research article a fault diagnosis scheme for rolling...

Research ArticleA Fault Diagnosis Scheme for Rolling Bearing Based on ParticleSwarm Optimization in Variational Mode Decomposition

Cancan Yi Yong Lv and Zhang Dang

School of Mechanical Engineering Wuhan University of Science and Technology Wuhan 430081 China

Correspondence should be addressed to Yong Lv lvyongwusteducn

Received 20 January 2016 Revised 15 May 2016 Accepted 23 May 2016

Academic Editor Athanasios Chasalevris

Copyright copy 2016 Cancan Yi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Variational mode decomposition (VMD) is a newmethod of signal adaptive decomposition In the VMD framework the vibrationsignal is decomposed intomultiplemode components byWiener filtering in Fourier domain and the center frequency of eachmodecomponent is updated as the center of gravity of themodersquos power spectrumTherefore each decomposedmode is compact arounda center pulsation and has a limited bandwidth In view of the situation that the penalty parameter and the number of componentsaffect the decomposition effect inVMDalgorithm a novelmethod of fault feature extraction based on the combination ofVMDandparticle swarm optimization (PSO) algorithm is proposed In this paper the numerical simulation and the measured fault signalsof the rolling bearing experiment system are analyzed by the proposed method The results indicate that the proposed method ismuchmore robust to sampling and noise Additionally the proposedmethod has an advantage over the EMD in complicated signaldecomposition and can be utilized as a potential method in extracting the faint fault information of rolling bearings compared withthe common method of envelope spectrum analysis

1 Introduction

Rolling bearing is one of the most commonly used parts inrotating machinery to support rotating shafts Due to the factthat its health state is directly related to the safety and a stableoperation of the machine the research of rolling bearingfault diagnosis has a great significance in actual application[1] However the early incipient fault feature is very faintand interfered by the strong background noise [2] Whenthe rolling bearing is in failure due to the influence of loadfriction noise and other factors the measured vibration sig-nal is a multicomponent amplitude-modulated-frequency-modulated (AM-FM) signal using systemrsquos natural frequencyas carrier frequency and the fault characteristic frequency asmodulation frequency respectivelyThus obtaining the AM-FM signal by the original signal decomposition and reducingthe effect of noise are the emphasized research content inextracting the early faint fault feature of the rolling bearing

Currently there aremanymethods used to fault diagnosisfor rolling bearings but these methods have some inherentlimitations For instance wavelet transform (WT) [3 4]analysis is not adaptive which is restricted by the selection of

wavelet basis function and the number of decompositionlevels empirical mode decomposition (EMD) is lack of theo-retical basis and there are some problems in its own algorithmsuch as the phenomenon of model mixing and the endeffect [5ndash7] which has been used to detect the bearing fault[8] Although the ensemble empirical mode decomposition(EEMD) [9] has some improvement in solving the problemof model mixing the actual effect is sensitive to the strongbackground noise Empirical wavelet transform (EWT) isintroduced by Gilles et al aimed at extracting a series of AM-FM signal from the original signal which largely depends onchoosing the boundaries of Fourier spectrum appropriately[10 11] The sparse decomposition theory [12] and manifoldlearning method [13] are proposed recently However thesparse decomposition relies on the design of the redundantatom library and decomposition algorithm which has anobvious problemof the large amount of calculationTheman-ifold learning result is also restricted by the parameter selec-tion of delay time and embedded dimension From anotheraspect the spectroscopy and ferrography of used grease areapplied to condition monitoring for rolling element bearing[14] which has a limited capacity in detecting the noniron

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 9372691 10 pageshttpdxdoiorg10115520169372691

2 Shock and Vibration

particle in lubricating oil and largely depends on the humanexperience

Lately Dragomiretskiy and Zosso proposed a new varia-tionalmode decomposition (VMD) [15]method whichwas anew advancedmultiresolution technique A series of iterativeupdating process was used to minimize the constrainedvariational model therefore the vibration signal was decom-posed into variousmodes or intrinsicmode functions (IMFs)using calculus of variation It overcame the disadvantageof lacking theoretical basis and noise sensitivity of EMDMoreover the VMD method could adaptively determine therelevant frequency band and estimated the correspondingmodel Based on the above-mentioned advantage it wasapplied to the early fault diagnosis of rolling bearing [1617] and economics field [18] From the theory of VMDalgorithm it can be known that the decomposed result ofVMD is restricted by the penalty parameter selection andthe number of components In order to extract the faintfault information of the bearing vibration signal effectivelythe parametersrsquo selection needs to be optimized Taking intoaccount the characteristics of fast convergence speed smallsetting parameters and easy to implement particle swarmoptimization (PSO) algorithm [19 20] the paper applies itinto the parameters optimization of VMD The optimizedtwo parameters are obtained by the PSO algorithm whichis used to the process of feature extraction for faint faultsignal After the original signal is decomposed by improvedVMD method the optimal component can be identified bythe principle of maximum correlated kurtosis [21] The indexof correlated kurtosis has integrated the characteristics of thekurtosis and correlation function which can represent thechanges of shock signal and avoid the problem of overfittingThe merits of the proposed method can be summarized thatthe parameter selection and decomposition result of VMDalgorithm are less affected by human experience Using thismethod to analyze the simulated signal and the measuredvibration signal from rolling bearing experiment system theresults indicate that the proposed method can accuratelyextract the early characteristic frequency of faint fault signalof the rolling bearing The flowchart about the proposedmethod is shown in Figure 1

The rest of the paper is organized as follows In Section 2the basic ideas of VMD and PSO are introduced Thesimulation signal analysis is described in Section 3 Themeasured signal in rolling bearing experiment system wasanalyzed in Section 4 The final conclusions are given inSection 5

2 Theories

21 Variational Mode Decomposition (VMD) Algorithm Var-iational mode decomposition (VMD) is a new method ofsignal decomposition based onWiener filtering one-dimen-sional Hilbert transform and heterodyne demodulation pro-posed lately by Konstantin Dragomiretskiy Different fromEMD it defines the mode component as amplitude-modu-lated-frequency-modulated (AM-FM) signals as follows

119906119896 (119905) = 119860119896 (119905) cos (120601119896 (119905)) (1)

The time series ofvibration signal

Initialization parameter

Optimal parametercombination obtained by

PSO method

The signal preprocessingby the proposed improved

VMD

Numerical simulationsignal analysis

Envelope spectrumanalysis

Fault featureextraction

The optimal modecomponent selection by

correlated kurtosis

Measured signal analysis

Figure 1 The flowchart about the proposed method

where 120601119896(119905) is a nondecreasing function thus 1206011015840119896(119905) ge 0

the envelope is nonnegative 119860119896(119905) ge 0 Additionally thechange of envelope 119860119896(119905) and instantaneous frequency 1206011015840

119896(119905)

are much slower than 120601119896(119905) Therefore the mode component119906119896(119905) can be regarded as a pure harmonic signal withamplitude 119860119896(119905) and instantaneous frequency 1206011015840

119896(119905)

The input signal is decomposed by VMD method intothe mode component of a specified scale according to thesubjective setting scale assuming that each mode is a finitebandwidth signal with a pulse as the center In order toevaluate the bandwidth of each mode and construct theconstraintmodel of the variational problemVMDfirstly usesthe Hilbert transform to obtain the single spectrum of eachmode and then transfers them to the fundamental frequencyby exponential correction The bandwidth of each mode isobtained through Gauss smooth demodulation signal finallywhich is called as 1198712 norm squared of the gradient Therebythe constrained variational problem is given as follows

min119906119896120596119896

sum

119896

10038171003817100381710038171003817100381710038171003817

120597119905 [(120575 (119905) +

119895

120587119905

) lowast 119906119896 (119905)] 119890minus11989512059611989611990510038171003817100381710038171003817100381710038171003817

2

2

(2)

Subject to sum

119896

119906119896 = 119891 (3)

where 119891 is the original signal 120575 is the Dirac distribution 119896is number of modes 119906119896 fl 1199061 1199062 119906119896 denotes each

Shock and Vibration 3

mode function 120596119896 fl 1205961 120596119896 indicates each centerfrequency sum

119896fl sum

119896

119896=1represents the sum of all mode

function and lowast denotes convolution Due to the difficulty ofsolving the constrained problem the penalty parameter120572 andthe Lagrange multiplication operator 120582(119905) are introduced toconvert the above constrained problem to the nonconstraintproblem Thereby obtaining a new solution expression

119871 (119906119896 120596119896 120582)

fl 120572sum119896

10038171003817100381710038171003817100381710038171003817

120597119905 [(120575 (119905) +

119895

120587119905

) lowast 119906119896 (119905)] 119890minus11989512059611989611990510038171003817100381710038171003817100381710038171003817

2

2

+

1003817100381710038171003817100381710038171003817100381710038171003817

119891 (119905) minus sum

119896

119906119896 (119905)

1003817100381710038171003817100381710038171003817100381710038171003817

2

2

+⟨120582 (119905) 119891 (119905) minussum

119896

119906119896 (119905)⟩

(4)

Therefore the Lagrangemultipliersmodal functions andtheir corresponding central frequency are iteratively updatedby using alternating directionmultipliermethod (ADMM) toobtain the saddle point in the expression Specific algorithmof classical VMD is given as the flowing expression

Step 1 Initialize 1119896 1205961

119896 1205821 119899 larr 0

Step 2 The value of 119906119896 120596119896 and 120582 is updated according to thefollowing formula

119899+1

119896(120596) =

119891 (120596) minus sum119894 =119896

119894 (120596) +120582 (120596) 2

1 + 2120572 (120596 minus 120596119896)2

120596119899+1

119896=

int

infin

01205961003816100381610038161003816119896 (120596)

1003816100381610038161003816

2119889120596

int

infin

0

1003816100381610038161003816119896 (120596)

1003816100381610038161003816

2119889120596

120582119899+1= 120582119899+ 120591(119891 minussum

119896

119906119899+1

119896)

(5)

Step 3 Repeat the iterative process of (2) until the functionconverges which is to satisfy the condition of sum

119896119899+1

119896minus

119899

1198962

2119899

1198962

2lt 119890 where 119890 is a given accuracy requirement

22 Particle Swarm Optimization (PSO) Algorithm Particleswarm optimization (PSO) is an intelligent algorithm to imi-tate birdsrsquo foraging behavior proposed by Kennedy throughreferring to the characteristics of all the individual bird in theprocess of feeding which is widely used in solving nonlinearproblems

Each particle in the PSO algorithm is used as the solutionof the optimization problem which has a position and thecorresponding speed determined by the optimization func-tionThe algorithm evaluates the pros and cons of all particlesby setting the appropriate fitness function In each iterationthe particles constantly update their speed and positionaccording to the fitness value of individual and the group

The updated particles continue to search the optimal valuein the search space

Specific configuration steps of PSO are as follows

Step 1 Establish the appropriate fitness function accordingto the actual problemThe iteration calculation is carried outby setting the number of iterations population number theinitial position and velocity of the particles

Step 2 Calculate the optimal values 119875 and 119866 respectivelycompared with the optimal solution of the current popula-tion and retain the better results where 119875 represents localextremum of individual particles and 119866 indicates globalextremum of group particles

Step 3 Update the position and speed of all particles in thepopulation according to the formula 120592119896+1

119894119889= 120596120592119896

119894119889+ 1198881120578(119901

119896

119894119889minus

119909119896

119894119889) + 1198882120578(119892

119896

119889minus 119909119896

119894119889) 119909119896+1119894119889= 119909119896

119894119889+ 120592119896+1

119894119889 where 120596 is the inertia

weight 1198881 1198882 are the learning factor 120578 is the random numberbetween 0sim1 and 120592119896

119894119889 119909119896119894119889are the speed and position of the

particle in the 119896th iteration of the 119889 dimension

Step 4 Repeat Steps 2 and 3 until meeting the maximumnumber of iterations

23 VariationalModeDecomposition Based on Particle SwarmOptimization Algorithm In the traditional algorithm ofvariational mode decomposition the user needs to set thepenalty parameter and the number of the components beforeprocessing the signal because of the theory limitation Fromthe theoretical study of VMD it can be known that thelarger penalty parameter indicates the smaller bandwidthof each component decomposed by source signal and viceversa Similarly inappropriate setting number of componentswill also result in some unacceptable mode compositionsTherefore selecting the appropriate parameter group of thecomponent number and the penalty parameter is the key toaccurately extract the fault information

PSO algorithm is a widely used intelligent optimizationalgorithmcomparedwith other optimization algorithms suchas genetic algorithm and artificial fish algorithm It is suitablefor the optimal selection of parameters in consideration ofits simple principle and mechanism fast convergence speedand meanwhile the good performance of global searchThe key part of the PSO algorithm based on the variablemode decomposition is the selection of fitness functionBecause of the incorrect settings of the penalty parameter andthe number of components some artifact components willgenerate which are independent with the source signal It isacceptable that the artifact components have less similaritywith the source signal Therefore the cross-correlation coef-ficient between the decomposed mode component and theoriginal signal is regarded as an evaluation index which isdefined in the following formula

119862 =

sum119879

119899=1(119903 (119899) minus 119903) (119910 (119899) minus 119910)

[sum119879

119899=1(119903 (119899) minus 119903)

2sum119879

119899=1(119910 (119899) minus 119910)

2]

12 (6)


where 119903(119899) 119910(119899) represent the original signal and the modecomponent respectively 119879 is the data length 119862 representsthe cross-correlation coefficient

From the above analysis it can be seen that the cross-correlation coefficient 119862 may fluctuate under the conditionof different parameters selection The largest mean value ofcross-correlation coefficient does not imply the best result ofmode decomposition The globally optimal value is achievedby considering the mean value and the variance of cross-correlation coefficientThe smaller value of variance indicatesthe less deviation from themean valueTherefore the optimalpenalty parameters and the number of components can bewell obtained by regarding the maximum ratio betweenthe mean value of 119862 and the variance as fitness functionThe detailed fitness function is expressed in the followingformula

fit fun = mean (119862)var (119862)

(7)

On the basis of the above theory analysis VMD based onPSO algorithm is applied to the analysis of simulated signaland the fault feature extraction of rolling bearing experimentsystem to verify the validity of the method in fault diagnosis

3 The Analysis of Simulated Signal

The measured rolling bearing vibration signal is alwaysconsisted of the amplitude-modulated-frequency-modulated(AM-FM) signals harmonic signal and noisy signal in actualapplication In order to verify the validity of the VMD basedon PSO the fault signal model is built by the followingsimulated signal

1199091 = sin (21205871198911119905)

1199092 = cos (21205871198912119905)

1199093 = sin (21205871198913119905 + cos (21205871198914119905))

(8)

and then

119904 = 1199091 + 1199092 + 1199093 (9)

where the frequencies of 1198911 1198912 1198913 and 1198914 are chosen as90Hz 150Hz 500Hz and 270Hz respectivelyThe syntheticsignal 119904 is composed by sinusoidal signal 1199091 cosine signal1199092 and frequency-modulated (FM) signal 1199093 The samplingfrequency is set as 1000Hz and the sampling point is 1000Thetime-domain graph of simulated signal 119904 is shown in Figure 2

The VMD based on particle swarm optimization algo-rithm is applied to decompose the above simulated signalThe number of iterations and the particles is 20 the inertiaweight linear decrease in the iterative process of the initialvalue is 09 and the final value is 04 The penalty parameterand the number of components optimized and selected byparticle swarm optimization (PSO) search algorithm are acollection of (2064 3) and the results of the decompositionare shown in Figure 3

In the three decomposed components shown in Figures3(b) 3(c) and 3(d) the blue line is on behalf of the original

0 02 04 06 08 1minus3

minus2

minus1

0

1

2

3

Time (s)

Am

plitu

de

Figure 2 The simulated signal 119904

Table 1 The cross-correlation coefficient of reconstruction signal

Sinusoidal signal Cosine signal FM signalCross-correlationcoefficient 09962 09985 09974

Table 2 The result of EMDmethod

Sinusoidal signal Cosine signal FM signalIMF1 02021 06132 05921IMF2 07983 02180 00032IMF3 00276 00014 0

signal and the red line represents the signal after the decom-position In order to more directly express the result ofdecomposition the similarity analysis is carried out betweenthe decomposed component and the original signal by theindex of cross-correlation coefficient Table 1 demonstratesthat the proposedmethod has a perfect ability of complicatedsignal decomposition

Empiricalmode decomposition (EMD) is used to decom-pose the above simulated signals and the results of EMDdecomposition are shown in Figure 4 There are eight IMFsdecomposed by EMD

Similarly the decomposition result also carried out simi-larity analysis with the original component Since the similar-ity of the three signal components (sinusoidal signal cosinesignal and FM signal) compared with mode componentsafter IMF3 is basically close to zero so there is only a list ofthe previous three components shown in Table 2

From the decomposition results of EMD it can be knownthat sine cosine and FM signal are partly mixed in IMF1 andIMF2 which cannot be separated well Compared with theproposed method in this paper the decomposition effect ofVMD is obviously better than EMD which can well separatethe components from the original signal

The above experimental results show that the proposedmethod can almost completely separate the components


0 5 10 15 20100

150

200

250

300

350

400

450

500

Number of iterations

Fitn

ess v

alue

(a) The fitness value during the iteration process

minus101

Original sinusoidal signal

0 005 01 015 02

0 005 01 015 02

minus101

Reconstructed sinusoidal signal

(b) The original and reconstructed sinusoidal signal

01

Original cosine signal

0 005 01 015 02

0 005 01 015 02

01

Reconstructed cosine signal

minus1

minus1

(c) The original and reconstructed cosine signal

01

Original FM signal

0 005 01 015 02

0 005 01 015 02

01

Reconstructed FM signal

minus1

minus1

(d) The original and reconstructed FM signal

Figure 3 The result of VMD decomposition

from the simulation signal free of noisy signal Howeverthe actual measured bearing vibration signal in the runtimeis often affected by the strong noise background Thus thefault feature information is usually submerged in the noiseenvironment In order to prove the validity of the methodwe discuss the feature extraction effect of this method underthe different noise level In the simulation signal the Gausswhite noise with standard variance being 01 02 04 0305 07 and 09 is added in turn The cross-correlationcoefficient between the decomposed mode components andthe original components in different noise conditions iscalculated Results of signal reconstruction by the proposedmethod are shown in Figure 5

The noise of different intensity is added in the simulationsignal and then the signal is processed by using the proposedmethod From Figure 5 it is obvious that the VMD algorithmbased on PSO has a good performance of denoising

4 Analysis of Measured Signal inRolling Bearing Experiment System

In order to verify that the proposed method is effective in theexperiment the vibration data of rolling bearing experimentsystem is used to be analyzed The experimental system isshown in Figure 6 The whole experimental device is driven

by a 550W (220Vsim50Hz) AC motor The yellow arrowpoints the position of the replaceable bearing in Figure 5In this experiment the electric spark machining methodis used to carry out pitting treatment on the outer ringof replaceable bearing to simulate the faults of the outerring of bearing The acceleration signal of the experimentis collected in the vertical direction of the bearing on theright side of the experimental platform using the CSI2130data analyzer of America The parameter failure frequenciesof fault simulation test-bed are shown in Table 3 It is worthmentioning that the rotating frequency 119891119888 and outer faultfrequency 119891119900 are 2417Hz and 8701Hz respectively

The time-domain graph of measured bearing fault signalis shown in Figure 7(a) The result of envelope spectrumanalysis demonstrates that it is difficult to extract the faultfeature due to the interference of noisy signal which ispresented in Figure 7(b)

From Figure 7(b) it is a fact that the 1 to 3multiplicationsrsquofrequency of bearing outer fault is interfered by other irrel-evant signals such as noisy signal Particularly the rotatingfrequency 119891119888 can hard be identification by the means ofenvelope spectrum analysis The proposed method in thispaper is introduced to optimize and analyze the bearing faultdata In the PSO algorithm the number of iterations is setas 50 and the population size is selected as 20 The fitness


minus2

0

2IM

F1

minus2

0

2

IMF2

minus05

0

05

IMF3

minus01

0

01

IMF4

minus02

0

02

IMF5

minus5

0

5

IMF6

0 05 1minus5

0

5

IMF7

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1minus001

0

001

IMF8

times10minus3

times10minus3

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)

Figure 4 The result of EMD decomposition

Table 3 The experimental parameters and fault frequency

Rotating speed 119903min Rotating frequencyHz Sampling frequencyHz Sampling times Outer fault frequencyHz1450 2417 16384 1 8701

value during the iteration process is shown in Figure 8 Theoptimized parameters of the penalty parameter and the num-ber of the components about VMD method are selected as acollection (1800 10) By using the optimized parameters thebearing fault data are decomposed into ten-mode componentand the selection of optimal mode component is carriedout Subsequently the best representative decompositioncomponent of eighth decomposed component was employedto verify the validity of themethod which was determined bythe criterion of maximum correlated kurtosis The correlatedkurtosis of different component was shown in Figure 9and we can draw a conclusion that the eighth decomposedcomponent has a lager correlated kurtosis value Figure 10indicates the result of envelope spectrum analysis for theeighth decomposed component

It can be seen from the decomposition results of opti-mized VMD algorithm that the identification accuracy offault feature frequency is improved compared with the

traditional envelope spectrum analysis shown in Figure 7(b)Only 3 multiplicationsrsquo frequency of the outer ring faultfrequency can be found in the envelope spectrum of theoriginal signal while after optimization it can find morethan 10 multiplications of the fault frequency and detect therotating frequency119891119888 Additionally the characteristic spectralline was obvious and less inferred by other spectral lineswhich confirms the validity of the proposed method to thefault feature extraction of rolling bearing The contrastiveanalysis of the EMD and the proposed method is also carriedout which indicates that the optimized VMD algorithm has abetter ability in fault feature identification shown in Figure 11

5 Conclusions

Anovelmethod of particle swarm optimization in variationalmode decomposition method was introduced in faint faultfeature extraction of rolling bearingThemain conclusions of


2 4 6 8 10 12 14 16 18 20095

0955

096

0965

097

0975

098

0985

099

0995

1Si

mila

rity

of co

nstr

ucte

d an

d or

igin

al si

gnal

SNR (dB)

(a) The similarity of sinusoidal component with different noise level

2 4 6 8 10 12 14 16 18 20094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)

(b) The similarity of cosine component with different noise level

2 4 6 8 10 12 14 16 18 20093

094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)

(c) The similarity of FM component with different noise level

Figure 5 The results of VMD decomposition with varied noise level

Figure 6 The rolling bearing experiment system

this paper include the following (1) The particle swarm opti-mization algorithm was applied to the parameter selectionof the optimal penalty parameter and the number of com-ponents which largely depends on the suitable fitness func-tion determined by the maximum ratio between the mean

value and the variance of cross-correlation coefficient More-over the maximum correlated kurtosis is used to selectthe optimal component It is significant that the proposedmethod can avoid the interference of human experience andthe diagnostic results are more reasonable (2) Simulated


0 02 04 06 08 1minus15

minus10

minus5

0

5

10

15

20

Am

plitu

de

Time (s)

(a) Time-domain graph of the measured signal

0 200 400 600 800 10000

005

01

015

02

025

03

035

04

045

05

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

fo

(b) Envelope spectrum graph of the measured signal

Figure 7 The time-frequency diagram of the measured signal

0 10 20 30 40 505514

5515

5516

5517

5518

5519

552

5521

5522

5523

5524


Fitn

ess v

alue

Figure 8 The fitness value during the iteration process

1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

The decomposed component by VMD method

The v

alue

of c

orre

lated

kur

tosis

times10minus28

Figure 9 The correlated kurtosis of different component


0 200 400 600 800 10000

005

01

015

02

025

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

5fo6fo

7fo

8fo

9fo 10fo

fc

fo

Figure 10 The envelope spectrum analysis of the eighth decom-posed component

0 200 400 600 800 10000

001

002

003

004

005

006

007

008

009

Am

plitu

de

Frequency (Hz)

2fofo

Figure 11 The decomposition results of EMD

signal and measured fault bearing signal measured from therolling bearing experiment system were used to verify thevalidity of the method The result demonstrated that theproposedmethod has an advantage over the traditional EMDmethod and envelope spectrum analysis in faint fault signalprocessing for rolling bearings whichmake it possible for theproposedmethod to be a powerful tool in solving the problemof signal channel bind source separation

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (nos 51475339 and 51405353) and the

Key Laboratory of Metallurgical Equipment and Controlof Education Ministry Wuhan University of Science andTechnology (2015B11)

References

[1] M Zeng Y Yang J Zheng and J Cheng ldquoMaximum marginclassification based on flexible convex hulls for fault diagnosis ofroller bearingsrdquo Mechanical Systems and Signal Processing vol66-67 pp 533ndash545 2016

[2] H Wang J Chen and G Dong ldquoFeature extraction of rollingbearingrsquos early weak fault based on EEMD and tunable Q-factorwavelet transformrdquo Mechanical Systems and Signal Processingvol 48 no 1-2 pp 103ndash119 2014

[3] X An D Jiang C Liu and M Zhao ldquoWind farm powerprediction based on wavelet decomposition and chaotic timeseriesrdquo Expert Systems with Applications vol 38 no 9 pp11280ndash11285 2011

[4] H Li and X Wang ldquoDetection of electrocardiogram character-istic points using lifting wavelet transform and Hilbert trans-formrdquo Transactions of the Institute of Measurement and Controlvol 35 no 5 pp 574ndash582 2013

[5] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of The RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[6] C-F Lin ldquoChaotic visual cryptosystem using empirical modedecomposition algorithm for clinical EEG signalsrdquo Journal ofMedical Systems vol 40 no 3 pp 1ndash10 2016

[7] Y Lei J Lin Z He and M J Zuo ldquoA review on empiricalmode decomposition in fault diagnosis of rotating machineryrdquoMechanical Systems and Signal Processing vol 35 no 1-2 pp108ndash126 2013

[8] P Nikolakopoulos and A Zavos ldquoSlew bearings damage detec-tion using hilbert huang transformation and acoustic methodsrdquoTribology in Industry vol 37 no 2 pp 170ndash175 2015

[9] H Aied A Gonzalez and D Cantero ldquoIdentification of suddenstiffness changes in the acceleration response of a bridge tomoving loads using ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 66-67 pp 314ndash338 2016

[10] J Gilles ldquoEmpirical wavelet transformrdquo IEEE Transactions onSignal Processing vol 61 no 16 pp 3999ndash4010 2013

[11] Y Jiang H Zhu and Z Li ldquoA new compound faults detectionmethod for A new compound faults detection method forrolling bearings based on empirical wavelet transform andchaotic oscillatorrolling bearings based on empirical wavelettransform and chaotic oscillatorrdquo Chaos Solitons amp Fractals2015

[12] X Chen Z Du J Li X Li and H Zhang ldquoCompressed sens-ing based on dictionary learning for extracting impulse compo-nentsrdquo Signal Processing vol 96 pp 94ndash109 2014

[13] Q He and X Wang ldquoTime-frequency manifold correlationmatching for periodic fault identification in rotatingmachinesrdquoJournal of Sound and Vibration vol 332 no 10 pp 2611ndash26262013

[14] R Liu ldquoCondition monitoring of low-speed and heavily loadedrolling element bearingrdquo Industrial Lubrication and Tribologyvol 59 no 6 pp 297ndash300 2007


[15] K Dragomiretskiy and D Zosso ldquoVariational mode decompo-sitionrdquo IEEE Transactions on Signal Processing vol 62 no 3 pp531ndash544 2014

[16] C Aneesh S Kumar P Hisham et al ldquoPerformance compari-son of variational mode decomposition over empirical wavelettransform for the classification of power quality disturbancesusing support vector machinerdquo Procedia Computer Science vol46 pp 372ndash380 2015

[17] Y Wang R Markert J Xiang and W Zheng ldquoResearch onvariationalmode decomposition and its application in detectingrub-impact fault of the rotor systemrdquo Mechanical Systems andSignal Processing vol 60 pp 243ndash251 2015

[18] S Lahmiri ldquoLong memory in international financial marketstrends and short movements during 2008 financial crisis basedon variational mode decomposition and detrended fluctuationanalysisrdquo Physica A Statistical Mechanics and its Applicationsvol 437 pp 130ndash138 2015

[19] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer US 2010

[20] M Yuwono Y Qin J Zhou Y Guo B G Celler and S W SuldquoAutomatic bearing fault diagnosis using particle swarm clus-tering and Hidden Markov Modelrdquo Engineering Applications ofArtificial Intelligence vol 47 pp 88ndash100 2016

[21] G L McDonald Q Zhao andM J Zuo ldquoMaximum correlatedKurtosis deconvolution and application on gear tooth chip faultdetectionrdquoMechanical Systems and Signal Processing vol 33 no1 pp 237ndash255 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



particle in lubricating oil and largely depends on the humanexperience

Lately Dragomiretskiy and Zosso proposed a new varia-tionalmode decomposition (VMD) [15]method whichwas anew advancedmultiresolution technique A series of iterativeupdating process was used to minimize the constrainedvariational model therefore the vibration signal was decom-posed into variousmodes or intrinsicmode functions (IMFs)using calculus of variation It overcame the disadvantageof lacking theoretical basis and noise sensitivity of EMDMoreover the VMD method could adaptively determine therelevant frequency band and estimated the correspondingmodel Based on the above-mentioned advantage it wasapplied to the early fault diagnosis of rolling bearing [1617] and economics field [18] From the theory of VMDalgorithm it can be known that the decomposed result ofVMD is restricted by the penalty parameter selection andthe number of components In order to extract the faintfault information of the bearing vibration signal effectivelythe parametersrsquo selection needs to be optimized Taking intoaccount the characteristics of fast convergence speed smallsetting parameters and easy to implement particle swarmoptimization (PSO) algorithm [19 20] the paper applies itinto the parameters optimization of VMD The optimizedtwo parameters are obtained by the PSO algorithm whichis used to the process of feature extraction for faint faultsignal After the original signal is decomposed by improvedVMD method the optimal component can be identified bythe principle of maximum correlated kurtosis [21] The indexof correlated kurtosis has integrated the characteristics of thekurtosis and correlation function which can represent thechanges of shock signal and avoid the problem of overfittingThe merits of the proposed method can be summarized thatthe parameter selection and decomposition result of VMDalgorithm are less affected by human experience Using thismethod to analyze the simulated signal and the measuredvibration signal from rolling bearing experiment system theresults indicate that the proposed method can accuratelyextract the early characteristic frequency of faint fault signalof the rolling bearing The flowchart about the proposedmethod is shown in Figure 1

The rest of the paper is organized as follows In Section 2the basic ideas of VMD and PSO are introduced Thesimulation signal analysis is described in Section 3 Themeasured signal in rolling bearing experiment system wasanalyzed in Section 4 The final conclusions are given inSection 5

2 Theories

21 Variational Mode Decomposition (VMD) Algorithm Var-iational mode decomposition (VMD) is a new method ofsignal decomposition based onWiener filtering one-dimen-sional Hilbert transform and heterodyne demodulation pro-posed lately by Konstantin Dragomiretskiy Different fromEMD it defines the mode component as amplitude-modu-lated-frequency-modulated (AM-FM) signals as follows

119906119896 (119905) = 119860119896 (119905) cos (120601119896 (119905)) (1)

The time series ofvibration signal

Initialization parameter

Optimal parametercombination obtained by

PSO method

The signal preprocessingby the proposed improved

VMD

Numerical simulationsignal analysis

Envelope spectrumanalysis

Fault featureextraction

The optimal modecomponent selection by

correlated kurtosis

Measured signal analysis

Figure 1 The flowchart about the proposed method

where 120601119896(119905) is a nondecreasing function thus 1206011015840119896(119905) ge 0

the envelope is nonnegative 119860119896(119905) ge 0 Additionally thechange of envelope 119860119896(119905) and instantaneous frequency 1206011015840

119896(119905)

are much slower than 120601119896(119905) Therefore the mode component119906119896(119905) can be regarded as a pure harmonic signal withamplitude 119860119896(119905) and instantaneous frequency 1206011015840

119896(119905)

The input signal is decomposed by VMD method intothe mode component of a specified scale according to thesubjective setting scale assuming that each mode is a finitebandwidth signal with a pulse as the center In order toevaluate the bandwidth of each mode and construct theconstraintmodel of the variational problemVMDfirstly usesthe Hilbert transform to obtain the single spectrum of eachmode and then transfers them to the fundamental frequencyby exponential correction The bandwidth of each mode isobtained through Gauss smooth demodulation signal finallywhich is called as 1198712 norm squared of the gradient Therebythe constrained variational problem is given as follows

min119906119896120596119896

sum

119896

10038171003817100381710038171003817100381710038171003817

120597119905 [(120575 (119905) +

119895

120587119905

) lowast 119906119896 (119905)] 119890minus11989512059611989611990510038171003817100381710038171003817100381710038171003817

2

2

(2)

Subject to sum

119896

119906119896 = 119891 (3)

where 119891 is the original signal 120575 is the Dirac distribution 119896is number of modes 119906119896 fl 1199061 1199062 119906119896 denotes each



119896fl sum

119896



119871 (119906119896 120596119896 120582)

fl 120572sum119896

10038171003817100381710038171003817100381710038171003817

120597119905 [(120575 (119905) +

119895

120587119905

) lowast 119906119896 (119905)] 119890minus11989512059611989611990510038171003817100381710038171003817100381710038171003817

2

2

+

1003817100381710038171003817100381710038171003817100381710038171003817

119891 (119905) minus sum

119896

119906119896 (119905)

1003817100381710038171003817100381710038171003817100381710038171003817

2

2

+⟨120582 (119905) 119891 (119905) minussum

119896

119906119896 (119905)⟩

(4)



119896 1205821 119899 larr 0


119899+1

119896(120596) =

119891 (120596) minus sum119894 =119896

119894 (120596) +120582 (120596) 2

1 + 2120572 (120596 minus 120596119896)2

120596119899+1

119896=

int

infin

01205961003816100381610038161003816119896 (120596)

1003816100381610038161003816

2119889120596

int

infin

0

1003816100381610038161003816119896 (120596)

1003816100381610038161003816

2119889120596

120582119899+1= 120582119899+ 120591(119891 minussum

119896

119906119899+1

119896)

(5)


119896119899+1

119896minus

119899

1198962

2119899

1198962









119894119889= 120596120592119896

119894119889+ 1198881120578(119901

119896

119894119889minus

119909119896

119894119889) + 1198882120578(119892

119896

119889minus 119909119896

119894119889) 119909119896+1119894119889= 119909119896

119894119889+ 120592119896+1








119862 =

sum119879

119899=1(119903 (119899) minus 119903) (119910 (119899) minus 119910)

[sum119879

119899=1(119903 (119899) minus 119903)

2sum119879

119899=1(119910 (119899) minus 119910)

2]

12 (6)




fit fun = mean (119862)var (119862)

(7)




1199091 = sin (21205871198911119905)

1199092 = cos (21205871198912119905)

1199093 = sin (21205871198913119905 + cos (21205871198914119905))

(8)

and then

119904 = 1199091 + 1199092 + 1199093 (9)




0 02 04 06 08 1minus3

minus2

minus1

0

1

2

3

Time (s)

Am

plitu

de












0 5 10 15 20100

150

200

250

300

350

400

450

500


Fitn

ess v

alue


minus101


0 005 01 015 02

0 005 01 015 02

minus101



01


0 005 01 015 02

0 005 01 015 02

01


minus1

minus1


01

Original FM signal

0 005 01 015 02

0 005 01 015 02

01


minus1

minus1











minus2

0

2IM

F1

minus2

0

2

IMF2

minus05

0

05

IMF3

minus01

0

01

IMF4

minus02

0

02

IMF5

minus5

0

5

IMF6

0 05 1minus5

0

5

IMF7

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1minus001

0

001

IMF8

times10minus3

times10minus3

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)







5 Conclusions



2 4 6 8 10 12 14 16 18 20095

0955

096

0965

097

0975

098

0985

099

0995

1Si

mila

rity

of co

nstr

ucte

d an

d or

igin

al si

gnal

SNR (dB)


2 4 6 8 10 12 14 16 18 20094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)


2 4 6 8 10 12 14 16 18 20093

094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)







0 02 04 06 08 1minus15

minus10

minus5

0

5

10

15

20

Am

plitu

de

Time (s)


0 200 400 600 800 10000

005

01

015

02

025

03

035

04

045

05

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

fo



0 10 20 30 40 505514

5515

5516

5517

5518

5519

552

5521

5522

5523

5524


Fitn

ess v

alue


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35


The v

alue

of c

orre

lated

kur

tosis

times10minus28



0 200 400 600 800 10000

005

01

015

02

025

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

5fo6fo

7fo

8fo

9fo 10fo

fc

fo


0 200 400 600 800 10000

001

002

003

004

005

006

007

008

009

Am

plitu

de

Frequency (Hz)

2fofo



Competing Interests


Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119896fl sum

119896



119871 (119906119896 120596119896 120582)

fl 120572sum119896

10038171003817100381710038171003817100381710038171003817

120597119905 [(120575 (119905) +

119895

120587119905

) lowast 119906119896 (119905)] 119890minus11989512059611989611990510038171003817100381710038171003817100381710038171003817

2

2

+

1003817100381710038171003817100381710038171003817100381710038171003817

119891 (119905) minus sum

119896

119906119896 (119905)

1003817100381710038171003817100381710038171003817100381710038171003817

2

2

+⟨120582 (119905) 119891 (119905) minussum

119896

119906119896 (119905)⟩

(4)



119896 1205821 119899 larr 0


119899+1

119896(120596) =

119891 (120596) minus sum119894 =119896

119894 (120596) +120582 (120596) 2

1 + 2120572 (120596 minus 120596119896)2

120596119899+1

119896=

int

infin

01205961003816100381610038161003816119896 (120596)

1003816100381610038161003816

2119889120596

int

infin

0

1003816100381610038161003816119896 (120596)

1003816100381610038161003816

2119889120596

120582119899+1= 120582119899+ 120591(119891 minussum

119896

119906119899+1

119896)

(5)


119896119899+1

119896minus

119899

1198962

2119899

1198962









119894119889= 120596120592119896

119894119889+ 1198881120578(119901

119896

119894119889minus

119909119896

119894119889) + 1198882120578(119892

119896

119889minus 119909119896

119894119889) 119909119896+1119894119889= 119909119896

119894119889+ 120592119896+1








119862 =

sum119879

119899=1(119903 (119899) minus 119903) (119910 (119899) minus 119910)

[sum119879

119899=1(119903 (119899) minus 119903)

2sum119879

119899=1(119910 (119899) minus 119910)

2]

12 (6)




fit fun = mean (119862)var (119862)

(7)




1199091 = sin (21205871198911119905)

1199092 = cos (21205871198912119905)

1199093 = sin (21205871198913119905 + cos (21205871198914119905))

(8)

and then

119904 = 1199091 + 1199092 + 1199093 (9)




0 02 04 06 08 1minus3

minus2

minus1

0

1

2

3

Time (s)

Am

plitu

de












0 5 10 15 20100

150

200

250

300

350

400

450

500


Fitn

ess v

alue


minus101


0 005 01 015 02

0 005 01 015 02

minus101



01


0 005 01 015 02

0 005 01 015 02

01


minus1

minus1


01

Original FM signal

0 005 01 015 02

0 005 01 015 02

01


minus1

minus1











minus2

0

2IM

F1

minus2

0

2

IMF2

minus05

0

05

IMF3

minus01

0

01

IMF4

minus02

0

02

IMF5

minus5

0

5

IMF6

0 05 1minus5

0

5

IMF7

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1minus001

0

001

IMF8

times10minus3

times10minus3

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)







5 Conclusions



2 4 6 8 10 12 14 16 18 20095

0955

096

0965

097

0975

098

0985

099

0995

1Si

mila

rity

of co

nstr

ucte

d an

d or

igin

al si

gnal

SNR (dB)


2 4 6 8 10 12 14 16 18 20094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)


2 4 6 8 10 12 14 16 18 20093

094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)







0 02 04 06 08 1minus15

minus10

minus5

0

5

10

15

20

Am

plitu

de

Time (s)


0 200 400 600 800 10000

005

01

015

02

025

03

035

04

045

05

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

fo



0 10 20 30 40 505514

5515

5516

5517

5518

5519

552

5521

5522

5523

5524


Fitn

ess v

alue


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35


The v

alue

of c

orre

lated

kur

tosis

times10minus28



0 200 400 600 800 10000

005

01

015

02

025

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

5fo6fo

7fo

8fo

9fo 10fo

fc

fo


0 200 400 600 800 10000

001

002

003

004

005

006

007

008

009

Am

plitu

de

Frequency (Hz)

2fofo



Competing Interests


Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












fit fun = mean (119862)var (119862)

(7)




1199091 = sin (21205871198911119905)

1199092 = cos (21205871198912119905)

1199093 = sin (21205871198913119905 + cos (21205871198914119905))

(8)

and then

119904 = 1199091 + 1199092 + 1199093 (9)




0 02 04 06 08 1minus3

minus2

minus1

0

1

2

3

Time (s)

Am

plitu

de












0 5 10 15 20100

150

200

250

300

350

400

450

500


Fitn

ess v

alue


minus101


0 005 01 015 02

0 005 01 015 02

minus101



01


0 005 01 015 02

0 005 01 015 02

01


minus1

minus1


01

Original FM signal

0 005 01 015 02

0 005 01 015 02

01


minus1

minus1











minus2

0

2IM

F1

minus2

0

2

IMF2

minus05

0

05

IMF3

minus01

0

01

IMF4

minus02

0

02

IMF5

minus5

0

5

IMF6

0 05 1minus5

0

5

IMF7

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1minus001

0

001

IMF8

times10minus3

times10minus3

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)







5 Conclusions



2 4 6 8 10 12 14 16 18 20095

0955

096

0965

097

0975

098

0985

099

0995

1Si

mila

rity

of co

nstr

ucte

d an

d or

igin

al si

gnal

SNR (dB)


2 4 6 8 10 12 14 16 18 20094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)


2 4 6 8 10 12 14 16 18 20093

094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)







0 02 04 06 08 1minus15

minus10

minus5

0

5

10

15

20

Am

plitu

de

Time (s)


0 200 400 600 800 10000

005

01

015

02

025

03

035

04

045

05

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

fo



0 10 20 30 40 505514

5515

5516

5517

5518

5519

552

5521

5522

5523

5524


Fitn

ess v

alue


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35


The v

alue

of c

orre

lated

kur

tosis

times10minus28



0 200 400 600 800 10000

005

01

015

02

025

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

5fo6fo

7fo

8fo

9fo 10fo

fc

fo


0 200 400 600 800 10000

001

002

003

004

005

006

007

008

009

Am

plitu

de

Frequency (Hz)

2fofo



Competing Interests


Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20100

150

200

250

300

350

400

450

500


Fitn

ess v

alue


minus101


0 005 01 015 02

0 005 01 015 02

minus101



01


0 005 01 015 02

0 005 01 015 02

01


minus1

minus1


01

Original FM signal

0 005 01 015 02

0 005 01 015 02

01


minus1

minus1











minus2

0

2IM

F1

minus2

0

2

IMF2

minus05

0

05

IMF3

minus01

0

01

IMF4

minus02

0

02

IMF5

minus5

0

5

IMF6

0 05 1minus5

0

5

IMF7

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1minus001

0

001

IMF8

times10minus3

times10minus3

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)







5 Conclusions



2 4 6 8 10 12 14 16 18 20095

0955

096

0965

097

0975

098

0985

099

0995

1Si

mila

rity

of co

nstr

ucte

d an

d or

igin

al si

gnal

SNR (dB)


2 4 6 8 10 12 14 16 18 20094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)


2 4 6 8 10 12 14 16 18 20093

094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)







0 02 04 06 08 1minus15

minus10

minus5

0

5

10

15

20

Am

plitu

de

Time (s)


0 200 400 600 800 10000

005

01

015

02

025

03

035

04

045

05

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

fo



0 10 20 30 40 505514

5515

5516

5517

5518

5519

552

5521

5522

5523

5524


Fitn

ess v

alue


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35


The v

alue

of c

orre

lated

kur

tosis

times10minus28



0 200 400 600 800 10000

005

01

015

02

025

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

5fo6fo

7fo

8fo

9fo 10fo

fc

fo


0 200 400 600 800 10000

001

002

003

004

005

006

007

008

009

Am

plitu

de

Frequency (Hz)

2fofo



Competing Interests


Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus2

0

2IM

F1

minus2

0

2

IMF2

minus05

0

05

IMF3

minus01

0

01

IMF4

minus02

0

02

IMF5

minus5

0

5

IMF6

0 05 1minus5

0

5

IMF7

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1minus001

0

001

IMF8

times10minus3

times10minus3

Time (s)

0 05 1Time (s)

0 05 1Time (s)

0 05 1Time (s)







5 Conclusions



2 4 6 8 10 12 14 16 18 20095

0955

096

0965

097

0975

098

0985

099

0995

1Si

mila

rity

of co

nstr

ucte

d an

d or

igin

al si

gnal

SNR (dB)


2 4 6 8 10 12 14 16 18 20094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)


2 4 6 8 10 12 14 16 18 20093

094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)







0 02 04 06 08 1minus15

minus10

minus5

0

5

10

15

20

Am

plitu

de

Time (s)


0 200 400 600 800 10000

005

01

015

02

025

03

035

04

045

05

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

fo



0 10 20 30 40 505514

5515

5516

5517

5518

5519

552

5521

5522

5523

5524


Fitn

ess v

alue


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35


The v

alue

of c

orre

lated

kur

tosis

times10minus28



0 200 400 600 800 10000

005

01

015

02

025

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

5fo6fo

7fo

8fo

9fo 10fo

fc

fo


0 200 400 600 800 10000

001

002

003

004

005

006

007

008

009

Am

plitu

de

Frequency (Hz)

2fofo



Competing Interests


Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2 4 6 8 10 12 14 16 18 20095

0955

096

0965

097

0975

098

0985

099

0995

1Si

mila

rity

of co

nstr

ucte

d an

d or

igin

al si

gnal

SNR (dB)


2 4 6 8 10 12 14 16 18 20094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)


2 4 6 8 10 12 14 16 18 20093

094

095

096

097

098

099

1

Sim

ilarit

y of

cons

truc

ted

and

orig

inal

sign

al

SNR (dB)







0 02 04 06 08 1minus15

minus10

minus5

0

5

10

15

20

Am

plitu

de

Time (s)


0 200 400 600 800 10000

005

01

015

02

025

03

035

04

045

05

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

fo



0 10 20 30 40 505514

5515

5516

5517

5518

5519

552

5521

5522

5523

5524


Fitn

ess v

alue


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35


The v

alue

of c

orre

lated

kur

tosis

times10minus28



0 200 400 600 800 10000

005

01

015

02

025

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

5fo6fo

7fo

8fo

9fo 10fo

fc

fo


0 200 400 600 800 10000

001

002

003

004

005

006

007

008

009

Am

plitu

de

Frequency (Hz)

2fofo



Competing Interests


Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 02 04 06 08 1minus15

minus10

minus5

0

5

10

15

20

Am

plitu

de

Time (s)


0 200 400 600 800 10000

005

01

015

02

025

03

035

04

045

05

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

fo



0 10 20 30 40 505514

5515

5516

5517

5518

5519

552

5521

5522

5523

5524


Fitn

ess v

alue


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35


The v

alue

of c

orre

lated

kur

tosis

times10minus28



0 200 400 600 800 10000

005

01

015

02

025

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

5fo6fo

7fo

8fo

9fo 10fo

fc

fo


0 200 400 600 800 10000

001

002

003

004

005

006

007

008

009

Am

plitu

de

Frequency (Hz)

2fofo



Competing Interests


Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 200 400 600 800 10000

005

01

015

02

025

Frequency (Hz)

Am

plitu

de

2fo

3fo

4fo

5fo6fo

7fo

8fo

9fo 10fo

fc

fo


0 200 400 600 800 10000

001

002

003

004

005

006

007

008

009

Am

plitu

de

Frequency (Hz)

2fofo



Competing Interests


Acknowledgments



References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article a fault diagnosis scheme for rolling...

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