sensors Review
Resonance-Based Sparse Signal Decomposition and Its Application in Mechanical Fault Diagnosis: A Review Wentao Huang *, Hongjian Sun and Weijie Wang School of Mechatronics Engineering, Harbin Institute of Technology, No. 92 Xidazhi Street, Harbin 150001, China;
[email protected] (H.S.);
[email protected] (W.W.) * Correspondence:
[email protected]; Tel.: +86-131-0158-4460 Academic Editor: Xue Wang Received: 7 April 2017; Accepted: 1 June 2017; Published: 3 June 2017
Abstract: Mechanical equipment is the heart of industry. For this reason, mechanical fault diagnosis has drawn considerable attention. In terms of the rich information hidden in fault vibration signals, the processing and analysis techniques of vibration signals have become a crucial research issue in the field of mechanical fault diagnosis. Based on the theory of sparse decomposition, Selesnick proposed a novel nonlinear signal processing method: resonance-based sparse signal decomposition (RSSD). Since being put forward, RSSD has become widely recognized, and many RSSD-based methods have been developed to guide mechanical fault diagnosis. This paper attempts to summarize and review the theoretical developments and application advances of RSSD in mechanical fault diagnosis, and to provide a more comprehensive reference for those interested in RSSD and mechanical fault diagnosis. Followed by a brief introduction of RSSD’s theoretical foundation, based on different optimization directions, applications of RSSD in mechanical fault diagnosis are categorized into five aspects: original RSSD, parameter optimized RSSD, subband optimized RSSD, integrated optimized RSSD, and RSSD combined with other methods. On this basis, outstanding issues in current RSSD study are also pointed out, as well as corresponding instructional solutions. We hope this review will provide an insightful reference for researchers and readers who are interested in RSSD and mechanical fault diagnosis. Keywords: resonance-based sparse signal decomposition; signal processing; mechanical fault diagnosis; feature extraction
1. Introduction With the rapid development of modern industry, mechanical equipment, the industrial heart, is developing towards the direction of large scale, high speed, high accuracy and system integration. Due to harsh working conditions, the faults of mechanical components may strike randomly, and more frequently in their later life. Even the fault of a single component is likely to result in the shutdown of an entire piece of mechanical equipment, especially when considering the chain effects. Thus, mechanical faults may cause huge economic costs and even catastrophic casualties [1,2]. For instance, in 2006, damage of the propulsion system was induced by a gearbox fault in the ship “Zhouying 4”, causing enormous pecuniary loss for the Chinese government [3]. Advanced fault diagnosis technology can not only detect mechanical faults as early as possible, before fatalities, but also fundamentally solve the problem of inadequate and excessive maintenance, which will be of great benefit to the safe operation of mechanical equipment. As a result, great attention has been paid to fault diagnosis technology. Mechanical fault diagnosis is a comprehensive and interdisciplinary study, since it combines monitoring, diagnosis, and prognostics. Its major research directions include signal acquisition and Sensors 2017, 17, 1279; doi:10.3390/s17061279
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sensing technologies, fault mechanisms and symptom relationships, signal processing and diagnostic Sensors 2017, 17, 1279 2 of 27 methods, and intelligent decision and diagnosis systems [4], as shown in Figure 1. Among these, signal processing andtechnologies, diagnosis methods are researched most extensively. It isprocessing acknowledged that vibration sensing fault mechanisms and symptom relationships, signal and diagnostic and effective intelligenttool decision and diagnosis systems [4], as[5,6]. shown in Figure 1. Among these, analysismethods, is the most for mechanical fault diagnosis Aiming at extracting information processing and diagnosis recognizing methods are researched most extensively. is acknowledged on faultsignal features and subsequently mechanical fault types, aItwealth of signal that processing vibration analysis is the most effective tool for mechanical fault diagnosis [5,6]. Aiming at extracting methods are continuously put forward and applied to fault vibration signal analysis. To date, fault information on fault features and subsequently recognizing mechanical fault types, a wealth of signal feature processing extractionmethods techniques based on signal processing include short-time transformation are continuously put forward and applied to fault vibration Fourier signal analysis. To (STFT) [7], transformation (WT) [8],based empirical mode decomposition (EMD) [2], resonance date,wavelet fault feature extraction techniques on signal processing include short-time Fourier transformation (STFT) [7], wavelet transformation (WT)etc. [8], empirical mode decomposition demodulation [9], and morphological operators [10], Although these techniques(EMD) have found [2], resonance demodulationfault [9], and morphological [10], etc. Althoughremaining these techniques wide application in mechanical diagnosis, there operators are still some problems to be settled. have found wide application in mechanical fault diagnosis, there are still some problems remaining For example, the basis function of wavelet transformation cannot be altered once selected; mode to be settled. For example, the basis function of wavelet transformation cannot be altered once mixing problems are common in EMD; and center frequency and bandwidth are difficult to determine selected; mode mixing problems are common in EMD; and center frequency and bandwidth are for resonance demodulation. to lacking effectiveDue solutions to effective the above problems, difficult to determine for Due resonance demodulation. to lacking solutions to themisdiagnosis above and even missedmisdiagnosis diagnosis emerge inmissed manydiagnosis cases, which become biggest in theory and problems, and even emerge in manythe cases, whichconstraints become the biggest constraints in theoryInand engineering applications. In method short, theoffeature extraction of signal engineering applications. short, the feature extraction mechanical faultmethod vibration mechanical fault vibration signal still needs to be further perfected. still needs to be further perfected. Fault mechanism & symptom relationship
Signal acquisition & sensing technology
Mechanical fault diagnosis
Signal processing & diagnosis method
Intelligent decision & diagnosis system
Figure 1. Research directions of fault diagnosis. Figure 1. Research directions of mechanical mechanical fault diagnosis.
Resonance-based sparse signal decomposition is a novel nonlinear signal processing method
Resonance-based sparse signalindecomposition is a novel signal processing method that that was proposed by Selesnick 2011 [11]. The method takesnonlinear resonance as an intrinsic property of was proposed by Selesnick in 2011 [11]. The method takes resonance as an intrinsic property of a signal a signal and evaluates resonance degree with a quality factor (defined as the ratio of the center and evaluates resonance degree with a quality the ratio of the center frequency and frequency and frequency bandwidth, denotedfactor Q). A (defined signal withasstronger resonance signifies that it possesses a higher Q, oscillates more times in astronger time domain, and its frequency banditispossesses distributeda higher frequency bandwidth, denoted Q). A signal with resonance signifies that more intensively; and versa.domain, Unlike frequency based signal methods, more such asintensively; WT Q, oscillates more times invice a time and its band frequency bandanalysis is distributed and EMD, RSSD realizes the nonlinear separation of each component according to its oscillation and vice versa. Unlike frequency band based signal analysis methods, such as WT and EMD, RSSD property (resonance) through morphological component analysis (MCA) [12], and obtains the realizessparsest the nonlinear separation each component itsbeen oscillation property (resonance) representation of eachof resonance component.according So far, RSSDto has widely applied in speech throughsignals morphological component analysis (MCA) [12], and obtains sparsest [13,14], biological signals [15,16], power systems [17–19], and otherthe fields [20–23], representation and the literature has mushroomed over the past years. of eachrelevant resonance component. So far, RSSD hasfew been widely applied in speech signals [13,14], The fault vibration signals collected[17–19], from mechanical systems—mainly bearings, gearboxes, biological signals [15,16], power systems and other fields [20–23], and the relevantand literature rotor systems—usually exhibit a superposition of periodic fault-induced vibration responses and has mushroomed over the past few years. random noise. Indeed, periodic signal components and background noise can correlate well with the The fault vibration signals collected from mechanical systems—mainly bearings, gearboxes, high- and low-resonance components in RSSD, especially from the viewpoint of resonance. Moreover, and rotor systems—usually exhibit a superposition of periodic fault-induced vibration responses since RSSD breaks through the limitations of traditional techniques where processing signals are and random noise. Indeed, periodic signal components and background noise signals can correlate well based on frequency band, it is more suitable to process mechanical fault vibration with nonlinear and non-stationary characteristics. Hence, related research in the mechanicaloffault with theobvious high- and low-resonance components in RSSD, especially from the viewpoint resonance. diagnosis field isbreaks increasingly heated. literaturetechniques survey reveals that processing a review on signals Moreover, since RSSD through the Unfortunately, limitations ofour traditional where the applications of RSSD in mechanical fault diagnosis has not been published. are based on frequency band, it is more suitable to process mechanical fault vibration signals with To fill up the review gap, this paper attempts to summarize the theoretical developments and obviousapplications nonlinear of and non-stationary characteristics. related research in the mechanical RSSD in mechanical fault diagnosis. ForHence, the purpose of centralizing, categorizing, and fault diagnosis field is increasingly heated. Unfortunately, our literature survey reveals that a review on the applications of RSSD in mechanical fault diagnosis has not been published. To fill up the review gap, this paper attempts to summarize the theoretical developments and applications of RSSD in mechanical fault diagnosis. For the purpose of centralizing, categorizing, and
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analyzing these individual studies, this paper provides a more comprehensive reference and promotes the Sensors 2017, 17, 1279 3 of 27 appearance of more advanced research about RSSD, which can make a great contribution to mechanical fault analyzing diagnosis.these The rest of this review organized as follows. In Section 2, a brief introduction individual studies,isthis paper provides a more comprehensive reference andabout theand appearance of more methods advancedofresearch which can make3apresents great an signalpromotes resonance two construction waveletabout bases RSSD, are presented. Section contributionalgorithm to mechanical fault diagnosis. rest of this review is organized as In Section implementation for RSSD, illustratesThe its performance, and investigates its follows. parameters’ influences 2, a brief introduction about signal resonance and two construction methods of wavelet bases are fault on decomposition results. Section 4 reviews recent application advances of RSSD in mechanical presented. Section 3 presents an implementation algorithm for RSSD, illustrates its performance, and diagnosis based on different optimization directions. In Section 5, these methodologies and applications investigates its parameters’ influences on decomposition results. Section 4 reviews recent application are synthesized in a flow chart and table. Outstanding issues and solutions of RSSD in mechanical fault advances of RSSD in mechanical fault diagnosis based on different optimization directions. In Section diagnosis are discussed in Section 6. Finally, concluding remarks are provided in Section 7. 5, these methodologies and applications are synthesized in a flow chart and table. Outstanding issues and solutions of RSSD in mechanical fault diagnosis are discussed in Section 6. Finally, concluding 2. Theoretical Foundation of Wavelet Bases remarks are provided in Section 7.
Due to the differences in signal resonance, RSSD is able to decompose complex signals into 2. Theoretical Foundation of Wavelet Bases high-resonance components comprising sustained oscillation signals, low-resonance components consistingDue of transient impulse signals, and residual Noteworthy, the core to the differences in signal resonance, RSSD iscomponents. able to decompose complex signals intosuperiority highresonance components comprising sustained oscillation low-resonance components of RSSD lies in the construction of wavelet bases. Thus, after signals, a brief explanation of signal resonance, consisting transient impulse and residual components. Noteworthy, the corerational-dilation superiority this section willofconcentrate on thesignals, construction methods of wavelet bases, including of RSSD lies in the construction wavelet bases. Thus,Q-factor after a brief explanation of signal resonance, wavelet transformation (RADWT)of [24,25] and tunable wavelet transformation (TQWT) [26]. this section will concentrate on the construction methods of wavelet bases, including rational-dilation
wavelet transformation (RADWT) [24,25] and tunable Q-factor wavelet transformation (TQWT) [26]. 2.1. Signal Resonance
The signalResonance resonance property can be described with the Q-factor, which is defined as the ratio of 2.1. Signal the centerThe frequency fc and bandwidth BW in frequency domain, as expressed in Equation (1): signal resonance property can be described with the Q-factor, which is defined as the ratio of the center frequency fc and bandwidth BW in frequency domain, as expressed in Equation (1):
Q=
fc BW f
(1)
(1) Q Figure 2 explains the resonance property inBW more detail. Note that the left section of this figure shows the2 explains time-domain waveforms, with spectra onfigure the right. Figure the resonance property in their more corresponding detail. Note that frequency the left section of this For Figure 2a,b, the time-domain signal resembles an impulse signal and exhibits no sustained shows the time-domain waveforms, with their corresponding frequency spectra on the right. For oscillatory thus it is defined as the low-resonance component. Fornothe signalsoscillatory in Figure 2c–f, Figurebehavior; 2a,b, the time-domain signal resembles an impulse signal and exhibits sustained behavior; thus it is defined as their the low-resonance component. For the signals in Figure 2c–f,and whose Qwhose Q-factors are equal to 4, oscillations are about eight times in time domain sustained factorsbehaviors are equal toare 4, their oscillations areare about eight times time domain andcomponents. sustained oscillatory oscillatory observed, so they defined as theinhigh-resonance Comparing behaviors observed, are defined the high-resonance Comparing Figure Figure 2b,d orare Figure 2f, itsoisthey evident that aashigher-Q signal hascomponents. a more concentrated frequency 2b,d or Figure 2f, it is evident that a higher-Q signal has a more concentrated frequency distribution distribution for a fixed frequency than a low-Q signal. Moreover, the signals in Figure 2c,e are likely for a fixed frequency than a low-Q signal. Moreover, the signals in Figure 2c,e are likely able to able to convert to the other one through compressed or stretched transformations in their time domains. convert to the other one through compressed or stretched transformations in their time domains. This This time-scaling transformation doesn’t change their resonance degrees, which explains why they time-scaling transformation doesn’t change their resonance degrees, which explains why they have have Q-factors. Q-factors. Also, their frequency distributions are nonoverlapping, in accordance the fact Also, their frequency distributions are nonoverlapping, in accordance with the factwith that the that the resonance property is independent time scale and frequency. resonance property is independent of time of scale and frequency. c
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Figure 2. Signals with differentQ-factors: Q-factors: (a,b) QQ = 4;=(e,f) Q =Q 4. = 4. Figure 2. Signals with different (a,b)QQ==1;1;(c,d) (c,d) 4; (e,f)
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Based on signal resonance, Selesnick successively put forward RADWT and TQWT to construct Based on signal resonance, Selesnick successively put forward RADWT and TQWT to construct the wavelet bases used in RSSD, and they are both overcomplete discrete wavelet transformations. the wavelet bases used in RSSD, and they are both overcomplete discrete wavelet transformations. 2.2. Basic Theory of RADWT 2.2. Basic Theory of RADWT Dyadic wavelet transformation is a common constant-Q transformation, but a low Q-factor Dyadic wavelet transformation is a common constant-Q transformation, but a low Q-factor limits the application in some conditions where a high frequency resolution is strictly requested [25]. limits the application in some conditions where a high frequency resolution is strictly requested [25]. To overcome this shortcoming, Selesnick developed RADWT in 2009 and successfully applied it to To overcome this shortcoming, Selesnick developed RADWT in 2009 and successfully applied it to RSSD in subsequent years [11]. RSSD in subsequent years [11]. On a theoretical level, like the tree-structure filter bank for Dyadic wavelet transform, RADWT On a theoretical level, like the tree-structure filter bank for Dyadic wavelet transform, RADWT is is also implemented by using the two-channel filter banks illustrated in Figure 3, except for the also implemented by using the two-channel filter banks illustrated in Figure 3, except for the difference difference that only one sampling process exists in the high-pass filter. that only one sampling process exists in the high-pass filter.
x ( n) p
H1 ( )
s
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H 1* ( )
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Figure 3. Two-channel filter banks of rational-dilation wavelet transformation (RADWT). Figure 3. Two-channel filter banks of rational-dilation wavelet transformation (RADWT).
Tunable parameters in RADWT include p, q, and s, and the dilation factor d is defined as p/q, Tunable parameters inp,RADWT s, and dilation factor d is defined as p/q, wherein co-prime numbers q shouldinclude satisfy pp, 1. In effect, the Q-factor is adjusted wherein co-prime numbers p, q should satisfy p < q and p/q + 1/s > 1. In effect, the Q-factor is adjusted with the combined action of p, q, and s, as in Equation (2): with the combined action of p, q, and s, as in Equation (2): 1 Q √d 1 (2) (2) Q = d 1 d 1−d It must be mentioned that this formula for the Q-factor is feasible only when the following It must be mentioned that this formula for the Q-factor is feasible only when the following equation is satisfied: equation is satisfied: 22 pp 11 (3) (3) 1 − q < ss q Without such a guarantee, it will be extremely difficult to find an accurate formula for calculating Without such a guarantee, it will be extremely difficult to find an accurate formula for Q-factors. Equation (3) enables the Q-factor to increase generally with increasing dilation factor d. calculating Q-factors. Equation (3) enables the Q-factor to increase generally with increasing dilation Particularly when the value of d gradually approaches zero, the Q-factor will increase rapidly according factor d. Particularly when the value of d gradually approaches zero, the Q-factor will increase rapidly to Equation (2), and the RADWT wavelet will have a more concentrated frequency, as well as a finer according to Equation (2), and the RADWT wavelet will have a more concentrated frequency, as well frequency resolution. as a finer frequency resolution. On the other hand, s with a value of 1 will lead to transient wavelet features in time domain, which On the other hand, s with a value of 1 will lead to transient wavelet features in time domain, usually indicates the low-resonance components. A value larger than 1 for s allows the constructed which usually indicates the low-resonance components. A value larger than 1 for s allows the wavelets to correlate well with high-resonance components, since the subband wavelets exhibit constructed wavelets to correlate well with high-resonance components, since the subband wavelets sustained oscillations in this case. Comparatively speaking, wavelet oscillatory characteristics are exhibit sustained oscillations in this case. Comparatively speaking, wavelet oscillatory characteristics more sensitive to s than to d, and increasing s can effectively strengthen the time-domain oscillation are more sensitive to s than to d, and increasing s can effectively strengthen the time-domain and frequency concentration. A sufficiently large s will avoid the “ringing” phenomenon, but result in oscillation and frequency concentration. A sufficiently large s will avoid the “ringing” phenomenon, a “brick wall” frequency distribution [25]. but result in a “brick wall” frequency distribution [25]. Assuming Equation (3) is satisfied, and given input signal sampling frequency fs , center frequency Assuming Equation (3) is satisfied, and given input signal sampling frequency fs, center fc and bandwidth BW of the level-j subband wavelet constructed by RADWT can be calculated: frequency fc and bandwidth BW of the level-j subband wavelet constructed by RADWT can be calculated: 1 f c = d j−3/2 (1 − + d) f s (4) s 1 fc d j 3/ 2 (1 d) fs (4) 1 s j −2 BW = 2d (1 − + d)(1 − d)π (5) s
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d)(1 d)
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Figure Figure 44 displays displays aa typical typical case case for for RADWT RADWT subband subband bases bases and and their their frequency frequency responses. responses. Note Note that q are specially chosen to make the Q-factor close to close 3, consistent with the following that values valuesofofp and p and q are specially chosen to make the Q-factor to 3, consistent with the TQWT. It can be seen thatbe allseen subbands show obvious oscillations the time domain, and the frequency following TQWT. It can that all subbands show obviousinoscillations in the time domain, and distributions first several subbands both have flat tops, which appear onlywhich when Equation (3) comes the frequencyofdistributions of first several subbands both have flat tops, appear only when into existence. In theinto meantime, theIn peak of each varyoffrom other vary and show Equation (3) comes existence. the amplitudes meantime, the peaksubband amplitudes eacheach subband from an overall trend. each othergrowth and show an overall growth trend. (a)
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Figure 4. 4. A A typical typical RADWT RADWT wavelet wavelet base base with with pp == 18, 18, qq == 25, 25, ss == 2. 2. (a) (a) Time-domain Time-domain waveforms; waveforms; Figure (b) Frequency Frequency responses. responses. (b)
Consequently, the proposal of RADWT greatly broadens the scope of Q-factors and improves Consequently, the proposal of RADWT greatly broadens the scope of Q-factors and improves the the flexibility of selecting Q-factors. Nevertheless, p = 18 and q = 25 can only yield a value flexibility of selecting Q-factors. Nevertheless, p = 18 and q = 25 can only yield a value approximating 3, approximating 3, rather than a precise Q. If pursuing a more accurate Q, larger values of p and q are rather than a precise Q. If pursuing a more accurate Q, larger values of p and q are unavoidable because unavoidable because they are co-prime. This will result in the length increase of the underthey are co-prime. This will result in the length increase of the under-decomposed signal, which blocks decomposed signal, which blocks RADWT’s applications to more general occasions. In terms of RADWT’s applications to more general occasions. In terms of Equation (3), an accurate calculation Equation (3), an accurate calculation for the Q-factor is also unreachable in some cases, which makes for the Q-factor is also unreachable in some cases, which makes it very difficult to select an exact it very difficult to select an exact Q-factor assisted with a priori knowledge. More unfortunately, Q-factor assisted with a priori knowledge. More unfortunately, without calculation formulas for fc and without calculation formulas for fc and BW like those given in Equations (4) and (5), subsequent BW like those given in Equations (4) and (5), subsequent analysis and optimization on subbands are analysis and optimization on subbands are enormously challenging, lacking theoretical foundation. enormously challenging, lacking theoretical foundation. Taking these restrictions into consideration, Taking these restrictions into consideration, a more advantages transformation needs to be a more advantages transformation needs to be developed. developed. 2.3. Basic Theory of TQWT 2.3. Basic Theory of TQWT As a significant breakthrough in constructing wavelet bases, TQWT overcomes the length As a significant breakthrough in constructing wavelet bases, TQWT overcomes the length limitation of the input signal excellently, and has a more extensive application prospects than RADWT. limitation of the input signal excellently, and has a more extensive application prospects than Via directly specifying the Q-factor, redundancy factor r, and decomposition level J, wavelet bases are RADWT. Via directly specifying the Q-factor, redundancy factor r, and decomposition level J, designed. Obviously, TQWT has more flexibility than RADWT; therefore, it is not only favored by wavelet bases are designed. Obviously, TQWT has more flexibility than RADWT; therefore, it is not RSSD researchers in the field of mechanical fault diagnosis, but also in most signal analysis fields [27]. only favored by RSSD researchers in the field of mechanical fault diagnosis, but also in most signal Briefly, the implementation of TQWT also depends on iterative analysis and synthesis filter banks, analysis fields [27]. Briefly, the implementation of TQWT also depends on iterative analysis and as shown in Figures 5 and 6, respectively. synthesis filter banks, as shown in Figures 5 and 6, respectively.
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HPS HPS
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Figure 5. The analysis filter banks of tunable Q-factor wavelet transformation (TQWT). Figure 5. The analysis filter banks of tunable Q-factor wavelet transformation (TQWT). Figure 5. The analysis filter banks of tunable Q-factor wavelet transformation (TQWT). d1 ( n ) n)) 2((n dd1··· dd J ((nn)) 2
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Figure 6. The synthesis filter banks of TQWT. Figure 6. The synthesis filter banks of TQWT. Figure 6. The synthesis filter banks of TQWT.
In Figures 5 and 6, HPS and LPS represent high- and low-pass scaling, and and are the In Figures 5 and 6, HPS and LPS represent high- 0and are the low-pass 1, 0 scaling, 1, and 1 and corresponding scaling parameters, which satisfy . H1 ( ) and In Figures 5 and 6, HPS and LPS represent high- and low-pass scaling, and β and α are the corresponding scaling parameters, which satisfy 0 1, 0 1, 1 . H1 ( ) and are the highand low-pass frequency responses, respectively. H ( ) corresponding scaling parameters, which satisfy 0 < α < 1, 0 < β < 1, α + β > 1. H1 (ω ) and H2 (ω ) 2 the highand low-pass frequency responses, respectively. (The ) are areHthe highand low-pass frequency responses, respectively. 2 parameters in TQWT consist of the Q-factor, redundancy factor r, and decomposition level The parameters TQWTconsist consist the Q-factor, redundancy factor r, and decomposition J, whose concrete descriptions are listed in Table 1. redundancy The parameters ininTQWT ofofthe Q-factor, decompositionlevel level J, J, whose concrete descriptions are listed in Table 1. whose concrete descriptions are listed in Table 1. Table 1. Summary of parameters in TQWT. Table Summary of of parameters in Table 1.1.Summary parameters inTQWT. TQWT. Parameter Symbol Function Description Parameter Symbol Function Description Quality factor Q Resonance degree Parameter Function Description Redundancy factor Symbol R Overlapping rate between subband frequency responses Quality factor Q Resonance degree Decomposition level frequency range responses Redundancy OverlappingDecomposition rate between subband frequency Quality factor factor QRJ Resonance degree Decomposition Decomposition frequency range Redundancy factor level RJ Overlapping rate between subband frequency responses Decomposition J Decomposition frequency range The Q-factorlevel describes the degree of signal resonance, and needs to be selected according to the
The Q-factor describessignals. the degree of signal resonance, and needs be selectedrate according to the characteristics of practical Redundancy factor r controls the to overlapping between characteristics of practical signals. Redundancy factor rQ-factor, controls an theincrease overlapping rate between the frequency responses of adjacent wavelets. With a fixed in redundancy factor The Q-factor describes the degree of signal resonance, and needs to be selected according to rthe frequency responses of adjacent wavelets. With a fixed Q-factor, an increase in redundancy factor will lead to a higher overlapping rate. Note that redundancy factor r should be strictly greater thanr characteristics of practical signals. Redundancy factor r controls the overlapping rate between the will lead togreater a higher overlapping rate. Note thatfor redundancy r should be strictly greater 1; a value than 3 is often recommended the perfectfactor reconstruction and sparsity. Oncethan the frequency responses of adjacent wavelets. With a fixed Q-factor, an increase in redundancy factor r 1; a valueand greater than 3 isfactor often rrecommended forthe thescaling perfectparameters reconstruction and sparsity. the and Q-factor redundancy are determined, can be Once obtained will lead to a higher overlapping rate. Note that redundancy factor r should be strictly greater than 1; Q-factor redundancy factor r are determined, the scaling parameters and can be obtained through and the following equation: a value greater than 3 is often recommended for the perfect reconstruction and sparsity. Once the through the following equation: 2 / (Q 1), the scaling 1 / rparameters β and α can be obtained Q-factor and redundancy factor r are determined, (6) 2 / ( Q 1), 1 /r (6) through the following equation: The decomposition level, J, adjusts the frequency coverage of the wavelets, with a higher J The decomposition the frequency of the 0wavelets, with aexcessive higher J enabling the wavelets tolevel, coverJ,a adjusts wider frequency rangecoverage and approach Hz. However, β = 2/( Q + 1), α = 1 − β/r (6) enabling the wavelets to coverJ)a will wider frequency and approach 0 Hz. However, excessive decomposition (an overlarge lead to poorrange computational efficiency. Specifically, TQWT decomposition overlarge J) will signal lead tointo poor computational efficiency.the Specifically, TQWT decomposes an (an n-point discrete-time J-level subbands, detail coefficients The decomposition level, J, adjusts the frequency coverage of the including wavelets, with a higher J enabling decomposes an n-point discrete-time signal intonumber J-level subbands, the detail and approximated coefficients. The maximum of levels isincluding set according to thecoefficients following the wavelets to cover a wider frequency range and approach 0 Hz. However, excessive decomposition and approximated coefficients. Thedown maximum numberwhole of levels is set according to the following equation ([·] represents the “round to the nearest unit“): (an overlarge J) will lead to poor computational efficiency. Specifically, TQWT decomposes an equation ([·] represents the “round down to the nearest whole unit“): n-point discrete-time signal into J-level subbands, * n / 8) the detail coefficients and approximated log(including J (7) max coefficients. The maximum number of levels isset log(according * n / 8) to the following equation ([·] represents J max log(1 / ) (7) the “round down to the nearest whole unit”): log(1 / ) The center frequency fc and bandwidth BW of the obtained level-j subband can be approximated: log ( β obtained ∗ n/8) level-j subband can be approximated: The center frequency fc and bandwidth BW of the Jmax = (7) ) j 2 (1/α fc log fs (8) 2 fc j 4 fs (8) The center frequency fc and bandwidth BW of the 4obtained level-j subband can be approximated:
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2−β fs 4α 1 BW 1 j 1 BW = 2 βα j−1 π 2
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fc = αj
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(9) (9)
Analyzing Equations (6), (8), and (9) jointly, it can be deduced that the tunable Q-factor wavelet canbe be entirely constructed with the determined Q-factor, redundancy factor r, and bases can entirely constructed with the determined Q-factor, redundancy factor r, and decomposition decomposition Figure wavelet 7 plots basis, a typical with 3,the r =decomposition 3, and J = 8. As the level J. Figure 7 level plots aJ. typical withwavelet Q = 3, rbasis, = 3, and J =Q8. =As levels decomposition levels increase, the corresponding frequency and bandwidth are both increase, the corresponding center frequency andcenter bandwidth are both lessened, while thelessened, Q-factor whileunchanged. the Q-factor stays unchanged. stays (a)
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Figure 7. A typical TQWT wavelet bases with Q = 3, r = 3, J = 8. (a) Time-domain waveforms; Figure 7. A typical TQWT wavelet bases with Q = 3, r = 3, J = 8. (a) Time-domain waveforms; (b) Frequency responses. (b) Frequency responses.
By comparing the TQWT and RADWT subbands, it can be observed that the frequency By comparing the TQWT and no RADWT it can be observed that level the frequency responses of the TQWT subbands longersubbands, have flat tops except for the first subband,responses and their of the TQWT subbands no longer have flat tops except for the first level subband, and theirparameters frequency frequency peaks each maintain a constant value. Instead of carefully choosing RADWT peaks each maintain constant value. Instead of carefully choosing parameters p, q, and s, p, q, and s, an explicit aQ-factor indicating resonance behavior is much RADWT more convenient to obtain with an explicit Q-factor indicating convenient to obtain formulas with TQWT, TQWT, especially when signal resonance oscillatory behavior behavior is is much knownmore in advance. Furthermore, for especially when signal oscillatory behavior is known in advance. Furthermore, formulas for calculating calculating TQWT subband center frequency fc and bandwidth BW greatly promote deeper TQWT subband frequency fc and of bandwidth BW subband greatly promote deepermakes developments for developments forcenter extracting information interest from signals, which TQWT more extracting of interest from subband signals, which makes TQWT more popular for popular forinformation many conditions. many conditions. 3. Resonance-Based Sparse Signal Decomposition 3. Resonance-Based Sparse Signal Decomposition 3.1. RSSD 3.1. RSSD Implementation Implementation Algorithm Algorithm The objective of RSSD is to to separate separate the the different different resonance resonance components components of given signal signal and and The objective of RSSD is of aa given realize the In In particular, forfor a given signal x= realize the sparsest sparsest representation representationofofeach eachresonance resonancecomponent. component. particular, a given signal x 1 + x2, via RSSD the signal is decomposed into the high-resonance component x1, low-resonance x = x1 + x2 , via RSSD the signal is decomposed into the high-resonance component x1 , low-resonance component xx2,, and the residual, assuming that x1 and x2 can be sparsely represented in bases S1, S2 component 2 and the residual, assuming that x1 and x2 can be sparsely represented in bases S1 , S2 (constructed by TQWT with high high and and low low Q-factors), Q-factors), respectively. respectively. Hence, Hence, aa suitable suitable optimization optimization (constructed by TQWT with problem for estimating coefficient matrixes W 1 and W2 under S1 and S2 is: problem for estimating coefficient matrixes W and W under S and S is:
1
2
1
2
2 n W1 , W2 arg min x S1 W1 S2 W22 2 1 W1 1 2 W2 1 o W1 , W2 = argmin W1 ,W2 kx − S1 W1 − S2 W2 k2 + λ1 kW1 k1 + λ2 kW2 k1
(10) (10)
W1 ,W2
where λ,1λ, are arethe thecorresponding correspondingweight weightcoefficients. coefficients.ItItisisworth worth mentioning mentioning that that relative relative values values where 1 2 2 of Withaafixed fixedλ1,, 1 ,,λ2 determine of λ determinethe theenergy energydistributions distributionsof ofthese these two two resonance resonance components. components. With 1
2
1
increasing 2 will increase the energy of x1 and decrease the energy of x2, and vice versa. Increasing both 1 , 2 will increase the energy of the residual and decrease that of the resonance components.
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increasing λ2 will increase the energy of x1 and decrease the energy of x2 , and vice versa. Increasing both λ1 , λ2 will increase the energy of the residual and decrease that of the resonance components. Although the optimization problem in Equation (10) is convex, a large number of variables and the non-differentiability of the l1 -norm make it difficult to solve [11]. Thankfully, the Split Augmented Lagrangian Shrinkage Algorithm (SALSA) [28,29] has been proven as a powerful tool to handle the optimization problem. With SALSA, RSSD iteratively updates coefficient matrixes W1 , W2 to achieve the minimization. At the end of all iterations, optimal coefficient matrixes W1∗ , W2∗ are gained, and the resonance components also reach a relatively considerable state for fault feature extraction, although they are not always sparse in terms of random noises. At this time, estimates of the highand low-resonance components are provided based on MCA: x1 = S1 W1∗ , x2 = S2 W2∗
(11)
More details about RSSD can be found in Reference [11]. Summarizing the contents of Sections 2.3 and 3.1, the concrete steps of RSSD are as follows: Step 1 Step 2
Step 3 Step 4 Step 5
Input a signal x for decomposition; Assisted with a priori information, select suitable Q-factors Q1 , and Q2, redundancy factors r1 , and r2 , and decomposition levels J1 , and J2 to construct corresponding wavelet bases S1 , S2 via TQWT; According to the observation signal noise level, determine suitable weight coefficients λ1 , λ2 and establish Equation (10); Solve the optimization problem with SALSA and obtain the optimal coefficient matrixes W1∗ , W2∗ ; Utilize W1∗ , W2∗ to represent the estimates of high- and low-resonance components with x1 = S1 W1∗ , x2 = S2 W2∗ .
To clearly demonstrate the superiority of RSSD in decomposing different resonance signals, a performance illustration is presented in the next subsection. 3.2. Illustration of RSSD Performance In this subsection, to verify the effectiveness of RSSD in processing mechanical vibration signals, a typical gear fault vibration signal is constructed as follows (the Morlet function): 5
y(t) =
∑e
√
ζ 1− ζ 2
[2π f (t−kT −τ )]2
cos[2π f (t − kT − τ )]
(12)
k =0
where damping ratio ζ = 0.01, frequency f = 100 Hz, time shift τ = 0.1 s, and cyclic period T = 0.2 s. Meanwhile, gaussian noise with a signal-noise ratio 4dB is also added, which makes the simulation signal closer to the practical one. In this example, the gear fault-induced signal components, in the form of the Morlet function, belong to the high-resonance component, and noises should fall into the low-resonance category for their non-oscillatory behaviors. Set decomposition parameters to Q1 = 4, r1 = 5, J1 = 105, λ1 = 0.3, Q2 = 1, r2 = 5, J2 = 10, and λ2 = 0.1, and employ standard RSSD to process the composite signal. The original signal, and obtained high- and low-resonance components, are displayed in Figure 8.
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4
Amplitude (0.1g)
3 2 1 0 -1 -2 -3 -4 0
0.2
0.4
0.6 Time (s)
0.8
1
1.2
x x1 x 2 Low-resonance component
High-resonance component 4
1
T = 0.2 s
3
Amplitude (0.1g)
Amplitude (0.1g)
0.5
0
2 1 0 -1 -2
-0.5
-3 -1 0
0.2
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0.6 Time (s)
0.8
1
1.2
-4 0
0.2
0.4
0.6 Time (s)
0.8
1
1.2
Figure8.8.Illustration Illustrationofofresonance-based resonance-basedsparse sparsesignal signaldecomposition decomposition(RSSD) (RSSD)performance. performance. Figure
It can be seen that fault-related waveforms are almost buried due to the presence of strong It can be seen that fault-related waveforms are almost buried due to the presence of strong background noises, and thus no obvious fault information can be found easily in original signal. background noises, and thus no obvious fault information can be found easily in original signal. Moreover, the oscillatory waveforms and noises overlap, either in time domain or in frequency Moreover, the oscillatory waveforms and noises overlap, either in time domain or in frequency domain, which will invalidate traditional frequency band based methods. In RSSD space, the original domain, which will invalidate traditional frequency band based methods. In RSSD space, the original signal can be decomposed into two parts: (1) gear fault-induced sustained waveforms; (2) noises with signal can be decomposed into two parts: (1) gear fault-induced sustained waveforms; (2) noises non-oscillatory features. With inspection, the results conform the theoretical analysis: the highwith non-oscillatory features. With inspection, the results conform the theoretical analysis: resonance component mainly consists of sustained oscillatory Morlet waveforms, while the transient the high-resonance component mainly consists of sustained oscillatory Morlet waveforms, while characteristic of the low-resonance component is apparent. Meanwhile, the period 0.2 s for the highthe transient characteristic of the low-resonance component is apparent. Meanwhile, the period 0.2 s resonance component further indicates an obvious gear fault. for the high-resonance component further indicates an obvious gear fault. In addition to validating RSSD’s effectiveness, we will also analyze the parameters’ influences In addition to validating RSSD’s effectiveness, we will also analyze the parameters’ influences on on decomposition results, especially on the high- and low-resonance waveforms. decomposition results, especially on the high- and low-resonance waveforms. 3.3.Effects EffectsofofRSSD RSSDParameters Parameters 3.3. Sincethe theredundancy redundancyfactor factorr rand anddecomposition decompositionlevel levelJ Jonly onlyaffect affectthe theoverlapping overlappingrate rateand and Since frequency coverage, they can cooperate with each other to achieve the frequency ranges requested frequency coverage, they can cooperate with each other to achieve the frequency ranges requested inin RSSD.As Assuch, such,only onlythe theeffects effectsofofthe theQ-factors Q-factorsand andweight weightcoefficients coefficientsare arefurther furtherinvestigated. investigated. RSSD. 3.3.1.Effects EffectsofofQ-Factors Q-Factors 3.3.1. Generally,the thelow lowQ-factor Q-factorisisset setasas1,1,which whichsatisfies satisfiesthe therequirement requirementofofrepresenting representingtransient transient Generally, noisesininaacomplex complexsignal. signal. However, However, aa high high Q-factor Q-factor needs needs to tobe beselected selectedassisted assistedwith withaapriori priori noises information. Therefore, we will further analyze the mechanical vibration signal simulated above information. Therefore, we will further analyze the mechanical vibration signal simulated above throughaltering alteringthe thehigh high Q-factor 1 and obtaining high-resonance components a different through Q-factor Q1Qand obtaining thethe high-resonance components for afor different Q1 , Q 1 , as shown in Figure 9. For the sake of clarity, Figure 9 only plots high-resonance components when as shown in Figure 9. For the sake of clarity, Figure 9 only plots high-resonance components when Q1 1 is equal 2, 3,5,4respectively. and 5, respectively. Noteworthy, the high-resonance component in blue isQequal to 2, 3,to4 and Noteworthy, the high-resonance component in blue corresponds to ofthe illustration of shown RSSD in performance shown in Section 3.2, with identical tocorresponds the illustration RSSD performance Section 3.2, with identical decomposition parameters. decomposition parameters. As seen in Figure 9, high-resonance waveforms seem not to exhibit As seen in Figure 9, high-resonance waveforms seem not to exhibit distinct differences though Q1 distinct differences varies from 2 to 5. though Q1 varies from 2 to 5.
Amplitude (0.1g)
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1 0 -1 5 4 3 Q1
2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
Figure Figure 9. 9. High-resonance High-resonance components components in in cases cases where where Q Q11 ==2,2,3,3,4,4,5.5.
To give a quantitative evaluation to this indistinctive variation, the degree of similarity between To give a quantitative evaluation to this indistinctive variation, the degree of similarity between resonance components and the original signal is utilized with the correlation coefficient equation resonance components and the original signal is utilized with the correlation coefficient equation [30,31] [30,31] N N mean(x1 ))( x (k) − mean(x)) ∑k=1 ( x(ix(k( k))− mean ( x ))( x ( k ) mean ( x )) i 1 k 1 i = 1, 2 (13) ρi = q i N ( x (k) − mean(x ))2 N ( x(k) − mean(x))i 2 1, 2 (13) ∑ k =1 N i 1 2 ∑N k =1 2 ( xi ( k ) mean( x1 )) ( x( k ) mean( x )) k 1 k 1
where xi (k) is the element in high- and low-resonance components respectively, x (k) is the element xi ( k ) is the where element highand low-resonance components is the element x( k ) the in the original signal, and in mean represents the averaging operation.respectively, Thus, we obtain correlation coefficients ρ1 , signal, ρ2 between the resonance components and operation. the originalThus, signal, as listed in in the original and mean represents the averaging werespectively, obtain the correlation Table 2. From Table 2, it is easy to discover that correlation coefficients change slightly when Q1 varies coefficients 1 , 2 between the resonance components and the original signal, respectively, as listed from 2 to 9 for both the high-resonance component and low-resonance component. The maximum in Table 2. of From 2, it is easy to components discover thatare correlation slightly when Q1 variations high-Table and low-resonance only 0.08%coefficients and 0.44%,change respectively. It is the tiny varies from 2 to 9 for both the high-resonance component and low-resonance component. The variations that explain why the high-resonance waveforms don’t show obvious differences. Meanwhile, maximum highand low-resonance components are only 0.08% and 0.44%, respectively. Table 2 alsovariations indicates of that the selection of a high Q-factor has some effects on the decomposition results, Itor is the tiny variations that explain why the high-resonance waveforms don’t show obvious rather an exacter description—‘limited effects’. differences. Meanwhile, Table 2 also indicates that the selection of a high Q-factor has some effects on the decomposition results, rather an coefficients exacter description—‘limited Table 2.orCorrelation when Q1 varies from 2effects’ to 9. Q1
Q 21 32 4 3 5 4 6 75 86 97 8 9
Table 2. Correlation coefficients when Q1 varies from 2Coefficient to 9. Correlation Coefficient Correlation Q2 (High) (Low) Correlation Coefficient Correlation Coefficient Q21 0.3067 0.9710 (High) (Low) 1 0.3072 0.9656 1 0.3067 0.9710 1 0.3072 0.9703 1 0.3072 0.9656 1 0.3068 0.9710 1 0.3072 0.9703 1 0.3065 0.9721 11 0.3068 0.9710 0.3063 0.9730 11 0.3065 0.9721 0.3063 0.9743 11 0.3063 0.9730 0.3059 0.9754 1 0.3063 0.9743 1 0.3059 0.9754
3.3.2. Effects of Weight Coefficients
analyzed, weight coefficients λ1 , λ2 control the energy distribution between each resonance 3.3.2. As Effects of Weight Coefficients component. To highlight the significant influence caused by variations of the weight coefficients, 1 , 2Figures As analyzed, weight that coefficients control the 11 energy distribution each we consider the condition only λ1 varies. 10 and plot the high- and between low-resonance resonance component. highlight the 0.3, significant caused bycorrelation variations coefficients of the weight components when λ1 isTo equal to 0.1, 0.2, and 0.4 influence respectively, whose are provided in Table 3. As is easily seen in Figures 10 and 11, as the weight coefficient λ changes, coefficients, we consider the condition that only 1 varies. Figures 10 and 11 plot the high-1 and lowboth high- and low-resonance components undergo considerable waveform changes, especially the resonance components when 1 is equal to 0.1, 0.2, 0.3, and 0.4 respectively, whose correlation high-resonance component. coefficients are provided in Table 3. As is easily seen in Figures 10 and 11, as the weight coefficient
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1 changes, changes,both both high- and low-resonance components undergo considerable waveform changes, 1 Sensors 2017, 17, 1279 high- and low-resonance components undergo considerable waveform changes, 11 of 27 Amplitude (0.1g) Amplitude (0.1g)
especiallythe thehigh-resonance high-resonancecomponent. component. especially 2 0
2 0 'Unwanted' noises 'Unwanted' noises
-2 -2 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1
Lambda1 Lambda1
0
0.2 0.2
0
0.6 0.4 0.6 0.4 Time (s) Time (s)
0.8 0.8
1
1.2 1.2
1
0.1, 0.2, 0.3, 0.4 . Figure10. 10.High-resonance High-resonancecomponents componentsin caseswhere whereλ = 1 0.1, 0.40.4. Figure 10. High-resonance components inincases cases where 0.1,0.2, 0.2,0.3, 0.3, Figure . 11
Amplitude (0.1g) Amplitude (0.1g)
5 0
5 0
-5 -5 0.4 0.4 0.3 0.3 0.2 0.2 Lambda1 Lambda1
0.1 0.1
0
0
0.2 0.2
0.6 0.4 0.6 0.4 Time (s) Time (s)
0.8 0.8
1
1
1.2 1.2
0.1, 0.1,0.2, 0.2, 0.3, 0.3, 0.4 Figure11. 11.Low-resonance Low-resonancecomponents componentsin caseswhere whereλ = Figure 11. Low-resonance components inincases cases where 1 0.1, 0.2, 0.3, 0.40.4. Figure . . 11
0.4, 0.3, the Indetail, detail,when whenλ1 1=0.4, , the thehighhigh-and andlow-resonance low-resonancecomponents componentscan canbebeclearly detail, when 0.4, 0.3, highand low-resonance components can clearlyseparated. separated. 0.3 InIn 1 Though amplitudes of both high-resonance components decay a little bit, their periodical oscillatory Thoughamplitudes amplitudesofofboth bothhigh-resonance high-resonancecomponents componentsdecay decaya alittle littlebit, bit,their theirperiodical periodicaloscillatory oscillatory Though features are fairly dominant, dominant,which whichindicates indicates a satisfactory decomposition result for mechanical features are fairly a satisfactory decomposition result for mechanical fault features are fairly dominant, which indicates a satisfactory decomposition result for mechanical fault fault diagnosis. diagnosis. diagnosis. When λ1 = 0.2, amplitude of the high-resonance waveform increases significantly, but slight 0.2, amplitude When 1 0.2 , amplitudeofofthe thehigh-resonance high-resonancewaveform waveformincreases increasessignificantly, significantly,but butslight slight When noise springs 1 up as well, compared with the condition when λ1 is 0.4 or 0.3, as described above. noisesprings springsup upasas well,compared compared withthe thecomponents conditionwhen when is 0.4 or 0.3,due as to described above. Consequently, the sparsity of high-resonance cannot be guaranteed the emergence 1 is 0.4 or 0.3, as described above. noise well, with condition 1 of unwanted noise. Consequently, thesparsity sparsityofofhigh-resonance high-resonancecomponents componentscannot cannotbebeguaranteed guaranteeddue duetotothe the Consequently, the Furthermore, the case where λ is equal to 0.1 is discussed. At this time, sustained oscillatory 1 emergenceofofunwanted unwantednoise. noise. emergence characteristics can hardly be distinguished in the high-resonance component, since noise is rather Furthermore,the the casewhere where 1 isisequal equaltoto0.1 0.1isisdiscussed. discussed.AtAtthis thistime, time,sustained sustainedoscillatory oscillatory Furthermore, abundant, which will case make few contributions to mechanical fault feature extraction. Reflecting on the 1 characteristicscan canhardly hardlybebe distinguished thehigh-resonance high-resonance sincenoise noise rather low-resonance components in the above cases, with an increasing component, λcomponent, evident variation lies characteristics distinguished ininthe isisrather 1 , the most since abundant, which will make few contributions to mechanical fault feature extraction. Reflecting on that their amplitudes increase Considering that there less fault information embedded abundant, which will make fewgradually. contributions to mechanical faultisfeature extraction. Reflecting on thelow-resonance low-resonancecomponents componentsinin theabove abovecases, cases, withananincreasing increasing , themost most evident variation in low-resonance components, high-resonance components are more suitable for the detection of the the with ,1 the evident variation 1 mechanical fault features. liesthat thattheir theiramplitudes amplitudesincrease increasegradually. gradually.Considering Consideringthat thatthere thereisisless lessfault faultinformation information lies A summary of the information in Table 3high-resonance shows that maximum correlation coefficient variations embedded in low-resonance components, components are more suitable forthe the embedded in low-resonance components, high-resonance components are more suitable for reach up to the surprising levels of 61.5% and 15.69% for the highand low-resonance components, detectionofofmechanical mechanicalfault faultfeatures. features. detection respectively. With a careful observation, the correlation coefficient of the high-resonance component undergoes a sharp variation when λ1 steps from 0.1 to 0.2, the primary cause of which is the rich presence of noise. In short, such a quantitative result demonstrates that weight coefficients have a remarkable effect on the decomposition results of RSSD, and they must be treated with caution.
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Table 3. Correlation coefficients in cases where λ1 varies from 0.1 to 0.8.
λ1
λ2
Correlation Coefficient (High)
Correlation Coefficient (Low)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.9182 0.3856 0.3072 0.3057 0.3053 0.3048 0.3041 0.3032
0.8351 0.9622 0.9703 0.9745 0.9792 0.9838 0.9882 0.9920
In spite of the outstanding performance of RSSD in processing nonlinear and non-stationary signals, whether RSSD can be conducted successfully relies not only on signal intrinsic characteristics, but also on the selection of ideal parameters to a great extent. By contrast, weight coefficients have a more significant and even decisive effect on decomposition results than Q-factors. As a result, to guarantee applied success, the optimized selection of Q-factors and weight coefficients is the most crucial issue for developing more advanced RSSD and promoting wider applications. 4. Applications of RSSD in Mechanical Fault Diagnosis Due to the rich fault-related information buried in noisy mechanical fault signals, fault diagnosis technique based on signal processing can extract sensitive features and identify fault types. RSSD separates complex signals based on a new perspective, resonance, overcomes the limitation of traditional frequency band based methods, and can reveal fault features from original mechanical vibration signals more effectively. This huge advantage allows RSSD to be extensively applied to the fault diagnosis of crucial industrial components, that is, bearings, gearboxes and rotors. To present an organized review, this section will provide a survey of the applications of RSSD in mechanical fault diagnosis based on different optimization directions, including original RSSD, parameter optimized RSSD, subband optimized RSSD, integrated optimized RSSD, and RSSD combined with other methods. 4.1. Original RSSD in Mechanical Fault Diagnosis Up to now, the original RSSD proposed by Selesnick in 2011 has been extensively applied in the field of mechanical fault diagnosis. This subsection will review the publications in which only original RSSD was used, without combining other techniques. In 2012, Yu et al. [32] pioneered the introduction of RSSD into the field of mechanical fault diagnosis and proposed an envelope demodulation method based on RSSD. This method separated transient impacts containing bearing fault information into their low-resonance components, and executed envelope demodulation analysis on the low-resonance components to detect the rolling bearing’s inner and outer fault features successfully. Afterwards, a similar method was also employed for the fault diagnosis of gears [33]. Xiang et al. [34,35] and Huang et al. [36,37] performed RSSD to decompose rolling bearing fault vibration signals into high- and low-resonance components, effectively reduced heavy background noise, and extracted weak rolling bearing fault information quickly and accurately. In consideration of the frequency overlap between gearbox fault vibration signal components, RSSD was utilized [38,39]. The obtained low-resonance component of the engineering gearbox fault signal, as well as its envelope spectrum, is illustrated in Figure 12. Since the gear fault feature frequency was 30.1 Hz, the peaks at 29.3 Hz and harmonics were cogent enough to indicate a gear fault. For the sake of safety in production, the factory checked and confirmed the broken gear teeth a few weeks later.
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envelope spectrum, is illustrated in Figure 12. Since the gear fault feature frequency was 30.1 Hz, the peaks2017, at 29.3 Hz and harmonics were cogent enough to indicate a gear fault. For the sake of safety in Sensors 17, 1279 13 of 27 production, the factory checked and confirmed the broken gear teeth a few weeks later. (a)
(b)
Figure 12. 12. Low-resonance Low-resonance component component of of gearbox gearbox fault fault vibration vibration signal. signal. (a) waveforms; Figure (a) Time-domain Time-domain waveforms; (b) Demodulation spectrum [38]. (b) Demodulation spectrum [38].
Zhang et al. [40] introduced an energy operator demodulating analysis method based on RSSD Zhang et al. [40] introduced an energy operator demodulating analysis method based on RSSD and and detected compound fault features of a gearbox. To segment the impacts from vibration signals detected compound fault features of a gearbox. To segment the impacts from vibration signals in rotor in rotor systems, Chen et al. [41,42] performed RSSD and diagnosed early rub-impact fault with the systems, Chen et al. [41,42] performed RSSD and diagnosed early rub-impact fault with the reassigned reassigned wavelet scalogram. In 2014, Wang et al. [43] used TQWT to construct high- and low-Qwavelet scalogram. In 2014, Wang et al. [43] used TQWT to construct high- and low-Q-factor wavelet factor wavelet bases to successfully separate the periodic component with rotor rotation rate from bases to successfully separate the periodic component with rotor rotation rate from transient component transient component induced by rub-impact fault. Accurate fault identification from the lowinduced by rub-impact fault. Accurate fault identification from the low-resonance component verified resonance component verified RSSD’s validity. In addition, a condition assessment system for an RSSD’s validity. In addition, a condition assessment system for an automatic tool changer in CNC automatic tool changer in CNC machining centers was established by Chen [44]. This monitoring machining centers was established by Chen [44]. This monitoring system, based on RSSD, could not system, based on RSSD, could not only detect the transient component caused by a globoidal only detect the transient component caused by a globoidal indexing cam fault, but also locate the indexing cam fault, but also locate the fault’s exact position. fault’s exact position. It is interesting that a group of scholars represented by Yu, Cui, and Chen thought the vibration It is interesting that a group of scholars represented by Yu, Cui, and Chen thought the vibration responses induced by mechanical fault impact exhibited transient characteristics, which should fall responses induced by mechanical fault impact exhibited transient characteristics, which should under the category of the low-resonance component, while the high-resonance component sparsely fall under the category of the low-resonance component, while the high-resonance component represented the high-frequency interferences. In sharp contrast, Huang’s team considered the fault sparsely represented the high-frequency interferences. In sharp contrast, Huang’s team considered impact as a damped oscillation response, whose amplitude diminished continuously but still the fault impact as a damped oscillation response, whose amplitude diminished continuously but exhibited sustained oscillation behaviors while the low-resonance component corresponded to still exhibited sustained oscillation behaviors while the low-resonance component corresponded to random interference in the original signal. Thus Huang preferred to discover fault information from random interference in the original signal. Thus Huang preferred to discover fault information from the high-resonance component rather than from the low-resonance component. In fact, all faultthe high-resonance component rather than from the low-resonance component. In fact, all fault-related related information cannot be completely decomposed into a single resonance component, and information cannot be completely decomposed into a single resonance component, and energy leakage energy leakage was inevitable due to their inherent mutual coherence. Yu et al. made use of the lowwas inevitable due to their inherent mutual coherence. Yu et al. made use of the low-resonance resonance component to seize the instantaneous peak induced by mechanical fault impact, while component to seize the instantaneous peak induced by mechanical fault impact, while Huang utilized Huang utilized the high-resonance component to catch the characteristics of damped oscillations. the high-resonance component to catch the characteristics of damped oscillations. These two detection These two detection methods were both reasonable and feasible in theory. Moreover, the fact that methods were both reasonable and feasible in theory. Moreover, the fact that they all extracted they all extracted the mechanical fault features in their own way confirms the energy leakage the mechanical fault features in their own way confirms the energy leakage phenomenon. In short, phenomenon. In short, mechanical fault-induced feature information will simultaneously lie in the mechanical fault-induced feature information will simultaneously lie in the high-resonance component, high-resonance component, the low-resonance component, and even the residual. the low-resonance component, and even the residual. The original RSSD has been proven effective for feature extraction, and has a promising The original RSSD has been proven effective for feature extraction, and has a promising application application in mechanical fault diagnosis. However, in consideration of the RSSD method and its in mechanical fault diagnosis. However, in consideration of the RSSD method and its application, application, some key points remain to be further investigated: some key points remain to be further investigated: (1) Construction of wavelet bases; (1) Construction of wavelet bases; (2) Optimized selection of numerous decomposition parameters; (2) Optimized selection of numerous decomposition parameters; (3) Reconstruction of subband signals; (3) Reconstruction of subband signals;resonances. (4) Sparse decomposition of multiple (4) Sparse decomposition of multiple resonances. At the present time, further research on RSSD mainly focuses on two aspects, the optimized selection of parameters (Point 2), and the reconstruction of subband signals (Point 3).
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4.2. Parameter Optimized RSSD in Mechanical Fault Diagnosis The analysis in Section 3.2 explains that parameters (Q-factors and weight coefficients) appreciably influence decomposition results, and that the effects of weight coefficients are much more predominant and crucial. Therefore, abundant literatures have paid much attention to selecting optimal decomposition parameters, seeking more effective application for mechanical fault diagnosis. Dai and Cui [45,46] comprehensively studied the effects of Q-factors, redundancy factors, and weight coefficients, and emphasized weight coefficients’ effects in RSSD. Some efficient suggestions were provided by Huang for selecting the decomposition level J in conjunction with a specific diagnosis objects: rolling bearings [47]. Coincidentally, both stressed the suitable selection of weight coefficients for feature extraction of early weak faults in rolling bearings. The conclusions from these studies are completely consistent with our analysis in Section 3.2. Regretfully, no index or indicator was adopted in these studies to evaluate the influences. Consequently, Cai et al. [48] offered a new method to select weight coefficients. In this method, three indexes were set up to assess the decomposition results, and weight coefficients corresponding to better results were adopted. Then RSSD with optimized parameters was used to successfully detect the compound fault features of a gearbox. To overcome the large subjective randomness when selecting RSSD parameters, an optimized RSSD based on the genetic algorithm (GA), and realizing the adaptive decomposition of high- and low-resonance components, was proposed in Reference [49] by concurrently selecting and optimizing the elements of coefficient matrixes. This method minimized information leakage in the process of signal decomposition and effectively extracted compound fault characteristics of rolling bearings. On this basis, Huang et al. [50] made full use of the global optimization ability of GA, and optimized Q-factors further. RSSD with the optimal Q-factors was used successfully to diagnose the composite faults of the planetary gear and bearing in a planetary gearbox. Similar work was also performed in Reference [51], though one difference that must be mentioned was that Li took the kurtosis of low-resonance component as the objective function used in GA, instead of a function like Equation (10), utilized in Huang’s research. Soon afterward, Zhang et al. [52] developed a composite weight function as the objective function in GA and adaptively optimized Q-factors and redundancy factors synchronously. Equations (14)–(16) provide more details about the smoothness SI of high-resonance component and kurtosis Kur of low resonance component used in GA, as well as the weight function F: exp
N
∑ ln x1 (i )/N
i =1 N
SI (x1 ) =
(14)
∑ x1 (i )/N
i =1
E Kur (x2 ) =
N
∑ ( x2 ( i ) − µ )
4
i =1
σ4
F = α1 · SI (x1 ) + β 2 · Kur (x2 )
(15) (16)
where σ and µ denote the standard deviation and the average of original signal x, E{ } represents the calculation of the expected value, and α1 and β 2 are the weight coefficients of SI and Kur, respectively. This method was executed to process the signal collected from a gearbox fault test rig with a faulty gear and an outer race faulty bearing simultaneously. The optimized parameters based on the above-mentioned GA were Q1 = 4.25, r1 = 8.06, Q2 = 1, and r2 = 6.68. Figure 13 displays the resulting high- and low-resonance components and their instantaneous amplitude spectra. It can be easily seen that clear peaks occurred at the rotation rate fr (10 Hz) in instantaneous amplitude spectrum of the optimal high-resonance component, which indicated a gear fault. Similarly, the dominant peaks at outer race fault feature frequency, and its harmonics shown in Figure 13d, verified the existence of the
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spectrum of the optimal high-resonance component, which indicated a gear fault. Similarly, the dominant peaks at outer race fault feature frequency, and its harmonics shown in Figure 13d, verified outer race Theofsame method wasThe alsosame applied to diagnose the compound of a gear tooth the fault. existence the outer race fault. method was also applied to diagnosefaults the compound faults of a gearouter tooth crack and a bearing outer race fault. crack and a bearing race fault.
13. RSSD results instantaneous amplitude spectra of resonance components: Figure Figure 13. RSSD results andandthetheinstantaneous amplitude spectra of resonance components: (a,c) Optimal high and low-resonance components, respectively; (b,d) Instantaneous amplitude (a,c) Optimal high and low-resonance components, respectively; (b,d) Instantaneous amplitude spectra spectra of (a,c), respectively [52]. of (a,c), respectively [52].
Since the single Q-factor pairs can hardly represent all resonances of interference or wanted components, some researchers have attempted solve this problem through the algorithm. Since the single Q-factor pairs can hardlytorepresent all resonances of iterative interference or wanted For example, Wang et al. [53] developed a comprehensive strategy based on RSSD and the iterative components, some researchers have attempted to solve this problem through the iterative algorithm. algorithm for the separation of full-oscillatory components from noisy machining vibration signals For example, Wang et al. [53] developed a comprehensive strategy based on RSSD and the iterative at the thin-walled parts machining process. When minimization of the Q-factor variation was algorithm for thechatter-related separation oflow-resonance full-oscillatory components from noisy machininghigh-resonance vibration signals at reached, component and periodic cutting-related the thin-walled parts process. When minimization Q-factor component weremachining separated successfully, providing a new means of forthe monitoring thevariation machiningwas state.reached, chatter-related low-resonance component and[54,55] periodic cutting-related high-resonance component Comparing to Reference [53], in 2016, Shi found that the oscillatory behaviors of highfrequency interferences in faulty rolling bearing vibration signals were manifold; thus a single high were separated successfully, providing a new means for monitoring the machining state. Comparing to Q-factor failed to eliminate all interference signals at once. Inspired by this, an iterative RSSD was Reference [53], in 2016, Shi [54,55] found that the oscillatory behaviors of high-frequency interferences put forward to “peel” the interferences step by step, and rolling bearing fault feature information was in faulty rolling bearing vibration signals were manifold; thus a single high Q-factor failed to eliminate successfully dug out from the final “purified” low-resonance component. It is important to note that all interference signals at once. Inspired by this, an iterative RSSD wasisput forward to “peel” the even though the iterated objects are different, the basic point of these studies identical. Specifically, interferences by step, and of rolling bearingcomponent fault feature wasbeforehand, successfully it is thestep oscillatory property one resonance that isinformation difficult to know and adug out single pair is low-resonance highly likely to cause the loss of useful such an iterative from the finalQ-factor “purified” component. It is information. important However, to note that even though the RSSD will inevitably suffer from low efficiency, especially when dealing with large-scale data. iterated objects are different, the basic point of these studies is identical. Specifically, it is the oscillatory property of one resonance component that is difficult to know beforehand, and a single Q-factor pair is 4.3. Subband Optimized RSSD in Mechanical Fault Diagnosis highly likely to cause the loss of useful information. However, such an iterative RSSD will inevitably Because there was still considerable noise in the resonance components obtained from RSSD, suffer from low efficiency, especially when dealing with large-scale data. some scholars continually exploited the approaches for reconstruction of subband signals to further highlight fault features. Huang et al. [47] first proposed the concept of the “main subband”, and 4.3. Subband Optimized RSSD in Mechanical Fault Diagnosis combined high- and low-resonance components to perform subband reconstruction in order to minimize thewas energy mentioned in Section As they described, for rolling bearing ER-12T, Because there stillleakage considerable noise in the4.1. resonance components obtained from RSSD, some the rotation rate and its harmonic components were primarily concentrated below 2 kHz, with two scholars continually exploited the approaches for reconstruction of subband signals to further highlight natural frequency bands occurring at approximately 3 and 10 kHz, and the rest belonging to all kinds of fault features. Huang et al. [47] first proposed the concept of the “main subband”, and combined highinterfering noise. Hence, the concrete frequency band distribution was gained, as viewed in Figure 14. and low-resonance components to subbands performapproaching subband reconstruction inbands, orderwhich to minimize the energy Using Equations (8) and (9), the natural frequency were defined leakageasmentioned in Section 4.1.calculated. As theyAfter described, rollingofbearing ER-12T, the rotation rate the main subbands, can be the mainfor subbands both highand low-resonance componentscomponents were obtained,were the original signal’s main subbands can be2 obtained through and its harmonic primarily concentrated below kHz, with two superposing natural frequency these two resonance components’ main subbands. The resulting main subband envelope spectra of bands occurring at approximately 3 and 10 kHz, and the rest belonging to all kinds of interfering the high-resonance component, low-resonance component, and original signal were illustrated in
noise. Hence, the concrete frequency band distribution was gained, as viewed in Figure 14. Using Equations (8) and (9), the subbands approaching natural frequency bands, which were defined as the main subbands, can be calculated. After the main subbands of both high- and low-resonance components were obtained, the original signal’s main subbands can be obtained through superposing these two resonance components’ main subbands. The resulting main subband envelope spectra of the high-resonance component, low-resonance component, and original signal were illustrated in Figure 15a–c. As can be seen in this figure, although obvious peaks all occurred at the feature frequency (163.5 Hz), the corresponding amplitudes implied that the main subband of original signal brought the
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Figure 15a–c. As can be seen in this figure, although obvious peaks all occurred at the feature Figure 15a–c. As can be seen in this figure, although obvious peaks all occurred at the feature frequency (163.5 Hz), the corresponding amplitudes implied that the main subband of original signal frequency (163.5 Hz), the corresponding amplitudes implied that the main subband of original signal richest faultthe information, thus it should be taken fullbe advantage particularly when diagnosing brought richest faultand information, and thus it should taken fullof, advantage of, particularly when brought the richest fault information, and thus it should be taken full advantage of, particularly when incipient bearing faults.bearing faults. diagnosing incipient diagnosing incipient bearing faults.
Frequency Frequency(Hz) (Hz)
High-Frequency random interfering noise High-Frequency random interfering noise 10000 10000 8000 8000
Second order natural frequency band of ER-12T bearing Second order natural frequency band of ER-12T bearing
High-Frequency random interfering noise High-Frequency random interfering noise 4000 4000 3000 3000 2000 2000 0 0
First order natural frequency band of ER-12T bearing First order natural frequency band of ER-12T bearing Rotation frequency and its harmonics Rotation frequency and its harmonics
Characteristics Characteristics Figure 14.Frequency Frequency distribution of of rolling rolling bearing ER-12T. Figure bearingER-12T. ER-12T. Figure14. 14. Frequency distribution distribution of rolling bearing
Amplitude Amplitude(0.1g) (0.1g)
-5
x 10 4x 10-5 4 2 2 0 0 -6 x 10 4x 10-6 4 2 2 0 0 -5 x 10 4x 10-5 4 2 2 0 0
(a) (a)
133.5 163.5 193.5 133.5 163.5 193.5 (b) (b)
327 327
133.5 163.5 193.5 133.5 163.5 193.5 (c) (c)
327 327
133.5 163.5 193.5 133.5 163.5 Frequency(Hz) 193.5 Frequency(Hz)
327 327
Figure 15. Main subband envelope spectra: (a) High-resonance component; (b) Low-resonance Figure Mainsubband subbandenvelope envelope spectra: spectra: (a) Figure 15.15.Main (a) High-resonance High-resonancecomponent; component;(b)(b)Low-resonance Low-resonance component; (c) Original signal [47]. component; (c) Original signal [47]. component; (c) Original signal [47].
Another subband reconstruction method was explored in Reference [56]. It merged the subbands Another subband reconstruction method was explored in Reference [56]. It merged the subbands of low-resonance component and method selected was the explored optimal signal with the biggest kurtosis for the subband reconstruction in Reference [56]. It merged the subbands of Another low-resonance component and selected the optimal signal with the biggest kurtosis for the extraction of fault features. Via this method, the fault features of a rolling bearing were prominent of low-resonance component the optimal signal withofthe biggestbearing kurtosiswere for the extraction extraction of fault features. and Via selected this method, the fault features a rolling prominent and easy to detect. As well as the direct mergence in Reference [56], Luo etwere al. [57] further combined of and fault features. ViaAs this method, the fault features of a rolling bearing and easy easy to detect. well as the direct mergence in Reference [56], Luo et al. [57]prominent further combined these merged signals with similar kurtosisinvalues and neighboring subbands. The final signal with to these detect. As well as the direct mergence Reference [56], Luo et al. [57] further combined these merged signals with similar kurtosis values and neighboring subbands. The final signal with the biggest kurtosis value was successfully used to diagnose a rolling bearing outer race fault. By merged signals with similar kurtosis valuesused and to neighboring The outer final race signal with the biggest kurtosis value was successfully diagnose a subbands. rolling bearing fault. Bythe selecting the components with extreme kurtosis values and removing those under the correlation selecting the components with extreme kurtosis valuesaand removing the correlation biggest kurtosis value was successfully used to diagnose rolling bearingthose outerunder race fault. By selecting coefficient threshold, Tang and Wang [58] successfully extracted weak characteristic frequency threshold, Tang and Wangvalues [58] successfully extracted weak characteristic frequency thecoefficient components with extreme kurtosis and removing those under the correlation coefficient components of rolling bearings. Motivated by the neighboring coefficient thresholding for wavelet components of rolling bearings. Motivated by the neighboring coefficient thresholding for wavelet of threshold, Tang and Wang [58] successfully extracted weak characteristic frequency components de-noising [59], He et al. [60] developed a neighboring coefficient de-noising (NCD) based RSSD. The de-noising [59], Motivated He et al. [60] developed a neighboring coefficient de-noising based RSSD. The rolling bearings. the neighboring coefficient thresholding for(NCD) wavelet [59], subbands were processedby through a neighboring coefficient thresholding scheme de-noising and then the subbands were processed through a neighboring coefficient thresholding scheme and then the He reconstructed et al. [60] developed a neighboring coefficient de-noising (NCD) based RSSD. The subbands were signal was obtained. The effective engineering applications, including bearing and reconstructed signal was obtained. The effective engineering applications, including bearing and processed through a neighboring coefficient thresholding scheme and then the reconstructed signal gearbox fault diagnosis, validated its practicability and effect on suppressing noise. Given the gearbox faultThe diagnosis, its applications, practicability including and effectbearing on suppressing noise. Given the was obtained. effectivevalidated engineering andand gearbox diagnosis, sparsity and clustering/grouping property of subband coefficients [61], He Zi [62]fault introduced a sparsity and clustering/grouping property of subband coefficients [61], He and Zi [62] introduced a validated its practicability and effect on suppressing noise. Given the sparsity and clustering/grouping property of subband coefficients [61], He and Zi [62] introduced a modified RSSD on the basis of
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modified RSSD on the basis of overlapping group shrinkage (OGS) to facilitate the sparsity of results, and thus accomplished early fault diagnosis of rolling bearings. overlapping group shrinkage (OGS) to facilitate the sparsity of results, and thus accomplished early fault diagnosis of rolling bearings. 4.4. Integrated Optimized RSSD in Mechanical Fault Diagnosis 4.4. Integrated Optimized RSSD in Mechanical Fault Diagnosis To pursue a perfect application of RSSD in mechanical fault diagnosis, numerous advanced pursue ahave perfect application of RSSD in diagnosis, numerous RSSDTo techniques recently been developed viamechanical combining fault parameter optimization andadvanced subband RSSD techniques have recently been developed via combining parameter optimization and subband reconstruction methods, which show greater accuracy and adaptability in the field of mechanical reconstruction which show greater accuracy and (ESW) adaptability in the field of mechanical fault diagnosis. methods, An ensemble super-wavelet transformation was proposed to investigate the fault diagnosis. An ensemble super-wavelet transformation proposed to investigate the vibration characteristics of bearing faults in temper mills and(ESW) wind was turbines [63]; the detailed flow vibration characteristics bearing in temper wind turbines the detailed flow chart is shown in Figureof16. With faults the fault featuremills ratioand Rf established by [63]; the Hilbert envelope chart is shown in Figurecan 16.adaptively With the fault feature ratio Rf Q-factor. established by the envelopeperformed spectrum, spectrum, this method select the optimal RSSD wasHilbert subsequently thisa method can adaptively select the optimal RSSD was subsequently performed on a faulty on faulty bearing’s vibration signal and theQ-factor. single optimal subband was extracted for subsequent bearing’s vibration andatthethe single optimal for subsequent reconstruction. reconstruction. As signal shown bottom of subband Figure was 16, extracted the impulses with period 0.0168 s As shown at thetobottom of Figure the impulses period 0.0168 s (corresponding to 59.52 Hz) (corresponding 59.52 Hz) in the 16, optimal subbandwith time-domain waveform, and the obvious peaks in the optimal subband time-domain waveform, and the obvious peaks at 60 Hz and 120 Hz in the at 60 Hz and 120 Hz in the frequency spectrum, all demonstrated that there was a defect on the outer frequency spectrum, all demonstrated that there was a defect on the outer race of bearing SKF-6232/C3. race of bearing SKF-6232/C3.
Input signal
Q-factor selection with Rf
Fault feature ratio
Single subband reconstruction
Single optimal subband
Figure Figure 16. 16. Example Exampleof ofbearing bearingfault faultdiagnosis diagnosisvia viaensemble ensemblesuper-wavelet super-wavelettransformation transformation (ESW) (ESW) [63]. [63].
In 2016, in response to the disadvantage in Reference [63] wherein the high-frequency In 2016, in response to the disadvantage in Reference [63] wherein the high-frequency interferences interferences were not taken into consideration and the reconstruction of a single subband might lead were not taken into consideration and the reconstruction of a single subband might lead to leakage of to leakage of useful fault features, He et al. [64] improved the ensemble super-wavelet useful fault features, He et al. [64] improved the ensemble super-wavelet transformation. By modifying transformation. By modifying the definition of fault feature ratio Rf and incorporating two optimal the definition of fault feature ratio Rf and incorporating two optimal subbands, the modified method subbands, the modified method exhibited stronger robustness to high-frequency interferences and exhibited stronger robustness to high-frequency interferences and preserved more fault-related preserved more fault-related information. According to a priori information on the intrinsic periodic information. According to a priori information on the intrinsic periodic features of impulsive bearing features of impulsive bearing faults, a periodic sparsity-based oriented super-wavelet transformation faults, a periodic sparsity-based oriented super-wavelet transformation (PSOSW) was proposed [65]. (PSOSW) was proposed [65]. An established weight index for the periodic sparsity feature energy ratio was adopted to guide the selection of RSSD parameters and reconstruction. The selected optimal
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Sensors 2017, 17, 1279 of 27 An established weight index for the periodic sparsity feature energy ratio was adopted to 18 guide the selection of RSSD parameters and reconstruction. The selected optimal super-wavelet bases super-wavelet bases were forRSSD RSSD. This superior RSSDincipient effectively discovered incipient were utilized for RSSD. Thisutilized superior effectively discovered weak fault features of a weak fault features of a motor bearing installed on wind-power generation equipment. With the motor bearing installed on wind-power generation equipment. With the research on bearing fault research on bearing fault mechanisms, Yu and Zhou [66] introduced an innovative method for mechanisms, Yu and Zhou [66] introduced an innovative method for diagnosing bearing faults using bearing faultsand using a combination and spectral kurtosis. In accordance with the adiagnosing combination of RSSD spectral kurtosis. ofInRSSD accordance with the spectral kurtosis maximum spectral kurtosis maximum principle, the optimal Q-factor and decomposition level were calculated. principle, the optimal Q-factor and decomposition level were calculated. Meanwhile, the neighboring Meanwhile, the neighboring de-noising method employed to eliminatesignal, the noise in coefficient de-noising methodcoefficient was employed to eliminate thewas noise in the reconstructed which the reconstructed whichfor resulted infault a better application for bearing fault diagnosis. resulted in a bettersignal, application bearing diagnosis.
4.5. RSSD Combined Combined with with Other Other Methods 4.5. RSSD Methods in in Mechanical Mechanical Fault Fault Diagnosis Diagnosis This This subsection subsection reviews reviews the the applications applications of of RSSD RSSD combined combined with with other other methods methods in in mechanical mechanical fault diagnosis. To solve the problem that gear fault features are difficult to reveal under rotation fault diagnosis. To solve the problem that gear fault features are difficult to reveal under rotation rate rate fluctuations, [67] investigated an order-domain analysis basedbased RSSD, RSSD, which used theused chirplet fluctuations, Sun Sunetetal.al. [67] investigated an order-domain analysis which the path pursuit algorithm to obtain rotation rate information. By performing order domain analysis on the chirplet path pursuit algorithm to obtain rotation rate information. By performing order domain low-resonance component obtained using RSSD along with the rotation rate information, the fault of analysis on the low-resonance component obtained using RSSD along with the rotation rate gear with rotation rate fluctuation was identified. 2014, Chen et al. [68] proposed an improved RSSD, information, the fault of gear with rotation rate In fluctuation was identified. In 2014, Chen et al. [68] based on ensemble empirical mode decomposition (EEMD) and TQWT, for the feature extraction of proposed an improved RSSD, based on ensemble empirical mode decomposition (EEMD) and aTQWT, rollingfor bearing’s early weak fault. Figures 17 and 18 presented a typical example of bearing fault the feature extraction of a rolling bearing’s early weak fault. Figures 17 and 18 presented identification adopting theirfault proposed method.adopting The original signal of amethod. bearingThe (type 6207) signal inner a typical example of bearing identification their proposed original race fault was shown in Figure 17, including timeand frequency-domain signals where nothing of a bearing (type 6207) inner race fault was shown in Figure 17, including time- and frequencyfault-related can where be readily observed. By applying several intrinsic functions (IMF) were domain signals nothing fault-related can EEMD, be readily observed. Bymode applying EEMD, several obtained. With afunctions calculation with were Equation (14), since owned thewith biggest kurtosis, it was handled intrinsic mode (IMF) obtained. WithIMF5 a calculation Equation (14), since IMF5 selectively using standard RSSD. The resulting envelope spectrum of the low-resonance component owned the biggest kurtosis, it was handled selectively using standard RSSD. The resulting envelope was shown Figure 18. The peaks at the inner race fault feature18. frequency (246 Hz)inner and sideband spectrum ofin the low-resonance component was shown in Figure The peaks at the race fault frequencies manifested the effectiveness of Chen’s method. feature frequency (246 Hz) and sideband frequencies manifested the effectiveness of Chen’s method.
Figure 17. 17. Rolling fault signal: signal: (a) Time-domain waveforms; waveforms; (b) (b) Envelope Envelope Figure Rolling bearing bearing early early weak weak fault (a) Time-domain demodulation spectrum [68]. demodulation spectrum [68].
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Figure 18. Envelope demodulation spectrum of IMF5 handled with RSSD [68]. Figure 18. Envelope demodulation spectrum of IMF5 handled with RSSD [68].
Aiming at the ill-posed problem of blind source separation, Mo et al. [69] introduced a fast Aiming at the ill-posed problem of blind sourceonseparation, Mo et al.which [69] introduced fast independent component analysis method (FastICA) the basis of RSSD, provided aa new independent analysis method on the basis of RSSD, which provided a new solution to thecomponent blind separation problem of (FastICA) the single-channel composite fault signal. The proposed solution to the blind separation problem of the single-channel composite fault signal. The proposed method was then carried out to extract roller bearing composite faults. By integrating RSSD and method then carried out[70] to extract roller composite integrating fractionalwas calculus, Yu et al. analyzed the bearing signal gathered byfaults. an oil By particle sensor.RSSD With and this fractional calculus, Yu et al. [70] analyzed the signal gathered by an oil particle sensor. With this method, the significant condition information of mechanical devices was provided in real time, owing method, the significant condition information of mechanical devices providedproblem in real time, owing to its advantageous computational efficiency and stability. To solve was the sparsity required in to its advantageous computational efficiency and stability. To solve the sparsity problem required compressed sensing (CS) and break through the limitation of signal length under the Shannon in compressed sensing (CS) and break through the limitation of signal lengthbased underonthe Shannon sampling theorem, a compressed fault-diagnosis method for roller bearings, RSSD, was sampling theorem, a compressed fault-diagnosis method for roller bearings, based on RSSD, was proposed by Wang et al. in 2016 [71]. The application cases powerfully demonstrated that the bearing proposed by Wang in 2016by[71]. Thetheory application caseseven powerfully that the fault features could et be al. extracted the CS with RSSD from thedemonstrated compressed samples. bearing fault features could be extracted by the CS theory with RSSD even from the compressed Combining RSSD analysis and manifold learning, an enhancement method was presented for samples. Combining analysis and manifold learning, an enhancement method was presented for periodic-impact faultRSSD features of rotating machinery [72]. Assisted by an impulse-enhanced signature periodic-impact faultthe features of rotating machinery Assisted by antwo impulse-enhanced signature index, it identified rotating machine’s practical[72]. faults, including defective bearing cases. index, it identified the rotating machine’s practical faults, including two defective bearing cases. Based Based on TQWT, Zhang et al. [73] established a kurtosis-based weighted sparse model utilizing two on TQWT, et al. [73] established a kurtosis-based weighted sparse utilizing twothe pieces of pieces of aZhang priori information. By means of convex optimization, Zhangmodel reliably extracted outer arace priori information. By means of convex optimization, Zhang reliably extracted the outer race fault fault information of a deep-groove ball bearing using the estimated wavelet coefficients. In information of a deep-groove ball proposed bearing using the estimated wavelet coefficients. In further research, further research, Wang et al. [74] an intelligent fault-diagnosis method for rolling bearings Wang et al.full [74]use proposed an intelligent fault-diagnosis method for rolling bearings classifiers that made(nearest full use that made of TQWT, principle component analysis (PCA), and intelligent of TQWT, principle component analysis (PCA), and intelligent classifiers (nearest neighbor classifier neighbor classifier and SVM classifier). The introduction of sparse wavelet energy (SWE) features and SVM classifier). The introduction of sparse wavelet energy (SWE) features this greatly cinched this method’s success, including both fault feature extraction andgreatly patterncinched recognition method’s success, including both fault feature extraction and pattern recognition of rolling bearings. of rolling bearings. 5. Summary and Discussion 5. Summary and Discussion In Section 4, we have reviewed the applications of RSSD in mechanical fault diagnosis. In light of In Section 4, we have reviewed the applications of RSSD in mechanical fault diagnosis. In light the numerous practical procedures and applications described, we believe that a clear and concrete of the numerous practical procedures and applications described, we believe that a clear and concrete flow chart is necessary, as shown in Figure 19. Note that both parameter optimization and subband flow chart is necessary, as shown in Figure 19. Note that both parameter optimization and subband reconstruction are included, so that such a flow chart describes a comprehensively optimized RSSD, reconstruction are included, so that such a flow chart describes a comprehensively optimized RSSD, which is of great value to guiding mechanical fault diagnosis. Meanwhile, due to the variety of which is of great value to guiding mechanical fault diagnosis. Meanwhile, due to the variety of approaches, the published research described above is also summarized in Table 4, for a one-page approaches, the published research described above is also summarized in Table 4, for a one-page overview. The category, reference numbers, diagnostic objects, and supporting techniques are listed in overview. The category, reference numbers, diagnostic objects, and supporting techniques are listed the table. in the table.
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Data acquisition
Objects
Input signal 1.5
1
0.5
Fault signal collection
0
-0.5
-1
-1.5
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Parameter optimization
Low-Q TQWT
MCA
High-Q TQWT
Objective function construction
SALSA
TQWT
Resonance-based sprase signal decomposition
0
Optimization problem solution
High-resonance component
Low-resonance component
Residual
Subband reconstruction Fault feature
Figure Flowchart chartof ofRSSD RSSD in diagnosis. Figure 19.19. Flow in mechanical mechanicalfault fault diagnosis. Table 4. Summary RSSDapplications applications in fault diagnosis. Table 4. Summary ofofRSSD inmechanical mechanical fault diagnosis. Category
Category Original RSSD
Objects Bearing Objects Gearbox
Original RSSD
Bearing Rotor Gearbox Others Rotor Parameter optimized RSSD Bearing Others
Parameter optimized RSSD
Bearing
Gearbox
Others Gearbox Subband optimized RSSD Bearing
Others Subband optimized RSSD
Bearing
Integrated optimized RSSD
Gearbox Bearing
Gearbox RSSD combined Bearing Integrated optimized RSSD with others Bearing
RSSD combined with others
Bearing Gearbox Oil
Gearbox Oil
Supporting Techniques References None Techniques [32,34–37] References Supporting None [33,38–40] None None [41–43] [32,34–37] None [33,38–40] None [44] None [41–43] Qualitative analysis [45–47] None [44] GA [49,51] Iteration [54,55] Qualitative analysis [45–47] Evaluation index [48] GA [49,51] GA [50,52] Iteration [54,55] Iteration [53] Evaluation [48] Main subbandindex [47] GA [50,52] Kurtosis [56–58] Iteration [53] NCD [60] OGS [62] Main subband [47] NCD [60] Kurtosis [56–58] ESW [63] NCD [60] Improved ESW [64] OGS [62] PSOSW [65] NCD [60] Kurtosis, NCD [66] EEMD [68] ESW [63] FastICA ESW [69] Improved [64] CS [71] PSOSW [65] Manifold learning [72] Kurtosis, NCD [66] Kurtosis [73] EEMD [68] PCA, SWE, Classifier [74] FastICA [69] Chirplet path pursuit [67] Fractional calculus [70] CS [71]
Manifold learning Kurtosis PCA, SWE, Classifier Chirplet path pursuit Fractional calculus
[72] [73] [74] [67] [70]
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From the application cases described in Section 4 and summarized in Table 4, the following tips are provided: (1)
(2)
(3)
(4)
(5)
At present, the applications of RSSD mainly focus on crucial industrial components like bearings, gearboxes, and rotors. Through RSSD, a mechanical fault signal is decomposed into the high-resonance component, low-resonance component, and residual, after which fault information can be extracted from these resonance components. In general, RSSD is considered as a powerful and excellent tool for the feature detection of mechanical faults. It must be stated that mechanical fault impact information will unavoidably scatter in high-resonance components, low-resonance components, and even residual components, due to the inherent coherence. This phenomenon can be thought as a kind of energy leakage. Different scholars are inclined to mine mechanical fault impact information from either highor low-resonance components. Thankfully, they all have gained satisfactory diagnosis results. In fact, since these two perspective-based mechanical fault-diagnosis methods are both feasible, it is difficult to say that one approach will always outperform the others. Therefore, it is highly advisable to pay closer attention to the high- and low-resonance components simultaneously, as well as to the residual component. Moreover, it is the energy leakage that facilitates our seeking an optimized RSSD to preserve and extract mechanical fault-related information as richly as possible, which will be of great significance for early mechanical fault identification. To date, one feasible solution to reduce energy leakage and preserve the richest fault features is to optimize RSSD decomposition parameters, mainly including Q-factors and weight coefficients. With a view to these parameters’ significant effects on decomposition results, an artificially selected parameter pair may be not capable of discovering sufficient fault features. Even worse, ill-selected decomposition parameters might yield misleading diagnosis results. To avoid these problems and pursue satisfactory diagnosis, many techniques such as genetic algorithm, kurtosis index, and iterative algorithms, are successively introduced to adaptively acquire optimal Q-factors, weight coefficients, redundancy factors, and decomposition levels. Compared to original RSSD, parameter optimized RSSD can generate resonance components with richer fault impulsive information and less energy leakage, which will make great contributions to diagnostic decision-making. On the other hand, taking inevitable noise interference into consideration, fault information in resonance components obtained by parameter optimized RSSD may still be masked. As a result, many researchers have applied themselves to the energy distribution and reconstruction of subband signals, for the purpose of suppressing noise. Guided by a common concept, these studies have introduced specific indexes that can reveal fault information richness, and some de-noising techniques, including main subband, NCD, OGS, etc. The subbands with the richest information have been chosen to reconstruct mechanical fault vibration signals. It has been found that reconstructed signals with optimized subbands show outstanding performance in clearly detecting mechanical fault features, because broadband noise is obviously weakened. Moreover, an advanced RSSD, combining both parameter and subband optimization, will certainly perform better in diagnosing incipient mechanical faults than unilateral optimized RSSD, let alone original RSSD. In early stages of identifying mechanical fault types assisted with RSSD, researchers employed RSSD alone, or optimized it with decomposition parameters and/or subband reconstruction. Recently, many studies attempt to develop superior fault diagnosis approaches by combining standard RSSD with additional techniques, which are attracting more and more attention. For instance, after pre-processing by EEMD, the IMF with the biggest kurtosis value is analyzed for fault feature extraction using RSSD; CS theory is utilized to obtain the sparsest representation of resonance components from compressed samples, which breaks through the limitation of signal length. Successful applications prove that these specific RSSD-based methods
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can outperform standard RSSD in many applications and provide constructive guidance for diagnosing mechanical faults. 6. Outstanding Issues and Solutions With RSSD’s swift advance, it has been proven fairly reliable in mechanical fault diagnosis. However, in addition to the above-described applications and developments, the following outstanding issues and solutions also deserve intensive discussion for advancing RSSD. 6.1. Construction of Wavelet Bases As explained in Section 2, there are only two available methods to construct wavelet bases, RADWT and TQWT. Unfortunately, the wavelet waveforms constructed by both methods are symmetric in the time domain (see Figure 2). Due to machine operation, fault-induced vibration signals of mechanical components (represented by rolling bearings) usually exhibit distinct single-side damped oscillation characteristics after being transmitted through several mechanical interfaces. Hence, apparent inconsistency lies on the fault signal waveforms and wavelets obtained from RADWT or TQWT. Wang uses the single-side characteristic of the simulation signal to explain why the amplitude of resulting resonance component is halved [53]. However, taking the effects of decomposition parameters into consideration, especially the weight coefficients’ (see Figures 10 and 11), conclusions drawn only from morphological characteristics aren’t convincing enough. Hence, when attempting to utilize a fixed method to construct wavelet bases with the aim of implementing all kinds of mechanical fault signal decomposition, wavelet bases will inevitably describe the morphological characteristics of fault signals inaccurately. To solve this problem, the following issues should be further investigated:
•
•
Establish the dynamic models of mechanical systems and study vibration excitation mechanisms of various mechanical fault types; seek quantitative parameters that can characterize the morphology of fault vibration waveforms so as to reveal the mapping relationships between fault-induced vibration signal waveforms and mechanical fault types; On this basis, explore new construction means and construct targeted wavelet bases according to specific mechanical fault types, to make wavelet morphological characteristics match better with practical fault signals.
6.2. Parameter Optimization and Subband Reconstruction Parameter optimization and subband reconstruction are two important topics of RSSD, as well as the fundamental assurance of RSSD’s successful application to mechanical fault diagnosis. State-of-the-art techniques primarily recur to the qualitative analysis, GA, and kurtosis index. However, qualitative analysis lacks a strong theoretical foundation, while objective function and prematurity in GA remain to be settled, and kurtosis is sensitive to noise interferences and unable to reflect the variation characteristics of fault impacts. All these shortcomings call for a more effective means in aspects of parameter optimization and subband reconstruction. Future work should pay close attention to the following aspects:
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•
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Thoroughly study parameters’ influences on decomposition results, mainly including the influences of Q-factors and weight coefficients, then set up reasonable indexes to quantitatively evaluate their influence levels; Based on quantitative influence levels, study parameters’ coupling effects on RSSD results; establish a multi-parameter fusion optimization problem and take full advantage of multiple optimization ideas to realize the highly efficient optimization of decomposition parameters; Research the intrinsic links between fault signal features and subband energy distribution of resulting high- and low-resonance components; then put forward new methods for subband reconstruction to guide the core feature extraction of mechanical fault vibration signals.
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6.3. Sparse Decomposition of Multiple Resonances Aware of the multiple resonances existing in fault signals, the authors attempted to separate different resonance components via the iterative algorithm [53–55]. However, the iterative algorithm requires that resonance differences of the hidden components be rather distinct. Moreover, the resulting burden on time and memory, as well as the establishment of convergence indexes, are also quite challenging. Therefore, it must break through these bottlenecks and exploit effective sparse decomposition of multiple resonances when dealing with multiple signal components that own indistinctive resonance differences. To this end, three important directions are highly suggested:
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Investigate methods that can reduce inherent mutual coherence between resonance components, to promote the effective separation of each resonance component. As is known, mechanical fault-induced periodic signals usually correlate to natural frequency bands of systems, and the energy of resulting resonance components will merely concentrate on some specific subband groups. Considering that different signal components may lead to different frequency bands, a potential means is to utilize the specific subbands corresponding to respective frequency bands for signal decomposition with several resonances, rather than all J-level subbands. In this way, mutual coherence can be effectively dealt with. Additionally, as shown in Figures 4 and 7, there is significant overlap of several adjacent subband responses: a latent scheme guides us to filter out some middle subbands whose frequency range can be filled up with their neighboring subbands. The vacant frequency ranges belonging to the filtered subbands can be prepared for another resonance component’s subbands. In this way, mutual incoherence between each resonance component can be feasibly enhanced. Notably, determining which subbands to filter relies heavily on the calculation of subband fc and BW with Equations (8) and (9), which needs a certain stock of a priori knowledge. Establish the optimization problem consisting of multiple resonances and develop a fast algorithm to seek its minimization; For potential mechanical fault types and resonances, seek characteristic variables or vectors that can describe mechanical fault vibration waveforms with several similar resonances. This direction offers an opportunity to further distinguish and separate resulting resonance components in view of signal structures.
7. Concluding Remarks In this paper, we attempt to synthesize and review the theoretical developments of RSSD and its applications in mechanical fault diagnosis. As is known, RSSD is a new nonlinear signal separation method depending on resonance rather than frequency or time scale, which solves the frequency overlapping problem of each component and promotes the sparsity of resonance components. Consequently, it has become a new hotspot for numerous fields closely related to signal processing, especially the field of mechanical fault diagnosis. So far, the main application objects of RSSD in mechanical fault diagnosis include bearings, gearboxes, and rotors, etc. In this review, based on the analysis of parameter effects and optimization directions, the application studies of RSSD have been classified into five categories: (1) original RSSD; (2) parameter optimized RSSD; (3) subband optimized RSSD; (4) integrated optimized RSSD; (5) RSSD combined with other methods. Despite preliminary achievements, there are some outstanding issues remaining to be discussed and solved, such as the construction of wavelet bases, suitable selection of parameters, subband reconstruction, and multi-resonance sparse decomposition. Specific approaches to these problems have been also pointed out. In conclusion, the intention of this paper is to synthesize and review the scattered research on the developments and applications of RSSD in mechanical fault diagnosis. This review expectantly provides an in-depth and comprehensive reference for researchers who are concerned with RSSD and
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mechanical fault diagnosis. Moreover, we hope more advantageous RSSD will be developed and play an increasingly significant role in mechanical fault diagnosis. Acknowledgments: The authors acknowledge financial support from the National Natural Science Foundation of China (51175102) and the Fundamental Research Funds for the Central Universities (HIT.NSRIF.201638). Conflicts of Interest: The authors declare no conflict of interest.
References 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12. 13.
14. 15. 16. 17. 18. 19.
20.
Feng, Z.; Zuo, M.J. Fault diagnosis of planetary gearboxes via torsional vibration signal analysis. Mech. Syst. Signal Process. 2013, 36, 401–421. [CrossRef] Lei, Y.G.; Lin, J.; He, Z.J.; Zuo, M.J. A review on empirical mode decomposition in fault diagnosis of rotating machinery. Syst. Signal Process. 2013, 35, 108–126. [CrossRef] Yang, D.S.; Kou, Y.Z.; Tian, H.N. Actuality and development of malfunction diagnosis technology about gear case. Mech. Eng. Autom. 2014, 2, 223–224. Wang, G.B.; He, Z.J.; Chen, X.F.; Lai, Y.N. Basic research on machinery fault diagnosis—What is the prescription. J. Mech. Eng. 2013, 49, 63–72. [CrossRef] Márquez, F.P.G.; Tobias, A.M.; Pérez, J.M.P.; Papaelias, M. Condition monitoring of wind turbines: Techniques and methods. Renew. Energy 2012, 46, 169–178. [CrossRef] Chen, B.Q.; Zhang, Z.S.; Zi, Y.Y.; He, Z.J.; Sun, C. Detecting of transient vibration signatures using an improved fast spatial–spectral ensemble kurtosis kurtogram and its applications to mechanical signature analysis of short duration data from rotating machinery. Mech. Syst. Signal Process. 2013, 40, 1–37. [CrossRef] Klein, R.; Ingman, D.; Braun, S. Non-stationary signals: Phase-energy approach—Theory and simulations. Mech. Syst. Signal Process. 2001, 15, 1061–1089. [CrossRef] Zhu, Z.K.; Yan, R.; Luo, L.; Feng, Z.H.; Kong, F.R. Detection of signal transients based on wavelet and statistics for machine fault diagnosis. Mech. Syst. Signal Process. 2009, 23, 1076–1097. [CrossRef] Wang, W.Y. Early detection of gear tooth cracking using the resonance demodulation technique. Mech. Syst. Signal Process. 2001, 15, 887–903. [CrossRef] Nikolaou, N.G.; Antoniadis, I.A. Application of morphological operators as envelope extractors for impulsive-type periodic signals. Mech. Syst. Signal Process. 2003, 17, 1147–1162. [CrossRef] Selesnick, I.W. Resonance-based signal decomposition: A new sparsity-enabled signal analysis method. Signal Process. 2011, 91, 2793–2809. [CrossRef] Bobin, J.; Starck, J.L.; Fadili, J.M.; Moudden, Y.; Donoho, D.L. Morphological component analysis: An adaptive thresholding strategy. IEEE Trans. Image Process. 2007, 16, 2675–2681. [CrossRef] [PubMed] Tan, C.T.; Guo, B.; Selesnick, I. Resonance-based decomposition for the manipulation of acoustic cues in speech: An assessment of perceived quality. IEEE Workshop Appl. Signal Process. Audio Acoust. 2011, 333–336. [CrossRef] Si, T.N.N.; Ng, W.H. Bi-orthogonal rational discrete wavelet transform with multiple regularity orders and application experiments. Signal Process. 2013, 93, 3014–3026. Chaibi, S.; Lajnef, T.; Kachour, A.; Samet, M. Separation of transient and oscillatory cereberal activities using over-complete rational dilation wavelt transforms. Int. Multi-Conf. Syst. Signals Devices 2011, 1–7. [CrossRef] Sahbi, C.; Tarek, L.; Zied, S.; Mounir, S.; Abdennaceur, K. A reliable approach to distinguish between transient with and without HFOs using TQWT and MCA. J. Neurosci. Methods 2014, 232, 36–46. Reddy, G.R.S.; Rao, R. Extraction of fundamental component in power quality application using tunable-Q wavelet transform. Int. J. Signal Process. Image Process. Pattern Recognit. 2016, 9, 355–380. [CrossRef] Zhao, Y.; Li, Z.M.; Li, T.Y. Low frequency oscillation modal parameter identification using resonance-based sparse signal decomposition and SSI method. Trans. China Electrotech. Soc. 2016, 31, 136–144. Zhao, Y.; Li, Z.M.; Nie, Y.H. A time-frequency analysis method for low frequency oscillation signals using resonance-based sparse signal decomposition and a frequency slice wavelet transform. Energies 2016, 9, 151. [CrossRef] Baradarani, A.; Maev, R.G.; Severin, F. Resonance based analysis of acoustic waves for 3D deep-layer fingerprint reconstruction. In Proceedings of the 2013 IEEE International Ultrasonics Symposium (IUS), Prague, Czech Republic, 21–25 July 2013; pp. 713–716. [CrossRef]
Sensors 2017, 17, 1279
21. 22. 23. 24. 25. 26. 27. 28. 29.
30. 31.
32. 33. 34. 35. 36.
37. 38. 39. 40. 41. 42. 43. 44. 45.
25 of 27
Riahi, M.A.; Goudarzi, A. TQWT and WDGA: Innovative methods for ground roll attenuation. J. Geophys. Eng. 2013, 10, 554–579. Si, T.N.N.; Al-Ashwal, W.A. Sea clutter mitigation using resonance-based signal decomposition. IEEE Geosci. Remote Sens. Lett. 2015, 12, 2257–2261. Woolford, S.; Burnett, I.S. Multiview 3d profilometry using resonance-based decomposition and three-phase shift profilometry. Int. Conf. Exp. Mech. 2014 2015. [CrossRef] Bayram, I.; Selesnick, I.W. Overcomplete discrete wavelet transforms with rational dilation factors. IEEE Trans. Signal Process. 2009, 57, 131–145. [CrossRef] Bayram, I.; Selesnick, I.W. Frequency-domain design of overcomplete rational-dilation wavelet transforms. IEEE Trans. Signal Process. 2009, 57, 2957–2972. [CrossRef] Selesnick, I.W. Wavelet transform with tunable Q-factor. IEEE Trans. Signal Process. 2011, 59, 3560–3575. [CrossRef] Selesnick, I.W. Sparse signal representations using the tunable Q-factor wavelet transform. Proc. SPIE Int. Soc. Opt. Eng. 2011, 8138, 815–822. Afonso, M.V.; Bioucas-Dias, J.M.; Figueiredo, M.A. Fast image recovery using variable splitting and constrained optimization. IEEE Trans. Image Process. 2010, 19, 2345. [CrossRef] [PubMed] Afonso, M.V.; Bioucas-Dias, J.M.; Figueiredo, M.A.T. An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Trans. Image Process. 2011, 20, 681–695. [CrossRef] [PubMed] Bendat, J.S.; Piersol, A.G. Random data: Analysis and measurement procedures. Meas. Sci. Technol. 2000, 1825–1826. Wang, S.B.; Huang, W.G.; Zhu, Z.K. Transient modeling and parameter identification based on wavelet and correlation filtering for rotating machine fault diagnosis. Mech. Syst. Signal Process. 2011, 25, 1299–1320. [CrossRef] Chen, X.M.; Yu, D.J.; Luo, J.S. Envelope demodulation method based on resonance-based sparse signal decomposition and its application in roller bearing fault diagnosis. J. Vib. Eng. 2012, 25, 628–636. Zhang, W.Y.; Yu, D.J.; Chen, X.M. Fault diagnosis of gears based on resonance-based sparse signal decomposition and envelope spectrum. China Mech. Eng. 2013, 24, 3349–3354. Xiang, W.W.; Cai, G.G.; Fan, W.; Huang, W.; Shang, L.; Zhu, Z. The research of the transient feature extraction by resonance-based method using double-TQWT. Lect. Notes Comput. Sci. 2014, 684–692. [CrossRef] Xiang, W.W.; Cai, G.G.; Fan, W.; Huang, W.G.; Li, S.; Zhu, Z.K. Transient feature extraction based on double-TQWT and its application in bearing fault diagnosis. J. Vib. Shock 2015, 34, 34–39. Huang, W.T.; Liu, Y.F.; Niu, P.L.; Wang, W.J. High resonance component of resonance-based sparse decomposition application in extraction of rolling bearing fault information. Adv. Mater. Res. 2013, 753–755, 2290–2296. [CrossRef] Huang, W.T.; Liu, Y.F.; Li, X.C. Resonance-based sparse decomposition application in extraction of rolling bearing weak fault information. Found. Intell. Syst. 2014, 823–831. [CrossRef] Cui, L.L.; Wu, N.; Mo, D.Y.; Wang, H.Q.; Chen, P. CQFB and PBP in diagnosis of local gear fault. Adv. Mech. Eng. 2014, 1–12. [CrossRef] Cui, L.L.; Mo, D.Y.; Wang, J. Composite Q-factor bases and applications to extraction of gear-box weak fault characteristics. J. Beijing Univ. Technol. 2014, 40, 1621–1626. Zhang, W.Y.; Yu, D.J.; Chen, X.M. Energy operator demodulating of signal’s resonance components for the compound fault diagnosis of gearbox. J. Vib. Eng. 2015, 28, 148–155. Chen, X.M.; Yu, D.J.; Luo, J.S. Early rub-impact diagnosis of rotors by using resonance-based sparse signal decomposition. China Mech. Eng. 2013, 24, 35–41. Chen, X.M.; Yu, D.J.; Li, R. Rub-impact diagnosis of rotors with resonance-based sparse signal decomposition and reassigned wavelet scalogram. J. Vib. Shock 2013, 32, 27–33. Wang, H.C.; Jin, C.; Dong, G.M.; Huo, B.Q.; Xu-Gang, H.U.; Miao, Z. Early rub-impact diagnosis of rotors based on tunable Q-factor wavelet transformation. J. Vib. Shock 2014, 33, 77–80. Chen, X.F.; Cai, G.G.; Cao, H.R.; Xin, W. Condition assessment for automatic tool changer based on sparsity-enabled signal decomposition method. Mechatronics 2015, 31, 50–59. [CrossRef] Mo, D.Y.; Cui, L.L.; Wang, J. Sparse signal decomposition method based on the dual Q-factor and its application to rolling bearing early fault diagnosis. J. Mech. Eng. 2013, 49, 37. [CrossRef]
Sensors 2017, 17, 1279
46. 47. 48. 49.
50.
51. 52. 53.
54.
55. 56. 57. 58. 59. 60.
61. 62.
63.
64. 65. 66. 67.
26 of 27
Cui, L.L.; Mo, D.Y.; Wang, H.Q.; Chen, P. Resonance-based nonlinear demodulation analysis method of rolling bearing fault. Adv. Mech. Eng. 2013, 5, 420694. [CrossRef] Huang, W.T.; Sun, H.J.; Liu, Y.F.; Wang, W.J. Feature extraction for rolling element bearing faults using resonance sparse signal decomposition. Exp. Tech. 2017, 41, 251–265. [CrossRef] Cai, G.G.; Chen, X.F.; He, Z.J. Sparsity-enabled signal decomposition using tunable Q-factor wavelet transform for fault feature extraction of gearbox. Mech. Syst. Signal Process. 2013, 41, 34–53. [CrossRef] Huang, W.T.; Fu, Q.; Dou, H.Y.; Dong, Z.Z. Resonance-based sparse signal decomposition based on genetic optimization and its application to composite fault diagnosis of rolling bearings. In Proceedings of the ASME 2015 International Mechanical Engineering Congress and Exposition, Houston, TX, USA, 13–19 November 2015. Huang, W.T.; Fu, Q.; Dou, H.Y. Resonance-based sparse signal decomposition based on the quality factors optimization and its application of composite fault diagnosis to planetary gearbox. J. Mech. Eng. 2016, 52, 44–51. [CrossRef] Li, X.; Yu, D.J.; Zhang, D.C. Fault diagnosis of rolling bearings based on the resonance-based sparse signal decomposition with optimal Q-factor. J. Vib. Eng. 2015, 28, 998–1005. Zhang, D.C.; Yu, D.J.; Zhang, W.Y. Energy operator demodulating of optimal resonance components for the compound faults diagnosis of gearboxes. Meas. Sci. Technol. 2015, 26, 115003. [CrossRef] Wang, Y.Q.; Bo, Q.L.; Liu, H.B.; Lian, M.; Wang, F.B.; Zhang, J. Full-oscillatory components decomposition from noisy machining vibration signals by minimizing the Q-factor variation. Trans. Inst. Meas. Control 2016. [CrossRef] Shi, J. Morphology-Based Fault Feature Extraction and Resampling-Free Fault Identification Techniques for Rolling Element Bearing Condition Monitoring. Ph.D. Thesis, University of Ottawa, Ottawa, ON, Canada, 2015. Shi, J.; Liang, M. Intelligent bearing fault signature extraction via iterative oscillatory behavior based signal decomposition (IOBSD). Expert Syst. Appl. 2016, 45, 40–55. [CrossRef] Zhang, D.C.; Yu, D.J.; Li, X. Fault diagnosis of rolling bearing based on tunable-Q wavelet reconstruction. J. Aerosp. Power 2015, 30, 3051–3057. Luo, J.S.; Yu, D.J.; Liang, M. A kurtosis-guided adaptive demodulation technique for bearing fault detection based on tunable-Q wavelet transform. Meas. Sci. Technol. 2013, 24, 055009. [CrossRef] Tang, G.J.; Wang, X.L. Application of tunable Q-factor wavelet transform to feature extraction of weak fault for rolling bearing. Proc. CSEE 2016, 36, 746–754. Cai, T.T.; Silverman, B.W. Incorporating information on neighbouring coefficients into wavelet estimation. Sankhy¯a Indian J. Stat. Ser. B 2001, 63, 127–148. He, W.P.; Zi, Y.Y.; Chen, B.Q.; Shuai, W.; He, Z.J. Tunable Q-factor wavelet transform denoising with neighboring coefficients and its application to rotating machinery fault diagnosis. Sci. China Technol. Sci. 2013, 56, 1956–1965. [CrossRef] Chen, P.Y.; Selesnick, I.W. Overlapping Group Shrinkage/Thresholding and Denoising; Technical Report; Polytechnic Institute of New York University: New York, NY, USA, 2012. He, W.P.; Zi, Y.Y. Sparsity-assisted signal representation for rotating machinery fault diagnosis using the tunable Q-factor wavelet transform with overlapping group shrinkage. In Proceedings of the 2014 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR), Lanzhou, China, 13–16 July 2014; pp. 18–23. [CrossRef] He, W.P.; Zi, Y.Y.; Chen, B.Q.; Wu, F.; He, Z.J. Automatic fault feature extraction of mechanical anomaly on induction motor bearing using ensemble super-wavelet transform. Mech. Syst. Signal Process. 2014, 54, 457–480. [CrossRef] He, W.P.; Zi, Y.Y.; Wan, Z.G.; Chen, B.Q. Improved ensemble super-wavelet transform for vibration-based machinery fault diagnosis. J. Manuf. Sci. Eng. 2016, 138, 071012. [CrossRef] He, W.P.; Zi, Y.Y.; Chen, B.Q.; Yao, B.; Zhang, Z.S. Periodic sparsity oriented super-wavelet analysis with application to motor bearing fault detection of wind turbine. J. Mech. Eng. 2016, 52, 41. [CrossRef] Yu, F.J.; Zhou, F.X. Bearing early faults diagnosis based on tunable Q-factor wavelet transform and spectral kurtosis. J. Central South Univ. Sci. Technol. 2015, 11, 4122–4128. Sun, Y.S.; Yu, D.J.; Chen, X.M.; Li, R. Order domain analysis based on resonance-based sparse signal decomposition and its application to gear fault diagnosis. J. Vib. Shock 2013, 32, 88–94.
Sensors 2017, 17, 1279
68. 69. 70. 71. 72. 73. 74.
27 of 27
Wang, H.C.; Chen, J.; Dong, G.M. Feature extraction of rolling bearing’s early weak fault based on EEMD and tunable Q-factor wavelet transform. Mech. Syst. Signal Process. 2014, 48, 103–119. [CrossRef] Mo, D.Y.; Cui, L.L.; Wang, J.; Gao, L.X. Application of parallel dual-Q-factors in extraction and blind separation of gearbox composite faults. Chin. J. Sci. Instrum. 2013, 34, 2013–2020. Luo, J.S.; Yu, D.J.; Liang, M. Enhancement of oil particle sensor capability via resonance-based signal decomposition and fractional calculus. Measurement 2015, 76, 240–254. [CrossRef] Wang, H.; Ke, Y.; Song, L.; Gang, T.; Peng, C. A sparsity-promoted decomposition for compressed fault diagnosis of roller bearings. Sensors 2016, 16, 1524. [CrossRef] [PubMed] Gan, M.; Wang, C.; Zhu, C.A. Fault feature enhancement for rotating machinery based on quality factor analysis and manifold learning. J. Intell. Manuf. 2015, 1–18. [CrossRef] Zhang, H.; Chen, X.; Du, Z.; Yan, R. Kurtosis based weighted sparse model with convex optimization technique for bearing fault diagnosis. Mech. Syst. Signal Process. 2016, 80, 349–376. [CrossRef] Wang, C.; Gan, M.; Zhu, C.A. Intelligent fault diagnosis of rolling element bearings using sparse wavelet energy based on overcomplete DWT and basis pursuit. J. Intell. Manuf. 2015. [CrossRef] © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
