## 4 Engineering Application  
The bearing inner race and rolling element fault data from the Western Reserve University Experimental Center were processed separately to illustrate the effectiveness of the CMSR system [48]. The experimental platform is shown in Figure 6. The 6205-2 RS JEM SKF deep groove ball bearing is employed in the experiment. The motor is connected to a dynamometer and torque sensor by a self-aligning coupling. The data were collected from accelerometers on the motor housing at the drive end of the motor. The vibration signals were measured under no-load (0 hp) at a rotating speed of 1797 r/min. Faults were introduced into the test bearings by using an electro-discharge machining method. The fault diameter was 0.007. The vibration signals of the motor bearing were sampled at the frequency of 12,800 Hz and the duration of each vibration signal was 10 s. The original vibration signals were divided into segmentations of samples, and each sample covered 2048 datapoints. According to the theoretical calculation values of fault characteristic frequency, the faults of the outer race, inner race, and rolling element of motor bearing are determined. As a comparison, the experimental signal is denoised by a MSR system. Size parameters of bearings are shown in Table 1. The bearing inner race and rolling element fault characteristic frequency is calculated as follows  
\[f_{BPFI}=\frac{n}{2}(1+\frac{d}{D}\cos\alpha)f_{r} \tag{14}\]  
\[f_{BSF}=\frac{D}{2d}\Bigg{[}1-\bigg{(}\frac{d}{D}\cos\alpha\bigg{)}^{2}\Bigg{]} f_{r} \tag{15}\]  
where \(f_{BPFI}\) and \(f_{BSF}\) are the fault characteristic frequencies, \(\alpha\) is the angular contact angle, \(d\) is the roller diameter, \(D\) is the bearing pitch diameter, the number of rollers is \(n\), and the frequency shift is \(f_{r}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Inner** & **Outer Ring** & **Pitch** & **Roller** & **Roller** & **Angular** \\
**Diameter** & **Diameter** & **Diameter** & **Diameter** & **Roller** & **Conact** \\
**(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **Angle** \\ \hline
25.001 & 51.999 & 39.04 & 7.94 & 9 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Size parameters of bearings. Figure 6: The experimental platform. Firstly, the bearing inner race fault is analyzed. The theoretical value of characteristic frequency is 162.2 Hz according to Equation (14) when motor speed is 1797 r/min, sampling length is 8192, and sampling frequency is 12 kHz. For the original vibration signal, Figure 7 is the time-domain waveform and power spectrum. In the power spectrum, there is a significant resonance band at 2000-4000 Hz, and no useful fault characteristic information can be obtained. After bandpass filtering (the filter segment is the resonance band) and envelope processing, the resulting envelope diagram is shown in Figure 8b. Although we can see the fault characteristic frequency from its amplitude spectrum, it cannot be effectively judged by the interference of surrounding noise, and the _SNR_ is \(-18.85\) dB. Figure 8c is the result of using the MSR to process the envelope signal. At this time, the output _SNR_ is improved to \(-16.3106\) dB, system parameters are \(a=84.6\), \(b=0.113\), \(R=1000\), and the high-frequency noise component is restrained, but the noise component of low-frequency is increased to affect the judgment. Figure 7 displays the output result of processing the envelope signal using the CMSR algorithm. The signal in time-domain waveform can be found to be periodic and characteristic frequency \(f_{BPFI}\) can be clearly discerned in the amplitude spectrum. The system parameters at this time are \(a=-0.19\), \(b=2.28\), \(r=-3.13\), \(R=545.96\), and output _SNR_ is \(-5.27\) dB. The feasibility of the CMSR algorithm can be proven by bearing inner race fault data analysis results. In order to continue to prove the effectiveness of the proposed method for bearing rolling element fault signal, the following experiment must be carried out.