research article bearing performance degradation...

Research ArticleBearing Performance Degradation Assessment UsingLifting Wavelet Packet Symbolic Entropy and SVDD

Jianmin Zhou Huijuan Guo Long Zhang Qingyao Xu and Hui Li

School of Mechatronic Engineering East China Jiaotong University Nanchang 330013 China

Correspondence should be addressed to Jianmin Zhou hotzjm163com

Received 16 June 2016 Accepted 28 September 2016

Academic Editor Ganging Song

Copyright copy 2016 Jianmin Zhou 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

Bearing performance degradation assessment is of great significance for proactive maintenance and near-zero downtime For thispurpose a novel assessmentmethod is proposed based on liftingwavelet packet symbolic entropy (LWPSE) and support vector datadescription (SVDD) LWPSE is presented for feature extraction by jointing use of lifting wavelet packet transform and symbolicentropy Firstly the LWPSEs of bearing signals from normal bearing condition are extracted to train an SVDD model by fittinga tight hypersphere around normal samples Then the relative distance from the LWPSEs of testing signals to the hypersphereboundary is calculated as a quantitative index for bearing performance degradation assessmentThe feasibility and efficiency of theproposed method were validated by the life-cycle data obtained from NASArsquos prognostics data repository and the comparison withHidden Markov Model (HMM) Finally the assessment results were verified by the envelope spectrum analysis method based onempirical mode decomposition and Hilbert envelope demodulation

1 Introduction

Bearings play an important role in rotating machineryThe performance of bearings generally affects the operationreliability of whole equipment directly [1] Actually bearingsalways undergo a degenerative process from normal states tofailure states There is possibility to make some proper main-tenance strategies to prevent the performance deteriorationof bearings if we can monitor the degenerative process ofbearings in time [2] Therefore degradation assessment ofbearings is significant to help reduce production downtimeand save maintenance costs [3]

Vibration signals are always being used to monitor therunning state of mechanical system because they carry alot of information which indicates the health condition ofmechanical equipment [4] Based on vibration signal analysislots of fault diagnosis methods combining advanced signalprocessing and pattern recognition techniques have beenproposed for bearings [5] Although bearing fault diagnosiscan be helpful for the indication of condition-based mainte-nance (CBM) they cannot reveal the degenerative trends of

bearings Given this problem bearing performance degrada-tion assessment has been a subject of extensive research inrecent years and more and more attention has been receivedfor its benefits in implementing CBM strategies [6]

Generally speaking performance degradation assess-ment of bearings mainly includes two steps namely featureextraction and degradation assessment Until now a lot ofwork covering these two aspects has been done Conventionalmonitor indexes such as root mean square (RMS) and kur-tosis have been frequently used as features for degradationassessment of bearings [7 8] Pan et al proposed spectralentropy as a complementary index to evaluate the degra-dation state of bearings and the results of both simulationsand experiments showed that spectral entropy can effectivelyreflect the degradation process of bearings [9] Yu proposeda method of locality preserving projection for feature extrac-tion and used a complementary index namely negative log-likelihood probability-based exponential weighted movingaverage statistic (NLLP-EWMA) to assess the performancedegradation process of bearings [10] Hong et al proposed amethod of wavelet packet-empirical mode decomposition for

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 3086454 10 pageshttpdxdoiorg10115520163086454

2 Shock and Vibration

feature extraction and then used self-organization mapping(SOM) for performance degradation assessment of bearings[3] Nelwamondo et al utilized GMM andHMM to diagnosefault in rolling bearings based on extracted features usingMultiscale Fractal Dimension (MFD) Mel Frequency Cep-stral Coefficients and kurtosis However themajor drawbackof HMM classifier is that it is costly and complex [11] Guoet al investigated the Hilbert envelope spectrum and SVMmethod to diagnose REB with ball fault inner race faultand outer race fault [12] The results show that the proposedmethod provides accurate diagnosis and good diagnosticresolution Zhang et al utilized particle swarm optimization-support vector machine (PSO-SVM) to realize the classifica-tion of fault location and degradation degree of rolling bear-ings [13] Dong and Luo used principal component analysis(PCA) to reduce the dimension of original features and thenconstructed the LS-SVMmodel for degradation evaluation ofbearings [14] Sun et al employed a kernel locality preservingprojection-based method to obtain an index for evaluatingthe degradation degree of a bearing [15] Liu et al extractedzero crossing features of vibration signals and then employeda coupled Hidden Markov Model for assessing the perfor-mance degradation of bearings quantitatively [16] Wang andChen used bilateral spectrum for feature extraction as it couldreveal nonlinearities and nonstationary characteristic andthen employed a support vector data description (SVDD)model to assess the degradation process of bearings [17]

Though a lot of work has been done there are stillmany challenges in assessing the performance degradation ofbearings effectively One of the challenges is how to extractmore consistent features for degradation assessment sincedifferent featuresmay only be useful at certain service stage orbe applicable for specific degradation modes of bearings Forexample RMS can correlate well with the fault developmentof bearings but it is not sensitive to incipient faults whileKurtosis Factor is sensitive to impulse faults (especiallyincipient faults) but it shows the poor stability as the damagegrows [7 8] Therefore a composite index which can satisfyboth the conditions of sensitivity and stability simultaneouslyis necessary Another challenge is how to build an intelligentassessment model based on the extracted features

To solve the abovementioned problems a novel assess-ment method combining lifting wavelet packet symbolicentropy (LWPSE) and support vector data description(SVDD) is proposed in this paper LWPSE is the com-bination of lifting wavelet packet transform (LWPT) andsymbolic entropy LWPT has some advantages over classicalwavelet packet transform including performing integer-to-integer wavelet transform and less computation andmemoryBesides LWPT can be reconstructed no matter how theprediction and update operators are designed Recentlythe applications of LWPT in fault diagnosis have achievedsome good results [18 19] Symbolic time series analysis(STSA) provides a coarse grained description of a dynamicalsystem based on a set of symbols It is not sensitive tothe measure noise and hence it has a good robustnessRecently it has been used for motoring degradation andfatigue damage using ultrasonic signals [20 21] SVDD isa single value classification method developed from SVM

Split

Split

Split

middot middot middot




P

P

U

U

P U

X(n)

X(2n)

X(2n + 1)

X12 (2n + 1)

X12

X11

X11(2n)

X11(2n + 1)

X12(2n)

X21

X22

X23

X24

Figure 1 Block diagram of the forward transform of LWPT

and the model can be established using normal data onlyMoreover it possesses the advantages of high computingefficiency and robustness Therefore SVDD has been widelyused in degradation assessment of bearings [8 17 22]

The following paper is organized as follows Sections 2and 3 are dedicated to the theories of lifting wavelet packetsymbolic entropy and SVDD respectively Section 4 presentsthe degradation assessment method based on LWPSE andSVDD and HMM Experimental validation and the relatedanalysis are provided in Section 5 The conclusions areprovided in Section 6

2 Lifting Wavelet Packet Symbolic Entropy

21 Lifting Wavelet Packet Transform Lifting wavelet packettransform (LWPT) is realized through the lifting scheme Itno longer relies on Fourier transform and all computationsare implemented in time domain Moreover it is possible toobtain the wavelets with some special properties through thedesign of predictor and updater [23] The forward transformprocess of LWPT is shown in Figure 1 We can run the liftingscheme backwards to derive the inverse transform from theforward transform The process of lifting scheme mainlyincludes three steps split predict and update The detailprocedure is as follows [24]

(1) Split Split the original signal 119883(119899) = 119909(119899) 119899 =0 1 119873 into two subsets namely even samples 119883(2119899) =119909(2119899) 119899 = 0 1 1198732 and odd samples 119883(2119899 + 1) =119909(2119899 + 1) 119899 = 0 1 1198732 where119873 is the length of119883(119899)(2) Predict and Update In this step each subband coefficientis calculated at level 119895

1198831198951 = 119883119895minus11 (2119899) + 119880 (119883119895minus11 (2119899 + 1))1198831198952 = 119883119895minus11 (2119899 + 1) minus 119875 (1198831198951 (2119899))

1198831198952119895minus1 = 119883119895minus12119895minus1 (2119899) + 119880 (119883119895minus12119895minus1 (2119899 + 1))1198831198952119895 = 119883119895minus12119895minus1 (2119899 + 1) minus 119875 (119883119895minus12119895minus1 (2119899))

(1)

Shock and Vibration 3

where 119875 is a prediction operator and119880 is an update operatorHere 119875 and 119880 are designed by interpolation subdivisionprinciple [25] In the context of interpolation subdivisionprinciple the orders of119875 and119880decide their valuesThereforeselecting different orders is equivalent to selecting differentbilateral orthogonal wavelet filters with different vanishingmoments In the current study the orders of 119875 and 119880 areboth selected as 12 since this can structure a wavelet functionwhich is closely similar tomechanical impulses [24]Thenwecan obtain the values of 119875 and 11988022 Symbolic Entropy Symbolic time series analysis (STSA)aims at symbolizing the time series data The basic processof STSA is converting the original time series signals intosequences of discrete symbols via partition function andthen we can use statistical features of the symbols to describethe dynamic statuses of a system [26] STSA includes thesymbolization of original time series and the quantitativeanalysis of symbolic series

Consider an original time series 119883 = 119909(119894) 119894 =1 2 119873 Converting 119883 into a binary symbol series 119878 =119904(119896) 119896 = 1 2 119873 is the most commonly used methodin the symbolization of original time series In this processpartitioning is the first step and it is also a key step Here thepartition function is obtained by finite difference method Itis defined as follows

119904 (119896) = 0 119909 (119894 + 1) minus 119909 (119894) le 01 119909 (119894 + 1) minus 119909 (119894) gt 0 (2)

where 119909(119894) (119894 = 1 2 119873) is the element of original timeseries signals 119883 and 119904(119896) is the binary symbol series which isequivalent to 0 or 1 After the symbolization a binary-codedsymbol series 119878 that only contains ldquo0rdquo and ldquo1rdquo can be gotAnd the next thing to do is dividing 119878 into decimal sequenceswith length 119871 namely words When the delay time 119879 and thelength of word 119871 are given 119878 can be cut into short symbolsequences

119875 (119896) = (119904 (119896) 119904 (119896 + 119879) 119904 (119896 + (119871 minus 1) 119879)) 119896 = 1 2 119873 minus (119871 minus 1) 119879 (3)

where119873 is the length of 119878Then119875(119896)needs to be transformedto the word119863(119896)

119863 (119896) = 119871sum119895=1

2119871minus119895119875119895 (119896)119875119895 (119896) = 119904 (119896 + (119895 minus 1) 119879)

(4)

Based on the work above a series of words can beobtained Here in order to reveal the intrinsic complexity ofthe original time series the histogram of symbol sequence asthe basis for quantitative statistics is introduced to measurethe occurrence number of each word in all the words [27]For example a series of words derived from a binary-codedsymbol series 119878 (the length of 119878 is 124) are 0 1 2 7 andthe occurrence numbers of each word are 27 16 2 18 15 4

0

5

10

15

20

25

30

Occ

urre

nce n

umbe

r

1 2 3 4 5 6 70Words

Figure 2 Histogram of symbol sequence word

1 2 3 4 5 6 70Words

0

005

01

015

02

025Pr

obab

ility

Figure 3 Probability of each word

and 17 then the histogram of symbol sequence can beshown in Figure 2 and the probability of each word (thenormalization of Figure 2) can be shown in Figure 3 Fromthe histogram of symbol sequence we can use the modifiedShannon entropy to describe the complexity of symbolsequence The modified Shannon entropy is defined as

119867(119879 119871) = minus 1log119872sum119901119894 log119901119894 (5)

where 119901119894 is the probability of the 119894th word and 119872 is thenumber of all the symbol sequences In the end symbolicentropy is equivalent to the modified Shannon entropy ofSTSA

Symbolic entropy is a complexity measurement it canmeasure the complexity of signals The vibration signals will


Outlier Target

Boundary

Figure 4 Schematic of the two-dimensional SVDD

be more andmore complex with the increase of fault severitythus causing the increase of the entropy Therefore symbolicentropy can reveal the complexity of vibration signals

23 Feature Extraction Using Lifting Wavelet Packet SymbolicEntropy The basic steps in lifting wavelet packet symbolicentropy (LWPSE) based feature extraction are as follows

(1) Employ LWPT to decompose bearing vibration sig-nals It has been illustrated in Section 21 that the bestchoices of the orders of 119875 and 119880 are both 12 Andin order to describe the features of bearing vibrationsignalsmore subtly the decomposition level is definedas 4 Then the decomposition coefficients can beobtained

(2) Choose the decomposition coefficients of each junc-tion in the last level to reconstruct the original signalsThen we can obtain the reconstructed signals

(3) Symbolize the reconstructed signals Here the valuesof 119879 and 119871 are suggested to be 5 and 15 based on a lotof trials since there is no theoretical rule to determinethem Then the symbolization of each reconstructedsignal can be done and the histogram of symbolsequence can be obtained

(4) Compute symbolic entropy of each reconstructedsignal by formula (5) Then a 16-dimensional featurevector 119881 = [1198671 1198672 11986716] of bearing vibrationsignals can be obtained

3 Support Vector Data Description

Support vector data description (SVDD)proposed byTax andDuin is inspired by the theory of support vector machine(SVM) proposed by Vapnik [28] The main idea of SVDDis to find an optimal hypersphere with minimal volumecontaining all or most targets as shown in Figure 4

Consider a training set 119909119894 119894 = 1 2 119899 119899 is the totalnumber of samples We try to find the optimal hyperspherewhich contains all or most normal samplesThis hypersphere

is described by center 119888 and radius 119877 and satisfies thefollowing optimization function

min 119871 (119877 119888 120585) = 1198772 + 119862 119899sum119894=1

120585119894st (119909119894 minus 119888)119879 (119909119894 minus 119888) le 1198772 + 120585119894

120585119894 ge 0 119894 = 1 2 119899(6)

where 119862 is a penalty parameter which controls the tradeoffbetween the volume of hypersphere and errors and 120585119894 are slackvariables which permit a few training data to be outside thehypersphere

Generally speaking (6) is solved by introducing Lagrangemultipliers and it can be transformed into the followingmax-imizing function 119871 with respect to the Lagrange multipliers120572119894

max 119871 = 119899sum119894=1

120572119894119909119894 sdot 119909119894 minus 119899sum119894119895=1

120572119894120572119895119909119894 sdot 119909119895st

119899sum119894=1

120572119894 = 1 0 le 120572119894 le 119862 forall119894(7)

Since the data in the input space are not always linearlypredicted we introduce a kernel function119870(119909119894 119909119895) = (Φ(119909119894) sdotΦ(119909119895)) to replace the inner product (119909119894 sdot 119909119895) where 119870 is aMercer kernel The kernel function 119870(119909119894 119909119895) can map thedata into a high-dimensional feature space and transform thenonlinear problem to a linear model Any function meetingMercerrsquos theorem can be employed as kernel function but notall of them are useful for SVDD Gaussian kernel is the mostcommonly used function It is defined as follows

119870(119909119894 119909119895) = exp(minus10038171003817100381710038171003817119909119894 minus 11990911989510038171003817100381710038171003817221205902 ) (8)

where 120590 is the width parameter Since Gaussian kernel canrestrain the growing distances for large feature spaces fordescribing the target data more compactly we employ it as119870(119909119894 119909119895) Then (7) becomes

max 119871 = 119899sum119894=1

120572119894119870(119909119894 119909119894) minus 119899sum119894119895=1

120572119894120572119895119870(119909119894 119909119895) (9)

All 120572119894 are got by solving (9) and only a few of them arenonzero The samples with 120572119894 gt 0 are called support vectorsThen the radius 119877 is obtained by any support vector 119909sv

1198772 = 119870 (119909sv sdot 119909sv) + 119899sum119894119895=1

120572119894120572119895119870(119909119894 sdot 119909119895)minus 2 119899sum119894=1

120572119894119870(119909119894 sdot 119909sv) (10)


Normalsignals

Testingsignals

Featureextraction

LWPSE

Degradationassessment DI

Offline modeling

Online assessment

Feature

LWPSEextraction SVDD

Modeling

Figure 5 Procedure of performance degradation assessment

For a new sample 119909119908 its distance to the center 119888 can bedescribed as follows

1198772119908 = 119870 (119909119908 sdot 119909119908) + 119899sum

119894119895=1

120572119894120572119895119870(119909119894 sdot 119909119895)minus 2 119899sum119894=1

120572119894119870(119909119908 sdot 119909119894) (11)

Here we introduce the theory of SVDD into bearing per-formance degradation assessmentThen the relative distancebetween the new sample 119909119908 and the hypersphere boundarycan be used as the degradation value (DV) of 119909119908 It is definedas follows

DV = (119877119908 minus 119877)119877 (12)

If DV le 0 119909119908 is accepted as a target which indicates thatthe bearing runs in a normal state Otherwise it is an outlierwhich indicates that the bearing runs in a degradation state

4 Degradation Assessment

41 Degradation Assessment Based on LWPSE and SVDD Inthis paper we proposed a bearing performance degradationassessment method based on LWPSE and SVDDThe frame-work for performance degradation assessment is shown inFigure 5 which includes two parts namely offline modelingand online assessmentThe steps of the proposed method areillustrated as follows

(1) The historical normal signals of a rolling bearing arecollected and their feature vectors are extracted usingLWPSE

(2) The feature vectors of normal state obtained from step(1) are used as training samples to establish an SVDDmodel Then the radius 119877 can be obtained by (10)

(3) As for the new testing signal 119909119908 its feature vectorsare extracted using LWPSE Then the generalizeddistance is calculated using (11) which is related to themodel established in step (2)

(4) The degradation value (DV) of 119909119908 is calculated using(12)Then the degradation index with a series of DVsof testing signals can be obtained Here we set all thevalues of DV to 0 when DV le 0

Bearing 1 Bearing 2 Bearing 3 Bearing 4

Radial loadAccelerometers Thermocouples

Motor

Figure 6 Bearing run-to-failure test rig

From the degradation index we can know that thebearing runs in a normal state while DV le 0 Otherwise itruns in a degradation state Moreover DV also reflects thedegree of fault severity of a bearing namely the larger DVmeans the larger degree of fault severity

42 Degradation Assessment Based on Hidden Markov Model(HMM) Hidden Markov model can be written as 120582 =(119873119872 120587 119860 119861) [29] Where119873 is the state number of Markovchain Normal initial failure deterioration deep degra-dation and failure are the five states of bearings 119872 isobservation that may occur in each state In the paper119872 is 11and 120587 is the vector of initial probability distribution 119860 is themodel of the state transition probability matrix Parameters119860 and 120587 are generally uniform or selected randomly 119861 is themodel of observation probability matrix

After model initialization is completed this paper usesthe first 200 sets of data to train model It is set whentwo adjacent output of log-likelihood probability value areless than 10minus3 and model training is completed Then enterthe 986 group under testing data into the model trainedabove When the program stops iterating the likelihoodprobability output value of rolling bearing is got Becausethe similar probability of their output is relatively small thispaper uses the log-likelihood probability log119875 to representthe index of performance degradation After getting its log-likelihood probability value the paper uses plot function todraw performance degradation curve of rolling bearing

5 Experimental Validation

51 Description of the Experiment The experimental datawere obtained from NASArsquos prognostics data repository [30]The schematic of bearing run-to-failure test rig is shownin Figure 6 During the experiment four Rexnord ZA-2115double row bearings were tested on one shaft The bearingshave 16 rollers in each row The roller diameter is 8407mmThe pitch diameter is 71501mm And the tapered contactangle is 1517∘ The shaft was driven by a motor The rotatingspeed was kept constant at 2000 rpm and a 6000 lb radialload was added to the shaft and bearings All the bearingswere lubricated The test stopped when the debris adhered to


533696

960

0

1

2

3

4

5

6

7

8D

V

200 400 600 800 10000Sample number

(a)

533696

960

0

1

2

3

4

5

6

7

8

DV

600 700 800 900 1000500Sample number

(b)

Figure 7 Assessment result using LWPSE and SVDD (a) Life-cycle assessment result of bearing 1 (b) Local enlargement of (a)

the magnetic plug exceeded a certain level On each bearingtwo accelerometers PCB 353B33 were installed to collectthe accelerations generated by the vibration signals with asampling rate of 20 kHz The interval time of each collectionwas 10 minutes and the data length of each collection was20480 pointsThe test was carried out for about 163 hours andit ended up with an outer race defect in bearing 1 Thereforethe life-cycle data of bearing 1 was employed to assess theperformance degradation of a bearing in this paper Fromthe geometric parameters of ZA-2115 bearings we can knowthat its ball pass frequency of the outer race (BPFO) is about2364Hz

52 Experimental Results Analysis

521 Assessment Results Using the ProposedMethod Thefirst200 sets of the life-cycle data of bearing 1 were used as normalsignals and their feature vectors were extracted using thefeature extraction method proposed in Section 23 Thenwe obtained a 200 lowast 16 matrix and the matrix was used toestablish an SVDD model After the model was establishedall the life-cycle data were used as testing signals and theirfeature vectors were extractedThen the feature vectors wereinputted to the trained SVDDmodel Finally the degradationindex DV of bearing 1 was obtained From the theory of timeseries analysis we can know that to certain extent the outputof a system at any time is determined by the output of theprevious few moments According to this theory we can dealwith the degradation index by five-point smoothing namelyDV(119905) = mean(DV(119905 minus 4 119905)) where mean mean(DV)meansgetting mean values The DV after five-point smoothing isshown in Figure 7 It is seen that the bearingrsquos performancedegradation process is clearly revealed by the DV Before 5330(the sample number is 533) minutes the DV is approximatelyequal to 0 which indicates that the rolling bearing runs in

the normal stage At 5330 minutes the DV has an obviousincrease and it indicates that the initial fault begins to occurAfter 6960 minutes the increasing trend of the DV is moreobvious and it indicates that the rolling bearing runs inthe fault progression stage After 9600 minutes the DV hasno more dramatic increase which indicates that the fault ofrolling bearing is developing to failure Also sometimes theDV has some abrupt fluctuation since the edge of crack maybe smoothed and rounded rapidly after their occurrence [31]In sum the performance degradation of bearing 1 could bedivided into four periods successfully normal stage from 0to 5330 minutes slight fault stage from 5330 to 6960 minutesfault progression stage from 6960 to 9600 minutes anddeveloping-to-failure stage from 9600 minutes to the end

In the proposed feature extraction method there are twoparameters to be predetermined namely the delay time 119879and the length of word 119871 which is described in the third stepof Section 23 Figure 8 plots the DV when the parameter 119879is equal to different values It can be seen that the increasingtrend of the DV is the best when 119879 is equal to 5 as it candescribe the degradation process more clearly Figure 9 plotsthe DV when the parameter 119871 is equal to different values Itcan be observed that the DV is the steadiest when 119871 is equalto 15 Therefore suitable 119879 and 119871 are meaningful for featureextraction

522 Assessment Results Using RMS RMS is one of thefrequently used monitoring indexes The RMS of bearing 1is shown in Figure 10 It can be seen that the increasing trendof RMS amplitudes is not obvious before 6990minutes whichindicates the difference of RMS in the normal stage and slightfault stage is not evident while the proposed degradationindex is evident In addition it decreases after 7040 minuteswhich is not consistent with the degradation process


02

46

8

0

500

1000

Sample number

0

2

4

6

DV

T

Figure 8 Assessment result with different values of 119879

010

2030

0

500

1000

Sample number

0

2

4

6

8

10

DV

L

Figure 9 Assessment result with different values of 119871523 Assessment Results Using Hidden Markov Model Per-formance degradation curve of the rolling bearing usingHMM is shown in Figure 11

The figure shows that probably in the 539th samplethe performance of the rolling bearing began to declineand early failure appeared Probably in the 699th samplethe rolling bearing degraded After the 820th sample therolling bearing exacerbated deeply After the 963rd samplethe rolling bearing failed completelyWe also can see that afterabout the 737th sample the performance degradation curveis inconsistent with its degree of fault The overall trend ofdegradation curve is falling but the curve still has relativelylarge fluctuations

524 Assessment Results Using LiftingWavelet Packet Entropyand SVDD For a comparison between different featureextraction methods the lifting wavelet packet entropy(LWPE) was used as feature to assess the performancedegradation of bearing 1 LWPE is the combination of liftingwavelet packet transform and energy entropy The basicsteps in LWPE are similar to the steps in LWPSE but

704

699

951

0

01

02

03

04

05

06

07

08

RMS

(g)

200 400 600 800 10000Sample number

(a)

699

704

951

0

01

02

03

04

05

06

07

08

RMS

(g)

600 700 800 900 1000500Sample number

(b)

Figure 10 Root mean square (RMS) (a) Life-cycle RMS of bearing1 (b) Local enlargement of (a)

LWPE extracts the energy entropy of each reconstructedsignal obtained by lifting wavelet packet transform [32] Theassessment result based on LWPE and SVDD is shown inFigure 12 It is observed that the DV has an obvious increaseat 5350 minutes and the increasing trend is more obviousafter 7000minutes which indicates the description of normalstage and slight fault stage is similar to the assessment resultbased on LWPSE and SVDD But the curve increases after7930 minutes which is not consistent with the degradationprocess By comparison among Figures 7 10 and 12 it isobvious that the degradation index proposed in this paper canreflect the bearing performance degradation process moreeffectively than the RMS and the degradation index based onLWPE and SVDD


539

737

820

963

699

minus120

minus100

minus80

minus60

minus40

minus20

0

200 400 600 800 10000Sample number

Log-

likel

ihoo

d pr

obab

ility

P

Figure 11 Assessment result using HMM

535

700

716

793

0

1

2

3

4

5

6

7

8

9

10

DV

200 400 600 800 10000Sample number

(a)

535 793

716

700

0

1

2

3

4

5

6

7

8

9

10

DV

600 700 800 900 1000500Sample number

(b)

Figure 12 Assessment result using LWPE and SVDD (a) Life-cycleassessment result of bearing 1 (b) Dramatic local enlargement of (a)

0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

231Hz

461Hz

691Hz

Figure 13 Demodulation result of the 533rd sample

0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

309 Hz

Figure 14 Demodulation result of the 532nd sample

53 Correctness Validation of Assessment Results To validatethe correctness of assessment results that the initial faultoccurs at 5330 minutes the envelope spectrum analysismethod based on empirical mode decomposition (EMD)and Hilbert envelope demodulation was employed in thispaper Here we analyze the 533rd sample and the 986thsample It was decomposed into several simple intrinsicmodefunctions (IMFs) by EMD first of all [33] Then we appliedthe Hilbert envelope demodulation analysis on IMF1 due tothe fact that it is the highest frequency signal which includesthe most detailed information of vibration signals [34] Thedemodulation result is shown in Figure 13 It can be seenthat there is an obvious spectrum peak at the frequency with231Hz which is close to the BPFO with 2364Hz Also thereexists obvious harmonic frequency characteristic Moreoverthe envelope spectrum of 532nd sample depicted in Figure 14shows no obvious spectrumpeak at the frequency close to theBPFO (the samples before the 532nd sample show the sameresults) The envelope spectrum of 986th sample depictedin Figure 15 shows that there is an obvious spectrum peakat the frequency with 231Hz which is close to the BPFO


Frequency (Hz)

0

01

02

03

04

05

Am

plitu

de (g

)

200 400 600 800 10000

231Hz461Hz 691Hz

Figure 15 Demodulation result of the 986th sample

with 2364Hz Also there exists obvious harmonic frequencycharacteristic It thus can be inferred that the initial fault ofthe outer race occurs at 5330 minutesThe analysis results areconsistent with the previous assessment results

6 Conclusions

In this paper the degradation index based on LWPSE andSVDD is proposed for bearing performance degradationassessment LWPSE is used to extract feature vectors andSVDD is employed to obtain the assessment results Theefficiency and validity of the proposed method are verifiedby the life-cycle data obtained from NASArsquos prognosticsdata repository Analysis results show that compared withthe RMS and the degradation index based on LWPE andSVDD the proposed degradation index is more sensitive toinitial fault and it has a consistent increasing trend with thedevelopment of bearing faultThemethod of HMMcan showthe overall performance degradation but its consistency ofperformance degradation is relatively poor Further analysisshows that the degradation index is affected by the parameters119879 and 119871 of the symbolic entropy thus suitable 119879 and 119871 aremeaningful for feature extraction Moreover the correctnessof assessment results is verified by the envelope spectrumanalysismethodbased onEMDandHilbert envelope demod-ulation It may be concluded that the proposed method isbetter than HMM in the paper and it is of great significancein guiding the maintenance of rotating machinery

Competing Interests

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

Acknowledgments

This work was funded under the Natural Science Foundationof China Grant no 51205130 The authors are grateful to allstudy participants

References

[1] W B Xiao J Chen G M Dong Y Zhou and Z Y WangldquoA multichannel fusion approach based on coupled hiddenMarkov models for rolling element bearing fault diagnosisrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 226 no 1 pp202ndash216 2012

[2] Y N Pan J Chen and G M Dong ldquoA hybrid model forbearing performance degradation assessment based on supportvector data description and fuzzy c-meansrdquo Proceedings of theInstitution ofMechanical Engineers Part C Journal ofMechanicalEngineering Science vol 223 no 11 pp 2687ndash2695 2009

[3] S Hong Z Zhou E Zio and K Hong ldquoCondition assessmentfor the performance degradation of bearing based on a com-binatorial feature extraction methodrdquo Digital Signal Processingvol 27 no 1 pp 159ndash166 2014

[4] S Hong B Wang G Li and Q Hong ldquoPerformance degra-dation assessment for bearing based on ensemble empiricalmode decomposition and gaussian mixture modelrdquo Journal ofVibration and Acoustics vol 136 no 6 article 061006 2014

[5] I El-Thalji and E Jantunen ldquoA summary of fault modellingand predictive health monitoring of rolling element bearingsrdquoMechanical Systems and Signal Processing vol 60 pp 252ndash2722015

[6] B Zhang L Zhang J Xu and P Wang ldquoPerformance degrada-tion assessment of rolling element bearings based on an indexcombining SVD and information exergyrdquo Entropy vol 16 no10 pp 5400ndash5415 2014

[7] B Tao L M Zhu H Ding and Y Xiong ldquoAn alternativetime-domain index for conditionmonitoring of rolling elementbearingsmdasha comparison studyrdquo Reliability Engineering amp Sys-tem Safety vol 92 no 5 pp 660ndash670 2007

[8] Z Shen Z He X Chen C Sun and Z Liu ldquoA monotonicdegradation assessment index of rolling bearings using fuzzysupport vector data description and running timerdquo Sensors vol12 no 8 pp 10109ndash10135 2012

[9] Y N Pan J Chen andX L Li ldquoSpectral entropy a complemen-tary index for rolling element bearing performance degradationassessmentrdquo Proceedings of the Institution of Mechanical Engi-neers Part C Journal of Mechanical Engineering Science vol223 no 5 pp 1223ndash1231 2009

[10] J B Yu ldquoLocal and nonlocal preserving projection for bearingdefect classification and performance assessmentrdquo IEEE Trans-actions on Industrial Electronics vol 59 no 5 pp 2363ndash23762012

[11] F V Nelwamondo T Marwala and U Mahola ldquoEarly clas-sifications of bearing faults using hidden Markov modelsGaussian mixture models mel-frequency cepstral coefficientsand fractalsrdquo International Journal of Innovative ComputingInformation amp Control vol 2 no 6 pp 1281ndash1299 2006

[12] L Guo J Chen and X I Li ldquoRolling bearing fault classificationbased on envelope spectrum and support vector machinerdquoJournal of Vibration and Control vol 15 no 9 pp 1349ndash13632009

[13] Y Zhang H Zuo and F Bai ldquoClassification of fault location andperformance degradation of a roller bearingrdquoMeasurement vol46 no 3 pp 1178ndash1189 2013

[14] S Dong and T Luo ldquoBearing degradation process predictionbased on the PCA and optimized LS-SVM modelrdquo Measure-ment vol 46 no 9 pp 3143ndash3152 2013


[15] C Sun Z Zhang Z He Z Shen B Chen andW Xiao ldquoNovelmethod for bearing performance degradation assessmentmdashakernel locality preserving projection-based approachrdquo Proceed-ings of the Institution of Mechanical Engineers Part C Journalof Mechanical Engineering Science vol 228 no 3 pp 548ndash5602014

[16] T Liu J Chen andGDong ldquoZero crossing and coupled hiddenMarkov model for a rolling bearing performance degradationassessmentrdquo Journal of Vibration and Control vol 20 no 16 pp2487ndash2500 2014

[17] H Wang and J Chen ldquoPerformance degradation assessment ofrolling bearing based on bispectrum and support vector datadescriptionrdquo Journal of Vibration and Control vol 20 no 13pp 2032ndash2041 2014

[18] Y Huang C Liu X F Zha and Y Li ldquoAn enhanced featureextraction model using lifting-based wavelet packet trans-form scheme and sampling-importance-resampling analysisrdquoMechanical Systems and Signal Processing vol 23 no 8 pp2470ndash2487 2009

[19] Z Wang S Bian M Lei C Zhao Y Liu and Z Zhao ldquoFeatureextraction and classification of load dynamic characteristicsbased on lifting wavelet packet transform in power system loadmodelingrdquo International Journal of Electrical Power and EnergySystems vol 62 pp 353ndash363 2014

[20] R A Gupta A K Wadhwani and S R Kapoor ldquoEarly esti-mation of faults in induction motors using symbolic dynamic-based analysis of stator current samplesrdquo IEEE Transactions onEnergy Conversion vol 26 no 1 pp 102ndash114 2011

[21] D S Singh S Gupta and A Ray ldquoIn-situ fatigue damagemonitoring using symbolic dynamic filtering of ultrasonicsignalsrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 223 no 6 pp 643ndash653 2009

[22] X Zhu Y Zhang and Y Zhu ldquoBearing performance degra-dation assessment based on the rough support vector datadescriptionrdquoMechanical Systems and Signal Processing vol 34no 1-2 pp 203ndash217 2013

[23] Y N Pan J Chen and X L Li ldquoBearing performance degrada-tion assessment based on lifting wavelet packet decompositionand fuzzy C-meansrdquoMechanical Systems and Signal Processingvol 24 no 2 pp 559ndash566 2010

[24] L Zhang G L Xiong H S Liu H Zou and W Guo ldquoFaultdiagnosis based on optimized node entropy using liftingwaveletpacket transform and genetic algorithmsrdquo Proceedings of theInstitution of Mechanical Engineers Part I Journal of Systemsand Control Engineering vol 224 no 5 pp 557ndash573 2010

[25] W Sweldens ldquoThe lifting scheme a custom-design constructionof biorthogonal waveletsrdquo Applied and Computational Har-monic Analysis vol 3 no 2 pp 186ndash200 1996

[26] R Li A Mita and J Zhou ldquoAbnormal state detection ofbuilding structures based on symbolic time series analysis andnegative selectionrdquo Structural Control and Health Monitoringvol 21 no 1 pp 80ndash97 2014

[27] T-W Chen and W-D Jin ldquoFeature extraction of radar emittersignals based on symbolic time series analysisrdquo in Proceedingsof the International Conference on Wavelet Analysis and Pat-tern Recognition (ICWAPR rsquo07) pp 1277ndash1282 Beijing ChinaNovember 2007

[28] D M J Tax and R P W Duin ldquoSupport vector domaindescriptionrdquo Pattern Recognition Letters vol 20 no 11ndash13 pp1191ndash1199 1999

[29] L Tao C Jin and D Guangming ldquoThe rolling bearing faultdiagnosis based on KPCA and coupled hidden Markov modelrdquoVibration and Shock vol 21 pp 85ndash89 2014

[30] J Lee H Qiu and G Yu ldquoNASA Ames Prognos-tics Data Repository-Bearing Data Setrdquo httpstiarcnasagovtechdashpcoeprognostic-data-repository

[31] R Rubini and U Meneghetti ldquoApplication of the envelope andwavelet transform analyses for the diagnosis of incipient faultsin ball bearingsrdquoMechanical Systems and Signal Processing vol15 no 2 pp 287ndash302 2001

[32] W-H Li B-X Dai and S-H Zhang ldquoBearing performancedegradation assessment based on Wavelet packet entropy andGaussian mixture modelrdquo Journal of Vibration and Shock vol32 no 21 pp 35ndash40 2013

[33] J B Ali N Fnaiech L Saidi B Chebel-Morello and F FnaiechldquoApplication of empirical mode decomposition and artificialneural network for automatic bearing fault diagnosis based onvibration signalsrdquo Applied Acoustics vol 89 pp 16ndash27 2015

[34] J Ma J Wu Y Fan and X Wang ldquoThe rolling bearing faultfeature extraction based on the LMD and envelope demodula-tionrdquo Mathematical Problems in Engineering vol 2015 ArticleID 429185 13 pages 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



feature extraction and then used self-organization mapping(SOM) for performance degradation assessment of bearings[3] Nelwamondo et al utilized GMM andHMM to diagnosefault in rolling bearings based on extracted features usingMultiscale Fractal Dimension (MFD) Mel Frequency Cep-stral Coefficients and kurtosis However themajor drawbackof HMM classifier is that it is costly and complex [11] Guoet al investigated the Hilbert envelope spectrum and SVMmethod to diagnose REB with ball fault inner race faultand outer race fault [12] The results show that the proposedmethod provides accurate diagnosis and good diagnosticresolution Zhang et al utilized particle swarm optimization-support vector machine (PSO-SVM) to realize the classifica-tion of fault location and degradation degree of rolling bear-ings [13] Dong and Luo used principal component analysis(PCA) to reduce the dimension of original features and thenconstructed the LS-SVMmodel for degradation evaluation ofbearings [14] Sun et al employed a kernel locality preservingprojection-based method to obtain an index for evaluatingthe degradation degree of a bearing [15] Liu et al extractedzero crossing features of vibration signals and then employeda coupled Hidden Markov Model for assessing the perfor-mance degradation of bearings quantitatively [16] Wang andChen used bilateral spectrum for feature extraction as it couldreveal nonlinearities and nonstationary characteristic andthen employed a support vector data description (SVDD)model to assess the degradation process of bearings [17]

Though a lot of work has been done there are stillmany challenges in assessing the performance degradation ofbearings effectively One of the challenges is how to extractmore consistent features for degradation assessment sincedifferent featuresmay only be useful at certain service stage orbe applicable for specific degradation modes of bearings Forexample RMS can correlate well with the fault developmentof bearings but it is not sensitive to incipient faults whileKurtosis Factor is sensitive to impulse faults (especiallyincipient faults) but it shows the poor stability as the damagegrows [7 8] Therefore a composite index which can satisfyboth the conditions of sensitivity and stability simultaneouslyis necessary Another challenge is how to build an intelligentassessment model based on the extracted features

To solve the abovementioned problems a novel assess-ment method combining lifting wavelet packet symbolicentropy (LWPSE) and support vector data description(SVDD) is proposed in this paper LWPSE is the com-bination of lifting wavelet packet transform (LWPT) andsymbolic entropy LWPT has some advantages over classicalwavelet packet transform including performing integer-to-integer wavelet transform and less computation andmemoryBesides LWPT can be reconstructed no matter how theprediction and update operators are designed Recentlythe applications of LWPT in fault diagnosis have achievedsome good results [18 19] Symbolic time series analysis(STSA) provides a coarse grained description of a dynamicalsystem based on a set of symbols It is not sensitive tothe measure noise and hence it has a good robustnessRecently it has been used for motoring degradation andfatigue damage using ultrasonic signals [20 21] SVDD isa single value classification method developed from SVM

Split

Split

Split





P

P

U

U

P U

X(n)

X(2n)

X(2n + 1)

X12 (2n + 1)

X12

X11

X11(2n)

X11(2n + 1)

X12(2n)

X21

X22

X23

X24

Figure 1 Block diagram of the forward transform of LWPT

and the model can be established using normal data onlyMoreover it possesses the advantages of high computingefficiency and robustness Therefore SVDD has been widelyused in degradation assessment of bearings [8 17 22]

The following paper is organized as follows Sections 2and 3 are dedicated to the theories of lifting wavelet packetsymbolic entropy and SVDD respectively Section 4 presentsthe degradation assessment method based on LWPSE andSVDD and HMM Experimental validation and the relatedanalysis are provided in Section 5 The conclusions areprovided in Section 6

2 Lifting Wavelet Packet Symbolic Entropy

21 Lifting Wavelet Packet Transform Lifting wavelet packettransform (LWPT) is realized through the lifting scheme Itno longer relies on Fourier transform and all computationsare implemented in time domain Moreover it is possible toobtain the wavelets with some special properties through thedesign of predictor and updater [23] The forward transformprocess of LWPT is shown in Figure 1 We can run the liftingscheme backwards to derive the inverse transform from theforward transform The process of lifting scheme mainlyincludes three steps split predict and update The detailprocedure is as follows [24]

(1) Split Split the original signal 119883(119899) = 119909(119899) 119899 =0 1 119873 into two subsets namely even samples 119883(2119899) =119909(2119899) 119899 = 0 1 1198732 and odd samples 119883(2119899 + 1) =119909(2119899 + 1) 119899 = 0 1 1198732 where119873 is the length of119883(119899)(2) Predict and Update In this step each subband coefficientis calculated at level 119895

1198831198951 = 119883119895minus11 (2119899) + 119880 (119883119895minus11 (2119899 + 1))1198831198952 = 119883119895minus11 (2119899 + 1) minus 119875 (1198831198951 (2119899))

1198831198952119895minus1 = 119883119895minus12119895minus1 (2119899) + 119880 (119883119895minus12119895minus1 (2119899 + 1))1198831198952119895 = 119883119895minus12119895minus1 (2119899 + 1) minus 119875 (119883119895minus12119895minus1 (2119899))

(1)




119904 (119896) = 0 119909 (119894 + 1) minus 119909 (119894) le 01 119909 (119894 + 1) minus 119909 (119894) gt 0 (2)




119863 (119896) = 119871sum119895=1

2119871minus119895119875119895 (119896)119875119895 (119896) = 119904 (119896 + (119895 minus 1) 119879)

(4)


0

5

10

15

20

25

30

Occ

urre

nce n

umbe

r

1 2 3 4 5 6 70Words


1 2 3 4 5 6 70Words

0

005

01

015

02

025Pr

obab

ility



119867(119879 119871) = minus 1log119872sum119901119894 log119901119894 (5)




Outlier Target

Boundary












min 119871 (119877 119888 120585) = 1198772 + 119862 119899sum119894=1

120585119894st (119909119894 minus 119888)119879 (119909119894 minus 119888) le 1198772 + 120585119894

120585119894 ge 0 119894 = 1 2 119899(6)



max 119871 = 119899sum119894=1

120572119894119909119894 sdot 119909119894 minus 119899sum119894119895=1

120572119894120572119895119909119894 sdot 119909119895st

119899sum119894=1

120572119894 = 1 0 le 120572119894 le 119862 forall119894(7)


119870(119909119894 119909119895) = exp(minus10038171003817100381710038171003817119909119894 minus 11990911989510038171003817100381710038171003817221205902 ) (8)


max 119871 = 119899sum119894=1

120572119894119870(119909119894 119909119894) minus 119899sum119894119895=1

120572119894120572119895119870(119909119894 119909119895) (9)


1198772 = 119870 (119909sv sdot 119909sv) + 119899sum119894119895=1

120572119894120572119895119870(119909119894 sdot 119909119895)minus 2 119899sum119894=1

120572119894119870(119909119894 sdot 119909sv) (10)


Normalsignals

Testingsignals

Featureextraction

LWPSE


Offline modeling

Online assessment

Feature


Modeling



1198772119908 = 119870 (119909119908 sdot 119909119908) + 119899sum

119894119895=1

120572119894120572119895119870(119909119894 sdot 119909119895)minus 2 119899sum119894=1

120572119894119870(119909119908 sdot 119909119894) (11)


DV = (119877119908 minus 119877)119877 (12)










Motor








533696

960

0

1

2

3

4

5

6

7

8D

V

200 400 600 800 10000Sample number

(a)

533696

960

0

1

2

3

4

5

6

7

8

DV

600 700 800 900 1000500Sample number

(b)









02

46

8

0

500

1000

Sample number

0

2

4

6

DV

T


010

2030

0

500

1000

Sample number

0

2

4

6

8

10

DV

L




704

699

951

0

01

02

03

04

05

06

07

08

RMS

(g)

200 400 600 800 10000Sample number

(a)

699

704

951

0

01

02

03

04

05

06

07

08

RMS

(g)

600 700 800 900 1000500Sample number

(b)




539

737

820

963

699

minus120

minus100

minus80

minus60

minus40

minus20

0

200 400 600 800 10000Sample number

Log-

likel

ihoo

d pr

obab

ility

P


535

700

716

793

0

1

2

3

4

5

6

7

8

9

10

DV

200 400 600 800 10000Sample number

(a)

535 793

716

700

0

1

2

3

4

5

6

7

8

9

10

DV

600 700 800 900 1000500Sample number

(b)


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

231Hz

461Hz

691Hz


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

309 Hz




Frequency (Hz)

0

01

02

03

04

05

Am

plitu

de (g

)

200 400 600 800 10000

231Hz461Hz 691Hz



6 Conclusions


Competing Interests


Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119904 (119896) = 0 119909 (119894 + 1) minus 119909 (119894) le 01 119909 (119894 + 1) minus 119909 (119894) gt 0 (2)




119863 (119896) = 119871sum119895=1

2119871minus119895119875119895 (119896)119875119895 (119896) = 119904 (119896 + (119895 minus 1) 119879)

(4)


0

5

10

15

20

25

30

Occ

urre

nce n

umbe

r

1 2 3 4 5 6 70Words


1 2 3 4 5 6 70Words

0

005

01

015

02

025Pr

obab

ility



119867(119879 119871) = minus 1log119872sum119901119894 log119901119894 (5)




Outlier Target

Boundary












min 119871 (119877 119888 120585) = 1198772 + 119862 119899sum119894=1

120585119894st (119909119894 minus 119888)119879 (119909119894 minus 119888) le 1198772 + 120585119894

120585119894 ge 0 119894 = 1 2 119899(6)



max 119871 = 119899sum119894=1

120572119894119909119894 sdot 119909119894 minus 119899sum119894119895=1

120572119894120572119895119909119894 sdot 119909119895st

119899sum119894=1

120572119894 = 1 0 le 120572119894 le 119862 forall119894(7)


119870(119909119894 119909119895) = exp(minus10038171003817100381710038171003817119909119894 minus 11990911989510038171003817100381710038171003817221205902 ) (8)


max 119871 = 119899sum119894=1

120572119894119870(119909119894 119909119894) minus 119899sum119894119895=1

120572119894120572119895119870(119909119894 119909119895) (9)


1198772 = 119870 (119909sv sdot 119909sv) + 119899sum119894119895=1

120572119894120572119895119870(119909119894 sdot 119909119895)minus 2 119899sum119894=1

120572119894119870(119909119894 sdot 119909sv) (10)


Normalsignals

Testingsignals

Featureextraction

LWPSE


Offline modeling

Online assessment

Feature


Modeling



1198772119908 = 119870 (119909119908 sdot 119909119908) + 119899sum

119894119895=1

120572119894120572119895119870(119909119894 sdot 119909119895)minus 2 119899sum119894=1

120572119894119870(119909119908 sdot 119909119894) (11)


DV = (119877119908 minus 119877)119877 (12)










Motor








533696

960

0

1

2

3

4

5

6

7

8D

V

200 400 600 800 10000Sample number

(a)

533696

960

0

1

2

3

4

5

6

7

8

DV

600 700 800 900 1000500Sample number

(b)









02

46

8

0

500

1000

Sample number

0

2

4

6

DV

T


010

2030

0

500

1000

Sample number

0

2

4

6

8

10

DV

L




704

699

951

0

01

02

03

04

05

06

07

08

RMS

(g)

200 400 600 800 10000Sample number

(a)

699

704

951

0

01

02

03

04

05

06

07

08

RMS

(g)

600 700 800 900 1000500Sample number

(b)




539

737

820

963

699

minus120

minus100

minus80

minus60

minus40

minus20

0

200 400 600 800 10000Sample number

Log-

likel

ihoo

d pr

obab

ility

P


535

700

716

793

0

1

2

3

4

5

6

7

8

9

10

DV

200 400 600 800 10000Sample number

(a)

535 793

716

700

0

1

2

3

4

5

6

7

8

9

10

DV

600 700 800 900 1000500Sample number

(b)


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

231Hz

461Hz

691Hz


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

309 Hz




Frequency (Hz)

0

01

02

03

04

05

Am

plitu

de (g

)

200 400 600 800 10000

231Hz461Hz 691Hz



6 Conclusions


Competing Interests


Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Outlier Target

Boundary












min 119871 (119877 119888 120585) = 1198772 + 119862 119899sum119894=1

120585119894st (119909119894 minus 119888)119879 (119909119894 minus 119888) le 1198772 + 120585119894

120585119894 ge 0 119894 = 1 2 119899(6)



max 119871 = 119899sum119894=1

120572119894119909119894 sdot 119909119894 minus 119899sum119894119895=1

120572119894120572119895119909119894 sdot 119909119895st

119899sum119894=1

120572119894 = 1 0 le 120572119894 le 119862 forall119894(7)


119870(119909119894 119909119895) = exp(minus10038171003817100381710038171003817119909119894 minus 11990911989510038171003817100381710038171003817221205902 ) (8)


max 119871 = 119899sum119894=1

120572119894119870(119909119894 119909119894) minus 119899sum119894119895=1

120572119894120572119895119870(119909119894 119909119895) (9)


1198772 = 119870 (119909sv sdot 119909sv) + 119899sum119894119895=1

120572119894120572119895119870(119909119894 sdot 119909119895)minus 2 119899sum119894=1

120572119894119870(119909119894 sdot 119909sv) (10)


Normalsignals

Testingsignals

Featureextraction

LWPSE


Offline modeling

Online assessment

Feature


Modeling



1198772119908 = 119870 (119909119908 sdot 119909119908) + 119899sum

119894119895=1

120572119894120572119895119870(119909119894 sdot 119909119895)minus 2 119899sum119894=1

120572119894119870(119909119908 sdot 119909119894) (11)


DV = (119877119908 minus 119877)119877 (12)










Motor








533696

960

0

1

2

3

4

5

6

7

8D

V

200 400 600 800 10000Sample number

(a)

533696

960

0

1

2

3

4

5

6

7

8

DV

600 700 800 900 1000500Sample number

(b)









02

46

8

0

500

1000

Sample number

0

2

4

6

DV

T


010

2030

0

500

1000

Sample number

0

2

4

6

8

10

DV

L




704

699

951

0

01

02

03

04

05

06

07

08

RMS

(g)

200 400 600 800 10000Sample number

(a)

699

704

951

0

01

02

03

04

05

06

07

08

RMS

(g)

600 700 800 900 1000500Sample number

(b)




539

737

820

963

699

minus120

minus100

minus80

minus60

minus40

minus20

0

200 400 600 800 10000Sample number

Log-

likel

ihoo

d pr

obab

ility

P


535

700

716

793

0

1

2

3

4

5

6

7

8

9

10

DV

200 400 600 800 10000Sample number

(a)

535 793

716

700

0

1

2

3

4

5

6

7

8

9

10

DV

600 700 800 900 1000500Sample number

(b)


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

231Hz

461Hz

691Hz


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

309 Hz




Frequency (Hz)

0

01

02

03

04

05

Am

plitu

de (g

)

200 400 600 800 10000

231Hz461Hz 691Hz



6 Conclusions


Competing Interests


Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Normalsignals

Testingsignals

Featureextraction

LWPSE


Offline modeling

Online assessment

Feature


Modeling



1198772119908 = 119870 (119909119908 sdot 119909119908) + 119899sum

119894119895=1

120572119894120572119895119870(119909119894 sdot 119909119895)minus 2 119899sum119894=1

120572119894119870(119909119908 sdot 119909119894) (11)


DV = (119877119908 minus 119877)119877 (12)










Motor








533696

960

0

1

2

3

4

5

6

7

8D

V

200 400 600 800 10000Sample number

(a)

533696

960

0

1

2

3

4

5

6

7

8

DV

600 700 800 900 1000500Sample number

(b)









02

46

8

0

500

1000

Sample number

0

2

4

6

DV

T


010

2030

0

500

1000

Sample number

0

2

4

6

8

10

DV

L




704

699

951

0

01

02

03

04

05

06

07

08

RMS

(g)

200 400 600 800 10000Sample number

(a)

699

704

951

0

01

02

03

04

05

06

07

08

RMS

(g)

600 700 800 900 1000500Sample number

(b)




539

737

820

963

699

minus120

minus100

minus80

minus60

minus40

minus20

0

200 400 600 800 10000Sample number

Log-

likel

ihoo

d pr

obab

ility

P


535

700

716

793

0

1

2

3

4

5

6

7

8

9

10

DV

200 400 600 800 10000Sample number

(a)

535 793

716

700

0

1

2

3

4

5

6

7

8

9

10

DV

600 700 800 900 1000500Sample number

(b)


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

231Hz

461Hz

691Hz


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

309 Hz




Frequency (Hz)

0

01

02

03

04

05

Am

plitu

de (g

)

200 400 600 800 10000

231Hz461Hz 691Hz



6 Conclusions


Competing Interests


Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










533696

960

0

1

2

3

4

5

6

7

8D

V

200 400 600 800 10000Sample number

(a)

533696

960

0

1

2

3

4

5

6

7

8

DV

600 700 800 900 1000500Sample number

(b)









02

46

8

0

500

1000

Sample number

0

2

4

6

DV

T


010

2030

0

500

1000

Sample number

0

2

4

6

8

10

DV

L




704

699

951

0

01

02

03

04

05

06

07

08

RMS

(g)

200 400 600 800 10000Sample number

(a)

699

704

951

0

01

02

03

04

05

06

07

08

RMS

(g)

600 700 800 900 1000500Sample number

(b)




539

737

820

963

699

minus120

minus100

minus80

minus60

minus40

minus20

0

200 400 600 800 10000Sample number

Log-

likel

ihoo

d pr

obab

ility

P


535

700

716

793

0

1

2

3

4

5

6

7

8

9

10

DV

200 400 600 800 10000Sample number

(a)

535 793

716

700

0

1

2

3

4

5

6

7

8

9

10

DV

600 700 800 900 1000500Sample number

(b)


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

231Hz

461Hz

691Hz


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

309 Hz




Frequency (Hz)

0

01

02

03

04

05

Am

plitu

de (g

)

200 400 600 800 10000

231Hz461Hz 691Hz



6 Conclusions


Competing Interests


Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










02

46

8

0

500

1000

Sample number

0

2

4

6

DV

T


010

2030

0

500

1000

Sample number

0

2

4

6

8

10

DV

L




704

699

951

0

01

02

03

04

05

06

07

08

RMS

(g)

200 400 600 800 10000Sample number

(a)

699

704

951

0

01

02

03

04

05

06

07

08

RMS

(g)

600 700 800 900 1000500Sample number

(b)




539

737

820

963

699

minus120

minus100

minus80

minus60

minus40

minus20

0

200 400 600 800 10000Sample number

Log-

likel

ihoo

d pr

obab

ility

P


535

700

716

793

0

1

2

3

4

5

6

7

8

9

10

DV

200 400 600 800 10000Sample number

(a)

535 793

716

700

0

1

2

3

4

5

6

7

8

9

10

DV

600 700 800 900 1000500Sample number

(b)


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

231Hz

461Hz

691Hz


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

309 Hz




Frequency (Hz)

0

01

02

03

04

05

Am

plitu

de (g

)

200 400 600 800 10000

231Hz461Hz 691Hz



6 Conclusions


Competing Interests


Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










539

737

820

963

699

minus120

minus100

minus80

minus60

minus40

minus20

0

200 400 600 800 10000Sample number

Log-

likel

ihoo

d pr

obab

ility

P


535

700

716

793

0

1

2

3

4

5

6

7

8

9

10

DV

200 400 600 800 10000Sample number

(a)

535 793

716

700

0

1

2

3

4

5

6

7

8

9

10

DV

600 700 800 900 1000500Sample number

(b)


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

231Hz

461Hz

691Hz


0

005

01

015

02

Am

plitu

de (g

)

200 400 600 800 10000Frequency (Hz)

309 Hz




Frequency (Hz)

0

01

02

03

04

05

Am

plitu

de (g

)

200 400 600 800 10000

231Hz461Hz 691Hz



6 Conclusions


Competing Interests


Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Frequency (Hz)

0

01

02

03

04

05

Am

plitu

de (g

)

200 400 600 800 10000

231Hz461Hz 691Hz



6 Conclusions


Competing Interests


Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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