## 5 Application to Rotating Machinery Fault Diagnosis
### Application to Fan Gearbox Fault Diagnosis  
Figure 15 shows the structure of the No.3 fan gearbox of a petrochemical company, in which the teeth number of gears 1, 2, 3, and 4 are 11, 31, 23, and 53, respectively. The rotational speed of shaft I is \(n_{\rm I}=980\) r/min, namely, the rotating frequency of shaft I is about \(f_{\rm I}=16.3\) Hz, the rotating frequency of shaft II is about \(f_{\rm II}=5.8\) Hz, and the rotating frequency of shaft III is about \(f_{\rm III}=2.5\) Hz. One day, the online monitoring system found that the vibration of the no.3 fan gearbox was abnormal. The root means square value of the vibration acceleration signal (on monitoring point A shown in Figure 15) was relatively bigger than the threshold value, but the fan running was not affected, and the date was close to the normal shutdown maintenance date, so keep observation. During the subsequent shutdown maintenance, the fan gearbox was examined. It was found that four gears of gearbox had different degrees of wear and tooth surface glue, especially the wear of gear 1 and gear 2 were very serious, which can be shown in Figure 16.  
In order to verify that Hilbert envelope spectrum based on RS-LOD can accurately identify the fault condition of the gearbox, a vibration acceleration signal (the sampling frequency is 1024 Hz) measured on the same day is selected for analysis, and which time-domain waveform and amplitude spectrum are as shown in Figures 17 and 18. By preliminary analysis, the main frequency components include the mesh frequency of gear 1 and gear 2, the mesh frequency of gear 3 and gear 4, and many other unknown frequencies. Moreover, the sidebands of the gear mesh frequency are very difficult to distinguish. Therefore, it is hard to estimate whether and where the fault has occurred in the gearbox from the amplitude spectrum in Figure 18. Further, the envelope demodulation technology based on the RS-LOD is used to identify the condition of the gearbox. According to the manner in Section 5.1, we can determine the value of \(p\) in this decomposition is 9. A total of ten MOC components are  
Figure 14: The Hilbert envelope spectrum of the \(\rm{MOC}_{1}(t)\) obtained by the RS-LOD.  
Figure 15: The no.3 fan gearbox structure schematic diagram.  
In Figure 20a, it can be seen that there is a spectral line with high amplitude at the double frequency of rotating frequency \(f_{1}\). In Figure 20b, there are several obvious spectral lines, which at the 1, 2, 3, 4 times of rotating frequency \(f_{1}\), and at the 1, 2 times of rotating frequency \(f_{11}\). In Figure 20c. It can be seen at the rotating frequency \(f_{1}\) and 1, 2 times of rotating frequency \(f_{11}\) there are clear spectral lines. According to the distribution and amplitude of above spectral lines, it can be determined that the spectral line width is \(\sim\)100 nm.  
Figure 19: RS-LOD decomposition results of the fan gearbox vibration acceleration signal (\(p=9\)).  
Figure 20: The Hilbert envelope spectrum of the MOC components: (**a**) Hilbert envelope spectrum of \(\text{MOC}_{1}(t)\); (**b**) Hilbert envelope spectrum of \(\text{MOC}_{2}(t)\); (**c**) Hilbert envelope spectrum of \(\text{MOC}_{3}(t)\).  
there are certain degree faults in all four gears on three shafts, and the faults on gear 1 and gear 2 are more serious, while the faults on gear 3 and gear 4 are relatively slight. It is consistent with the actual gearbox check, which confirmed the effectiveness of the proposed diagnosis method.