# Mechanical Incipient Fault Detection and Performance Analysis Using Adaptive Teager-VMD Method
## 4 Fault Experiment Analysis  
In this section, the feasibility of the introduced method on the actual measurement rig is further verified. As illustrated in Figure 12a, the bench simulates the radial load by applying different radial load forces to the rolling bearing through a hydraulic lever. In addition, the rolling bearing operating speed is adjusted by controlling the motor speed. Figure 11: Processing results of ICFOA-based CTEO-VMD method. (**a**) Convergence curves for the searching process by five algorithms. (**b**) The envelope spectrum of the reconstructed signal. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{5}{c}{**Method**} \\ \cline{2-6}  & **PSO** & **FOA** & **CPSO** & **CFOA** & **ICFOA** \\ \hline Fitness & 3,318 & 2.055 & 1.656 & 0.7217 & 0.2852 \\ \([K,\,\alpha]\) & \([4,1714]\) & \([4,2045]\) & \([5,2865]\) & \([5,3065]\) & \([6,3207]\) \\ iterations & 85 & 38 & 37 & 44 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results obtained by different optimization algorithms. The rolling bearings are shown in Figure 12b,c. Among them, the diameter of the rolling bearing is 39.5 mm, the diameter of the rolling element is 7.5 mm, the number of rolling elements is 12, and the angle of the internal contact surface is 0 degrees. In order to effectively evaluate the operating conditions of low speed and heavy load, the radial load force applied here was set at 5000 N. The rotation speed of the inner ring is 60 r/min, and the vibration signal sampling rate is 10 kHz. Depending on the parameter calculation reference, the theoretical values of outer rings and inner rings can be calculated, and the fault signatures are listed in Table 3. The measured vibration signals of the bearing inner ring fault (BIRF) and bearing outer ring fault (BORF) are shown in Figure 13.  
Figure 12: The experimental rig and defect diagrams. (**a**) Experimental rig. (**b**) Inner ring defect. (**c**) Outer ring defect. Figure 13: Vibration signals from rolling bearing fault. (**a**) BIRF. (**b**) BORF. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Speed (r/min)** & **BIRF** & **BORF** \\
**(f/Hz)** & **(f/Hz)** & **(f/Hz)** \\ \hline
60 & 7.14 & 4.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Rolling bearing fault characteristic frequencies. To evaluate the availability of our method, the signals of identical bearings with inner and outer ring faults were analyzed using standard CEEMD, LMD, ITD, and VMD methods. Figures 14 and 15 show the spectra of BIRF and BORF processed by these methods, respectively. As shown in Figures 14 and 15, the spectral lines of fault characteristics (outer ring \(f_{\rm outer}\) and inner ring \(f_{inner}\) are 4.88 Hz and 7.17 Hz, respectively) obtained by CEEMD, LMD, ITD, and VMD are fully overwhelmed by additional irrelevant interference spectral lines. Due to the sparseness and weakness of bearings and early fault signals under low-speed and heavy-load conditions, it is hard to use the above signal processing methods to find useful information for diagnosis. In this study, CTEO was used to preprocess the early fault signal of the bearing to increase the weak impulse component in the raw signal. From Figure 16a,b, it can be observed that the CTEO effectively enhances the impulse components in the fault signal. Then, standard CEEMD, LMD, ITD, and VMD were applied to analyze the enhanced signal, and the envelope spectra are presented in Figures 17 and 18. Among them, Figure 17a-d show the envelope spectra of the inner ring fault signal processed by CEEMD, LMD, ITD, and VMD, respectively. As seen from the spectrum magnitude, the amplitudes of inner ring fault characteristic frequency in Figure 17 are obviously larger than those in Figure 14. Figure 18a-d demonstrate the envelope spectra of the outer ring fault signal with CEEMD, LMD, ITD, and VMD, respectively. The amplitude of each outer ring fault characteristic frequency in Figure 18 is significantly larger than that in Figure 15, too.