sensors Article
Weak Fault Feature Extraction of Rolling Bearings Based on an Improved Kurtogram Xianglong Chen 1, *, Fuzhou Feng 1 and Bingzhi Zhang 2 1 2
*
Department of Mechanical Engineering, Academy of Armored Forces Engineering, Beijing 100072, China;
[email protected] Beijing Special Vehicle Research Institute, Beijing 100072, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-10-6671-8514
Academic Editor: Gangbing Song Received: 24 May 2016; Accepted: 3 September 2016; Published: 13 September 2016
Abstract: Kurtograms have been verified to be an efficient tool in bearing fault detection and diagnosis because of their superiority in extracting transient features. However, the short-time Fourier Transform is insufficient in time-frequency analysis and kurtosis is deficient in detecting cyclic transients. Those factors weaken the performance of the original kurtogram in extracting weak fault features. Correlated Kurtosis (CK) is then designed, as a more effective solution, in detecting cyclic transients. Redundant Second Generation Wavelet Packet Transform (RSGWPT) is deemed to be effective in capturing more detailed local time-frequency description of the signal, and restricting the frequency aliasing components of the analysis results. The authors in this manuscript, combining the CK with the RSGWPT, propose an improved kurtogram to extract weak fault features from bearing vibration signals. The analysis of simulation signals and real application cases demonstrate that the proposed method is relatively more accurate and effective in extracting weak fault features. Keywords: correlated kurtosis; redundant second generation wavelet package transform; kurtogram; weak fault diagnosis
1. Introduction Rolling bearings are one of the most common but the most vulnerable parts in mechanical systems. In order to ensure uninterrupted operation and avoid unnecessary losses caused by sudden failure, extraction of weak fault failures of rolling bearings has become a key factor to condition monitoring and fault diagnosis concerning mechanical systems [1,2]. Influenced by heavy background noise and signal transmission paths, fault features of rolling bearings can be greatly compromised. Consequently, it becomes difficult to extract bearing weak fault features, not to mention to diagnose such faults accurately. This issue had been studied by some signal processing methods, such as Wavelet Transform (WT) [3], Empirical Mode Decomposition (EMD) [4], and Local Mean Decomposition (LMD) [5]. By utilizing noise to enhance signal weak features, stochastic resonance (SR) [6] can effectively process signals with low signal-to-noise ratio (SNR). Lei [6] proposed an adaptive stochastic resonance (ASR) method which performs well in extracting weak characteristics for fault diagnosis. By using the sparsity measurements, Tse [7] proposed the sparsogram to determine the resonant frequency bands quickly and detect fault signals with low SNR. Tse [8] further studied the sparsogram with a joint algorithm based on the complex Morlet wavelet filter and genetic algorithm for maximizing the sparsity measurement value. Tse [7,8] verified the superiority of the sparsogram in bearing fault diagnosis. Later, Antoni [9] proposed the infograms, including the SE infogram and the SES infogram, which can capture the signature of repetitive transients in both time and frequency domains. Starting with Bayesian inference, Wang [10]
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proposed an extended infogram to optimize wavelet filtering for the identification of bearing fault features, and which extract more signatures than the SE infogram. Antoni [11,12] put forward the method of spectral kurtosis (or SK) for non-stationary feature extraction. Due to its superiority, several in-depth studies have been done in the bearing vibration monitoring and fault diagnosis [13–16]. Based on short-time Fourier Transform (STFT), Antoni [17] later built a faster computation method for kurtogram to extract transient features in vibration signals. Wang [18] creatively combined the inherent nonlinear structure with kurtogram to extract weak fault features from low SNR bearing vibration signals. However, kurtogram is conditioned by STFT and FIR [19,20], either of which limits its accuracy in extracting transient features from noise signals. To address the above limitations, Lei [20] suggested that wavelet package transform (WPT) be used to construct kurtogram. Wang [21] tentatively experimented the calculation of the envelope power spectrum of WPT nodes to enhance the kurtogram theories. With the help of the periodicity feature of impulses, McDonald [22] proposed the correlated kurtosis (CK) to detect cyclic transients more effectively than the kurtosis. Based on redundant lifting scheme, redundant second generation wavelet package transform (RSGWPT) [23] doesn’t depend on Fourier transform. It possesses time invariant properties, so it can not only afford more detailed local time-frequency description of the signals, but also restrict the frequency aliasing components of the analysis. Therefore, it is helpful to identify the bearing fault-sensitive frequency bands and extract weak fault features. Combining CK and RSGWPT, this manuscript proposes an improved kurtogram, which can remedy the drawbacks of STFT and the kurtosis. The effectiveness of this method is demonstrated by simulation and experimental analysis. The rest of the manuscript includes: correlated theoretical background in Section 2, the methods proposed by the authors in Section 3, the simulation analysis and test applications in Sections 4 and 5, respectively, and our conclusions in Section 6. 2. Theoretical Background 2.1. Spectral Kurtosis and Kurtogram As a complement to the method of power spectral density in indicating transients of signals, spectral kurtosis (SK), which can indicate both the presence and location of transients in frequency domain, was first introduced by Dwyer [24] and further studied by Antoni [11,12,17]. Given the Wold-Cramér decomposition of a non-stationary signal, signal y(n) as the response of a system with time varying impulse response can be expressed as [2,18,21,22]: y(n) =
Z +1/2 −1/2
e j2π f n H(n, f )dX( f )
(1)
where dX( f ) is an orthonormal spectral increment and H(n, f ) is the complex envelope of y(n) y(n) at frequency f . The fourth-order spectral cumulant of y(n) is defined as: 2 C4y ( f ) = S4y ( f ) − 2S2y (f)
(2)
where S2y ( f ) is the time-averaged result of S2y (n, f ), S2y (n, f ) is an instantaneous moment and measures the energy of the complex envelope. The SK can be formally defined as: Ky ( f ) =
C4y ( f ) 2 (f) S2y
=
S4y ( f ) 2 (f) S2y
−2
Considering the presence of added noise addn(n), the SK can be further expressed as:
(3)
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K y addn ( f )
Ky ( f )
(4) [1 ( f )] 2 Ky ( f ) (4) ( f ) =added noise2 addn( n) and ( f ) is the noise-to( naddn ) with where K y addn ( f ) is the SK of signal Ky+ [1 + ρ( f )] signal ratio. where Ky+ addn ( f ) is the SK of signal y(n) with added noise addn(n) and ρ( f ) is the noise-to-signal ratio. The SK of a stationary signal is a constant function of frequency and the SK of a stationary The SK of a stationary signal is a constant function of frequency and the SK of a stationary Gaussian signal is identically zero. Where there are signal transients, there will be a high spectral Gaussian signal is identically zero. Where there are signal transients, there will be a high spectral kurtosis peak value to be recorded in both time and location, therefore, the SK possesses the dual kurtosis peak value to be recorded in both time and location, therefore, the SK possesses the dual abilities to detect and localize transients from a signal. abilities to detect and localize transients from a signal. In order to improve the calculation efficiency of spectral kurtosis, Antoni [18] utilized binary In order to improve the calculation efficiency of spectral kurtosis, Antoni [18] utilized binary tree and 1/3-binary tree algorithms to split frequency bands. Based on these calculations, the tree and 1/3-binary tree algorithms to split frequency bands. Based on these calculations, the implementation constructs the kurtogram which is a 2D map and presents values of SK calculated implementation constructs the kurtogram which is a 2D map and presents values of SK calculated for various frequency band parameters of depth k and bandwidth （f ）k . Antoni [20] proposed the for various frequency band parameters of depth k and bandwidth (∆ f )k . Antoni [20] proposed the original kurtogram paving paving as as shown shown in in Figure Figure 1. 1. original kurtogram （f）k
Levels
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Figure 1. Paving of the original kurtogram proposed by Antoni [20].
2.2. Correlated Kurtosis Spectral kurtosis kurtosis has hasbeen beenwidely widelyapplied appliedininrotating rotatingmachine machinefaults faults diagnosis superiority diagnosis forfor itsits superiority in in detecting transients of no-stationary signals. However, McDonald found that when compared detecting transients of no-stationary signals. However, McDonald [22] [22] found that when compared with with a signal containing consecutive periodicity of impulses, signal with impulse a single will impulse will a signal containing consecutive periodicity of impulses, a signalawith a single generate generate a higher value. kurtosisIt value. can be concluded the kurtosis more effective in detecting a a higher kurtosis can beItconcluded that thethat kurtosis is moreiseffective in detecting a single single impulse than consecutive periodicity of impulses. Rotating machines’ vibration signals usually impulse than consecutive periodicity of impulses. Rotating machines’ vibration signals usually contain contain consecutive periodicity of impulses, which are produced by mechanical faults, than rathera than a consecutive periodicity of impulses, which are produced by mechanical faults, rather single single impulse, which may be incurred heavy background noise. Consequently, the kurtosis impulse, which may be incurred by heavy by background noise. Consequently, the kurtosis interfered by interfered by heavy noise may indicate wrong transients or wrong locations or both of transients [25], heavy noise may indicate wrong transients or wrong locations or both of transients [25], the simulation the simulation analysis in verifies Section this 4 also verifies this possibility. address the above problem, analysis in Section 4 also possibility. To address the To above problem, McDonald [22] McDonaldthe [22] proposed the correlated the helpfeature of theofperiodicity feature of proposed correlated kurtosis (CK) withkurtosis the help(CK) of thewith periodicity impulses which exist impulses which exist in signals. in signals. The CK of zero-mean signal YYcan can formally defined [22]: bebe formally defined as as [22]:
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CK1 ( T ) =
CK M ( T ) =
∑nN=1 (yn · yn−T )
2
(5)
2 (∑nN=1 y2n )
M ∑nN=1 (∏m =0 yn−mT )
(∑nN=1 y2n )
2
M +1
(6)
where N is the number of samples in the input signal Y, T is the periodicity of impulses, and M is the CK shift. Equation (5) is the first shift CK and Equation (6) is the M-th shift CK. Equation (6) is more general than Equation (5). And especially when T = 0 and M = 1, Equation (6) is equal to the norm of kurtosis. Higher shift CK can be used to test a larger series of impulses in a signal. M can be selected based on different fault signals. McDonald [22] verified that CK approached a maximum about the specified period as opposed to the kurtosis which tended to a maximum with a single impulse. CK takes advantage of the periodical feature of the faults as well as the impulse-like vibration behavior associated with most types of faults. It also can decrease the inference of heavy noise and is more effective in detecting signal cyclic transients. 2.3. Redundant Second Generation Wavelet Package Transform Second generation wavelet transform (SGWT) is a new wavelet construction method using lifting scheme in time domain. It does not depend on Fourier transform and it can construct bi-orthogonal wavelet basis function flexibly. By designing prediction operator and update operator, it can adaptively achieve no-linear wavelet transform [3]. Compared with traditional wavelet transform, it possesses time invariant property and, therefore, can not only afford more detailed local time-frequency description of the signal, but also restrict the frequency aliasing components of the analysis owing to the negligence of the split and merge steps in the decomposition and reconstruction stage [3]. Zhou [23] proposed redundant second generation wavelet package transform (RSGWPT). The construct process is shown as follows. (1)
The prediction step and update step of RSGWPT at level l are performed by Pl and U l , which are expressed as follows: Xl +1,2 = Xl,1 − Pl +1 ( Xl,1 ) l +1 Xl +1,1 = Xl,1 + U ( Xl +1,2 ) ... = Xl,2l − Pl +1 ( Xl,2l ) X l + 1 l +1,2 Xl +1,2l +1 −1 = Xl,2l + U l +1 ( Xl +1,2l +1 )
(2)
where Xl,k is the k-th wavelet node coefficients at level l, Pl is the redundant prediction operator at level l and U l is the redundant update operator at level l. The reconstruction stage of RSGWPT can be obtained from its decomposition stage, which is expressed as follows: 1 l +1 l +1 l +1 Xl,2l = 2 (Xl +1,2l +1 −1 − U ( Xl +1,2l +1 ) + Xl +1,2l +1 + P (Xl +1,2l +1 −1 − U ( Xl +1,2l +1 ))) ... X = 1 (X l +1 ( X l +1 (X l +1 ( X l,1 l +1,1 − U l +1,2 ) + Xl +1,2 + P l +1,1 − U l +1,2 ))) 2
(3)
(7)
(8)
The redundant prediction operator Pl and the redundant update operator U l at level l are expressed as follows: ( Pjl
=
p m , j = 2l m 0, j 6= 2l m
( j = 1, 2, . . . , 2l N )
(9)
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p , j 2llm PPjll pmm , j l2 m j 00 ,, jj 22lm m ( l uunnu,, ,jjj=22l2nnl n l n U l jl U Uj j= 0 , j 2llnl 0 0,, j 6=22nn
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l ((jj 1, 1,2,..., 2,...,22l N N))
(9) 5 of (9)20
ll ((jj (1, 2,...,2 . ).) , 2l N e) 1, 2,...,2 j= 1, 2, N .N
(10) (10) (10)
=are eprediction ,,N where ,, m and initial PP {{ppmP}}= {and 2, · · ·U ,N 1, 2,initial ··· ,N are initial operator prediction operator { pm1,2, }, m {1,u2, }, ,n,N m 1,2, = N1, and n wherewhere and are initial prediction operator and initial 2, N U {uunn}} ,,Unn= 1, m e and initial of update operator of SGWT, N and N are their length. are their N and length. update operator SGWT, N update operator of SGWT, N and N are their length. An example of two levels RSGWPT decomposition stage are shown in An example example of of two two levels levels RSGWPT RSGWPT decomposition decomposition stage stage and and reconstruction reconstruction stage stage are are shown shown in in An stage and reconstruction Figures 2 and 3. Figures 22 and and 3. 3. Figures X 1 X22，， 1
X 1 X11，， 1 XX0，1 0， 1
PP11
2 2
2 U U2
PP
X X22，，22
1 U U1
X X11，，22
X X22，，44
PP22
2 2
U U
X X22，，33 Figure Figure 2. 2. An An example example of of two two levels levels RSGWPT RSGWPT decomposition decomposition stage. stage.
X X22，， 1 1 2 U U2
PP22
X X11，， 1 1
X X22，，22 1 U U1
X X22，，44 X X22，，33
2 U U2
PP22
PP11
X X00，， 1 1
X X11，，22
Figure Figure 3. 3. An An example example of of two two levels levels RSGWPT RSGWPT reconstruction reconstruction stage. stage.
Based on redundant lifting scheme, RSGWPT possesses time property which can Basedon onredundant redundant lifting scheme, RSGWPT possesses time invariant invariant property which can Based lifting scheme, RSGWPT possesses time invariant property which can acquire acquire richer feature information and more precise frequency localization information [23,26]. It can acquire richer feature information and more precise frequency localization information [23,26]. It can richer feature information and more precise frequency localization information [23,26]. It can restrict restrict the frequency aliasing components of the analysis. Therefore, RSGWPT is superior to WPT restrict the frequency aliasing components of the analysis. to STFT WPT the frequency aliasing components of the analysis. Therefore,Therefore, RSGWPT RSGWPT is superioristosuperior WPT and and STFT in time-frequency analysis, and itit is helpful to identify fault-sensitive frequency bands and STFT in time-frequency analysis, and is helpful to identify fault-sensitive frequency bands in time-frequency analysis, and it is helpful to identify fault-sensitive frequency bands which can be which can be to extract weak which be used used tofault extract weak fault fault features. features. used tocan extract weak features. 3. 3. Proposed Proposed Method 3. ProposedMethod Method
Under the influence of factors such as heavy background noise and signal transmission Under the the influence influence of of factors factors such such as as heavy heavy background background noise noise and and signal signal transmission transmission paths, Under paths, fault-sensitive frequency bands will be easily masked. It causes serious challenges on the detection fault-sensitive frequency frequency bands bands will will be be easily easily masked. masked. It It causes causes serious serious challenges challenges on on the the detection detection fault-sensitive and diagnosis of bearing faults. The original kurtogram put forward Antoni [17] can adaptive and diagnosis diagnosisof ofbearing bearingfaults. faults.The Theoriginal originalkurtogram kurtogramput put forward by Antoni [17] can be adaptive and forward byby Antoni [17] can bebe adaptive in in identifying fault features when the bearing failure occurs. Therefore, it becomes a useful to in identifying fault features when the bearing failure occurs. Therefore, it becomes a useful tool to identifying fault features when the bearing failure occurs. Therefore, it becomes a useful tool totool detect detect and bearings’ fault. Yet, with regard to in the detect and diagnose diagnose bearings’ fault. Yet, withto regard to capabilities capabilities in time-frequency time-frequency analysis, the and diagnose bearings’ fault. Yet, with regard capabilities in time-frequency analysis,analysis, the original original employing either FIR, WPT. WPT, original kurtogram, kurtogram, employing either STFT orcomparatively FIR, is is comparatively comparatively inferior to WPT. Based on WPT, kurtogram, employing either STFT or STFT FIR, isor inferior inferior to WPT.to Based onBased WPT,on Lei [20] Lei [20] and Wang [21] put forward an improved kurtogram and an enhanced kurtogram. However, Lei [20] and Wang [21] put an forward an improved kurtogram and an enhanced kurtogram. and Wang [21] put forward improved kurtogram and an enhanced kurtogram. However, However, WPT also has innate limitations such as frequency aliasing, fault features unidentifiable and bearing failures difficult to detect. Nevertheless, the original kurtogram is more effective to a single impulse and more
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WPT also has innate limitations such as frequency aliasing, fault features unidentifiable and bearing failures difficult to detect. Nevertheless, the original kurtogram is more effective to a single impulse vulnerable to the interference of noise signal It could precisely transients and more vulnerable to the interference of[26]. noise signalnot [26]. It couldand notaccurately preciselydetect and accurately detect positions domain. in the frequency domain. Allrestrain of thesekurtogram reasons restrain kurtogram or theirtransients positionsor in their the frequency All of these reasons application in the application the detection of under bearings’ under the circumstances detection ofin bearings’ failures the failures circumstances of heavy noise. of heavy noise. As is mentioned above, CK approaches a maximum for consecutive impulses about the specified maximum with with just just single single impulse. impulse. It can also period as opposed to the kurtosis which tends to a maximum reduce the theinterference interference noise in detecting impulse RSGWPT possesses time invariant of of noise in detecting impulse series.series. RSGWPT possesses time invariant property. property. It can the frequency aliasing components of the analysis. It also can acquire richer It can restrict therestrict frequency aliasing components of the analysis. It also can acquire richer feature feature information and more frequency precise frequency localization information. As RSGWPT a result, RSGWPT is information and more precise localization information. As a result, is superior superior to WPT STFT in time-frequency analysis, and it to is helpful identify fault-sensitive to WPT and STFTand in time-frequency analysis, and it is helpful identifyto fault-sensitive frequency frequency bands, canthe facilitate the of extraction of weak faultOwing features. Owing to the superiority bands, which can which facilitate extraction weak fault features. to the superiority of CK and of CK andby RSGWPT, bykurtosis replacing kurtosis with kurtosis correlation andRSGWPT, STFT withthis RSGWPT, this RSGWPT, replacing with correlation andkurtosis STFT with manuscript manuscript improved kurtogram, which can steadily identify fault-sensitive frequency proposes anproposes improvedankurtogram, which can steadily identify fault-sensitive frequency bands and bands andextract efficiently weak fault The proposed method, withanalysis, the envelop efficiently weakextract fault features. Thefeatures. proposed method, coupled withcoupled the envelop can analysis, can be to extract bearings’ weak fault features andbearings’ diagnosefaults, bearings’ as in is be employed toemployed extract bearings’ weak fault features and diagnose as isfaults, shown shown in Figure 4. Figure 4. Start Sample rolling bearing vibration signal Perform RSGWPT on the signal Calculate correlated kurtosis of RSGWPT nodes Generate improved kurtogram w ith CK values Select the RSGWPT node w ith the maximum CK Demodulate the node w ith envelope analysis Calculate envelope spectrum and extract fault features End Figure Figure 4. 4. Flow Flow chart chart of of the the proposed proposed method. method.
4. 4. Simulation SimulationAnalysis Analysis Rolling bearing vibrationsignals signals usually contain consecutive periodicity of impulses, when Rolling bearing vibration usually contain consecutive periodicity of impulses, when failure failure occurs inner outer ring,element rolling and element and the cage. to In verify order the to verify the occurs at inneratring, thering, outerthe ring, rolling the cage. In order proposed proposed method, a rolling bearing vibration model referred to [27,28] is used to construct bearing method, a rolling bearing vibration model referred to [27,28] is used to construct bearing failure failure simulation simulation signals.signals. Rolling vibration model model is is expressed expressed as as follows: follows: Rolling bearing bearing vibration x( t ) s( t ) n( t ) Ai h( t iT i ) n( t ) i x (t) = s(t) + n(t) = ∑i Ai h(t − iT − τi ) + n(t) h ( t ) exp( Ct ) cos(2 f t) h(t) = exp(−Ct)cos(2π f nn t) A0 cos(2 fr t ) A A i i = A0 cos(2π f r t )
(11) (11)
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where T is the average period of impulse series. f is the fault feature frequency which equals to where T is the average period of impulse series. f i i is the fault feature frequency which equals to therotating rotating frequency frequency which which equals the the reciprocal reciprocal of of T T and and is the is set set to to 100 100 Hz, Hz, ffr isisthe equals to to 20 20 Hz, Hz, ffnn isisthe fault-sensitive 4000 Hz, Hz, ffss isisthe thesampling samplingfrequency frequencywhich whichisis fault-sensitive resonance resonance frequency which equals to 4000 set to 12,800 Hz, τ is the tiny random fluctuation of the i-th impulse and τ ∼ N ( 0, 0.05/ f ) , C isisthe r the set to 12,800 Hz, i i is the tiny random fluctuation of the i-th impulse and ~N(0,0.05 / fr ) , C damping coefficient which equals to 900, and the sampling points are set to 12,800. damping coefficient which equals to 900, and the sampling points are set to 12,800. In line with Equation (11), simulation signals with Gaussian noise are generated as is shown in In line with Equation (11), simulation signals with Gaussian noise are generated as is shown in Figure 5. When noise variation D ≥ 0.6, both the impulse series and the fault-sensitive resonance Figure 5. When noise variation D 0.6 , both the impulse series and the fault-sensitive resonance frequency band are totally masked by heavy background noise, there are no certain resonance frequency frequency band are totally masked by heavy background noise, there are no certain resonance bands that can be directly detected from frequency spectrums. Heavy noise leads to difficulties in frequency bands that can be directly detected from frequency spectrums. Heavy noise leads to extracting fault features from raw signals. difficulties in extracting fault features from raw signals.
D=1
2 0 -2
D=0.8
2 0 -2
D=0.6
2 0 -2
D=0.4
2 0 -2
D=0.2
2 0 -2
D=0
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2 0 -2 0
(b)
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0.2 0.3 Time [s]
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0.01 0.005 0 0.01 0.005 0 0.01 0.005 0 0.01 0.005 0 0.01 0.005 0 0.01 0.005 0
0
2000 4000 Frequency [H z]
6000
Figure 5. (a) Six simulation signals in time domain; (b) six simulation signals in frequency domain. Figure 5. (a) Six simulation signals in time domain; (b) six simulation signals in frequency domain.
In verify the effectiveness of theof proposed method method in identifying fault-sensitive resonance Inorder orderto to verify the effectiveness the proposed in identifying fault-sensitive frequency and extracting weak fault features background noise, a simulationnoise, signala resonancebands frequency bands and extracting weakwith faultheavy features with heavy background with noise variation D =noise 0.8 isvariation designated testisthe proposedtomethods. The simulation signal’s designated test the proposed methods. The simulation signal with D to0.8 transient features overwhelmed heavy background its fault-sensitive resonance simulation signal’sare transient featuresbyare overwhelmed by noise heavyand background noise and its faultfrequency band is obscure, so the simulation signal is appropriate to verify the effectiveness of the sensitive resonance frequency band is obscure, so the simulation signal is appropriate to verify the proposed method. effectiveness of the proposed method. Analysis Analysisof ofthe thesimulation simulationsignal signalperformed performedby bythe theproposed proposedmethod methodisisshown shownin inFigure Figure6.6.From From Figure Figure 6a, 6a, one one can can find find that that the the maximum maximum correlated correlated kurtosis kurtosis occurs occurs at at level level 44 with with aa frequency frequency bandwidth 400 Hz Hz and and a band central 4000 Hz Hz which is inconsistent bandwidthfRCK_bw fRCK_bw ==400 central frequency frequencyfRCK_c fRCK_c == 4000 inconsistent with with the the fault-sensitive fault-sensitive resonance resonance frequency frequency of of the the simulation simulation signal. signal. From From Figure Figure 6b, 6b, the the fault fault feature feature frequency frequencyfifi== 100 Hz and its harmonic frequencies frequencies are are easy easy to to identify. identify. In In the the meanwhile, meanwhile, fault fault feature feature frequencies frequenciesare aredominant dominantin infrequency frequencydomain. domain. Analysis Analysisof ofthe thesame same simulation simulation signal signal obtained obtained by by the the original original kurtogram kurtogram is is shown in Figure 7. With interference of ofheavy heavynoise, noise, one from Figure thatmaximum the maximum kurtosis With the the interference one cancan findfind from Figure 7a,b 7a,b that the kurtosis occurs occurs at level 6 with a band central frequency f = 2200 Hz. The frequency band extracted via at level 6 with a band central frequency fFK_c = 2200 Hz. The frequency band extracted via maximum FK_c maximum is from different the fault-sensitive resonance frequency fn = 4000 Hz. The width Hz. The width of the kurtosis iskurtosis different the from fault-sensitive resonance frequency fn = 4000 of the extracted frequency is f=FK_bw = 100 By contrast, the original kurtogram indicates the extracted frequency band band is fFK_bw 100 Hz. ByHz. contrast, the original kurtogram indicates the faultfault-sensitive resonance frequency improperly andresults the results hence obtained incorrect. sensitive resonance frequency improperly and the hence obtained incorrect.
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(a)
RSGWPT+CK CKmax =9e-005 @N w =2 , fc=4000H z
(a)
RSGWPT+CK max(CK)=9e-005 @ Nw=24, fc=4000Hz
-5
x 10 9 -5 x 10 9 8.5
Levels Levels
1 21
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(b)
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i
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i
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Figure 6. (a) The results of the improved kurtogram; (b) the results of the corresponding envelope spectrum. Figure 6. (a) The results of the improved kurtogram; (b)6.the results of the corresponding envelope spectrum. Figure stft+kurt Kmax =0 @N w =26, fc=2200H z stft+kurt max(Kurt)=3.8 @ Nw=24.6, fc=5625Hz
Level: level: log2(Nw) log2(N w )
(a) (a)
0 0 3 3 3.6 3.6 4 4 4.6 4.6 5 55.6
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1 0 00 0
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100 150 200 250 300 100Squared 150envelope 200frequency 250 [H z] 300 Squared envelope frequency [Hz]
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Figure 7.7.(a) (b) the thereuslts reusltsofofsquared squaredenvelope envelope spectrum. Figure (a)The Theresults resultsof oforiginal original kurtogram; kurtogram; (b) spectrum. Figure 13.
ordertotoillustrate illustratethe the necessity necessity and and effect effect of a a InInorder of RSGWPT RSGWPTused usedininthe theproposed proposedmethod, method, kurtogram is constructed only by RSGWPT and the kurtosis, which is different from the original kurtogram is constructed only by RSGWPT and the kurtosis, which is different from the original kurtogram. The analysis is shown in Figure 8. The maximum kurtosis node occurs at level 6 of kurtogram. The analysis is shown in Figure 8. The maximum kurtosis node occurs at level 6 of RSGWPT RSGWPT with a central frequency fRK_c = 3000 Hz and a bandwidth fRK_bw = 100 Hz, and its with a central frequency fRK_c = 3000 Hz and a bandwidth fRK_bw = 100 Hz, and its corresponding corresponding envelope spectrum doesn’t contain any noticeable fault feature frequencies. However, envelope spectrum doesn’t contain any noticeable fault feature frequencies. However, one can learn one can learn from Figure 8a that the constructed kurtogram contains an identifiable kurtosis peak from Figure 8a that the constructed kurtogram contains an identifiable kurtosis peak value at level 4 around the fault-sensitive resonance frequency fn 1= 4000 Hz. The RSGWPT is usually believed to be superior to either WPT or STFT in time-frequency analysis but not to such a degree that the effect of
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value at level 4 around the fault-sensitive resonance frequency fn = 4000 Hz. The RSGWPT is usually believed to be superior to either WPT or STFT in time-frequency analysis but not to such a degree that the of kurtosis in detecting transient be free from interference of heavythe kurtosis in effect detecting transient components mightcomponents be free frommight interference of heavy noise, making noise, making the RSGWPT fail fault-sensitive to identify appropriate frequencythe band or to RSGWPT fail to identify appropriate frequency fault-sensitive band or to demodulate frequencies demodulate the frequencies fault features. fault features. RSGWPT+kurt Kmax =0.25506 @N w =26, fc=3000H z
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Figure 8. (a) The results of kurtogram; (b) trhe results of envelope spectrum. Figure 8. (a) The results of kurtogram; (b) trhe results of envelope spectrum.
To further the previous studies, the proposed method, by combining RSGWPT and correlated To further previous studies, the proposedand method, by combining and correlated kurtosis, can the facilitate the detection of transients the extraction of weakRSGWPT fault features, just as is kurtosis, facilitate the detection of transients and the between extraction of weakand fault just as is showncan in Figure 6. Thus, the necessity of the combination RSGWPT CKfeatures, in the proposed method has been proved, the effect of the proposed method extractingand weak features of shown in Figure 6. Thus, theand necessity of the combination betweeninRSGWPT CKfault in the proposed signals with heavy background noise has been verified in turn. method has been proved, and the effect of the proposed method in extracting weak fault features of Similarly, thebackground manuscript noise selectshas a set of 20 simulation signals generated by Equation (11) with signals with heavy been verified in turn. a Gaussian mean of 0 and different (varying from 0.05 to 1by with step length Similarly,noise the manuscript selects a setnoise of 20variation simulation signals generated Equation (11) 0.05) with a to verify the effect of the proposed method in extracting weak fault features. Both the proposed Gaussian noise mean of 0 and different noise variation (varying from 0.05 to 1 with step length 0.05) to method and the original kurtogram areinutilized to analyze these signals, Both their the central frequencies, verify the effect of the proposed method extracting weak fault features. proposed method frequency bandwidth and decomposition levels of the identified fault-sensitive frequency are and the original kurtogram are utilized to analyze these signals, their central frequencies,bands frequency shown in Figures 9 and 10, respectively. bandwidth and decomposition levels of the identified fault-sensitive frequency bands are shown in One can learn that, from Figure 9a, within the range of the noise variation between 0.15 and 1, Figures 9 and 10, respectively. the indicated maximum correlated kurtosis frequency bands are adjacent to the real fault-sensitive One can learn that, from Figure 9a, within the range of the noise variation between 0.15 and 1, frequency fn = 4000 Hz. From Figure 9b, the indicated maximum correlated kurtosis frequency bands the indicated maximum correlated kurtosis frequency bands are adjacent to the real fault-sensitive with adequate bandwidth fall between appropriate range scales. From Figure 9c, the indicated central frequency fn = 4000 Hz. From Figure 9b, the indicated maximum correlated kurtosis frequency bands frequencies of maximum correlated kurtosis frequency bands are more concentrated, neighboring on with bandwidthfrequency. fall between appropriatethe range scales.method From Figure 9c, theidentify indicated theadequate real fault-sensitive Consequently, proposed can steadily the central fault frequencies of maximum correlated kurtosis frequency bands are more concentrated, neighboring on sensitive resonance frequency band, which is helpful to demodulate fault features exactly. the real From fault-sensitive frequency. Consequently, the proposed method can steadily identify the fault Figure 10a, the maximum kurtosis frequency bands indicated by the original kurtogram sensitive resonance frequency which is helpful faultrange features exactly. are also close to the real faultband, sensitive frequency into thedemodulate noise variation between 0.2 and 0.6. From Figure 10a, the maximum kurtosis frequency bands indicated by the original Nevertheless, from Figure 10b, the indicated maximum kurtosis frequency bands havekurtogram a larger arefluctuation also close range to theofreal fault sensitive frequency in bandwidths. the noise variation range10c, between 0.2 and decomposition levels and narrow From Figure the indicated 0.6.maximum Nevertheless, from Figure 10b, the indicated kurtosis and frequency bands have afault larger kurtosis frequency bands have deviatedmaximum central frequencies it is useless to extract features. range Moreover, within the noise variation rangebandwidths. from 0.65 toFrom 1, theFigure original is fluctuation of decomposition levels and narrow 10c,kurtogram the indicated maximum kurtosis frequency bands have deviated central frequencies and it is useless to extract fault features. Moreover, within the noise variation range from 0.65 to 1, the original kurtogram is
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interpreted frequency domain. domain. interpreted by by heavy heavy noise noise and and indicates indicates an an incorrect incorrect location location of of the the transients transients in in frequency Consequently, their envelope spectrums are hard to extract correct weak fault features. Consequently, their envelope spectrums are hard to extract correct weak fault features. Consequently, their envelope spectrums are hard to extract correct weak fault features. Fault-sensitive Fault-sensitivefrequency frequency band band identified identified by by improved improved kurtogram kurtogram N Number umber of of times times (c)
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Figure (a) The The central central frequencies; (b) the the frequency bandwidth 9. decomposition Figure 9. 9. (a) (a) The central frequencies; frequencies; (b) (b) the frequency frequency bandwidth bandwidth and and decomposition decomposition levels; levels; (c) distribution of central frequencies. distribution of central frequencies. (c) distribution of central frequencies.
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Figure 10. (a) (a) The central central frequencies; (b) (b) the frequency frequency bandwidth and and decomposition levels; levels; Figure Figure 10. 10. (a) The The central frequencies; frequencies; (b) the the frequency bandwidth bandwidth and decomposition decomposition levels; (c) distribution of central frequencies. (c) (c) distribution distribution of of central central frequencies. frequencies.
As As mentioned mentioned above, above, itit cannot cannot identify identify fault-sensitive fault-sensitive frequency frequency bands bands effectively effectively or or extract extract weak fault features correctly because the original kurtogram is restricted by limitations of STFT weak fault features correctly because the original kurtogram is restricted by limitations of STFT and and kurtosis kurtosis in in detecting detecting consectutive consectutive impulses. impulses. However, However, the the proposed proposed method method based based on on RSGWPT RSGWPT and and
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As mentioned above, it cannot identify fault-sensitive frequency bands effectively or extract weak Sensorsfeatures 2016, 16, correctly 1482 11 of 20 fault because the original kurtogram is restricted by limitations of STFT and kurtosis in detecting consectutive impulses. However, the proposed method based on RSGWPT and CK can be CK can be in identifying fault-sensitive frequency bands. Accordingly, it can correctly extract steadier in steadier identifying fault-sensitive frequency bands. Accordingly, it can correctly extract weak fault weak fault features of simulation signals with heavy background noise. features of simulation signals with heavy background noise.
5. Applications 5. Applications Data sets sets from from the the Case Case Western Western Reserve Reserve University University (CWRU) (CWRU) Bearing Bearing Data Data Center Center and and vibration vibration Data signals from a rolling bearing of a real transmission are utilized to verify the validity of the proposed signals from a rolling bearing of a real transmission are utilized to verify the validity of the method. method. proposed 5.1. Case 1: Extraction Test Test Based Based on on Bearing Bearing Data Data from from CWRU CWRU The bearing bearing test rig of the CWRU Bearing Data Center [29] is shown in Figure 11, which consists of Reliance Electric motor, a torque transducer/encoder, a dynamometer and control of aatwo twohorsepower horsepower Reliance Electric motor, a torque transducer/encoder, a dynamometer and electronics. Motor bearings are seeded faults using electro-discharge machining. Faults ranging control electronics. Motor bearings arewith seeded with faults using electro-discharge machining. Faults from 0.007 inches 0.040 to inches diameter are introduced separately on the inner rolling ranging from 0.007toinches 0.040ininches in diameter are introduced separately on thering, inner ring, element and outer ring. Faulty are thenare reinstalled into the into test rig loads ranging rolling element and outer ring.bearings Faulty bearings then reinstalled theand testmotor rig and motor loads from 0 tofrom 3 horsepower and motor speeds rotating from 1797 to 1720 RPM are applied. ranging 0 to 3 horsepower and motor speedsatrotating at from 1797 to 1720 RPM areVibration applied. data are collected accelerometers, which are mounted at the 12 o’clock position at both the drive Vibration data areusing collected using accelerometers, which are mounted at the 12 o’clock position at end endend of the housing. data sets are recorded a data acquisition bothand thefan drive andmotor fan end of theVibration motor housing. Vibration data using sets are recorded usingsystem a data and the sampling is set to 12frequency KHz for some andsome 48 KHz for others.and The48 details acquisition systemfrequency and the sampling is setexperiments to 12 KHz for experiments KHz of andoffrequencies of faults are shown in test forfaulted others. bearing The details faulted bearing and frequencies ofTable faults1. areFurther showndetails in Tableregarding 1. Furtherthe details setup can be on the CWRU Bearing Data Center Website [1,29]. regarding thefound test setup can be found on the CWRU Bearing Data Center Website [1,29].
Figure11. 11.The Thebearing bearingtest testrig rigofofthe theCWRU CWRUBearing Bearing Data Data Center Figure Center[29]. [29]. Table Table 1. 1. Bearing Bearing details details and and fault fault frequencies. frequencies.
Position on Rig Position on Rig Drive end Drive end Fan end Fan end
Bearing Model Bearing Model SKF 6205-2 RS JEM SKF SKF6205-2 6203-2RS RSJEM JEM SKF 6203-2 RS JEM
Fault Frequencies Frequencies (Multiple (Multiple of of Running Running Speed Speed in in Hz) Hz) Fault Inner Ring Outer Ring Cage Train Rolling Element Inner Ring Outer Ring Cage Train Rolling Element 5.415 3.585 0.398 4.714 5.415 3.585 0.398 4.714 4.947 3.053 0.382 3.987 4.947 3.053 0.382 3.987
Data sets from CWRU Bearing Data Center have been a standard reference used to test diagnosis algorithms in the field of bearing diagnosis. In order to examine the CWRU data sets thoroughly and to classify them appropriately, Smith [1] firstly achieved a benchmark study by using three established benchmark fault diagnostic methods which respectively are envelope analysis of the raw signal (Method 1), cepstrum pre-whitening (Method 2) and the original kurtogram (Method 3). With classifications and recommendations given by Smith, both data X171_FE and X224_DE cannot
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Data sets from CWRU Bearing Data Center have been a standard reference used to test diagnosis algorithms in the field of bearing diagnosis. In order to examine the CWRU data sets thoroughly and to classify them appropriately, Smith [1] firstly achieved a benchmark study by using three established benchmark fault diagnostic methods which respectively are envelope analysis of the raw signal (Method 1), cepstrum pre-whitening (Method 2) and the original kurtogram (Method Sensors 2016, 16, 1482 12 of 3). 20 With classifications and recommendations given by Smith, both data X171_FE and X224_DE cannot diagnose clearly by three established benchmark methods. So, data X171_FE and data X224_DE are of the the proposed proposed method. method. used to verify the validity and superiority of Data X171_FE (inner ring faults, sampling frequency is set to 12 KHz, rotating speed equals to 1752 RPM, inner ring fault frequency equals to 158.1 Hz) are used as benchmark data to compare between the proposed and the original kurtogram method. X171_FE and its frequency spectrum are shown in Figure 12. The results obtained by the original kurtogram are shown in Figure 13. One can fromFigure Figure the squared envelope spectrum not only obvious exhibits rotating obviousfrequency rotating learn from 13b13b thatthat the squared envelope spectrum not only exhibits frequency and its harmonics, but alsofault indicates fault frequency fBFPI. However, as the benchmark and its harmonics, but also indicates frequency fBFPI . However, just as thejust benchmark study of study of Smith found the squared envelope spectrum discrete components the expected Smith found that the that squared envelope spectrum shows shows discrete components at the at expected fault fault frequencies but they are not dominant in the spectrum. The results obtained by the proposed frequencies but they are not dominant in the spectrum. The results obtained by the proposed method method are in Figure One canFigure learn 14b from Figure 14bfrequency that the fault frequency its are shown in shown Figure 14. One can14. learn from that the fault and its harmonicsand of the harmonicsspectrum of the envelope are those more in obvious those inthe Figure and that fault envelope are morespectrum obvious than Figure than 13b and that fault13b frequency hasthe a much frequency has a much higher value while than its while13b. it isHowever, not true influenced with Figure higher value than its side frequencies it isside notfrequencies true with Figure by13b. the However, influenced by theits obvious rotating its harmonics side frequencies, the fault obvious rotating frequency, harmonics andfrequency, side frequencies, the faultand frequency and its harmonics frequency andare itsnot harmonics in Figure 14b are in not extraordinarily in theenhanced spectrum,results but it in Figure 14b extraordinarily dominant the spectrum, but dominant it really obtains reallythose obtains enhanced than in Figure 13b.results than those in Figure 13b.
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Figure 12. 12. (a) (a) Data Data X171_FE X171_FE in in time time domain; domain; (b) (b) the the spectrum spectrum of Figure of X171_FE X171_FE in in frequency frequency domain. domain.
Data X224_DE (rolling element faults, sampling frequency is set to 12 KHz, rotating speed equals 1754 RPM, rolling element fault frequency equals 137.8 Hz) are also used as benchmark data to compare between the proposed and the original kurtogram method. X224_DE and its frequency spectrum are shown in Figure 15. As the benchmark study of Smith mentions, data X224_DE are not diagnosable for the specified bearing fault by three established methods, including the original kurtogram method. The results obtained by the original kurtogram are shown in Figure 16. One can learn from Figure 16b that, with the influences of STFT and heavy noise, the maximum kurtosis frequency band occurs at level 6, the obtained squared envelope spectrum doesn’t contain any clearly fault frequencies except rotating frequency and its harmonics. The results obtained by the proposed method are shown in Figure 17. One can learn from Figure 17b that the envelope spectrum exhibits
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Figure 13. (a) The kurtogram of data X171_FE; (b) the Figure 13.squared envelope spectrum of the frequency band with the maximum kurtosis. RSGWPT+CK max(CK)=0.000337 @ Nw=23, fc=6000Hz
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Figure 14. (a) The improved kurtogram of data X171_FE; Figure 14. (b) the envelope spectrum of the frequency band with the maximum correlated kurtosis. stft+kurt
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Data X224_DE (rolling element faults, sampling frequency is set to 12 KHz, rotating speed equals 0 1754 RPM, rolling element fault frequency equals 137.8 Hz) are also used as benchmark data0.5 to compare 3 between the proposed and the original kurtogram method. X224_DE and its frequency spectrum are 0.4 3.6 15. As the benchmark study of Smith mentions, data X224_DE are not diagnosable for shown in Figure the specified 4bearing fault by three established methods, including the original kurtogram 0.3 method. 4.6 5 5.6
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The results obtained by the original kurtogram are shown in Figure 16. One can learn from Figure 16b that, with the influences of STFT and heavy noise, the maximum kurtosis frequency band occurs at level 6, the obtained squared envelope spectrum doesn’t contain any clearly fault frequencies except rotating frequency and its harmonics. The results obtained by the proposed method are shown in Figure 17. One can learn from Figure 17b that the envelope spectrum exhibits clearly fault frequency, Sensors 2016, 16, 1482 14 of 20 its harmonics and side frequencies. RSGWPT+CK max(CK)=0.000337 @ Nw=23, fc=6000Hz
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Figure 15. (a) Data X224_DE in time domain; (b) the spectrum of X224_DE in frequency domain. Figure 14.spectrum Figure 15. (a) Data X224_DE in time domain; (b) the of X224_DE in frequency domain. stft+kurt Kmax =0.5 @N w =26, fc=3750H z
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Figure 16. (a) The kurtogram of data X224_DE; (b) the squared envelope spectrum of the frequency Figure 16. (a) The kurtogram of data X224_DE; (b) the Figure 16. squared envelope spectrum of the frequency band with the maximum kurtosis. band with the maximum kurtosis.
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Figure 17. Figure 17. (a) The improved kurtogram of data X224_DE; (b) the envelope spectrum of the frequency band with the maximum correlated kurtosis. -4 RSGWPT+CK max(CK)=0.000411 @ Nw=24, fc=12500Hz (a) x 10 4
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As is mentioned above, the proposed method can effectively highlight fault frequencies and extract correct1bearing fault features. CWRU data has verified the validity and superiority of the 3 proposed method. 2
5.2. Case 2: Extraction Test Based on Data from a Real Transmission Rolling Bearing 3
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4 A real transmission experiment system is shown in Figure 18, which consists of an automobile transmission of5BJ2020S, an electric motor, a generator and a data-acquisition module, as exactly shown 1 in Figure 18a. The electric motor is used to drive the transmission and the generator to simulate system 6 load. Vibration data are collected by using accelerometers, which are mounted at locations neighboring 0 0.2Locations 0.4 0.6 0.8 in 1Figure 1.218b.1.4 1.6 1.8 on the faulted bearing. are shown The bearing of the2output shaft (Shaft-2) 4 (b) frequency [Hz] x 10 is seeded with 0.04an outer ring fault using electro-discharge machining, as exactly shown in Figure 18c. 2fBPFO using a PXI data acquisition module and the sample frequency is set Vibration data setsf are recorded 3fBPFO BPFO to 40 KHz. The faulted bearing details and outer ring fault frequencies are shown in Table 2. 0.02
Table 2. The faulted bearing details and outer ring fault frequency. 0
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Figure 18. 18. (a) (a) Structure Structure of of the the experiment experiment system; system; (b) (b) locations locations of of sensors, sensors, bearing bearing and and shaft; shaft; (c) (c) the the Figure faulted bearing. bearing. faulted
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Figure a recorded vibration signal, the corresponding frequency spectrum and 2 (a)19 displays envelope spectrum. With the influence of heavy noise, one cannot find any impulse components from Figure 18. (a) Structure of the experiment system; (b) locations of sensors, bearing and shaft; (c) the 0 And the fault feature frequencies are completely masked by heavy noise in the the vibration signal. faulted bearing. envelope spectrum. 0
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Firstly, the original kurtogram extract500 fault features by analyzing the signal 0 100 200 is used 300 to400 600 700 800 900 1000 which is Envelope frequency [H z] shown in Figure 19a, the results are shown in Figure 20. According to the kurtogram presented in Figure 10a, the maximum kurtosis is calculated at the 5.6th decomposition level and the Figure 19. (a) (a) Vibration signal; (b) frequencyfFK_c spectrum; (c) envelope spectrum. fFK_bw = 625 Hz. 19. Vibration frequency spectrum; envelope spectrum. correspondingFigure frequency band has a signal; central(b) frequency = 19375(c) Hz and a bandwidth The squared envelope spectrum of the frequency band with maximum kurtosis is presented in Figure 20b, Firstly, the original kurtogram is used to extract fault features by analyzing the signal which is shown in Figure 19a, the results are shown in Figure 20. According to the kurtogram presented in Figure 10a, the maximum kurtosis is calculated at the 5.6th decomposition level and the corresponding frequency band has a central frequency fFK_c = 19375 Hz and a bandwidth fFK_bw = 625 Hz.
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Firstly, the original kurtogram is used to extract fault features by analyzing the signal which is shown in Figure 19a, the results are shown in Figure 20. According to the kurtogram presented in Figure 10a, the maximum kurtosis is calculated at the 5.6th decomposition level and the corresponding Sensors 2016,band 16, 1482 17 of 20 frequency has a central frequency fFK_c = 19375 Hz and a bandwidth fFK_bw = 625 Hz. The squared envelope spectrum of the frequency band with maximum kurtosis is presented in Figure 20b, which which has ascope largeof scope of masked frequencies, therefore it becomes more difficult to identify has a large masked frequencies, therefore it becomes more difficult to identify correctcorrect weak weakfrequencies. fault frequencies. fault stft+kurt
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Figure 20. 20. (a) (a) The The kurtogram; kurtogram; (b) (b) the the squared squared envelope envelope spectrum spectrum of of the the frequency frequency band band with with the the Figure maximum kurtosis. maximum kurtosis. RSGWPT+CK CK
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max x 10 (a) as the CWRU case, the effectiveness The same of the proposed method is to be demonstrated in 4 actual test of fault features extraction. The results of the same signal are shown in Figure 21. According to the proposed kurtogram presented in Figure 21a, the maximum correlated kurtosis is calculated at 1 3 the 4th RSGWPT decomposition level and its corresponding frequency band has a central frequency 2 fRCK_c = 12500 Hz and bandwidth fRCK_bw = 1250 Hz. It can be observed that the fault frequency and its 3 2 harmonics are quite efficiently extracted. At the same time, they are dominant in the frequency domain. Therefore, it may4be concluded that, by comparison with the original kurtogram, the proposed method can successfully 5extract fault features from signals despite the heavy background noise. 1 In order to further demonstrate the effectiveness of the proposed method, the original signal, the 6 filtered signal with maximum kurtosis via original kurtogram and the filtered signal with maximum 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 correlated kurtosis via the proposed method are separately presented in Figure 22a–c. 4 [Hfind z] any impulses from Figure (b) of heavy background noise, frequency With influence one cannot x 10 22a. Several 0.04 impulses can be found in Figure 22b, but there is no sign of periodicity. Besides, the weak impulses However, are not clear enough, either. in Figure 22c, the filtered signal via the proposed method has clearly exhibit consecutive periodicity of impulses and there is significant sign of periodicity which 0.02 coincides with the fault feature frequency. Thus, it can be concluded that the proposed method is more efficient in extracting fault feature despite heavy noise. And this application test with real case verifies 0 the validity and superiority of the proposed method.
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Figure 17. max(CK)=0.000411 @ Nw=24, fc=12500Hz
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original kurtogram; (c) (c) signal filtered by the Figure 22. (a) The original original signal; signal;(b) (b)signal signalfiltered filteredbybythe the original kurtogram; signal filtered by proposed method. the proposed method.
6. Conclusions With influence of heavy background noise, one cannot find any impulses from Figure 22a. Several impulses can be found in Figure 22b, but there is no sign of periodicity. Besides, the weak For the inherent limitations of the kurtogram, this manuscript put forward an improved version impulses are not clear enough, either. However, in Figure 22c, the filtered signal via the proposed of kurtogram, which has been proved more effective in the extraction of weak fault features of rolling method has clearly exhibit consecutive periodicity of impulses and there is significant sign of bearing signals. Regarding the proposed approach, the CK can reduce the noise impact on the detection periodicity which coincides with the fault feature frequency. Thus, it can be concluded that the of impulse consequences and approach a maximum for a periodic impulse about the specified period proposed method is more efficient in extracting fault feature despite heavy noise. And this application test with real case verifies the validity and superiority of the proposed method.
6. Conclusions
For the inherent limitations of the kurtogram, this manuscript put forward an improved version
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as opposed to the kurtosis which tends to a maximum with a single impulse; the RSGWPT, based on redundant lifting scheme and possessing time invariant property, can not only acquire richer feature information and more precise frequency localization information, but also restrict the frequency aliasing components of the analysis results, the RSGWPT is confirmed to be superior to WPT and STFT in time-frequency analysis, and hence helpful to detect fault-sensitive frequency bands which materialize the extraction of weak fault features. Combining the advantages of CK with those of RSGWPT, this manuscript proposes an improved kurtogram. The proposed method, coupled with the envelop analysis, enable the researchers extract bearing weak fault features from signals against heavy background noise. Analysis results of simulation signals, CWRU data and a real transmission bearing vibration signals demonstrate that, the proposed method is effective in extracting weak fault features from rolling bearing vibration signals. Acknowledgments: This research is supported by National Natural Science Foundation of China (51205427), Equipment Advanced Research Foundation (9140A27020115JB35071). Author Contributions: X.C. and B.Z. proposed the method. X.C. contributed to the data acquisition, analyzed the data and wrote the draft manuscript, B.Z. provided some valuable advices, F.F. contribute to data acquisition, participated in manuscript writing and revised the manuscript. Conflicts of Interest: The authors declare no conflicts of interest.
References 1. 2. 3. 4.
5. 6. 7.
8. 9. 10. 11. 12. 13.
14.
Smith, W.A.; Randall, R.B. Rolling element bearing diagnostics using the Case Western Reserve University data: A benchmark study. Mech. Syst. Signal Process. 2015, 64–65, 100–131. [CrossRef] Jia, F.; Lei, Y.; Shan, H.; Lin, J. Early Fault Diagnosis of Bearings Using an Improved Spectral Kurtosis by Maximum Correlated Kurtosis Deconvolution. Sensors 2015, 15, 29363–29377. [CrossRef] [PubMed] Chen, J.; Li, Z.; Pan, J.; Chen, G.; Zi, Y.; Yuan, J.; Chen, B.; He, Z. Wavelet transform based on inner product in fault diagnosis of rotating machinery: A review. Mech. Syst. Signal Process. 2016, 70–71, 1–35. [CrossRef] Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.; Yen, N.C.; Tung, C.C.; Liu, H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 1998, 454, 903–995. [CrossRef] Smith, J.S. The local mean decomposition and its application to EEG perception data. J. R. Soc. Interface 2005, 2, 443–454. [CrossRef] [PubMed] Lei, Y.; Han, D.; Lin, J.; He, Z. Planetary gearbox fault diagnosis using an adaptive stochastic resonance method. Mech. Syst. Signal Process. 2013, 38, 113–124. [CrossRef] Tse, P.W.; Wang, D. The design of a new sparsogram for fast bearing fault diagnosis: Part 1 of the two related manuscripts that have a joint title as “Two automatic vibration-based fault diagnostic methods using the novel sparsity measurement—Parts 1 and 2”. Mech. Syst. Signal Process. 2013, 40, 499–519. [CrossRef] Tse, P.W.; Wang, D. The automatic selection of an optimal wavelet filter and its enhancement by the new sparsogram for bearing fault detection. Mech. Syst. Signal Process. 2013, 40, 520–544. [CrossRef] Antoni, J. The infogram: Entropic evidence of the signature of repetitive transients. Mech. Syst. Signal Process. 2016, 74, 73–94. [CrossRef] Wang, D. An extension of the infograms to novel Bayesian inference for bearing fault feature identification. Mech. Syst. Signal Process. 2016, 80, 19–30. [CrossRef] Antoni, J. The spectral kurtosis: A useful tool for characterizing non-stationary signals. Mech. Syst. Signal Process. 2006, 20, 282–307. [CrossRef] Antoni, J.; Randall, R.B. The spectral kurtosis: Application to the vibratory surveillance and diagnostics of rotating machines. Mech. Syst. Signal Process. 2006, 20, 308–331. [CrossRef] De la Rosa, J.J.G.; Agüera Pérez, A.; Palomares Salas, J.C.; Sierra Fernández, J.M. A novel measurement method for transient detection based in wavelets entropy and the spectral kurtosis: An application to vibrations and acoustic emission signals from termite activity. Measurement 2015, 68, 58–69. [CrossRef] Barszcz, T.; Randall, R.B. Application of spectral kurtosis for detection of a tooth crack in the planetary gear of a wind turbine. Mech. Syst. Signal Process. 2009, 23, 1352–1365. [CrossRef]
Sensors 2016, 16, 1482
15.
16. 17. 18.
19. 20. 21. 22. 23. 24.
25.
26.
27.
28. 29.
20 of 20
Chen, B.; Zhang, Z.; Zi, Y.; He, Z.; 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] Wang, Y.; Xiang, J.; Markert, R.; Liang, M. Spectral kurtosis for fault detection, diagnosis and prognostics of rotating machines: A review with applications. Mech. Syst. Signal Process. 2016, 66, 679–698. [CrossRef] Antoni, J. Fast computation of the kurtogram for the detection of transient faults. Mech. Syst. Signal Process. 2007, 21, 108–124. [CrossRef] Wang, Y.; Xu, G.; Liang, L.; Jiang, K. Detection of weak transient signals based on wavelet packet transform and manifold learning for rolling element bearing fault diagnosis. Mech. Syst. Signal Process. 2015, 54, 259–276. [CrossRef] Rother, A.; Jelali, M.; Söffker, D. A brief review and a first application of time-frequency-based analysis methods for monitoring of strip rolling mills. J. Process Control 2015, 35, 65–79. [CrossRef] Lei, Y.; Lin, J.; He, Z.; Zi, Y. Application of an improved kurtogram method for fault diagnosis of rolling element bearings. Mech. Syst. Signal Process. 2011, 25, 1738–1749. [CrossRef] Wang, D.; Tse, P.W.; Tsui, K.L. An enhanced Kurtogram method for fault diagnosis of rolling element bearings. Mech. Syst. Signal Process. 2013, 35, 176–199. [CrossRef] McDonald, G.L.; Zhao, Q.; Zuo, M.J. Maximum correlated Kurtosis deconvolution and application on gear tooth chip fault detection. Mech. Syst. Signal Process. 2012, 33, 237–255. [CrossRef] Zhou, R.; Bao, W.; Li, N.; Huang, X.; Yu, D. Mechanical equipment fault diagnosis based on redundant second generation wavelet packet transform. Digit. Signal Process 2010, 20, 276–288. [CrossRef] Dwyer, R. Detection of non-Gaussian signals by frequency domain Kurtosis estimation. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Boston, MA, USA, 14–16 April 1983; pp. 607–610. Zhang, X.; Kang, J.; Hao, L.; Cai, L.; Zhao, J. Bearing fault diagnosis and degradation analysis based on improved empirical mode decomposition and maximum correlated kurtosis deconvolution. J. Vibroeng. 2015, 17, 243–260. Li, N.; Zhou, R.; Hu, Q.; Liu, X. Mechanical fault diagnosis based on redundant second generation wavelet packet transform, neighborhood rough set and support vector machine. Mech. Syst. Signal Process. 2012, 28, 608–621. [CrossRef] Randall, R.B.; Antoni, J.; Chobsaard, S. The Relationship between spectral correlation and envelope analysis in the diagnostics of bearing faults and other cyclostationary machine signals. Mech. Syst. Signal Process. 2001, 15, 945–962. [CrossRef] Antoni, J.; Bonnardot, F.; Raad, A.; El Badaoui, M. Cyclostationary modelling of rotating machine vibration signals. Mech. Syst. Signal Process. 2004, 18, 1285–1314. [CrossRef] Case Western Reserve University Bearing Data Center Website. Available online: http://csegroups.case. edu/bearingdatacenter/ (accessed on 9 April 2016). © 2016 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/).
