Wavelets for fault diagnosis of rotary machines_ A review with

Signal Processing 96 (2014) 1–15

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Signal Processing journal homepage: www.elsevier.com/locate/sigpro

Wavelets for fault diagnosis of rotary machines: A review with applications Ruqiang Yan a,n, Robert X. Gao b, Xuefeng Chen c a

School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269, USA The State Key Laboratory for Manufacturing Systems Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China b c

a r t i c l e i n f o

abstract

Article history: Received 29 January 2013 Received in revised form 10 April 2013 Accepted 15 April 2013 Available online 22 April 2013

Over the last 20 years, particularly in last 10 years, great progress has been made in the theory and applications of wavelets and many publications have been seen in the field of fault diagnosis. This paper attempts to provide a review on recent applications of the wavelets with focus on rotary machine fault diagnosis. After brief introduction of the theoretical background on both classical wavelet transform and second generation wavelet transform, applications of wavelets in rotary machine fault diagnosis are summarized according to the following categories: continuous wavelet transform-based fault diagnosis, discrete wavelet transform-based fault diagnosis, wavelet packet transform-based fault diagnosis, and second generation wavelet transform-based fault diagnosis. In addition, some new research trends, including wavelet finite element method, dual-tree complex wavelet transform, wavelet function selection, new wavelet function design, and multi-wavelets that advance the development of wavelet-based fault diagnosis are also discussed. & 2013 Elsevier B.V. All rights reserved.

Keywords: Classical wavelet transform Second generation wavelet transform Wavelet finite element method Wavelet selection Wavelet design Multi-wavelets

1. Introduction Increasing need for high-quality, low-cost products and safe production has accelerated the change of machine maintenance strategy from corrective over preventive to condition-based maintenance, for which real-time fault diagnosis and prognosis are needed. As one of the major equipments in modern industry, increased installation and usage of rotary machines have been seen in many industry branches, such as manufacturing. Accordingly, fault diagnosis of rotary machines has taken on new significance. These efforts have promoted continued advancement of measurement as well as signal-processing technologies.

n Corresponding author. Tel.: +86 25 83790692 (office), +86 13584054760 (mobile); fax: +86 25 83794158. E-mail address: [email protected] (R. Yan).

0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.04.015

As an example, the world has witnessed a tremendous growth in the theory and practice of wavelets [1–10], and many publications have been seen, describing advancement in wavelet theory and its successful applications in various fields of engineering. The adaptive, multiresolution capability of the wavelet transform has also made it a powerful mathematical tool for diagnostics of machine operation conditions in manufacturing. In 2004, Peng and Chu provided a review and summarized the development and applications of wavelet transform on machine condition monitoring and fault diagnosis over the past years [11]. Since then, this technique has been extensively studied and utilized by various researchers, new development and applications of the wavelet transform emerged from the areas of fault diagnosis significantly impact the state-of-the-art technology. This paper attempts to provide an overview of some of the latest efforts in the development and applications of

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R. Yan et al. / Signal Processing 96 (2014) 1–15

wavelet transform for fault diagnosis of rotary machines. The purpose is to provide researchers and engineers, who are working in this vibrant field, a comprehensive knowledge, and help them better applying the theory of wavelet transform to solve problems related to fault diagnosis of rotary machines. The paper is divided into five parts, including this introduction section. Section 2 presents the basic introduction on classical wavelet transform, which includes continuous wavelet transform (CWT), discrete wavelet transform (DWT), and wavelet packet transform (WPT). Also in this section we will provide a brief introduction on the so-called second generation wavelet transform (SGWT) that is based on lifting scheme. Then, Section 3 reviews applications of wavelets in rotary machine fault diagnosis over the past ten years. After that, Section 4 discusses some new trends of wavelets in rotary machine fault diagnosis. Finally, concluding remarks are drawn in Section 5. 2. Theoretical background of wavelet transform Wavelet transform can be considered as a mathematical tool that converts a signal in time domain into a different form, i.e. a series of wavelet coefficients, in timescale domain. A wavelet function, which is a small wave, possesses oscillating wavelike characteristics and concentrates its energy short in time, is needed to implement the wavelet transform. Traditionally, the wavelet transform can be categorized as CWT, DWT, and WPT. In the following subsections, the basic theory on each of them will be introduced. 2.1. Continuous wavelet transform (CWT) The CWT of a signal x(t) can be performed through a convolution operation between the signal x(t) and complex conjugate of a family of wavelets, which is expressed as [12] Z 1 t−τ cwtðs; τÞ ¼ pffiffi xðtÞψ n dt ð1Þ s s where the symbols s and τ denotes scale and translation parameters, respectively. ψ*() is the complex conjugate of the scaled and shifted wavelet function ψ(). Eq. (1) indicates that the CWT is similar to the Fourier transform except that the family of wavelets is used as the basis functions to replace the sine and cosine functions. Since the family of wavelets contains two parameters (scale parameter s and translation parameter τ), transforming a signal with the family of wavelets means such a signal will be projected into a two-dimensional, time-scale plane, instead of the one-dimensional plan in the Fourier transform. Furthermore, computation of Eq. (1) can be achieved by a pair of Fourier transform and inverse Fourier transform as below. The frequency domain equivalent of Eq. (1) is expressed as CWTðs; f Þ ¼ F cwtðs; τÞ Z ∞ Z ∞ 1 t−τ ¼ pffiffi dt e−j2πf τ dτ xðtÞψ n s s2π −∞ −∞ pffiffi ¼ sXðf ÞΨ n ðsf Þ ð2Þ

where the symbol F[] denotes the operator of Fourier transform. X(f) denotes the Fourier transform of x(t) and Ψ* () denotes the Fourier transform of ψ*(). By taking the inverse Fourier transform, Eq. (2) is converted back into time domain as pffiffi ð3Þ cwtðs; tÞ ¼ F −1 fCWTðs; f Þg ¼ sF −1 fXðf ÞΨ n ðsf Þg where the symbol F−1[] denotes the operator of inverse Fourier transform. Eq. (3) indicates the CWT of a signal x(t) can work as a band-pass filter with the wavelet function itself and its scale parameter s controls the filtering performance. 2.2. Discrete wavelet transform (DWT) Performing CWT on a signal will lead to redundant information generation, as the scale parameter s and translation parameter τ are changed continuously. Although redundancy is useful in some applications such as signal denoising and feature extraction where the desired performance is achieved at the cost of increased computational time, other applications may need to emphasize computational efficiency. It turns out that dyadic scales (i.e., s¼2, τ¼k2j) can achieve such requirement, while at the same time avoiding to sacrifice information contained in the signal. Mathematically, after such scale discretization, the DWT can be realized as ! Z 1 t−k2j dwtðj; kÞ ¼ pffiffiffiffi xðtÞψ n dt ð4Þ 2j 2j Practically, DWT can be implemented by means of a pair of low-pass and high-pass wavelet filters, denoted as h (k) and g(k)¼(−1)kh(1−k), respectively. These filters, also known as Quadrature Mirror Filters (QMF), are constructed from the selected wavelet function ψ(t) and its corresponding scaling function ϕ(t), expressed as [13] pffiffiffi 8 > < ϕðtÞ ¼ 2∑ hðkÞϕð2t−kÞ k

> : ψðtÞ ¼

pffiffiffi 2∑ gðkÞϕð2t−kÞ

ð5Þ

k

with ∑hðkÞ ¼ k

pffiffiffi 2 and ∑gðkÞ ¼ 0. Using the wavelet filters, k

the signal is decomposed into a set of low and highfrequency components as [13] 8 < aj;k ¼ ∑ hð2k−mÞaj−1;m m ð6Þ : dj;k ¼ ∑ gð2k−mÞaj−1;m m

In Eq. (6), aj,k is the approximation coefficient, which represents the signal's low-frequency components, and dj,k is the detail coefficient, which corresponds to the signal's high-frequency components. The approximation and detail coefficients at wavelet scale 2j (with j denoting the level) are obtained by convolving the approximation coefficients at the previous level (j−1) with the low-pass and high-pass filter coefficients, respectively. 2.3. Wavelet packet transform (WPT) The WPT can further decompose the detail information of the signal in the high frequency region. To perform WPT


of a signal at a certain level (e.g. level 3), the functions in Eq. (5) are unified as pffiffiffi 8 > < u2n ðtÞ ¼ 2∑ hðkÞun ð2t−kÞ k ð7Þ pffiffiffi > : u2nþ1 ðtÞ ¼ 2∑ gðkÞun ð2t−kÞ k

where u0 ðtÞ ¼ ϕðtÞ, and u1 ðtÞ ¼ ψðtÞ. Correspondingly, the signal is decomposed as [14] 8 < djþ1;2n ¼ ∑ hðm−2kÞdj;n m ð8Þ : djþ1;2nþ1 ¼ ∑ gðm−2kÞdj;n m

where dj,n denotes the wavelet coefficients at the j level, n sub-band, dj+1,2n and dj+1,2n+1 denotes the wavelet coefficients at the j+1 level, 2n and 2n+1 sub-bands, respectively, and m is the number of the wavelet coefficients. As illustrated in Fig. 1, a 3-level WPT generates a total of eight sub-bands, and each sub-band covers one eighth of the frequency information successively.

(n is a natural number), it will be split as ( xodd ¼ fxð2t−1Þg t ¼ 1; 2; 3; :::; n xeven ¼ fxð2tÞg

3

ð9Þ

After the splitting operation is completed, the odd and even sub-samples are obtained and the signal is subsampled by factor of two. Then the prediction operation is executed, which predicts the odd data sample with the even data sample as xodd ¼ Pðxeven Þ

ð10Þ

In Eq. (10), P is the prediction operator that is independent of the signal. The difference, denoted as d, between the predicted results and the odd samples is considered as the detail coefficients of the signal x(t), and it is expressed as d ¼ xodd −xodd ¼ xodd −Pðxeven Þ

ð11Þ

Knowing the even sample xeven and the detail coefficients d, the approximation coefficients are calculated using the updating operator U as

2.4. Second generation wavelet transform (SGWT)

a ¼ xeven þ UðdÞ

SGWT is considered as an alternative implementation of the classical DWT. Specifically, the mechanism of constructing a wavelet function from the translation and dilation of a fixed function is replaced by using lifting scheme [15]. Fig. 2 illustruates how the lifting scheme works, where the decomposition part (left part of Fig. 2) of the lifting scheme, similar to its counterpart in the classical DWT, is to obtain both the approximation and detail coefficients of the original signal. It mainly includes three operational steps: (1) splitting, (2) prediction, and (3) updating. When the lifting scheme is performed, the signal x(t) is firstly split into two subsets, the odd sample xodd and the even sample xeven, by means of a sample operation. For example, given a signal x(t), where t ¼ 1; 2; 3; :::2n

Similar to the prediction operation, the updating operation is also independent of the signal. The functions of prediction and updating operators are similar to that of a pair of h(k) and g(k) filters in the classical wavelet transform, and they can be derived from the scaling function ϕðtÞ and wavelet function ψðtÞ by iteration algorithm. It should be noted that the prediction and updating operators can be optimized using different algorithms, such as the Claypoole's optimization algorithm [16]. After the three operation steps are executed, the signal is decomposed into two parts: approximation and detail. This process can be iterated by taking the approximation part as the input signal to continue the decomposition. Furthermore, by iterated decomposition of the detail and

Fig. 1. Illustration of wavelet packet transform.

Fig. 2. Illustration of signal decomposition and reconstruction using lifting scheme.

ð12Þ

4


the approximation parts together, second generation WPT can be realized with the lifting scheme. The SGWT is invertible, and the signal reconstruction process is illustrated in the right part of Fig. 2. Similar to the decomposition process, the reconstruction process involves both the prediction operator and the updating operator. This means that the following relationship exists: ( xodd ¼ d þ Pðxeven Þ ð13Þ xeven ¼ a−UðdÞ The signal can then be reconstructed by merging xeven and xodd. 3. Applications of wavelet transform in rotary machine fault diagnosis Applications of wavelet transform have been seen in various engineering fields, such as biomedical engineering, civil engineering, manufacturing engineering, and so on. This section summarizes utilization of wavelet transform in fault diagnosis of rotary machines, widely used equipment in manufacturing, with emphasis on their key components, such as bearings, gearbox, and rotors. 3.1. CWT for rotary machine fault diagnosis Applying CWT to a signal produces a time-scale representation of that signal, as indicated by a series of wavelet coefficients at different scales. By utilizing these wavelet coefficients as input, Zuo et al. enabled application of independent component analysis (ICA) in gear fault diagnosis when the data collection was only performed on a single sensor [17]. Zhu et al. mapped the wavelet coefficients into a polar diagram to enhance periodic transients caused by gearbox and bearing faults [18]. Meltzer and Dien also used the polar wavelet amplitude maps to improve detection capability of faulted gears, which were operated under non-stationary rotating speed [19]. In another study, phase angle information was added into a 3D CWT plot for improving the detectability of rotor cracks [20]. Rafiee and Tse approximated the autocorrelation function of the wavelet coefficients as simple sinusoidal function for feature extraction in gearbox fault diagnosis [21]. Furthermore, Tse et al. optimized the wavelet coefficients by designing an exact wavelet analysis based on genetic algorithm [22], and helped to detect faults in a motor pump rotary machine. Various parameters extracted from wavelet coefficients of the CWT were often used as indictors to characterize machine health status. Zhang et al. explored the wavelet gray moment and the first order wavelet gray moment vector from scalogram, which is defined as the square of the modulus of the wavelet coefficients, of vibration signals to quantify eight different faults, such as imbalance, misalignment shaft crack, etc., in an experimental rotor test system [23]. The mean frequency of the scalogram was also extracted to detect presence of pitting faults as well as its progression in gears [24]. A fault growth parameter derived from amplitude of the wavelet coefficients was proposed to quantitatively assess gear fault severity under

varying load conditions [25]. It was found that such fault growth parameter is insensitive to varying load and can detect early gear fault. Lipschitz exponents calculated using the wavelet transforms modulus maximal (WTMM) method was introduced by Peng et al. to characterize a signal's singularity [26], and three parameters (the number of Lipschitz exponents per rotation, the mean value and the relative standard deviation of Lipschitz exponents) extracted from them were used to differentiate four typical rotary machine faults: rub-impact between stator and rotor, oil whirl, coupling misalignment, and imbalance. Utilizing the CWT, Rafiee et al. extracted several statistical parameters (i.e., standard deviation, variance, fourth central moment, and kurtosis) from the wavelet coefficients of synchronized vibration signals with different wavelet functions for both bearings and gearbox fault diagnosis [27]. Kankar et al. also extracted statistical features from wavelet coefficients to classify bearing faults with the assistance of artificial intelligent techniques [28]. Making use of band-pass filtering characteristic of the CWT, Yang et al. developed a CWT-based adaptive filter to track energy in the time-varying fault related frequency bands from the power signal for monitoring wind turbine conditions [29]. Al-Raheem et al. diagnosed bearing faults using autocorrelation of denoised vibration signal through optimized CWT [30]. Qiu et al. designed a wavelet filter to extract weak fault signatures from rolling bearings, where the Shannon entropy is used to optimize the shape of the Morlet wavelet, and singular value decomposition (SVD) is performed to detect periodicity of the signal for appropriate decomposition scale selection [31]. Jafarizadeh et al. combined the time-averaging-based noise cancellation with Morlet wavelet filtering to diagnose gear damage from asynchronous input signals [32]. The Morlet wavelet filtering was also used with sparse code shrinkage to enhance the impulsive features and suppress residual noise for bearing fault detection [33]. Lin and Zuo tuned the Morlet wavelet function using Kurtosis maximum principle and then extracted periodic impulses through an adaptive wavelet filtering [34]. Noise contaminated in gearbox vibration signals is reduced through a CWT-based filtering process [35], where the parameters of Morlet wavelet and the decomposition scale were optimized using cross-validation method and minimum Shannon entropy, respectively. The gearbox fault feature was found to be much clear with this hybrid method. Yang and Ren optimized the parameters of Morlet wavelet and then applied CWT-based envelope analysis to detect impulsive features in mechanical signals [36]. A peak energy criterion was used to guide the selection of both center frequency and bandwidth of the Morlet wavelet for demodulating the defective bearing vibration signals at the resonance region [37]. The parameters of the Morlet wavelet was also optimized using genetic algorithm in reference [38], and with enhancement of autocorrelation to the wavelet coefficients, it was found that it is very effective to identify bearing faults [38]. A new denoising scheme that combines resonance frequency estimation with in-band noise reduction was also proposed for enhancing detectability of bearing fault signals [39], where Gabor wavelet was used for wavelet filtering, and a smoothness index defined as the ratio of the geometric mean to the


arithmetic mean of the wavelet coefficient moduli was applied to adjust the shape of Gabor wavelet function [40]. Taking advantage of abundance of wavelet functions, Li et al. selected the Hermitian wavelet as the basis function to perform CWT for detecting singularity characteristic of a signal [41], which is often generated by localized defects in a mechanical system, such as gear crack fault. In addition, the amplitude and phase map of the Hermitian wavelet transform are used together to identify bearing faults [42]. The Laplace wavelet was also used to preprocess the vibration signals of rolling bearings with different fault conditions [43]. The features extracted from wavelet coefficients in both time and frequency domains were input to artificial neural networks for fault classification. Feng et al. constructed an anti-symmetric real Laplace wavelet filter to extract weak fault features from rolling bearing vibration signals [44]. A Laplace wavelet envelope power spectrum was utilized by Al-Raheem et al. to identify bearing fault characteristic frequencies [45]. By exploring the good lowpass filter characteristic of the Haar wavelet, Li et al. applied the Haar wavelet-based CWT to detect low frequency signals for both gears and bearings [46]. Hou et al. proposed a resonance demodulation scheme for bearing fault diagnosis using harmonic wavelet [47], where the proper resonance region was chosen by calculating the relative wavelet energy of each potential sub-frequency band determined by the harmonic wavelet parameters. The harmonic wavelet was also utilized to design an ideal band-pass filter to demodulate the vibration signals for enhanced bearing defect identification [48]. Integrating CWT with other techniques, various hybrid approaches have been developed for fault diagnosis of rotary machines. Combet et al. investigated wavelet bicoherence technique to detect early differential local damage in a gearbox [49]. Zhu et al. combined the CWT with Kolmogorov–Smirnov (K–S) test to develop a transient detection method and used it to identify the existence of machine faults [50]. Hong and Liang jointly applied the CWT with Lempel–Ziv complexity to characterize bearing fault severities [51]. Wang et al. presented a method integrating the Morlet wavelet with correlation filtering for identifying both the parameters of defect-induced impulse response and cyclic period between successive impulses [52], which provide an effective way of defect feature detection for gearbox and bearings [53]. Konar and Chattopadhyay proposed a hybrid CWT-Support Vector Machine (SVM) approach for bearing fault detection in induction motor [54]. Saravanan et al. also combined CWT with SVM/Proximal SVM for spur bevel gearbox fault diagnosis [55,56]. Liu and Tang presented a hybrid time– frequency method to achieve effective gear fault diagnosis, based on improved Morlet wavelet and auto-term windows [57]. Halim et al. combined the time synchronous average and CWT to extract periodic component at various wavelet scales, and identified gear faults [58]. The CWT, together with autocorrelation enhancement, was presented to enable small defect detection of bearings [59]. Kankar et al. used the similar approach for bearing fault diagnosis [60]. Jiang et al. applied adaptive Morlet wavelet and singular value decomposition (SVD) to process wind turbine vibration signals, where modified Shannon

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wavelet entropy was used to tune center frequency and bandwidth of the Morlet wavelet for better extracting impulsive features hidden in the signals [61]. Li et al. designed a multi-scale autocorrelation approach via morphological wavelet slice to analyze the bearing signals [62]. The morphological wavelet theory was also studied to design undecimated wavelet decomposition scheme for extracting impulse features and smoothing noise components [63]. Hong and Liang developed a signal-channel signal separation method by using CWT and Fourier transform [64], where mixed bearing signals containing both the inner and outer race defects were separated and identified. Assisted with multi-level exponential moving average power filtering, Liu detected machine incipient fault by applying Shannon wavelet spectrum analysis to vibration signals [65]. Liu et al. proposed a wavelet crossspectrum technique to extract features from vibration signals for bearing fault diagnosis [66]. They also synthesized wavelet coefficients from different bandwidths using extended Shannon function to enhance feature characteristics [67]. A Jarque–Bera statistic index is utilized to select the bandwidth of the Morlet wavelet for demodulating defect-induced vibrations, which was further enhanced by cross-correlating the wavelet coefficients from several sub-frequency bands that contribute to the defectinduced vibrations. Yan and Gao designed a multi-scale enveloping spectrogram (MuSEnS) technique by integrating CWT with envelope analysis to process vibration signals for bearing fault diagnosis [68,69]. Fig. 3 shows an example of bearing defect identification by applying the MuSEnS technique to vibration signals measured from a bearing test system, where a seeded defect with 0.1 mm diameter hole was introduced into the outer raceway of a SKF N205 bearing. The bearing was subject to a radial load at 1833 N, and running under 1200 rpm. Based on the geometrical parameter of the bearing and its rotating speed, a defect characteristic frequency at 105 Hz can be analytically calculated. Such frequency component was clearly seen across various scales in the spectrogram. The technique was also applied to monitor spindle conditions [70]. It was further improved by adding computed order tracking to eliminate the effect of speed variation [71]. In addition to above applications, the CWT was also applied to estimate instantaneous rotating speed of rotary machines [72], and isolate random vibrations from measured signals for extracting useful information about bearing health [73]. More interesting, the Morlet wavelet was used as kernel function to form a wavelet support vector machine (WSVM) for gearbox fault diagnosis [74]. The similar WSVM approach was designed by Widodo and Yang to process transient current signals for induction machine diagnosis [75]. The Morlet wavelet was also used as activation function to construct a wavelet neural network (WNN) for fault diagnosis of locomotive roller bearings [76]. 3.2. DWT for rotary machine fault diagnosis The multi-resolution analysis ability of the DWT make it suitable for revealing fault-related information from non-stationary signals sampled on rotary machines. As an example, Kim et al. conducted a comparative study on

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Fig. 3. Example of bearing defect identification through multi-scale enveloping spectrogram.

damage detection of rotor systems [77]. It was found that among various time–frequency techniques including STFT, WVD and DWT, the DWT is the most effective one in extracting features related to shaft-crack condition from vibration signals collected during acceleration and deceleration process. Ohue et al. evaluated the dynamic characteristics of gear sets in time–frequency domain using DWT and CWT [78], where it was found that failed tool can be identified by looking at intensity changes of wavelet coefficient of the vibration signals. Omar and Gaouda designed a dynamic windowed wavelet multi-resolution analysis approach to detect and localize gear tooth defects in a noisy environment [79]. Djebala optimized the wavelet multi-resolution analysis using kurtosis as a measure to get clear signal for early defect detection [80]. Kumar and Singh used the Symlet wavelet as the wavelet function to perform DWT on the bearing vibration signal for measuring its out race defect width [81]. Overcomplete rational dilation DWT was also studied to extract periodical impulses for incipient bearing fault identification [82]. DWT-based denoising has been widely adopted in many studies to remove strong background noise and enhance fault-related information contained in measured signals. For example, a block boot-strapping with white noise test was used to find optimal decomposition level to perform an enhanced DWT-based threshold denoising for

bearing condition monitoring [83]. Li et al. developed a gear multi-fault diagnosis method based on integration of the DWT, autoregressive (AR) model and principal component analysis (PCA), where the DWT was performed to denoise the raw vibration signals [84]. Kwak applied DWT to detect machine tool failure and denoise the cutting force signal in a turning process [85]. By analyzing wavelet coefficients of the cutting force signal, occurrence of tool failure and chatter vibration can be identified. Mohanty and Kar used the DWT to demodulate motor current signal for removing noise and unrelated intervening neighboring feature [86], and then spectrum containing gear mesh frequencies at certain decomposition level was selected for gear fault diagnosis. They further integrated motor current signal analysis with DWT to monitor gear vibrations [87]. Extracting a good set of fault-related features from wavelet coefficients helps to identify machine defects in a much effective way. Li et al. extracted multiscale slope features from slope of logarithmic variances calculated from wavelet coefficients of the DWT, and used them for classifying both bearing and gearbox faults with high accuracy as well as high stability [88]. Wang et al. extracted a frequency spectrum growth index (FSGI) from optimal detail coefficients of the DWT and used it to characterize machine health condition quantitatively [89]. Purushotham et al. studied multi-fault diagnosis of


Fig. 4. Example bearing defect identification through unified time-scalefrequency analysis.

rolling bearing by extracting defect-related impulses from wavelet coefficients at Mel-frequency scales [90]. Wang and Jiang performed a DWT-based adaptive filtering operation on vibration signals, which were measured from an aircraft engine rotor experiment, to enhance signal features, and then extracted the correlation dimension parameter to characterize rotor rub-impact fault deterioration grade [91]. Yu et al. clustered the coefficients of the DWT based on entropy value of coefficients from a library of representative signals, and extracted features from energy content of the clusters as input to a probabilistic neural network for bearing fault diagnosis [92]. Zhou et al. presented a wavelet correlation modeling approach to monitor the status of fluid dynamic bearings in brushless DC motors, where the correlation model was established between the wavelet coefficient features and the bearing wear through multivariable regression [93]. Niu et al. applied the DWT to start-up transient current signals of the induction motor for extracting fault-related features [94], such as RMS, shape factor, and kurtosis, etc., which are classified by several classifiers. The classification results from all the classifiers are fused using Bayesian belief fusion and multi-agent fusion to achieve improved accuracy of fault diagnosis. With the help of other techniques, the ability of DWT in fault diagnosis has been improved. For example, Castejon et al. developed a two-stage bearing fault diagnosis method, where the first stage is to use DWT-based

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multi-resolution analysis for extracting interesting features from signals and the second stage is to use a supervised neural network for classifying different bearing conditions in a very incipient stage [95]. Lou and Laparo combined the DWT and an adaptive neural fuzzy inference system (ANFIS) to separate different bearing fault conditions subject to varying loads [96]. Wu et al. applied the DWT and the ANFIS to classifying gear faults [97]. Fuzzy logic inference or neural network alone was also combined with DWT by Wu and Chan for gear fault identification [98,99]. Xian integrated the DWT with SVM to classify mechanical failures of spherical roller bearing in a high performance hydraulic injection molding machine [100]. Sanz et al. made use of the capability of DWT in dealing with transient signals and the ability of unsupervised auto-associate neural networks in extracting features to diagnose rotating machine faults [101]. Sanz et al. also analyzed coefficients obtained from DWT of the signals and then used a multi-layer perceptron neural network to monitor gear dynamics [102]. Saravanan and Ramachandran investigated the usage of the DWT for feature extraction and a decision tree for spur bevel gearbox fault classification [103]. They also combined DWT with ANN for diagnosing incipient gearbox faults [104]. Li et al. proposed an intelligent method for bearing fault diagnosis by using DWT and ant colony optimization (ACO) [105]. Combining the strength of both the time-scale and frequency domain techniques, a unified time-scale-frequency analysis (TSFA) technique that integrates the DWT with spectral analysis was developed to enhance the overall effectiveness of feature extraction and defect extraction [106–108]. Fig. 4 shows an example of using the TSFA technique to identify the bearing defect, where the vibration signal measured on a SKF 6220 ball bearing with a 0.25 diameter hole on its inner raceway is first processed by the DWT. A Fourier transform is subsequently applied to the wavelet coefficients, as seen in the middle of the Fig. 4. The spectrum of the wavelet coefficients shown in the bottom of Fig. 4 indicates the existence of an inner raceway defect, as there is an appreciable frequency peak identified at the corresponding location. Furthermore, frequency features extracted using the TSFA technique was used as input to a multi-layer perceptron neural network for machine defect severity classification [109]. 3.3. WPT for rotary machine fault diagnosis The enhanced signal decomposition capability in high frequency region makes WPT an attractive tool for detecting and differentiating transient components with high frequency characteristics. As an example, the WPT was comparatively studied with other time–frequency analysis techniques in non-stationary signal processing used for bearing health monitoring [110]. It was found that the WPT was able to identify defect-induced transient components embedded within the bearing vibration signal. Furthermore, it was also able to identify the frequency shifts associated with the phase transitions of defect growth. As shown in Fig. 5, the vibration signal was measured on a Timken 1100KR bearing from a run-tofailure test. The time-frequency analysis results using WPT

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Fig. 5. WPT result of a bearing vibration signal at defect growth phase transition.

has revealed the frequency change of the defect-induced transient vibrations with time varying when the bearing defect size increases. The WPT was also used as a preprocessor to decompose the original signal into narrow band signals and helped to improve the performance of Hilbert–Huang transform [111]. The results showed that the WPT-assisted HHT is more effective in detecting bearing defect-induced impact vibrations than scalogram. Tian et al. used the WPT as a preprocessing tool to reduce the cross-term interference that is normally existed in Wigner–Ville distribution (WVD), and enhanced the ability of WVD in bearing fault diagnosis [112]. Similar to the DWT, features extracted from wavelet coefficients of the WPT have also been widely used for characterizing machine faults. For example, Zarei and Poshtan applied the WPT to induction motor stator current and extracted sub-frequency band energy as fault index to detect bearing faults [113]. Boskoski and Juricic extracted Renyi entropy values from coefficients of vibration signals to detect mechanical faults in rotational drives [114]. Such features were found to be sensitive to fault occurrence and robust to varying operating conditions. Li et al. developed a wavelet-based higher-order statistics approach for fault diagnosis in rolling bearings, where Kurtosis values extracted from wavelet coefficients of both the WPT and DWT were used as a measure for damage detection and classification [115]. Feng et al. proposed two normalized wavelet packets quantifiers: the wavelet packets relative energy and the total wavelet packets entropy, extracted from acoustic emission signals of faulty bearings, to detect localized bearing defects [116]. Cao and Qiao developed a nonlinear damage diagnosis strategy by using WPT and fractals [117], where the wavelet packet component signal assurance criterion and correlation-integral based damage indictor were constructed to describe damage information.

While various features can be extracted from each subfrequency band of the WPT-based signal decomposition, defect-related signatures are mostly concentrated in only a few sub-frequency bands. Selecting significant features from the pool of WPT-based feature set has been investigated to improve efficiency of fault diagnosis by some researchers. By utilizing advantage of the simplicity expression of the harmonic wavelet, Yan and Gao designed an efficient harmonic WPT-based feature selection approach to evaluate the machine health status [118], where the Fisher linear discriminant criterion was used to guide selection of key sub-frequency bands’ energy features. Hong et al. also proposed a diagnosis method based on WPT and Fisher's linear discriminant [119]. By finding the best sub-frequency band from the energy map of the whole time–frequency domain, the normal, cracked and broken conditions of the bevel gear were clearly classified, and the fault positions were accurately detected. Yan and Gao proposed a WPTbased principal feature analysis (PFA) to select the most representative features for spindle bearing defect severity evaluation [120]. The PFA, as an extension of the PCA, took all the principal components into account to reduce information redundancy, while at the same time achieving improved defect classification rate with a few representative sub-frequency band features. There are multiple ways to decompose a signal using WPT, making it possible to optimize the process of signal decomposition for improved machine fault diagnosis. Liu selected the orthogonal basis from two sets of wavelet packet basis functions for rotary machine fault diagnosis, where one set represents localized fault-related transients chosen by using kurtosis measure, and the other contains other components selected by employing the Coifman and Wickerhauser's best basis algorithm [121]. By identifying the local discriminant bases from the WPT results, Wu et al. identified an aeroengine rub-impact fault [122]. The same approach was also used for gearbox defect severity assessment [123]. In addition, Yang et al. used the maximal overlap WPT to analyze gear fault vibration signals and concluded that when integrating with Hilbert spectrum, it gave excellent results [124]. Ocak et al. designed a new scheme that combines WPT and Hidden Markov modeling (HMM) to track bearing wear as well as its severity [125]. After DWTbased denoising, Zhao and Yan performed an ant colony clustering analysis on WPT-based band energy features for characterizing machine tool degradation from a drill test bed [126]. The WPT was also used with hybrid SVM to form an intelligent method for machine fault diagnosis [127]. Al-Badour et al. combined WPT and CWT to detect impulsive faults like rubbing from a custom-built rotor kit [128]. Using the genetic algorithm, Rafiee et al. optimized the decomposition level of the WPT, order of the Daubechies wavelet function, and number of neurons in hidden layer of an ANN to identify slight-worn, medium-worn and broken-tooth of gears faults perfectly [129]. 3.4. SGWT for rotary machine fault diagnosis Due to its fast calculation and easy implementation, applications of the SGWT for fault diagnosis of rotary machines have been seen increasingly in recent years.


Using SGWT, Chen et al. extracted features from vibration signals of the water hydraulic motor to classify different piston conditions [130]. Gao et al. applied redundant SGWT to acoustic emission signals collected on lowspeed heavy-duty gears and identified tooth-face abrasion and peel-off faults [131]. He et al. detected milling cutter breakage damage by applying the SGWT to acoustic emission signals measured from the milling process of a CNC machine [132,133]. Duan and Gao applied a lifting-based undecimated wavelet transform to signals obtained from an air compressor and a generator for the purpose of getting more diagnostic information [134]. To solve the problem of frequency aliasing in fault feature extraction, Bao et al. transformed the signal using an anti-aliasing lifting scheme to achieve high fault classification accuracy [135]. Using adaptive redundant lifting scheme, Jiang et al. analyzed vibration signals from a gearbox with wear fault and extracted the impulse components from the complex background [136]. The self-adaptive redundant SGWT was used to decompose a vibration signal measured on a 552732QT antifriction bearing with outer race slight fault, as shown in Fig. 6. Based on its rotating speed at 385 rpm and its geometrical parameters, characteristic frequency of the antifriction bearing with outer race slight fault is calculated as f0 ¼ 43.6 Hz.

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The prediction and updating operators of the second generation wavelet were devised to self-adaptively match fault characteristic of the antifriction bearing outer raceway. The prediction operator coefficients are calculated as [0.1302, −0.0934, −0.0159, 0.4791, 04791, −0.0159, −0.0934, 0.1302], and the updating operator coefficients are calculated as [0.1306, 0.1194, 0.1194, 0.1306]. By adopting the two operators, the vibration signal is redundantly decomposed. The approximation signal (a1) and detail signal (d1) are illustrated in Fig. 7. As it can be seen in Fig. 7, the detail signal clearly reveals periodic impulses hidden in the vibration signal. Its period is about 21.7 ms, which corresponds to a 46.10 Hz frequency component. This is close to fault characteristic frequency on antifriction bearing outer raceway. As a result, the antifriction bearing outer raceway fault in its early stage is identified. Derived from adaptive lifting scheme, Li et al. devised an adaptive morphological gradient lifting wavelet to enhance impulsive feature raised by bearing defects while at the same time depressing the noise contained in the original signal [137]. A customized wavelet denoising approach was developed by considering the intra- and inter-scale dependency for effective bearing fault detection [138]. Li et al. proposed a redundant nonlinear SGWT-based denoising method to remove noise contained in mechanical faulty

Fig. 6. An acceleration signal of antifriction bearing with outer race slight fault.

Fig. 7. The approximation signal (a1) and detail signal (d1) of self-adaptive redundant second generation wavelet decomposition.

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signals [139]. In addition, the SGWT was combined with independent component analysis to detect weak signatures of bearing faults [140]. Gao et al. used lifting scheme based wavelet transform, together with SVM and rule-based reasoning, to identify broken cog in a gearbox clearly [141]. Chen et al. combined the adaptive redundant SGWT and the Hilbert transform to analyze vibration signals from a machine tool spindle and identified bearing faults [142]. The improved redundant lifting scheme realized through normalization, together with the shock pulse method (SPM), was utilized by Li et al. for monitoring bearing health status [143]. The lifting-based WPT has also been investigated to process signals for characterizing machine health conditions. Lei et al. used this method to decompose vibration signals and extracted both time- and frequency-domain features from the decomposed sub-frequency band signals as well as the original signals [144]. Then sensitive features selected from the feature sets through a distance evaluation technique are input to a weighted K nearest neighbor classifier for bearing fault type and severity level classification. The improved WPT extended from SGWT was used to decompose vibration signals, and node energy features were extracted to fit a support vector data description (SVDD) model for generating a health index, which characterized the bearing performance degradation effectively [145]. Hu et al. integrated the lifting-based WPT with SVMs ensembles to separate different bearing faults conditions and assess incipient fault severity [146]. A mean model that combines the lifting-based WPT, Fisher criterion, and Fuzzy c-means clustering was constructed by Huang et al. to assess rotary machine performance with reduced time consumption and improved efficiency [147]. Pan et al. also used the lifting-based WPT and Fuzzy c-means to build an assessment model to track the performance degradation of bearings [148]. In addition, the redundant lifting-based WPT was applied to vibration signals for achieving high classification performance in mechanical equipment fault diagnosis [149,150]. Huang et al. proposed an enhanced model by using lifting-based WPT and sampling-importance-resampling methods to improve the efficiency of feature extraction [151]. Zhang et al. also applied the lifting-based WPT to extract features for characterizing various bearing faults and a range of fault severities in an induction motor [152]. 4. Research trends of wavelets in rotary machine fault diagnosis With the development for many years, the wavelets have been widely applied to process signals in the area of fault diagnosis. Besides those summarized applications in previous section, there are some new research trends in using wavelets for faults diagnosis, as briefly discussed below. Wavelets can represent mathematical functions as a set of components that represent constituent features at different scales. Wavelet numerical methods, including wavelet finite element method (WFEM) [153], have thus been studied by many researchers. In traditional finite element method (FEM), many basis functions used for structural analysis can be obtained by substituting scaling

functions or wavelet functions for conventional polynomials and taking advantage of multi-resolution property of the wavelet. As a result, basis functions with different resolutions and scales can be adopted for the requirement of precision, and this provides an alternative way to identify structural parameters for detecting machine faults. In recently years, Finite element of B-Spline wavelet on the interval was formulated by Chen et al. [154], and successfully used to study wave propagation in onedimension structures [155] and identify both the crack location and size of rotors [156–158]. Shift-variant property is inherent in DWT/WPT and SGWT because of down-sampling or splitting operation. This means a small shift in the input signal will cause large variations in energy distribution of the wavelet coefficients at different scales [159]. This may affect their effectiveness in detecting transients related to machine faults. The dualtree complex wavelet transform (DT-CWT), due to its approximately shift-invariance property, has attracted some researchers’ attention. As an example, a signal denoising method based on DT-CWT was introduced to reveal periodic impulses in gearbox vibration signals [160]. The DT-CWT was also studied to enhance signal denoising performance and detect multiple fault signatures [161,162]. In some circumstance, it was found that it is powerful to diagnose multiple faults of bearings and outperforms SGWT and other techniques. Abundance of wavelet functions, which can be used in wavelet transform for signal analysis, has been developed over the past decades. Since the choice of wavelet function in the first place may affect the result of wavelet transform at the end, such abundance raises a natural question as to how to choose a wavelet function that is best suited for analyzing a specific signal. To tackle this question, Jiang and Liu applied t-test to validate the correlation between features extracted from DWT coefficients and the original signal, and then applied the estimated probability from the t-test to guide selection of wavelet functions [163]. Schukin et al. used the minimum total error and time– frequency resolution to evaluate different wavelet functions for impulsive parameter identification of a single-degree-offreedom system model [164]. Yan and Gao studied the energy measure [165] and cross-correlation measure [166] for wavelet selection in bearing vibration analysis. They also designed a maximum energy-to-Shannon entropy ratio criterion for quantitatively evaluating performance of five complex-valued wavelet functions in rolling bearing fault diagnosis [167]. Such criterion, together with maximum relative wavelet energy, was used by Kankar et al. to select appropriate wavelet function from six candidate wavelets for extracting bearing defect-induced features [168]. Wavelet functions were generally developed primarily from a mathematical standpoint (e.g. orthonormal and biorthonormal wavelet functions with compact support), without a direct link to physical systems that would be analyzed. On the other hand, signals to be analyzed in engineering are generally associated with a machine or a structure. This has motivated the research in constructing new wavelet functions for engineering applications. For example, an impulse response wavelet was formulated from unit-impulse response function to detect bearing


defects [169]. The impulse wavelet was also designed from the impulse response of a hammer-strike on a rolling bearing to enhance the performance of bearing defect detection [170,171]. Li et al. designed a new signaladapted lifting scheme, in which the wavelet function was directly constructed from the statistics of the signal itself, for fault diagnosis of rotary machines [172]. Duan et al. constructed a wavelet with impact characteristics using the lifting scheme to extract transients for predicting incipient fault in rotary machines [173]. A new family of biorthogonal wavelet was designed to detect the changes and the presents of gear faults [174]. Using lifting scheme, a new set of biorthogonal wavelet was also designed to extract weak fault features from a heavy oil catalytic cracking unit [175]. In addition, as we all know, the wavelet-based diagnostics is realized by matching and extracting fault-related features that are most similar to a given wavelet function. However, different faults may result in various dynamic responses or even the same type of fault generates diverse signatures in different machine structures. Traditionally, both the classical WT and SGWT only use a single wavelet function to capture fault-related features. To overcome such a limitation, multi-wavelet, that offers multiple wavelet functions, has been introduced to match one or more faults for diagnosis purpose. For example, a multiwavelet system was chosen to analyze vibration signal of a gearing system with cracked tooth. With the tachometer signal as a reference, location of the cracked tooth was identified [176]. Using two-scale similarity transform, Yuan et al. constructed a library of multi-wavelets [177], from which the one that is the most similar to fault features of vibration signals was chosen, based on kurtosis maximum principle, and then applied to diagnose outer raceway defect in a bearing and rub-impact fault in a gas turbine unit. The new multiwavelet constructed via adaptive lifting was also applied to analyze vibration signals from gearbox [178], and used together with sliding window denoising to enable effective bearing fault detection [179]. Furthermore, the multiwavelet denoising with improved neighboring coefficients was developed for diagnosing locomotive rolling bearing faults [180]. Adaptive redundant multiwavelet packet was formulated to facilitate compound faults detection [181], and used together with spectral kurtosis to enhance fault detection capability in rotary machines [182]. 5. Concluding remarks In this paper, we have provided a review on utilizing wavelets as a powerful tool for signal analysis with the purpose of fault diagnosis in rotary machines. We know that a topic review on all of the literatures published in archival journals and proceedings is not possible, and what we have done is to summarize typical applications appeared in the past 10 years based on the following four categories: CWT-based fault diagnosis, DWT-based fault diagnosis, WPT-based fault diagnosis, and SGWT-based fault diagnosis. Besides, some new research trends, such as WFEM, DT-CWT, wavelet selection, wavelet design, and multi-wavelet, that help to improve the effectiveness of

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fault diagnosis in rotary machines, have also been discussed. It should be noted that there still exist some challenges in using wavelets for rotary machine fault diagnosis. For example, the essence of wavelet transform is a kind of correlation calculation between the signal to be analyzed and the wavelet function in different scales. The more similar the signal is to the wavelet function, the better the defect-related features will be extracted. Therefore, constructing new wavelet functions that adaptively match the defect-related signal characteristics is one of the key issues that need further exploration. However, with the wavelet transform becoming more and more mature and new theoretical contributions being made, it is believed that wavelets will continue to be one of the most appealing techniques that dominate the field of rotary machine fault diagnosis.

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Wavelets for fault diagnosis of rotary machines_ A review with

Wavelets for fault diagnosis of rotary machines_ A review with

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