Morphological undecimated wavelet decomposition for fault diagnostics of rolling element bearings
Introduction
Rolling element bearings are among the most important and frequently encountered components in the vast majority of rotating machines, their carrying capacity and reliability being prominent for the overall machine performance. Therefore, quite naturally, the fault identification of rolling element bearings has been the subject of extensive research.
When a fault in one surface of a bearing strikes another surface, a force impulse is generated which excites resonances in the bearing and the machine. The successive impacts produce a series of impulse responses which may be amplitude modulated as a result of the passage of the fault through the load zone or of the varying transmission path between the impact point and the vibration measurement point. This physical effect has been exploited by several vibration analysis methods [1], based either on detailed models of the vibration response, or on signal processing methods. A comprehensive model for the nature of vibrations induced by the fault in a rolling element bearing, taking the detailed account of imperfections, wear and lubrication, has been proposed in Refs. [2], [3]. Several frequency domain signal processing methods have been developed to extract the useful information contained in the signals from the overall response, the envelope analysis being the most widely accepted one [4]. Improvement in this direction has been recently proposed, based on joint time–frequency domain methods, mainly on wavelet transforms (WTs), Hilbert–Huang transforms, as well as on cyclostationary analysis. This paper considers as a possible alternative, the application of a purely time domain analysis procedure, based on morphological wavelet (MW) decomposition concepts.
Morphological signal processing comprises a broad collection of theoretical concepts and mathematical tools for signal analysis, nonlinear signal operators, design methodologies and application systems that are related to mathematical morphology (MM) [5], [6]. Morphological signal processing was firstly used to analyze binary image data and was then extended to gray-level images [7]. The traditional tools of linear systems and Fourier analysis are of limited use for solving geometry-based problems because they do not directly address the issues of how to quantify the shape and the size of the signals. Contrarily, morphological signal processing is perfectly able to quantify all aspects. However, applications of morphological filters in one-dimensional time series have been quite limited, restricted practically to biomedical EEG signals [8], [9]. It has been recognized that multi-resolution signal decomposition schemes provide convenient and effective ways to process information. Most of the modern multi-resolution decomposition schemes are based on the theories of pyramid and wavelet, using the convolution and time–frequency domain transformations. However, the linear filtering approaches to multi-resolution signal decomposition have not been theoretically justified. In particular, the operators used for generating various levels of signal components in a pyramid must crucially depend on an application. Therefore, in recent years, a number of researchers have proposed nonlinear multi-resolution signal decomposition schemes based on morphological operators. However, until Goutsias and Heijmans presented a set of fundamental theories named morphological pyramid (MP) and MW, which were derived from traditional wavelet and pyramid theories, there are not a unified standpoint and framework for nonlinear pyramids, filter banks and wavelets, including MPs and wavelets construction [10], [11]. MP and MW extend the original wavelet and pyramid from the linear domain to the nonlinear domain. Moreover, they do not require the time–frequency domain analysis.
Based on the MW theory, a multi-resolution signal decomposition scheme, the morphological undecimated wavelet (MUDW) decomposition scheme is presented in this paper. The analysis operators and synthesis operators of the MUDW scheme are constructed according to the morphological coupled wavelet theories. Such a scheme, composed of morphological operators, totally inherits the simple computation property of MM operators. One of the analysis operators in the MUDW scheme is constructed from two parts, one extracts the impulsive components and the other fulfills the noise reduction. Such a construction is efficient for extracting features from the defective bearing signals with noise disturbance. The characteristic frequency of the bearing defect is very obvious by the frequency spectrum analysis of the approximate signals. Compared with the enveloping method, the MUDW decomposition method is more valuable, as demonstrated by the results using the simulated data and measured signals in the rolling element bearing test rig.
This paper is organized as follows. In Section 2, we briefly introduce the concepts of the morphological operators, the MP condition and the morphological coupled wavelet. The construction of the proposed MUDW decomposition scheme is discussed and its properties are analyzed in Section 3. In Section 4, the effects of the MUDW decomposition scheme are examined using simulated impulsive signals with noises and harmonic components. The comparison is made with the current enveloping demodulation method. The proposed procedure is evaluated in Section 5, using three vibration signals from defective rolling element bearings measured in the bearing test rig, which presents an outer race fault, an inner race fault and a rolling element fault, respectively. Finally, Section 6 lays out the conclusive remarks.
Section snippets
Basic concepts of MUDW decomposition scheme
In contrast with Fourier transform and wavelet analysis, MM is developed from set theory and integral geometry, and is concerned with the shape of a signal waveform in the complete time domain rather than the frequency domain. MM is a nonlinear approach and has been widely used in the areas of image processing, machine vision and pattern recognition, due to its robustness in preserving the shape while suppressing noise. The mathematical calculation involved in MM includes only addition,
MUDW decomposition scheme based on morphological gradient (MG)
The proposed MUDW decomposition scheme is based on the synthesis and analysis operators. It adopts not only multi-stage and varying-scale coupled wavelet, but also the undecimated algorithm. The undecimated algorithm is based on the idea of no decimation. It applies the WT and omits both down-sampling in the forward and up-sampling in the inverse transform. More precisely, it applies the transform at each point of the signal. In signal processing, this algorithm may give the best results, in
Simulation analysis
In order to verify the effectiveness of the MUDW decomposition scheme on noise suppression and impulsive feature extraction, a simulated signal is processed using the aforementioned method firstly. The simulated signal is formulated as follows (the sampling frequency is 1024 Hz and the sampling time is 1 s):where x1(t) is the sum of two harmonic waves: ; x2(t) is a typical series of exponentially decaying impulses with the impulse function of
Application on defective rolling element bearings
When a fault in one surface of a rolling element bearing strikes another surface, it produces an impact, which excites natural frequencies of the bearing and of the entire machine. Therefore, the typical response resulting from these periodic impacts, which are produced by bearing faults, usually comprises a sharp rise that corresponds to the impact between the rolling surfaces at the location of the defect and a gradual decay that corresponds to the vibration damping of the bearing outer ring.
Conclusions
In this paper, a MM-based MUDW decomposition scheme has been proposed to effectively smooth noise and extract the impulse components in the vibration signals of defective rolling element bearings. The development of its analysis operator has been discussed in detail and the multi-stage and varying-scale MUDW decomposition procedure has been analyzed. The efficiency of the MUDW decomposition scheme has been evaluated in simulation studies and the experimental signals measured in the bearing test
Acknowledgments
This research is supported by Natural Science Foundation of China (Grant No. 50425516 and 10732060) and the “863” High-Tech Scheme (2006AA04Z438) by the Ministry of Science and Technology. Comments and suggestions from the referees and editors are very much appreciated.
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