Recent advances in time–frequency analysis methods for machinery fault diagnosis: A review with application examples
Highlights
► We present a systematic review of recent developments in time–frequency analysis methods. ► With a focus on nonstationary signal analysis, we revisited more than 100 representative articles published since 1990. ► More than 20 major methods, classified into six categories, have been examined in the context of machinery fault diagnosis. ► The principle, illustration, application review and remarks are provided for each of these methods. ► Application cases have also been presented to demonstrate the applications of several of the reviewed methods.
Introduction
The intrinsic dynamic nature and/or the external excitations, especially under time-varying operational conditions during machine operations often lead to nonstationary signals. Such nonstationary signals contain rich information about machinery health conditions. Therefore, important fault features can be extracted from these signals for fault detection and diagnosis, if proper analysis methods are applied [1].
The statistics of nonstationary signals change with time. However, most of the traditional signal analysis methods are based on the stationary assumption. Such methods can only provide analyses in terms of the statistical average in the time or frequency domain, but cannot reveal the local features in both time and frequency domains simultaneously. Thus it is not appropriate to apply such methods to nonstationary signal analysis in engineering applications. The development of signal analysis methods suited to extracting the time-varying features from nonstationary signals has become increasingly important for machinery fault diagnosis [1], [2], [3].
Joint time–frequency analysis is an effective approach to addressing these issues. With this method, signals are presented in a time–frequency–amplitude/energy density 3D space. Hence, both the constituent frequency components and their time variation features can be revealed [4], [5], [6], [7], [8], [9]. To date, various time–frequency analysis methods have been proposed. The traditional methods include linear time–frequency representations such as short time Fourier transform and wavelet transform [10], [11], [12], [13] and bilinear/quadratic time–frequency distributions, e.g., Cohen and affine class distributions based on Wigner–Ville distribution [14], [15], [16]. Adaptive optimal kernel methods can modify the kernel adaptively, thus making the time–frequency distribution suitable to identify signal structures [17], [18]. To suppress the cross-terms and improve time–frequency resolution, the reassignment method has been proposed [19]. Considering the nonlinearity and non-Gaussianity of signals, researchers have integrated higher order statistics with time–frequency analysis to construct time-varying higher order spectrum methods such as Wigner higher order spectrum [20], [21], L-class [22] and S-class distributions [23]. In order to match various types of structures in nonstationary signals, researchers have extended basis expansion to atomic decomposition [7]. Atomic decomposition can represent arbitrary signals using some optimal functions adaptively chosen from a library of functions which are not limited to orthogonal basis. Such atomic decomposition based methods lead to adaptive parametric time–frequency analysis because they involve optimization in signal decomposition. The method of frame [24], best orthogonal basis [25], matching pursuit [26], [27], [28], [29] and basis pursuit [30] are some good examples. The performance of atomic decomposition methods depends on the diversity of atoms in the dictionary employed. To match the complex structures of signals, researchers have developed various dictionaries such as Gabor, wavelet, wavelet packets, chirplets, FMmlets, as well as Dopplerlets [30], [31], [32], [33], [34]. Recently, the time invariant auto-regressive moving average (ARMA) models have been extended to time–frequency analysis to handle nonstationary signals [35]. Another key issue in nonstationary signal analysis is the accurate estimation of the instantaneous frequency. To this end, several adaptive non-parametric methods have been developed recently. These include empirical mode decomposition (EMD) [36], [37], [38], [39], [40], its variant ensemble empirical mode decomposition (EEMD) [41], and local mean decomposition (LMD) [42]. These methods are used to decompose signals thus satisfying the mono-component requirement by instantaneous frequency estimation. In addition, the energy separation method [43], [44], [45], [46] has also been suggested to estimate both instantaneous frequency and envelope amplitude in a fine resolution.
Most of the above mentioned time–frequency analysis methods have been applied to machinery fault diagnosis. An overview of the related studies would motivate or inspire researchers and engineers to improve the existing signal feature extraction methods and explore new ones. The existence of the large body of literature developed over the past few decades makes it unrealistic to review each and every article published in this field. Therefore, we will focus on recent key advances in time–frequency analysis methods and their typical applications in machinery fault diagnosis. We further limit the articles to those published after the 1990s. This review is structured as follows. We first revisit the traditional methods including linear and bilinear time–frequency representations in 2 Linear time–frequency representation, 3 Bilinear time–frequency distribution, respectively. We then review the time-varying higher order spectra developed based on quadratic distributions in Section 4. In 5 Adaptive parametric time–frequency analysis, 6 Time–frequency ARMA models, we present an overview of the newly developed parametric methods, i.e. time–frequency analysis based on atomic decomposition and time-varying ARMA models, followed by a review of some data driven methods including EMD, EEMD, LMD, and energy separation in Section 7. In the next section, we present some application examples to illustrate the applications of a few recently developed methods in machinery fault diagnosis. Finally, in Section 9, we summarize the pros and cons of these time–frequency analysis methods, and point out some application prospects in machinery fault diagnosis.
Section snippets
Linear time–frequency representation
Linear time–frequency representation is essentially a process to decompose signals into a weighted sum of a series of bases localized in both time and frequency domains, for example, short time Fourier transform and wavelet transform [10], [11]. They are free from the cross term interferences which are inherent with bilinear time–frequency distributions (to be reviewed in Section 3), but the time–frequency resolution is governed by the Heisenberg uncertainty principle. Due to the trade-off
Bilinear time–frequency distribution
Bilinear time–frequency distribution represents the signal energy distribution in the joint time–frequency domain. The Wigner–Ville distribution is the basis of almost all the bilinear time–frequency distributions. It has the highest time–frequency resolution. However, for multi-component signals, it suffers from the inevitable cross-term interferences, thus is not suitable for many real applications. In order to maintain a higher time–frequency resolution, to obtain a non-negative
Time-varying higher order spectra
The non-Gaussian nature of signals can be characterized by higher order statistics. Such statistics are effective for suppressing Gaussian white noise or colored noise with improved signal-to-ratio. This method can identify the information of signal bias from the Gaussian distribution. They can extract the higher order correlation reflecting the nonlinearity of a system. However, as the higher order statistics and higher order spectra are based on the stationarity assumption, it is
Adaptive parametric time–frequency analysis
The methods of adaptive parametric time–frequency analysis are based on atomic decomposition [5]. They represent a signal by a model to best match the time–frequency features according to the signal structural characteristics. This model describes signals in a simple yet sparse way, and can condense the information contained in signals. From the model, we can find the true constituent signal components, thus reveal the auto-terms on the time–frequency plane, completely remove the cross-terms,
Time–frequency ARMA models
Parametric models for random processes, such as auto-regressive (AR), moving average (MA), and ARMA, are well known and have been applied to signal analysis. Recently, Jachan et al. [35] extended these time invariant parametric models to the time–frequency domain, and proposed time–frequency auto-regressive moving average (TFARMA) models for describing underspread nonstationary processes. They added both time delays and frequency shifts to time invariant ARMA models to capture the nonstationary
Adaptive non-parametric time–frequency analysis
Adaptive non-parametric approaches include empirical mode decomposition and local mean decomposition based time–frequency analysis methods. They extract the intrinsic oscillating mode by data fitting or data smoothing. Hence these approaches are completely signal-driven and there is no need to construct any basis to match the signal components.
Application examples
Conventional time–frequency analysis methods (such as wavelet transform and bilinear time–frequency distributions) have been widely used in machinery fault diagnosis. Considering some key issues like the contradiction between time localization and frequency resolution, cross-term interferences, the multi-component features, nonstationarity, nonlinearity and non-Gaussianity of signals, we focus on time-varying higher order spectra, adaptive parametric time–frequency analysis and adaptive
Summary and prospects
Nonstationary signal analysis is a common yet key issue for machinery fault diagnosis, especially when machinery is running under time-varying conditions. Proper time–frequency analysis can be used to identify the constituent components of signals and their time variation, and thus to analyze nonstationary signals. The methods reviewed in this paper have their respective advantages and disadvantages, as summarized in Table 1. In real applications, one should select a proper method according to
Acknowledgment
This work is supported by the National Natural Science Foundation of China (51075028 and 11272047), the Beijing Natural Science Foundation (3102022), the Program for New Century Excellent Talents in University (NCET-12–0775), the Natural Sciences and Engineering Research Council of Canada (I2IPJ 387179 and RGPIN 121433-2011) and the Ontario Centers of Excellence (OT-SE-E50622). The authors are grateful to the two anonymous reviewers for their valuable and detailed comments and suggestions which
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