A novel correntropy-based band selection method for the fault diagnosis of bearings under fault-irrelevant impulsive and cyclostationary interferences

Highlights

• An indicator to calculate the targeted fault energy based on correntropy is proposed.

• A band selection method that is insensitive to impulsive and cyclostationary noise is developed.

• Comparisons with classical band selection methods are conducted on simulation and experimental data.

• Bearing faults are successfully diagnosed in three challenging datasets.

Abstract

Demodulation analysis is one of the most effective methods for bearing fault diagnosis. However, in practical applications, the interferences from ambient noises or other rotating components may create great challenges to demodulation analysis and thus decrease its effectiveness. Generally, a selection procedure for the most informative frequency band (IFB) is usually implemented in advance to extract the fault features that are hidden by the interferences. The fast kurtogram (FK) has been utilized as a benchmark for the IFB selection. Although designed to identify the most impulsive part of the signal, the FK is inevitably affected by the fault-irrelevant impulsive and cyclostationary interferences due to the dual sensitiveness to the impulsiveness and cyclostationarity of the kurtosis, and thus it may produce a misleading band for demodulation. To address this issue, a novel and robust IFB selection method based on the fault energy of correntropy (named FECgram) is proposed in this paper to replace the FK, through which the IFB can capture the fault symptom without being influenced by the fault-irrelevant impulsive and cyclostationary interferences. The superiority of the FECgram in combination with the squared envelope spectrum (SES) is validated on both simulation data and three different challenging experimental datasets.

Introduction

As one of the most vital components in rotating machinery, bearings are widely utilized in various industrial applications, including helicopters, railway axles, turbine machinery, etc. Harsh operational conditions such as continuously long work duration and heavy dynamic loads from the rotating components often make the bearings prone to failures. The unexpected failures can result in the performance degradation and safety-critical failures of the whole machinery, leading to unplanned shutdowns and increase in maintenance costs, or even fatal accidents. Therefore, the diagnostics of bearings is of great importance in practical engineering applications, through which the bearing faults can be diagnosed opportunely, necessary maintenance can be scheduled in advance, and the health integrity of the machines can be guaranteed. Consequently, over the last decades, abundant studies have been conducted towards the fault diagnosis of bearings [1], [2].

Envelope analysis, as one of the demodulation approaches, is the most commonly used methodology for bearing fault diagnosis, given the fact that the bearing fault feature is embodied in the almost-periodic repetitive impulses. These impulses, which emerged by the collisions between the faulty and healthy parts of bearings, repeat themselves periodically using the reciprocal of fault characteristic frequency (FCF) and then modulate the resonance frequency. Therefore, the fault information can be extracted by the envelope techniques [3]. To strengthen the bearing fault feature surrounded by interferences, a band that contains the richest fault information should always be selected prior to the envelope analysis. Nevertheless, the selection of the most informative frequency band (IFB) which contributes to revealing the bearing fault symptom submerged by interferences sometimes remains a challenging issue [4].

There exist extensive studies on how to accurately obtain the IFB before the implementation of demodulation analysis. The kurtogram [5], as the first systematic method for IFB selection, is a pioneering solution for the band selection issue. Spectral kurtosis (SK), with the definition of the normalized fourth-order cumulant of the real or imaginary part of each spectral line after the short-time Fourier transform (STFT) of the signal [6], is the theoretical basis of the kurtogram. By taking advantage of SK, the impulsiveness can be well-described in different frequency ranges. Then in Antoni’s work [7], the SK was systematically interpreted with the theory of Wold-Cramer decomposition, making it appropriate for nonstationary signal analysis. Subsequently, the practical applications of SK were further explored in the study [5], where the kurtogram based on STFT was proposed. The SK at each frequency range under different window lengths can be expressed by kurtogram. However, theoretically, there are infinite combinations of window lengths and frequencies, thus resulting in excessive computation burden and limiting the industrial uptake of kurtogram. Accordingly, the fast kurtogram (FK) was proposed by Antoni [8] to reduce the computation effort. By using finite impulse response (FIR) filters with 1/3-binary tree structure, SK values can be calculated only in a limited number of bandwidths and center frequencies, which significantly improves the computational efficiency of the FK and facilitates it to be a benchmark for IFB selection in bearing fault diagnosis.

Since then, a series of FK-based selection algorithms have been proposed for the improvement of IFB selection. For example, Lei and Lin [9] developed a wavelet packet transform (WPT)-based IFB selection method, where the STFT or FIR filter was replaced by the WPT. The decomposition effect was improved significantly owing to the improved performance of WPT over the STFT or FIR filter under non-stationary conditions. In Ref. [10], the spectrum was decomposed into bands with different initial resolutions using multi-scale clustering, through which bandwidth optimization was achieved. The results of the above-mentioned methods and other related research which focus on improving the performance of the filters or frequency bands division procedure are, unfortunately, affected by the fault-irrelevant impulsive interferences due to their impulsive-based nature [11]. In addition, these impulsive-based methods are vulnerable under fault-irrelevant cyclostationary interferences, because of the dual sensitivity to non-Gaussianity and non-stationarity of kurtosis. In study [12], Borghesani et al. demonstrated the relationship between the kurtosis and SES, and found that the fault-irrelevant cyclostationary interferences could make the kurtosis-based tools prone to fail. The fault-irrelevant impulsive and/or cyclostationary interferences, therefore, have significant impacts on the IFB selection procedure, and may ultimately lead to the misdiagnosis of bearing faults.

To suppress the impulsive interferences, additional methods such as the Autogram [13], alpha stable distribution-based gram [14], and protrugram [15] were recently developed. Among these methods, the protrugram gained the most attention, as it first shifted the focus from impulsiveness to cyclostationarity — one of the other essential symptoms in faulty bearings. Compared to FK, kurtosis of the time-domain envelope is replaced by kurtosis of the envelope spectrum (ES) in protrugram analysis. Moreover, the protrugram changes the variable bandwidths in FK to a fixed band which is usually applied as approximately 3–5 times the bearing FCF. Another remarkable work, coined as infogram, was proposed by Antoni [16]. This thermodynamics-based method is constituted by the SE-infogram and SES-infogram. In the infogram, the SE-infogram is calculated through the negative entropy of the squared envelope (SE) in time-domain, with the capability of measuring the impulsiveness of the signal. Similarly, the SES-infogram is defined as the negative entropy of the SES, indicating the cyclostationaity of the analyzed signal. Upon integrating the two grams, a high impulsiveness band with repetitive behavior can be selected.

The protrugram and infogram, on the one hand, alleviate the influence of fault-irrelevant impulsive noises, on the other hand, still perform poorly when fault-irrelevant cyclostationary noise exists as a result of ignoring the realistic cyclic periods induced by bearing faults. To address this issue, the ratio of cyclic content (RCC) [12] and indicator of second-order cyclostationarity (ICS2) [17] can both be utilized to estimate the cyclostationarity of specific frequencies. The RCC and ICS2 are both calculated by summing the targeted components, but normalized by the sum of all components or the zero-frequency component in SES respectively. By considering the realistic cyclic periods, the influence from the fault-irrelevant cyclostationary noise can be alleviated to a great extent and the bearing’s faults can subsequently be diagnosed correctly. However, it was demonstrated that external impulsive components with high-level energy can compromise SES [18], invalidating the RCC or ICS2-based IFB selection methods.

Recently, the correntropy has been adopted to suppress impulsive noise, thus providing the theoretical feasibility to develop a robust indicator for the signal in the presence of high impulsive noises. The correntropy proposed by Santamaria et al. [19] in 2006, which can be regarded as the correlation function defined in the reproducing kernel Hilbert space, has shown its strong ability to deal with signals under impulsive noise by adjusting its kernel function. Since then, numerous studies have been conducted focusing on the development of theory and application of correntropy, including local similarity measure [20], non-linear test [21], time delay estimation [22], and image processing [23]. However, to the best of our knowledge, only a few studies relevant to correntropy have been published in the area of machinery condition monitoring. For example, the cyclic correntropy spectrum was used to diagnose the bearing fault under the impulsive noise environment in Ref. [24]. The correntropy of the intrinsic mode function was used for demodulation in Ref. [25]. Unfortunately, the use of correntropy in condition monitoring is still limited and needs further exploration.

By taking advantage of the anti-impulsive property of correntropy, a novel indicator named fault energy based correntropy (FEC) is proposed in this paper. The FEC, calculating the energy at the targeted FCFs based on correntropy, can solely represent the bearing fault feature and is not affected by the impulsive interference. Furthermore, an IFB selection method called FECgram is established to overcome the drawbacks of the traditional band selection methods. In the FECgram, the signal is decomposed into sub-frequency-bands using a FK-identical 1/3-binary tree at first, then the FEC of each sub-signal is computed. By using the proposed indicator FEC, the FECgram can detect the energy of the targeted fault frequency without being influenced by fault-irrelevant impulsive and cyclostationary interferences. This renders the FECgram to be an optimal method that can accurately diagnose the bearing’s faults in most complex working environments.

The paper is organized as follows: the FK and protrugram techniques are reviewed in Section 2, which aims to provide the theoretical basis to interpret two typical (impulsive-based and cyclostationary-based) IFB selection methods. Section 3 first introduces the fundamentals of correntropy, and then proposes a robust indicator FEC to characterize the energy of targeted FCFs without the influence of fault-irrelevant impulsive and cyclostationary interferences. Furthermore, a novel band selection method named FECgram is established to select the optimal IFB. Section 4 uses simulation data to validate the effectiveness of FECgram under different levels of impulsive and cyclostationary noise, thereby presenting a comprehensive study of the performance of the FECgram. In Section 5, three sets of challenging experimental data from three different test rigs, including bearing signals with high-level impulsive noise and outer/inner race faulty planet bearings, are used to demonstrate the superiority of the FECgram. Conclusion and further work are summarized in Section 6.