# 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  
For further validation, this case study focuses on an inter-shaft bearing vibration dataset published by HIT [1]. The experimental setup associated with this dataset is illustrated in Figure 15. This setup consists of three primary systems: a modified aero-engine, a motor drive system, and a lubricant system. The modified aero-engine functions as the central component of the setup and generates vibration signals during operation. Meanwhile, the motor drive system supplies the required driving force to drive the aero-engine under varying speeds and load conditions. The lubricant system ensures the smooth and efficient functioning of this engine. The dual-rotor configuration is of particular significance, serving as the key structural element of the aero-engine. This dual-rotor system comprises an LP compressor, an HP compressor, an LP turbine, and an HP turbine, with the inter-shaft bearing positioned between the LP and HP rotors. They are illustrated in Figure 16. The structural parameters of the inter-shaft bearing are detailed in Table 2.  
Figure 16: Schematic diagram of an aero-engine support structure. Figure 14: Incipient bearing fault case: the FFT spectrum of the original signal after processing with the ITD-SVD method. Figure 15: Experiment platform of the HIT inter-shaft bearing dataset [1]. The experiment data is collected by K9000 XL acceleration sensors (Jinan Kedong Instrument Co., Ltd., Jinan, China). The dataset consists of five vibration signal files, where each file contains 20,480 sampling points collected over 0.8 s. The sampling frequency is set to 25 kHz. The sampling frequency needs to be more than twice the maximum frequency. This ensures coverage of the fault characteristic frequencies we care about, as well as their higher harmonics [1]. For analysis purposes, the signals are segmented into samples with a fixed length of 20,480 points. In data1 and data2, the vibration signals are collected from a healthy inter-shaft bearing. In data3 and data4, the inter-shaft fault types are injected with inner race faults. In data5, the inter-shaft fault type is an injected outer race fault [1]. Physical images of these injected inner race and outer race faults of the inter-shaft bearing are shown in Figure 17. This paper investigates vibration signals from datasets data4 and data5. The objective is to evaluate the diagnostic capability of DIES for inter-shaft bearing faults. For the inter-shaft bearing, the inner fault characteristic frequency \(f_{i}\) and the outer fault characteristic frequency \(f_{o}\) are computed as follows [32]:  
\[f_{i}=\frac{1}{2}\times\frac{|n_{i}-n_{e}|}{60}\bigg{(}1+\frac{d\cos\alpha}{D} \bigg{)}Z \tag{18}\]  
\[f_{o}=\frac{1}{2}\times\frac{|n_{i}-n_{e}|}{60}\bigg{(}1-\frac{d\cos\alpha}{D} \bigg{)}Z \tag{19}\]  
where \(n_{i}\) denotes the rotational speed of the low-pressure rotor, \(n_{e}\) denotes the rotational speed of the high-pressure rotor, \(d\) denotes the rolling element diameter, \(D\) denotes the diameter of pitch circle, \(Z\) denotes the number of rolling elements, and \(\alpha\) denotes the nominal pressure angle. In this case, the proposed DIES method is applied to deal with the inter-shaft bearing outer race fault signal and inner race fault signal. The original signals are presented in \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Number of rolling elements \(Z\) & 15 pc. \\ Diameter of inner ring \(D_{i}\) & 30 mm \\ Diameter of outer ring \(D_{e}\) & 65 mm \\ Diameter of pitch circle \(D\) & 55 mm \\ Rolling element diameter \(d\) & 7.5 mm \\ Nominal pressure angle \(\alpha\) & 0\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Parameter information of the inter-shaft bearing in HIT dataset. Figure 17: The inter-shaft bearings with artificial (**a**) outer ring fault, (**b**) inner ring fault, (**c**) inner ring fault [1]. Figure 18. The outer race fault signal was recorded at a HP rotor speed of 3600 rpm and an LP rotor speed of 3000 rpm.