# Mechanical Incipient Fault Detection and Performance Analysis Using Adaptive Teager-VMD Method
## 3 Simulated Signal Evaluation  
Bearings, as a key part of heavy-load mechanical equipment, are prone to pitting, corrosion, or cracking in long-term operation. When other parts make contact with or impact the defect location, a periodic shock signal is generated. The signals of rotating machinery measured by acceleration sensors are analyzed. When it fails, the fault signal is a sinusoidal signal with exponential decay. The sparsity and weakness are the significant characteristics of early faults in vibration signals. It is particularly susceptible to interference from strong noise or other ineffective signals, and no useful characteristic information reflecting the fault can be identified for fault diagnosis. In order to evaluate the reliability and superiority of our proposed method, a simulation case was used for analysis and verification as follows:  
\[y(t)=y_{s}(t)+y_{n}(t) \tag{14}\]  
where \(y_{s}(t)\) A is a sinusoidal decaying signal with a pulse period of 8 Hz, which is used to simulate the pulse response signal caused by the shock. \(y_{n}(t)\) denotes a white Gaussian noise. The SNR of \(y(t)\) is \(X_{SNR}\) = \(-\)20 dB. The simulated fault impulse \(y_{s}(t)\) of the bearing is defined by Equation (4). The parameters for fault simulation signal are specified as \(f_{r}\) = 1 Hz, \(f_{n}\) = 3000 Hz, \(C\) = 750 and \(A_{0}\) = 0.2. Moreover, the fault characteristic frequency \(f_{inner}\) = 1/\(T\) = 8 Hz, sampling frequency \(f_{s}\) = 12 kHz and data length L = 12,000. The original vibration signal and noise-added signal are shown in Figure 7a,b, respectively. When Gaussian white noise was applied to the pure pulses shown in Figure 7a, the simulated impulse signal was completely submerged by the interferences. The construct signal shown in Figure 7b was simulated to evaluate the actual early fault signal of the bearing properly. Then, the mixed signal was evaluated by CEEMD, LMD, VMD, and ITD, separately. The operation parameters of these algorithms were set according to the corresponding literature. Based on these methods, the fault characteristic frequency (theoretical value 8 Hz, actual value \(f_{1}\) = 8.057 Hz) and its harmonic in the frequency domain is shown in Figure 8.  
As shown in Figure 8a-d, the fault representation \(f_{1}\) and its doubling frequency \(2f_{1}\) corresponding to CEMD, LMD, ITD, and VMD are very blurry and low amplitude. Some larger interference frequency components are distributed in the dual frequency band. Compared with other methods, VMD has some advantages in characteristic amplitude, but it is also very weak. Moreover, some signal spectra are completely submerged. The experiments show that when the SNR is low, the above methods lack the ability to extract fault characterization directly. According to Equation (3), the mixed simulation signal is enhanced by CTEO (shown in Figure 9). The envelope spectrum corresponding to the enhanced signal is analyzed using the same signal processing method as in Figure 8. From the analysis results of the envelope spectrum in Figure 10, it can be seen that the amplitude of the envelope spectrum of fault signal enhanced by the energy operator is obviously improved. In particular, when the VMD method is used, the magnitudes of fundamental and double frequencies of fault  
Figure 7: Simulated signal of bearing failure. (**a**) Raw impulse. (**b**) The impulse with noise. representation are improved noticeably, and the effect is superior to other signal processing methods. However, interference signals in the spectrum are very clear. As shown by the analysis of the principle of VMD, its decomposition performance is significantly affected by parameters \(K\) and \(\alpha\). However, there is some blindness and contingency in selecting parameters by manual experience, and the performance of VMD is not guaranteed to be optimal. Hence, in order to obtain an optimal VMD and eliminate the invalid signal components in fault signals effectively, the proposed ICFOA is used to optimize the parameters.