Research ArticleRolling Bearing Fault Diagnosis Based on CEEMD andTime Series Modeling

Liye Zhao12 Wei Yu12 and Ruqiang Yan1

1 School of Instrument Science and Engineering Southeast University Nanjing 210096 China2 Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology Ministry of Education Nanjing 210096 China

Correspondence should be addressed to Ruqiang Yan ruqiangseueducn

Received 2 May 2014 Accepted 19 June 2014 Published 7 July 2014

Academic Editor Xuefeng Chen

Copyright copy 2014 Liye Zhao 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

Accurately identifying faults in rolling bearing systems by analyzing vibration signals which are often nonstationary is challengingTo address this issue a new approach based on complementary ensemble empirical mode decomposition (CEEMD) and timeseries modeling is proposed in this paper This approach seeks to identify faults appearing in a rolling bearing system using properautoregressive (AR) model established from the nonstationary vibration signal First vibration signals measured from a rollingbearing test system with different defect conditions are decomposed into a set of intrinsic mode functions (IMFs) by means ofthe CEEMD method Second vibration signals are filtered with calculated filtering parameters Third the IMF which is closelycorrelated to the filtered signal is selected according to the correlation coefficient between the filtered signal and each IMF and thenthe ARmodel of the selected IMF is established Subsequently the ARmodel parameters are considered as the input feature vectorsand the hidden Markov model (HMM) is used to identify the fault pattern of a rolling bearing Experimental study performed ona bearing test system has shown that the presented approach can accurately identify faults in rolling bearings

1 Introduction

Rolling element bearing failure is one of the foremost causesof failures in rotating machinery and such failure mayresult in costly production loss and catastrophic accidentsEarly detection and diagnosis of bearing faults while themachine is still in operation can help to avoid abnormalevent progression and to reduce productivity loss [1] Sincestructural defects can cause changes of the bearing dynamiccharacteristics as manifested in vibrations vibration-basedanalysis has long been established as a commonly usedtechnique for diagnosing bearing faults [2] However somenonlinear factors such as clearance friction and stiffnessaffect complexity of the vibration signals thus it is difficultto make an accurate evaluation on the working condition ofrolling bearings only through analysis in time or frequencydomain as it does traditionally [3]

In order to overcome limitations of the traditional tech-niques autoregressive (AR) model has been successfullyapplied to extracting features from vibration signals for fault

diagnosis in recent years [4ndash6] This is because AR modelis a time series analysis method whose parameters compriseimportant information of the system condition and an accu-rate AR model can reflect the characteristics of a dynamicsystem [7] For example AR model was combined with afuzzy classifier for fault diagnosis in vehicle transmissiongear [8] Three distinct techniques of autoregressive mod-eling were compared for their performance and reliabilityunder conditions of various bearings signal lengths [9] Adiagnosis method based on the AR model and continuousHMM has also been used to monitor and diagnose therolling bearing working conditions [10] However when theAR model is applied directly to the nonstationary bearingvibration signals the analysis results are imperfect since theestimation method of the autoregression parameters of theAR model is no longer applicable Because the vibrationsignal is nonstationary whereas the AR model is suitablefor stationary signal processing it is therefore necessaryto preprocess the vibration signals before the AR model isgenerated

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 101867 13 pageshttpdxdoiorg1011552014101867

2 Mathematical Problems in Engineering

Empirical mode decomposition (EMD) is an adaptivetime-frequency signal processing method [11] With EMD asignal is decomposed into a series of intrinsicmode functions(IMFs) according to its own characteristics [12] For examplea new fault feature extraction approach based on EMDmethod and AR model was used to process vibration signalsof roller bearings [3] However when the EMD method isapplied to the nonstationary signals containing intermittentsignal components the original signal cannot be decomposedaccurately because of the problem of mode mixing [13] Toalleviate mode mixing Wu and Huang developed ensembleempiricalmode decomposition (EEMD) to improve EMDByadding noise to the original signal and calculating the meansof IMFs repeatedly EEMD is more accurate and effectivefor signal decomposition [13] Although the EEMD methodhas effectively resolved the mode-mixing problem it is timeconsuming for implementing the large enough ensemblemean That is to say the algorithm efficiency will be greatlyreduced Aiming at solving this problem the complementaryensemble EMD (CEEMD) method is proposed [14] In thisapproach the residue of added white noises can be extractedfrom the mixtures of data and white noises via pairs ofcomplementary ensemble IMFs with positive and negativeadded white noises The CEEMD method has the sameperformance as the EEMD but the computational efficiencyis greatly improved

In this paper we combine the advantages of CEEMDand time series model and propose a new method based onCEEMD and AR model for rolling bearing fault diagnosisThe CEEMD is used as the pretreatment to filter the signaland extract the IMF which is closely correlated to thefiltered signal and then the AR model of the selected IMFis established The AR model parameters are used as thefeature vectors to a classifier where the hiddenMarkovmodel(HMM) is used to identify the fault pattern of a rollingbearing The rest of this paper is organized as follows InSection 2 the review of the fault diagnosis method based onAR model is presented and the proposed method for rollingbearing fault diagnosis is discussed The evaluations andexperiments are presented in Section 3 Finally concludingremarks are drawn in Section 4

2 Theoretical Framework

21 Time Series Modeling Autoregressive moving average(ARMA)model is the representative time seriesmodel whichcan be expressed in linear difference equation form as

119909119905+ 1205931119909119905minus1

+ sdot sdot sdot + 120593119899119909119905minus119899

= 119886119905+ 1205791119886119905minus1

+ sdot sdot sdot + 120579119898

119886119905minus119898

(1)

where 119899 and 119898 are the parameters of the ARMA (119899 119898)model 119909

119905is zero mean stationary random sequence 119886

119905is

white noise sequence and 120593119894and 120579

119895are model parameters

to be estimated The parameters of 120593119894and 120579

119895are estimated

by the time sequence of 119909119905(119905 = 1 2 3 ) which is called

the time series modeling If 120593119894

= 0 the ARMA (119899 119898) modelwill degrade to 119898 order MA(119898) model and if 120579

119894= 0 the

ARMA (119899 119898) model will degrade as 119899 order AR (119899) model in(1) The AR model is stable and its structure is simpler than

ARMA model Therefore the AR model will be establishedfor characterizing the rolling bearing vibration signal if theprecision of the model is enough for expressing the systemwhich is expressed as

119909119905

= 1205931119909119905minus1

+ sdot sdot sdot + 120593119899119909119905minus119899

+ 119886119905 (2)

where 119905 = 1 2 119873 119873 is the length of the time series 119909119905 119899

is the order number and 119886119905

sim NID(0 1205902

119886)The 120590

2

119886is expressed

as

1205902

120572=

1

119873 minus 119899

119899

119899

sum

119905=119899+1

(119909119905minus

119899

sum

120580=1

120593120580119909119905minus1

)

2

(3)

It is critical to determine the order number of the ARmodel because the accuracy of the order not only affects theaccuracy of identification of the system but also influencesthe stability of the system In order to estimate the order oftheARmodel correctly FPE criterion BIC criterion andAICcriterion are usually used [15] and they are expressed as

FPE criterion

FPE (119899) =

119873 + 119899

119873 minus 119899

1205902

119886 (4)

BIC criterion

AIC (119899) = 119873 ln1205902

119886+ 2119899 (5)

AIC criterion

BIC (119899) = 119873 ln1205902

119886+ 119899 ln119873 (6)

After the model order is determined the nonlinear leastsquares method can be used to estimate model parametersand then the AR model with specific parameters is estab-lished

22 Complementary Ensemble Empirical Mode Decomposi-tion Complementary ensemble empirical mode decomposi-tion (CEEMD) is an improved algorithm based on empiricalmode decomposition (EMD) Through EMD process anycomplex time series can be decomposed into finite numbersof intrinsic mode functions (IMFs) and each IMF reflectsthe dynamic characteristic of the original signal The IMFcomponent must satisfy two conditions (a) the number ofpoles and zeros is either equal to each other or differs at mostby one (b) the upper and lower envelopes must be locallysymmetric about the timeline The basic principle of EMDmethod is to decompose the original signal 119909(119905) into the formas shown in (7) by continuously eliminating the mean of theupper and lower envelope connected with the minimum andmaximum of the signal [16] Consider

119909119905

=

119899

sum

119894=1

imf119894(119905)

+ 119903119899(119905)

(7)

where 119909119905is the vibration signal imf

119894(119905)is the IMF component

including different frequency bands ranging fromhigh to low

Mathematical Problems in Engineering 3

0 0002 0004 0006 0008 001

01

Signals (V)

0 0002 0004 0006 0008 001

02

0 0002 0004 0006 0008 0010

05

0 0002 0004 0006 0008 001

05

Time (s)

s(t)

s1(t)

s2(t)

s3(t)

minus1

minus2

minus5

Figure 1 Signal waveforms

and 119903119899(119905)

is the residue of the decomposition process which isthe mean trend of 119909

119905

The EMD method is a kind of adaptive local analysismethod with each IMF highlighting the local features ofthe data However EMD decomposition results often sufferfrom mode mixing which is defined as either a single IMFconsisting of widely disparate scales or a signal residing indifferent IMF components [17] To make it clear a simulatedsignal 119904(119905) consists of a Gaussian-type impulse interference1199041(119905) and a cosine component with 500Hz frequency 119904

2(119905)

and a trend term 1199043(119905) is used as an example The equation of

the simulated signal is expressed as

119904 (119905) = sin (2120587120572119905) 119890minus((119905minus1199050)

2120590)

+ cos (2120587120573119905) + 50119905 (8)

where 120572 = 3000 120573 = 500 and 120590 = 106

Thewaveformof the simulated signal is shown in Figure 1and the corresponding EMD results for the signal 119904(119905) areshown in Figure 2 where the mode mixing happens

To overcome the problem of mode mixing the ensembleempirical mode decomposition (EEMD) was proposed [18]where Gaussian white noises with finite amplitude are addedto the original signal during the entire decomposition pro-cess Due to the uniform distribution statistical characteris-tics of the white noise the signal with white noise becomescontinuous in different time scales and no missing scales arepresent As a result mode mixing is effectively eliminated bythe EEMD process [18] The EEMD decomposition result ofsignal 119904(119905) is shown in Figure 3 where the added white noiseamplitude is 025 times the original signal standard deviationand the number of decompositions is 200 times

It should be noted that during the EEMD process eachindividual trial may produce noisy results but the effectof the added noise can be suppressed by large numberof ensemble mean computations This would be too timeconsuming to implement An improved algorithm namedcomplementary ensemble mode decomposition (CEEMD)

0 0002 0004 0006 0008 001

0

1

IMF1

Signals (V)

0 0002 0004 0006 0008 001

0

5

IMF2

0 0002 0004 0006 0008 001

0

5

IMF3

0 0002 0004 0006 0008 001

0

1

IMF4

Time (s)

minus1

minus1

minus5

minus5

Figure 2 The decomposition result by EMD

0 0002 0004 0006 0008 001

0

1

IMF1

Signals (V)

0 0002 0004 0006 0008 001

0

02

IMF2

0 0002 0004 0006 0008 001

0

01

IMF3

0 0002 0004 0006 0008 001

0

2

IMF4

Time (s)

minus1

minus02

minus01

minus2

Figure 3 The decomposition result by EEMD

is suggested to improve the computation efficiency In thisalgorithm the residue of the added white noises can beextracted from themixtures of data and white noises via pairsof complementary ensemble IMFs with positive and negativeadded white noises Although this new approach yields IMFwith a similar RMSnoise to EEMD it eliminates residue noisein the IMFs and overcomes the problem ofmodemixing withmuch more efficiency [14] The procedure on implementingCEEMD is shown below

(a) 1199091and 1199092are constructed by adding a pair of opposite

phase Gaussian white noises 119909119899with the same ampli-

tude Then 1199091

= 119909 + 119909119899and 119909

2= 119909 minus 119909

119899

4 Mathematical Problems in Engineering

(b) 1199091and 119909

2are decomposed by EMD only a few times

and IMF1199091

and IMF1199092

are ensemble means of thecorresponding IMF generated from each trial

(c) the average of corresponding component in IMF1199091

and IMF1199092

is calculated as the CEEMD decomposi-tion results that is

IMF =

(IMF1199091

+ IMF1199092

)

2

(9)

The flow chart of CEEMD is shown in Figure 4 where 119899 is thedecomposition trials

Figure 5 is the decomposition result by CEEMD for thesignal 119904(119905) As compared to the result shown in Figure 3the decomposition accuracies of EEMD and CEEMD areconsistent while EEMD takes 162 s and CEEMD only needs013 s

23 Fault Diagnosis Based on CEEMD and Time Series ModelBased on CEEMD and time series model a hybrid faultdiagnosis approach can be designed The hybrid approachcombines the advantages of CEEMD method in the non-stationary signal decomposition with the ability of timeseries modeling in feature extraction The flow chart of thedeveloped approach is shown in Figure 6

The main steps are as follows

Step 1 The rolling bearing vibration signal is sampled andthen decomposed by CEEMD with the process shown inFigure 4

Step 2 The product of energy density and average periodof the IMFs which is a constant value according to [19] iscalculated using (10) and parameter 119877119875

119895is calculated using

(11) Then the signal is filtered by comparing the parameter119877119875119895and the given threshold value that is to say when 119877119875

119895⩾

1 the previous 119895 minus 1 IMFs with the trend term need to beremoved as noise and to rebuild the residual IMFs as filteredsignal [19 20]

119875119895

= 119864119895

times 119879119895 (10)

119877119875119895

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119875119895

minus ((1 (119895 minus 1)) sum119895minus1

119894=1119875119895)

(1 (119895 minus 1)) sum119895minus1

119894=1119875119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119895 ge 2) (11)

where 119864119895

= (1119873) sum119873

119894=1[119860119895(119894)]2 is the energy density of the

119895th IMF 119879119895

= 2119873119874119895is the average period of the 119895th IMF 119873

is the length of each IMF 119860119895is the amplitude of the 119895th IMF

and 119874119895is the total number of extreme points of 119895th IMF

Step 3 Equation (12) is used to calculate the correlationcoefficient between the filtered signal and each IMF and theIMFwhich is closely correlated to the filtered signal is selectedfor AR modeling [21]

120588119909119910

=

sum119873

119896=1119909 (119896) 119910 (119896)

[sum119873

119896=1119909(119896)2

sum119873

119896=1119910(119896)2]

12 (12)

Input signal

Add two reverse white noises

final IMFs of CEEMD

Calculate IMFs with EMD

Calculate the means of above two IMFs

No

Yes

M = n

Calculate (IMF1+ IMF2)2 as the

Figure 4 Decomposition flow chart of CEEMD

0 0002 0004 0006 0008 001

0

1

IMF1

Signals (V)

0 0002 0004 0006 0008 001

0

05

IMF2

0 0002 0004 0006 0008 001

0

02

IMF3

0 0002 0004 0006 0008 001

0

2

IMF4

Time (s)

minus1

minus05

minus02

minus2

Figure 5 The decomposition result by CEEMD

Step 4 The least square method is used to estimate theparameters vectors of theARmodel established in Step 3 andthe parameters vectors are considered as the model featurevector

Step 5 After scalar quantization by index calculation formulaof Lloyds algorithm in (13) [22] the feature vector is used totrain the HMM of each bearing working condition

indx (119909) =

1 119909 le partition (119894)

119894 + 1 partition (119894) lt 119909 le partition (119894 + 1)

119873 partition (119873 minus 1) lt 119909

(13)

Mathematical Problems in Engineering 5

HMM1

HMM training

HMM2 Comparisonprobability

Classification result

Vectorsquantization

Vibration signals

Signal decomposed by CEEMD

Calculate correlation coefficient

Feature vectors

IMF selected for modeling

Model parameter estimation

Feature vectors extracted

Signal filtered

Calculate parameter RP

Feature vectors extraction Fault diagnosis classification

Sample feature vectors

HMMn

Figure 6 The flow chart of the proposed method

where 119873 is the length of the codebook vector partition (119894)is the partition vector with the length of 119873 minus 1 and 119909 is thefeature vector for scalar quantization

Step 6 A test vibration signal can then be acquired fordiagnosis and themodel feature vector is first extractedAfterscalar quantization the feature vector is put into the well-trained HMMs and the corresponding HMM which has themaximum probability is regarded as the classification result[23]

3 Evaluation of the Method Based onCEEMD and AR Model

31 Evaluation Using Simulated Signals To demonstrate thevalidity of the method proposed in this study three signals1199091(119905) 1199092(119905) and 119909

3(119905) are simulated as shown in Figure 7The

signal 1199091(119905) consists of a Gaussian-type impulse interference

a cosine component with 10Hz frequency a trend term andwhite noise The signal 119909

2(119905) consists of a Gaussian-type

impulse interference a square wave with 65 duty ratio atrend term and white noise The signal 119909

3(119905) consists of a

Gaussian-type impulse interference a sawtooth wave with15Hz frequency a trend term and white noise

Figure 8 shows the results of the CEEMD of signals 1199091(119905)

1199092(119905) and 119909

3(119905) Correlation coefficients between filtered

signal and each IMF are illustrated in Table 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0

4Signals (V)

0

4

0

4

Time (s)

x1(t)

x2(t)

x3(t)

minus4

minus4

minus4

Figure 7 Signal waveforms of 1199091(119905) 1199092(119905) and 119909

3(119905)

It can be seen in Table 1 that the IMF which is closelycorrelated to the filtered signal is IMF5 for both signal1199091(119905) and signal 119909

3(119905) and IMF6 for signal 119909

2(119905) They are

used to construct the AR models and the correspondingfeature vectors are estimated as shown in Table 2 After scalarquantization the feature vectors are used to train the HMMfor signal classification

6 Mathematical Problems in Engineering

02

IMF1

Signals (V)

01

IMF2

005

IMF3

01

IMF4

02

IMF5

002

IMF6

001

IMF7

00204

Res

Time (s)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

minus2

minus1

minus05

minus1

minus2

minus01

minus02

(a) 1199091(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

05

IMF6

01

IMF7

005

1

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus5

minus1

(b) 1199092(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

002

IMF6

001

IMF7

002

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus02

minus01

minus02

(c) 1199093(119905)

Figure 8 The decomposition results by CEEMD

Mathematical Problems in Engineering 7

Table 1 Correlation coefficients between filtered signal and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

1199091(119905) minus00031 minus00009 00371 04096 09668 02428 01273 minus00448

1199092(119905) 00051 00004 00435 02111 04695 08887 07214 minus00201

1199093(119905) minus00234 minus00154 00286 05900 08953 01649 01887 minus00210

Table 2 Model parameter estimation results

Signal Model parameter1205931

1205932

1205933

1205934

1205935

1205936

1199091(119905) 47183 minus91103 92034 minus51408 15207 minus01914

1199092(119905) 48894 minus98269 103945 minus61194 19153 minus02531

1199093(119905) 48718 minus99955 111616 minus72529 26430 minus04282

Table 3 Signal classification results

Signal type Test sample Classification results Classification rate [] Overall classificationrate []

1199091(119905) 119909

2(119905) 119909

3(119905)

1199091(119905) 20 19 1 0 95

9671199092(119905) 20 0 19 1 95

1199093(119905) 20 0 0 20 100

A total of 90 feature vectors were collected from threegroups of signals using the proposed approach One-third ofthe feature vectors in each condition were used for trainingthe classifier and others were used for testing The results ofthe signal classification are listed in Table 3

Results in Table 3 indicate that the presented methodbased on CEEMD and time series modeling can effectivelyidentify different signals and the overall classification rate is967 For the purpose of comparison the signal classifica-tion rates use themethod based on time seriesmodeling onlyand the method based on EMD and time series modelingis also calculated 883 and 933 classification rates areobtained respectively It is obvious that efficiency of the signalclassification method proposed in this paper is improved to acertain extent

32 Evaluation Using Experimental Data In order to illus-trate the practicability and effectiveness of the proposedmethod a bearing fault data set from the electrical engi-neering laboratory of Case Western Reserve University isanalyzed [24] The data set is acquired from the test standshown in Figure 9 where it consists of a 2 hp motor atorque transducer a dynamometer and control electronicsThe test bearings support the motor shaft which is the deepgroove ball bearings with the type of 6205-2RS JEMSKFVibration data was collected at 12000 samples per secondusing accelerometers which are attached to the housing withmagnetic bases The motor load level was controlled by thefan in the right side of Figure 9

2 hp motor Torque

transducerDynamometer

Figure 9 Bearing test stand

Figure 10 illustrates representative waveforms of the sam-ple vibration signals measured from the test bearings underfour initial conditions (a) signal from a healthy bearing (b)signal from a bearing with inner ring defect (c) signal froma bearing with rolling element defect and (d) signal from abearing with outer ring defect These signals were measuredunder 0 hpmotor load with themotor speed of 1797 rpmThedecomposed IMFs of these signals are shown in Figure 11

Correlation coefficients calculated between the filteredsignal and each IMF are shown in Table 4

The IMF which is closely correlated to the filtered signalis IMF2 for signal (a) and IMF1 for signals (b) (c) and (d)respectivelyThese IMFs are used for ARmodel constructionThe model order estimation curves of the four conditionsbased on the principle of FPE criterion are shown in Figure 12We can see that when the model order is 6 each modelrsquos

8 Mathematical Problems in Engineering

0 005 01 015 02 025

0

05

minus05

Signals (V)

(a)

0 005 01 015 02 025

0

2

minus2

(b)

0 005 01 015 02 025

0

1

minus1

(c)

0 005 01 015 02 025

0

5

Time (s)

minus5

(d)

Figure 10 Vibration signal waveforms of different conditions

Table 4 Correlation coefficients between filtered signals and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

(a) 04135 07538 04381 04880 04356 01792 00971 minus00056(b) 08794 04275 02583 01337 00421 00285 minus00009 minus00074(c) 09509 02180 02325 01337 00821 00350 minus00017 00009(d) 09878 01267 00636 00509 00136 00060 minus00008 minus00068

Table 5 Model parameter estimation results

Signal Model parameter1205931 1205932 1205933 1205934 1205935 1205936

(a) 31280 minus47797 42245 minus21489 04241 00356(b) 02084 minus13585 05142 minus06356 03471 minus00422(c) 01335 minus16472 03941 minus08473 02142 minus01011(d) minus01172 minus12159 01178 minus01283 01467 02533

residual tends to be stable Therefore the model order isselected as 6 and the results of parameters estimation arelisted in Table 5

The parameters in Table 5 were quantified by Lloydsalgorithm in (12) as feature vectors for training the HMMs ofdifferent conditions The results of quantization are revealedin Figure 13

A total of 160 feature vectors were collected from thefour conditions half of the feature vectors were used fortraining the classifier and others for signal classification andthe classification results are listed in Table 6 Out of 80 testfeature vectors just two cases were not correctly classifiedand the overall classification rate is 975

For comparison Tables 7 and 8 list classification resultsbased on time series modeling usingmeasured signal directlyand based on EMD and time series model method From

the comparison results the proposed method is efficient forrolling bearing fault diagnosis and the overall classificationrate of the proposedmethod is higher to a certain extent thanthe other two methods mentioned above

4 Conclusions

Aiming at diagnosing rolling bearing faults a hybrid ap-proach based on CEEMD and time series modeling is pro-posed in this paperThe CEEMDmethod can decompose thenonstationary signal into a series of IMFs with low compu-tation AR model is an effective approach to extract thefault feature of the vibration signals and the fault patterncan be identified directly by the extracted fault featureswithout establishing the mathematical model and studying

Mathematical Problems in Engineering 9

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

002

IMF1

Signals (V)

002

IMF2

0005

IMF3

001

IMF4

001

IMF5

0005

IMF6

0002

IMF7

0002004

Res

Time (s)

minus02

minus02

minus005

minus01

minus01

minus005

minus002

(a) No defect

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

02

IMF1

Signals (V)

01

IMF2

002

IMF3

002

IMF4

0005

IMF5

0002

IMF6

0001

IMF7

0002004

Res

Time (s)

minus1

minus2

minus02

minus02

minus005

minus002

minus001

(b) Inner ring defect

005

IMF1

Signals (V)

001

IMF2

001

IMF3

0005

IMF4

0005

IMF5

0001

IMF6

05

IMF7

0002004

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus05

minus01

minus01

minus005

minus005

minus001

minus5

times10minus3

(c) Rolling element defect

05

IMF1

Signals (V)

005

IMF2

002

IMF3

002

IMF4

0005

IMF5

0005

IMF6

0005

IMF7

002004006

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus5

minus05

minus02

minus02

minus005

minus005

minus005

(d) Outer ring defect

Figure 11 The decomposition results by CEEMD under different conditions

10 Mathematical Problems in Engineering

0 5 10 15 20 25 300

1

2

3

4

5

6

Model order

Mod

el re

sidua

l

DeltaFPE

times10minus4

(a) No defect

0 5 10 15 20 25 300

001

002

003

004

005

006

007

Model order

Mod

el re

sidua

l

DeltaFPE

(b) Inner ring defect

0 5 10 15 20 25 300

0005

001

0015

002

Model order

Mod

el re

sidua

l

DeltaFPE

(c) Rolling element defect

0 5 10 15 20 25 300

01

02

03

04

05

Model order

Mod

el re

sidua

l

DeltaFPE

(d) Outer ring defect

Figure 12 The model order estimation curves

Table 6 Fault diagnosis using CEEMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 20 0 0 0 100

975Inner ringdefect 20 0 19 1 0 95

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 0 20 100

Mathematical Problems in Engineering 11

1 2 3 4 5 6

0

5

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus5

(a) No defect

1 2 3 4 5 6

0

1

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

(b) Inner ring defect

1 2 3 4 5 6

0

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

minus2

(c) Rolling element defect

1 2 3 4 5 6

0

05A

mpl

itude

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus05

minus1

minus15

(d) Outer ring defect

Figure 13 The results of quantization

Table 7 Fault diagnosis using time series model only

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

900Inner ringdefect 20 1 17 2 0 85

Rolling elementdefect 20 0 2 17 1 85

Outer ringdefect 20 0 0 1 19 95

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Empirical mode decomposition (EMD) is an adaptivetime-frequency signal processing method [11] With EMD asignal is decomposed into a series of intrinsicmode functions(IMFs) according to its own characteristics [12] For examplea new fault feature extraction approach based on EMDmethod and AR model was used to process vibration signalsof roller bearings [3] However when the EMD method isapplied to the nonstationary signals containing intermittentsignal components the original signal cannot be decomposedaccurately because of the problem of mode mixing [13] Toalleviate mode mixing Wu and Huang developed ensembleempiricalmode decomposition (EEMD) to improve EMDByadding noise to the original signal and calculating the meansof IMFs repeatedly EEMD is more accurate and effectivefor signal decomposition [13] Although the EEMD methodhas effectively resolved the mode-mixing problem it is timeconsuming for implementing the large enough ensemblemean That is to say the algorithm efficiency will be greatlyreduced Aiming at solving this problem the complementaryensemble EMD (CEEMD) method is proposed [14] In thisapproach the residue of added white noises can be extractedfrom the mixtures of data and white noises via pairs ofcomplementary ensemble IMFs with positive and negativeadded white noises The CEEMD method has the sameperformance as the EEMD but the computational efficiencyis greatly improved

In this paper we combine the advantages of CEEMDand time series model and propose a new method based onCEEMD and AR model for rolling bearing fault diagnosisThe CEEMD is used as the pretreatment to filter the signaland extract the IMF which is closely correlated to thefiltered signal and then the AR model of the selected IMFis established The AR model parameters are used as thefeature vectors to a classifier where the hiddenMarkovmodel(HMM) is used to identify the fault pattern of a rollingbearing The rest of this paper is organized as follows InSection 2 the review of the fault diagnosis method based onAR model is presented and the proposed method for rollingbearing fault diagnosis is discussed The evaluations andexperiments are presented in Section 3 Finally concludingremarks are drawn in Section 4

2 Theoretical Framework

21 Time Series Modeling Autoregressive moving average(ARMA)model is the representative time seriesmodel whichcan be expressed in linear difference equation form as

119909119905+ 1205931119909119905minus1

+ sdot sdot sdot + 120593119899119909119905minus119899

= 119886119905+ 1205791119886119905minus1

+ sdot sdot sdot + 120579119898

119886119905minus119898

(1)

where 119899 and 119898 are the parameters of the ARMA (119899 119898)model 119909

119905is zero mean stationary random sequence 119886

119905is

white noise sequence and 120593119894and 120579

119895are model parameters

to be estimated The parameters of 120593119894and 120579

119895are estimated

by the time sequence of 119909119905(119905 = 1 2 3 ) which is called

the time series modeling If 120593119894

= 0 the ARMA (119899 119898) modelwill degrade to 119898 order MA(119898) model and if 120579

119894= 0 the

ARMA (119899 119898) model will degrade as 119899 order AR (119899) model in(1) The AR model is stable and its structure is simpler than

ARMA model Therefore the AR model will be establishedfor characterizing the rolling bearing vibration signal if theprecision of the model is enough for expressing the systemwhich is expressed as

119909119905

= 1205931119909119905minus1

+ sdot sdot sdot + 120593119899119909119905minus119899

+ 119886119905 (2)

where 119905 = 1 2 119873 119873 is the length of the time series 119909119905 119899

is the order number and 119886119905

sim NID(0 1205902

119886)The 120590

2

119886is expressed

as

1205902

120572=

1

119873 minus 119899

119899

119899

sum

119905=119899+1

(119909119905minus

119899

sum

120580=1

120593120580119909119905minus1

)

2

(3)

It is critical to determine the order number of the ARmodel because the accuracy of the order not only affects theaccuracy of identification of the system but also influencesthe stability of the system In order to estimate the order oftheARmodel correctly FPE criterion BIC criterion andAICcriterion are usually used [15] and they are expressed as

FPE criterion

FPE (119899) =

119873 + 119899

119873 minus 119899

1205902

119886 (4)

BIC criterion

AIC (119899) = 119873 ln1205902

119886+ 2119899 (5)

AIC criterion

BIC (119899) = 119873 ln1205902

119886+ 119899 ln119873 (6)

After the model order is determined the nonlinear leastsquares method can be used to estimate model parametersand then the AR model with specific parameters is estab-lished

22 Complementary Ensemble Empirical Mode Decomposi-tion Complementary ensemble empirical mode decomposi-tion (CEEMD) is an improved algorithm based on empiricalmode decomposition (EMD) Through EMD process anycomplex time series can be decomposed into finite numbersof intrinsic mode functions (IMFs) and each IMF reflectsthe dynamic characteristic of the original signal The IMFcomponent must satisfy two conditions (a) the number ofpoles and zeros is either equal to each other or differs at mostby one (b) the upper and lower envelopes must be locallysymmetric about the timeline The basic principle of EMDmethod is to decompose the original signal 119909(119905) into the formas shown in (7) by continuously eliminating the mean of theupper and lower envelope connected with the minimum andmaximum of the signal [16] Consider

119909119905

=

119899

sum

119894=1

imf119894(119905)

+ 119903119899(119905)

(7)

where 119909119905is the vibration signal imf

119894(119905)is the IMF component

including different frequency bands ranging fromhigh to low

Mathematical Problems in Engineering 3

0 0002 0004 0006 0008 001

01

Signals (V)

0 0002 0004 0006 0008 001

02

0 0002 0004 0006 0008 0010

05

0 0002 0004 0006 0008 001

05

Time (s)

s(t)

s1(t)

s2(t)

s3(t)

minus1

minus2

minus5

Figure 1 Signal waveforms

and 119903119899(119905)

is the residue of the decomposition process which isthe mean trend of 119909

119905

The EMD method is a kind of adaptive local analysismethod with each IMF highlighting the local features ofthe data However EMD decomposition results often sufferfrom mode mixing which is defined as either a single IMFconsisting of widely disparate scales or a signal residing indifferent IMF components [17] To make it clear a simulatedsignal 119904(119905) consists of a Gaussian-type impulse interference1199041(119905) and a cosine component with 500Hz frequency 119904

2(119905)

and a trend term 1199043(119905) is used as an example The equation of

the simulated signal is expressed as

119904 (119905) = sin (2120587120572119905) 119890minus((119905minus1199050)

2120590)

+ cos (2120587120573119905) + 50119905 (8)

where 120572 = 3000 120573 = 500 and 120590 = 106

Thewaveformof the simulated signal is shown in Figure 1and the corresponding EMD results for the signal 119904(119905) areshown in Figure 2 where the mode mixing happens

To overcome the problem of mode mixing the ensembleempirical mode decomposition (EEMD) was proposed [18]where Gaussian white noises with finite amplitude are addedto the original signal during the entire decomposition pro-cess Due to the uniform distribution statistical characteris-tics of the white noise the signal with white noise becomescontinuous in different time scales and no missing scales arepresent As a result mode mixing is effectively eliminated bythe EEMD process [18] The EEMD decomposition result ofsignal 119904(119905) is shown in Figure 3 where the added white noiseamplitude is 025 times the original signal standard deviationand the number of decompositions is 200 times

It should be noted that during the EEMD process eachindividual trial may produce noisy results but the effectof the added noise can be suppressed by large numberof ensemble mean computations This would be too timeconsuming to implement An improved algorithm namedcomplementary ensemble mode decomposition (CEEMD)

0 0002 0004 0006 0008 001

0

1

IMF1

Signals (V)

0 0002 0004 0006 0008 001

0

5

IMF2

0 0002 0004 0006 0008 001

0

5

IMF3

0 0002 0004 0006 0008 001

0

1

IMF4

Time (s)

minus1

minus1

minus5

minus5

Figure 2 The decomposition result by EMD

0 0002 0004 0006 0008 001

0

1

IMF1

Signals (V)

0 0002 0004 0006 0008 001

0

02

IMF2

0 0002 0004 0006 0008 001

0

01

IMF3

0 0002 0004 0006 0008 001

0

2

IMF4

Time (s)

minus1

minus02

minus01

minus2

Figure 3 The decomposition result by EEMD

is suggested to improve the computation efficiency In thisalgorithm the residue of the added white noises can beextracted from themixtures of data and white noises via pairsof complementary ensemble IMFs with positive and negativeadded white noises Although this new approach yields IMFwith a similar RMSnoise to EEMD it eliminates residue noisein the IMFs and overcomes the problem ofmodemixing withmuch more efficiency [14] The procedure on implementingCEEMD is shown below

(a) 1199091and 1199092are constructed by adding a pair of opposite

phase Gaussian white noises 119909119899with the same ampli-

tude Then 1199091

= 119909 + 119909119899and 119909

2= 119909 minus 119909

119899

4 Mathematical Problems in Engineering

(b) 1199091and 119909

2are decomposed by EMD only a few times

and IMF1199091

and IMF1199092

are ensemble means of thecorresponding IMF generated from each trial

(c) the average of corresponding component in IMF1199091

and IMF1199092

is calculated as the CEEMD decomposi-tion results that is

IMF =

(IMF1199091

+ IMF1199092

)

2

(9)

The flow chart of CEEMD is shown in Figure 4 where 119899 is thedecomposition trials

Figure 5 is the decomposition result by CEEMD for thesignal 119904(119905) As compared to the result shown in Figure 3the decomposition accuracies of EEMD and CEEMD areconsistent while EEMD takes 162 s and CEEMD only needs013 s

23 Fault Diagnosis Based on CEEMD and Time Series ModelBased on CEEMD and time series model a hybrid faultdiagnosis approach can be designed The hybrid approachcombines the advantages of CEEMD method in the non-stationary signal decomposition with the ability of timeseries modeling in feature extraction The flow chart of thedeveloped approach is shown in Figure 6

The main steps are as follows

Step 1 The rolling bearing vibration signal is sampled andthen decomposed by CEEMD with the process shown inFigure 4

Step 2 The product of energy density and average periodof the IMFs which is a constant value according to [19] iscalculated using (10) and parameter 119877119875

119895is calculated using

(11) Then the signal is filtered by comparing the parameter119877119875119895and the given threshold value that is to say when 119877119875

119895⩾

1 the previous 119895 minus 1 IMFs with the trend term need to beremoved as noise and to rebuild the residual IMFs as filteredsignal [19 20]

119875119895

= 119864119895

times 119879119895 (10)

119877119875119895

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119875119895

minus ((1 (119895 minus 1)) sum119895minus1

119894=1119875119895)

(1 (119895 minus 1)) sum119895minus1

119894=1119875119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119895 ge 2) (11)

where 119864119895

= (1119873) sum119873

119894=1[119860119895(119894)]2 is the energy density of the

119895th IMF 119879119895

= 2119873119874119895is the average period of the 119895th IMF 119873

is the length of each IMF 119860119895is the amplitude of the 119895th IMF

and 119874119895is the total number of extreme points of 119895th IMF

Step 3 Equation (12) is used to calculate the correlationcoefficient between the filtered signal and each IMF and theIMFwhich is closely correlated to the filtered signal is selectedfor AR modeling [21]

120588119909119910

=

sum119873

119896=1119909 (119896) 119910 (119896)

[sum119873

119896=1119909(119896)2

sum119873

119896=1119910(119896)2]

12 (12)

Input signal

Add two reverse white noises

final IMFs of CEEMD

Calculate IMFs with EMD

Calculate the means of above two IMFs

No

Yes

M = n

Calculate (IMF1+ IMF2)2 as the

Figure 4 Decomposition flow chart of CEEMD

0 0002 0004 0006 0008 001

0

1

IMF1

Signals (V)

0 0002 0004 0006 0008 001

0

05

IMF2

0 0002 0004 0006 0008 001

0

02

IMF3

0 0002 0004 0006 0008 001

0

2

IMF4

Time (s)

minus1

minus05

minus02

minus2

Figure 5 The decomposition result by CEEMD

Step 4 The least square method is used to estimate theparameters vectors of theARmodel established in Step 3 andthe parameters vectors are considered as the model featurevector

Step 5 After scalar quantization by index calculation formulaof Lloyds algorithm in (13) [22] the feature vector is used totrain the HMM of each bearing working condition

indx (119909) =

1 119909 le partition (119894)

119894 + 1 partition (119894) lt 119909 le partition (119894 + 1)

119873 partition (119873 minus 1) lt 119909

(13)

Mathematical Problems in Engineering 5

HMM1

HMM training

HMM2 Comparisonprobability

Classification result

Vectorsquantization

Vibration signals

Signal decomposed by CEEMD

Calculate correlation coefficient

Feature vectors

IMF selected for modeling

Model parameter estimation

Feature vectors extracted

Signal filtered

Calculate parameter RP

Feature vectors extraction Fault diagnosis classification

Sample feature vectors

HMMn

Figure 6 The flow chart of the proposed method

where 119873 is the length of the codebook vector partition (119894)is the partition vector with the length of 119873 minus 1 and 119909 is thefeature vector for scalar quantization

Step 6 A test vibration signal can then be acquired fordiagnosis and themodel feature vector is first extractedAfterscalar quantization the feature vector is put into the well-trained HMMs and the corresponding HMM which has themaximum probability is regarded as the classification result[23]

3 Evaluation of the Method Based onCEEMD and AR Model

31 Evaluation Using Simulated Signals To demonstrate thevalidity of the method proposed in this study three signals1199091(119905) 1199092(119905) and 119909

3(119905) are simulated as shown in Figure 7The

signal 1199091(119905) consists of a Gaussian-type impulse interference

a cosine component with 10Hz frequency a trend term andwhite noise The signal 119909

2(119905) consists of a Gaussian-type

impulse interference a square wave with 65 duty ratio atrend term and white noise The signal 119909

3(119905) consists of a

Gaussian-type impulse interference a sawtooth wave with15Hz frequency a trend term and white noise

Figure 8 shows the results of the CEEMD of signals 1199091(119905)

1199092(119905) and 119909

3(119905) Correlation coefficients between filtered

signal and each IMF are illustrated in Table 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0

4Signals (V)

0

4

0

4

Time (s)

x1(t)

x2(t)

x3(t)

minus4

minus4

minus4

Figure 7 Signal waveforms of 1199091(119905) 1199092(119905) and 119909

3(119905)

It can be seen in Table 1 that the IMF which is closelycorrelated to the filtered signal is IMF5 for both signal1199091(119905) and signal 119909

3(119905) and IMF6 for signal 119909

2(119905) They are

used to construct the AR models and the correspondingfeature vectors are estimated as shown in Table 2 After scalarquantization the feature vectors are used to train the HMMfor signal classification

6 Mathematical Problems in Engineering

02

IMF1

Signals (V)

01

IMF2

005

IMF3

01

IMF4

02

IMF5

002

IMF6

001

IMF7

00204

Res

Time (s)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

minus2

minus1

minus05

minus1

minus2

minus01

minus02

(a) 1199091(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

05

IMF6

01

IMF7

005

1

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus5

minus1

(b) 1199092(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

002

IMF6

001

IMF7

002

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus02

minus01

minus02

(c) 1199093(119905)

Figure 8 The decomposition results by CEEMD

Mathematical Problems in Engineering 7

Table 1 Correlation coefficients between filtered signal and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

1199091(119905) minus00031 minus00009 00371 04096 09668 02428 01273 minus00448

1199092(119905) 00051 00004 00435 02111 04695 08887 07214 minus00201

1199093(119905) minus00234 minus00154 00286 05900 08953 01649 01887 minus00210

Table 2 Model parameter estimation results

Signal Model parameter1205931

1205932

1205933

1205934

1205935

1205936

1199091(119905) 47183 minus91103 92034 minus51408 15207 minus01914

1199092(119905) 48894 minus98269 103945 minus61194 19153 minus02531

1199093(119905) 48718 minus99955 111616 minus72529 26430 minus04282

Table 3 Signal classification results

Signal type Test sample Classification results Classification rate [] Overall classificationrate []

1199091(119905) 119909

2(119905) 119909

3(119905)

1199091(119905) 20 19 1 0 95

9671199092(119905) 20 0 19 1 95

1199093(119905) 20 0 0 20 100

A total of 90 feature vectors were collected from threegroups of signals using the proposed approach One-third ofthe feature vectors in each condition were used for trainingthe classifier and others were used for testing The results ofthe signal classification are listed in Table 3

Results in Table 3 indicate that the presented methodbased on CEEMD and time series modeling can effectivelyidentify different signals and the overall classification rate is967 For the purpose of comparison the signal classifica-tion rates use themethod based on time seriesmodeling onlyand the method based on EMD and time series modelingis also calculated 883 and 933 classification rates areobtained respectively It is obvious that efficiency of the signalclassification method proposed in this paper is improved to acertain extent

32 Evaluation Using Experimental Data In order to illus-trate the practicability and effectiveness of the proposedmethod a bearing fault data set from the electrical engi-neering laboratory of Case Western Reserve University isanalyzed [24] The data set is acquired from the test standshown in Figure 9 where it consists of a 2 hp motor atorque transducer a dynamometer and control electronicsThe test bearings support the motor shaft which is the deepgroove ball bearings with the type of 6205-2RS JEMSKFVibration data was collected at 12000 samples per secondusing accelerometers which are attached to the housing withmagnetic bases The motor load level was controlled by thefan in the right side of Figure 9

2 hp motor Torque

transducerDynamometer

Figure 9 Bearing test stand

Figure 10 illustrates representative waveforms of the sam-ple vibration signals measured from the test bearings underfour initial conditions (a) signal from a healthy bearing (b)signal from a bearing with inner ring defect (c) signal froma bearing with rolling element defect and (d) signal from abearing with outer ring defect These signals were measuredunder 0 hpmotor load with themotor speed of 1797 rpmThedecomposed IMFs of these signals are shown in Figure 11

Correlation coefficients calculated between the filteredsignal and each IMF are shown in Table 4

The IMF which is closely correlated to the filtered signalis IMF2 for signal (a) and IMF1 for signals (b) (c) and (d)respectivelyThese IMFs are used for ARmodel constructionThe model order estimation curves of the four conditionsbased on the principle of FPE criterion are shown in Figure 12We can see that when the model order is 6 each modelrsquos

8 Mathematical Problems in Engineering

0 005 01 015 02 025

0

05

minus05

Signals (V)

(a)

0 005 01 015 02 025

0

2

minus2

(b)

0 005 01 015 02 025

0

1

minus1

(c)

0 005 01 015 02 025

0

5

Time (s)

minus5

(d)

Figure 10 Vibration signal waveforms of different conditions

Table 4 Correlation coefficients between filtered signals and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

(a) 04135 07538 04381 04880 04356 01792 00971 minus00056(b) 08794 04275 02583 01337 00421 00285 minus00009 minus00074(c) 09509 02180 02325 01337 00821 00350 minus00017 00009(d) 09878 01267 00636 00509 00136 00060 minus00008 minus00068

Table 5 Model parameter estimation results

Signal Model parameter1205931 1205932 1205933 1205934 1205935 1205936

(a) 31280 minus47797 42245 minus21489 04241 00356(b) 02084 minus13585 05142 minus06356 03471 minus00422(c) 01335 minus16472 03941 minus08473 02142 minus01011(d) minus01172 minus12159 01178 minus01283 01467 02533

residual tends to be stable Therefore the model order isselected as 6 and the results of parameters estimation arelisted in Table 5

The parameters in Table 5 were quantified by Lloydsalgorithm in (12) as feature vectors for training the HMMs ofdifferent conditions The results of quantization are revealedin Figure 13

A total of 160 feature vectors were collected from thefour conditions half of the feature vectors were used fortraining the classifier and others for signal classification andthe classification results are listed in Table 6 Out of 80 testfeature vectors just two cases were not correctly classifiedand the overall classification rate is 975

For comparison Tables 7 and 8 list classification resultsbased on time series modeling usingmeasured signal directlyand based on EMD and time series model method From

the comparison results the proposed method is efficient forrolling bearing fault diagnosis and the overall classificationrate of the proposedmethod is higher to a certain extent thanthe other two methods mentioned above

4 Conclusions

Aiming at diagnosing rolling bearing faults a hybrid ap-proach based on CEEMD and time series modeling is pro-posed in this paperThe CEEMDmethod can decompose thenonstationary signal into a series of IMFs with low compu-tation AR model is an effective approach to extract thefault feature of the vibration signals and the fault patterncan be identified directly by the extracted fault featureswithout establishing the mathematical model and studying

Mathematical Problems in Engineering 9

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

002

IMF1

Signals (V)

002

IMF2

0005

IMF3

001

IMF4

001

IMF5

0005

IMF6

0002

IMF7

0002004

Res

Time (s)

minus02

minus02

minus005

minus01

minus01

minus005

minus002

(a) No defect

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

02

IMF1

Signals (V)

01

IMF2

002

IMF3

002

IMF4

0005

IMF5

0002

IMF6

0001

IMF7

0002004

Res

Time (s)

minus1

minus2

minus02

minus02

minus005

minus002

minus001

(b) Inner ring defect

005

IMF1

Signals (V)

001

IMF2

001

IMF3

0005

IMF4

0005

IMF5

0001

IMF6

05

IMF7

0002004

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus05

minus01

minus01

minus005

minus005

minus001

minus5

times10minus3

(c) Rolling element defect

05

IMF1

Signals (V)

005

IMF2

002

IMF3

002

IMF4

0005

IMF5

0005

IMF6

0005

IMF7

002004006

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus5

minus05

minus02

minus02

minus005

minus005

minus005

(d) Outer ring defect

Figure 11 The decomposition results by CEEMD under different conditions

10 Mathematical Problems in Engineering

0 5 10 15 20 25 300

1

2

3

4

5

6

Model order

Mod

el re

sidua

l

DeltaFPE

times10minus4

(a) No defect

0 5 10 15 20 25 300

001

002

003

004

005

006

007

Model order

Mod

el re

sidua

l

DeltaFPE

(b) Inner ring defect

0 5 10 15 20 25 300

0005

001

0015

002

Model order

Mod

el re

sidua

l

DeltaFPE

(c) Rolling element defect

0 5 10 15 20 25 300

01

02

03

04

05

Model order

Mod

el re

sidua

l

DeltaFPE

(d) Outer ring defect

Figure 12 The model order estimation curves

Table 6 Fault diagnosis using CEEMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 20 0 0 0 100

975Inner ringdefect 20 0 19 1 0 95

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 0 20 100

Mathematical Problems in Engineering 11

1 2 3 4 5 6

0

5

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus5

(a) No defect

1 2 3 4 5 6

0

1

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

(b) Inner ring defect

1 2 3 4 5 6

0

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

minus2

(c) Rolling element defect

1 2 3 4 5 6

0

05A

mpl

itude

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus05

minus1

minus15

(d) Outer ring defect

Figure 13 The results of quantization

Table 7 Fault diagnosis using time series model only

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

900Inner ringdefect 20 1 17 2 0 85

Rolling elementdefect 20 0 2 17 1 85

Outer ringdefect 20 0 0 1 19 95

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

0 0002 0004 0006 0008 001

01

Signals (V)

0 0002 0004 0006 0008 001

02

0 0002 0004 0006 0008 0010

05

0 0002 0004 0006 0008 001

05

Time (s)

s(t)

s1(t)

s2(t)

s3(t)

minus1

minus2

minus5

Figure 1 Signal waveforms

and 119903119899(119905)

is the residue of the decomposition process which isthe mean trend of 119909

119905

The EMD method is a kind of adaptive local analysismethod with each IMF highlighting the local features ofthe data However EMD decomposition results often sufferfrom mode mixing which is defined as either a single IMFconsisting of widely disparate scales or a signal residing indifferent IMF components [17] To make it clear a simulatedsignal 119904(119905) consists of a Gaussian-type impulse interference1199041(119905) and a cosine component with 500Hz frequency 119904

2(119905)

and a trend term 1199043(119905) is used as an example The equation of

the simulated signal is expressed as

119904 (119905) = sin (2120587120572119905) 119890minus((119905minus1199050)

2120590)

+ cos (2120587120573119905) + 50119905 (8)

where 120572 = 3000 120573 = 500 and 120590 = 106

Thewaveformof the simulated signal is shown in Figure 1and the corresponding EMD results for the signal 119904(119905) areshown in Figure 2 where the mode mixing happens

To overcome the problem of mode mixing the ensembleempirical mode decomposition (EEMD) was proposed [18]where Gaussian white noises with finite amplitude are addedto the original signal during the entire decomposition pro-cess Due to the uniform distribution statistical characteris-tics of the white noise the signal with white noise becomescontinuous in different time scales and no missing scales arepresent As a result mode mixing is effectively eliminated bythe EEMD process [18] The EEMD decomposition result ofsignal 119904(119905) is shown in Figure 3 where the added white noiseamplitude is 025 times the original signal standard deviationand the number of decompositions is 200 times

It should be noted that during the EEMD process eachindividual trial may produce noisy results but the effectof the added noise can be suppressed by large numberof ensemble mean computations This would be too timeconsuming to implement An improved algorithm namedcomplementary ensemble mode decomposition (CEEMD)

0 0002 0004 0006 0008 001

0

1

IMF1

Signals (V)

0 0002 0004 0006 0008 001

0

5

IMF2

0 0002 0004 0006 0008 001

0

5

IMF3

0 0002 0004 0006 0008 001

0

1

IMF4

Time (s)

minus1

minus1

minus5

minus5

Figure 2 The decomposition result by EMD

0 0002 0004 0006 0008 001

0

1

IMF1

Signals (V)

0 0002 0004 0006 0008 001

0

02

IMF2

0 0002 0004 0006 0008 001

0

01

IMF3

0 0002 0004 0006 0008 001

0

2

IMF4

Time (s)

minus1

minus02

minus01

minus2

Figure 3 The decomposition result by EEMD

is suggested to improve the computation efficiency In thisalgorithm the residue of the added white noises can beextracted from themixtures of data and white noises via pairsof complementary ensemble IMFs with positive and negativeadded white noises Although this new approach yields IMFwith a similar RMSnoise to EEMD it eliminates residue noisein the IMFs and overcomes the problem ofmodemixing withmuch more efficiency [14] The procedure on implementingCEEMD is shown below

(a) 1199091and 1199092are constructed by adding a pair of opposite

phase Gaussian white noises 119909119899with the same ampli-

tude Then 1199091

= 119909 + 119909119899and 119909

2= 119909 minus 119909

119899

4 Mathematical Problems in Engineering

(b) 1199091and 119909

2are decomposed by EMD only a few times

and IMF1199091

and IMF1199092

are ensemble means of thecorresponding IMF generated from each trial

(c) the average of corresponding component in IMF1199091

and IMF1199092

is calculated as the CEEMD decomposi-tion results that is

IMF =

(IMF1199091

+ IMF1199092

)

2

(9)

The flow chart of CEEMD is shown in Figure 4 where 119899 is thedecomposition trials

Figure 5 is the decomposition result by CEEMD for thesignal 119904(119905) As compared to the result shown in Figure 3the decomposition accuracies of EEMD and CEEMD areconsistent while EEMD takes 162 s and CEEMD only needs013 s

23 Fault Diagnosis Based on CEEMD and Time Series ModelBased on CEEMD and time series model a hybrid faultdiagnosis approach can be designed The hybrid approachcombines the advantages of CEEMD method in the non-stationary signal decomposition with the ability of timeseries modeling in feature extraction The flow chart of thedeveloped approach is shown in Figure 6

The main steps are as follows

Step 1 The rolling bearing vibration signal is sampled andthen decomposed by CEEMD with the process shown inFigure 4

Step 2 The product of energy density and average periodof the IMFs which is a constant value according to [19] iscalculated using (10) and parameter 119877119875

119895is calculated using

(11) Then the signal is filtered by comparing the parameter119877119875119895and the given threshold value that is to say when 119877119875

119895⩾

1 the previous 119895 minus 1 IMFs with the trend term need to beremoved as noise and to rebuild the residual IMFs as filteredsignal [19 20]

119875119895

= 119864119895

times 119879119895 (10)

119877119875119895

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119875119895

minus ((1 (119895 minus 1)) sum119895minus1

119894=1119875119895)

(1 (119895 minus 1)) sum119895minus1

119894=1119875119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119895 ge 2) (11)

where 119864119895

= (1119873) sum119873

119894=1[119860119895(119894)]2 is the energy density of the

119895th IMF 119879119895

= 2119873119874119895is the average period of the 119895th IMF 119873

is the length of each IMF 119860119895is the amplitude of the 119895th IMF

and 119874119895is the total number of extreme points of 119895th IMF

Step 3 Equation (12) is used to calculate the correlationcoefficient between the filtered signal and each IMF and theIMFwhich is closely correlated to the filtered signal is selectedfor AR modeling [21]

120588119909119910

=

sum119873

119896=1119909 (119896) 119910 (119896)

[sum119873

119896=1119909(119896)2

sum119873

119896=1119910(119896)2]

12 (12)

Input signal

Add two reverse white noises

final IMFs of CEEMD

Calculate IMFs with EMD

Calculate the means of above two IMFs

No

Yes

M = n

Calculate (IMF1+ IMF2)2 as the

Figure 4 Decomposition flow chart of CEEMD

0 0002 0004 0006 0008 001

0

1

IMF1

Signals (V)

0 0002 0004 0006 0008 001

0

05

IMF2

0 0002 0004 0006 0008 001

0

02

IMF3

0 0002 0004 0006 0008 001

0

2

IMF4

Time (s)

minus1

minus05

minus02

minus2

Figure 5 The decomposition result by CEEMD

Step 4 The least square method is used to estimate theparameters vectors of theARmodel established in Step 3 andthe parameters vectors are considered as the model featurevector

Step 5 After scalar quantization by index calculation formulaof Lloyds algorithm in (13) [22] the feature vector is used totrain the HMM of each bearing working condition

indx (119909) =

1 119909 le partition (119894)

119894 + 1 partition (119894) lt 119909 le partition (119894 + 1)

119873 partition (119873 minus 1) lt 119909

(13)

Mathematical Problems in Engineering 5

HMM1

HMM training

HMM2 Comparisonprobability

Classification result

Vectorsquantization

Vibration signals

Signal decomposed by CEEMD

Calculate correlation coefficient

Feature vectors

IMF selected for modeling

Model parameter estimation

Feature vectors extracted

Signal filtered

Calculate parameter RP

Feature vectors extraction Fault diagnosis classification

Sample feature vectors

HMMn

Figure 6 The flow chart of the proposed method

where 119873 is the length of the codebook vector partition (119894)is the partition vector with the length of 119873 minus 1 and 119909 is thefeature vector for scalar quantization

Step 6 A test vibration signal can then be acquired fordiagnosis and themodel feature vector is first extractedAfterscalar quantization the feature vector is put into the well-trained HMMs and the corresponding HMM which has themaximum probability is regarded as the classification result[23]

3 Evaluation of the Method Based onCEEMD and AR Model

31 Evaluation Using Simulated Signals To demonstrate thevalidity of the method proposed in this study three signals1199091(119905) 1199092(119905) and 119909

3(119905) are simulated as shown in Figure 7The

signal 1199091(119905) consists of a Gaussian-type impulse interference

a cosine component with 10Hz frequency a trend term andwhite noise The signal 119909

2(119905) consists of a Gaussian-type

impulse interference a square wave with 65 duty ratio atrend term and white noise The signal 119909

3(119905) consists of a

Gaussian-type impulse interference a sawtooth wave with15Hz frequency a trend term and white noise

Figure 8 shows the results of the CEEMD of signals 1199091(119905)

1199092(119905) and 119909

3(119905) Correlation coefficients between filtered

signal and each IMF are illustrated in Table 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0

4Signals (V)

0

4

0

4

Time (s)

x1(t)

x2(t)

x3(t)

minus4

minus4

minus4

Figure 7 Signal waveforms of 1199091(119905) 1199092(119905) and 119909

3(119905)

It can be seen in Table 1 that the IMF which is closelycorrelated to the filtered signal is IMF5 for both signal1199091(119905) and signal 119909

3(119905) and IMF6 for signal 119909

2(119905) They are

used to construct the AR models and the correspondingfeature vectors are estimated as shown in Table 2 After scalarquantization the feature vectors are used to train the HMMfor signal classification

6 Mathematical Problems in Engineering

02

IMF1

Signals (V)

01

IMF2

005

IMF3

01

IMF4

02

IMF5

002

IMF6

001

IMF7

00204

Res

Time (s)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

minus2

minus1

minus05

minus1

minus2

minus01

minus02

(a) 1199091(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

05

IMF6

01

IMF7

005

1

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus5

minus1

(b) 1199092(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

002

IMF6

001

IMF7

002

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus02

minus01

minus02

(c) 1199093(119905)

Figure 8 The decomposition results by CEEMD

Mathematical Problems in Engineering 7

Table 1 Correlation coefficients between filtered signal and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

1199091(119905) minus00031 minus00009 00371 04096 09668 02428 01273 minus00448

1199092(119905) 00051 00004 00435 02111 04695 08887 07214 minus00201

1199093(119905) minus00234 minus00154 00286 05900 08953 01649 01887 minus00210

Table 2 Model parameter estimation results

Signal Model parameter1205931

1205932

1205933

1205934

1205935

1205936

1199091(119905) 47183 minus91103 92034 minus51408 15207 minus01914

1199092(119905) 48894 minus98269 103945 minus61194 19153 minus02531

1199093(119905) 48718 minus99955 111616 minus72529 26430 minus04282

Table 3 Signal classification results

Signal type Test sample Classification results Classification rate [] Overall classificationrate []

1199091(119905) 119909

2(119905) 119909

3(119905)

1199091(119905) 20 19 1 0 95

9671199092(119905) 20 0 19 1 95

1199093(119905) 20 0 0 20 100

A total of 90 feature vectors were collected from threegroups of signals using the proposed approach One-third ofthe feature vectors in each condition were used for trainingthe classifier and others were used for testing The results ofthe signal classification are listed in Table 3

Results in Table 3 indicate that the presented methodbased on CEEMD and time series modeling can effectivelyidentify different signals and the overall classification rate is967 For the purpose of comparison the signal classifica-tion rates use themethod based on time seriesmodeling onlyand the method based on EMD and time series modelingis also calculated 883 and 933 classification rates areobtained respectively It is obvious that efficiency of the signalclassification method proposed in this paper is improved to acertain extent

32 Evaluation Using Experimental Data In order to illus-trate the practicability and effectiveness of the proposedmethod a bearing fault data set from the electrical engi-neering laboratory of Case Western Reserve University isanalyzed [24] The data set is acquired from the test standshown in Figure 9 where it consists of a 2 hp motor atorque transducer a dynamometer and control electronicsThe test bearings support the motor shaft which is the deepgroove ball bearings with the type of 6205-2RS JEMSKFVibration data was collected at 12000 samples per secondusing accelerometers which are attached to the housing withmagnetic bases The motor load level was controlled by thefan in the right side of Figure 9

2 hp motor Torque

transducerDynamometer

Figure 9 Bearing test stand

Figure 10 illustrates representative waveforms of the sam-ple vibration signals measured from the test bearings underfour initial conditions (a) signal from a healthy bearing (b)signal from a bearing with inner ring defect (c) signal froma bearing with rolling element defect and (d) signal from abearing with outer ring defect These signals were measuredunder 0 hpmotor load with themotor speed of 1797 rpmThedecomposed IMFs of these signals are shown in Figure 11

Correlation coefficients calculated between the filteredsignal and each IMF are shown in Table 4

The IMF which is closely correlated to the filtered signalis IMF2 for signal (a) and IMF1 for signals (b) (c) and (d)respectivelyThese IMFs are used for ARmodel constructionThe model order estimation curves of the four conditionsbased on the principle of FPE criterion are shown in Figure 12We can see that when the model order is 6 each modelrsquos

8 Mathematical Problems in Engineering

0 005 01 015 02 025

0

05

minus05

Signals (V)

(a)

0 005 01 015 02 025

0

2

minus2

(b)

0 005 01 015 02 025

0

1

minus1

(c)

0 005 01 015 02 025

0

5

Time (s)

minus5

(d)

Figure 10 Vibration signal waveforms of different conditions

Table 4 Correlation coefficients between filtered signals and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

(a) 04135 07538 04381 04880 04356 01792 00971 minus00056(b) 08794 04275 02583 01337 00421 00285 minus00009 minus00074(c) 09509 02180 02325 01337 00821 00350 minus00017 00009(d) 09878 01267 00636 00509 00136 00060 minus00008 minus00068

Table 5 Model parameter estimation results

Signal Model parameter1205931 1205932 1205933 1205934 1205935 1205936

(a) 31280 minus47797 42245 minus21489 04241 00356(b) 02084 minus13585 05142 minus06356 03471 minus00422(c) 01335 minus16472 03941 minus08473 02142 minus01011(d) minus01172 minus12159 01178 minus01283 01467 02533

residual tends to be stable Therefore the model order isselected as 6 and the results of parameters estimation arelisted in Table 5

The parameters in Table 5 were quantified by Lloydsalgorithm in (12) as feature vectors for training the HMMs ofdifferent conditions The results of quantization are revealedin Figure 13

A total of 160 feature vectors were collected from thefour conditions half of the feature vectors were used fortraining the classifier and others for signal classification andthe classification results are listed in Table 6 Out of 80 testfeature vectors just two cases were not correctly classifiedand the overall classification rate is 975

For comparison Tables 7 and 8 list classification resultsbased on time series modeling usingmeasured signal directlyand based on EMD and time series model method From

the comparison results the proposed method is efficient forrolling bearing fault diagnosis and the overall classificationrate of the proposedmethod is higher to a certain extent thanthe other two methods mentioned above

4 Conclusions

Aiming at diagnosing rolling bearing faults a hybrid ap-proach based on CEEMD and time series modeling is pro-posed in this paperThe CEEMDmethod can decompose thenonstationary signal into a series of IMFs with low compu-tation AR model is an effective approach to extract thefault feature of the vibration signals and the fault patterncan be identified directly by the extracted fault featureswithout establishing the mathematical model and studying

Mathematical Problems in Engineering 9

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

002

IMF1

Signals (V)

002

IMF2

0005

IMF3

001

IMF4

001

IMF5

0005

IMF6

0002

IMF7

0002004

Res

Time (s)

minus02

minus02

minus005

minus01

minus01

minus005

minus002

(a) No defect

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

02

IMF1

Signals (V)

01

IMF2

002

IMF3

002

IMF4

0005

IMF5

0002

IMF6

0001

IMF7

0002004

Res

Time (s)

minus1

minus2

minus02

minus02

minus005

minus002

minus001

(b) Inner ring defect

005

IMF1

Signals (V)

001

IMF2

001

IMF3

0005

IMF4

0005

IMF5

0001

IMF6

05

IMF7

0002004

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus05

minus01

minus01

minus005

minus005

minus001

minus5

times10minus3

(c) Rolling element defect

05

IMF1

Signals (V)

005

IMF2

002

IMF3

002

IMF4

0005

IMF5

0005

IMF6

0005

IMF7

002004006

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus5

minus05

minus02

minus02

minus005

minus005

minus005

(d) Outer ring defect

Figure 11 The decomposition results by CEEMD under different conditions

10 Mathematical Problems in Engineering

0 5 10 15 20 25 300

1

2

3

4

5

6

Model order

Mod

el re

sidua

l

DeltaFPE

times10minus4

(a) No defect

0 5 10 15 20 25 300

001

002

003

004

005

006

007

Model order

Mod

el re

sidua

l

DeltaFPE

(b) Inner ring defect

0 5 10 15 20 25 300

0005

001

0015

002

Model order

Mod

el re

sidua

l

DeltaFPE

(c) Rolling element defect

0 5 10 15 20 25 300

01

02

03

04

05

Model order

Mod

el re

sidua

l

DeltaFPE

(d) Outer ring defect

Figure 12 The model order estimation curves

Table 6 Fault diagnosis using CEEMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 20 0 0 0 100

975Inner ringdefect 20 0 19 1 0 95

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 0 20 100

Mathematical Problems in Engineering 11

1 2 3 4 5 6

0

5

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus5

(a) No defect

1 2 3 4 5 6

0

1

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

(b) Inner ring defect

1 2 3 4 5 6

0

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

minus2

(c) Rolling element defect

1 2 3 4 5 6

0

05A

mpl

itude

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus05

minus1

minus15

(d) Outer ring defect

Figure 13 The results of quantization

Table 7 Fault diagnosis using time series model only

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

900Inner ringdefect 20 1 17 2 0 85

Rolling elementdefect 20 0 2 17 1 85

Outer ringdefect 20 0 0 1 19 95

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

(b) 1199091and 119909

2are decomposed by EMD only a few times

and IMF1199091

and IMF1199092

are ensemble means of thecorresponding IMF generated from each trial

(c) the average of corresponding component in IMF1199091

and IMF1199092

is calculated as the CEEMD decomposi-tion results that is

IMF =

(IMF1199091

+ IMF1199092

)

2

(9)

The flow chart of CEEMD is shown in Figure 4 where 119899 is thedecomposition trials

Figure 5 is the decomposition result by CEEMD for thesignal 119904(119905) As compared to the result shown in Figure 3the decomposition accuracies of EEMD and CEEMD areconsistent while EEMD takes 162 s and CEEMD only needs013 s

23 Fault Diagnosis Based on CEEMD and Time Series ModelBased on CEEMD and time series model a hybrid faultdiagnosis approach can be designed The hybrid approachcombines the advantages of CEEMD method in the non-stationary signal decomposition with the ability of timeseries modeling in feature extraction The flow chart of thedeveloped approach is shown in Figure 6

The main steps are as follows

Step 1 The rolling bearing vibration signal is sampled andthen decomposed by CEEMD with the process shown inFigure 4

Step 2 The product of energy density and average periodof the IMFs which is a constant value according to [19] iscalculated using (10) and parameter 119877119875

119895is calculated using

(11) Then the signal is filtered by comparing the parameter119877119875119895and the given threshold value that is to say when 119877119875

119895⩾

1 the previous 119895 minus 1 IMFs with the trend term need to beremoved as noise and to rebuild the residual IMFs as filteredsignal [19 20]

119875119895

= 119864119895

times 119879119895 (10)

119877119875119895

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119875119895

minus ((1 (119895 minus 1)) sum119895minus1

119894=1119875119895)

(1 (119895 minus 1)) sum119895minus1

119894=1119875119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119895 ge 2) (11)

where 119864119895

= (1119873) sum119873

119894=1[119860119895(119894)]2 is the energy density of the

119895th IMF 119879119895

= 2119873119874119895is the average period of the 119895th IMF 119873

is the length of each IMF 119860119895is the amplitude of the 119895th IMF

and 119874119895is the total number of extreme points of 119895th IMF

Step 3 Equation (12) is used to calculate the correlationcoefficient between the filtered signal and each IMF and theIMFwhich is closely correlated to the filtered signal is selectedfor AR modeling [21]

120588119909119910

=

sum119873

119896=1119909 (119896) 119910 (119896)

[sum119873

119896=1119909(119896)2

sum119873

119896=1119910(119896)2]

12 (12)

Input signal

Add two reverse white noises

final IMFs of CEEMD

Calculate IMFs with EMD

Calculate the means of above two IMFs

No

Yes

M = n

Calculate (IMF1+ IMF2)2 as the

Figure 4 Decomposition flow chart of CEEMD

0 0002 0004 0006 0008 001

0

1

IMF1

Signals (V)

0 0002 0004 0006 0008 001

0

05

IMF2

0 0002 0004 0006 0008 001

0

02

IMF3

0 0002 0004 0006 0008 001

0

2

IMF4

Time (s)

minus1

minus05

minus02

minus2

Figure 5 The decomposition result by CEEMD

Step 4 The least square method is used to estimate theparameters vectors of theARmodel established in Step 3 andthe parameters vectors are considered as the model featurevector

Step 5 After scalar quantization by index calculation formulaof Lloyds algorithm in (13) [22] the feature vector is used totrain the HMM of each bearing working condition

indx (119909) =

1 119909 le partition (119894)

119894 + 1 partition (119894) lt 119909 le partition (119894 + 1)

119873 partition (119873 minus 1) lt 119909

(13)

Mathematical Problems in Engineering 5

HMM1

HMM training

HMM2 Comparisonprobability

Classification result

Vectorsquantization

Vibration signals

Signal decomposed by CEEMD

Calculate correlation coefficient

Feature vectors

IMF selected for modeling

Model parameter estimation

Feature vectors extracted

Signal filtered

Calculate parameter RP

Feature vectors extraction Fault diagnosis classification

Sample feature vectors

HMMn

Figure 6 The flow chart of the proposed method

where 119873 is the length of the codebook vector partition (119894)is the partition vector with the length of 119873 minus 1 and 119909 is thefeature vector for scalar quantization

Step 6 A test vibration signal can then be acquired fordiagnosis and themodel feature vector is first extractedAfterscalar quantization the feature vector is put into the well-trained HMMs and the corresponding HMM which has themaximum probability is regarded as the classification result[23]

3 Evaluation of the Method Based onCEEMD and AR Model

31 Evaluation Using Simulated Signals To demonstrate thevalidity of the method proposed in this study three signals1199091(119905) 1199092(119905) and 119909

3(119905) are simulated as shown in Figure 7The

signal 1199091(119905) consists of a Gaussian-type impulse interference

a cosine component with 10Hz frequency a trend term andwhite noise The signal 119909

2(119905) consists of a Gaussian-type

impulse interference a square wave with 65 duty ratio atrend term and white noise The signal 119909

3(119905) consists of a

Gaussian-type impulse interference a sawtooth wave with15Hz frequency a trend term and white noise

Figure 8 shows the results of the CEEMD of signals 1199091(119905)

1199092(119905) and 119909

3(119905) Correlation coefficients between filtered

signal and each IMF are illustrated in Table 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0

4Signals (V)

0

4

0

4

Time (s)

x1(t)

x2(t)

x3(t)

minus4

minus4

minus4

Figure 7 Signal waveforms of 1199091(119905) 1199092(119905) and 119909

3(119905)

It can be seen in Table 1 that the IMF which is closelycorrelated to the filtered signal is IMF5 for both signal1199091(119905) and signal 119909

3(119905) and IMF6 for signal 119909

2(119905) They are

used to construct the AR models and the correspondingfeature vectors are estimated as shown in Table 2 After scalarquantization the feature vectors are used to train the HMMfor signal classification

6 Mathematical Problems in Engineering

02

IMF1

Signals (V)

01

IMF2

005

IMF3

01

IMF4

02

IMF5

002

IMF6

001

IMF7

00204

Res

Time (s)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

minus2

minus1

minus05

minus1

minus2

minus01

minus02

(a) 1199091(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

05

IMF6

01

IMF7

005

1

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus5

minus1

(b) 1199092(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

002

IMF6

001

IMF7

002

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus02

minus01

minus02

(c) 1199093(119905)

Figure 8 The decomposition results by CEEMD

Mathematical Problems in Engineering 7

Table 1 Correlation coefficients between filtered signal and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

1199091(119905) minus00031 minus00009 00371 04096 09668 02428 01273 minus00448

1199092(119905) 00051 00004 00435 02111 04695 08887 07214 minus00201

1199093(119905) minus00234 minus00154 00286 05900 08953 01649 01887 minus00210

Table 2 Model parameter estimation results

Signal Model parameter1205931

1205932

1205933

1205934

1205935

1205936

1199091(119905) 47183 minus91103 92034 minus51408 15207 minus01914

1199092(119905) 48894 minus98269 103945 minus61194 19153 minus02531

1199093(119905) 48718 minus99955 111616 minus72529 26430 minus04282

Table 3 Signal classification results

Signal type Test sample Classification results Classification rate [] Overall classificationrate []

1199091(119905) 119909

2(119905) 119909

3(119905)

1199091(119905) 20 19 1 0 95

9671199092(119905) 20 0 19 1 95

1199093(119905) 20 0 0 20 100

A total of 90 feature vectors were collected from threegroups of signals using the proposed approach One-third ofthe feature vectors in each condition were used for trainingthe classifier and others were used for testing The results ofthe signal classification are listed in Table 3

Results in Table 3 indicate that the presented methodbased on CEEMD and time series modeling can effectivelyidentify different signals and the overall classification rate is967 For the purpose of comparison the signal classifica-tion rates use themethod based on time seriesmodeling onlyand the method based on EMD and time series modelingis also calculated 883 and 933 classification rates areobtained respectively It is obvious that efficiency of the signalclassification method proposed in this paper is improved to acertain extent

32 Evaluation Using Experimental Data In order to illus-trate the practicability and effectiveness of the proposedmethod a bearing fault data set from the electrical engi-neering laboratory of Case Western Reserve University isanalyzed [24] The data set is acquired from the test standshown in Figure 9 where it consists of a 2 hp motor atorque transducer a dynamometer and control electronicsThe test bearings support the motor shaft which is the deepgroove ball bearings with the type of 6205-2RS JEMSKFVibration data was collected at 12000 samples per secondusing accelerometers which are attached to the housing withmagnetic bases The motor load level was controlled by thefan in the right side of Figure 9

2 hp motor Torque

transducerDynamometer

Figure 9 Bearing test stand

Figure 10 illustrates representative waveforms of the sam-ple vibration signals measured from the test bearings underfour initial conditions (a) signal from a healthy bearing (b)signal from a bearing with inner ring defect (c) signal froma bearing with rolling element defect and (d) signal from abearing with outer ring defect These signals were measuredunder 0 hpmotor load with themotor speed of 1797 rpmThedecomposed IMFs of these signals are shown in Figure 11

Correlation coefficients calculated between the filteredsignal and each IMF are shown in Table 4

The IMF which is closely correlated to the filtered signalis IMF2 for signal (a) and IMF1 for signals (b) (c) and (d)respectivelyThese IMFs are used for ARmodel constructionThe model order estimation curves of the four conditionsbased on the principle of FPE criterion are shown in Figure 12We can see that when the model order is 6 each modelrsquos

8 Mathematical Problems in Engineering

0 005 01 015 02 025

0

05

minus05

Signals (V)

(a)

0 005 01 015 02 025

0

2

minus2

(b)

0 005 01 015 02 025

0

1

minus1

(c)

0 005 01 015 02 025

0

5

Time (s)

minus5

(d)

Figure 10 Vibration signal waveforms of different conditions

Table 4 Correlation coefficients between filtered signals and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

(a) 04135 07538 04381 04880 04356 01792 00971 minus00056(b) 08794 04275 02583 01337 00421 00285 minus00009 minus00074(c) 09509 02180 02325 01337 00821 00350 minus00017 00009(d) 09878 01267 00636 00509 00136 00060 minus00008 minus00068

Table 5 Model parameter estimation results

Signal Model parameter1205931 1205932 1205933 1205934 1205935 1205936

(a) 31280 minus47797 42245 minus21489 04241 00356(b) 02084 minus13585 05142 minus06356 03471 minus00422(c) 01335 minus16472 03941 minus08473 02142 minus01011(d) minus01172 minus12159 01178 minus01283 01467 02533

residual tends to be stable Therefore the model order isselected as 6 and the results of parameters estimation arelisted in Table 5

The parameters in Table 5 were quantified by Lloydsalgorithm in (12) as feature vectors for training the HMMs ofdifferent conditions The results of quantization are revealedin Figure 13

A total of 160 feature vectors were collected from thefour conditions half of the feature vectors were used fortraining the classifier and others for signal classification andthe classification results are listed in Table 6 Out of 80 testfeature vectors just two cases were not correctly classifiedand the overall classification rate is 975

For comparison Tables 7 and 8 list classification resultsbased on time series modeling usingmeasured signal directlyand based on EMD and time series model method From

the comparison results the proposed method is efficient forrolling bearing fault diagnosis and the overall classificationrate of the proposedmethod is higher to a certain extent thanthe other two methods mentioned above

4 Conclusions

Aiming at diagnosing rolling bearing faults a hybrid ap-proach based on CEEMD and time series modeling is pro-posed in this paperThe CEEMDmethod can decompose thenonstationary signal into a series of IMFs with low compu-tation AR model is an effective approach to extract thefault feature of the vibration signals and the fault patterncan be identified directly by the extracted fault featureswithout establishing the mathematical model and studying

Mathematical Problems in Engineering 9

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

002

IMF1

Signals (V)

002

IMF2

0005

IMF3

001

IMF4

001

IMF5

0005

IMF6

0002

IMF7

0002004

Res

Time (s)

minus02

minus02

minus005

minus01

minus01

minus005

minus002

(a) No defect

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

02

IMF1

Signals (V)

01

IMF2

002

IMF3

002

IMF4

0005

IMF5

0002

IMF6

0001

IMF7

0002004

Res

Time (s)

minus1

minus2

minus02

minus02

minus005

minus002

minus001

(b) Inner ring defect

005

IMF1

Signals (V)

001

IMF2

001

IMF3

0005

IMF4

0005

IMF5

0001

IMF6

05

IMF7

0002004

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus05

minus01

minus01

minus005

minus005

minus001

minus5

times10minus3

(c) Rolling element defect

05

IMF1

Signals (V)

005

IMF2

002

IMF3

002

IMF4

0005

IMF5

0005

IMF6

0005

IMF7

002004006

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus5

minus05

minus02

minus02

minus005

minus005

minus005

(d) Outer ring defect

Figure 11 The decomposition results by CEEMD under different conditions

10 Mathematical Problems in Engineering

0 5 10 15 20 25 300

1

2

3

4

5

6

Model order

Mod

el re

sidua

l

DeltaFPE

times10minus4

(a) No defect

0 5 10 15 20 25 300

001

002

003

004

005

006

007

Model order

Mod

el re

sidua

l

DeltaFPE

(b) Inner ring defect

0 5 10 15 20 25 300

0005

001

0015

002

Model order

Mod

el re

sidua

l

DeltaFPE

(c) Rolling element defect

0 5 10 15 20 25 300

01

02

03

04

05

Model order

Mod

el re

sidua

l

DeltaFPE

(d) Outer ring defect

Figure 12 The model order estimation curves

Table 6 Fault diagnosis using CEEMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 20 0 0 0 100

975Inner ringdefect 20 0 19 1 0 95

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 0 20 100

Mathematical Problems in Engineering 11

1 2 3 4 5 6

0

5

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus5

(a) No defect

1 2 3 4 5 6

0

1

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

(b) Inner ring defect

1 2 3 4 5 6

0

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

minus2

(c) Rolling element defect

1 2 3 4 5 6

0

05A

mpl

itude

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus05

minus1

minus15

(d) Outer ring defect

Figure 13 The results of quantization

Table 7 Fault diagnosis using time series model only

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

900Inner ringdefect 20 1 17 2 0 85

Rolling elementdefect 20 0 2 17 1 85

Outer ringdefect 20 0 0 1 19 95

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

HMM1

HMM training

HMM2 Comparisonprobability

Classification result

Vectorsquantization

Vibration signals

Signal decomposed by CEEMD

Calculate correlation coefficient

Feature vectors

IMF selected for modeling

Model parameter estimation

Feature vectors extracted

Signal filtered

Calculate parameter RP

Feature vectors extraction Fault diagnosis classification

Sample feature vectors

HMMn

Figure 6 The flow chart of the proposed method

where 119873 is the length of the codebook vector partition (119894)is the partition vector with the length of 119873 minus 1 and 119909 is thefeature vector for scalar quantization

Step 6 A test vibration signal can then be acquired fordiagnosis and themodel feature vector is first extractedAfterscalar quantization the feature vector is put into the well-trained HMMs and the corresponding HMM which has themaximum probability is regarded as the classification result[23]

3 Evaluation of the Method Based onCEEMD and AR Model

31 Evaluation Using Simulated Signals To demonstrate thevalidity of the method proposed in this study three signals1199091(119905) 1199092(119905) and 119909

3(119905) are simulated as shown in Figure 7The

signal 1199091(119905) consists of a Gaussian-type impulse interference

a cosine component with 10Hz frequency a trend term andwhite noise The signal 119909

2(119905) consists of a Gaussian-type

impulse interference a square wave with 65 duty ratio atrend term and white noise The signal 119909

3(119905) consists of a

Gaussian-type impulse interference a sawtooth wave with15Hz frequency a trend term and white noise

Figure 8 shows the results of the CEEMD of signals 1199091(119905)

1199092(119905) and 119909

3(119905) Correlation coefficients between filtered

signal and each IMF are illustrated in Table 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0

4Signals (V)

0

4

0

4

Time (s)

x1(t)

x2(t)

x3(t)

minus4

minus4

minus4

Figure 7 Signal waveforms of 1199091(119905) 1199092(119905) and 119909

3(119905)

It can be seen in Table 1 that the IMF which is closelycorrelated to the filtered signal is IMF5 for both signal1199091(119905) and signal 119909

3(119905) and IMF6 for signal 119909

2(119905) They are

used to construct the AR models and the correspondingfeature vectors are estimated as shown in Table 2 After scalarquantization the feature vectors are used to train the HMMfor signal classification

6 Mathematical Problems in Engineering

02

IMF1

Signals (V)

01

IMF2

005

IMF3

01

IMF4

02

IMF5

002

IMF6

001

IMF7

00204

Res

Time (s)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

minus2

minus1

minus05

minus1

minus2

minus01

minus02

(a) 1199091(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

05

IMF6

01

IMF7

005

1

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus5

minus1

(b) 1199092(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

002

IMF6

001

IMF7

002

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus02

minus01

minus02

(c) 1199093(119905)

Figure 8 The decomposition results by CEEMD

Mathematical Problems in Engineering 7

Table 1 Correlation coefficients between filtered signal and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

1199091(119905) minus00031 minus00009 00371 04096 09668 02428 01273 minus00448

1199092(119905) 00051 00004 00435 02111 04695 08887 07214 minus00201

1199093(119905) minus00234 minus00154 00286 05900 08953 01649 01887 minus00210

Table 2 Model parameter estimation results

Signal Model parameter1205931

1205932

1205933

1205934

1205935

1205936

1199091(119905) 47183 minus91103 92034 minus51408 15207 minus01914

1199092(119905) 48894 minus98269 103945 minus61194 19153 minus02531

1199093(119905) 48718 minus99955 111616 minus72529 26430 minus04282

Table 3 Signal classification results

Signal type Test sample Classification results Classification rate [] Overall classificationrate []

1199091(119905) 119909

2(119905) 119909

3(119905)

1199091(119905) 20 19 1 0 95

9671199092(119905) 20 0 19 1 95

1199093(119905) 20 0 0 20 100

A total of 90 feature vectors were collected from threegroups of signals using the proposed approach One-third ofthe feature vectors in each condition were used for trainingthe classifier and others were used for testing The results ofthe signal classification are listed in Table 3

Results in Table 3 indicate that the presented methodbased on CEEMD and time series modeling can effectivelyidentify different signals and the overall classification rate is967 For the purpose of comparison the signal classifica-tion rates use themethod based on time seriesmodeling onlyand the method based on EMD and time series modelingis also calculated 883 and 933 classification rates areobtained respectively It is obvious that efficiency of the signalclassification method proposed in this paper is improved to acertain extent

32 Evaluation Using Experimental Data In order to illus-trate the practicability and effectiveness of the proposedmethod a bearing fault data set from the electrical engi-neering laboratory of Case Western Reserve University isanalyzed [24] The data set is acquired from the test standshown in Figure 9 where it consists of a 2 hp motor atorque transducer a dynamometer and control electronicsThe test bearings support the motor shaft which is the deepgroove ball bearings with the type of 6205-2RS JEMSKFVibration data was collected at 12000 samples per secondusing accelerometers which are attached to the housing withmagnetic bases The motor load level was controlled by thefan in the right side of Figure 9

2 hp motor Torque

transducerDynamometer

Figure 9 Bearing test stand

Figure 10 illustrates representative waveforms of the sam-ple vibration signals measured from the test bearings underfour initial conditions (a) signal from a healthy bearing (b)signal from a bearing with inner ring defect (c) signal froma bearing with rolling element defect and (d) signal from abearing with outer ring defect These signals were measuredunder 0 hpmotor load with themotor speed of 1797 rpmThedecomposed IMFs of these signals are shown in Figure 11

Correlation coefficients calculated between the filteredsignal and each IMF are shown in Table 4

The IMF which is closely correlated to the filtered signalis IMF2 for signal (a) and IMF1 for signals (b) (c) and (d)respectivelyThese IMFs are used for ARmodel constructionThe model order estimation curves of the four conditionsbased on the principle of FPE criterion are shown in Figure 12We can see that when the model order is 6 each modelrsquos

8 Mathematical Problems in Engineering

0 005 01 015 02 025

0

05

minus05

Signals (V)

(a)

0 005 01 015 02 025

0

2

minus2

(b)

0 005 01 015 02 025

0

1

minus1

(c)

0 005 01 015 02 025

0

5

Time (s)

minus5

(d)

Figure 10 Vibration signal waveforms of different conditions

Table 4 Correlation coefficients between filtered signals and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

(a) 04135 07538 04381 04880 04356 01792 00971 minus00056(b) 08794 04275 02583 01337 00421 00285 minus00009 minus00074(c) 09509 02180 02325 01337 00821 00350 minus00017 00009(d) 09878 01267 00636 00509 00136 00060 minus00008 minus00068

Table 5 Model parameter estimation results

Signal Model parameter1205931 1205932 1205933 1205934 1205935 1205936

(a) 31280 minus47797 42245 minus21489 04241 00356(b) 02084 minus13585 05142 minus06356 03471 minus00422(c) 01335 minus16472 03941 minus08473 02142 minus01011(d) minus01172 minus12159 01178 minus01283 01467 02533

residual tends to be stable Therefore the model order isselected as 6 and the results of parameters estimation arelisted in Table 5

The parameters in Table 5 were quantified by Lloydsalgorithm in (12) as feature vectors for training the HMMs ofdifferent conditions The results of quantization are revealedin Figure 13

A total of 160 feature vectors were collected from thefour conditions half of the feature vectors were used fortraining the classifier and others for signal classification andthe classification results are listed in Table 6 Out of 80 testfeature vectors just two cases were not correctly classifiedand the overall classification rate is 975

For comparison Tables 7 and 8 list classification resultsbased on time series modeling usingmeasured signal directlyand based on EMD and time series model method From

the comparison results the proposed method is efficient forrolling bearing fault diagnosis and the overall classificationrate of the proposedmethod is higher to a certain extent thanthe other two methods mentioned above

4 Conclusions

Aiming at diagnosing rolling bearing faults a hybrid ap-proach based on CEEMD and time series modeling is pro-posed in this paperThe CEEMDmethod can decompose thenonstationary signal into a series of IMFs with low compu-tation AR model is an effective approach to extract thefault feature of the vibration signals and the fault patterncan be identified directly by the extracted fault featureswithout establishing the mathematical model and studying

Mathematical Problems in Engineering 9

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

002

IMF1

Signals (V)

002

IMF2

0005

IMF3

001

IMF4

001

IMF5

0005

IMF6

0002

IMF7

0002004

Res

Time (s)

minus02

minus02

minus005

minus01

minus01

minus005

minus002

(a) No defect

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

02

IMF1

Signals (V)

01

IMF2

002

IMF3

002

IMF4

0005

IMF5

0002

IMF6

0001

IMF7

0002004

Res

Time (s)

minus1

minus2

minus02

minus02

minus005

minus002

minus001

(b) Inner ring defect

005

IMF1

Signals (V)

001

IMF2

001

IMF3

0005

IMF4

0005

IMF5

0001

IMF6

05

IMF7

0002004

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus05

minus01

minus01

minus005

minus005

minus001

minus5

times10minus3

(c) Rolling element defect

05

IMF1

Signals (V)

005

IMF2

002

IMF3

002

IMF4

0005

IMF5

0005

IMF6

0005

IMF7

002004006

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus5

minus05

minus02

minus02

minus005

minus005

minus005

(d) Outer ring defect

Figure 11 The decomposition results by CEEMD under different conditions

10 Mathematical Problems in Engineering

0 5 10 15 20 25 300

1

2

3

4

5

6

Model order

Mod

el re

sidua

l

DeltaFPE

times10minus4

(a) No defect

0 5 10 15 20 25 300

001

002

003

004

005

006

007

Model order

Mod

el re

sidua

l

DeltaFPE

(b) Inner ring defect

0 5 10 15 20 25 300

0005

001

0015

002

Model order

Mod

el re

sidua

l

DeltaFPE

(c) Rolling element defect

0 5 10 15 20 25 300

01

02

03

04

05

Model order

Mod

el re

sidua

l

DeltaFPE

(d) Outer ring defect

Figure 12 The model order estimation curves

Table 6 Fault diagnosis using CEEMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 20 0 0 0 100

975Inner ringdefect 20 0 19 1 0 95

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 0 20 100

Mathematical Problems in Engineering 11

1 2 3 4 5 6

0

5

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus5

(a) No defect

1 2 3 4 5 6

0

1

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

(b) Inner ring defect

1 2 3 4 5 6

0

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

minus2

(c) Rolling element defect

1 2 3 4 5 6

0

05A

mpl

itude

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus05

minus1

minus15

(d) Outer ring defect

Figure 13 The results of quantization

Table 7 Fault diagnosis using time series model only

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

900Inner ringdefect 20 1 17 2 0 85

Rolling elementdefect 20 0 2 17 1 85

Outer ringdefect 20 0 0 1 19 95

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

02

IMF1

Signals (V)

01

IMF2

005

IMF3

01

IMF4

02

IMF5

002

IMF6

001

IMF7

00204

Res

Time (s)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

minus2

minus1

minus05

minus1

minus2

minus01

minus02

(a) 1199091(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

05

IMF6

01

IMF7

005

1

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus5

minus1

(b) 1199092(119905)

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

0 02 04 06 08 1

02

IMF1

Signals (V)

01

IMF2

01

IMF3

01

IMF4

01

IMF5

002

IMF6

001

IMF7

002

Res

Time (s)

minus2

minus1

minus1

minus1

minus1

minus02

minus01

minus02

(c) 1199093(119905)

Figure 8 The decomposition results by CEEMD

Mathematical Problems in Engineering 7

Table 1 Correlation coefficients between filtered signal and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

1199091(119905) minus00031 minus00009 00371 04096 09668 02428 01273 minus00448

1199092(119905) 00051 00004 00435 02111 04695 08887 07214 minus00201

1199093(119905) minus00234 minus00154 00286 05900 08953 01649 01887 minus00210

Table 2 Model parameter estimation results

Signal Model parameter1205931

1205932

1205933

1205934

1205935

1205936

1199091(119905) 47183 minus91103 92034 minus51408 15207 minus01914

1199092(119905) 48894 minus98269 103945 minus61194 19153 minus02531

1199093(119905) 48718 minus99955 111616 minus72529 26430 minus04282

Table 3 Signal classification results

Signal type Test sample Classification results Classification rate [] Overall classificationrate []

1199091(119905) 119909

2(119905) 119909

3(119905)

1199091(119905) 20 19 1 0 95

9671199092(119905) 20 0 19 1 95

1199093(119905) 20 0 0 20 100

A total of 90 feature vectors were collected from threegroups of signals using the proposed approach One-third ofthe feature vectors in each condition were used for trainingthe classifier and others were used for testing The results ofthe signal classification are listed in Table 3

Results in Table 3 indicate that the presented methodbased on CEEMD and time series modeling can effectivelyidentify different signals and the overall classification rate is967 For the purpose of comparison the signal classifica-tion rates use themethod based on time seriesmodeling onlyand the method based on EMD and time series modelingis also calculated 883 and 933 classification rates areobtained respectively It is obvious that efficiency of the signalclassification method proposed in this paper is improved to acertain extent

32 Evaluation Using Experimental Data In order to illus-trate the practicability and effectiveness of the proposedmethod a bearing fault data set from the electrical engi-neering laboratory of Case Western Reserve University isanalyzed [24] The data set is acquired from the test standshown in Figure 9 where it consists of a 2 hp motor atorque transducer a dynamometer and control electronicsThe test bearings support the motor shaft which is the deepgroove ball bearings with the type of 6205-2RS JEMSKFVibration data was collected at 12000 samples per secondusing accelerometers which are attached to the housing withmagnetic bases The motor load level was controlled by thefan in the right side of Figure 9

2 hp motor Torque

transducerDynamometer

Figure 9 Bearing test stand

Figure 10 illustrates representative waveforms of the sam-ple vibration signals measured from the test bearings underfour initial conditions (a) signal from a healthy bearing (b)signal from a bearing with inner ring defect (c) signal froma bearing with rolling element defect and (d) signal from abearing with outer ring defect These signals were measuredunder 0 hpmotor load with themotor speed of 1797 rpmThedecomposed IMFs of these signals are shown in Figure 11

Correlation coefficients calculated between the filteredsignal and each IMF are shown in Table 4

The IMF which is closely correlated to the filtered signalis IMF2 for signal (a) and IMF1 for signals (b) (c) and (d)respectivelyThese IMFs are used for ARmodel constructionThe model order estimation curves of the four conditionsbased on the principle of FPE criterion are shown in Figure 12We can see that when the model order is 6 each modelrsquos

8 Mathematical Problems in Engineering

0 005 01 015 02 025

0

05

minus05

Signals (V)

(a)

0 005 01 015 02 025

0

2

minus2

(b)

0 005 01 015 02 025

0

1

minus1

(c)

0 005 01 015 02 025

0

5

Time (s)

minus5

(d)

Figure 10 Vibration signal waveforms of different conditions

Table 4 Correlation coefficients between filtered signals and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

(a) 04135 07538 04381 04880 04356 01792 00971 minus00056(b) 08794 04275 02583 01337 00421 00285 minus00009 minus00074(c) 09509 02180 02325 01337 00821 00350 minus00017 00009(d) 09878 01267 00636 00509 00136 00060 minus00008 minus00068

Table 5 Model parameter estimation results

Signal Model parameter1205931 1205932 1205933 1205934 1205935 1205936

(a) 31280 minus47797 42245 minus21489 04241 00356(b) 02084 minus13585 05142 minus06356 03471 minus00422(c) 01335 minus16472 03941 minus08473 02142 minus01011(d) minus01172 minus12159 01178 minus01283 01467 02533

residual tends to be stable Therefore the model order isselected as 6 and the results of parameters estimation arelisted in Table 5

The parameters in Table 5 were quantified by Lloydsalgorithm in (12) as feature vectors for training the HMMs ofdifferent conditions The results of quantization are revealedin Figure 13

A total of 160 feature vectors were collected from thefour conditions half of the feature vectors were used fortraining the classifier and others for signal classification andthe classification results are listed in Table 6 Out of 80 testfeature vectors just two cases were not correctly classifiedand the overall classification rate is 975

For comparison Tables 7 and 8 list classification resultsbased on time series modeling usingmeasured signal directlyand based on EMD and time series model method From

the comparison results the proposed method is efficient forrolling bearing fault diagnosis and the overall classificationrate of the proposedmethod is higher to a certain extent thanthe other two methods mentioned above

4 Conclusions

Aiming at diagnosing rolling bearing faults a hybrid ap-proach based on CEEMD and time series modeling is pro-posed in this paperThe CEEMDmethod can decompose thenonstationary signal into a series of IMFs with low compu-tation AR model is an effective approach to extract thefault feature of the vibration signals and the fault patterncan be identified directly by the extracted fault featureswithout establishing the mathematical model and studying

Mathematical Problems in Engineering 9

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

002

IMF1

Signals (V)

002

IMF2

0005

IMF3

001

IMF4

001

IMF5

0005

IMF6

0002

IMF7

0002004

Res

Time (s)

minus02

minus02

minus005

minus01

minus01

minus005

minus002

(a) No defect

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

02

IMF1

Signals (V)

01

IMF2

002

IMF3

002

IMF4

0005

IMF5

0002

IMF6

0001

IMF7

0002004

Res

Time (s)

minus1

minus2

minus02

minus02

minus005

minus002

minus001

(b) Inner ring defect

005

IMF1

Signals (V)

001

IMF2

001

IMF3

0005

IMF4

0005

IMF5

0001

IMF6

05

IMF7

0002004

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus05

minus01

minus01

minus005

minus005

minus001

minus5

times10minus3

(c) Rolling element defect

05

IMF1

Signals (V)

005

IMF2

002

IMF3

002

IMF4

0005

IMF5

0005

IMF6

0005

IMF7

002004006

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus5

minus05

minus02

minus02

minus005

minus005

minus005

(d) Outer ring defect

Figure 11 The decomposition results by CEEMD under different conditions

10 Mathematical Problems in Engineering

0 5 10 15 20 25 300

1

2

3

4

5

6

Model order

Mod

el re

sidua

l

DeltaFPE

times10minus4

(a) No defect

0 5 10 15 20 25 300

001

002

003

004

005

006

007

Model order

Mod

el re

sidua

l

DeltaFPE

(b) Inner ring defect

0 5 10 15 20 25 300

0005

001

0015

002

Model order

Mod

el re

sidua

l

DeltaFPE

(c) Rolling element defect

0 5 10 15 20 25 300

01

02

03

04

05

Model order

Mod

el re

sidua

l

DeltaFPE

(d) Outer ring defect

Figure 12 The model order estimation curves

Table 6 Fault diagnosis using CEEMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 20 0 0 0 100

975Inner ringdefect 20 0 19 1 0 95

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 0 20 100

Mathematical Problems in Engineering 11

1 2 3 4 5 6

0

5

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus5

(a) No defect

1 2 3 4 5 6

0

1

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

(b) Inner ring defect

1 2 3 4 5 6

0

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

minus2

(c) Rolling element defect

1 2 3 4 5 6

0

05A

mpl

itude

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus05

minus1

minus15

(d) Outer ring defect

Figure 13 The results of quantization

Table 7 Fault diagnosis using time series model only

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

900Inner ringdefect 20 1 17 2 0 85

Rolling elementdefect 20 0 2 17 1 85

Outer ringdefect 20 0 0 1 19 95

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 Correlation coefficients between filtered signal and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

1199091(119905) minus00031 minus00009 00371 04096 09668 02428 01273 minus00448

1199092(119905) 00051 00004 00435 02111 04695 08887 07214 minus00201

1199093(119905) minus00234 minus00154 00286 05900 08953 01649 01887 minus00210

Table 2 Model parameter estimation results

Signal Model parameter1205931

1205932

1205933

1205934

1205935

1205936

1199091(119905) 47183 minus91103 92034 minus51408 15207 minus01914

1199092(119905) 48894 minus98269 103945 minus61194 19153 minus02531

1199093(119905) 48718 minus99955 111616 minus72529 26430 minus04282

Table 3 Signal classification results

Signal type Test sample Classification results Classification rate [] Overall classificationrate []

1199091(119905) 119909

2(119905) 119909

3(119905)

1199091(119905) 20 19 1 0 95

9671199092(119905) 20 0 19 1 95

1199093(119905) 20 0 0 20 100

A total of 90 feature vectors were collected from threegroups of signals using the proposed approach One-third ofthe feature vectors in each condition were used for trainingthe classifier and others were used for testing The results ofthe signal classification are listed in Table 3

Results in Table 3 indicate that the presented methodbased on CEEMD and time series modeling can effectivelyidentify different signals and the overall classification rate is967 For the purpose of comparison the signal classifica-tion rates use themethod based on time seriesmodeling onlyand the method based on EMD and time series modelingis also calculated 883 and 933 classification rates areobtained respectively It is obvious that efficiency of the signalclassification method proposed in this paper is improved to acertain extent

32 Evaluation Using Experimental Data In order to illus-trate the practicability and effectiveness of the proposedmethod a bearing fault data set from the electrical engi-neering laboratory of Case Western Reserve University isanalyzed [24] The data set is acquired from the test standshown in Figure 9 where it consists of a 2 hp motor atorque transducer a dynamometer and control electronicsThe test bearings support the motor shaft which is the deepgroove ball bearings with the type of 6205-2RS JEMSKFVibration data was collected at 12000 samples per secondusing accelerometers which are attached to the housing withmagnetic bases The motor load level was controlled by thefan in the right side of Figure 9

2 hp motor Torque

transducerDynamometer

Figure 9 Bearing test stand

Figure 10 illustrates representative waveforms of the sam-ple vibration signals measured from the test bearings underfour initial conditions (a) signal from a healthy bearing (b)signal from a bearing with inner ring defect (c) signal froma bearing with rolling element defect and (d) signal from abearing with outer ring defect These signals were measuredunder 0 hpmotor load with themotor speed of 1797 rpmThedecomposed IMFs of these signals are shown in Figure 11

Correlation coefficients calculated between the filteredsignal and each IMF are shown in Table 4

The IMF which is closely correlated to the filtered signalis IMF2 for signal (a) and IMF1 for signals (b) (c) and (d)respectivelyThese IMFs are used for ARmodel constructionThe model order estimation curves of the four conditionsbased on the principle of FPE criterion are shown in Figure 12We can see that when the model order is 6 each modelrsquos

8 Mathematical Problems in Engineering

0 005 01 015 02 025

0

05

minus05

Signals (V)

(a)

0 005 01 015 02 025

0

2

minus2

(b)

0 005 01 015 02 025

0

1

minus1

(c)

0 005 01 015 02 025

0

5

Time (s)

minus5

(d)

Figure 10 Vibration signal waveforms of different conditions

Table 4 Correlation coefficients between filtered signals and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

(a) 04135 07538 04381 04880 04356 01792 00971 minus00056(b) 08794 04275 02583 01337 00421 00285 minus00009 minus00074(c) 09509 02180 02325 01337 00821 00350 minus00017 00009(d) 09878 01267 00636 00509 00136 00060 minus00008 minus00068

Table 5 Model parameter estimation results

Signal Model parameter1205931 1205932 1205933 1205934 1205935 1205936

(a) 31280 minus47797 42245 minus21489 04241 00356(b) 02084 minus13585 05142 minus06356 03471 minus00422(c) 01335 minus16472 03941 minus08473 02142 minus01011(d) minus01172 minus12159 01178 minus01283 01467 02533

residual tends to be stable Therefore the model order isselected as 6 and the results of parameters estimation arelisted in Table 5

The parameters in Table 5 were quantified by Lloydsalgorithm in (12) as feature vectors for training the HMMs ofdifferent conditions The results of quantization are revealedin Figure 13

A total of 160 feature vectors were collected from thefour conditions half of the feature vectors were used fortraining the classifier and others for signal classification andthe classification results are listed in Table 6 Out of 80 testfeature vectors just two cases were not correctly classifiedand the overall classification rate is 975

For comparison Tables 7 and 8 list classification resultsbased on time series modeling usingmeasured signal directlyand based on EMD and time series model method From

the comparison results the proposed method is efficient forrolling bearing fault diagnosis and the overall classificationrate of the proposedmethod is higher to a certain extent thanthe other two methods mentioned above

4 Conclusions

Aiming at diagnosing rolling bearing faults a hybrid ap-proach based on CEEMD and time series modeling is pro-posed in this paperThe CEEMDmethod can decompose thenonstationary signal into a series of IMFs with low compu-tation AR model is an effective approach to extract thefault feature of the vibration signals and the fault patterncan be identified directly by the extracted fault featureswithout establishing the mathematical model and studying

Mathematical Problems in Engineering 9

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

002

IMF1

Signals (V)

002

IMF2

0005

IMF3

001

IMF4

001

IMF5

0005

IMF6

0002

IMF7

0002004

Res

Time (s)

minus02

minus02

minus005

minus01

minus01

minus005

minus002

(a) No defect

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

02

IMF1

Signals (V)

01

IMF2

002

IMF3

002

IMF4

0005

IMF5

0002

IMF6

0001

IMF7

0002004

Res

Time (s)

minus1

minus2

minus02

minus02

minus005

minus002

minus001

(b) Inner ring defect

005

IMF1

Signals (V)

001

IMF2

001

IMF3

0005

IMF4

0005

IMF5

0001

IMF6

05

IMF7

0002004

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus05

minus01

minus01

minus005

minus005

minus001

minus5

times10minus3

(c) Rolling element defect

05

IMF1

Signals (V)

005

IMF2

002

IMF3

002

IMF4

0005

IMF5

0005

IMF6

0005

IMF7

002004006

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus5

minus05

minus02

minus02

minus005

minus005

minus005

(d) Outer ring defect

Figure 11 The decomposition results by CEEMD under different conditions

10 Mathematical Problems in Engineering

0 5 10 15 20 25 300

1

2

3

4

5

6

Model order

Mod

el re

sidua

l

DeltaFPE

times10minus4

(a) No defect

0 5 10 15 20 25 300

001

002

003

004

005

006

007

Model order

Mod

el re

sidua

l

DeltaFPE

(b) Inner ring defect

0 5 10 15 20 25 300

0005

001

0015

002

Model order

Mod

el re

sidua

l

DeltaFPE

(c) Rolling element defect

0 5 10 15 20 25 300

01

02

03

04

05

Model order

Mod

el re

sidua

l

DeltaFPE

(d) Outer ring defect

Figure 12 The model order estimation curves

Table 6 Fault diagnosis using CEEMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 20 0 0 0 100

975Inner ringdefect 20 0 19 1 0 95

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 0 20 100

Mathematical Problems in Engineering 11

1 2 3 4 5 6

0

5

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus5

(a) No defect

1 2 3 4 5 6

0

1

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

(b) Inner ring defect

1 2 3 4 5 6

0

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

minus2

(c) Rolling element defect

1 2 3 4 5 6

0

05A

mpl

itude

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus05

minus1

minus15

(d) Outer ring defect

Figure 13 The results of quantization

Table 7 Fault diagnosis using time series model only

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

900Inner ringdefect 20 1 17 2 0 85

Rolling elementdefect 20 0 2 17 1 85

Outer ringdefect 20 0 0 1 19 95

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 005 01 015 02 025

0

05

minus05

Signals (V)

(a)

0 005 01 015 02 025

0

2

minus2

(b)

0 005 01 015 02 025

0

1

minus1

(c)

0 005 01 015 02 025

0

5

Time (s)

minus5

(d)

Figure 10 Vibration signal waveforms of different conditions

Table 4 Correlation coefficients between filtered signals and each IMF

Signal Correlation coefficientIMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8

(a) 04135 07538 04381 04880 04356 01792 00971 minus00056(b) 08794 04275 02583 01337 00421 00285 minus00009 minus00074(c) 09509 02180 02325 01337 00821 00350 minus00017 00009(d) 09878 01267 00636 00509 00136 00060 minus00008 minus00068

Table 5 Model parameter estimation results

Signal Model parameter1205931 1205932 1205933 1205934 1205935 1205936

(a) 31280 minus47797 42245 minus21489 04241 00356(b) 02084 minus13585 05142 minus06356 03471 minus00422(c) 01335 minus16472 03941 minus08473 02142 minus01011(d) minus01172 minus12159 01178 minus01283 01467 02533

residual tends to be stable Therefore the model order isselected as 6 and the results of parameters estimation arelisted in Table 5

The parameters in Table 5 were quantified by Lloydsalgorithm in (12) as feature vectors for training the HMMs ofdifferent conditions The results of quantization are revealedin Figure 13

A total of 160 feature vectors were collected from thefour conditions half of the feature vectors were used fortraining the classifier and others for signal classification andthe classification results are listed in Table 6 Out of 80 testfeature vectors just two cases were not correctly classifiedand the overall classification rate is 975

For comparison Tables 7 and 8 list classification resultsbased on time series modeling usingmeasured signal directlyand based on EMD and time series model method From

the comparison results the proposed method is efficient forrolling bearing fault diagnosis and the overall classificationrate of the proposedmethod is higher to a certain extent thanthe other two methods mentioned above

4 Conclusions

Aiming at diagnosing rolling bearing faults a hybrid ap-proach based on CEEMD and time series modeling is pro-posed in this paperThe CEEMDmethod can decompose thenonstationary signal into a series of IMFs with low compu-tation AR model is an effective approach to extract thefault feature of the vibration signals and the fault patterncan be identified directly by the extracted fault featureswithout establishing the mathematical model and studying

Mathematical Problems in Engineering 9

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

002

IMF1

Signals (V)

002

IMF2

0005

IMF3

001

IMF4

001

IMF5

0005

IMF6

0002

IMF7

0002004

Res

Time (s)

minus02

minus02

minus005

minus01

minus01

minus005

minus002

(a) No defect

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

02

IMF1

Signals (V)

01

IMF2

002

IMF3

002

IMF4

0005

IMF5

0002

IMF6

0001

IMF7

0002004

Res

Time (s)

minus1

minus2

minus02

minus02

minus005

minus002

minus001

(b) Inner ring defect

005

IMF1

Signals (V)

001

IMF2

001

IMF3

0005

IMF4

0005

IMF5

0001

IMF6

05

IMF7

0002004

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus05

minus01

minus01

minus005

minus005

minus001

minus5

times10minus3

(c) Rolling element defect

05

IMF1

Signals (V)

005

IMF2

002

IMF3

002

IMF4

0005

IMF5

0005

IMF6

0005

IMF7

002004006

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus5

minus05

minus02

minus02

minus005

minus005

minus005

(d) Outer ring defect

Figure 11 The decomposition results by CEEMD under different conditions

10 Mathematical Problems in Engineering

0 5 10 15 20 25 300

1

2

3

4

5

6

Model order

Mod

el re

sidua

l

DeltaFPE

times10minus4

(a) No defect

0 5 10 15 20 25 300

001

002

003

004

005

006

007

Model order

Mod

el re

sidua

l

DeltaFPE

(b) Inner ring defect

0 5 10 15 20 25 300

0005

001

0015

002

Model order

Mod

el re

sidua

l

DeltaFPE

(c) Rolling element defect

0 5 10 15 20 25 300

01

02

03

04

05

Model order

Mod

el re

sidua

l

DeltaFPE

(d) Outer ring defect

Figure 12 The model order estimation curves

Table 6 Fault diagnosis using CEEMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 20 0 0 0 100

975Inner ringdefect 20 0 19 1 0 95

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 0 20 100

Mathematical Problems in Engineering 11

1 2 3 4 5 6

0

5

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus5

(a) No defect

1 2 3 4 5 6

0

1

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

(b) Inner ring defect

1 2 3 4 5 6

0

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

minus2

(c) Rolling element defect

1 2 3 4 5 6

0

05A

mpl

itude

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus05

minus1

minus15

(d) Outer ring defect

Figure 13 The results of quantization

Table 7 Fault diagnosis using time series model only

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

900Inner ringdefect 20 1 17 2 0 85

Rolling elementdefect 20 0 2 17 1 85

Outer ringdefect 20 0 0 1 19 95

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

002

IMF1

Signals (V)

002

IMF2

0005

IMF3

001

IMF4

001

IMF5

0005

IMF6

0002

IMF7

0002004

Res

Time (s)

minus02

minus02

minus005

minus01

minus01

minus005

minus002

(a) No defect

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

02

IMF1

Signals (V)

01

IMF2

002

IMF3

002

IMF4

0005

IMF5

0002

IMF6

0001

IMF7

0002004

Res

Time (s)

minus1

minus2

minus02

minus02

minus005

minus002

minus001

(b) Inner ring defect

005

IMF1

Signals (V)

001

IMF2

001

IMF3

0005

IMF4

0005

IMF5

0001

IMF6

05

IMF7

0002004

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus05

minus01

minus01

minus005

minus005

minus001

minus5

times10minus3

(c) Rolling element defect

05

IMF1

Signals (V)

005

IMF2

002

IMF3

002

IMF4

0005

IMF5

0005

IMF6

0005

IMF7

002004006

Res

Time (s)

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

0 005 01 015 02 025

minus5

minus05

minus02

minus02

minus005

minus005

minus005

(d) Outer ring defect

Figure 11 The decomposition results by CEEMD under different conditions

10 Mathematical Problems in Engineering

0 5 10 15 20 25 300

1

2

3

4

5

6

Model order

Mod

el re

sidua

l

DeltaFPE

times10minus4

(a) No defect

0 5 10 15 20 25 300

001

002

003

004

005

006

007

Model order

Mod

el re

sidua

l

DeltaFPE

(b) Inner ring defect

0 5 10 15 20 25 300

0005

001

0015

002

Model order

Mod

el re

sidua

l

DeltaFPE

(c) Rolling element defect

0 5 10 15 20 25 300

01

02

03

04

05

Model order

Mod

el re

sidua

l

DeltaFPE

(d) Outer ring defect

Figure 12 The model order estimation curves

Table 6 Fault diagnosis using CEEMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 20 0 0 0 100

975Inner ringdefect 20 0 19 1 0 95

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 0 20 100

Mathematical Problems in Engineering 11

1 2 3 4 5 6

0

5

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus5

(a) No defect

1 2 3 4 5 6

0

1

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

(b) Inner ring defect

1 2 3 4 5 6

0

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

minus2

(c) Rolling element defect

1 2 3 4 5 6

0

05A

mpl

itude

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus05

minus1

minus15

(d) Outer ring defect

Figure 13 The results of quantization

Table 7 Fault diagnosis using time series model only

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

900Inner ringdefect 20 1 17 2 0 85

Rolling elementdefect 20 0 2 17 1 85

Outer ringdefect 20 0 0 1 19 95

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 5 10 15 20 25 300

1

2

3

4

5

6

Model order

Mod

el re

sidua

l

DeltaFPE

times10minus4

(a) No defect

0 5 10 15 20 25 300

001

002

003

004

005

006

007

Model order

Mod

el re

sidua

l

DeltaFPE

(b) Inner ring defect

0 5 10 15 20 25 300

0005

001

0015

002

Model order

Mod

el re

sidua

l

DeltaFPE

(c) Rolling element defect

0 5 10 15 20 25 300

01

02

03

04

05

Model order

Mod

el re

sidua

l

DeltaFPE

(d) Outer ring defect

Figure 12 The model order estimation curves

Table 6 Fault diagnosis using CEEMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 20 0 0 0 100

975Inner ringdefect 20 0 19 1 0 95

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 0 20 100

Mathematical Problems in Engineering 11

1 2 3 4 5 6

0

5

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus5

(a) No defect

1 2 3 4 5 6

0

1

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

(b) Inner ring defect

1 2 3 4 5 6

0

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

minus2

(c) Rolling element defect

1 2 3 4 5 6

0

05A

mpl

itude

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus05

minus1

minus15

(d) Outer ring defect

Figure 13 The results of quantization

Table 7 Fault diagnosis using time series model only

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

900Inner ringdefect 20 1 17 2 0 85

Rolling elementdefect 20 0 2 17 1 85

Outer ringdefect 20 0 0 1 19 95

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

1 2 3 4 5 6

0

5

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus5

(a) No defect

1 2 3 4 5 6

0

1

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

(b) Inner ring defect

1 2 3 4 5 6

0

Am

plitu

de

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus1

minus2

(c) Rolling element defect

1 2 3 4 5 6

0

05A

mpl

itude

1 2 3 4 5 60

10

20

Am

plitu

de

Parameter

minus05

minus1

minus15

(d) Outer ring defect

Figure 13 The results of quantization

Table 7 Fault diagnosis using time series model only

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

900Inner ringdefect 20 1 17 2 0 85

Rolling elementdefect 20 0 2 17 1 85

Outer ringdefect 20 0 0 1 19 95

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Table 8 Fault diagnosis using EMD and time series model

Fault type Test sample Classification results Classification rate [] Overall classificationrate []

Nodefect

Innerringdefect

Rollingelementdefect

Outerringdefect

No defect 20 19 1 0 0 95

9375Inner ringdefect 20 0 18 2 0 90

Rolling elementdefect 20 0 1 19 0 95

Outer ringdefect 20 0 0 1 19 95

the fault mechanism of the system In this paper the CEEMDmethod is used as a pretreatment which can increase theaccuracy of the AR model for the measured signal andthe AR model of the IMF which is closely correlated tothe filtered signal is established to extract the fault featureparameters Comparing to the EMD-AR approach and thedirect modeling approach where raw signals are directlyused as input for AR modeling a higher classification ratewas shown to be achieved by using the new approach (eg967 for simulated signals and 975 for experimental data)Meanwhile we anticipate that the proposed method can alsobe used for incipient fault diagnosis in rolling bearing wherefurther experiments are needed to verify the accuracy Sincethe approach presented in this study is generic in nature itcan be readily adapted to a broad range of applications formachine fault diagnosis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been supported in part by the National NaturalScience Foundation of China (no 61101163 and no 51175080)and the Nature Science Foundation of Jiangsu Province ofChina (no BK2012739)

References

[1] B Li P L Zhang ZWang SMi andD Liu ldquoAweightedmulti-scale morphological gradient filter for rolling element bearingfault detectionrdquo ISA Transactions vol 50 no 4 pp 599ndash6082011

[2] R Yan and R X Gao ldquoWavelet domain principal feature anal-ysis for spindle health diagnosisrdquo Structural Health Monitoringvol 10 no 6 pp 631ndash642 2011

[3] C Junsheng Y Dejie and Y Yu ldquoA fault diagnosis approachfor roller bearings based on EMD method and AR modelrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 350ndash362 2006

[4] J L Wang and G Wang ldquoGear box fault diagnosis based onsystemmodelrdquoMechanical EngineeringampAutomation no 1 pp118ndash119 2012

[5] D J Yu M F Chen J S Cheng and Y Y Yang ldquoFault diagnosisapproach for rotor systems based on AR model and supportvector machinerdquo System Engineering Theory amp Practice vol 27no 5 pp 152ndash157 2007

[6] X J Zeng and Y J Huang ldquoFault diagnosis based on ARmodel and support vector machinerdquo Mechanical Science andTechnology for Aerospace Engineering vol 29 no 7 pp 972ndash9752010

[7] H Ding Y Wu and S Z Yang Fault Diagnosis by TimeSeries Analysis Applied Time Series Analysis World ScientificPublishing Singapore 1989

[8] Z M Yang and A D Yin ldquoFault recognition of vehicle trans-mission gear based on time series analysis and fuzzy clusterrdquoTransactions of the Chinese Society for Agricultural Machineryvol 35 no 2 pp 129ndash133 2004

[9] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[10] X M Liu J Qiu and G J Liu ldquoA diagnosis model based onAR-continuous HMM and its applicationrdquo Mechanical Scienceand Technology vol 24 no 3 pp 350ndash352 2005

[11] 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 of A Mathematical Physical amp Engineering Sciences vol454 no 1971 pp 903ndash995 1998

[12] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 no 1 pp 417ndash457 1999

[13] Z H Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[14] J Yeh J Shieh and N E Huang ldquoComplementary ensembleempirical mode decomposition a novel noise enhanced dataanalysis methodrdquo Advances in Adaptive Data Analysis vol 2no 2 pp 135ndash156 2010

[15] C Liu X Zhou S Yang W Liang and Q Miao ldquoCoolingfan bearing diagnosis based on ARamp MEDrdquo in Proceedingsof the International Conference on Quality Reliability RiskMaintenance and Safety Engineering (ICQR2MSE rsquo12) pp 622ndash626 Chengdu China June 2012

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[16] W C Chen and F Liu ldquoHybrid process data denoising methodbased on EEMDrdquo Application Research of Computer vol 29 no4 pp 1368ndash1370 2012

[17] J M Lu and F L Meng ldquoRolling bearing fault diagnosis withhilbert spectrum based on EEMDrdquo in Proceedings of the 4thInternational Conference on Signal Processing Systems pp 118ndash122 Kuala Lumpur Malaysia December 2012

[18] J Zhang R Q Yan R X Gao and Z Feng ldquoPerformanceenhancement of ensemble empirical mode decompositionrdquoMechanical Systems and Signal Processing vol 24 no 7 pp2104ndash2123 2010

[19] Z H Wu and N E Huang ldquoA study of the characteristics ofwhite noise using the empirical mode decomposition methodrdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 460 no 2046 pp 1597ndash1611 2004

[20] R X Chen B P Tang and J H Ma ldquoAdaptive de-noisingmethod based on ensemble empirical mode decomposition forvibration signalrdquo Journal of Vibration and Shock vol 31 no 15pp 82ndash86 2012

[21] N Rong H Song R Liu X J Peng and K Xu ldquoTrackingcoherence for inter-area oscillation using masking signal EMDand correlation analysisrdquo in Proceedings of the 11th IEEE Inter-national Conference on Electronic Measurement amp Instrumentspp 58ndash63 Harbin China August 2013

[22] L LWinger ldquoLinearly constrained generalized Lloyd algorithmfor reduced codebook vector quantizationrdquo IEEE Transactionson Signal Processing vol 49 no 7 pp 1501ndash1509 2001

[23] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[24] httpcsegroupscaseedubearingdatacenter

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of