Classiﬁcation of ball bearing faults using a hybrid intelligent model Applied Soft Computing

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Applied Soft Computing

jo u r n al hom e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / a s o c

Full Length Article

Classiﬁcation of ball bearing faults using a hybrid intelligent model

Manjeevan Seera

^a,∗

, M.L. Dennis Wong

^a,b

, Asoke K. Nandi

^c,d

aFacultyofEngineering,ComputingandScience,SwinburneUniversityofTechnology(SarawakCampus),Sarawak,Malaysia

bHeriot-WattUniversityMalaysia,Putrajaya,Malaysia

cDepartmentofElectronicandComputerEngineering,BrunelUniversityLondon,UxbridgeUB83PH,UnitedKingdom

dTheKeyLaboratoryofEmbeddedSystemsandServiceComputing,CollegeofElectronicandInformationEngineering,TongjiUniversity,Shanghai,China

a r t i c l e i n f o

Articlehistory:

Received28March2016

Receivedinrevisedform28January2017 Accepted18April2017

Availableonline21April2017

Keywords:

Conditionmonitoring Ballbearing Electricalmotor

Fuzzymin-maxneuralnetwork Randomforest

a b s t r a c t

Inthispaper,classiﬁcationofballbearingfaultsusingvibrationsignalsispresented.Areviewofcondition monitoringusingvibrationsignalswithvariousintelligentsystemsisﬁrstpresented.Ahybridintelligent model,FMM-RF,consistingoftheFuzzyMin-Max(FMM)neuralnetworkandtheRandomForest(RF) model,isproposed.AbenchmarkproblemistestedtoevaluatethepracticalityoftheFMM-RFmodel.

Theproposedmodelisthenappliedtoareal-worlddataset.Inbothcases,powerspectrumandsample entropyfeaturesareusedforclassiﬁcation.Resultsfrombothexperimentsshowgoodaccuracyachieved bytheproposedFMM-RFmodel.Inaddition,asetofexplanatoryrulesintheformofadecisiontree isextractedtojustifythepredictions.TheoutcomesindicatetheusefulnessofFMM-RFinperforming classiﬁcationofballbearingfaults.

1. Introduction

Conditionmonitoringofmachinery isbecoming increasingly important in modern maintenance. There is a need to reduce unscheduleddowntimeinordertomaintaincorporatecompeti- tiveness.Thecostofmaintenancecanalsobereducedbyconstantly monitoring thehealth of the machines [1]. In this way,disas- trousfaultsthatcouldpotentiallyhappencanbedetectedearly, whichwillreducethetotaldowntimeofthemachineandentire operations.Predictive maintenancetechniqueshave beeneffec- tivelyutilizedinreducing unexpectedmachinefailures[2].One ofthemostcommonly usedpredictivemaintenance technology isvibrationmonitoring,duetotheamountofmachineconditions informationthatisprovided[2].

Inthepasttwodecades,anumberofresearchershavereported theirachievementsonconditionmonitoringofrotatingmachinery.

Conditionmonitoringintherotatingmachinesoftheindustryuses accelerometersandvibrationtransmittersinordertoacquiredata [3–5].Oncedataisacquired,itisthenvitaltoprocessthesedata.

Patternrecognitionisthecentraltaskin themachine condition monitoring,withvarioussolutionsreported[6–9].Itﬁrstlooksat informationfrommultitudeofsources,suchastransducersignals

∗Correspondingauthor.

E-mailaddress:[email protected](M.Seera).

fromthemachine[9].Featureextractionisthenusedtoextract usefulfeaturesfromthecollectedinformation.

Selectingtherightfeaturesisthekeytosolveaccuratelythe classiﬁcationproblem,andthechoiceoffeaturescangreatlyaffect theclassiﬁcationperformance[1].Ingeneral,time-domainfeatures arecommonlyusedinmachineconditionmonitoring.Thefeatures commonlyused,butnotlimitedtoaretherootmeansquared(RMS) voltage,thepeakvoltage,theX-Yplot,andthecrestfactor(theratio ofthepeakvoltageovertheRMSvoltage).Formoreadvancedmeth- ods,thevibrationdataisoftentransformedtoitsfrequencydomain equivalent,whichisthePowerSpectrumorFFT.Withtheincreased computingpoweranddigitalstorageinrecentyears,theuseof waterfalldiagramanddiscretewavelettransformhasincreased.

Thecontributionsofthispaperaretwo-fold:theuseofahybrid intelligent model for detection and classiﬁcation of real-world rollerballbearingfaultsaswellasdetailedinvestigationsinthe useofasetofpowerspectrumandsampleentropy-basedfeatures for performingthistask. For validationpurposes,a well-known benchmarkdatabaseisﬁrstusedintheexperimentalworks.Then, areal-worlddatasetwithnewfeaturesextractedusingentropy isusedtofurthervalidatethedata.It isworthmentioningthat thehybridintelligentmodeldeliversa simpleyetusefultreein classifyingtheoutputsfromthedata.

Thispaperisorganizedasfollows.Aliteraturereviewoncon- ditionmonitoringusingvibrationsignalswithvariousintelligent systemsisﬁrstpresentedinSection2.Detailsofthepowerspec- trum and sample entropy feature extractions are presented in http://dx.doi.org/10.1016/j.asoc.2017.04.034

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Section3.ThehybridFMM-RFmodelisdetailed inSection4.A benchmarkstudytoevaluateFMM-RFeffectivenessisdetailedin Section5.ApplicabilityofFMM-RFisreal-worlddatasetisshown inSection6.ConcludingremarksareﬁnallyofferedinSection7.

2. Literaturereview

Aliteraturereviewforconditionmonitoringusingvibrationsig- nalswithvariousintelligentsystemsispresentedinthissection.

Theempiricalmodedecomposition(EMD)energyentropyisused toextractfeaturesfromvibrationsignalsin[10].Featuresarethen selectedusingtheintrinsicmodefunctions(IMFs)method,which arethenfedtoanartiﬁcialneuralnetwork(ANN)withbackprop- agation(BP)toclassifybearingsdefects[10].Resultsindicatethe proposedmethodcanaccuratelydeterminebearingdefectsusing run-to-failurevibrationsignals [10].Hilbert Transformand Fast FourierTransformaretwofeatureextractionmethodsin[11]for vibrationsignals.TheANN-basedfaultestimationalgorithmwith theuseofgeneticalgorithm (GA) is usedfor fault diagnosisof rollingbearings[11].Improvedclassiﬁcationresultsareseenfrom theexperiments[11].

Sevendecomposed signals withvaryingfrequency range for kurtosisofbearingvibrationsignalispresentedin[12].Ahybrid empirical mode decomposition and relevance vector machine (RVM)withartificialbeecolonyalgorithmmodelisusedtopredict thesignals[12].Resultsindicatethehybridmodelin[12]improves theaccuracyratesascomparedtoRVMinkurtosisofbearingvibra- tionsignal.Remainingusefullife(RUL)ofbearingsisofinterestin [13]and[14].Asimplifiedfuzzyadaptiveresonancetheorymap (SFAM)neuralnetworkisusedtopredicttheRULofrollingelement bearings[13].TheWeibulldistributionisusedtofitmeasurements, basedonthesevendefinedclasses[13].Experimentalresultsin[13]

indicatethereliabilityoftheRULprediction.Ageneralisedfunction fromWeibullfunctionisusedtoﬁtmeasurementsin[14],similar to[13].AnANNisusedforclassiﬁcation,withavalidationmech- anismusedtoimproveANNperformance[14].Usingreal-world vibrationdatafrompumpbearings,goodresultsareachieved[14].

A total of sevenbearing states from theEMD are produced in [15]. For feature reduction, the principal component analy- sisandlinear discriminantanalysisare used[15].Classification is then carried out using the probabilistic neural network and SFAM[15].Resultsshowbettergeneralizationcapabilityascom- paredwith othermethods [15]. Vibrationsignals from bearing arepre-processedusingthede-trendedfluctuationanalysisand rescaled-rangeanalysistechniquesin [16]. Signalsare acquired fromdifferentfrequencyandloadconditions[16].Usingprincipal componentanalysisandANN,theclassificationyieldedfairlygood results[16].Ahardcompetitivegrowingneuralnetwork(HC-GNN) withshrinkagelearningisusedin[17]forfaultdetectionanddiag- nosisofsmallbearingfaults.Wavelettransformisusedinfeature extraction[17].TheHC-GNNcreatessmallernetworkscompared toothernetworks[17].Resultsonamachinerysystemwithvar- ioussmallbearingfaultsindicategoodresultsfromtheproposed network[17].

Predictionoffaultybearingconditionsusingtheintervaltype-2 fuzzyneuralnetworkisdetailedin[18].Atotalofthreedifferent featuresareextracted[18].Thefaultybearingsareusedforvalida- tion,withresultscomparedwiththosefromadaptiveneuro-fuzzy inferencesystem(ANFIS)[18].Theproposedmethodyieldsbetter predictionaccuracyascomparedtoANFIS[18].Frequency-domain featuresofbearingvibrationsignalsareextractedin[19].Iniden- tifyingfaulttypes,asequentialdiagnosistechniquethroughthe partially-linearizedneuralnetwork(PLNN)isdone[19].ThePLNN canautomaticallydeterminefault typesin rollingbearingwith goodaccuracyrates[19].Time-domaindataisextractedfromvibra-

tionsignalsinarotor-bearingsystemin[20].Themeasurements aredoneatfivedifferentrotatingspeeds[20].Forclassification,a supportvectormachineisutilizedwithgoodclassificationaccuracy achieved[20].

Thedependentfeaturevectorisfirstusedin[21]forfaultdiag- nosisofrollingelementbearings.Thefeatures arethenfedtoa probabilityneuralnetwork [21].Experimentalresultsshowthe proposedmethodachievesan efficientaccuracy inanalysisthe bearingfaults[21].Vibrationsignalsfromarotor-bearingsystem areanalyzedin[22].Thekeykernels(KK)andparticleswarmopti- mization(PSO),knownasKK-PSOmethodisproposedforVolterra seriesidentificationinfeatureextraction[22].Usingsimulationand real-data,resultsshowtheKK-PSOmethodoutperformstheleast squareandtraditionalPSOmethod[22].Inashaft-bearingmech- anism,both vibrationandcurrent signaldataareacquired[23].

Thetime-domainandfrequency-domainparametersareextracted frombothsignals[23].Amulti-stagealgorithmbasedonANNand ANFISmodelisusedfor classiﬁcation,withresultsshowingthe proposedmethodiseffective[23].

Various vibration conditions are extracted from drilling machinesin[24].Theradialbasisneuralnetworkisthenusedto analyzetheacquiredsignals[24].Comparedtoradialbasisnet- works,theproposednetworkhasbetterperformanceinadapting toreal-timeparametersofthedrillingmachines[24].Incondition diagnosisofvariousbearingsystems,vibrationsignalsareusedas inputsin[25].Tenstatisticalfeaturesareextractedfromthesignals.

AhybridtechniquecombiningGAwithadaptiveoperatorprobabil- ities(AGAs)andbackpropagationneuralnetworks(BPNNs),named AGAs-BPNNsisproposed[25].Resultsfromtheexperimentshow theproposedAGAs-BPNNsmethodacquiredhigherclassiﬁcation accuracy[25].

3. Featureextraction

In the general pattern classification framework, the feature extractionisakeystageforextractingthesalientinformationfrom therawsignalsandreducingthedimensionalityoftheinputvector totheclassification engine.Thefeatureextractionmethodcho- senisdependentonthespecifictask.Inthissection,wepresent twodifferenttypesoffeatureextractionmethodforclassifyingball bearingfaults:(1)theconventionalpowerspectrumand(2)the sampleentropy[26].Theformerisacommonlyusedfeaturefor classifyingballbearingfaultsandthelatterwasrecentlyintroduced in[27].

3.1. Powerspectrum(PS)

Theexistenceofdefectsinballbearingswillexhibitashighfre- quencyspikesandotherfaultpatternsinthevibrationtimeseries.

Inthefrequencydomain,thistranslatestotheadditionofnewhar- monicsinthepowerspectrum(PS).Assuch,PSisoftenchosen forconditionmonitoringproblemsasitcompactlyrepresentsthe timevaryingtimedomainsignalintoasetofﬁxedlengthvector representingthesquaremagnitudeofthefrequencycomponents (harmonics).Therearemanymethodsforestimatingthesignal’s powerspectrum.In thiswork,we adoptedthecommonlyused Welch’smethodtoperformthistask.TheWelch’smethodisessen- tiallyanon-parametricmethodwhichcomputethePSthroughan averagingprocess.

Given the time series dataXn=

x₀,x₁,x₂,x₃···,x_N−1

, the Welch’sPSestimatecanbewrittenas:

PX_i

e^jω

= 1 KLU

K−1

i=0

|

L−1

n=0

Wnx_n+iDe⁻^jnω|

2

(1)

(3)

whereW_iisthewindowsfunctionofLengthL-1;Kisthenumberof segmentstheXnisdividedintowithDpointsoverlappingbetween twoconsecutivesegments;andUisanormalizingfactordeﬁnedas

1 L

L−1 n=0

|w_n|².

Inthiswork,wehavechosentouseHanningWindowwitha windowlengthof1024samplesandtheoverlappingfactorisset to50%ofthewindowlength.FollowingthecomputationofthePS, weextractthePSvaluesfromDCto12KHzwith500Hzintervals, whichresultedinavectorof25PSfeatures.

3.2. Sampleentropy(SampEn)

Shannon’sentropy is a measure of information content and morespeciﬁcallyitmeasuresthelevelofunpredictabilityofagiven sampledtimeseries.Asfaultsintheballbearingwillintroducenew patternsintothevibrationsignals,e.g.,spikesatdifferentintervals, andbroadenenvelopetheinformationcontentofthevibrationsig- nalswillchange.Therefore,itisintuitiveforonetocapturethese changesthroughcomputingthesignals’entropyvalues.However, itisdifﬁculttoestimateShannon’sentropydirectlyforatimeseries signal.

Theapproximateentropy(ApEn)isproposedin[28]fornoisy andshorttimeseries,inordertoestimatetherateofgenerating newinformation.Inreducingthebiasproducedbypatternself- matchinginApEn,theSampleEntropy(SampEn)isproposedin [26]forasampledtimeseriesdatafromacontinuousprocess.This providesanaccuratenegativelogarithmintendedfor ApEn.We brieﬂypresentthecomputationofSampEnasfollows.

Givenatime-seriesdataXn=

x₀,x₁,x₂,x₃···,x_N−1

oflength Nasabovewithasamplinginterval,,letm<<Nbeaconstant, then Xn can be divided into (N–m+1) template vectors X_n+t^’ =

xn+0,xn+1,···xn+m−1

,each of length m and for all n=0, ..., (N–m).Let

X_i^’,X_j’

betheChebyshevdistancefunctionandif foranytwotemplatevectors,where

X’_i,X’_j

<rthenamatch isrecorded.Thetoleranceparameter,r,byconvention,issettoa fractionofthestandarddeviationofthesequenceforconvenience.

Ingeneral,thesettingisusuallyaﬁfthofthestandarddeviationof thegiventimeseries.

Furthermore,letAmdenotesthenumberofmatchesoflength m,andBm-1denotesthenumberofmatchesoflengthmexceptat theendofthesequence.Thesampleentropycanbecomputedas [29]:

SampEn (Xn)=−ln A_m B_m₋₁

:Am=/0andBm−1=/0 (2) InthecasethateitherAmorBm-1iszero,then,

SampEn (Xn)=−ln N−m N−m−1

:Am=0orB_m−1=0 (3) form=0,Bm-1issetto^N(N₂⁻¹⁾.

Inthispaper,SampEnwithm=0,1,2(labelledasm0,m1,m2) foreachoftheacquiredvibrationsignalareextractedandrwasset totherecommendedone-ﬁfthofthestandarddeviation.Therefore, foreachofthevibrationsignals,wehavethreeSampEnfeatures.

4. Hybridintelligentmodel

Thedetailsofthehybridintelligentmodel,FMM-RF,areout- linedinthefollowingsubsections.DetailsoftheClassiﬁcationand RegressionTree(CART)aregiven,beingapartofRF.Inaddition, themodiﬁcationsofFMMandCARTaregivenin therespective subsections.Theprocedureofthehybridmodelisgivenasfollows, inFig.1.

4.1. FuzzyMin-Max

FMMuseshyperboxfuzzysetsforlearning.Toregulateahyper- boxsize,theexpansionparameter(user-defined)of ∈ [0,1]is used.Themin(minimum)andmax(maximum)pointsinahyper- boxareusedinmeasuringhowaninputpatternfitsinthehyperbox fromafuzzymembershipfunction.Ahyperboxfuzzyset(B_j)with V_jbeingtheminpoint,W_jbeingthemaxpoints,andIⁿbeingaunit hypercubeisdefinedasfollow[30]:

B_j=

X,V_j,W_j,f

X,V_j,W_j

∀X∈Iⁿ (4)

Thejointfuzzysetthatcategorisestheoutputclassk^this:

C_k= ∪

j∈KB_j (5)

wherehyperboxesbelongtoclasskisdenotedbyK.

ThelearningalgorithminFMMconstructsnon-linearbound- aries for each output class. As such, overlapping between hyperboxesisonlyallowedforthesameclass.Amembershipfunc- tioncanbecomputedusing[30]:

b_j(A_h)= 1 2n

n

i=1

max(0,1−max

0,␥min

1,a_hi−w_ji

+max(0,1−max(0,␥min

1,v_ji₋a_hi

(6) where beingthesensitivity parameterregulatedthespeedof membershipfunctionandA_h=(a_h1,a_h2,.,a_hn)istheh^thinputpat- tern.

TherearethreenodelayersinFMM,consistingoftheinput(F_A), hidden(FB),andoutput(F_C)layers.F_Acorrespondstonumberof inputdimension,FBbeingthehyperboxlayer,andFC correspond- ingtothenumberofoutputclasses.Everyhyperboxsetismarked withoneF_Bnodewhilemintomaxpointsarecontainedwithinthe connectionsofFAtoFB.ConnectionbetweenthenodesofFBandFC

is:

u_jk=

1 0

ifb_jisahyperboxforclassC_k otherwise

(7)

whereC_kbeingk^thtargetclassinF_Cwhileb_jbeingj^thhiddennode inFB.AfuzzyunionisdoneineveryFCnode:

c_k=max

j=1b_ju_jk (8)

TheF_Cnodescanbeusedintwoways.Theﬁrstoneistheoutputs useddirectly,whichproducesasoftdecision,orthesecondone calledwinner-take-allwhereitusesaharddecision.

TointegrateFMMwithCARTandRF,ahyperboxB_jisﬁrsttagged withCF_j,aconﬁdencefactor,whichiscalculatedas:

CF_j=(1−n)U_j+nA_j (9)

wheren∈[0,1]beingtheweightingfactor,U_jbeingusageofhyper- box,andA_jbeingaccuracyofhyperbox.

Theconﬁdencefactorcanidentifythehyperboxesthatareused regularlyandfairlyaccurate,andalsothosenotbeingusedregularly buthighlyaccurate.Inaddition,thecentroidsofhyperboxesare calculatedasfollows,astheoriginalFMMonlycontainsthemin andmaxpoints:

C_ji^new=C_ji+|a_hi−C_ji|

N_j (10)

whereC_jibeingthecentroidofhyperbox,N_jbeingnumberofcon- taineddatainhyperbox,anda_hiistheh-thinputdata.

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Fig.1. ProcedureoftheproposedFMM-RFmodel.

4.2. Classiﬁcationandregressiontree

Inbuildingadecisiontree,atrainingdataset,whichconsistsof inputdatawithitsrespectiveclassesisneeded.Thedatafortrain- ingconsistsofcentroidsoftheFMMhyperboxes(asinEq.(6)),are partitionedintoanumberofsmallergroups.Basedoneinputsam- ples,theprocessofbuildingthetreestartsattherootnodeinwhich alldatasamplesaretakenintoaccount.Splittingoftreehappens whenthedatasamplesarenotpure,ithappenswhentheyarenot fromthesameclass.Whenthishappens,twoleafnodesaregener- atedfromthemostnotablefeaturefromsamplesofdata.Thissame tree-splittingtechniqueisusedtillafulldecisiontreeisgenerated.

Inprinciple,theGiniimpurityindexisusedtodeterminewhen tree splitting should occur, starting with the measurement of degreeofimpurityfromsamplesofdata,G[31]:

Gini(G)=1−

i

g²(i) (11)

whereg(i),wherei=1,...,e,isthefraction(probabilityofinstance) ofthei-thinputsampleatnodetosplit,inregardstoallminput samples.

Inmeasuringthegoodness-of-split,p,theimpurityfunctionof everyleafnodeisutilized.Inanidealcase,everyleafnodecontains datasamplesonlyfromasingleclass.Tree-splittingstopswhenthis occurs;else,thegoodness-of-splitatthespittingnode(indicated asnodel)subjecttothei-thinputsampleiscalculated[32]:

i(p,l)=i(l)−d_L[i(l_L)]−d_R[i(l_R)] (12) whered_Landd_Rshowsthedatasamplefractionatnodelthatmoves totheleft(dL)andright(dR)childnodeswhilei(dL)andi(dR)show theimpuritymeasuresoftheleftandrightchildnodes[32].

Duringtreebuilding,itisplausibleforasampleofdataintak- inganincorrectbranchinCART.Intacklingthisissue,thecentroid fromeachprototypenodeinFAMisgivenaweight,alsoknown asconﬁdencefactor,whichiscomputedusingEq.(9).Usingthis weightinformation,Eq.(13)replacesEq.(11):

Gini(G)=1−

i

v²(i) (13)

wherev(i)istheweightofthei-thinputsampleatnodel,i=1,...,e.

Thesigniﬁcanceofeveryprototypenodeisshownbytheconﬁdence factor,orweightintheproposedequation.

4.3. Randomforest

Therandomforest(RF)structureisdisplayedinFig.2.Classes arelistedaskandnumberoftreesasT[33].Theconstructionof RFisbasedonthebaggingmethod,usingrandomattributeselec- tion.Usingadataset(D)withtuples(t)andCARTtrees(k)inthe ensemble,ineveryiterationD_iisformedusingdtuplesfromsam- plereplacementmethod[31].TheCARTisthenappliedingrowing theRFtreeuntilitreachesitsmaximalsize.Pruningisthendone tolocatearobustsubsetofensemblemembers.

Pruningshrinksthetreebyeitherturningbranchnodestoleaf nodesorremovingleafnodesundertheoriginalbranch.Thecost- complexitypruningalgorithm[31]isutilized,whereitstartsfrom bottomofthetreeandcost-complexityataninternalnodeisthen counted.Ifthesub-treeresultsinasmallercostcomplexity,itis pruned;elseitremains[31].Themajorityvotingmethodisthen usedin combiningpredictionsfromtheensemble,asshown in Fig.2.

5. Experiments:benchmark

Inthebenchmarkexperiment,thetestsetupismadeupofa 3-phasemotor,a torqueencoder/transducer,and adynamome- ter.Differentloadlevelsweremeasuredwiththedynamometer.

Inacquiringthevibrationsignalsfromthemotorbearingsmanu- facturedbySKF,anaccelerometerwasﬁttedontopofdrive-end ofmotor.Thevibrationsampleswerethensampledat12kHzand savedusinga16-channeldigitalaudiotaperecorder.Faultsinsin- glepointswithdiametersof7,14,21,and28milswereinserted usingelectro-dischargemachining.Operatingconditionsofnormal (N),outerring(OR)racefault,innerring(IR)racefault,andballfault (BF)werecreatedatfourloadlevelsfrom0to3Hp.

InadditiontoFMM-RF,fourothermodels,i.e.FMM,CART,RF, andFMM-CART[34] wereusedforcomparison purposes.FMM, CART,andRFarestandalonemodels,withtheirdetailsgivenin Sections4.1,4.2,and4.3,respectively.FMM-CARTisacombination ofFMMandCART,withtheuseofcentroidsandconﬁdencefactorin FMMandamodiﬁedGiniimpurityindexinCART.Tocomparethe resultswith[35],the5-foldcrossvalidationmethodisused.Atotal of10testrunswereconductedintotal,withtheresultscomputed usingthebootstrapmethod.Theaveragesandstandarddeviations (StdDev)werecomputedwitharesamplingrateof5000forareli- ableperformance[36].TheexperimentswererunusingMATLAB^® R2014aonanIntelCorei52.60GHzprocessorwith8GBofRAM.

Thebenchmarkexperimentsweresplitintothree,usingSam- pleEntropy(SampEn)features,PowerSpectrum(PS)features,and

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Fig.2. Therandomforeststructure(adoptedfrom[33]).

Table1

Resultsofbenchmarkexperiments.

Model Features Accuracy(%) StdDev Complexity

FMM SampEn 98.29 1.15 41hyperboxes

PS 94.48 3.28 169hyperboxes

SampEn+PS 94.65 2.47 173hyperboxes

CART SampEn 82.51 6.53 15leafnodes

PS 88.30 3.07 12leafnodes

SampEn+PS 89.21 2.25 13leafnodes

RF SampEn 96.95 0.03 16leafnodes

FMM-CART SampEn 93.20 1.42 10leafnodes

FMM-RF SampEn 99.84 0.02 9leafnodes

thecombinationofbothsetsoffeatures.Theresultsareshownin Table1.ItcanbeseenthatFMM-RFacquiredthehighestaccu- racyrateat99.89%usingthecombinedSampEnandPSfeatures, whileCARTacquiredthelowestaccuracyrateusingSampEnfea- turesalone.FMM-RFhadtheleastcomplexnetworkwhileFMM hadthemostcomplexnetworkwith173hyperboxes.Thestandard deviationofFMM-RFwasthelowest,at0.02.

Oneofthemainadvantagesofthehybridintelligentmodelisthe abilitytoexplainitspredictionsusingadecisiontree.Thedecision treeishelpfulforitsinterpretability,wherebyknowledgelearned canberevealedandrepresentedintermsofarulesettousers.

WithreferencetothedecisiontreeforCWRUdatainFig.3,the mostimportantfeaturefromFMM-RFis“f₁₃”.

Whenthevalueis<0.10,theinputiscategorizedasOR,elseit thetreesplitsat“f1”.Whenthevalueof“f1”is<0.08,theinputis categorizedasNO,elsethetreesplitsagain.Whenthevalueof“f₂₀” is≥0.62,theinputiscategorizedasIR,elsethetreetakesasplit at“f9”,wherethetreesplitsintotwobranches.Whenthevalue is<0.36,itsplitsto“f₂₀”,whereifthevalueis≥0.20,theinputis categorizedasIR,elseitisBF.Ontheotherhand,whenthevalueis

≥0.36,the“m0”ischecked,whereifthevalueis≥0.12,theinputis categorizedasBF.Thetreethenmakesitﬁnalsplitat“f₆”,whereif thevalueis≥0.34,itiscategorizedasIR,elseitisBF.

AsshowninTable2,comparisonofresultswith[35]isshown.

Atotalof3modelswereused,consistingofmultilayerperceptron (MLP),FMM,andCART.Resultsin[35]werecomputedusinga5- foldcrossvalidationmethod.Thefeaturesusedin[35]aredifferent

Fig.3. DecisiontreeforCWRUdatasetusingthecombinationofSampEnandPS features.

Table2

ResultscomparisonofFMM-RFwith[35].

Model Accuracy(%) StdDev Complexity

MLP[35] 85.23 6.86 50hiddennodes

FMM[35] 96.37 2.87 25hyperboxes

CART[35] 99.02 0.98 5leafnodes

FMM-RF 99.89 0.53 8leafnodes

fromthoseinthispaper,wheretheyconsistedofninetime-domain featuresandseven-frequencydomainfeatures.Whiletheresults ofFMM-RFacquiredthehighestaccuracyratewiththesmallest standarddeviation,CART[35]ontheotherhandhadthesimplest networkwithﬁveleafnodes.

6. Experiments:real-world

Realdatawasacquiredfromasmalltestrig[5,37,38],asdepicted inFig.4thatemulatesarunningrollerbearingsenvironment.The testrigconsistsofaDCmotorwhichdrivestheshaftthroughaﬂex- iblecoupling.Twoplummerbearingblocksthensupporttheshaft.

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Fig.4. Thesetupofthedataacquisitiontestrig(reproducedfrom[5]).

0 0.01 0.02 0.03 0.04

-30 -20 -10 0 10 20 30 40 50

t/ms

Amplitude/mV

NO

0 0.01 0.02 0.03 0.04

-50 -40 -30 -20 -10 0 10 20 30 40 50

t/ms

Amplitude/mV

NW

0 0.01 0.02 0.03 0.04

-250 -200 -150 -100 -50 0 50 100 150 200

t/ms

Amplitude/mV

IR

0 0.01 0.02 0.03 0.04

-30 -20 -10 0 10 20 30 40

t/ms

Amplitude/mV

OR

0 0.01 0.02 0.03 0.04

-300 -200 -100 0 100 200

t/ms

Amplitude/mV

RE

0 0.01 0.02 0.03 0.04

-40 -30 -20 -10 0 10 20 30 40

t/ms

Amplitude/mV

CA

Fig.5.Sixsamplevibrationsignalswithrespectivefaulttypes.

Sixconditionsweretestedandrecorded.Twonormalconditions;

abrandnewcondition(NO)andawornbutundamagedcondition (NW);fourfaultconditions;outerrace(OR),cage(CA),innerrace (IR),androllingelement(RE)faults.Themachinewasoperatedat arangeofspeeds,from25to75rev/s,andtentime-serieswere takenateachspeed.Thisresultedin960samples,with160exam- pletime-serieseachfromtheconditions.Forthiswork,thedata wasacquiredatsixteendifferentspeedswhichaddnon-linearity ontothisproblem.

Fig.5depictssamplevibrationsignalsforthesixdifferentfault types.Dependingonthefaulttypes,thedefectinthebearingmod- ulatesthevibrationsignalsandsomewithdistinctivespikes.Two faultconditions,innerandouterracehavereasonablyperiodicsig- nalascomparedtotherollingelementwhichmayormaynotbe

periodic.Thisdependsonanumberoffactorswhichincludethe severityofdamagetorollingelement,bearingloading,andball trackwithintheraceway.Thecagefaultcreatesarandomdistor- tion,againdependingonseverityofdamageandbearingloading.

ThefeaturespaceforthesethreefeaturesisshowninFig.6.

SimilartothebenchmarkexperimentsinSection5,theFMM, CART, RF, and FMM-CART [34] were used for comparison purposes.Tocomparetheresultswith[27],the10-foldcrossvalidation methodisused.Atotalof10testrunswereconductedintotal,with theresultscalculatedwiththebootstrapmethod.Theresultsofthe 2-classproblemareshowninTable3.FMM-RFacquiredthehigh- estaccuracyrateat99.82%usingSampEn+PSfeatureswhileCART usingPSonlyfeaturesacquiredthelowestaccuracyrate.FMM-RF at5leafnodeshadtheleastcomplexnetworkwhileFMMhadthe

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Fig.6. Featurespacefortheentropicfeaturesofm0,m1,andm2.

Table3

Resultsof2-classproblem.

mostcomplexnetworkwith171hyperboxes.Thestandarddevia- tionofFMM-RFwasthelowest,at0.02.

Withreferencetothedecisiontreefor2-classprobleminFig.7, themostimportantfeaturefromFMM-RFis“f₂₁”.Thetreesplits intotwomain parts.Whenthevalueis ≥0.32,“f₁₃”ischecked, whereifthevalueis<0.62,theinputiscategorizedashealthy,else thetreesplitsagainto“m₀”.Whenthevalueis≥0.92,theinput iscategorizedasfaulty,elseitishealthy.Whenthevalueof“f₂₁” is<0.32,thetreesplitsto“f18”,whereifthevalueis≥0.18,“f19” ischecked.Whenthevalueis≥0.11,theinputiscategorizedas faulty,elseitishealthy.Whenthevalueis<0.18,“f₂₂”ischecked, whereifthevalueis≥0.09,theinputiscategorizedasfaulty,else itishealthy.

Table4

Resultsof6-classproblem.

Fig.8. Decisiontreefor6-classproblemusingentropicfeaturesusingSampEnand PSfeatures.

Inadditionto2-classproblem,the6-classproblemisconducted, withresultsshowninTable4.Thesamesetupwasusedforthe2- classproblemisusedinthisexperiment.Theresultsaresimilar tothoseofthe2-classproblem,withFMM-RFacquiringthehigh- estaccuracyrateandCARTthelowest.Again,FMMhadthemost complexnetworkwith82hyperboxeswhileFMM-RFhad6–10leaf nodes,withFMM-CARTcominginsecondwithmaximumof11leaf nodes.FMM-RFhadtheloweststandarddeviationat0.02.

Fig.7. Decisiontreefor2-classproblemusingSampEnandPSfeatures.

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Table5

ResultscomparisonofFMM-RFwith[27].

Model Accuracy(%) StdDev Complexity

SVM[27] 93.50 0.50 20hiddennodes

MLP[27] 93.70 0.21 –

FMM-RF 99.81 0.41 8leafnodes

Withreferencetothedecisiontreefor6-classprobleminFig.8, themostimportantfeaturefromFMM-RFis“f₁₈”.Thetreesplits intotwomainbranches,oneontheleftanotherontheright.When thevalueof“f18”is≥0.23,thetreesplitsto“f11”,whereifthevalue is≥0.29,theinputiscategorizedasIR,elseitisRE.Whenthevalue of“f₁₈”is<0.23,thetreesplitsto“f₁₁”.Whenthevalueof“f₁₁”is

≥0.24,theinputiscategorizedasIR,elseitsplitsto“f3”.When“f3” is<0.35,theinputiscategorizedasNO,elseitsplitsto“f₁₆”.When thevalueof“f₁₆”is<0.25,theinputiscategorizedasOR,elseitsplits again.Whenthevalueof“f1”is<0.11,theinputiscategorizedas NW,elsethetreetakestheﬁnalsplit.Whenthevalueof“m2”is

≥0.78,theinputiscategorizedasCA,elseitisRE.

Thecomparisonsofresultsaredonewiththosefrom[27],as showninTable5.Twodifferentmodelsareusedin[27],consisting ofsupportvectormachine(SVM)andMLP.AlinearSVMclassiﬁes linearlyseparableinputdatabyusingahyperplanedetermined throughtrainingwithasetoflabelledtrainingdata.SVM,asamem- berofthekernelmachinefamily,canbegeneralisedtonon-linearly separabledatathroughtheuseofthekerneltrick.Inanutshell, thenon-linearlyseparabledataisprojectedtoalinearlysepara- blespacethroughachosenkernelbeforeapplyingtheusualSVM classiﬁcationprocedures.

Ontheotherhand,theMLPisaclassicalfeedforwardneural networkwheretheneuronsarearrangedinatwolayersconﬁg- urationconnected through individualweights. The weights are obtainedbyback-propagationtrainingalgorithms.Theindividual neuron(perceptron)consistsofmultipleinputsandanon-linear outputactivatedthroughanactivationfunction.Acommonactiva- tionfunctionofchoicewouldbethehyperbolictangentfunction.

Duringthetrainingstage,SVMisusuallyslowertotrainthana MLPgiventhesamedatasetanditrequiresfurtheradaptationfor multiclassclassiﬁcation.However,duringtheclassiﬁcationstage, SVMismuchfasterthantheMLPasitrequiresonlyacosineproduct.

Intermsofpredictionaccuracy,SVMisreportedtohavesuperior accuracythanMLPinmanyliterature,althoughtheperformance willalsodependonthenatureoftheproblem,dataconﬁguration andotherconstraints.

The10-foldcrossvalidationmethodwasusedtogettheresults in[27].TheSVMusestheradialbasisfunction(RBF)kernelwhile theMLPhas20hiddennodes.Featuresusedin[27]consistedof thethreeentropyfeatures.TheresultsofFMM-RFarethehighest, withtheloweststandarddeviation.Withthesethreefeatures,SVM [27]acquiredthelowestaccuracyrate,withalmost6%lowerthan thatofFMM-RF.TheFMM-RFachievedbetteraccuracywiththe samesampleentropyfeaturesetandwithmuchreducedstructure complexityandtrainingefforts.Theclassiﬁcationrulesobtained fromFMM-RFisalsoeasilycomprehensible.

7. Conclusions

Theclassiﬁcationresultsofballbearingfaultsusingvibration signalshavebeenpresentedinthispaper.Variousconditionmon- itoringtechniqueswithvibrationsignalsusingintelligentsystems aredetailed.ThehybridFMM-RFmodelhasbeenproposedand usedintheexperiments,whichweredividedintobenchmarkand real-worlddata.Powerspectrumandsampleentropyfeatureswere used in the feature extraction, where important features were extracted.Boththebenchmarkandreal-worlddatasetshowed

accurateperformancesusingtheFMM-RFmodel.Thebestresults ofbenchmarkandreal-worlddatasetswereat99.9%and99.8%

respectively.Inadditiontoaccurateresults,explanatoryrulesfrom adecisiontreegeneratedbyFMM-RF,whichexplainedtheresults, arepresented.Thisstudydoesindicatetheusefulnessofthepro- posedhybridFMM-RFmodelforclassiﬁcationofballbearingfaults.

Acknowledgements

ProfessorNandiisaDistinguishedVisitingProfessoratTongji University,Shanghai,China.Thisworkwaspartlysupportedbythe NationalScienceFoundationofChinagrantnumber61520106006 andtheNationalScienceFoundationof Shanghaigrant number 16JC1401300.

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