## 2 Vibration Signal Processing Method Based on VMD
### Fault Simulation Experiment  
The real vibration signal of gear bearing usually contains three signal components--impact signal, noise signal, and harmonic interference signal--in order to verify the effectiveness of VMD decomposition for fault signals. The simulation signal \(f(t)\) of rolling bearing outer ring local damage fault is constructed as follows:  
\[\left\{\begin{array}{l}f(t)=x_{1}(t)+x_{2}(t)+x_{3}(t)\\ x_{1}(t)=2 exp(-at_{0})cos(2\pi f_{1}t)\\ x_{2}(t)=sin(2\pi f_{2}t)+sin(2\pi f_{3}t)\end{array}\right. \tag{10}\]  
It can be observed from the formula that the simulation signal \(f(t)\) is composed of \(x_{1}(t)\), \(x_{2}(t)\), and \(x_{3}(t)\), where \(x_{1}(t)\) is the simulation signal of impact attenuation caused by fault, which is represented by a pulse sequence with periodicity that obeys the exponential attenuation law. The amplitude of \(x_{1}(t)\) is 2, \(t_{0}=mod(k/f_{s},1/f_{m}),k=0,1,\cdots,2047\), the sampling frequency \(f_{s}\)= 2048 Hz, the number of sampling points \(N=2048\), the attenuation coefficient \(a=100\), the carrier frequency \(f_{1}=200\) Hz, and the fault feature frequency \(f_{m}=16\) Hz. \(x_{2}(t)\) is the sine superposition signal of frequency \(f_{2}=20\) Hz and \(f_{3}=30\) Hz, which is used to simulate harmonic interference. \(x_{3}(t)\) is the Gaussian white noise of mean value \(\overline{x}=0\) and standard deviation \(\sigma=0.1\), which is used to simulate the background noise. As shown in Figure 3, the time-domain waveforms of \(f(t)\), \(x_{1}(t)\), \(x_{2}(t)\), and \(x_{3}(t)\) are given, respectively.  
Figure 4 shows the time-domain waveform and spectrum of simulation signal \(f(t)\), and the obvious fault feature frequencies cannot be obtained from both the time-domain and frequency-domain perspectives. In the spectrum diagram, obvious harmonic interference components of 20 Hz and 30 Hz can be observed while the fault feature frequency \(f_{m}=16\) Hz is submerged in the interference signal and noise. The envelope waveform and envelope spectrum of simulation signal \(f(t)\) obtained based on Hilbert transform are shown in Figure 5. It is impossible to identify accurately whether there is a fault from the envelope waveform. In the envelope spectrum, there are both 16 Hz feature frequency with small amplitude and 10 Hz interference component with large amplitude in the envelope spectrum. The low amplitude spike around 200 Hz in the figure is a resonance frequency band.  
Figure 3: Time-domain waveform of rolling bearing fault simulation signal \(f(t)\) and its components.  
VMD decomposition result of simulation signal \(f(t)\) (the best parameter combination obtained after optimization by IPSO-VMD is \([\hat{K},\hat{\pi}]=[3,5500]\)), as shown in Figure 6, IMF1 perfectly corresponds to the harmonic interference \(x_{2}(t)\) in the original signal, and the periodic shock attenuation can be observed in IMF2. The corresponding fault characteristic frequency 16 Hz and its frequency multiplication components (32 Hz, 48 Hz, etc.) can be clearly identified from the IMF2 and IMF3 components. Therefore, VMD can effectively decompose the \(x_{1}(t)\) and \(x_{2}(t)\) components of the rolling bearing fault simulation signal.  
Figure 4: Simulation signal \(f(t)\) time-domain waveform and its spectrum.  
Figure 5: Envelope waveform and envelope spectrum of simulation signal \(f(t)\).  
Figure 6: VMD decomposition results of rolling bearing fault simulation signal \(f(t)\).