## 4 Engineering Applications  
By analyzing the bearing experimental data of Case Western Reserve University [58], we further confirmed the feasibility of the SCSSR method. The experiment uses a deep groove ball bearing, whose type is 6205-2 RS JEM SKF (SKY FU (SHANGHAI) RESPONSIBILITY LIMITED, Shanghai, China). Table 1 shows the dimensional parameters of the bearing. In comparison, since the FSR needs to set the filter, the operation is cumbersome. This paper only analyzes the data with CSR and USSSR.  
The acceleration sensor is used to gather a vibration signal; the sampling number is 4096, the sampling frequency is 12 kHz, and 1730 r/min is the motor speed. The inner ring theory fault characteristic frequency is 156.14 Hz. The time-domain waveform and power spectrum of the bearing inner ring fault signal collected by the experiment are shown in Figure 4a. We are not able to discover the periodic component in the time-domain waveform, and from the power spectrum, we are not able to observe the inner ring fault characteristic frequency \(f_{BPFI}\). By bandpass-filtering the original signal, the filtered signal is subjected to the HT transform to acquire an envelope signal and an envelope  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Inner Diameter** & \
\begin{tabular}{c} **Outer Ring** \\ **Diameter** \\ \end{tabular} & **Number of Rollers** &
\begin{tabular}{c} **Rolling Element** \\ **Diameter** \\ \end{tabular} & **Pitch Distance** \\ \hline
2.5001 cm & 5.1999 cm & 9 & 0.794 cm & 3.904 cm \\ \hline \hline \end{tabular}
\end{table}
Table 1: Size parameters of 6205-2 RS.  
Figure 3: Analysis results for the simulation signal: (**a**) original signal; (**b**) classical system parameters (CSR) output signal; (**c**) underdamped step-varying second-order (USSSR) output signal; (**d**) second-order coupled step-varying stochastic resonance (SCSSR) output signal; (**e**) SCSSR output signal when a is changed.  
amplitude spectrum, as displayed in Figure 4b. We can see the inner ring fault characteristic frequency \(f_{BPFI}\), from the envelope amplitude spectrum, but it is difficult to judge the fault due to the interference of the surrounding noise and the frequency conversion \(f_{r}\). The SNR is \(-17.53\) dB from Equation (6). Figure 4c is the result of using the CSR system to process the envelope signal. In this figure, the energy of high frequency is lower, and the feature of the inner ring fault frequency \(f_{BPFI}\) and the frequency conversion \(f_{r}\) is amplified. The system parameters are shown below: \(a\) is 0.0015; \(b\) is 5235.68; \(R\) is 672.536; the output SNR is \(-15.511\) dB. After the data are processed by the USSSR system, the system parameters are \(a\) = 0.908, \(b\) = 12476.8, \(R\) = 304.395, \(k\) = 0.222; the output results are shown in Figure 4d, the SNR is reduced to \(-10.42\) dB, and \(f_{BPFI}\) can be clearly found in the amplitude spectrum. Meanwhile, the output SNR comparison CSR system is further enhanced. If the SCSSR algorithm is used to process the envelope, the outputs are shown in Figure 4e. The optimal parameters are \(a\) = 4.5, \(b\) = 1.6, \(k\) = 0.9, \(r\) = \(-4.3\), \(R\) = 556.51, and the output SNR is \(-2.636\) dB. The output signal of the \(y\) channel effectively enhances the \(x\) channel output, and the frequency \(f_{BPFI}\) is clearly visible. Once again, the bearing fault data analysis results demonstrate the feasibility and superiority of the SCSSR method.  
Figure 4: \(Cont.\)