# Denoised Improved Envelope Spectrum for Fault Diagnosis of Aero-Engine Inter-Shaft Bearing
## 3 Experimental Validation
### Case Study on HIT Inter-Shaft Bearing Fault Diagnosis
The inner race fault signal was recorded at an HP rotor speed of 6000 rpm and an LP rotor speed of 5000 rpm. For comparison, the SK-ES and IES methods are also implemented. The FFT spectrum of the original signals, along with the spectra obtained using SK-ES, IES, and the proposed DIES methods, are depicted in Figures 19a-d and 20a-d, respectively. Figure 19: Inter-shaft bearing fault case: (**a**) the FFT spectrum of the outer race fault signal from data5, (**b**) the spectrum generated using the SK-ES method, (**c**) the spectrum generated using the IES method, (**d**) the spectrum generated using the proposed DIES method. Figure 18: Inter-shaft bearing signal: (**a**) outer race fault signal from data5, (**b**) inner race fault signal from data4. For an inter-shaft bearing outer race fault signal, the FFT spectrum identifies the HP rotor frequency \(f_{hp}\) (61.22 Hz), as well as additional frequency components (646.973 Hz) that appear after the fault occurs. It is displayed in Figure 19a. As shown in Figure 19b, the SK-ES method cannot extract useful frequency information, as the fault characteristic frequency is masked by interference components and strong noise. Notably, the IES method effectively extracts multiple harmonics related to the characteristic frequency of the bearing outer race fault \(f_{o}\). The result is shown in Figure 19c. However, it also captures the high-pressure rotor frequency \(f_{hp}\) (61.22 Hz) and combination frequency \(10\times f_{lp}-f_{hp}\), both of which are considered interference components. Furthermore, the IES captures many random noise components, which might complicate the fault identification. Before applying the proposed method to this same outer race fault signal, it is worth noting that the baseline data under the same operating conditions are rarely available in the field of aviation. To solve this problem, the baseline healthy signal is generated by combining healthy signals from multiple operating conditions in this case. It can be observed in Figure 19d that the proposed DIES provides a clearer representation of the harmonics associated with the outer race FCF \(f_{o}\) compared to the IES method. The results demonstrate the superior performance of the proposed DIES in isolating and highlighting the fault-related harmonics from the strong background noise and dual-frequency excitation. Additionally, the DIES suppresses the frequencies of the deterministic interferential components that may also be present in the spectrum of a healthy inter-shaft bearing signal. For an inter-shaft bearing inner race fault signal, the FFT spectrum (Figure 20a) identifies the HP rotor frequency \(f_{hp}\) (100 Hz), its harmonics, and several prominent interfering frequency components. As shown in Figure 20b, the SK-ES method cannot extract useful  
Figure 20: Inter-shaft bearing fault case: (**a**) the FFT spectrum of the inner race fault signal from data4, (**b**) the spectrum generated using the SK-ES method, (**c**) the spectrum generated using the IES method, (**d**) the spectrum generated using the proposed DIES method. frequency information, as the fault characteristic frequency is masked by interference components and strong noise. Notably, the IES method (Figure 20c) effectively extracts the fifth harmonic \(5\times f_{i}\) related to the characteristic frequency of the bearing inner race fault. However, the IES also includes the interfering frequency components and random noise elements, which could hinder accurate fault diagnosis. Moreover, under the interference of dual-frequency excitation, the interference components exhibit a broadband distribution with relatively large amplitudes. In contrast, Figure 20d illustrates that the proposed DIES method outperforms the IES method. The DIES effectively identifies the fifth harmonic \(5\times f_{i}\) related to the bearing inner race FCF, while suppressing noise and interference.