Detection of Bearing Faults Using a Novel Adaptive Morphological Update Lifting Wavelet
- Yi-Fan Li^{1, 2},
- MingJian Zuo^{1, 3}Email authorView ORCID ID profile,
- Ke Feng^{1} and
- Yue-Jian Chen^{3}
https://doi.org/10.1007/s10033-017-0186-1
© The Author(s) 2017
Received: 17 November 2016
Accepted: 29 September 2017
Published: 4 November 2017
Abstract
The current morphological wavelet technologies utilize a fixed filter or a linear decomposition algorithm, which cannot cope with the sudden changes, such as impulses or edges in a signal effectively. This paper presents a novel signal processing scheme, adaptive morphological update lifting wavelet (AMULW), for rolling element bearing fault detection. In contrast with the widely used morphological wavelet, the filters in AMULW are no longer fixed. Instead, the AMULW adaptively uses a morphological dilation-erosion filter or an average filter as the update lifting filter to modify the approximation signal. Moreover, the nonlinear morphological filter is utilized to substitute the traditional linear filter in AMULW. The effectiveness of the proposed AMULW is evaluated using a simulated vibration signal and experimental vibration signals collected from a bearing test rig. Results show that the proposed method has a superior performance in extracting fault features of defective rolling element bearings.
Keywords
1 Introduction
Rolling element bearings, one of the most important and frequently used components in engineering machinery, play a critical role in system performances [1]. Effectively detecting the defects of rolling element bearings can provide an assurance for the reliability of machine sets.
When a localized fault occurs on the surface of the inner race, outer race or rolling element, the vibration signal would present repetitive peaks which are further modulated by rotational frequencies of machine components. The impulses contain important information about the bearing health status. Therefore, the extraction of cyclic faulty intervals is the essential task in bearing fault detection.
Many methods, such as wavelet transform [2], empirical mode decomposition [3], higher order spectrum [4], morphology filter [5] and order tracking [6] have been applied successfully in bearing fault detection and fault diagnosis. Wavelet transform (WT) is one of the most popular signal processing technologies among them. However, the classical WT, both continuous and discrete, are linear [7]. Because of the fact that a signal often contains information at many scales or resolutions, multi-resolution approaches are indispensable for a thorough understanding of such a signal. Therefore, it is desired to extend WT to nonlinear area.
The lifting scheme, proposed by Sweldens [8], has provided a useful way to design nonlinear wavelets. The flexibility and freedom offered by the lifting scheme have attracted researchers to develop various nonlinear WTs, including morphological ones. Not until 2000, Heijmans and Goutsias [9, 10] firstly gave the theoretical presentation of a general framework for constructing morphological wavelet (MW). The theoretical foundation of MW is extending the classic wavelet from the linear domain, which is based on convolution, to the nonlinear domain, which is based on morphological operations. In this way, the MW does inherit the multi-dimension and multi-level analysis of wavelet while only involving the purely time domain analysis. As a consequence, a very high computational efficiency is achieved.
Following the work of Heijmans and Goutsias [9, 10], morphological gradient wavelet [11, 12], morphological undecimated wavelet (MUDW) [13–15] and morphological undecimated wavelet slices [16] were proposed and applied for fault detection and fault diagnosis. A major disadvantage of MW and the extended version is that the filter structure is fixed in the whole analysis process, which cannot cope with the sudden impulses and stationary data accurately in one signal at the same time. In many applications, it is desirable to have a filterbank that somehow determines how to shape itself according to the signal being analyzed. Based on this consideration, Piella and Heijmans [17] proposed an adaptive update lifting wavelet (AULW). The basic idea underlying AULW is to employ different update lifting filters to modify the approximation signal according to the signal local gradient information. However, the essence of AULW is still linear wavelet decomposition. The effectiveness of employing AULW to process the real mechanical vibration signals is actually compromised.
A new method of morphological lifting scheme, an adaptive morphological update lifting wavelet (AMULW) is proposed in this study. The aim of AMULW is to address the disadvantages both from the linear wavelet decomposition and from the fixed filter in AULW and MW. In AMULW, the nonlinear morphological dilation-erosion filter, as well as the average filter, is adaptively adopted in the update lifting scheme according to the geometry of the signal. Consequently, the impulsive features would be strengthened and the noise would be suppressed effectively.
The rest of this paper is organized as follows: Section 2 briefly introduces the fundamentals of MW; Section 3 proposes the AMULW; the advantage of AMULW over the AULW and MUDW is demonstrated by using a simulated vibration signal in Section 4; Section 5 applies the proposed AMULW technique to experimental signals of rolling element bearings and demonstrates the detection ability of AMULW for an inner race fault bearing and an outer race fault bearing; Conclusions are drawn in Section 6.
2 Morphological Wavelet
In MW, the analysis operator \(\psi_{j}^{ \uparrow }\) and the synthesis operator \(\psi_{j}^{ \downarrow }\) are morphological operators. Some MWs have been established, such as morphological Haar wavelet [18].
3 Adaptive Morphological Update Lifting Wavelet
3.1 Principles of AMULW
On account of the presence of the decision map D, the update lifting operator U can be adaptively selected to modify the approximation signal. The morphological dilation-erosion filter is adopted to strengthen the impulsive features when there is an impulse in a signal, while the average filter is employed to smooth a signal at the time when the signal amplitudes vary weakly.
3.2 The Selection of the Threshold T
The CFIC represents the portion of the fault frequency amplitude to the overall frequency amplitude. A larger value of CFIC implies a better fault detection performance. Other statistic criterion, such as kurtosis [21], smoothness index [22], crest factor [23], peak energy [24] and fusion criterion of kurtosis, smoothness index and crest factor [25], can also be used as a guide to select the fault relevant information. In this study, CFIC is employed. The signal with the largest CFIC value is picked out from the whole 11 candidates and only this signal will be utilized in the fault detection.
3.3 Perfect Reconstruction
In Eq. (11), \(\alpha = \beta = \gamma = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}\), which meets the requirement of Eq. (18). For Eq. (9), we discuss the influence of the value of y _{1}(n−1) and y _{1}(n) on perfect reconstruction. In summary, there are three situations:
Regardless the value of y _{1}(n − 1) and y _{1}(n), the situations in Eqs. (19)–(21) accords with the requirement of Eq. (18). Therefore, the proposed AMULW fits the perfect reconstruction condition.
4 Simulated Signal Analysis
Parameters used in the simulation of signal s _{1}(t)
i | A | T | \(\tau_{i}\) | n(t) |
---|---|---|---|---|
16 | [3, 4] | 1/16 | 0 | 0 |
a | f _{0} | ϕ | ϕ _{ A } |
---|---|---|---|
100 | 600 | 0 | 0 |
In Eq. (22), S _{2}(t) = 0.2sin(2π10t) + 0.4 cos(2π25t) simulates two interference frequencies, s _{3}(t) represents a Gaussian white noise. The signal-to-noise ratio of s(t) is 0 dB.
5 Experimented Lab Signal Analysis
5.1 Analysis of the Vibration Signal of Bearing with an Inner Race Fault
5.2 Analysis of the Vibration Signal of Bearing with Outer Race Fault
6 Conclusions
In this paper, an AMULW with perfect reconstruction is developed to detect rolling element bearing faults. Compared with Fourier transforms using the same filter and wavelets being translation and dilation of one given function, lifting scheme adapts local data irregularities in the transform process. In the proposed AMULW, two filters are adaptively employed. The nonlinear morphological dilation-erosion filter is effective to extract impulses while the average filter is suitable for removing noise. Therefore, AMULW can reasonably process a non-stationary mechanical vibration signal comprising of impulses and interferences. The experimental evaluation results have shown that the proposed AMULW is capable of extracting the impulsive features of the bearing vibration signals. It outperforms AULW in detection both of an inner race fault and an outer race fault of a rolling element bearing and it outperforms MUDW in detection of an outer race fault of a rolling element bearing.
Notes
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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