## 3 Simulation and Experimental Verification
### Case 2: A High-Speed Locomotive Wheelset Bearing Fault Signal under Variable Load Conditions  
In this case, the vibration signal has been collected from a high-speed locomotive wheelset bearing under variable load conditions, which is carried out to further testify the validity of this proposed method. The test rig structure is presented in Figure 8. The basic experimental setups and the test bearing are depicted in Figure 9. The sensor is located at the top of the end-shield of the test bearing in Figure 9b and the vibration signal is collected by a PCB 356A25 accelerometer. The dynamic loads can be obtained by the radical and axial actuators. The structure parameters of the bearing are listed in Table 1. There is an artificial local defect in the outer race of test bearing as plotted in Figure 9d, of which the width is 1 mm and length is 5 mm. It can be noted that the artificial defect is relatively slight in comparison with its geometries. To simulate the load changes during the operation process of a wheelset bearing, the random force is added to the vertical direction with a mean force of approximately 80 kN and a frequency range among 0.2 to 20 Hz. A random force with a maximum of 50 kN and a frequency range among 0.2 to 20 Hz is applied on the axis. The dynamic loading forces generated by the actuator are depicted in Figure 10. The sampling frequency \(F_{\text{s}}\) is set as 51.2 kHz and the set speed is 2 000 r/min. The measured time domain signal of the axle-box under the variable load conditions is presented in Figure 11. It can be seen that the amplitudes of the vibration signal change dramatically due to the variable loads. Therefore, this example can be used as an excellent experimental case to testify the effectiveness of \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Spaces** & **Roller Diameter** & **Pitch Diameter** & **Contact Angle** & **Roller Number** \\ \hline FAG F-80781109 & 26.5 mm & 185 mm & 10 deg & 17 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the bearing. Figure 7: Case1: The results of Autogram (**a**) the Autogram; (**b**) the filtered signal obtained by Autogram; (**c**) the SES of (**b**). the Mkurtogram for the selection of IFB under the influence of variable load conditions. The outer race FCF \(f_{\rm o}\), inner race FCF \(f_{\rm i}\) and rolling element FCF \(f_{\rm fb}\) can be calculated as 243.4 Hz, 323.3 Hz, and 114 Hz, respectively. Based on the speed, its corresponding outer race fault period \(T_{\rm o}\) = 210.5 can be calculated. The Mkurtogram is presented in Figure 12a, where optimal center frequency \(f_{\rm c}\) = 13.2 kHz and bandwidth \(B_{w}\) = 800 Hz are found. Then, the optimal IFB is taken to filter the vibration signal. From the SES depicted in Figure 12c, we can find that the outer race FCF \(f_{\rm o}\) and its multiple harmonics are clearly, which indicates that the test wheelset bearing has an outer race fault. Hence, in this case, the result indicates that the Mkurtogram has the ability to identify weak outer race fault even when the amplitudes of the vibration signal change dramatically due to the applied variable loads. Figure 8: Case 2: The structure of the test bench. Figure 9: Case 2: (**a**) railway bearing test bench overview; (**b**) the local enlarged test point; (**c**) the test wheelset bearing; (**d**) the partial enlarged view of fault. For better comparison, the same vibration data is processed by the FK and the Autogram, and the results are given in Figures 13 and 14, respectively. The center frequency \(f_{\rm c}\) = 12 kHz and bandwidth \(B_{w}\) = 1600 Hz are determined from FK as displayed in Figure 13a. There is  
Figure 11: Case 2: Time domain signal. Figure 10: Case 2: Dynamic loading forces; (**a**) the axis direction; (**b**) the vertical direction. an obvious interfering impulse in the filtered signal as plotted in Figure 13b. From the SES in Figure 13c, we can see that only the outer race \(\text{FCF}\,f_{\text{o}}\) is found and the background noise interference is obvious.