Denoised Improved Envelope Spectrum for Fault Diagnosis of Aero-Engine Inter-Shaft Bearing

Abstract

The inter-shaft bearing is an important component of aero-engine rotor systems. It works between a high-pressure rotor and a low-pressure rotor. Effective fault diagnosis of it is significant for an aero-engine. The casing vibration signals can promptly and intuitively reflect changes in the operational health status of an aero-engine’s support system. However, affected by a complex vibration transmission path and vibration of the dual-rotor, the intrinsic vibration information of the inter-shaft bearing is faced with strong noise and a dual-frequency excitation problem. This excitation is caused by the wide span of vibration source frequency distribution that results from the quite different rotational speeds of the high-pressure rotor and low-pressure rotor. Consequently, most existing fault diagnosis methods cannot effectively extract inter-shaft bearing characteristic frequency information from the casing signal. To solve this problem, this paper proposed the denoised improved envelope spectrum (DIES) method. First, an improved envelope spectrum generated by a spectrum subtraction method is proposed. This method is applied to solve the multi-source interference with wide-band distribution problem under dual-frequency excitation. Then, an improved adaptive-thresholding approach is subsequently applied to the resultant subtracted spectrum, so as to eliminate the influence of random noise in the spectrum. An experiment on a public run-to-failure bearing dataset validates that the proposed method can effectively extract an incipient bearing fault characteristic frequency (FCF) from strong background noise. Furthermore, the experiment on the inter-shaft bearing of an aero-engine test platform validates the effectiveness and superiority of the proposed DIES method. The experimental results demonstrate that this proposed method can clearly extract fault-related information from dual-frequency excitation interference. Even amid strong background noise, it precisely reveals the inter-shaft bearing’s fault-related spectral components.

1. Introduction

An aero-engine is the core component of an aircraft. Its stable operation guarantees the safe flight of the aircraft [1]. The rolling bearing within this system is deemed one of the most critical elements. Its unexpected failure could cause major economic losses. And it might even lead to catastrophic incidents. [2]. Among the rolling bearings of an aero-engine, the inter-shaft bearing is a significant component in the support system of a dual-rotor aero-engine [3]. A tough working environment is extremely likely to cause various faults, such as wear, scratches, and peeling off of oil film [4]. Once a fault occurs in an inter-shaft bearing, it will weaken the stability of an engine in minor cases and even cause the shutdown of the engine in the air [5,6]. Therefore, conducting fault diagnosis for the inter-shaft bearing is of great significance.

In the current body of research, fault diagnosis of inter-shaft bearings is predominantly performed using vibration signal analysis. After all, the casing signals measured by vibration sensors are able to reveal the health status of an aero-engine’s transmission chain. Moreover, the dynamic degradation process of the inter-shaft bearing often leads to variations or abnormalities in vibration signals. Through vibration signal analysis, the spectral characteristics of vibration signals can be obtained. By extracting the fault characteristic frequencies from the spectrum, it is possible to determine whether an inter-shaft bearing has failed and, if it has, what type of fault has occurred. The typical vibration-based diagnosis methods mainly include narrowband-pass filtering, signal decomposition, spectral correlation analysis, and so on. For example, Liao et al. [7] proposed a rotary speed–difference–domain spectrum and envelope spectrum method for fault detection in inter-shaft bearings. The rotary speed difference between the high-pressure (HP) rotor and the low-pressure (LP) rotor is used as a triggering signal. The vibration signals were sampled for the whole cycles of the rotary speed–difference–domain. However, this signal acquisition process is very complex and not suitable for practical conditions. Ai et al. [8] treat the peak factor of a partial envelope spectrum as a new optimization parameter of the frequency band and apply it to resonance demodulation [9]. When a single inner or outer race fault occurs in an inter-shaft bearing, the method proposed by Ai et al. [8] can effectively identify the type of fault. However, the core of this method is still based on the bandpass-filtering principle, which complicates the extraction of weak fault signals from a background of wide frequency distribution noise.

As for signal decomposition-based diagnosis methods, they have been widely utilized for fault component extraction. The representative approach includes variational mode decomposition (VMD) [10], wavelet transform (WT) [11], and so on. However, these approaches have a high parameter sensitivity. For example, when the VMD is used for fault component isolation, its two parameters, named mode number and penalty factor, need to be carefully optimized [12]. To effectively extract the fault information of the inter-shaft bearing from vibration signals, Yu et al. [6] proposed an decomposition–reconstruction-based diagnostic method. This method integrates information fusion, WT, singular value decomposition (SVD), Katz fractal dimension, and cross-correlation function (CCF). It is termed as CCF–WT–SVD–Katz. Each component of this algorithm facilitates signal denoising, enhanced reconstruction, and fault feature extraction. Moreover, Pan et al. [13] introduced a method for diagnosing bearing faults by combining intrinsic time-scale decomposition (ITD) [14] and SVD. The results demonstrated that the ITD-SVD method exhibits higher accuracy in noisy conditions compared to ITD and principal component analysis methods. However, these signal decomposition-based diagnosis methods are still unable to tackle the multi-source interference with a wide-band distribution problem caused by the dual-frequency excitation of an aero-engine. As shown in Figure 1, it displays an inter-shaft bearing vibration signal under dual-frequency excitation. The fast Fourier transform (FFT) spectrum more clearly reveals the rotation frequencies of both the high-pressure and low-pressure rotors. As illustrated in Figure 2, the dual-frequency excitation makes the collected casing signal contain multiple deterministic interferential components. These components include rotation frequency and the harmonics of the HP rotor and LP rotor, as well as numerous combination frequencies [15,16]. They occupy a very wide frequency band.

An alternative and classical approach for identifying bearing faults is the spectral correlation (SC) analysis [17]. By projecting the vibration signal onto a two-dimensional plane of spectral and cyclic frequencies, SC effectively captures the second-order cyclostationary characteristics associated with bearing faults. Antoni [18] identified that incipient bearing faults exhibit distinct random cyclostationary characteristics within the vibration signals of the rolling elements. This observation highlighted the importance of cyclic spectral coherence, which is considered a critical tool for the early diagnosis of such faults. This technique not only confirms the presence of fault-related information amidst significant background noise but also serves as an indicator of relative fault severity. Despite its versatility in analyzing cyclostationary signals, the widespread application of cyclic spectral coherence has been limited by its computational intensity. To mitigate this limitation, Antoni et al. [19] introduced the Fast-SC algorithm. It is designed to accelerate the computation of spectral coherence. Additionally, they suggested novel methods for calculating the squared envelope spectrum (SES) and the enhanced envelope spectrum (EES) derived from SC. Following these advancements, a variety of fault diagnosis methods have been proposed by leveraging the spectral coherence [20]. Expanding upon this foundation, Abboud et al. [21] developed an improved envelope spectrum (IES), achieved by integrating SC across spectral frequency variables. Experimental validation demonstrated that this approach significantly enhances the diagnostic performance of the SES. Abboud et al. [22] further compared the performance of two advanced bearing fault diagnosis methodologies: an MED-SK-based approach and a CS-IES-based method. The experimental findings conclusively revealed that the CS-IES-based strategy exhibits superior capability in detecting the incipient faults of rotating machinery. Mauricio et al. [23] proposed the improved envelope spectrum via the feature optimization-gram (IESFOgram) method, which selects the optimal frequency band and extracts the corresponding improved envelope spectrum (IES) for bearing fault diagnosis. Chen et al. [24] introduced a method based on blind and targeted features to identify the optimal spectral frequency band of spectral coherence for bearing diagnostics. The results demonstrate that the IES-based methods can accurately discriminate the failure-related spectral frequency bands of spectral coherence, even in the presence of noise and stronger interfering components.

Although IES can effectively extract weak faults under high noise conditions, its performance diminishes when addressing the multi-source interference with wide-band distribution problem under dual-frequency excitation [16,25]. After all, the frequency distribution of vibration sources is very large since the high-pressure rotor and low-pressure rotor operate with significant differences in rotational speed. Then, the dual-frequency excitation problem is easily formed in the monitoring signal of inter-shaft bearings. Consequently, the vibration information excited by the inter-shaft bearing is easily overwhelmed by background noise of the same frequency.

To effectively identify inter-shaft bearing faults from casing signals of an aero-engine test platform, a novel method referred to as DIES is proposed in this paper. First, this method computes a subtracted normalized improved envelope spectrum (NIES) employing a spectrum subtraction technique. This technique is applied to subtract the healthy signal spectrum from the faulty signal spectrum [26]. In this process, NIES is used to extract weak fault information on the inter-shaft bearing under the interference of dual-frequency excitation. Then, the spectrum subtraction technique is used to suppress the frequencies of deterministic interferential components in the NIES of the fault vibration signal. Subsequently, an improved adaptive-thresholding method is proposed and applied to the subtracted NIES, in order to eliminate its high background noise. Compared to these methods mentioned above, the DIES can not only diagnose a bearing fault accurately under high background noise but also be more effective in handling the multi-source interference with a wide-band distribution problem under dual-frequency excitation. To demonstrate the effectiveness and superiority of the proposed method, comparative evaluations are conducted against existing classical methods. The classical methods used for comparison in this study include spectral kurtosis–envelope spectrum (SK-ES), IES, and ITD-SVD. Experimental results show that the proposed DIES method is more robust than IES, SK-ES, and ITD-SVD in handling the fault diagnosis problem of inter-shaft bearings. The reasons are as follows. While IES can extract cycle frequencies to a certain extent, it lacks the capability to eliminate interference frequency components. SK-ES essentially extracts target frequency bands through narrowband filtering, so it cannot effectively extract hidden cyclostationary periodic signals. Although the ITD-SVD method has strong denoising ability, it fails to effectively extract weak fault signal components. In contrast, the proposed DIES method is capable of extracting hidden cycle frequency components. Moreover, it can eliminate interference frequency components through spectrum subtraction and an improved adaptive thresholding technique. These advantages render it more effective than other comparative methods.

The remainder of this paper is organized as follows. Section 2 presents the proposed DIES framework in detail. Section 3 evaluates the effectiveness and generalizability of the proposed method through experiments on two real-world datasets. Finally, conclusions are provided in Section 4.

2. Methods

This section presents the proposed DIES method in detail. To address the multi-source interference with wide-band distribution problem under dual-frequency excitation, the DIES method operates through three key technical aspects. First, the IES method based on cyclic spectral analysis is applied to amplify fault-related spectral components. The resultant IES provides the critical basis for subsequent aspects. Furthermore, to eliminate the influence of inherent interference frequencies, a spectrum subtraction method is established. This method utilizes both healthy and faulty signals to derive the subtracted spectrum. Subsequently, an improved adaptive threshold denoising method is applied to further suppress background noise in the subtracted spectrum. Finally, the fault-related information of the inter-shaft bearing could be mined clearly.

2.1. IES Algorithm

With the help of the Fast-SC algorithm, the spectral correlation and a normalized spectral correlation can be computed efficiently. It provides a powerful tool for analyzing cyclic frequency and spectral frequency. The IES is determined using spectral coherence (SCoh) [18] and a normalized form of SC [27]. The SC is mathematically expressed as the double discrete Fourier transform of the instantaneous autocorrelation function [19]. By integrating the SC over a spectral frequency axis within a specific spectral band containing diagnostic information, the IES is derived to amplify fault-related spectral components, providing the basis for solving the multi-source interference with wide-band distribution problem under dual-frequency excitation.

Consider a discrete vibration signal represented as $x\left(t_N\right)$, where $t_N=N/F_s$, $N=0,1,\dots ,L-1$ and $F_s$ denotes sampling frequency. The variable $L$ denotes the length of the vibration signal. In cyclostationary theory, the instantaneous autocorrelation function $R_x\left(t_N, {\tau }_M\right)$ is essential for calculating the SCoh of cyclostationary processes. This function, related to the bearing vibration signal, includes period $T$, which is defined as follows:

$$\begin{array}{c}{R}_x\left(t_N, {\tau }_M\right)=E\left(x\left(t_N\right)x{\left(t_N-{\tau }_M\right)}^{*}\right)=R_x\left(t_N+T, {\tau }_M\right)\end{array}$$

(1)

where $E$ denotes the ensemble average operator, * represents the complex conjugate, ${\tau }_M=M/F_s$ is the time delay, and $M$ corresponds to the discrete time points. The spectral correlation $S{C}_x\left(\alpha ,f\right)$ is then defined as the double discrete Fourier transform of Equation (1). Its mathematical expression is provided as follows:

$$S{C}_x\left(\alpha ,f\right)=\underset{N\to \infty }{\lim }{\displaystyle \frac{1}{\left(2N+1\right)F_s}}{\displaystyle \sum _{n=-N}^N{\displaystyle \sum _{m=-\infty }^{\infty }{R}_x\left(t_n,{\tau }_m\right)e^{-j2\pi n\frac{\alpha }{F_s}}{e}^{-j2\pi m\frac{f}{F_s}}}}$$

(2)

The function $S{C}_x\left(\alpha ,f\right)$ measures the correlation between two frequency components of a signal located at $f$ and $f+\alpha$. It can be interpreted as the decomposition of the analyzed signal with respect to the “modulation frequency” $\alpha$ and “carrier frequency” $f$. SCoh${\gamma }_x(\alpha ,f)$ is the normalized form of SC, where the amplitude is scaled between 0 and 1:

$$\begin{array}{c}{\gamma }_x\left(\alpha ,f\right)={\displaystyle \frac{S{C}_x\left(\alpha ,f\right)}{\sqrt{S{C}_x\left(0,f\right)S{C}_x\left(0,f-\alpha \right)}}}\end{array}$$

(3)

where $S{C}_x\left(0,f\right)$ represents the classic power spectral density.

Based on the Scoh${\gamma }_x(\alpha ,f)$, the IES method [28] utilizes the local extrema of each cyclic frequency spectrum slice (CFSS) to identify candidate fault frequencies (CFFs). A CFSS is defined as the SCoh${\gamma }_x(\alpha ,f)$, in which the cyclic frequency $\alpha$ exhibits non-zero values along lines parallel to the spectral frequency axis in the $(\alpha ,f)$ plane. Certain vibration signals demonstrate second-order cyclostationarity. Their spectral correlation structure shows clear local maxima, which are distributed along CFSSs. These CFSSs are oriented parallel to the circle frequency $\alpha$ axis. Consequently, CFSSs with a higher number of local maxima are more likely to correspond to cyclic frequencies caused by localized defects in rolling element bearings.

Assume that the cyclic frequencies of ${\gamma }_x\left(\alpha ,f\right)$ are equidistantly discretized as $\left\{{\alpha }_n\right|{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N={\alpha }_{\max }\}$, where $N$ denotes the number of the CFSSs. The key information can be captured by defining $\eta ={[{\eta }_{\left(1\right)},{\eta }_{\left(2\right)},\dots ,{\eta }_{\left(N\right)}]}^T$ to qualify the number of the local maxima in each CFSS. The $n$th element ${\eta }_{\left(n\right)}$ represents the number of local maxima in the $n$th $CFSS{\gamma }_x\left(\alpha ,f\right)$. By sorting ${\eta }_{\left(n\right)}$ in descending order, $\tilde{\eta }={[{\tilde{\eta }}_{\left(1\right)},{\tilde{\eta }}_{\left(2\right)},\dots ,{\tilde{\eta }}_{\left(N\right)}]}^T$ is obtained [28]. The cyclic frequencies corresponding to the first $K$ CFSSs with the most local maxima are expressed as:

$$\begin{array}{c}\begin{array}{c}\widehat{\alpha }\left(k\right)=\left\{{\alpha }_n|{\eta }_{\left(n\right)}={\tilde{\eta }}_{\left(k\right)},n=1,2,\dots N\right\},k=1,2,\dots ,K\end{array}\end{array}$$

(4)

The cyclic frequencies $\widehat{\alpha }\left(k\right),k=1,2,\dots K$ are defined as CFFs.

The IES method employs a 1/3-binary tree filter structure to divide the full spectral frequency band. For the $i$-th ($i=0,1,1.6,2,2.6,3,\dots$) narrowband at the $l$-th level, the resulting $IE{S}_{l,i}\left(\alpha \right)$ is formulated as follows:

$$\begin{array}{c}IE{S}_{l,i}\left(\alpha \right)={\displaystyle \frac{1}{F_s/2^{l+1}}} {{\displaystyle \int }}_{{\displaystyle \frac{F_s\cdot \left(i-1\right)}{2^{l+1}}}}^{{\displaystyle \frac{F_s\cdot i}{2^{l+1}}}}\left|{\gamma }_x\left(\alpha ,f\right)\right|\end{array}df$$

(5)

An energy indicator is then applied to qualify the level of fault information contained in $IE{S}_{l,i}\left(\alpha \right)$. This indicator is referred to as a diagnostic index for all CFFs. The ratio of the energy $E{R}_{l,i}$ of $IE{S}_{l,i}\left(\alpha \right)$ is defined as [28]:

$$\begin{array}{c}E{R}_{l,i}={\displaystyle \frac{{{\displaystyle \sum }}_{n=1}^N{I}_{\left\{\widehat{\alpha }\left(k\right),d=1,2,\dots ,K\right\}}\left({\alpha }_n\right)\mid IE{S}_{l,i}\left({\alpha }_n\right){\mid }^2}{{{\displaystyle \sum }}_{n=l}^N\mid IE{S}_{l,i}\left({\alpha }_n\right){\mid }^2}}\end{array}$$

(6)

where $I\left({\alpha }_n\right)$ is defined as:

$$I_{\left\{\widehat{\alpha }\left(k\right),k=1,2,\dots ,K\right\}}\left({\alpha }_n\right)=\left\{\begin{array}{c}1,if{\alpha }_n\in \left\{\widehat{\alpha }\left(k\right),k=1,2,\dots ,K\right\}\\ 0,if{\alpha }_n\notin \left\{\widehat{\alpha }\left(k\right),k=1,2,\dots ,K\right\}\end{array}\right.$$

(7)

Finally, the $IE{S}_{l,i}\left({\alpha }_n\right)$ with the maximum ER value is selected as the optimal spectrum tool for further analysis.

2.2. Improved Adaptive Denoising for Subtracted NIES

The IES is able to efficiently extract the weak fault components of a bearing under high background noise. However, when the IES encounters both strong noise and a dual-frequency excitation problem, its performance diminishes. Specifically, the spectrum obtained by the use of IES retains interfering frequency components and random noise, complicating the identification of FCFs and their harmonics. For instance, Figure 3 illustrates the IES of an inter-shaft bearing vibration signal with an inner race fault. In the figure, the frequency response at BPFI (ball pass frequency inner race) and its i harmonics (i = {1st, 2nd, 3rd, 4th, 5th, …}) are considered as the frequencies related to the inner race fault of the bearing. Furthermore, it can be observed that the deterministic interference and noise components obscure the fault components. Notably, some interfering components exhibit higher amplitudes than the fault-related frequencies.

To address these limitations, an improved adaptive denoising method is proposed. The core principle of this method lies in that the deterministic interfering components are generally considered to exist in both healthy and faulty signals within the frequency domain. Then, to mitigate their impact on fault diagnosis, an improved noise subtraction (INS) method is introduced to eliminate these interfering components.

Initially, the normalized IES (NIES) for both faulty and healthy signals is computed. A subtraction operation is further performed between the two NIES to derive a subtracted NIES. For an IES comprising $N_{IES}$ spectral lines with amplitudes $[A_1,A_2,\dots ,A_N]$, the NIES is defined as follows:

$$\begin{array}{c} NIES={\displaystyle \frac{IES}{{‖IES‖}_{L1}}} \end{array}$$

(8)

where ${‖IES‖}_{L1}={{\displaystyle \sum }}_{i=1}^N\left|A_i\right|$ for $i=1,2,\dots ,N$. Using this formula, the NIES for both faulty and healthy signals are calculated. The subtraction operation is then applied, yielding the analytic solution:

$$\begin{array}{c} NIE{S}_{sub}={\displaystyle \frac{NIE{S}_F-NIE{S}_H}{{‖NIE{S}_F-NIE{S}_H‖}_{L2}}}\end{array}$$

(9)

where $NIE{S}_F$ and $NIE{S}_H$ represent the normalized IES of the faulty and healthy signals, respectively. Let

$$\left(NIE{S}_F-NIE{S}_H\right)={\left[A_1^{*},\dots A_2^{*},\dots A_N^{*}\right]}^T$$

(10)

where $A_n^{*}$ denotes the spectral line of the subtraction result. The $L_2$ norm is computed as:

$${‖NIE{S}_F-NIE{S}_H‖}_{L2}=\sqrt{{\displaystyle \sum _{i=1}^N{\left(A_i^{*}\right)}^2}}$$

(11)

Following spectral subtraction, the $NIE{S}_{sub}$ method facilitates the identification of fault-related components, random noise, and deterministic interference components. The latter may also appear in the spectrum of healthy signals. An illustration of the spectral subtraction process is shown in Figure 4. The red lines represent the frequencies of faulty components, while the black lines represent the frequencies of deterministic interference components. Since the spectral energy ratio of deterministic interference components in the fault spectrum is smaller than that in the healthy spectrum, spectral components can be distinguished by symbols after the normalization and subtraction process. As illustrated in Figure 5, the $NIE{S}_{sub}$ is obtained through spectral subtraction using the real healthy and faulty inter-shaft bearing signal spectra. In the figure, the frequency response at BPFO (ball pass frequency outer race) and its i harmonics (i = {1st, 2nd, 3rd, 4th, 5th,…}) are considered as the frequencies related to the outer race fault of the bearing. In the $NIE{S}_{sub}$, frequencies with positive amplitudes correspond to fault-related frequencies. Those with negative amplitudes reflect deterministic interference components.

However, in addition to deterministic interferential components, the subtracted spectrum still contains much random noise. These noises typically vary across different sampled signals, and strong noise may hinder the recognition of fault information, as depicted in Figure 3. To mitigate the impact of random and strong noise components effectively, an improved adaptive denoising technique is proposed to accentuate the fault frequencies. The proposed improved adaptive threshold is defined as follows:

First, the amplitude percentile points (APPs) of all the spectral lines of the $NIE{S}_{sub}$ are computed and sorted in descending order of their amplitudes:

$$\begin{array}{c}APPs=\left(A_i,{\displaystyle \frac{N_i}{N}}\right),i=1,2,\dots ,N\end{array}$$

(12)

where $N_i\in \left[0,N-1\right],N_i\in {\mathbb{Z}}^{+}$. Here, $N_i$ represents the number of spectral lines with amplitudes smaller than $A_i$. As shown in Figure 6, the horizontal axis corresponds to amplitude $A_i$, while the vertical axis represents the percentile ${\displaystyle \frac{N_i}{N}}$.

To determine an appropriate threshold for spectral denoising, the first D% of the APPs are selected for further analysis. As shown in Figure 6, the amplitude percentile point sequence is used to identify the optimal threshold point that distinguishes random noise from meaningful data. This is achieved through a line-based change point detection model. It utilizes a linear approach to detect significant changes in slope. Figure 6 illustrates this model. Two lines are fitted to different segments of the data, and the threshold point is identified where the slope change is most pronounced.

Once the adaptive threshold point is determined, the frequencies associated with random noise can be effectively eliminated. As shown in Figure 7a, the positive amplitudes of the $NIE{S}_{sub}$ are extracted to illustrate the denoising effect of adaptive thresholding. This threshold is represented by a red dotted line. However, although this operation eliminates some random noise with low amplitude, the spectrum still contains high-amplitude noise, as shown in Figure 7b. According to the method’s principle, the parameter $D$ is critical. Based on the principle stated in the study of B. Hou et al. [29], the parameter $D$ was empirically set to 2. Subsequently, the $NIE{S}_{sub}$ is denoised using a computed adaptive threshold, producing the DIES. The result of DIES is shown in Figure 7c. In summary, the calculation process of the adaptive threshold is as follows:

(1)

Compute amplitude percentile points of all the spectral lines of the $NIE{S}_{sub}$;

(2)

The first D% of the data is selected in descending order;

(3)

Compute line-based change point using the selected data;

(4)

Calculate the threshold for denoising.

2.3. The Proposed DIES for Improving SC Analysis

In general, the route of the proposed DIES for diagnosing an inter-shaft bearing fault is depicted in Figure 8. Specifically, the flowchart of the proposed DIES method is depicted in Figure 9. It includes 5 steps that are stated as follows:

Step 1: Perform fast SC analysis on both the faulty and healthy signals to generate the improved envelope spectrum;

Step 2: Compute NIES, which is utilized for spectral subtraction;

Step 3: Derive the subtracted NIES by subtracting the healthy NIES from the faulty NIES;

Step 4: Calculate an adaptive threshold for subtracted NIES by using a two-straight-line-based change point detection model;

Step 5: Determine the frequencies of the fault components by using the subtracted NIES and its adaptive threshold. Frequencies of spectral lines whose amplitudes are larger than the threshold can be ascertained as frequencies of fault components.

From these steps, it can be concluded that the core thought of the proposed DIES method is as follows. First, the improved envelope spectrum module demodulates the hidden periodic fault signal components. The resultant IES provides the critical basis for solving the multi-source interference with a wide-band distribution problem under dual-frequency excitation. Subsequently, the spectral subtraction module removes the influence of inherent interference frequencies in the spectrum. Finally, an improved adaptive threshold denoising module is applied to further eliminate noise and interference components from the spectrum, allowing for the extraction of fault characteristic frequency components more clearly.

Overall, the proposed method focuses on efficiently reducing noise and enhancing the clarity of fault signatures, thereby improving its effectiveness in diagnosing inter-shaft bearing faults.

3. Experimental Validation

In order to demonstrate the effectiveness and superiority of the proposed DIES method, this section utilizes two experimental datasets. That is, the IMS Incipient Bearing Faults dataset [30] and the HIT Inter-shaft Bearing Faults dataset [1]. The former is primarily employed as a benchmark to assess the effectiveness of the proposed method in extracting weak fault signals obscured by strong background noise. The latter is utilized to verify the effectiveness and generalization of the proposed method in addressing challenges posed by high background noise and dual-frequency excitation problems.

3.1. 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. Parameter information of the ZA-2115 bearings in IMS dataset.

Structural Parameter	Value
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 $\varphi$	15.17°
OperationParameter	Frequency
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 $f_C$	14.8 Hz

[22]. According to the following calculation formulas for bearing characteristic frequencies:

$$f_r={\displaystyle \frac{RPM}{60}}$$

(13)

$$f_I={\displaystyle \frac{n}{2}}\left(1+{\displaystyle \frac{d}{D}}\cos \varphi \right)f_r$$

(14)

$$f_o={\displaystyle \frac{n}{2}}\left(1-{\displaystyle \frac{d}{D}}\cos \varphi \right)f_r$$

(15)

$$f_B={\displaystyle \frac{D}{2d}}\left(1-{\left({\displaystyle \frac{d}{D}}\cos \varphi \right)}^2\right)f_r$$

(16)

$$f_C={\displaystyle \frac{f_r}{2}}\left(1-{\displaystyle \frac{d}{D}}\cos \varphi \right)$$

(17)

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 D. Abboud 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.

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. The spectrum generated using the SK-ES method reveals the FCF of the bearing’s outer race fault $f_o$, which is shown in Figure 13b. However, it also contains multiple obvious interfering frequency components and random noise. The spectrum generated using the IES method includes multiple higher-order harmonics associated with the FCF of the bearing’s outer race fault $f_o$, as shown in Figure 13c. However, it still contains significant interfering frequency components and random noise, which may compromise the accuracy of the fault diagnosis. Compared to the SK-ES method and IES method, the proposed DIES method exhibits superior performance, as illustrated in Figure 13d. It can be clearly observed that the proposed DIES method effectively highlights the FCF $f_o$ and its harmonics corresponding to the bearing’s outer race fault. Additionally, the spectrum processed by the DIES is devoid of noise and interfering frequency components, making it significantly less intrusive.

To further validate the effectiveness of the proposed DIES method, an effective denoising technique, ITD-SVD, was applied to the same raw signal presented in Figure 12. The results are illustrated in Figure 14. It is obvious that ITD-SVD is capable of detecting the bearing outer race FCF $f_o$ and its second harmonic $2\times f_o$. However, the performance of ITD-SVD is not as effective as DIES in identifying these features. It failed to extract the higher-order harmonics associated with the outer race fault frequency. Furthermore, the results exhibit several noticeable interfering frequency components that could hinder the bearing fault diagnosis.

3.2. 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. Parameter information of the inter-shaft bearing in HIT dataset.

Parameter	Value
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°

.

The experiment data is collected by K9000XL 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]:

$$f_i={\displaystyle \frac{1}{2}}\times {\displaystyle \frac{\left|n_i-n_e\right|}{60}}\left(1+{\displaystyle \frac{d\cos \alpha }{D}}\right)Z$$

(18)

$$f_o={\displaystyle \frac{1}{2}}\times {\displaystyle \frac{\left|n_i-n_e\right|}{60}}\left(1-{\displaystyle \frac{d\cos \alpha }{D}}\right)Z$$

(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 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. 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 Figure 19a–d and Figure 20a–d, respectively.

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 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.

Overall, the results highlight the superior performance of the proposed DIES method in diagnosing an inter-shaft bearing fault under strong noise and dual-frequency excitation interference.

In this diagnostic case for an inter-shaft bearing, the ITD-SVD method is also employed for a performance comparison. ITD-SVD is applied to the same original signal shown in Figure 18a,b. The corresponding frequency spectra are illustrated in Figure 21a,b. As depicted in Figure 21a, ITD-SVD extracts only the HP rotor frequency $f_{hp}$ and the 10th harmonic of the outer race FCF $10\times f_o$. Similarly, Figure 21b shows that this method just isolates the HP rotor frequency, along with several prominent interfering frequency components, but fails to extract the inner race FCF.

Overall, the ITD-SVD method is effective in noise control [13], enabling the extraction of the characteristic frequency of the high-pressure rotor in an aero-engine. While the method is effective in signal decomposition and noise reduction, it still fails to extract the bearing FCF component when addressing dual-frequency excitation problems.

Finally, we have summarized the comprehensive analysis results of the comparative methods in

Table 3. Comprehensive analysis table of several comparative methods.

Method	Whether Fault-Related Frequency Components Are Extracted	Whether Interference Frequency Components Are Effectively Eliminated	Whether Noise Components Are Effectively Eliminated
SK-ES	IMS-Yes; HIT-No	No	No
IES	Yes	No	No
ITD-SVD	Yes	No	Yes
DIES	Yes	Yes	Yes

. The summary will be mainly carried out from the following aspects: whether the fault-related frequency components are extracted, whether the interference frequency components are effectively eliminated, and whether the noise components are effectively eliminated.

4. Conclusions

This article proposes a DIES-based diagnosis method for an inter-shaft bearing of an aero-engine. This method is developed by extending the improved envelope spectrum. It is fundamentally based on spectral correlation analysis. By employing the spectral noise subtraction method on the faulty NIES and the baseline healthy NIES, a subtracted NIES signature can be derived. Furthermore, an adaptive threshold-based denoising strategy is introduced to mitigate the impact of prominent interfering frequency components and random noise on the spectral line features within the subtracted NIES. The effectiveness and generalization capability of the proposed DIES method are validated through experimental studies on both the inter-shaft bearing vibration dataset and the incipient fault bearing vibration dataset. Comparative analyses reveal that the DIES method outperforms the SK-ES and IES methods, as well as a more recent ITD-SVD denoising method. Additionally, the proposed DIES method provides a clearer visualization of fault-related spectral components, whereas the spectra produced by the compared methods appear more cluttered.