## 3 Simulation and Experimental Verification
### Case 3: A Freight Locomotive Wheelset Bearing Signal with Compound Faults  
To further demonstrate the performance of the Mkurtogram, a compound fault experiment is carried out in this section. It is worth noting that the wheelset bearing defections  
Figure 14: Case 2: The results of Autogram (**a**) the Autogram; (**b**) the filtered signal; (**c**) the SES of (**b**). Figure 13: Case2: The results of FK (**a**) the FK; (**b**) the filtered signal obtained by FK; (**c**) the SES of (**b**). are not manufactured manually. They are natural damages generated during the operation of the railway freight vehicles, which are located in the outer and inner raceway, respectively. The experimental setup and the tested wheelset bearing are shown in Figure 15, respectively. Parameters of the bearing are listed in Table 2.  
The sampling frequency \(F_{\text{s}}\) is 25.6 kHz and the set speed is 465 r/min. Then, the bearing fault characteristic frequencies are given in Table 3. The measured time domain waveform is shown in Figure 16. In this case, the corresponding outer race fault period is \(T_{\text{o}}\) = 383.5 and the inner race fault period is \(T_{i}\) = 290, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline
**Bearing Spaces** & **Roller Diameter** & **Pitch Diameter** & **Contact Angle** & **Roller Number** \\ \hline
197,726 & 24.74 mm & 176.29 mm & 8.833 deg & 20 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of the wheelset bearing. Figure 15: Case 3: (**a**) experimental setup; (**b**) a fatigue spall fault on the inner raceway; (**c**) the slight dent fault on the outer raceway. \begin{table}
\begin{tabular}{c c c c} \hline \(f_{\text{r}}\) & \(f_{\text{o}}\) & \(f_{i}\) & \(f_{\text{rb}}\) \\ \hline
7.75 Hz & 66.75 Hz & 88.25 Hz & 27.08 Hz \\ \hline \end{tabular}
\end{table} 
Table 3: Bearing characteristic frequencies. In this case, we use the proposed method to identify the wheelset bearing compound faults. Firstly, based on the outer race fault period \(T_{\rm o}\), the Mkurtogram is carried to detect the outer race fault. The IFB identified by Mkurtogram is displayed in Figure 17a. It can be seen that the center frequency \(f_{\rm c}=10\) kHz with bandwidth \(B_{w}=800\) Hz has the highest Mkurt. Then, the filtered signal generated from the IFB and its corresponding SES are calculated as plotted in Figure 17b,c. As shown in Figure 17b, the fault-induced repetitive transients can be observed obviously in the filtered signal. From Figure 17c, it can be found that the outer race \(\mathrm{FCF}f_{\rm o}\) is dominated in the SES. Therefore, the proposed Mkurtogram has successfully detected the outer race fault. Then, based on inner race fault period \(T_{i}\), the Mkurtogram is taken to detect the inner race fault. The IFB determined by Mkurtogram is plotted in Figure 18a, where the center frequency \(f_{\rm c}=4800\) Hz and bandwidth \(B_{w}=3200\) Hz are determined. Then, the filtered signal generated from the IFB and its corresponding SES are plotted in Figure 18b,c. From the filtered signal in Figure 18b, the repetitive transients generated by the bearing defect are seen clearly. In addition, from the SES presented in Figure 18c, the first seven harmonics of inner race \(\mathrm{FCF}f_{\rm i}\) can be identified and their multiple side-bands are also dominated in the SES. As a result, the inner race fault is also identified by Mkurtogram. Figure 16: Case 3: The bearing time domain waveform. Figure 17: Case 3: The results of Mkurtogram based on the outer race fault period \(T_{\rm o}\). (**a**) the Mkurtogram; (**b**) the filtered signal; (**c**) the SES of (**b**). To compare the proposed Mkurtogram with FK and the Autogram, the above vibration signal is processed by the two approaches, and the results are given in Figure 19. In this case, we can see that the FK and the Autogram find a same IFB (\(f_{\rm c}\) = 10 kHz and \(B_{w}\) = 800 Hz). By contrast, it can be seen that the IFB determined by FK and Autogram is also identical with Mkurtogram as displayed in Figure 17a.