical CEEMD, LMD, and ITD methods, it can be seen that the spectral lines representing the fault feature in the envelope spectra obtained with the VMD method are significantly larger than those obtained with the remaining three methods. Considering the above evaluation and comparison, the VMD method outperforms mechanical vibration signal processing compared to other commonly used methods. Moreover, these comparison results prove that it is a feasible and effective strategy to enhance and preprocess the original fault signal using the energy operator. As shown in Figure 17d, by combining CTEO and VMD methods, the fault feature of the BIRF can be effectively identified. However, it is difficult to distinguish the fundamental  
Figure 16: Bearing fault signal enhanced by CTEO. (**a**) Inner ring. (**b**) Outer ring. Figure 15: Envelope spectrum of BORF signal. (**a**) CEEMD. (**b**) LMD. (**c**) ITD. (**d**) VMD. frequency from the bi-frequency of this feature, and there is still a large amount of the interference component mixed in the spectrum. Figure 18d shows a similar spectrum distribution to Figure 17d. The reasons for this mainly involve two aspects: (1) The early fault features are severely sparse and faint, nearly overwhelmed by background noise and interference components; (2) The VMD parameters are not optimal. Therefore, in this study, the ICFOA was used to obtain the most suitable combined values \([K,\alpha]\) of VMD, following the same procedure as described in Section 3. In addition, the parameters related to the chaotic mapping and FOA remained unchanged, except that the VMD update step needed to be manually set based on the standard deviation of the raw fault. Firstly, the optimal combination values of VMD searched by the ICFOA were \([K_{i},\alpha_{i}]=\) [13, 3175] and \([K_{o},\alpha_{o}]=\) [11, 2406], and the obtained parameters were saved for each. Then, the optimal approach was used to analyze the BIRF and BORF signals excited by the bearing in Figure 16a and b, respectively. The BIRF signal was decomposed into 13 IMFs (Figure 19a), and the corresponding kurtosis values of each IMF component were 20.1, 22.6, 20.8, 22.7, 22.2, 21.1, 23.1, 16.9, 18.7, 21.6, 24.2, 22.3, and 31.4. The mean kurtosis value is 22.1. The BORF signal was decomposed into 11 IMF components (as shown in Figure 20a), and the corresponding kurtosis values for each IMF component were 14.6, 18.7, 16.4, 20.8, 17.5, 16.2, 18.2, 16.5, 15.2, 17.2, and 24.5, with a mean kurtosis of 17.8. Then, the kurtosis of each IMF component was calculated, the corresponding IMF component with a kurtosis larger than the mean value was selected, and a fresh signal was reconstructed. The reconstructed representations are shown in Figures 19b and 20b. Finally, the envelope spectra of the two reconstructed signals were solved. The final effects are illustrated in Figures 19c and 20c. As shown in Figure 19c, the fault representation (7.17 Hz) and its second harmonics (14.34 Hz) of the BIRF reconstructed signal are effectively extracted, and the interference components are obviously reduced. Similarly, from Figure 20c, the fundamental frequency (4.88 Hz) and double frequency (9.76 Hz) of the reconstructed fault signal can be effectively observed, and the interference spectrum is also significantly reduced. The experimental evaluations demonstrate that the proposed approach can efficiently identify fault features under excessive load and low-speed operation. Additionally, it offers a fruitful means for incipient fault diagnosis of heavy-duty and low-speed rotation equipment. Figure 17: _Cont._  
Figure 17: The envelope spectrum of BIRF enhanced by CTEO. (**a**) CEEMD. (**b**) LMD. (**c**) ITD. (**d**) VMD. Figure 18: The envelope spectrum of BORF enhanced by CTEO.