## 4 Experimental Data Analysis
### Case.2 Application to Time-varying Feature Extraction of Bearing  
The experimental data are collected from the rolling bearing fault simulation bench on the Spectra Quest machinery fault simulator (MFS-PK5M) [34; 35], and the whole device is driven by a variable frequency speed-regulating motor, as shown in Figure 13. The test bench mainly includes tachometers, the drive control system and the rotor-bearing system. There are two ER16 K rolling bearings at each end of the rotor. A special note is that the left bearing is in good condition while the right one is replaceable. Thus, the right one is selected as faulty bearings with inner fault. Structural parameters of the researched rolling bearing are described in Table 2. Theoretically, the Fault Characteristic Coefficient (FCC) corresponding to the outer race fault (FCC\({}_{0}\)) and the inner race fault (FCC\({}_{\text{I}}\)) can be respectively calculated as 3.57 and 5.43. To verify the effectiveness of this method in time-varying conditions, the operating rotational speed is increased from 13 Hz to 25.7 Hz. Therefore, according to the rotation frequency and FCC, we can accurately calculate the fault characteristic frequency. Figure 11: The result provided by proposed method using RV. Figure 12: The result provided by EMD. Then, the parametric time-frequency analysis methods, such as STFT and SET, are used to process the fault bearing signal for time-varying fault feature identification. Obviously, Figure 16 shows the TF analysis result generated by STFT, which has a low TF resolution. Fuzzy TF plane directly affects the readability. The main reason for this is the lack of post-processing for the TF transformation coefficient. Based on the TF reassignment framework and the classical SST theory, SET has an improvement in the performance of TF representations compared with STFT. The trend of time-varying characteristic changing can be obtained in Figure 17. However, the energy of the characteristic curve is not concentrated enough, and many independent components exist in the TF plane. Finally, the proposed method in this paper is conducted to bearing fault signal analysis and the corresponding result is drawn in Figure 18. It is evident that the proposed method based on TF denoising scheme and reassignment vector has a better performance. The time-varying rotation frequency \(f_{i}\), \(2f_{i}\) and \(3f_{i}\) can be identified in the TF plane. From this, the conclusion can be drawn that the rolling bearing has an inner fault. Meanwhile, the result provided by EMD is plotted in Figure 19 and the concerned time-varying characteristic cannot be detected. In a summary, through the analysis of rolling bearing fault data, the validity of the proposed signal decomposition algorithm from the TF plane is illustrated. Moreover, to further illustrate the effectiveness of the proposed method in dealing with various types of faults, fault signal analysis of bearing outer race under time-varying speed in the above Spectra Quest machinery fault simulator has also been conducted. The vibration signal is measured from a faulty bearing with an outer race defect and the operating rotational frequency \(f_{\tau}\) increases from 14.0 Hz to 21.7 Hz. Similarly, according to the rotation frequency \(f_{\tau}\) and Fault Characteristic Coefficient (FCC), we can accurately calculate the time-varying outer race fault characteristic frequency \(f_{\circ}\)[36]. Figure 20 shows the TFR obtained by STFT and the proposed method. Due to deficiency of STFT in dealing with strong frequency modulation signal (especially in the noisy environment), time-frequency blurs occur in the result generated by STFT in Figure 20a. Fortunately, the proposed method has suggested higher resolution of ridges, compared with STFT in time-frequency plane in Figure 20b. The varied rotational frequency \(f_{\tau}\) and outer race fault characteristic frequency \(f_{\circ}\) can both be identified in the TF plane. Therefore, we can diagnose the outer race fault, which is consistent with the actual situation.