# Mechanical Incipient Fault Detection and Performance Analysis Using Adaptive Teager-VMD Method
## 2 Proposed Methods
### Fault Feature Extraction  
The critical component bearings of large, low-speed, and heavily loaded machinery are vulnerable to damage. Therefore, when an early micro-fault occurs in a bearing (i.e., an inner ring defect and an outer ring defect), the energy in the collected vibrational signal representing the shock component of the fault is severely inconspicuous. Effective information is difficult to obtain using common time-frequency analysis methods from raw vibrational signals. Therefore, CTEO was used to enhance the raw vibrational signal in this study. Then, the optimal VMD was used to decompose it to obtain several IMF components, and the kurtosis value of each IMF component was calculated. The mean kurtosis criterion was then used to screen the IMF components, and the selected IMF components were reconstructed into the current signal [40]. Finally, if the bearing of the mechanical equipment fails (such as the inner ring or the outer ring), the vibration signal  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**Algorithm**} & **Local Optimal** & **Global Optimal** & **Iteration Number** & **Repeat Times** \\ \hline \multirow{4}{*}{\(f_{1}(x)\)} & PSO & 26 & 74 & \multirow{4}{*}{200} & \multirow{4}{*}{100} \\  & FOA & 19 & 81 & \\ \cline{1-1} \cline{3-5}  & CPSO & 13 & 88 & \\ \cline{1-1} \cline{3-5}  & CFOA & 12 & 88 & \\ \cline{1-1} \cline{3-5}  & ICFOA & 2 & 98 & \\ \hline \multirow{4}{*}{\(f_{2}(x)\)} & PSO & 34 & 66 & \multirow{4}{*}{200} & \multirow{4}{*}{100} \\  & FOA & 32 & 68 & \\ \cline{1-1} \cline{3-5}  & CPSO & 26 & 74 & \\ \cline{1-1} \cline{3-5}  & CFOA & 22 & 78 & \\ \cline{1-1} \cline{3-5}  & ICFOA & 7 & 93 & \\ \hline \multirow{4}{*}{\(f_{3}(x)\)} & PSO & 28 & 72 & \multirow{4}{*}{15} \\  & FOA & 24 & 76 & \\ \cline{1-1} \cline{3-5}  & CPSO & 17 & 83 & \\ \cline{1-1} \cline{3-5}  & CFOA & 14 & 86 & \\ \cline{1-1} \cline{3-5}  & ICFOA & 4 & 96 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Statistical results of global optimal solutions on three different benchmark functions.  
collected will contain a specific frequency impact component. The proposed method is then used to decompose and reconstruct the vibration signal. Then, the envelope spectrum of the reconstructed signal is analyzed, and the characteristic frequency of the corresponding fault is extracted from the envelope spectrum. The fault characteristics and extraction flow is demonstrated in Figure 6. Moreover, the specific steps are listed below:  
Step 1: The impact component of the fault signal is enhanced by CTEO.  
Step 2: The relevant parameters of the 3D-LSCCM are initialized and an appropriate indicator function is designed.  
Step 3: The preset values of the fruit fly population are initialized. The binding group \([K,\alpha]\) of the VMD corresponds to the location of the individual fly.  
Step 4: Decompose the signal by VMD under different individual positions of flies, and calculate the fitness of each location.  
Step 5: The fitness values of the individuals in the population are compared, and the optimal evaluation values of the individuals and the population are updated.  
Step 6: Update the position of the individual flies by using Equations (7) and (8).  
Step 7: Repeat the iterative procedure of steps 4-6. When the iterative number reaches the maximum set value, the optimal parameters determined by the evaluation function are recorded.  
Step 8: Use the obtained parameter combination \([K,\alpha]\) to construct the optimal VMD.  
Step 9: Calculate the kurtosis of every independent mode and find the mean of all their kurtosis values. Based on the mean value, IMF components with kurtosis larger than the mean are chosen and reconstructed to obtain fresh representation.  
Step 10: The envelope spectrum of the new representation is calculated and analyzed. Afterward, it is matched to the fault feature to determine the corresponding defect type.  
Figure 6: Fault diagnosis process using proposed CTeager-VMD.