Adaptive Mode Decomposition Methods and Their Applications in Signal Analysis for Machinery Fault Diagnosis: A Review With Examples
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paper_title: ON INSTANTANEOUS FREQUENCY

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Instantaneous frequency (IF) is necessary for understanding the detailed mechanisms for nonlinear and nonstationary processes. Historically, IF was computed from analytic signal (AS) through the Hilbert transform. This paper offers an overview of the difficulties involved in using AS, and two new methods to overcome the difficulties for computing IF. The first approach is to compute the quadrature (defined here as a simple 90° shift of phase angle) directly. The second approach is designated as the normalized Hilbert transform (NHT), which consists of applying the Hilbert transform to the empirically determined FM signals. Additionally, we have also introduced alternative methods to compute local frequency, the generalized zero-crossing (GZC), and the teager energy operator (TEO) methods. Through careful comparisons, we found that the NHT and direct quadrature gave the best overall performance. While the TEO method is the most localized, it is limited to data from linear processes, the GZC method is the m...
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paper_title: A review on empirical mode decomposition in fault diagnosis of rotating machinery

paper_content:

Abstract Rotating machinery covers a broad range of mechanical equipment and plays a significant role in industrial applications. It generally operates under tough working environment and is therefore subject to faults, which could be detected and diagnosed by using signal processing techniques. Empirical mode decomposition (EMD) is one of the most powerful signal processing techniques and has been extensively studied and widely applied in fault diagnosis of rotating machinery. Numerous publications on the use of EMD for fault diagnosis have appeared in academic journals, conference proceedings and technical reports. This paper attempts to survey and summarize the recent research and development of EMD in fault diagnosis of rotating machinery, providing comprehensive references for researchers concerning with this topic and helping them identify further research topics. First, the EMD method is briefly introduced, the usefulness of the method is illustrated and the problems and the corresponding solutions are listed. Then, recent applications of EMD to fault diagnosis of rotating machinery are summarized in terms of the key components, such as rolling element bearings, gears and rotors. Finally, the outstanding open problems of EMD in fault diagnosis are discussed and potential future research directions are identified. It is expected that this review will serve as an introduction of EMD for those new to the concepts, as well as a summary of the current frontiers of its applications to fault diagnosis for experienced researchers.
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paper_title: Application of EMD method and Hilbert spectrum to the fault diagnosis of roller bearings

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Based upon empirical mode decomposition (EMD) method and Hilbert spectrum, a method for the fault diagnosis of roller bearings is proposed in this paper. The local Hilbert spectrum and local Hilbert marginal spectrum are introduced. The orthogonal wavelet bases are used to translate vibration signals of a roller bearing into time-scale representation. Then, an envelope signal can be obtained by envelope spectrum analysis of wavelet coefficients of high scales. By applying EMD method and Hilbert transform to the envelope signal, we can get the local Hilbert marginal spectrum from which the faults in a roller bearing can be diagnosed and fault patterns can be identified. Practical vibration signals measured from roller bearings with out-race faults or inner-race faults are analysed by the proposed method. The results show that the proposed method is superior to the traditional envelope spectrum method in extracting the fault characteristics of roller bearings.
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paper_title: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

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A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...
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paper_title: Time-varying vibration decomposition and analysis based on the Hilbert transform

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This paper introduces a simple method for time-varying vibration decomposition based on the Hilbert transform. The non-stationary frequency of the largest component is estimated as an average function of the instantaneous frequency of the composition, and the corresponding envelope is estimated according to synchronous demodulation. The method is demonstrated using computer simulation of different types of non-stationary and nonlinear vibration.
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paper_title: On amplitude and frequency demodulation using energy operators

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It is shown that the nonlinear energy-tracking signal operator Psi (x)=(dx/dt)/sup 2/-xd/sup 2/x/dt/sup 2/ and its discrete-time counterpart can estimate the AM and FM modulating signals. Specifically, Psi can approximately estimate the amplitude envelope of AM signals and the instantaneous frequency of FM signals. Bounds are derived for the approximation errors, which are negligible under general realistic conditions. These results, coupled with the simplicity of Psi , establish the usefulness of the energy operator for AM and FM signal demodulation. These ideas are then extended to a more general class of signals that are sine waves with a time-varying amplitude and frequency and thus contain both an AM and an FM component; for such signals it is shown that Psi can approximately track the product of their amplitude envelope and their instantaneous frequency. The theoretical analysis is done for both continuous- and discrete-time signals. >
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paper_title: The local mean decomposition and its application to EEG perception data

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This paper describes the local mean decomposition (LMD), a new iterative approach to demodulating amplitude and frequency modulated signals. The new method decomposes such signals into a set of functions, each of which is the product of an envelope signal and a frequency modulated signal from which a time-varying instantaneous frequency can be derived. The LMD method can be used to analyse a wide variety of natural signals such as electrocardiograms, functional magnetic resonance imaging data, and earthquake data. The paper presents the results of applying LMD to a set of scalp electroencephalogram (EEG) visual perception data. The LMD instantaneous frequency and energy structure of the EEG is examined, and compared with results obtained using the spectrogram. The nature of visual perception is investigated by measuring the degree of EEG instantaneous phase concentration that occurs following stimulus onset over multiple trials. The analysis suggests that there is a statistically significant difference between the theta phase concentrations of the perception and no perception EEG data.
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paper_title: Application of the wavelet transform in machine condition monitoring and fault diagnostics: a review with bibliography

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Abstract The application of the wavelet transform for machine fault diagnostics has been developed for last 10 years at a very rapid rate. A review on all of the literature is certainly not possible. The purpose of this review is to present a summary about the application of the wavelet in machine fault diagnostics, including the following main aspects: the time–frequency analysis of signals, the fault feature extraction, the singularity detection for signals, the denoising and extraction of the weak signals, the compression of vibration signals and the system identification. Some other applications are introduced briefly as well, such as the wavelet networks, the wavelet-based frequency response function, etc. In addition, some problems in using the wavelet for machine fault diagnostics are analysed. The prospects of the wavelet analysis in solving non-linear problems are discussed.
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paper_title: A rolling bearing fault diagnosis approach based on LCD and fuzzy entropy

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Abstract When the fault happens, the complexity of the intrinsic oscillation from the mechanical system will change. Fuzzy entropy (FuzzyEn), which is defined to measure the complexity and self-similarity of the time series, can be utilized to measure the complexity of vibration signal and reflect changes of complexity of the intrinsic oscillation. Since the changes distribute in different scales, a new non-stationary signal analysis method, local characteristic-scale decomposition (LCD), is proposed and used to decompose the vibration signal adaptively into series of intrinsic scale components (ISC) in different scales. And then a new rolling bearing fault diagnosis approach based on LCD and FuzzyEn is proposed. Namely, firstly, by using the LCD rolling bearing vibration signal is decomposed into numbers of ISCs; then the FuzzyEns of the first few ISCs that contain main failure information are extracted. Thirdly, the FuzzyEns obtained are taken as the inputs to the adaptive neuro-fuzzy inference systems (ANFIS) classifier. Finally, the proposed method is applied to experimental data, and the analysis results show that the proposed method performs effectively for the rolling bearing fault diagnosis.
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paper_title: Nonlinear vibrating system identification via Hilbert decomposition

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Abstract This paper deals with the identification of nonlinear vibration systems, based on measured signals for free and forced vibration regimes. Two categories of time domain signal are analyzed, one of a fast inter-modulation signal and a second as composed of several mono-components. To some extent, this attempts to imitate analytic studies of such systems, with its two major analysis groups – the perturbation and the harmonic balance methods. Two appropriate signal processing methods are then investigated, one based on demodulation and the other on signal decomposition. The Hilbert Transform (HT) has been shown to enable effective and simple methods of analysis. We show that precise identification of the nonlinear parameters can be obtained, contrary to other average HT based methods where only approximation parameters are obtained. The effectiveness of the proposed methods is demonstrated for the precise nonlinear system identification, using both the signal demodulation and the signal decomposition methods. Following the exposition of the tools used, both the signal demodulation as well as decomposition are applied to classical examples of nonlinear systems. Cases of nonlinear stiffness and damping forces are analyzed. These include, among other, an asymmetric Helmholtz oscillator, a backlash with nonlinear turbulent square friction, and a Duffing oscillator with dry friction.
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paper_title: Energy separation in signal modulations with application to speech analysis

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An efficient solution to the fundamental problem of estimating the time-varying amplitude envelope and instantaneous frequency of a real-valued signal that has both an AM and FM structure is provided. Nonlinear combinations of instantaneous signal outputs from the energy operator are used to separate its output energy product into its AM and FM components. The theoretical analysis is done first for continuous-time signals. Then several efficient algorithms are developed and compared for estimating the amplitude envelope and instantaneous frequency of discrete-time AM-FM signals. These energy separation algorithms are used to search for modulations in speech resonances, which are modeled using AM-FM signals to account for time-varying amplitude envelopes and instantaneous frequencies. The experimental results provide evidence that bandpass-filtered speech signals around speech formants contain amplitude and frequency modulations within a pitch period. >
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paper_title: ITERATIVE FILTERING AS AN ALTERNATIVE ALGORITHM FOR EMPIRICAL MODE DECOMPOSITION

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The empirical mode decomposition (EMD) was a method pioneered by (N. Huang et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear nonstationary time series analysis, Proc. Roy. Soc. Lond. A454 (1998) 903–995) as an alternative technique to the traditional Fourier and wavelet techniques for studying signals. It decomposes a signal into several components called intrinsic mode functions (IMFs), which have shown to admit better behaved instantaneous frequencies via Hilbert transforms. In this paper, we propose an alternative algorithm for EMD based on iterating certain filters, such as Toeplitz filters. This approach yields similar results as the more traditional sifting algorithm for EMD. In many cases the convergence can be rigorously proved.
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paper_title: Adaptive Local Iterative Filtering for Signal Decomposition and Instantaneous Frequency analysis

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Abstract Time–frequency analysis for non-linear and non-stationary signals is extraordinarily challenging. To capture features in these signals, it is necessary for the analysis methods to be local, adaptive and stable. In recent years, decomposition based analysis methods, such as the empirical mode decomposition (EMD) technique pioneered by Huang et al., were developed by different research groups. These methods decompose a signal into a finite number of components on which the time–frequency analysis can be applied more effectively. In this paper we consider the Iterative Filtering (IF) approach as an alternative to EMD. We provide sufficient conditions on the filters that ensure the convergence of IF applied to any L 2 signal. Then we propose a new technique, the Adaptive Local Iterative Filtering (ALIF) method, which uses the IF strategy together with an adaptive and data driven filter length selection to achieve the decomposition. Furthermore we design smooth filters with compact support from solutions of Fokker–Planck equations (FP filters) that can be used within both IF and ALIF methods. These filters fulfill the derived sufficient conditions for the convergence of the IF algorithm. Numerical examples are given to demonstrate the performance and stability of IF and ALIF techniques with FP filters. In addition, in order to have a complete and truly local analysis toolbox for non-linear and non-stationary signals, we propose new definitions for the instantaneous frequency and phase which depend exclusively on local properties of a signal.
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paper_title: Intrinsic time-scale decomposition: time–frequency–energy analysis and real-time filtering of non-stationary signals

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We introduce a new algorithm, the intrinsic time-scale decomposition (ITD), for efficient and precise time–frequency–energy (TFE) analysis of signals. The ITD method overcomes many of the limitations of both classical (e.g. Fourier transform or wavelet transform based) and more recent (empirical mode decomposition based) approaches to TFE analysis of signals that are nonlinear and/or non-stationary in nature. The ITD method decomposes a signal into (i) a sum of proper rotation components, for which instantaneous frequency and amplitude are well defined, and (ii) a monotonic trend. The decomposition preserves precise temporal information regarding signal critical points and riding waves, with a temporal resolution equal to the time-scale of extrema occurrence in the input signal. We also demonstrate how the ITD enables application of single-wave analysis and how this, in turn, leads to a powerful new class of real-time signal filters, which extract and utilize the inherent instantaneous amplitude and frequency/phase information in combination with other relevant morphological features.
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paper_title: Variational Mode Decomposition

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During the late 1990s, Huang introduced the algorithm called Empirical Mode Decomposition, which is widely used today to recursively decompose a signal into different modes of unknown but separate spectral bands. EMD is known for limitations like sensitivity to noise and sampling. These limitations could only partially be addressed by more mathematical attempts to this decomposition problem, like synchrosqueezing, empirical wavelets or recursive variational decomposition. Here, we propose an entirely non-recursive variational mode decomposition model, where the modes are extracted concurrently. The model looks for an ensemble of modes and their respective center frequencies, such that the modes collectively reproduce the input signal, while each being smooth after demodulation into baseband. In Fourier domain, this corresponds to a narrow-band prior. We show important relations to Wiener filter denoising. Indeed, the proposed method is a generalization of the classic Wiener filter into multiple, adaptive bands. Our model provides a solution to the decomposition problem that is theoretically well founded and still easy to understand. The variational model is efficiently optimized using an alternating direction method of multipliers approach. Preliminary results show attractive performance with respect to existing mode decomposition models. In particular, our proposed model is much more robust to sampling and noise. Finally, we show promising practical decomposition results on a series of artificial and real data.
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paper_title: Empirical Wavelet Transform

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Some recent methods, like the empirical mode decomposition (EMD), propose to decompose a signal accordingly to its contained information. Even though its adaptability seems useful for many applications, the main issue with this approach is its lack of theory. This paper presents a new approach to build adaptive wavelets. The main idea is to extract the different modes of a signal by designing an appropriate wavelet filter bank. This construction leads us to a new wavelet transform, called the empirical wavelet transform. Many experiments are presented showing the usefulness of this method compared to the classic EMD.
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paper_title: Nonlinear Mode Decomposition: a new noise-robust, adaptive decomposition method

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We introduce a new adaptive decomposition tool, which we refer to as Nonlinear Mode Decomposition (NMD). It decomposes a given signal into a set of physically meaningful oscillations for any waveform, simultaneously removing the noise. NMD is based on the powerful combination of time-frequency analysis techniques - which together with the adaptive choice of their parameters make it extremely noise-robust - and surrogate data tests, used to identify interdependent oscillations and to distinguish deterministic from random activity. We illustrate the application of NMD to both simulated and real signals, and demonstrate its qualitative and quantitative superiority over the other existing approaches, such as (ensemble) empirical mode decomposition, Karhunen-Loeve expansion and independent component analysis. We point out that NMD is likely to be applicable and useful in many different areas of research, such as geophysics, finance, and the life sciences. The necessary MATLAB codes for running NMD are freely available at http://www.physics.lancs.ac.uk/research/nbmphysics/diats/nmd/.
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paper_title: Recent advances in time–frequency analysis methods for machinery fault diagnosis: A review with application examples

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Abstract Nonstationary signal analysis is one of the main topics in the field of machinery fault diagnosis. Time–frequency analysis can identify the signal frequency components, reveals their time variant features, and is an effective tool to extract machinery health information contained in nonstationary signals. Various time–frequency analysis methods have been proposed and applied to machinery fault diagnosis. These include linear and bilinear time–frequency representations (e.g., wavelet transform, Cohen and affine class distributions), adaptive parametric time–frequency analysis (based on atomic decomposition and time–frequency auto-regressive moving average models), adaptive non-parametric time–frequency analysis (e.g., Hilbert–Huang transform, local mean decomposition, and energy separation), and time varying higher order spectra. This paper presents a systematic review of over 20 major such methods reported in more than 100 representative articles published since 1990. Their fundamental principles, advantages and disadvantages, and applications to fault diagnosis of machinery have been examined. Some examples have also been provided to illustrate their performance.
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paper_title: A NEW VIEW OF NONLINEAR WATER WAVES: The Hilbert Spectrum

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We survey the newly developed Hilbert spectral analysis method and its applications to Stokes waves, nonlinear wave evolution processes, the spectral form of the random wave field, and turbulence. Our emphasis is on the inadequacy of presently available methods in nonlinear and nonstationary data analysis. Hilbert spectral analysis is here proposed as an alternative. This new method provides not only a more precise definition of particular events in time-frequency space than wavelet analysis, but also more physically meaningful interpretations of the underlying dynamic processes.
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paper_title: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

paper_content:

A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...
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paper_title: A confidence limit for the empirical mode decomposition and Hilbert spectral analysis

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The confidence limit is a standard measure of the accuracy of the result in any statistical analysis. Most of the confidence limits are derived as follows. The data are first divided into subsections and then, under the ergodic assumption, the temporal mean is substituted for the ensemble mean. Next, the confidence limit is defined as a range of standard deviations from this mean. However, such a confidence limit is valid only for linear and stationary processes. Furthermore, in order for the ergodic assumption to be valid, the subsections have to be statistically independent. For non‐stationary and nonlinear processes, such an analysis is no longer valid. The confidence limit of the method here termed EMD/HSA (for empirical mode decomposition/Hilbert spectral analysis) is introduced by using various adjustable stopping criteria in the sifting processes of the EMD step to generate a sample set of intrinsic mode functions (IMFs). The EMD technique acts as a pre‐processor for HSA on the original data, producing a set of components (IMFs) from the original data that equal the original data when added back together. Each IMF represents a scale in the data, from smallest to largest. The ensemble mean and standard deviation of the IMF sample sets obtained with different stopping criteria are calculated, and these form a simple random sample set. The confidence limit for EMD/HSA is then defined as a range of standard deviations from the ensemble mean. Without evoking the ergodic assumption, subdivision of the data stream into short sections is unnecessary; hence, the results and the confidence limit retain the full‐frequency resolution of the full dataset. This new confidence limit can be applied to the analysis of nonlinear and non‐stationary processes by these new techniques. Data from length‐of‐day measurements and a particularly violent recent earthquake are used to demonstrate how the confidence limit is obtained and applied. By providing a confidence limit for this new approach, a stable range of stopping criteria for the decomposition or sifting phase (EMD) has been established, making the results of the final processing with HSA, and the entire EMD/HSA method, more definitive.
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paper_title: A study of the characteristics of white noise using the empirical mode decomposition method

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Based on numerical experiments on white noise using the empirical mode decomposition (EMD) method, we find empirically that the EMD is effectively a dyadic filter, the intrinsic mode function (IMF) components are all normally distributed, and the Fourier spectra of the IMF components are all identical and cover the same area on a semi–logarithmic period scale. Expanding from these empirical findings, we further deduce that the product of the energy density of IMF and its corresponding averaged period is a constant, and that the energy–density function is chi–squared distributed. Furthermore, we derive the energy–density spread function of the IMF components. Through these results, we establish a method of assigning statistical significance of information content for IMF components from any noisy data. Southern Oscillation Index data are used to illustrate the methodology developed here.
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paper_title: COMPLEMENTARY ENSEMBLE EMPIRICAL MODE DECOMPOSITION: A NOVEL NOISE ENHANCED DATA ANALYSIS METHOD

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The phenomenon of mode-mixing caused by intermittence signals is an annoying problem in Empirical Mode Decomposition (EMD) method. The noise assisted method of Ensemble EMD (EEMD) has not only effectively resolved this problem but also generated a new one, which tolerates the residue noise in the signal reconstruction. Of course, the relative magnitude of the residue noise could be reduced with large enough ensemble, it would be too time consuming to implement. An improved algorithm of noise enhanced data analysis method is suggested in this paper. In this approach, the residue of added white noises can be extracted from the mixtures of data and white noises via pairs of complementary ensemble IMFs with positive and negative added white noises. Though this new approach yields IMF with the similar RMS noise as EEMD, it effectively eliminated residue noise in the IMFs. Numerical experiments were conducted to demonstrate the new approach and also illustrate the problems of mode splitting and translation.
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paper_title: On empirical mode decomposition and its algorithms

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Huang’s data-driven technique of Empirical Mode Decomposition (EMD) is presented, and issues related to its effective implementation are discussed. A number of algorithmic variations, including new stopping criteria and an on-line version of the algorithm, are proposed. Numerical simulations are used for empirically assessing performance elements related to tone identification and separation. The obtained results support an interpretation of the method in terms of adaptive constant-Q filter banks.
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paper_title: ENSEMBLE EMPIRICAL MODE DECOMPOSITION: A NOISE-ASSISTED DATA ANALYSIS METHOD

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A new Ensemble Empirical Mode Decomposition (EEMD) is presented. This new approach consists of sifting an ensemble of white noise-added signal (data) and treats the mean as the final true result. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As EEMD is a time–space analysis method, the added white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the component of the signal (original data), which is then treated as the true and more physical meaningful answer. The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF. With this ensemble mean, one can separate scales naturall...
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paper_title: On empirical mode decomposition and its algorithms

paper_content:

Huang’s data-driven technique of Empirical Mode Decomposition (EMD) is presented, and issues related to its effective implementation are discussed. A number of algorithmic variations, including new stopping criteria and an on-line version of the algorithm, are proposed. Numerical simulations are used for empirically assessing performance elements related to tone identification and separation. The obtained results support an interpretation of the method in terms of adaptive constant-Q filter banks.
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paper_title: ENSEMBLE EMPIRICAL MODE DECOMPOSITION: A NOISE-ASSISTED DATA ANALYSIS METHOD

paper_content:

A new Ensemble Empirical Mode Decomposition (EEMD) is presented. This new approach consists of sifting an ensemble of white noise-added signal (data) and treats the mean as the final true result. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As EEMD is a time–space analysis method, the added white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the component of the signal (original data), which is then treated as the true and more physical meaningful answer. The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF. With this ensemble mean, one can separate scales naturall...
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paper_title: An enhanced empirical mode decomposition method for blind component separation of a single-channel vibration signal mixture

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Blind component separation aims to decompose a single-channel vibration signal mixture into periodic components and random transient components. In addition to periodic components, random transient components with a high degree of impulsiveness are signals of interest in practice. An adaptive signal processing method called empirical mode decomposition (EMD) decomposes a nonlinear and non-stationary signal into the sum of simple components termed intrinsic mode functions (IMFs). Ensemble empirical mode decomposition (EEMD) is an improvement of EMD and aims to relieve a mode mixing problem that exists in EMD. However, there is no universal standard formula that can be used to select appropriate parameters of EEMD. Improper parameters of EEMD still cause a mode mixing problem that makes a signal of a similar scale reside in some successive IMFs. An enhanced EEMD for the purpose of blind component separation is developed in this paper to respectively extract periodic components and random transient component...
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paper_title: A new wind turbine fault diagnosis method based on ensemble intrinsic time-scale decomposition and WPT-fractal dimension

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In this paper an ensemble intrinsic time-scale decomposition (EITD) method based on the cubic spline interpolation and linear transformation of intrinsic time-scale decomposition (ITD) was proposed, which can restrain the end effect and avoid the signal distortion. Combining ensemble intrinsic time-scale decomposition (EITD) with wavelet packet transform (WPT) and correlation dimension (CD), a novel method for decomposing nonstationary vibration signal and diagnosing wind turbine faults is presented. In implementation of the method, wavelet packet transform is employed to denoise raw vibration signals. Some important influencing factors relating directly to the computational precision of correlation dimension are discussed. The advantage of combining EITD and fractal dimension is that it does recognize the wind turbine gearbox fault types, and can solve the difficulty of recognizing fault conditions when two or more fractal dimensions are close to each other. To verify the effectiveness of the EITD-WPT-CD in detecting the faults, their induced vibrations are collected from high speed shaft gear under normal and faulty conditions through acceleration measurement. The results show that this method is capable of extracting the signal features and identifying the working conditions. The fault diagnosis application in a wind turbine gearbox indicates that the proposed method improved the accuracy of fault diagnosis.
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paper_title: A review on empirical mode decomposition in fault diagnosis of rotating machinery

paper_content:

Abstract Rotating machinery covers a broad range of mechanical equipment and plays a significant role in industrial applications. It generally operates under tough working environment and is therefore subject to faults, which could be detected and diagnosed by using signal processing techniques. Empirical mode decomposition (EMD) is one of the most powerful signal processing techniques and has been extensively studied and widely applied in fault diagnosis of rotating machinery. Numerous publications on the use of EMD for fault diagnosis have appeared in academic journals, conference proceedings and technical reports. This paper attempts to survey and summarize the recent research and development of EMD in fault diagnosis of rotating machinery, providing comprehensive references for researchers concerning with this topic and helping them identify further research topics. First, the EMD method is briefly introduced, the usefulness of the method is illustrated and the problems and the corresponding solutions are listed. Then, recent applications of EMD to fault diagnosis of rotating machinery are summarized in terms of the key components, such as rolling element bearings, gears and rotors. Finally, the outstanding open problems of EMD in fault diagnosis are discussed and potential future research directions are identified. It is expected that this review will serve as an introduction of EMD for those new to the concepts, as well as a summary of the current frontiers of its applications to fault diagnosis for experienced researchers.
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paper_title: Application of EMD method and Hilbert spectrum to the fault diagnosis of roller bearings

paper_content:

Based upon empirical mode decomposition (EMD) method and Hilbert spectrum, a method for the fault diagnosis of roller bearings is proposed in this paper. The local Hilbert spectrum and local Hilbert marginal spectrum are introduced. The orthogonal wavelet bases are used to translate vibration signals of a roller bearing into time-scale representation. Then, an envelope signal can be obtained by envelope spectrum analysis of wavelet coefficients of high scales. By applying EMD method and Hilbert transform to the envelope signal, we can get the local Hilbert marginal spectrum from which the faults in a roller bearing can be diagnosed and fault patterns can be identified. Practical vibration signals measured from roller bearings with out-race faults or inner-race faults are analysed by the proposed method. The results show that the proposed method is superior to the traditional envelope spectrum method in extracting the fault characteristics of roller bearings.
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paper_title: A rotating machinery fault diagnosis method based on local mean decomposition

paper_content:

Local mean decomposition (LMD) is a novel self-adaptive time-frequency analysis method, which is particularly suitable for the processing of multi-component amplitude-modulated and frequency-modulated (AM-FM) signals. By using LMD, any complicated signal can be decomposed into a number of product functions (PFs), each of which is the product of an envelope signal and a purely frequency modulated signal from which physically meaningful instantaneous frequencies can be obtained. In fact, each PF is just a mono-component AM-FM signal. Therefore, the procedure of LMD may be regarded as the process of demodulation. While fault occurs in gear or roller bearing, the vibration signals picked up would exactly display AM-FM characteristics. So it is possible to diagnose gear and roller bearing fault by LMD. Targeting the modulation features of the gear or roller bearing fault vibration signal, a rotating machinery fault diagnosis method based on LMD is proposed. In this paper, firstly the LMD method is introduced; secondly, the LMD method is compared with another competing time-frequency analysis approach, namely, empirical mode decomposition (EMD) method and the results show the superiority of the LMD method; finally, the LMD method is applied to the gear and roller bearing fault diagnosis. The analysis results from the practical gearbox vibration signal demonstrate that the diagnosis approach based on LMD could identify gear and roller bearing work condition accurately and effectively.
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paper_title: Multivariate and multiscale monitoring of large-size low-speed bearings using Ensemble Empirical Mode Decomposition method combined with Principal Component Analysis

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Abstract With a view to detecting incipient failures in large-size low-speed rolling bearings and ensuring minimal effect of subjectivity on the process, a new data-driven multivariate and multiscale statistical monitoring method is proposed. The proposed method which combines the Principal Component Analysis (PCA) multivariate monitoring approach and the Ensemble Empirical Mode Decomposition (EEMD) method, which adaptively decomposes signals into various time scales, was called the EEMD-based multiscale PCA (EEMD–MSPCA). The method is very general in nature, which is why it could also be used in different areas and for various tasks. It can be used for controlling each time scale of decomposition or only the selected ones, for multivariate and multiscale filtering or for monitoring system operation on the basis of reconstructed i.e. filtered signals. The efficiency of the proposed EEMD–MSPCA method for the task of bearing condition monitoring and signal filtering was evaluated on simulated as well as on actual vibration and Acoustic Emission (AE) signals measured on a purpose built test stand. The fact that the proposed method is able to identify the local bearing defect of a very small size indicates that AE and vibration signals carry sufficient information on the bearing condition and that the proposed EEMD–MSPCA method ensures high-reliability bearing fault detection.
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paper_title: Mono-component feature extraction for mechanical fault diagnosis using modified empirical wavelet transform via data-driven adaptive Fourier spectrum segment

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Abstract Due to the multi-modulation feature in most of the vibration signals, the extraction of embedded fault information from condition monitoring data for mechanical fault diagnosis still is not a relaxed task. Despite the reported achievements, Wavelet transform follows the dyadic partition scheme and would not allow a data-driven frequency partition. And then Empirical Wavelet Transform (EWT) is used to extract inherent modulation information by decomposing signal into mono-components under an orthogonal basis and non-dyadic partition scheme. However, the pre-defined segment way of Fourier spectrum without dependence on analyzed signals may result in inaccurate mono-component identification. In this paper, the modified EWT (MEWT) method via data-driven adaptive Fourier spectrum segment is proposed for mechanical fault identification. First, inner product is calculated between the Fourier spectrum of analyzed signal and Gaussian function for scale representation. Then, adaptive spectrum segment is achieved by detecting local minima of the scale representation. Finally, empirical modes can be obtained by adaptively merging mono-components based on their envelope spectrum similarity. The adaptively extracted empirical modes are analyzed for mechanical fault identification. A simulation experiment and two application cases are used to verify the effectiveness of the proposed method and the results show its outstanding performance.
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paper_title: Decomposition of non-stationary signals into varying time scales: Some aspects of the EMD and HVD methods

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Abstract The Empirical Mode Decomposition has become very popular since its first introduction. Its suitability and expected performance for specific signal processing task is however somewhat open ended. Addressed are basic questions concerning the decomposition of signals according to different time scales, from noise sensitivity to frequency resolution. Comparing it to the recently introduced Hilbert Vibration Decomposition, signals simulating moving mechanical systems are analyzed. It is argued that in spite of signal adaptive properties, a universal fully automatic decomposition is still beyond our capabilities.
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paper_title: Joint amplitude and frequency demodulation analysis based on local mean decomposition for fault diagnosis of planetary gearboxes

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Abstract The vibration signals of faulty planetary gearboxes have complicated spectral structures due to the amplitude modulation and frequency modulation (AMFM) nature of gear damage induced vibration and the additional multiplicative amplitude modulation (AM) effect caused by the time-varying vibration transfer paths (for local gear damage case) and the passing planets (for distributed gear damage case). The spectral complexity leads to the difficulty in fault diagnosis of planetary gearboxes. Observing that both the amplitude envelope and the instantaneous frequency of planetary gearbox vibration signals are associated with the characteristic frequency of the faulty gear, a joint amplitude and frequency demodulation method is proposed for fault diagnosis of planetary gearboxes. In order to satisfy the mono-component requirement by instantaneous frequency estimation, a signal is firstly decomposed into product functions (PF) using the local mean decomposition (LMD) method. Then, the earliest extracted PF that has an instantaneous frequency fluctuating around the gear meshing frequency or its harmonics is chosen for further analysis, because it contains most of the information about the gear fault. The amplitude demodulation analysis can be accomplished through Fourier transforming the amplitude envelope of the chosen PF. For the frequency demodulation analysis, Fourier transform is applied to the estimated instantaneous frequency of the chosen PF to reveal its fluctuating frequency, thus obtaining the spectrum of the instantaneous frequency. By joint application of the amplitude and frequency demodulation methods, planetary gearbox faults can be diagnosed by matching the dominant peaks in the envelope spectrum and the spectrum of instantaneous frequency with the theoretical characteristic frequencies of faulty gears. The performance of the proposed method is illustrated by simulated signal analysis, and is validated by experimental signal analysis of a lab planetary gearbox with intentionally created pitting and naturally developed wear.
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paper_title: Partly ensemble empirical mode decomposition: An improved noise-assisted method for eliminating mode mixing

paper_content:

The problem of mode mixing caused by intermittence or noise signal is irritating in empirical mode decomposition (EMD) method. A noise-assisted data analysis method called ensemble EMD (EEMD) was proposed recently to resolve this problem. However, EEMD will generate a new problem that tolerates the residue noise of the signal reconstruction. As an enhanced EEMD, complementary EEMD (CEEMD), is developed to restrain the reconstruction residue and yields IMFs with the similar RMS noise as EEMD and effectively eliminates residue noise in the IMFs. However, whether EEMD or CEEMD, they both generate false components when the chosen parameters are inappropriate and the final obtained IMFs are not really meeting the definition of IMF. In this paper a partly ensemble EMD (PEEMD) method is proposed to resolve the mode mixing problem. In PEEMD method, after the intermittency or noise signal is obtained by an ensemble way and is detected by permutation entropy, the residual signal is decomposed directly by using the EMD method. By analyzing simulation and real signal, the analysis results indicate that PEEMD can eliminate the residue noise in the IMFs effectively and generates IMFs with better performance, and represents a sound improvement over the original EMD, EEMD and CEEMD.
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paper_title: Time-varying vibration decomposition and analysis based on the Hilbert transform

paper_content:

This paper introduces a simple method for time-varying vibration decomposition based on the Hilbert transform. The non-stationary frequency of the largest component is estimated as an average function of the instantaneous frequency of the composition, and the corresponding envelope is estimated according to synchronous demodulation. The method is demonstrated using computer simulation of different types of non-stationary and nonlinear vibration.
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paper_title: A Fault Diagnosis Scheme for Rolling Bearing Based on Particle Swarm Optimization in Variational Mode Decomposition

paper_content:

Variational mode decomposition (VMD) is a new method of signal adaptive decomposition. In the VMD framework, the vibration signal is decomposed into multiple mode components by Wiener filtering in Fourier domain, and the center frequency of each mode component is updated as the center of gravity of the mode’s power spectrum. Therefore, each decomposed mode is compact around a center pulsation and has a limited bandwidth. In view of the situation that the penalty parameter and the number of components affect the decomposition effect in VMD algorithm, a novel method of fault feature extraction based on the combination of VMD and particle swarm optimization (PSO) algorithm is proposed. In this paper, the numerical simulation and the measured fault signals of the rolling bearing experiment system are analyzed by the proposed method. The results indicate that the proposed method is much more robust to sampling and noise. Additionally, the proposed method has an advantage over the EMD in complicated signal decomposition and can be utilized as a potential method in extracting the faint fault information of rolling bearings compared with the common method of envelope spectrum analysis.
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paper_title: A hybrid fault diagnosis method based on second generation wavelet de-noising and local mean decomposition for rotating machinery

paper_content:

Abstract In order to extract fault features of large-scale power equipment from strong background noise, a hybrid fault diagnosis method based on the second generation wavelet de-noising (SGWD) and the local mean decomposition (LMD) is proposed in this paper. In this method, a de-noising algorithm of second generation wavelet transform (SGWT) using neighboring coefficients was employed as the pretreatment to remove noise in rotating machinery vibration signals by virtue of its good effect in enhancing the signal–noise ratio (SNR). Then, the LMD method is used to decompose the de-noised signals into several product functions (PFs). The PF corresponding to the faulty feature signal is selected according to the correlation coefficients criterion. Finally, the frequency spectrum is analyzed by applying the FFT to the selected PF. The proposed method is applied to analyze the vibration signals collected from an experimental gearbox and a real locomotive rolling bearing. The results demonstrate that the proposed method has better performances such as high SNR and fast convergence speed than the normal LMD method.
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paper_title: Filter bank property of variational mode decomposition and its applications

paper_content:

The variational mode decomposition (VMD) was proposed recently as an alternative to the empirical mode decomposition (EMD). To shed further light on its performance, we analyze the behavior of VMD in the presence of irregular samples, impulsive response, fractional Gaussian noise as well as tones separation. Extensive numerical simulations are conducted to investigate the parameters mentioned in VMD on these filter bank properties. It is found that, unlike EMD, the statistical characterization of the obtained modes reveals a different equivalent filter bank structure, robustness with respect to the nonuniformly sampling and good resolution in spectrum analysis. Moreover, we illustrate the influences of the main parameters on these properties, which provides a guidance on tuning them. Based on these findings, three potential applications in extracting time-varying oscillations, detrending as well as detecting impacts using VMD are presented. HighlightsAn in-depth elaboration on the inherent characteristics of VMD is investigated through extensive numerical experiments.A well-controlled use of the VMD technique for specific application is addressed.Three potential applications of VMD are illustrated.
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paper_title: Bearing fault detection based on hybrid ensemble detector and empirical mode decomposition

paper_content:

Abstract Aiming at more efficient fault diagnosis, this research work presents an integrated anomaly detection approach for seeded bearing faults. Vibration signals from normal bearings and bearings with three different fault locations, as well as different fault sizes and loading conditions are examined. The Empirical Mode Decomposition and the Hilbert Huang transform are employed for the extraction of a compact feature set. Then, a hybrid ensemble detector is trained using data coming only from the normal bearings and it is successfully applied for the detection of any deviation from the normal condition. The results prove the potential use of the proposed scheme as a first stage of an alarm signalling system for the detection of bearing faults irrespective of their loading condition.
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paper_title: Amplitude and frequency demodulation analysis for fault diagnosis of planet bearings

paper_content:

Abstract The vibration signals of planet bearings feature more complex amplitude modulation and frequency modulation (AM-FM) than those of fixed-axis bearings, due to the effects of time-varying vibration transfer path, load zone passing and time-varying angles between gear pair mesh lines of action and impact force vector, in addition to that of bearing fault. Their Fourier spectra have a complex sideband structure, leading to difficulty in feature extraction and fault diagnosis. In order to address this issue, a joint amplitude and frequency demodulation analysis method is proposed to reveal the fault features, by considering the modulation characteristics. To thoroughly understand planet bearing vibration characteristics, the explicit equations for amplitude and frequency demodulated spectra of outer race, rolling element and inner race fault cases are derived, and the fault symptoms are summarized respectively. The signal is firstly decomposed into intrinsic mode functions (IMFs) by empirical mode decomposition (EMD), thus meeting the mono-component requirement by instantaneous frequency calculation. Then a sensitive component with the instantaneous frequency fluctuating around the resonance frequency is chosen for further frequency demodulation analysis. Finally, planet bearing fault can be diagnosed by matching the peaks identified in amplitude and frequency demodulated spectra with the theoretical fault characteristic frequencies. The proposed method is validated with both numerical simulated and lab experimental signal analyses.
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paper_title: Fault diagnosis of a wind turbine rolling bearing using adaptive local iterative filtering and singular value decomposition

paper_content:

Adaptive local iterative filtering (ALIF) is a new signal decomposition method that uses the iterative filters strategy together with an adaptive and data-driven filter length selection to achieve the decomposition. The complexity of wind power generation systems means that the randomness and kinetic mutation behaviour of their vibration signals are demonstrated at different scales. Thus it is necessary to analyse the vibration signal across multiple scales. A method based on ALIF and singular value decomposition (SVD) was used for the fault diagnosis of a wind turbine roller bearing. The ALIF method is used to decompose the bearing vibration signal into several stable components. The components, which contain major fault information, are selected to build an initial feature vector matrix. The singular value of the matrix is computed as the feature vectors of each bearing fault. The feature vectors embody the characteristics of the vibration signal. The nearest neighbour algorithm is used as a classifier ...
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paper_title: A comparative study between Empirical Wavelet Transforms and Empirical Mode Decomposition Methods: Application to bearing defect diagnosis

paper_content:

Abstract The Ensemble Empirical Mode Decomposition (EEMD) is a noise assisted method that may sometimes provide a significant improvement on Empirical Mode Decomposition (EMD). However, the amplitude and number of added noise need to be selected when applying the EEMD method. Furthermore, the computation time which depends on the number of ensemble trails is very high compared to the EMD. In this paper, a new way for choosing the appropriate added noise is presented. Conversely, a recently-developed method called the Empirical Wavelet Transform (EWT) is investigated. A comparative study between the EMD and EWT methods is conducted. The results show that the EWT is better than the EEMD and EMD on mode estimates and computation time is significantly reduced. An experimental study on bearing diagnosis is conducted. The EWT is applied to experimental data coming from damaged bearings. In the paper, an index selection is introduced that allows for the automatic selection of the Intrinsic Mode Functions (IMF) that should be used to perform the envelope spectrum. It is shown that choosing all the IMF selected by the index is more efficient than only choosing the best one. The envelope of the sum of the selected IMF clearly reveals the bearing frequencies and its harmonics which are excited by the defect. This approach seems to be an effective and efficient method for processing bearing fault signals.
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paper_title: A rolling bearing fault diagnosis approach based on LCD and fuzzy entropy

paper_content:

Abstract When the fault happens, the complexity of the intrinsic oscillation from the mechanical system will change. Fuzzy entropy (FuzzyEn), which is defined to measure the complexity and self-similarity of the time series, can be utilized to measure the complexity of vibration signal and reflect changes of complexity of the intrinsic oscillation. Since the changes distribute in different scales, a new non-stationary signal analysis method, local characteristic-scale decomposition (LCD), is proposed and used to decompose the vibration signal adaptively into series of intrinsic scale components (ISC) in different scales. And then a new rolling bearing fault diagnosis approach based on LCD and FuzzyEn is proposed. Namely, firstly, by using the LCD rolling bearing vibration signal is decomposed into numbers of ISCs; then the FuzzyEns of the first few ISCs that contain main failure information are extracted. Thirdly, the FuzzyEns obtained are taken as the inputs to the adaptive neuro-fuzzy inference systems (ANFIS) classifier. Finally, the proposed method is applied to experimental data, and the analysis results show that the proposed method performs effectively for the rolling bearing fault diagnosis.
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paper_title: A new approach based on OMA-empirical wavelet transforms for bearing fault diagnosis

paper_content:

Abstract Amplitude demodulation is a key means of diagnosing bearing faults. The quality of demodulation determines the effectiveness of spectrum analysis in detecting defects. However, the quality of the demodulated signal depends on the frequency band selected for demodulation. In this paper, a new method combining the empirical wavelet transform (EWT) and operational modal analysis (OMA) is proposed. EWT acts like a filter bank in which the support boundaries of the filter are defined using OMA. The proposed method (OMA–EWT) decomposes the signal into multiple components and kurtosis values are used to select automatically the components for performing the envelope spectrum in order to extract the frequency related to the defect. The method is validated on two test benches and a comparative study is conducted with the kurtogram. The results show that the combined OMA–EWT method can improve EWT for decomposing the signal into multiple components. Using OMA–EWT, a selection of all the components excited by the defect gives more accurate diagnostic results.
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paper_title: Local rub-impact fault diagnosis of the rotor systems based on EMD

paper_content:

Aiming at the characteristics of the vibration signal of a rotor system with local rub-impact fault, a local rub-impact fault diagnosis method of rotor system based on EMD (empirical mode decomposition) is proposed in this paper. When the local rub-impact fault occurs in the rotor system, the vibration signals of the rotor system will present amplitude-modulated feature because of the periodic rub-impact between the stator and the rotor. However, it is hard to extract the rub-impact feature for the rub-impact signal is always submerged in the background and noise signals. By using EMD, any complicated signal can be decomposed into a finite and often small number of IMFs (intrinsic mode functions), and each IMF may be amplitude-modulated or frequency-modulated signal. Therefore, the rub-impact signal, background signal and noise signal can be separated efficiently and the fault feature of vibration signal of a rotor system with local rub-impact fault can be extracted after the vibration signal of the rotor system is decomposed by EMD method. The analysis results from the experimental signal show that the intrinsic mode functions of the vibration signal of a rotor system with local rub-impact fault obviously own the amplitude-modulated characteristics, while those under other work conditions own no distinct amplitude-modulated characteristics. Therefore, the EMD method can be applied to the local rub-impact fault diagnosis of the rotor systems efficiently.
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paper_title: Generator bearing fault diagnosis for wind turbine via empirical wavelet transform using measured vibration signals

paper_content:

The implementation of condition monitoring and fault diagnosis system (CMFDS) on wind turbine is significant to lower the unscheduled breakdown. Generator is one of the most important components in wind turbine, and generator bearing fault identification always draws lots of attention. However, non-stationary vibration signal of weak fault and compound fault with a large amount of background noise makes this task challenging in many cases. So, effective signal processing method is essential in the accurate diagnosis step of CMFDS. As a novel signal processing method, empirical Wavelet Transform (EWT) is used to extract inherent modulation information by decomposing signal into mono-components under an orthogonal basis, which is seen as a powerful tool for mechanical fault diagnosis. Moreover, in order to avoid the inaccurate identification the internal modes caused by the heavy noise, wavelet spatial neighboring coefficient denoising with data-driven threshold is applied to increase Signal to Noise Ratio (SNR) before EWT. The effectiveness of the proposed technique on weak fault and compound fault diagnosis is first validated by two experimental cases. Finally, the proposed method has been applied to identify fault feature of generator bearing on wind turbine in wind farm successfully.
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paper_title: Detection of gear faults in variable rotating speed using variational mode decomposition (VMD)

paper_content:

The ensemble empirical mode decomposition (EEMD) was largely used in the diagnosis of the rotating machines, this method could detect the defect at an early stage in the case of non variable speed or slightly variable, but when the speed of the machine varies in acceleration or deceleration the use of the EEMD under these conditions shows a limitation with the detection of the impulses, that are influenced by the presence of the mode mixing, and the end effect. To detect the shocks due to the defect where the variation of speed is forced by the working conditions, we propose to use the Variational Mode Decomposition (VMD) which was recently proposed by Konstantin Dragomiretskiy. This method gave promising results in the detection of the defects on machine elements under non stationary conditions imposed by the variation of speed and torque. In this work, first we show by simulated signal the advantage of VMD compared to the EEMD in the detection of impulses in the case of variable speed and load. Then, we analyze vibration signals given by a dynamic modeling of a gear transmission in the case of non stationary load and speed, for healthy gear and two different of localized faults (early and advanced). The modes are extracted using VMD and followed by calculation of spectrogram and statistics values, which give more information about the defect and allow us to detect it at an early stage.
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paper_title: Rolling bearing fault diagnosis based on LCD–TEO and multifractal detrended fluctuation analysis

paper_content:

Abstract A rolling bearing vibration signal is nonlinear and non-stationary and has multiple components and multifractal properties. A rolling-bearing fault-diagnosis method based on Local Characteristic-scale Decomposition–Teager Energy Operator (LCD–TEO) and multifractal detrended fluctuation analysis (MF-DFA) is first proposed in this paper. First, the vibration signal was decomposed into several intrinsic scale components (ISCs) by using LCD, which is a newly developed signal decomposition method. Second, the instantaneous amplitude was obtained by applying the TEO to each major ISC for demodulation. Third, the intrinsic multifractality features hidden in each major ISC were extracted by using MF-DFA, among which the generalized Hurst exponents are selected as the multifractal feature in this paper. Finally, the feature vectors were obtained by applying principal components analysis (PCA) to the extracted multifractality features, thus reducing the dimension of the multifractal features and obtaining the fault feature insensitive to variation in working conditions, further enhancing the accuracy of diagnosis. According to the extracted feature vector, rolling bearing faults can be diagnosed under variable working conditions. The experimental results demonstrate its desirable diagnostic performance under both different working conditions and different fault severities. Simultaneously, the results of comparison show that the performance of the proposed diagnostic method outperforms that of Hilbert–Huang transform (HHT) combined with MF-DFA or LCD–TEO combined with mono-fractal analysis.
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paper_title: Bearing fault diagnosis of wind turbine based on intrinsic time-scale decomposition frequency spectrum

paper_content:

In order to better identify the complex running conditions of wind turbine main bearings, we developed a bearing fault diagnosis method based on intrinsic time-scale decomposition frequency spectrum. The main bearing acceleration vibration signal from wind turbine is captured under four conditions—good bearing, outer race fault, inner race fault and roller fault. The proposed method consists of the following steps. First, the main bearing acceleration vibration signal is decomposed into several proper rotation components by using the intrinsic time-scale decomposition method. Second, the frequency spectrum of the first few proper rotation components (containing dominant fault features) is analyzed. The dominant resonant frequency range of each analyzed rotation component is derived, and then, the sum of frequency amplitude in said frequency range can be obtained. This sum is regarded as the fault feature vectors. Finally, the fault feature vectors are input to the least square support vectors machine, and the faults of wind turbine main bearing then can be diagnosed. The experiment results show that the proposed method can diagnose failures of wind turbine bearings quickly and more accurately.
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paper_title: Nonlinear vibrating system identification via Hilbert decomposition

paper_content:

Abstract This paper deals with the identification of nonlinear vibration systems, based on measured signals for free and forced vibration regimes. Two categories of time domain signal are analyzed, one of a fast inter-modulation signal and a second as composed of several mono-components. To some extent, this attempts to imitate analytic studies of such systems, with its two major analysis groups – the perturbation and the harmonic balance methods. Two appropriate signal processing methods are then investigated, one based on demodulation and the other on signal decomposition. The Hilbert Transform (HT) has been shown to enable effective and simple methods of analysis. We show that precise identification of the nonlinear parameters can be obtained, contrary to other average HT based methods where only approximation parameters are obtained. The effectiveness of the proposed methods is demonstrated for the precise nonlinear system identification, using both the signal demodulation and the signal decomposition methods. Following the exposition of the tools used, both the signal demodulation as well as decomposition are applied to classical examples of nonlinear systems. Cases of nonlinear stiffness and damping forces are analyzed. These include, among other, an asymmetric Helmholtz oscillator, a backlash with nonlinear turbulent square friction, and a Duffing oscillator with dry friction.
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paper_title: Performance enhancement of ensemble empirical mode decomposition

paper_content:

Ensemble empirical mode decomposition (EEMD) is a newly developed method aimed at eliminating mode mixing present in the original empirical mode decomposition (EMD). To evaluate the performance of this new method, this paper investigates the effect of two parameters pertinent to EEMD: the amplitude of added white noise and the number of ensemble trials. A test signal with mode mixing that mimics realistic bearing vibration signals measured on a bearing test bed was developed to enable quantitative evaluation of the EEMD and provide guidance on how to choose the two parameters appropriately for bearing signal decomposition. Subsequently, a modified EEMD (MEEMD) method is proposed to reduce the computational cost of the original EEMD method as well as improving its performance. Numerical evaluation and systematic study using vibration data measured on an experimental bearing test bed verified the effectiveness and computational efficiency of the proposed MEEMD method for bearing defect diagnosis.
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paper_title: Diagnostics of gear faults based on EMD and automatic selection of intrinsic mode functions

paper_content:

Abstract Signal processing is an important tool for diagnostics of mechanical systems. Many different techniques are available to process experimental signals, among others: FFT, wavelet transform, cepstrum, demodulation analysis, second order ciclostationarity analysis, etc. However, often hypothesis about data and computational efforts restrict the application of some techniques. In order to overcome these limitations, the empirical mode decomposition has been proposed. The outputs of this adaptive approach are the intrinsic mode functions that are treated with the Hilbert transform in order to obtain the Hilbert–Huang spectrum. Anyhow, the selection of the intrinsic mode functions used for the calculation of Hilbert–Huang spectrum is normally done on the basis of user’s experience. On the contrary, in the paper a merit index is introduced that allows the automatic selection of the intrinsic mode functions that should be used. The effectiveness of the improvement is proven by the result of the experimental tests presented and performed on a test-rig equipped with a spiral bevel gearbox, whose high contact ratio made difficult to diagnose also serious damages of the gears. This kind of gearbox is normally never employed for benchmarking diagnostics techniques. By using the merit index, the defective gearbox is always univocally identified, also considering transient operating conditions.
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paper_title: Fault diagnosis for wind turbine planetary gearboxes via demodulation analysis based on ensemble empirical mode decomposition and energy separation

paper_content:

Planetary gearboxes play an important role in wind turbine drive trains. Fault diagnosis of planetary gearboxes is a key topic for maintenance of wind turbines. Considering the spectral complexity of planetary gearbox vibration signals as well as their amplitude modulation and frequency modulation (AMFM) nature, we propose a simple yet effective method to diagnose planetary gearbox faults based on amplitude and frequency demodulations. We use the energy separation algorithm to estimate the amplitude envelope and instantaneous frequency of modulated signals for further demodulation analysis, by exploiting the adaptability of Teager energy operator to instantaneous changes in signals and the fine time resolution. However, the energy separation algorithm requires signals to be mono-components. To satisfy this requirement, we decompose signals into intrinsic mode functions (IMFs) using the ensemble empirical mode decomposition (EEMD) method as it can decompose any signal into mono-components. We further propose a criterion to guide the selection of the most relevant IMF for demodulation analysis according to the wavelet-like filter nature of EEMD and the AMFM characteristics of the planetary gearbox vibration signals. By matching the dominant peaks in the Fourier spectra of the obtained amplitude envelope and instantaneous frequency with the theoretical characteristic frequency of each gear, we can diagnose planetary gearbox faults. The principle and effectiveness of the proposed method are illustrated by simulation and the experimental analysis of signals from a planetary gearbox of a wind turbine test rig. With the proposed method, both the wear and chipping faults can be detected and located for a sun gear of the planetary gearbox test rig.
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paper_title: Pressure fluctuation signal analysis of a hydraulic turbine based on variational mode decomposition

paper_content:

Due to the nonstationarity of the pressure fluctuation signal from a hydraulic turbine draft tube, this research proposes a pressure fluctuation signal analysis method based on variational mode decomposition (VMD). Firstly, the research was used to compare VMD with empirical mode decomposition (EMD) by using a simulation signal, which shows that the VMD method could effectively inhibit mode mixing, but EMD method could not. This could improve the accuracy of such analysis. Then, the VMD method was applied to analyze the real pressure fluctuation signal from a hydraulic turbine. The results showed that VMD produced clearer time–frequency spectra. This allowed accurate analysis of pressure fluctuation signals. This research provided a new method of pressure fluctuation signal analysis for hydraulic turbine draft tubes.
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paper_title: Adaptive signal decomposition based on wavelet ridge and its application

paper_content:

Signal decomposition is a widely-used approach for multicomponent signal processing. To improve the accuracy and anti-noise performance of multicomponent decomposition, this paper proposes a novel multicomponent signal decomposition method based on wavelet ridge extraction, called the Wavelet ridge signal decomposition (WRSD). A wavelet ridge extraction algorithm is introduced. We find that this algorithm can obtain the wavelet ridge of one component in a multicomponent signal and the initial scale will determine the wavelet ridge of which component is extracted. Since the instantaneous frequency obtained by wavelet ridge has small frequency fluctuation, low-pass filtering is used to increase the accuracy of instantaneous frequency estimation. With the improved instantaneous frequency, the synchronous demodulation method is used to separate the corresponding component from the signal composition. By repeating this process, all components can be adaptively and automatically obtained. This method is employed to analyze three typical simulated vibration signals and compared with Hilbet vibration decomposition and empirical mode decomposition. The comparison results demonstrate its superiority in decomposition accuracy and noise insensitivity. Finally, the proposed WRSD method is successfully applied to the diagnosis of a shaft misalignment fault and a gearbox wear fault. An adaptive and automatic multicomponent signal decomposition method is proposed.The initial scale determines the wavelet ridge of which component is extracted.This method has higher accuracy and is less sensitive to noise than HVD and EMD.The method can be effectively applied to the weak fault feature extraction.
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paper_title: Wheel-bearing fault diagnosis of trains using empirical wavelet transform

paper_content:

Abstract Rolling bearings are used widely as wheel bearing in trains. Fault detection of the wheel-bearing is of great significance to maintain the safety and comfort of train. Vibration signal analysis is the most popular technique that is used for rolling element bearing monitoring, however, the application of vibration signal analysis for wheel bearings is quite limited in practice. In this paper, a novel method called empirical wavelet transform (EWT) is used for the vibration signal analysis and fault diagnosis of wheel-bearing. The EWT method combines the classic wavelet with the empirical mode decomposition, which is suitable for the non-stationary vibration signals. The effectiveness of the method is validated using both simulated signals and the real wheel-bearing vibration signals. The results show that the EWT provides a good performance in the detection of outer race fault, roller fault, and the compound fault of outer race and roller.
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paper_title: Underdetermined Blind Source Separation with Variational Mode Decomposition for Compound Roller Bearing Fault Signals

paper_content:

In the condition monitoring of roller bearings, the measured signals are often compounded due to the unknown multi-vibration sources and complex transfer paths. Moreover, the sensors are limited in particular locations and numbers. Thus, this is a problem of underdetermined blind source separation for the vibration sources estimation, which makes it difficult to extract fault features exactly by ordinary methods in running tests. To improve the effectiveness of compound fault diagnosis in roller bearings, the present paper proposes a new method to solve the underdetermined problem and to extract fault features based on variational mode decomposition. In order to surmount the shortcomings of inadequate signals collected through limited sensors, a vibration signal is firstly decomposed into a number of band-limited intrinsic mode functions by variational mode decomposition. Then, the demodulated signal with the Hilbert transform of these multi-channel functions is used as the input matrix for independent component analysis. Finally, the compound faults are separated effectively by carrying out independent component analysis, which enables the fault features to be extracted more easily and identified more clearly. Experimental results validate the effectiveness of the proposed method in compound fault separation, and a comparison experiment shows that the proposed method has higher adaptability and practicability in separating strong noise signals than the commonly-used ensemble empirical mode decomposition method.
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paper_title: A new wind turbine fault diagnosis method based on the local mean decomposition

paper_content:

This paper proposed a novel wind turbine fault diagnosis method based on the local mean decomposition (LMD) technology. Wind energy is a renewable power source that produces no atmospheric pollution. The condition monitoring and fault diagnosis in wind turbine system are important in avoiding serious damage. Vibration analysis is a normal and useful technology in wind turbine condition monitoring and fault diagnosis. However, the relatively slow speed of the wind turbine components set a limitation in early fault diagnosis using vibration monitoring method. The traditional time-frequency analysis techniques have some drawbacks which make them not suitable for the nonlinear, non-Gaussian signal analysis. LMD is a new iterative approach to demodulate amplitude and frequency modulated signals, which is suitable for obtaining instantaneous frequencies in wind turbine condition monitoring and fault diagnosis. The experiment analysis of the wind turbine vibration signal proves the validity and availability of the new method.
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paper_title: Damage detection in gear systems using empirical mode decomposition

paper_content:

A method for monitoring the evolution of gear faults based on the newly developed empirical mode decomposition scheme is presented. A theoretical model for a gear pair with a tooth root crack was developed. Experimental vibration signals from a test rig were decomposed into oscillatory functions called intrinsic mode functions. An empirical law, which relates the energy content of the intrinsic modes to the crack magnitude, was established. The modal energy is thus associated with the deterioration in gear condition and can be utilized for system failure prediction. In addition, it is shown that the instantaneous frequency of the vibration signal is a sensitive indicator of the existence of damage in the gear pair.
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paper_title: A Novel Method for Mechanical Fault Diagnosis Based on Variational Mode Decomposition and Multikernel Support Vector Machine

paper_content:

A novel fault diagnosis method based on variational mode decomposition (VMD) and multikernel support vector machine (MKSVM) optimized by Immune Genetic Algorithm (IGA) is proposed to accurately and adaptively diagnose mechanical faults. First, mechanical fault vibration signals are decomposed into multiple Intrinsic Mode Functions (IMFs) by VMD. Then the features in time-frequency domain are extracted from IMFs to construct the feature sets of mixed domain. Next, Semisupervised Locally Linear Embedding (SS-LLE) is adopted for fusion and dimension reduction. The feature sets with reduced dimension are inputted to the IGA optimized MKSVM for failure mode identification. Theoretical analysis demonstrates that MKSVM can approximate any multivariable function. The global optimal parameter vector of MKSVM can be rapidly identified by IGA parameter optimization. The experiments of mechanical faults show that, compared to traditional fault diagnosis models, the proposed method significantly increases the diagnosis accuracy of mechanical faults and enhances the generalization of its application.
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paper_title: Research on variational mode decomposition and its application in detecting rub-impact fault of the rotor system

paper_content:

Abstract Multi-component extraction is an available method for vibration signal analysis of rotary machinery, so a novel method of rubbing fault diagnosis based on variational mode decomposition (VMD) is proposed. VMD is a newly developed technique for adaptive signal decomposition, which can non-recursively decompose a multi-component signal into a number of quasi-orthogonal intrinsic mode functions. The equivalent filtering characteristics of VMD are investigated, and the behavior of wavelet packet-like expansion is first found based on fractional Gaussian noise via numerical simulations. VMD is then applied to detect multiple rubbing-caused signatures for rotor–stator fault diagnosis via numerical simulated response signal and practical vibration signal. A comparison has also been conducted to evaluate the effectiveness of identifying the rubbing-caused signatures by using VMD, empirical wavelet transform (EWT), EEMD and EMD. The analysis results of the rubbing signals show that the multiple features can be better extracted with the VMD, simultaneously.
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paper_title: Gear fault diagnosis method based on local mean decomposition and generalized morphological fractal dimensions

paper_content:

Abstract Aiming at gear fault diagnosis, a fusion method of local mean decomposition (LMD) and generalized morphological fractal dimensions (GMFDs) is proposed. Firstly, a signal is decomposed by LMD into several product functions (PFs) which have physical meanings. Secondly, mutual information entropy value between each PF and original signal can be computed, and the PF corresponding to the maximum value is considered as containing the richest feature information of original signal, thus the PF is used as data source. Lastly, GMFDs are extracted from the data source, and some GMFDs which can quantitatively and comprehensively characterize nonlinear information of gear running states are adopted as feature vectors, hence gear faults can be diagnosed by kernel fuzzy c-means (KFCM). In order to demonstrate superiority of the proposed method, the GMFDs are extracted from signals of different lengths, ones sampled under three different working conditions of load and speed, ones without decomposition of LMD. The gear signals are tested and verified, and the result demonstrates that the proposed method is superior and can diagnose gear faults accurately.
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paper_title: A new compound faults detection method for rolling bearings based on empirical wavelet transform and chaotic oscillator

paper_content:

Abstract The rolling bearings often suffer from compound faults in practice. The concurrence of different faults increases the fault detection difficulty and the decoupling detection of compound faults is attracting considerable attentions. Recent publications report the application of the multiwavelets and empirical mode decomposition (EMD) for compound faults decoupling. However, due to limited adaptability they would induce mode mixing or/and overestimation problems in the signal processing. Particularly, the mode mixing would greatly degrade their performance on compound faults detection. To address this issue, this work presents a new method based on the empirical wavelet transform-duffing oscillator (EWTDO) for compound faults decoupling diagnosis of rolling bearings. The empirical wavelet transform (EWT) is able to extract intrinsic modes of a signal by fully adaptive wavelet basis. Hence, the mode mixing and overestimation can be resolved in decoupling processing and the compound faults can be correctly decomposed into different single faults in the form of empirical modes. Then, each single fault frequency was incorporated into a duffing oscillator to establish its corresponding fault isolator. By directly observing the chaotic motion from the Poincar mapping of the isolator outputs the single faults were identified one by one from the empirical modes. Experimental tests were carried out on a rolling bearing fault tester to examine the efficacy of the proposed EWTDO method on compound faults detection. The analysis results show attractive performance with respect to existing decoupling approaches based on the multiwavelets and EMD. In particular, our proposed method is much more reliable in decoupling the compound faults. Hence, the proposed method has practical importance in compound faults decoupling diagnosis for rolling bears.
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paper_title: An Improved Time-Frequency Representation based on Nonlinear Mode Decomposition and Adaptive Optimal Kernel

paper_content:

The Foundation of Key Laboratory ::: of China’s Education Ministry (UASP1201) and A Project Funded by the ::: Priority Academic Program Development of Jiangsu Higher Education ::: Institutions.
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paper_title: Fault Diagnosis of Rotating Machinery Based on an Adaptive Ensemble Empirical Mode Decomposition

paper_content:

The vibration based signal processing technique is one of the principal tools for diagnosing faults of rotating machinery. Empirical mode decomposition (EMD), as a time-frequency analysis technique, has been widely used to process vibration signals of rotating machinery. But it has the shortcoming of mode mixing in decomposing signals. To overcome this shortcoming, ensemble empirical mode decomposition (EEMD) was proposed accordingly. EEMD is able to reduce the mode mixing to some extent. The performance of EEMD, however, depends on the parameters adopted in the EEMD algorithms. In most of the studies on EEMD, the parameters were selected artificially and subjectively. To solve the problem, a new adaptive ensemble empirical mode decomposition method is proposed in this paper. In the method, the sifting number is adaptively selected, and the amplitude of the added noise changes with the signal frequency components during the decomposition process. The simulation, the experimental and the application results demonstrate that the adaptive EEMD provides the improved results compared with the original EEMD in diagnosing rotating machinery.
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paper_title: A comparison study of improved Hilbert-Huang transform and wavelet transform : Application to fault diagnosis for rolling bearing

paper_content:

Abstract For rolling bearing fault detection, it is expected that a desired time–frequency analysis method should have good computation efficiency, and have good resolution in both time domain and frequency domain. As the best available time–frequency method so far, the wavelet transform still cannot fulfill the rolling bearing fault detection task very well since it has some inevitable deficiencies. The recent popular time–frequency analysis method, Hilbert–Huang transform (HHT), has good computation efficiency and does not involve the concept of the frequency resolution and the time resolution. So the HHT seems to have potential to become a perfect tool for rolling bearing fault detection. However, in practical applications, the HHT also suffers from some unsolved deficiencies. To ameliorate these deficiencies, by seeking help from the wavelet packet transform (WPT) and a simple but effective method for intrinsic mode function (IMF) selection, an improved HHT is put forward in this studying. Several numerical study cases will be used to validate the capabilities of the improved HHT. Finally, the improved HHT's performance in rolling bearing fault detection is compared with that of the wavelet based scalogram through experimental case studies. The comparison results have shown that (1) the improved HHT has better resolution both in time domain and in frequency domain than the scalogram; (2) the improved HHT has better computing efficiency than scalogram; (3) the HHT spectrum also has one unresolved and maybe inevitable deficiency—ripple phenomenon in its estimated frequency, which would mislead our analysis.
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paper_title: Demodulation analysis based on adaptive local iterative filtering for bearing fault diagnosis

paper_content:

Abstract The vibration signals from defective rolling bearings are multi-component amplitude modulation (AM)-frequency modulation (FM) signals. Traditional envelope analysis method is based on a filter. The centre frequency and bandwidth of the filter are set according to experience. So the filter method will induce a demodulation error. This research proposes a rolling bearing fault diagnosis method based on adaptive local iterative filtering (ALIF) and envelope spectrum. The ALIF method is a new method for the analysis of non-stationary signals. It uses an iterative filters strategy together with an adaptive, data-driven filter length selection to achieve the necessary decomposition. Smooth filters with compact support from the solutions of the Fokker–Planck equations are used within the ALIF method. The ALIF method offers good performance in obtaining more accurate components of non-stationary signals and in suppressing mode mixing. The ALIF method can decompose a multi-component AM-FM signal into a number of stationary components. The envelope demodulation method is used to analyse those components containing fault information which, in turn, can reveal bearing fault features. The vibration signals from a rolling bearing with an outer race fault and an inner race fault are used to verify the proposed method. The results show that this method can effectively extract bearing fault features.
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paper_title: Application of local mean decomposition to the surveillance and diagnostics of low-speed helical gearbox

paper_content:

Abstract Gears are common power transmission elements and are frequently responsible for transmission failures. Instantaneous time-frequency spectrum (ITFS) resulted from local mean decomposition is applied to the surveillance and early fault diagnosis of a finishing rolling mill in this paper. Results of practical signals demonstrate that ITFS is effective and reliable for the early detection of gear local fault. In addition, a new parameter to evaluate the damage severity of the gearbox is also developed based on the marginal spectrum derived from ITFS. The utility of the new gear fault symptom has been investigated using practical vibration signals. Results show that the new parameter is only sensitive to the changes caused by the deterioration of a monitored unit and insensitive to the influence of the variable non-deterioration factors such as varying speed and loads. This new index may thus find its wide applications for machine prognostics in the near future.
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paper_title: Application of the EEMD method to rotor fault diagnosis of rotating machinery

paper_content:

Empirical mode decomposition (EMD) is a self-adaptive analysis method for nonlinear and non-stationary signals. It may decompose a complicated signal into a collection of intrinsic mode functions (IMFs) based on the local characteristic time scale of the signal. The EMD method has attracted considerable attention and been widely applied to fault diagnosis of rotating machinery recently. However, it cannot reveal the signal characteristic information accurately because of the problem of mode mixing. To alleviate the mode mixing problem occurring in EMD, ensemble empirical mode decomposition (EEMD) is presented. With EEMD, the components with truly physical meaning can be extracted from the signal. Utilizing the advantage of EEMD, this paper proposes a new EEMD-based method for fault diagnosis of rotating machinery. First, a simulation signal is used to test the performance of the method based on EEMD. Then, the proposed method is applied to rub-impact fault diagnosis of a power generator and early rub-impact fault diagnosis of a heavy oil catalytic cracking machine set. Finally, by comparing its application results with those of the EMD method, the superiority of the proposed method based on EEMD is demonstrated in extracting fault characteristic information of rotating machinery.
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paper_title: A demodulation method based on improved local mean decomposition and its application in rub-impact fault diagnosis

paper_content:

Demodulation is an available method for mechanical diagnoses, and a demodulation technique based on improved local mean decomposition (LMD) is proposed in this paper. A method of boundary process and a strategy for determining the step size of moving average are presented to improve the LMD algorithm. Instantaneous amplitude (IA) and instantaneous frequency (IF) of the signal can be computed independently of Hilbert transform using the improved LMD method. A well-constructed description of the derived IA and IF is given in the form of instantaneous time–frequency spectrum (ITFS) which preserves both the time and frequency information simultaneously. Results of three synthetic signals indicate that this proposed method is the best demodulation approach to extracting the all-round carrier and modulated components as well as the accurate IF, compared with Hilbert–Huang transform and stationary wavelet transform. The validity of the technique is then demonstrated on a real rotor system of a gas turbine with rub-impact fault. Due to the opposite friction during operation, the transient fluctuations of the IF of the fundamental harmonic component are successfully identified in the ITFS. In addition, we find that the proposed technique is more effective and sensitive than other methods in detecting sub-harmonics and FM components contained in the rub-impact signals. Thus the present method is powerful in the analysis of modulated signals and is an effective tool for the detection of rub-impact faults.
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paper_title: Application of the intrinsic time-scale decomposition method to fault diagnosis of wind turbine bearing

paper_content:

A fault diagnosis method of wind turbine bearing based on intrinsic time-scale decomposition (ITD) is put forward. In the proposed method, the vibration signal of the main bearing is decomposed into several proper rotation components by the ITD method. The frequency centers of the proper rotation components that contain predominant energy are computed and considered as fault feature vectors. The nearest neighbor algorithm is applied to identify the fault types of the wind turbine bearing. The experimental data of the wind turbine spherical roller bearing in four conditions (normal, outer race fault, inner race fault and roller fault) are applied to evaluate the performance of the proposed method. The results demonstrate the feasibility and accuracy of this approach for the diagnosis of the wind turbine bearing faults under uncertain conditions.
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paper_title: An ensemble local means decomposition method and its application to local rub-impact fault diagnosis of the rotor systems

paper_content:

Abstract Targeting the shock characteristics of the vibration signal of a rotor system with local rub-impact fault, a local rub-impact fault diagnosis method of rotor system based on ELMD (ensemble local means decomposition) is proposed in this paper. The local mean decomposition (LMD) is a newly self-adaptive time–frequency analysis method, by which any complicated multi-component signal could be decomposed into a set of product functions (PFs) whose instantaneous frequencies in theory have physical significance. Unfortunately, mode mixing phenomenon which makes the decomposition results devoid of physical meaning is common when LMD is performed in practice. Targeting this shortcoming, the filter bank structure of white noise by LMD is obtained by numerical experiments, and then an improved method based upon noise-assisted analysis, ensemble local mean decomposition, is put forward. In ELMD, firstly, different white noise is added to the targeted signal; secondly, LMD is used to decompose the noise-added signal into product functions (PFs); finally, the ensemble means of corresponding PF components derived from LMD is regarded as the final decomposition result. The analytical results from simulation signal and experimental rotor local rub-impact signal demonstrate that the ELMD approach can be used to overcome the mode mixing of the original LMD method effectively.
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paper_title: Gearbox fault diagnosis using empirical mode decomposition and Hilbert spectrum

paper_content:

The empirical mode decomposition (EMD) and Hilbert spectrum are a new method for adaptive analysis of non-linear and non-stationary signals. This paper applies this method to vibration signal analysis for localised gearbox fault diagnosis. We first study the properties of the recently developed B-spline EMD as a filter bank, which is helpful in understanding the mechanisms behind EMD. Then we investigate the effectiveness of the original and the B-spline EMD as well as their corresponding Hilbert spectrum in the fault diagnosis. Vibration signals collected from an automobile gearbox with an incipient tooth crack are used in the investigation. The results show that the EMD algorithms and the Hilbert spectrum perform excellently. They are found to be more effective than the often used continuous wavelet transform in detection of the vibration signatures.
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paper_title: An improved Hilbert–Huang transform and its application in vibration signal analysis

paper_content:

Abstract The vibration generated by industrial machines always contains nonlinear and non-stationary signals. Recently, a number of new methods have been proposed to analyse these signals. One of the promising methods is the Hilbert–Huang Transform (HHT). The HHT is derived from the principals of empirical mode decomposition (EMD) and the Hilbert Transform. When applying the HHT, first, the EMD will decompose the acquired signal into a collection of intrinsic mode functions (IMF). The IMF is a kind of complete, adaptive and almost orthogonal representation for the analysed signal. Since the IMF is almost monocomponent, it can determine all the instantaneous frequencies from the nonlinear or non-stationary signal. Second, the local energy of each instantaneous frequency can be derived through the Hilbert Transform. Hence, the result is an energy–frequency–time distribution of the signal. Since applying the process of HHT is not computational intensive, the HHT becomes a promising method to extract the properties of nonlinear and non-stationary signal. However, after the completion of a thorough experiment, the result generated by the HHT has its deficiency. First, the EMD will generate undesirable IMFs at the low-frequency region that may cause misinterpretation to the result. Second, depends on the analysed signal, the first obtained IMF may cover too wide a frequency range such that the property of monocomponent cannot be achieved. Third, the EMD operation cannot separate signals that contain low-energy components. In this study, new techniques have been applied to improve the result of HHT. In the improved version of HHT, the wavelet packet transform (WPT) is used as preprocessing to decompose the signal into a set of narrow band signals prior to the application of EMD. With the help from WPT, each IMF derived from the EMD can truly become monocomponent. Then, a screening process is conducted to remove unrelated IMFs from the result. Both simulated and experimental vibration signals of having a rotary system with the fault of rubbing occurred have proven that the improved HHT does show the rubbing symptoms more clear and accurate than the original HHT. Hence, the improved HHT is a precise method for nonlinear and non-stationary signal analysis.
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paper_title: Rotating machine fault diagnosis using empirical mode decomposition

paper_content:

Abstract In this paper, an empirical mode decomposition (EMD) based approach for rotating machine fault diagnosis is investigated. EMD is a new time–frequency analyzing method for nonlinear and non-stationary signals. By using EMD a complicated signal can be decomposed into a number of intrinsic mode functions (IMFs) based on the local characteristic time scale of the signal. The IMFs, working as the basis functions, represent the intrinsic oscillation modes embedded in the signal. However, our research shows that IMFs sometimes fail to reveal the signal characteristics due to the effect of noises. Hence, combined mode function (CMF) is presented. With CMF, the neighboring IMFs are combined to obtain an oscillation mode depicting signal features more precisely. The adaptive filtering features of EMD and CMF are discussed, and the simulation signals are applied to test their performance. Finally, a practical fault signal of a power generator from a thermal-electric plant is analyzed to diagnose the fault by using EMD and CMF. The results show that EMD and CHF can extract the rotating machine fault characteristics and identify the fault patterns effectively.
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paper_title: Diagnostics of gear faults based on EMD and automatic selection of intrinsic mode functions

paper_content:

Abstract Signal processing is an important tool for diagnostics of mechanical systems. Many different techniques are available to process experimental signals, among others: FFT, wavelet transform, cepstrum, demodulation analysis, second order ciclostationarity analysis, etc. However, often hypothesis about data and computational efforts restrict the application of some techniques. In order to overcome these limitations, the empirical mode decomposition has been proposed. The outputs of this adaptive approach are the intrinsic mode functions that are treated with the Hilbert transform in order to obtain the Hilbert–Huang spectrum. Anyhow, the selection of the intrinsic mode functions used for the calculation of Hilbert–Huang spectrum is normally done on the basis of user’s experience. On the contrary, in the paper a merit index is introduced that allows the automatic selection of the intrinsic mode functions that should be used. The effectiveness of the improvement is proven by the result of the experimental tests presented and performed on a test-rig equipped with a spiral bevel gearbox, whose high contact ratio made difficult to diagnose also serious damages of the gears. This kind of gearbox is normally never employed for benchmarking diagnostics techniques. By using the merit index, the defective gearbox is always univocally identified, also considering transient operating conditions.
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paper_title: Application of the empirical mode decomposition to the analysis of esophageal manometric data in gastroesophageal reflux disease

paper_content:

The empirical mode decomposition (EMD) is a general signal processing method for analyzing nonlinear and non-stationary time series. The central idea of EMD is to decompose a time series into a finite and often small number of intrinsic mode functions (IMFs). An IMF is defined as any function having the number of extrema and the number of zero-crossings equal (or differing at most by one), and also having symmetric envelopes defined by the local minima, and maxima respectively. The decomposition procedure is adaptive, data-driven, therefore, highly efficient The EMD is first described, and its performance is validated by simulations. The EMD is then applied to the analysis of esophageal manometric time series in gastroesophageal reflux disease. The results show that the EMD may prove to be a vital technique for the analysis of esophageal manometric data.
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paper_title: A CRITERION FOR SELECTING RELEVANT INTRINSIC MODE FUNCTIONS IN EMPIRICAL MODE DECOMPOSITION

paper_content:

Information extraction from time series has traditionally been done with Fourier analysis, which use stationary sines and cosines as basis functions. However, data that come from most natural phenomena are mostly nonstationary. A totally adaptive alternative method has been developed called the Hilbert–Huang transform (HHT), which involves generating basis functions called the intrinsic mode functions (IMFs) via the empirical mode decomposition (EMD). The EMD is a numerical procedure that is prone to numerical errors that may persist in the decomposition as extra IMFs. In this study, results of numerical experiments are presented, which would establish a stringent threshold by which relevant IMFs are distinguished from IMFs that may have been generated by numerical errors. The threshold is dependent on the correlation coefficient between the IMFs and the original signal. Finally, the threshold is applied to IMFs of earthquake signals from five accelerometers located in a building.
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paper_title: A comparison study of improved Hilbert-Huang transform and wavelet transform : Application to fault diagnosis for rolling bearing

paper_content:

Abstract For rolling bearing fault detection, it is expected that a desired time–frequency analysis method should have good computation efficiency, and have good resolution in both time domain and frequency domain. As the best available time–frequency method so far, the wavelet transform still cannot fulfill the rolling bearing fault detection task very well since it has some inevitable deficiencies. The recent popular time–frequency analysis method, Hilbert–Huang transform (HHT), has good computation efficiency and does not involve the concept of the frequency resolution and the time resolution. So the HHT seems to have potential to become a perfect tool for rolling bearing fault detection. However, in practical applications, the HHT also suffers from some unsolved deficiencies. To ameliorate these deficiencies, by seeking help from the wavelet packet transform (WPT) and a simple but effective method for intrinsic mode function (IMF) selection, an improved HHT is put forward in this studying. Several numerical study cases will be used to validate the capabilities of the improved HHT. Finally, the improved HHT's performance in rolling bearing fault detection is compared with that of the wavelet based scalogram through experimental case studies. The comparison results have shown that (1) the improved HHT has better resolution both in time domain and in frequency domain than the scalogram; (2) the improved HHT has better computing efficiency than scalogram; (3) the HHT spectrum also has one unresolved and maybe inevitable deficiency—ripple phenomenon in its estimated frequency, which would mislead our analysis.
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paper_title: An improved Hilbert–Huang transform and its application in vibration signal analysis

paper_content:

Abstract The vibration generated by industrial machines always contains nonlinear and non-stationary signals. Recently, a number of new methods have been proposed to analyse these signals. One of the promising methods is the Hilbert–Huang Transform (HHT). The HHT is derived from the principals of empirical mode decomposition (EMD) and the Hilbert Transform. When applying the HHT, first, the EMD will decompose the acquired signal into a collection of intrinsic mode functions (IMF). The IMF is a kind of complete, adaptive and almost orthogonal representation for the analysed signal. Since the IMF is almost monocomponent, it can determine all the instantaneous frequencies from the nonlinear or non-stationary signal. Second, the local energy of each instantaneous frequency can be derived through the Hilbert Transform. Hence, the result is an energy–frequency–time distribution of the signal. Since applying the process of HHT is not computational intensive, the HHT becomes a promising method to extract the properties of nonlinear and non-stationary signal. However, after the completion of a thorough experiment, the result generated by the HHT has its deficiency. First, the EMD will generate undesirable IMFs at the low-frequency region that may cause misinterpretation to the result. Second, depends on the analysed signal, the first obtained IMF may cover too wide a frequency range such that the property of monocomponent cannot be achieved. Third, the EMD operation cannot separate signals that contain low-energy components. In this study, new techniques have been applied to improve the result of HHT. In the improved version of HHT, the wavelet packet transform (WPT) is used as preprocessing to decompose the signal into a set of narrow band signals prior to the application of EMD. With the help from WPT, each IMF derived from the EMD can truly become monocomponent. Then, a screening process is conducted to remove unrelated IMFs from the result. Both simulated and experimental vibration signals of having a rotary system with the fault of rubbing occurred have proven that the improved HHT does show the rubbing symptoms more clear and accurate than the original HHT. Hence, the improved HHT is a precise method for nonlinear and non-stationary signal analysis.
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paper_title: The local mean decomposition and its application to EEG perception data

paper_content:

This paper describes the local mean decomposition (LMD), a new iterative approach to demodulating amplitude and frequency modulated signals. The new method decomposes such signals into a set of functions, each of which is the product of an envelope signal and a frequency modulated signal from which a time-varying instantaneous frequency can be derived. The LMD method can be used to analyse a wide variety of natural signals such as electrocardiograms, functional magnetic resonance imaging data, and earthquake data. The paper presents the results of applying LMD to a set of scalp electroencephalogram (EEG) visual perception data. The LMD instantaneous frequency and energy structure of the EEG is examined, and compared with results obtained using the spectrogram. The nature of visual perception is investigated by measuring the degree of EEG instantaneous phase concentration that occurs following stimulus onset over multiple trials. The analysis suggests that there is a statistically significant difference between the theta phase concentrations of the perception and no perception EEG data.
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paper_title: An ensemble local means decomposition method and its application to local rub-impact fault diagnosis of the rotor systems

paper_content:

Abstract Targeting the shock characteristics of the vibration signal of a rotor system with local rub-impact fault, a local rub-impact fault diagnosis method of rotor system based on ELMD (ensemble local means decomposition) is proposed in this paper. The local mean decomposition (LMD) is a newly self-adaptive time–frequency analysis method, by which any complicated multi-component signal could be decomposed into a set of product functions (PFs) whose instantaneous frequencies in theory have physical significance. Unfortunately, mode mixing phenomenon which makes the decomposition results devoid of physical meaning is common when LMD is performed in practice. Targeting this shortcoming, the filter bank structure of white noise by LMD is obtained by numerical experiments, and then an improved method based upon noise-assisted analysis, ensemble local mean decomposition, is put forward. In ELMD, firstly, different white noise is added to the targeted signal; secondly, LMD is used to decompose the noise-added signal into product functions (PFs); finally, the ensemble means of corresponding PF components derived from LMD is regarded as the final decomposition result. The analytical results from simulation signal and experimental rotor local rub-impact signal demonstrate that the ELMD approach can be used to overcome the mode mixing of the original LMD method effectively.
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paper_title: Intrinsic time-scale decomposition: time–frequency–energy analysis and real-time filtering of non-stationary signals

paper_content:

We introduce a new algorithm, the intrinsic time-scale decomposition (ITD), for efficient and precise time–frequency–energy (TFE) analysis of signals. The ITD method overcomes many of the limitations of both classical (e.g. Fourier transform or wavelet transform based) and more recent (empirical mode decomposition based) approaches to TFE analysis of signals that are nonlinear and/or non-stationary in nature. The ITD method decomposes a signal into (i) a sum of proper rotation components, for which instantaneous frequency and amplitude are well defined, and (ii) a monotonic trend. The decomposition preserves precise temporal information regarding signal critical points and riding waves, with a temporal resolution equal to the time-scale of extrema occurrence in the input signal. We also demonstrate how the ITD enables application of single-wave analysis and how this, in turn, leads to a powerful new class of real-time signal filters, which extract and utilize the inherent instantaneous amplitude and frequency/phase information in combination with other relevant morphological features.
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paper_title: A review on empirical mode decomposition in fault diagnosis of rotating machinery

paper_content:

Abstract Rotating machinery covers a broad range of mechanical equipment and plays a significant role in industrial applications. It generally operates under tough working environment and is therefore subject to faults, which could be detected and diagnosed by using signal processing techniques. Empirical mode decomposition (EMD) is one of the most powerful signal processing techniques and has been extensively studied and widely applied in fault diagnosis of rotating machinery. Numerous publications on the use of EMD for fault diagnosis have appeared in academic journals, conference proceedings and technical reports. This paper attempts to survey and summarize the recent research and development of EMD in fault diagnosis of rotating machinery, providing comprehensive references for researchers concerning with this topic and helping them identify further research topics. First, the EMD method is briefly introduced, the usefulness of the method is illustrated and the problems and the corresponding solutions are listed. Then, recent applications of EMD to fault diagnosis of rotating machinery are summarized in terms of the key components, such as rolling element bearings, gears and rotors. Finally, the outstanding open problems of EMD in fault diagnosis are discussed and potential future research directions are identified. It is expected that this review will serve as an introduction of EMD for those new to the concepts, as well as a summary of the current frontiers of its applications to fault diagnosis for experienced researchers.
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paper_title: Wavelet transform based on inner product in fault diagnosis of rotating machinery: A review

paper_content:

Abstract As a significant role in industrial equipment, rotating machinery fault diagnosis (RMFD) always draws lots of attention for guaranteeing product quality and improving economic benefit. But non-stationary vibration signal with a large amount of noise on abnormal condition of weak fault or compound fault in many cases would lead to this task challenging. As one of the most powerful non-stationary signal processing techniques, wavelet transform (WT) has been extensively studied and widely applied in RMFD. Numerous publications about the study and applications of WT for RMFD have been presented to academic journals, technical reports and conference proceedings. Many previous publications admit that WT can be realized by means of inner product principle of signal and wavelet base. This paper verifies the essence on inner product operation of WT by simulation and field experiments. Then the development process of WT based on inner product is concluded and the applications of major developments in RMFD are also summarized. Finally, super wavelet transform as an important prospect of WT based on inner product are presented and discussed. It is expected that this paper can offer an in-depth and comprehensive references for researchers and help them with finding out further research topics.
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paper_title: A rolling bearing fault diagnosis approach based on LCD and fuzzy entropy

paper_content:

Abstract When the fault happens, the complexity of the intrinsic oscillation from the mechanical system will change. Fuzzy entropy (FuzzyEn), which is defined to measure the complexity and self-similarity of the time series, can be utilized to measure the complexity of vibration signal and reflect changes of complexity of the intrinsic oscillation. Since the changes distribute in different scales, a new non-stationary signal analysis method, local characteristic-scale decomposition (LCD), is proposed and used to decompose the vibration signal adaptively into series of intrinsic scale components (ISC) in different scales. And then a new rolling bearing fault diagnosis approach based on LCD and FuzzyEn is proposed. Namely, firstly, by using the LCD rolling bearing vibration signal is decomposed into numbers of ISCs; then the FuzzyEns of the first few ISCs that contain main failure information are extracted. Thirdly, the FuzzyEns obtained are taken as the inputs to the adaptive neuro-fuzzy inference systems (ANFIS) classifier. Finally, the proposed method is applied to experimental data, and the analysis results show that the proposed method performs effectively for the rolling bearing fault diagnosis.
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paper_title: A rolling bearing fault diagnosis approach based on LCD and fuzzy entropy

paper_content:

Abstract When the fault happens, the complexity of the intrinsic oscillation from the mechanical system will change. Fuzzy entropy (FuzzyEn), which is defined to measure the complexity and self-similarity of the time series, can be utilized to measure the complexity of vibration signal and reflect changes of complexity of the intrinsic oscillation. Since the changes distribute in different scales, a new non-stationary signal analysis method, local characteristic-scale decomposition (LCD), is proposed and used to decompose the vibration signal adaptively into series of intrinsic scale components (ISC) in different scales. And then a new rolling bearing fault diagnosis approach based on LCD and FuzzyEn is proposed. Namely, firstly, by using the LCD rolling bearing vibration signal is decomposed into numbers of ISCs; then the FuzzyEns of the first few ISCs that contain main failure information are extracted. Thirdly, the FuzzyEns obtained are taken as the inputs to the adaptive neuro-fuzzy inference systems (ANFIS) classifier. Finally, the proposed method is applied to experimental data, and the analysis results show that the proposed method performs effectively for the rolling bearing fault diagnosis.
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paper_title: Recent advances in time–frequency analysis methods for machinery fault diagnosis: A review with application examples

paper_content:

Abstract Nonstationary signal analysis is one of the main topics in the field of machinery fault diagnosis. Time–frequency analysis can identify the signal frequency components, reveals their time variant features, and is an effective tool to extract machinery health information contained in nonstationary signals. Various time–frequency analysis methods have been proposed and applied to machinery fault diagnosis. These include linear and bilinear time–frequency representations (e.g., wavelet transform, Cohen and affine class distributions), adaptive parametric time–frequency analysis (based on atomic decomposition and time–frequency auto-regressive moving average models), adaptive non-parametric time–frequency analysis (e.g., Hilbert–Huang transform, local mean decomposition, and energy separation), and time varying higher order spectra. This paper presents a systematic review of over 20 major such methods reported in more than 100 representative articles published since 1990. Their fundamental principles, advantages and disadvantages, and applications to fault diagnosis of machinery have been examined. Some examples have also been provided to illustrate their performance.
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paper_title: Time-varying vibration decomposition and analysis based on the Hilbert transform

paper_content:

This paper introduces a simple method for time-varying vibration decomposition based on the Hilbert transform. The non-stationary frequency of the largest component is estimated as an average function of the instantaneous frequency of the composition, and the corresponding envelope is estimated according to synchronous demodulation. The method is demonstrated using computer simulation of different types of non-stationary and nonlinear vibration.
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paper_title: Time-varying vibration decomposition and analysis based on the Hilbert transform

paper_content:

This paper introduces a simple method for time-varying vibration decomposition based on the Hilbert transform. The non-stationary frequency of the largest component is estimated as an average function of the instantaneous frequency of the composition, and the corresponding envelope is estimated according to synchronous demodulation. The method is demonstrated using computer simulation of different types of non-stationary and nonlinear vibration.
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paper_title: Empirical Wavelet Transform

paper_content:

Some recent methods, like the empirical mode decomposition (EMD), propose to decompose a signal accordingly to its contained information. Even though its adaptability seems useful for many applications, the main issue with this approach is its lack of theory. This paper presents a new approach to build adaptive wavelets. The main idea is to extract the different modes of a signal by designing an appropriate wavelet filter bank. This construction leads us to a new wavelet transform, called the empirical wavelet transform. Many experiments are presented showing the usefulness of this method compared to the classic EMD.
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paper_title: Partly ensemble empirical mode decomposition: An improved noise-assisted method for eliminating mode mixing

paper_content:

The problem of mode mixing caused by intermittence or noise signal is irritating in empirical mode decomposition (EMD) method. A noise-assisted data analysis method called ensemble EMD (EEMD) was proposed recently to resolve this problem. However, EEMD will generate a new problem that tolerates the residue noise of the signal reconstruction. As an enhanced EEMD, complementary EEMD (CEEMD), is developed to restrain the reconstruction residue and yields IMFs with the similar RMS noise as EEMD and effectively eliminates residue noise in the IMFs. However, whether EEMD or CEEMD, they both generate false components when the chosen parameters are inappropriate and the final obtained IMFs are not really meeting the definition of IMF. In this paper a partly ensemble EMD (PEEMD) method is proposed to resolve the mode mixing problem. In PEEMD method, after the intermittency or noise signal is obtained by an ensemble way and is detected by permutation entropy, the residual signal is decomposed directly by using the EMD method. By analyzing simulation and real signal, the analysis results indicate that PEEMD can eliminate the residue noise in the IMFs effectively and generates IMFs with better performance, and represents a sound improvement over the original EMD, EEMD and CEEMD.
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paper_title: Local rub-impact fault diagnosis of the rotor systems based on EMD

paper_content:

Aiming at the characteristics of the vibration signal of a rotor system with local rub-impact fault, a local rub-impact fault diagnosis method of rotor system based on EMD (empirical mode decomposition) is proposed in this paper. When the local rub-impact fault occurs in the rotor system, the vibration signals of the rotor system will present amplitude-modulated feature because of the periodic rub-impact between the stator and the rotor. However, it is hard to extract the rub-impact feature for the rub-impact signal is always submerged in the background and noise signals. By using EMD, any complicated signal can be decomposed into a finite and often small number of IMFs (intrinsic mode functions), and each IMF may be amplitude-modulated or frequency-modulated signal. Therefore, the rub-impact signal, background signal and noise signal can be separated efficiently and the fault feature of vibration signal of a rotor system with local rub-impact fault can be extracted after the vibration signal of the rotor system is decomposed by EMD method. The analysis results from the experimental signal show that the intrinsic mode functions of the vibration signal of a rotor system with local rub-impact fault obviously own the amplitude-modulated characteristics, while those under other work conditions own no distinct amplitude-modulated characteristics. Therefore, the EMD method can be applied to the local rub-impact fault diagnosis of the rotor systems efficiently.
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paper_title: Variational Mode Decomposition

paper_content:

During the late 1990s, Huang introduced the algorithm called Empirical Mode Decomposition, which is widely used today to recursively decompose a signal into different modes of unknown but separate spectral bands. EMD is known for limitations like sensitivity to noise and sampling. These limitations could only partially be addressed by more mathematical attempts to this decomposition problem, like synchrosqueezing, empirical wavelets or recursive variational decomposition. Here, we propose an entirely non-recursive variational mode decomposition model, where the modes are extracted concurrently. The model looks for an ensemble of modes and their respective center frequencies, such that the modes collectively reproduce the input signal, while each being smooth after demodulation into baseband. In Fourier domain, this corresponds to a narrow-band prior. We show important relations to Wiener filter denoising. Indeed, the proposed method is a generalization of the classic Wiener filter into multiple, adaptive bands. Our model provides a solution to the decomposition problem that is theoretically well founded and still easy to understand. The variational model is efficiently optimized using an alternating direction method of multipliers approach. Preliminary results show attractive performance with respect to existing mode decomposition models. In particular, our proposed model is much more robust to sampling and noise. Finally, we show promising practical decomposition results on a series of artificial and real data.
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paper_title: An improved Hilbert–Huang transform and its application in vibration signal analysis

paper_content:

Abstract The vibration generated by industrial machines always contains nonlinear and non-stationary signals. Recently, a number of new methods have been proposed to analyse these signals. One of the promising methods is the Hilbert–Huang Transform (HHT). The HHT is derived from the principals of empirical mode decomposition (EMD) and the Hilbert Transform. When applying the HHT, first, the EMD will decompose the acquired signal into a collection of intrinsic mode functions (IMF). The IMF is a kind of complete, adaptive and almost orthogonal representation for the analysed signal. Since the IMF is almost monocomponent, it can determine all the instantaneous frequencies from the nonlinear or non-stationary signal. Second, the local energy of each instantaneous frequency can be derived through the Hilbert Transform. Hence, the result is an energy–frequency–time distribution of the signal. Since applying the process of HHT is not computational intensive, the HHT becomes a promising method to extract the properties of nonlinear and non-stationary signal. However, after the completion of a thorough experiment, the result generated by the HHT has its deficiency. First, the EMD will generate undesirable IMFs at the low-frequency region that may cause misinterpretation to the result. Second, depends on the analysed signal, the first obtained IMF may cover too wide a frequency range such that the property of monocomponent cannot be achieved. Third, the EMD operation cannot separate signals that contain low-energy components. In this study, new techniques have been applied to improve the result of HHT. In the improved version of HHT, the wavelet packet transform (WPT) is used as preprocessing to decompose the signal into a set of narrow band signals prior to the application of EMD. With the help from WPT, each IMF derived from the EMD can truly become monocomponent. Then, a screening process is conducted to remove unrelated IMFs from the result. Both simulated and experimental vibration signals of having a rotary system with the fault of rubbing occurred have proven that the improved HHT does show the rubbing symptoms more clear and accurate than the original HHT. Hence, the improved HHT is a precise method for nonlinear and non-stationary signal analysis.
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paper_title: Nonlinear Mode Decomposition: a new noise-robust, adaptive decomposition method

paper_content:

We introduce a new adaptive decomposition tool, which we refer to as Nonlinear Mode Decomposition (NMD). It decomposes a given signal into a set of physically meaningful oscillations for any waveform, simultaneously removing the noise. NMD is based on the powerful combination of time-frequency analysis techniques - which together with the adaptive choice of their parameters make it extremely noise-robust - and surrogate data tests, used to identify interdependent oscillations and to distinguish deterministic from random activity. We illustrate the application of NMD to both simulated and real signals, and demonstrate its qualitative and quantitative superiority over the other existing approaches, such as (ensemble) empirical mode decomposition, Karhunen-Loeve expansion and independent component analysis. We point out that NMD is likely to be applicable and useful in many different areas of research, such as geophysics, finance, and the life sciences. The necessary MATLAB codes for running NMD are freely available at http://www.physics.lancs.ac.uk/research/nbmphysics/diats/nmd/.
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paper_title: Nonlinear Mode Decomposition: a new noise-robust, adaptive decomposition method

paper_content:

We introduce a new adaptive decomposition tool, which we refer to as Nonlinear Mode Decomposition (NMD). It decomposes a given signal into a set of physically meaningful oscillations for any waveform, simultaneously removing the noise. NMD is based on the powerful combination of time-frequency analysis techniques - which together with the adaptive choice of their parameters make it extremely noise-robust - and surrogate data tests, used to identify interdependent oscillations and to distinguish deterministic from random activity. We illustrate the application of NMD to both simulated and real signals, and demonstrate its qualitative and quantitative superiority over the other existing approaches, such as (ensemble) empirical mode decomposition, Karhunen-Loeve expansion and independent component analysis. We point out that NMD is likely to be applicable and useful in many different areas of research, such as geophysics, finance, and the life sciences. The necessary MATLAB codes for running NMD are freely available at http://www.physics.lancs.ac.uk/research/nbmphysics/diats/nmd/.
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paper_title: ITERATIVE FILTERING AS AN ALTERNATIVE ALGORITHM FOR EMPIRICAL MODE DECOMPOSITION

paper_content:

The empirical mode decomposition (EMD) was a method pioneered by (N. Huang et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear nonstationary time series analysis, Proc. Roy. Soc. Lond. A454 (1998) 903–995) as an alternative technique to the traditional Fourier and wavelet techniques for studying signals. It decomposes a signal into several components called intrinsic mode functions (IMFs), which have shown to admit better behaved instantaneous frequencies via Hilbert transforms. In this paper, we propose an alternative algorithm for EMD based on iterating certain filters, such as Toeplitz filters. This approach yields similar results as the more traditional sifting algorithm for EMD. In many cases the convergence can be rigorously proved.
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paper_title: Adaptive Local Iterative Filtering for Signal Decomposition and Instantaneous Frequency analysis

paper_content:

Abstract Time–frequency analysis for non-linear and non-stationary signals is extraordinarily challenging. To capture features in these signals, it is necessary for the analysis methods to be local, adaptive and stable. In recent years, decomposition based analysis methods, such as the empirical mode decomposition (EMD) technique pioneered by Huang et al., were developed by different research groups. These methods decompose a signal into a finite number of components on which the time–frequency analysis can be applied more effectively. In this paper we consider the Iterative Filtering (IF) approach as an alternative to EMD. We provide sufficient conditions on the filters that ensure the convergence of IF applied to any L 2 signal. Then we propose a new technique, the Adaptive Local Iterative Filtering (ALIF) method, which uses the IF strategy together with an adaptive and data driven filter length selection to achieve the decomposition. Furthermore we design smooth filters with compact support from solutions of Fokker–Planck equations (FP filters) that can be used within both IF and ALIF methods. These filters fulfill the derived sufficient conditions for the convergence of the IF algorithm. Numerical examples are given to demonstrate the performance and stability of IF and ALIF techniques with FP filters. In addition, in order to have a complete and truly local analysis toolbox for non-linear and non-stationary signals, we propose new definitions for the instantaneous frequency and phase which depend exclusively on local properties of a signal.
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paper_title: Recent advances in time–frequency analysis methods for machinery fault diagnosis: A review with application examples

paper_content:

Abstract Nonstationary signal analysis is one of the main topics in the field of machinery fault diagnosis. Time–frequency analysis can identify the signal frequency components, reveals their time variant features, and is an effective tool to extract machinery health information contained in nonstationary signals. Various time–frequency analysis methods have been proposed and applied to machinery fault diagnosis. These include linear and bilinear time–frequency representations (e.g., wavelet transform, Cohen and affine class distributions), adaptive parametric time–frequency analysis (based on atomic decomposition and time–frequency auto-regressive moving average models), adaptive non-parametric time–frequency analysis (e.g., Hilbert–Huang transform, local mean decomposition, and energy separation), and time varying higher order spectra. This paper presents a systematic review of over 20 major such methods reported in more than 100 representative articles published since 1990. Their fundamental principles, advantages and disadvantages, and applications to fault diagnosis of machinery have been examined. Some examples have also been provided to illustrate their performance.
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paper_title: ON INSTANTANEOUS FREQUENCY

paper_content:

Instantaneous frequency (IF) is necessary for understanding the detailed mechanisms for nonlinear and nonstationary processes. Historically, IF was computed from analytic signal (AS) through the Hilbert transform. This paper offers an overview of the difficulties involved in using AS, and two new methods to overcome the difficulties for computing IF. The first approach is to compute the quadrature (defined here as a simple 90° shift of phase angle) directly. The second approach is designated as the normalized Hilbert transform (NHT), which consists of applying the Hilbert transform to the empirically determined FM signals. Additionally, we have also introduced alternative methods to compute local frequency, the generalized zero-crossing (GZC), and the teager energy operator (TEO) methods. Through careful comparisons, we found that the NHT and direct quadrature gave the best overall performance. While the TEO method is the most localized, it is limited to data from linear processes, the GZC method is the m...
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paper_title: A review on empirical mode decomposition in fault diagnosis of rotating machinery

paper_content:

Abstract Rotating machinery covers a broad range of mechanical equipment and plays a significant role in industrial applications. It generally operates under tough working environment and is therefore subject to faults, which could be detected and diagnosed by using signal processing techniques. Empirical mode decomposition (EMD) is one of the most powerful signal processing techniques and has been extensively studied and widely applied in fault diagnosis of rotating machinery. Numerous publications on the use of EMD for fault diagnosis have appeared in academic journals, conference proceedings and technical reports. This paper attempts to survey and summarize the recent research and development of EMD in fault diagnosis of rotating machinery, providing comprehensive references for researchers concerning with this topic and helping them identify further research topics. First, the EMD method is briefly introduced, the usefulness of the method is illustrated and the problems and the corresponding solutions are listed. Then, recent applications of EMD to fault diagnosis of rotating machinery are summarized in terms of the key components, such as rolling element bearings, gears and rotors. Finally, the outstanding open problems of EMD in fault diagnosis are discussed and potential future research directions are identified. It is expected that this review will serve as an introduction of EMD for those new to the concepts, as well as a summary of the current frontiers of its applications to fault diagnosis for experienced researchers.
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paper_title: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

paper_content:

A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...
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paper_title: ITERATIVE FILTERING AS AN ALTERNATIVE ALGORITHM FOR EMPIRICAL MODE DECOMPOSITION

paper_content:

The empirical mode decomposition (EMD) was a method pioneered by (N. Huang et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear nonstationary time series analysis, Proc. Roy. Soc. Lond. A454 (1998) 903–995) as an alternative technique to the traditional Fourier and wavelet techniques for studying signals. It decomposes a signal into several components called intrinsic mode functions (IMFs), which have shown to admit better behaved instantaneous frequencies via Hilbert transforms. In this paper, we propose an alternative algorithm for EMD based on iterating certain filters, such as Toeplitz filters. This approach yields similar results as the more traditional sifting algorithm for EMD. In many cases the convergence can be rigorously proved.
---
paper_title: On amplitude and frequency demodulation using energy operators

paper_content:

It is shown that the nonlinear energy-tracking signal operator Psi (x)=(dx/dt)/sup 2/-xd/sup 2/x/dt/sup 2/ and its discrete-time counterpart can estimate the AM and FM modulating signals. Specifically, Psi can approximately estimate the amplitude envelope of AM signals and the instantaneous frequency of FM signals. Bounds are derived for the approximation errors, which are negligible under general realistic conditions. These results, coupled with the simplicity of Psi , establish the usefulness of the energy operator for AM and FM signal demodulation. These ideas are then extended to a more general class of signals that are sine waves with a time-varying amplitude and frequency and thus contain both an AM and an FM component; for such signals it is shown that Psi can approximately track the product of their amplitude envelope and their instantaneous frequency. The theoretical analysis is done for both continuous- and discrete-time signals. >
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paper_title: ON INSTANTANEOUS FREQUENCY

paper_content:

Instantaneous frequency (IF) is necessary for understanding the detailed mechanisms for nonlinear and nonstationary processes. Historically, IF was computed from analytic signal (AS) through the Hilbert transform. This paper offers an overview of the difficulties involved in using AS, and two new methods to overcome the difficulties for computing IF. The first approach is to compute the quadrature (defined here as a simple 90° shift of phase angle) directly. The second approach is designated as the normalized Hilbert transform (NHT), which consists of applying the Hilbert transform to the empirically determined FM signals. Additionally, we have also introduced alternative methods to compute local frequency, the generalized zero-crossing (GZC), and the teager energy operator (TEO) methods. Through careful comparisons, we found that the NHT and direct quadrature gave the best overall performance. While the TEO method is the most localized, it is limited to data from linear processes, the GZC method is the m...
---
paper_title: ON INSTANTANEOUS FREQUENCY

paper_content:

Instantaneous frequency (IF) is necessary for understanding the detailed mechanisms for nonlinear and nonstationary processes. Historically, IF was computed from analytic signal (AS) through the Hilbert transform. This paper offers an overview of the difficulties involved in using AS, and two new methods to overcome the difficulties for computing IF. The first approach is to compute the quadrature (defined here as a simple 90° shift of phase angle) directly. The second approach is designated as the normalized Hilbert transform (NHT), which consists of applying the Hilbert transform to the empirically determined FM signals. Additionally, we have also introduced alternative methods to compute local frequency, the generalized zero-crossing (GZC), and the teager energy operator (TEO) methods. Through careful comparisons, we found that the NHT and direct quadrature gave the best overall performance. While the TEO method is the most localized, it is limited to data from linear processes, the GZC method is the m...
---
paper_title: ON INSTANTANEOUS FREQUENCY

paper_content:

Instantaneous frequency (IF) is necessary for understanding the detailed mechanisms for nonlinear and nonstationary processes. Historically, IF was computed from analytic signal (AS) through the Hilbert transform. This paper offers an overview of the difficulties involved in using AS, and two new methods to overcome the difficulties for computing IF. The first approach is to compute the quadrature (defined here as a simple 90° shift of phase angle) directly. The second approach is designated as the normalized Hilbert transform (NHT), which consists of applying the Hilbert transform to the empirically determined FM signals. Additionally, we have also introduced alternative methods to compute local frequency, the generalized zero-crossing (GZC), and the teager energy operator (TEO) methods. Through careful comparisons, we found that the NHT and direct quadrature gave the best overall performance. While the TEO method is the most localized, it is limited to data from linear processes, the GZC method is the m...
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paper_title: Application of the wavelet transform in machine condition monitoring and fault diagnostics: a review with bibliography

paper_content:

Abstract The application of the wavelet transform for machine fault diagnostics has been developed for last 10 years at a very rapid rate. A review on all of the literature is certainly not possible. The purpose of this review is to present a summary about the application of the wavelet in machine fault diagnostics, including the following main aspects: the time–frequency analysis of signals, the fault feature extraction, the singularity detection for signals, the denoising and extraction of the weak signals, the compression of vibration signals and the system identification. Some other applications are introduced briefly as well, such as the wavelet networks, the wavelet-based frequency response function, etc. In addition, some problems in using the wavelet for machine fault diagnostics are analysed. The prospects of the wavelet analysis in solving non-linear problems are discussed.
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paper_title: On amplitude and frequency demodulation using energy operators

paper_content:

It is shown that the nonlinear energy-tracking signal operator Psi (x)=(dx/dt)/sup 2/-xd/sup 2/x/dt/sup 2/ and its discrete-time counterpart can estimate the AM and FM modulating signals. Specifically, Psi can approximately estimate the amplitude envelope of AM signals and the instantaneous frequency of FM signals. Bounds are derived for the approximation errors, which are negligible under general realistic conditions. These results, coupled with the simplicity of Psi , establish the usefulness of the energy operator for AM and FM signal demodulation. These ideas are then extended to a more general class of signals that are sine waves with a time-varying amplitude and frequency and thus contain both an AM and an FM component; for such signals it is shown that Psi can approximately track the product of their amplitude envelope and their instantaneous frequency. The theoretical analysis is done for both continuous- and discrete-time signals. >
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paper_title: Energy separation in signal modulations with application to speech analysis

paper_content:

An efficient solution to the fundamental problem of estimating the time-varying amplitude envelope and instantaneous frequency of a real-valued signal that has both an AM and FM structure is provided. Nonlinear combinations of instantaneous signal outputs from the energy operator are used to separate its output energy product into its AM and FM components. The theoretical analysis is done first for continuous-time signals. Then several efficient algorithms are developed and compared for estimating the amplitude envelope and instantaneous frequency of discrete-time AM-FM signals. These energy separation algorithms are used to search for modulations in speech resonances, which are modeled using AM-FM signals to account for time-varying amplitude envelopes and instantaneous frequencies. The experimental results provide evidence that bandpass-filtered speech signals around speech formants contain amplitude and frequency modulations within a pitch period. >
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paper_title: Defect diagnostics of roller bearing using instantaneous frequency normalization under fluctuant rotating speed

paper_content:

We investigated the feasibility of utilizing the normalized characteristic frequencies for diagnosing the defective roller bearings in case of fluctuant rotating speeds. The time-frequency distributions of the envelope signals of the vibration data were constructed through the Empirical mode decomposition (EMD) as well as the instantaneous frequency calculation. The bearing defect-related frequencies were then normalized with respect to the instantaneous rotation frequency of the shaft so that the factor of the rotating speed fluctuation was removed; thus the characteristic frequencies of bearing malfunctions could be observed in terms of constant values. The magnitude distributions of the marginal envelope spectra at the corresponding normalized bearing defect-related frequencies were extracted as the feature vectors. The Support vector machine (SVM) was used to classify the extracted feature vectors of different bearing fault classes. A test rig of roller bearing system was performed to illustrate the different bearing faults, including different levels of inner race defect, outer race defect and roller defect. The analysis results demonstrate the capability and effectiveness of the proposed approach for accurately identifying the bearing defects in case of fluctuant rotating speed.
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paper_title: Teager energy operator for multi-modulation extraction and its application for gearbox fault detection

paper_content:

This paper presents a parameter-free and broadband approach to detecting gear faults based on vibration signals. The technique is implemented using the Teager energy operator (TEO). It is shown that this operator can extract amplitude, phase and frequency modulations that are associated with various gear faults. Spectral analysis of the TEO-transformed signal provides the necessary information for fault detection. To improve the effectiveness of the proposed technique, we also devised a wavelet de-noising step based on online threshold estimation. In the de-noising step, the threshold estimation is performed through a frequency domain median absolute deviation (FMAD) scheme. The proposed fault detection technique is tested on simulated as well as experimental data acquired from a single-stage bevel gearbox and a two-stage parallel gearbox.
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paper_title: Comparative study of instantaneous frequency based methods for leak detection in pipeline networks

paper_content:

Abstract Methods of pressure transient analysis can be seen as a promising, accurate and low-cost tool for leak and feature detection in pipelines. Various systems have been developed by several groups of researchers in recent years. Such techniques have been successfully demonstrated under laboratory conditions but are not yet established for use with real field test data. The current paper presents a comparative study of instantaneous frequency analysis techniques based on pressure transients recorded within a live distribution network. The instantaneous frequency of the signals are analysed using the Hilbert transform (HT), the Normalised Hilbert transform (NHT), Direct Quadrature (DQ), Teager Energy Operator (TEO) and Cepstrum. This work demonstrates the effectiveness of the instantaneous frequency analysis in detecting a leaks and other features within the network. NHT and DQ allowed for the identification of the approximate location of leaks. The performance TEO is moderate, with Cepstrum being the worst performing method.
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paper_title: Characterization of gear faults in variable rotating speed using Hilbert-Huang Transform and instantaneous dimensionless frequency normalization

paper_content:

Abstract The objective of this research is to investigate the feasibility of utilizing the instantaneous dimensionless frequency (DLF) normalization and Hilbert-Huang Transform (HHT) to characterize the different gear faults in case of variable rotating speed. The normalized DLF of the vibration signals are calculated based on the rotating speed of shaft and the instantaneous frequencies of Intrinsic Mode Functions (IMFs) which are decomposed by Empirical Mode Decomposition (EMD) process. The faulty gear features on DLF-energy distribution of vibration signal can be extracted without the presence of shaft rotating speed, so that the proposed approach can be applied for characterizing the malfunctions of gearbox system under variable shaft rotating speed. A test rig of gear transmission system is performed to illustrate the gear faults, including worn tooth, broken tooth and gear unbalance. Different methods to determine the instantaneous frequency are employed to verify the consistence of characterization results. The DLF-energy distributions of vibration signals are investigated in different faulty gear conditions. The analysis results demonstrate the capability and effectiveness of the proposed approach for characterizing the gear malfunctions at the DLFs corresponding to the meshing frequency as well as the shaft rotating frequency. The support vector machine (SVM) is then employed to classify the vibration patterns of gear transmission system at different malfunctions. Using the energy distribution at the characteristic DLFs as the features, the different fault types of gear can be identified by SVM with high accuracy.
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paper_title: Nonlinearity and non-stationarity analysis of dynamic response of vehicle–track coupling system enhanced by Huang transform

paper_content:

Abstract A method is proposed to analysis the nonlinearity and non-stationarity of dynamic response of high speed vehicle–track coupling system, to understand the properties of signal and its underlying physical meaning. First, a space coupling model of high speed vehicle–track system is built according to the theory of coupling dynamics of vehicle–track system. Then, the dynamic response of vehicle–track system is processed by Ensemble Empirical Mode Decomposition (EEMD) to generate the intrinsic mode functions (IMFs). Next, the “Direct Quadrature” (DQ) method is used to calculate the local instantaneous frequency and instantaneous amplitude of the IMFs, the generalized zero-crossing method and the full wave zero-crossing method are applied to calculate the mean local frequency of the IMFs. Finally, the index and degree of nonlinearity and non-stationarity based on intra-wave frequency modulation (the deviation of the instantaneous frequency from the generalized zero-crossing frequency) and inter-wave frequency modulation (the deviation of the instantaneous frequency from the full wave zero-crossing frequency), are defined to quantify the nonlinearity and non-stationarity of dynamic response. The usefulness and capability of the proposed method are displayed through two cases, which are considered general enough to be the representatives of nonlinear and non-stationary dynamic response of vehicle–track coupling system.
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paper_title: An energy operator approach to joint application of amplitude and frequency-demodulations for bearing fault detection

paper_content:

Bearings are among the most frequently used components. Bearing failure could lead to complete stall of a mechanical system, unpredicted productivity loss for production facilities or catastrophic consequence for mission-critical equipment. As such, bearing fault detection and diagnosis is an imperative part of most of preventive maintenance procedures. This paper presents a parameter independent yet simple to implement fault detection technique. The Teager energy operator is tailored to extract both the amplitude and frequency modulations of the vibration signals measured from mechanical systems. The incorporation of the frequency modulation information into the proposed bearing fault detection method has eliminated the need for interference removal steps. As the amplitude demodulation (AD) is also inherent in the energy operator, the fault frequency can be detected from the spectrum of the energy-transformed signal. The effectiveness of the proposed method has been validated using both simulated and experimental data.
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paper_title: Time–frequency analysis of time-varying modulated signals based on improved energy separation by iterative generalized demodulation

paper_content:

Energy separation algorithm is good at tracking instantaneous changes in frequency and amplitude of modulated signals, but it is subject to the constraints of mono-component and narrow band. In most cases, time-varying modulated vibration signals of machinery consist of multiple components, and have so complicated instantaneous frequency trajectories on time–frequency plane that they overlap in frequency domain. For such signals, conventional filters fail to obtain mono-components of narrow band, and their rectangular decomposition of time–frequency plane may split instantaneous frequency trajectories thus resulting in information loss. Regarding the advantage of generalized demodulation method in decomposing multi-component signals into mono-components, an iterative generalized demodulation method is used as a preprocessing tool to separate signals into mono-components, so as to satisfy the requirements by energy separation algorithm. By this improvement, energy separation algorithm can be generalized to a broad range of signals, as long as the instantaneous frequency trajectories of signal components do not intersect on time–frequency plane. Due to the good adaptability of energy separation algorithm to instantaneous changes in signals and the mono-component decomposition nature of generalized demodulation, the derived time–frequency energy distribution has fine resolution and is free from cross term interferences. The good performance of the proposed time–frequency analysis is illustrated by analyses of a simulated signal and the on-site recorded nonstationary vibration signal of a hydroturbine rotor during a shut-down transient process, showing that it has potential to analyze time-varying modulated signals of multi-components. & 2010 Elsevier Ltd. All rights reserved.
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paper_title: Joint amplitude and frequency demodulation analysis based on intrinsic time-scale decomposition for planetary gearbox fault diagnosis

paper_content:

Abstract Planetary gearbox vibration signals feature complex modulations, thus leading to intricate sideband structure and resulting in difficulty in fault characteristic frequency identification. Intrinsic time-scale decomposition has unique merits, such as high adaptability to changes in signals, low computational complexity, good capability to suppress mode mixing and to preserve temporal information of transients, and excellent suitability for mono-component decomposition of complex multi-component signals. In order to address the issue with planetary gearbox fault diagnosis due to the multiple modulation sources, a joint amplitude and frequency demodulation analysis method is proposed, by exploiting the merits of intrinsic time-scale decomposition. The signal is firstly decomposed into a series of mono-component proper rotational components. Then the one with its instantaneous frequency fluctuating around the gear meshing frequency or its harmonics is selected as the sensitive component. Next, Fourier transformation is applied to the instantaneous amplitude and instantaneous frequency of the sensitive component to obtain the amplitude and frequency demodulated spectra respectively. Finally, a planetary gearbox fault is diagnosed by matching the peaks in the amplitude and frequency demodulated spectra with the theoretical gear fault characteristic frequencies. The proposed method is illustrated by a numerical simulated signal, and further validated by lab experimental signals of a planetary gearbox. The localized faults of sun, planet and ring gears are diagnosed, showing the effectiveness of the method.
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Title: Adaptive Mode Decomposition Methods and Their Applications in Signal Analysis for Machinery Fault Diagnosis: A Review With Examples

Section 1: INTRODUCTION
Description 1: This section covers the importance of signal analysis in machinery fault diagnosis and introduces adaptive mode decomposition methods as an alternative to traditional signal processing techniques.

Section 2: GENERAL PRINCIPLE
Description 2: This section explains the general principle and theoretical background of adaptive mode decomposition methods, including the modeling of multi-component signals and the conditions for mono-component decomposition.

Section 3: ADAPTIVE MODE DECOMPOSITION ALGORITHMS
Description 3: This section reviews various adaptive mode decomposition algorithms, such as EMD, LMD, ITD, LCD, HVD, EWT, VMD, NMD, and ALIF, including their principles, illustrations, application reviews, and remarks.

Section 4: INSTANTANEOUS FREQUENCY ESTIMATION APPROACHES
Description 4: This section discusses different approaches for estimating the instantaneous frequency, including Hilbert Transform, Empirical AM-FM Decomposition, Direct Quadrature, Normalized Hilbert Transform, Generalized Zero-Crossing, and Energy Separation approaches.

Section 5: APPLICATION EXAMPLES
Description 5: This section presents examples of the application of adaptive mode decomposition methods in machinery fault diagnosis, focusing on rotor vibration analysis, planetary gear fault diagnosis, and rolling bearing fault diagnosis.

Section 6: SUMMARY AND PROSPECTS
Description 6: This section summarizes the key points covered in the review and discusses the remaining challenges and future research directions in the field of adaptive mode decomposition for machinery fault diagnosis.