# Denoised Improved Envelope Spectrum for Fault Diagnosis of Aero-Engine Inter-Shaft Bearing
## 3 Experimental Validation
### Case Study on Incipient Bearing Fault Diagnosis  
In the first case study, a publicly accessible run-to-failure vibration dataset was employed for validation purposes. The experimental setup is depicted in Figure 10. The rotation speed was kept constant at 2000 RPM by an AC motor coupled to the shaft via rub belts [30]. The bearing type is ZA-2115 (Regal Rexnord Corporation, Milwaukee, USA). The installation location of this bearing is displayed in Figure 10. The structural and operational parameters of it are presented in Table 1[22]. According to the following calculation formulas for bearing characteristic frequencies:  
\[f_{r}=\frac{RPM}{60} \tag{13}\]  
\[f_{I}=\frac{n}{2}\bigg{(}1+\frac{d}{D}\cos\phi\bigg{)}f_{r} \tag{14}\]  
\[f_{o}=\frac{n}{2}\bigg{(}1-\frac{d}{D}\cos\phi\bigg{)}f_{r} \tag{15}\]  
\[f_{B}=\frac{D}{2d}\bigg{(}1-\bigg{(}\frac{d}{D}\cos\phi\bigg{)}^{2}\bigg{)}f_{r} \tag{16}\]  
\[f_{\rm C}=\frac{f_{r}}{2}\bigg{(}1-\frac{d}{D}\cos\phi\bigg{)} \tag{17}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Structural Parameter** & **Value** \\ \hline Number of rolling elements \(n\) & 16 pc. \\ Rolling element diameter \(d\) & 8.4 mm \\ Diameter of pitch circle \(D\) & 71.5 mm \\ Nominal pressure angle \(\phi\) & 15.17\({}^{\circ}\) \\ \hline
**OperationParameter** & **Frequency** \\ \hline Rotation frequency \(f_{r}\) & 33 Hz \\ Outer race fault frequency \(f_{o}\) & 236.4 Hz \\ Inner race fault frequency \(f_{I}\) & 296 Hz \\ Rolling element fault frequency \(f_{B}\) & 139.6 Hz \\ Cage fault frequency \(f_{\rm C}\) & 14.8 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter information of the ZA-2115 bearings in IMS dataset. Figure 10: Experimental platform related to IMS bearing fault dataset [30]. These frequencies can be calculated separately, including the rotation frequency \(f_{r}\), the outer race fault frequency \(f_{o}\), the inner race fault frequency \(f_{I}\), the rolling element fault frequency \(f_{B}\), and the cage fault frequency \(f_{C}\). The rotating shaft operated at a steady speed of 2000 rpm. A PCB 353B33 High Sensitivity Quartz ICP accelerometer (PCB Piezotronics, Depew, NY, USA) is installed on the housing of each bearing. Throughout the run-to-failure experiment, vibration signals were recorded at 10 min intervals by the use of NI DAQ Card 6062 (Emerson Electric Co., St. Louis, MO, USA). The sampling frequency was set to be 20 kHz, with each signal containing 20,480 data points. In total, 984 signal files were acquired. Additionally, based on the detection results in reference [31], it is confirmed that the bearing has developed an incipient outer race fault. To diagnose an incipient bearing fault in this experiment, the signal recorded in Set No. 2 is analyzed. It corresponds to the run-to-failure operation of bearing 1. At the end of the experiment, this bearing developed an outer race fault. Figure 11 presents the trend in healthy status of this bearing throughout its entire operational life cycle. As shown in Figure 11, the range of the file index containing the incipient bearing fault is from 534 to 701. It is confirmed by [PERSON] et al. [22]. One signal with an outer race fault is shown in Figure 12, whose file index is 534. This figure clearly shows that strong noise obscures the weak fault information. The proposed DIES method is applied to the incipient fault signal shown in Figure 12. For benchmarking purposes, the IES method [21] is also employed. Figure 13a-d displays the FFT spectrum of the original signal, the spectrum generated using the SK-ES method, the spectrum calculated using the IES method, and the spectrum obtained through the proposed DIES method. Figure 11: Degeneration curve of bearing 1 in IMS dataset. Figure 12: An incipient bearing outer race fault signal. As shown in Figure 13a, the FFT spectrum contains several frequency components that appear after an incipient fault occurs, as well as interfering frequency components.