# Research on a Noise Reduction Method Based on Multi-Resolution Singular Value Decomposition
## 3 Analysis
### Instance Verification  
This paper takes the rolling bearing experiment of Case Western Reserve University as an example to diagnose the fault bearing. The bearing used in the experiment is 6205-2 RS JEM SKF deep groove ball bearing, the motor power is 2 HP, the rotating frequency is 1730 r/min, the sampling frequency is 12 KHz, and the number of sampling points is 1024. Without affecting the normal service performance of the bearing, a small groove with a diameter of 0.007 inches is machined on the bearing outer ring to simulate a local bearing crack, the acceleration sensor at the driving end collects the signals of the bearing outer ring.  
The formula for calculating the defect frequency of outer ring of rolling bearing is:  
\[f_{\infty}=\frac{1}{2}Z(1-\frac{d}{D}cos\alpha)f_{r} \tag{6}\]  
It can be calculated from Equation (6), where \(Z\) is the number of rolling elements, d is the diameter of rolling elements, D is the pitch diameter, \(\alpha\) is the contact angle of bearing, and is the rotation frequency. The signal and power spectrum of the bearing outer ring during faulty operation are shown in Figure 9.  
It can be seen from Figure 9 that the signal time-domain waveform is very complex, and it is difficult to see the existence of periodic shock. From the frequency domain, the frequency band is mainly distributed in the middle frequency band, and it is difficult to determine the occurrence of shock.  
MRSVD is used to decompose this signal into 5 layers, and the results are shown in Figure 10. It can be seen that in the detail signal, especially in the first two detailed signals, a very obvious periodic impact was obtained. Based on this, it can be determined that the bearing's raceway is damaged. However, since the damage is not serious, the impact in the original signal is not obvious and it is difficult to confirm.  
Figure 8: Index changes of various models under different noise powers.  
In order to more clearly and intuitively see the frequency of outer ring faults, an envelope spectrum analysis is required. The above-mentioned decomposition methods are used to process the outer ring vibration signal of the rolling bearing, and the component with the largest kurtosis value is selected for envelope spectrum analysis to obtain Figure 11. Table 6 shows the component selection when the kurtosis of each model is maximum.  
As shown in Figure 11, it can be seen that several signal processing methods approximately locate the outer ring defect frequency of the bearing at 105.5 hz (theoretical value is 107.36 hz), EMD and VMD at the basic frequency, EEMD at the second octave, wavelet decomposition, SVD and MRSVD  
\begin{table}
\begin{tabular}{c c c c c c c} \hline Noise reduction model & EMD & EEMD & VMD & Wavelet decomposition & SVD & MRSVD \\ \hline Maximum kurtosis component & IMF1 & IMF3 & IMF3 & d1 & / & D2 \\ \hline \end{tabular}
\end{table}
Table 6: Component selection for maximum kurtosis of various noise reduction models.  
Figure 10: MRSVD breakdown detail signal.  
Figure 9: Fault operation signal and spectrum of bearing outer ring.  
not only at the basic frequency but also at the seventh octave. The occurrence of high-order harmonics may be the asymmetry or looseness of the inner and outer rings in the bearing Situation. Although the amplitude of the basic frequency extracted by other methods is large, it is still greatly disturbed by noise. Only the envelope spectrum of MRSVD effectively reduces the noise interference, highlights the impact characteristics of the fault, and has the highest diagnosis accuracy. In conclusion, MRSVD can be used to accurately identify the fault type of bearing after noise reduction.