FAULTPROGNOSISOFBEARINGSINELECTRICALDRIVESANDMOTORSByRodneyK.SingletonIIADISSERTATIONSubmittedtoMichiganStateUniversityinpartialfulllmentoftherequirementsforthedegreeofElectricalEngineeringŒDoctorofPhilosophy2016ABSTRACTFAULTPROGNOSISOFBEARINGSINELECTRICALDRIVESANDMOTORSByRodneyK.SingletonIIInrecentyears,therehasbeenagrowinginterestindiagnosisandprognosisofmotorsandelec-tricaldrives.Effectiveandaccuratediagnosisandprognosisofsystemswilleventuallyleadtoconditionbasedmaintenance,whichwilldecreasemaintenancecostsandsystemdowntime,im-provingthereliabilityofelectricaldrives.Morethan50%ofmotorfailuresareduetoballbearings.Assuch,theareaofbearingfaultdiagnosisandprognosishasattractedalotofattentioninrecentyears.Althoughmanytechniqueshavebeensuccessfullyappliedforbearingfaultdiagnosis,prog-nosisoffaultsandespeciallypredictingtheremainingusefullife(RUL)ofbearingsisaremainingchallenge.Themainreasonsforthisarealackofaccuratephysicaldegradationmodels,limitedlabeledtrainingdata,andthelackofaprioriknowledgeofthedifferenthealthstatesofbear-ings.Thereareseveralfactorsthatcontributetobearingfailure,includingthemechanicalstressofaloadandtheelectricalstressofbearingcurrents.Duetotheintrinsicpropertiesofmotorsdrivenbypulse-widthmodulation(PWM)operation,therearecurrentpathsthatformfromthemotorshaftthroughtheracesofthebearingandbacktoground.Thesecurrentpathsarecausedbyvoltagedivisioninteractionwiththecommonmodevoltageandstraycapacitanceswithinthemotor.Onetypeofbearingcurrent,electricdischargemachining(EDM)current,causesasignif-icantamountofdamagetobearings.ThepresenceofEDMcurrentscausespittingintherotatingelementsofthebearingandultimatelyleadstobearingfailure.Althoughthisrelationshipiswellknownandstudied,littleworkhasbeendonetorelatebearingcurrentdischargeeventstobearingvibrationsforfailureprognosis.Inthiswork,weproposebothcomputationalandexperimentalapproachesforRULestimationofbearings.InChapter2,wepresenttwoplatformswhichwereusedtoacceleratetheagingprocessofbearings.Therst,thePRONOSTIAPlatform,acceleratedbearingdegradationviaexcessiveloads,whilecollectingvibrationandtemperaturedataoverthecourseofarun.Thesecondplatformisanewtestbedweconstructedtobetterunderstandtherelationshipbetweenbearingcurrents,vibrationsandfailure.Thistestbedappliesanelectricalstressontestbearingstoinduceacceleratedaging.Overthecourseoftheexperiments,wecollectmultiplesensordataincludingcurrent,temperature,andvibrationfromstarttofailureinordertocorrelatecurrentdataaswellasvibrationdatatobearingfailure.InChapter3,weintroduceanapproachforlearningthehiddenhealthstatesofabearingfromvibrationsignals.Thisproposedapproachisbasedonextractingmultiplefeaturesfromsensorsignalsandidentifyingchangepointsinthestateofthesystembasedonthesefeatures.WealsoproposeaframeworkbasedontemporalHiddenMarkovModelforunsupervisedclusteringofbearingvibrationdatainordertoidentifyhiddenhealthstatesinthedata.InChapter4,weintroduceadata-drivenmethodology,whichreliesonbothtimeandtime-frequencydomainfeaturestotracktheevolutionofbearingfaultsbasedonvibrationsignals.AnextendedKalmanlterisappliedtothesefeaturestopredicttheremainingusefullifeandtoprovideacondenceintervaltotheRULestimates.PerformanceoftheproposedmethodsareevaluatedonthePRONOSTIAexperimentaltestbeddata.InChapter5,weproposeacomputationalframeworkthatrelatesthecurrentdischargeeventswiththeevolutionofvibrationdataforamoreaccurateRULestimation.WeuseacurrentdischargeinuxeventasatriggertoperformRULestimationonbearingsusingvibrationdata,resultinginhigheraccuracyandefciency.Thisdissertationisdedicatedtomygrandmother,MattieMaeTaylor.ivACKNOWLEDGEMENTSThismaterialisbasedinpartuponworksupportedbytheNationalScienceFoundationunderGrantNo.EECS-1102316andbytheNationalScienceFoundationGraduateResearchFellowshipunderGrantNo.DGE-0802267.vTABLEOFCONTENTSLISTOFTABLES.......................................viiiLISTOFFIGURES.......................................ixCHAPTER1INTRODUCTION...............................11.1StateoftheFieldofBearingFailurePrognosis....................31.2ContributionsoftheThesis..............................71.3Background......................................91.3.1Time-FrequencyDistributions........................91.3.2Time-FrequencyFeatureExtraction.....................12CHAPTER2EXPERIMENTALDATA...........................132.1Background......................................132.1.1ReviewofSomePreviousPlatforms.....................132.1.2BearingCurrentFormation..........................152.1.2.1CirculatingBearingCurrents....................172.1.2.2ShaftGroundingCurrent......................172.1.2.3ElectricDischargeMachiningCurrents..............172.2PRONOSTIAPLATFORM..............................202.3AcceleratedBearingDegradationPlatformviaElectricalStress...........232.4Conclusions......................................25CHAPTER3DISCOVERINGTHEHIDDENHEALTHSTATESFROMBEARINGVIBRATIONDATA..............................263.1Introduction......................................263.2HiddenHealthStateIdenticationviaEventDetection...............273.2.1FeatureExtraction..............................273.2.2EventDetection................................293.2.3Results....................................303.2.3.1EstimatingtheHealthStates....................323.2.3.2PerformanceofMultipleFeatures.................333.3HiddenHealthStateIdenticationviaEventDetection...............353.3.1TemporalHiddenMarkovModels......................353.3.2FeatureExtraction..............................383.3.3CalculatingtheHMMparameters......................423.3.4UnsupervisedClusteringviaTemporalHMM................443.4Conclusions......................................48CHAPTER4FAULTPROGNOSISANDRULESTIMATIONONBEARINGSVIAEXTENDEDKALMANFILTER.......................514.1Background......................................514.1.1EKFParameterLearning...........................52vi4.1.2RULPrediction................................554.1.3RULCondenceIntervals..........................564.1.4RULEstimationviaExtendedKalmanFilter.................564.1.4.1FeatureExtractionandCurveFitting...............564.1.4.2RULEstimation..........................58
4.1.4.3ComparisonofEKFvs.KF....................61
4.1.4.4CondenceIntervalEstimation..................624.2Conclusions......................................62CHAPTER5THEUSEOFBEARINGCURRENTSANDVIBRATIONSINLIFE-TIMEESTIMATIONOFBEARINGS....................645.1Background......................................645.1.1BearingCharacteristicFrequencies......................645.1.2Data......................................645.2Methodology.....................................655.2.1FeatureExtraction..............................655.2.1.1BearingVibrationFeatures.....................655.2.2DetectionandTrackingofEDMCurrents..................665.2.3RULPredictionviaEKF...........................705.3ExperimentalResults.................................715.3.1TemperatureAnalysis.............................715.3.2ComparisonwithConventionalVibrationAnalysis.............725.3.3Event-triggeredRULEstimationsusingCurrentDischargeInux......745.4Conclusions......................................77CHAPTER6CONCLUSIONS................................786.1FutureWork......................................796.1.1UsingtheHiddenHealthStatesofBearingsforEffectiveFaultPrognosis.796.1.2RFDetectionofBearingDischargeEvents..................80BIBLIOGRAPHY........................................82viiLISTOFTABLESTable2.1OperatingConditionSpecicationsforthePRONOSTIAPlatform........21Table4.1NMSEBetweenCurveFitsandFeatures......................57Table4.2CurveFittingParametersforTrainingData....................57Table4.3ComparisonofRULusingVariancevs.Entropy..................59Table4.4ComparisonofRULestimationsusingEKFvsKF................62Table5.1BearingCharacteristicFrequencies........................64Table5.2ComparisonofRULaccuracyfortrainingacrossalltimeversustrainingaftertheinuxevent...................................75viiiLISTOFFIGURESFigure1.1Diagramofaballbearing..............................6Figure1.2Flutingintheouterraceofadamagedbearing...................6Figure1.3Diagramofadiscretewaveletdecomposition[1].................11Figure2.1ThreephasevoltagesofanACdriveandtheaverageofallthree,orthecommonmodevoltage[2].............................16Figure2.2Straycapacitancesofaninductionmotor[3]....................18Figure2.3Bearingelectricloadstates.Thetoprowshows3specicvoltageprolesandtheircorrespondingcurrentresponsesareshowndirectlybelow.(a)In-sulated.(b)Discharge.(c)Ohmic[4].......................20Figure2.4OverviewofPRONOSTIAsetup[5]........................22Figure2.5BearingsupportshaftofPRONOSTIAplatform[5]................22Figure2.6Comparisonofnewvs.degradedbearing[5]....................23Figure2.7Overviewofacceleratedbearingdegradationplatformduetoinducedelec-tricalstress.....................................24Figure3.1RawDataofInitialVibrationSignalforBearing11...............27Figure3.2RawDataofFinalVibrationSignalforBearing11................28Figure3.3Choi-WilliamsTransformationofInitialHorizontalVibrationSignalwith˙=10forBearing11...............................28Figure3.4Choi-WilliamsTransformationofFinalHorizontalVibrationSignalwith˙=10forBearing11.................................29Figure3.5FeaturesforBearing11acrosstime........................31Figure3.6Change-pointgroupingintotransitionstages(shaded)foroperatingcondi-tions1and2....................................33Figure3.7HealthstateprogressionforBearing12......................34Figure3.8HealthstateprogressionforBearing21......................34Figure3.9Z-Scorecomputationusing2featuresonBearing12...............35ixFigure3.10Z-Scorecomputationusing4featuresonBearing12...............36Figure3.11Z-Scorecomputationusing6featuresonBearing12...............36Figure3.12AccelerometerresultsforBearing2fromstarttofailure.(a)Showsthevibra-tionsfromthehorizontalaccelerometerforBearing2and(b)thevibrationsfromtheverticalaccelerometerforBearing2...................39Figure3.13Varianceofthehorizontalvibrationdata......................39Figure3.14RawDataofInitialVibrationSignalforBearing2.................40Figure3.15RawDataofFinalVibrationSignalforBearing2.................40Figure3.16Choi-WilliamsTransformationofInitialHorizontalVibrationSignalofBear-ing2with˙=10..................................41Figure3.17Choi-WilliamsTransformationofFinalHorizontalVibrationSignalofBear-ing2with˙=10..................................41Figure3.18ProgressionofthefaultforBearing2intheTFdomain..............43Figure3.19FeaturesusedintemporalHMMclusteringforBearing1acrosstime.......45Figure3.19cont'd.......................................46Figure3.20TemporalHMMClusteringresultsforBearing1.................47Figure3.21TemporalHMMClusteringresultsforBearing2.................48Figure3.22TemporalHMMClusteringresultsforBearing4.................49Figure4.1Medianlteredtime-domainvarianceacrossall6trainingsetsfortheFEMTOdata.........................................53Figure4.2Curvettingtovarianceandentropyfeatures...................54Figure4.3RULEstimationforBearing13withthevariancefeature.............60Figure4.4RULEstimationforBearing33withthevariancefeature.............60Figure4.5RULEstimationforBearing24..........................61Figure4.6RULEstimationforBearing33withdifferentEKFtrackingstarttimes.....61xFigure5.1AccelerometerrecordingsforBearing1fromstarttofailure.(a)Thevi-brationsfromthehorizontalaccelerometerand(b)thevibrationsfromtheverticalaccelerometer................................65Figure5.2BearingcurrentsamplesfromBearing1.(a)Currentsamplefromabearingundernormalconditionand(c)acloseupofthissample.(b)Currentsampleinwhichadischargeeventhasoccurredand(d)acloseupofthisdischargeevent.67Figure5.3Currentsamplew/dischargeeventandcorrespondingreconstructedsignalusingthelevel8detailcoefcientsfromaHaarwaveletdecomposition......69Figure5.4Normalcurrentsampleandcorrespondingreconstructedsignalusingthelevel8detailcoefcientsfromaHaarwaveletdecomposition...........69Figure5.5RMSFrequencyfeatureforBearings1,2,3,4and5...............71Figure5.6TemperaturesignalforBearings1,2,3,4and5..................72Figure5.7Relationshipbetweenbearingcurrentdischargesandvibrationsforthe5testbearings.Therstrowshowsthecumulativebearingdischargesacrosstheentirerun.ThesecondrowshowstheRMSFrequencyofthevibrations,extractedfromthefrequencydomain........................73Figure5.8Magnitudeofthefrequencyspectrumateachbearingcharacteristicfrequencytrackedintimeforthe5testbearings.......................74Figure5.9Detectionofthecurrentdischargeinuxevent.Thetopplotshowsthenum-berofdischargeeventsacrosstime.ThebottomplotshowstheNMSEbe-tweenthettedlineandthedatapoints,witheachpointrepresentingtheerroroverthepreviousmminutes.........................76Figure5.10RULEstimationsforBearings1and2.EachplotshowstheresultsofstartingRULestimationsfromthebeginningandfromthecurrentdischargeinuxevent.CondenceintervalsaroundthetrueRULareshowntoevaluatetheaccuracyoftheestimations.............................76Figure6.1BearingCurrentDischargeEvent[6]........................81xiCHAPTER1INTRODUCTIONCommonpracticeinindustryistoperformxedintervalmaintenanceasasolutiontomaintenanceofelectromechanicalsystems.However,thereareseveralproblemsthatariseusingthispractice.First,thereisthepossibilitythatfailurecouldoccurbetweenscheduledmaintenances,whichcouldresultinacatastrophicaccident.Second,performingthesescheduledmaintenancechecksincurshighcosts,eveninthecasewherethereisnofaultdetected.Third,xedintervalmaintenancerequiresthemachinetobeunnecessarilyoutofuseandunabletoperformitsusualfunction,whichiscostlytotheuser.Condition-basedmaintenance,includingeffectivediagnosticandprognostictools,providesasolutiontothisproblemasmaintenanceonlyoccurswhentheuserisalertedtoanimpendingfailure,providedbyaremainingusefullifeestimation[7Œ9].TheRULofasystemisdenedasthetimebetweenthecurrenttimeinstanttotheendoftheusefullife.TheconceptofRULhasbeenwidelyusedinreliabilityanalysis,manufacturingsys-temsandoperationalresearch[10,11].AccurateRULpredictionsofelectromechanicalsystemswillprovidetheuserwithtimetogetthedefectivepartxedorreplaced.Thiswillreducemain-tenancecosts,systemdowntime,andmoreimportantlyincreasesystemsafetyandreliability[7].Althoughtherehasbeenalotofprogressintheareaoffaultdiagnosisusingsignalprocessingandpatternrecognitiontechniques[12Œ15],wellunderstoodsystematicmethodologiesforprognosisandRULpredictionfromlimitedamountoftrainingdataarestillnotavailable.ThecurrentapproachesusedforRULestimationincludemodel-basedanddata-drivenmeth-ods[16,17].Themodel-basedapproachestoprognosisusemathematicalrepresentationstoincor-porateaphysicalunderstandingofthesystem,andincludebothsystemmodelingandphysics-of-failure(PoF)modeling.Inthesystemmodelingapproach,mathematicalfunctionsormappings,suchasdifferentialequations,areusedtorepresentthesystem.Statisticalestimationtechniquesbasedonresidualsandparityrelations(thedifferencebetweenthemodelpredictionsandsystemobservations)arethenusedtodetect,isolateandpredictdegradationandremainingusefullife.1EstimationtechniquessuchasKalmanltersandparticlelters,arecommonlyusedtocalculatetheresiduals[18].Forexample,thisapproachtoprognosticswasdemonstratedforlithiumionbat-terieswherealumpedparametermodelwasusedalongwithextendedKalmanlterandparticlelteralgorithmstoestimateremainingusefullife(RUL)[19].Physics-basedfailuremodels[20]relyonthephysicsoftheunderlyingdegradationprocesstobeabletopredicttheonsetoffailuresandareapplicableinsituationswhereaccuratemathematicalmodelscanbeconstructedfromrstprinciples.Forexample,theYu-Harrisbearinglifeequation[21]iscommonlyusedtopredictspallinitiation.Developmentofthemodelsrequiresdetailedknowledgeoftheunderlyingphysicalpro-cessesthatleadtosystemfailure,andincomplexsystems,itisdifculttocreatedynamicmodelsrepresentingthemultiplephysicalprocessesoccurringinthesystem.Thisisoneofthelimitationsofmodel-basedapproaches.Data-drivenapproaches,ontheotherhand,useconditionmonitoringdatacoupledwitharti-cialintelligence,e.g.,neuro-fuzzysystems[22,23],orstatisticallearningandpatternrecognitiontools[24Œ26],e.g.,Markovchains,hiddenMarkovmodels(HMMs)[23,27,28],totrainasys-tem,anduseittoestimatetheRUL.Mostofthesetechniquesconsistofanofinelearningstagethroughhistoricaldata,whichincludesfeatureextractionanddegradationstatelearning,followedbyanonlinestagethatcontinuallyupdatesthepredictionofRUL,andprovidesanestimateofthepredictionuncertainty.Inthelaststageofdata-drivenmethodologies,thelearnedmodelsareappliedtotestdatatodeterminethetimetothenextdegradationstateorprovideaprobabilityoffailure.Oneoftheadvantagesofdata-drivenapproachesisthattheycanbeusedasblack-boxmodelsastheylearnthebehaviorofthesystembasedonmonitoreddataandhencedonotrequiresystem-specicknowledge.Further,data-drivenapproachescanbeappliedtocomplexsystemssincedata-drivenapproachescanbeusedtomodelthecorrelationbetweenparametersandinter-actionsbetweensubsystemsaswellaseffectsofenvironmentalparametersusinginsitudatafromthesystem.Oneofthelimitationsofdata-drivenapproachesliesintherequirementoftrainingdata.Data-drivenapproachesdependonhistorical,e.g.,training,systemdatatodeterminecorre-lations,establishpatterns,andevaluatedatatrendsleadingtofailure.Inmanycases,therewillbe2insufcienthistoricaloroperationaldatatoobtainhealthestimatesanddeterminetrendthresholdsforfailureprognostics.Asolutiontothisproblemistofusesystemmodels,suchasPoFmodels,withthedata-drivenmodels.Inbothmodel-basedanddata-driventechniques,workhasbeendonetoperformprognosisanddiagnosisthroughtheuseofintermediaryhealthstates.Inmostcases,thereisanactualphysicalmeaningtotheunderlyinghealthstates,asin[29],wherethehealthstatescorrespondtothenumberofdamagedormissingteethinagear.Similarly,in[30],thenumberofbrokenrotorbarsininductionmachines,whichcanincurmanysecondaryeffectssuchasmechanicalvibrations,increasesintemperature,andstatorwindingdamage[31],determinesthehealthstateofthemotor.However,problemsarisewhendealingwithacomponentwhichdoesnothavewell-denedhealthstatesthroughoutitsdegradationprocess[28].Inthiscase,healthstatesneedtobelearnedfromthedataovertimethrougheventorchangepointdetection[32].Recently,therehasbeenanincreasedinterestinevent,orchange-point,detectionduetoitsabilitytocapturetrendchangesorinterestingpatternsintimeseriesdata[32Œ34].Moreover,theseapproachescanbeusedtopartitionagiventimeseriesintodifferenteventintervals,especiallywhentheseintervalsarenotknownapriori.1.1StateoftheFieldofBearingFailurePrognosis
Bearingsareoneofthemostwidelyencounteredmechanicalpartsinrotationalequipmentandconstitutealargeportionoffailures.Motorfailuresareoftenlinkedtobearingfailure.Therefore,bearingconditionmonitoringcanbeverycosteffectiveandreducethemaintenancedowntimebyprovidinganadvancewarningsystemthatallowsfortheschedulingoftimelycorrectiveandrepairactions[35Œ37].TraditionalestimationofthelifetimeofabearingisbasedontheANSI/AFBMAStandardliferatingformula[27].However,theactuallifetimeofabearingcandiffersignicantlyfromthetheoreticaloneduetotheoperatingconditions.Therefore,thereisaneedforbearingfaultprognosisandremainingusefullifetimeestimationfromvibrationsignalanalysis.Althoughthebearingvibrationsignalscontainveryspecicinformationaboutthebearing'sfaultconditions,3itisquitedifculttodetectandtrackthesignatureofthefaultsatanearlystage.Moreover,thecharacteristicsofthefaultdonotnecessarilyprogressmonotonicallyovertimewhichmakesithardforstandarddata-drivenpatternrecognitionapproachestosucceed.Thus,oneoftheremainingchallengesinprognosticsofbearingsishowtoextractthefeaturesandconstructstatisticaltrackingalgorithmsfromthevibrationsignals.Mostofthecurrentbearingprognosisalgorithmsrelyondifferentsignaltransformstoex-tractrelevantfeaturesfromthevibrationsignalsinconjunctionwithmachinelearningandneuralnetworkapproaches[22,25,38].However,amajorityofthesemethodsincludelabeledtrainingdatatoidentifythedifferenthealthstatesduringthelifetimeofabearingandthenbuildstatisticalmodelssuchasHiddenMarkovModels(HMMs)withparameterslearnedfromthisdata[27,28].Moreover,mostofthecurrentprognosisalgorithmsthatrelyonprobabilisticmodelsyieldtheprob-abilityoftransitioningtofailurestateinthenexttimestepratherthananestimateofRUL[39].Inmostreallifeconditions,thereisalackoflabeleddatacorrespondingtothedifferenthealthstatesduringthelifetimeofthebearing.Therefore,thereisnogroundtruthinformationaboutthetimingofthetransitionsbetweenstatesandthetotalnumberofstates.Thecurrentstate-of-theartindeterminingthelifetimeorconditionofbearingisbearingvi-brationanalysis[40,41].Thisapproachisbasedonspectralanalysisofthevibrationdatawhichsearchesforthemostlikelyfrequenciespresentinvibrationdatabasedonthebearing'sgeometry.Thisanalysisisusedtodetermineabearingfaultaswellastodistinguishbetweenthedifferentpossiblefaultlocations,suchastheinnerrace,outerrace,etc.Itcanalsobeusedasanearlywarningdetectionmethod.Theshortcomingofvibrationanalysisisthatbearingvibrationdataisoftennoisy,whichleadstoacomplicatedfrequencyspectrumanddifcultiesinanalysis.In[40],waveletlteringisusedtoextractbearingcharacteristicfrequencyinformationfromnoisyvibra-tiondata,howeverthisapproachonlyprovidesearlywarningandnotprognosis.BearingvibrationshavebeenusedforRULestimation,butmuchlessworkhasusedthecharacteristicbearingfre-quenciesextractedfromvibrationanalysisasfeatures.In[41],particlelteringwasusedtotrackbearingvibrationfeaturesextractedfromrecurrencequanticationanalysisacrosstime.However,4themajordisadvantageofusingvibrationsforbearingfaultclassicationandRULestimationisthatthereislittletonosignicantinformationpresentinthevibrationsintheearlystagesoftherun.Asthebearingdegrades,thevibrationsnallystarttoshowsignicantchangeswhichcanbeusedforaccurateRULestimations.Thereisalsosignicantnoiseinbearingvibrationread-ingsduetoothercomponentsinthemotorwhichcannotalwaysbeseparatedout,especiallyatthebeginningstagesoftherunofabearing.Althoughmuchworkhasbeendonewithbearingvibrationsasanindicatortobearingfailure,muchlessworkhasbeendonetouseothertypesofsignalssuchascurrentorvoltage.Bearingcurrentsandtheireffectsonbearingshavebeenrecognizedasaproblemformanyyears[42Œ44].Thesebearingcurrentsappearwhenamotorisunderinverteroperationandarefoundinoneofthreeforms:circulatingcurrents,shaftgroundingcurrents,orEDMcurrents[2,3,45].ThemostdamagingtypeofthesecurrentsisEDM.EDMcurrentsoccurwhenahighvoltageacrossthebear-ingbreaksdownthelubricationlmsurroundingtherotatingelements(seeFigure1.1).Theresultisacurrentdischargeeventbetweentheouterandinnerracesofthebearing.Thesedischargeeventscarryenoughenergyinthemtocausepitsandcratersontheballsandtheracewaysofthebearing.Thesecraterseventuallyleadtouting(seeFigure1.2),whichiswhentheasymmetryintherotatingelementscausedbythecratersleadstotheballsdiggingdeepgroovesonthebearingraceways,andthelifeofthebearingbecomessignicantlyreduced[46,47].Whilethisrelation-shipbetweenbearingcurrentsandbearingfailureiswellknown,directlymeasuringthesebearingcurrentsisphysicallyimpossibleandbearingcurrentmeasurementsrequirespecialequipmentandpersonnelinnormalmotoroperation[2,48].Severaltechniqueshavebeendevelopedtoindirectlymeasurebearingcurrents,includingdetectingthemfrombearingvibrations[4]andusingaradio-frequency(RF)measurementsetuptodetectbearingdischargeevents[47].However,thechallengeofestimatingordetectingbearingcurrentsstillremains.5(a)Bearingdiagram(b)Crosssection
ofBearingFigure1.1Diagramofaballbearing.Figure1.2Flutingintheouterraceofadamagedbearing.61.2ContributionsoftheThesis
InChapter2,wepresenttwodifferentplatformsfromwhichdataiscollectedandusedinthiswork.TherstplatformisthePRONOSTIAplatform[5],whichhasprovidedanextensivevibrationdatasetforthreetypesofoperationforbearingsloadedwithradialforces.ThisdatasethasbeenusedbyotherinvestigatorstoevaluatefailureprognosisandRULestimationalgorithms[36,38,49].Thesecondplatformpresentedisoneweconstructedthatacceleratestheagingprocessofabearingbyapplyingavoltagetothebearingshaft.Thisinducedvoltageisdesignedinsuchawaythatitemulatescommonmodevoltagefromaninverter-drivenmotor,thusallowingEDMcurrentstoowthroughthebearing.TheseEDMcurrentscauseirreparabledamagetothebearing[43,50]andaidintheaccelerationoftheirdegradation.Theexperimentissetupasfollows.First,weconstructtheplatformtohavethehighestprobabilitytoexhibitseverelydamagingbearingcurrentdischargeevents.Thisworstcasescenarioforbearingoperationentailsapplyingahighdv/dt,square-wavevoltagetothebearingshaftwithnoloadattachedandathighspeed[42].Second,weacquirevibration,currentandtemperaturedatafromthestartoftherununtilthebearingreachesitsfailurestate.InChapter3,weaddresstheproblemsofhealthstateestimationfromvibrationdatacollectedfrombearings.Duetothestochasticnatureofbearingfailures,vibrationdataisverynoisy.More-over,previousresearchhasshownthatbearingsdonotnecessarilyfollowamonotonicdegradationpatternwhichmakesidenticationofhealthstatesevenmorechallengingandimportant[51].ThispartofthethesisprovidestwocomplementaryapproachestoextractingunderlyinghealthstatesusingeventdetectionandtemporalHiddenMarkovModeltechniques.First,weproposeanewhealthstateestimationprocessforbearingsusingchange-pointdetectioninvibrationdata.Thesechange-pointsareassumedtocorrespondtobetransitionarystagesbetweenthehiddenhealthstatesofabearing.Next,weuseatemporalHiddenMarkovModelforunsupervisedclusteringofbear-ingvibrationdatatogainabetterunderstandingofhowabearingtransitionsthroughintermediarystagestofailure.7InChapter4,weintroduceastochasticdata-drivenapproachthatisindependentfromfaultseveritydiagnosisandthatcontinuouslyupdatestheRULestimateasnewdatasamplescomein.Forthispurpose,weuseanextendedKalmanlteringbasedapproachtorstlearnthedegradationtrendoftheextractedfeaturesfromthetrainingdata,thentoapplythistrendtotestingdatatopre-dictRULandnally,toprovideacondenceboundaroundtheestimatedRUL.WefollowcloselytheframeworkproposedbyLalletal.[52,53]forimplementingEKFforbearingRULestimationandofferseveralimprovementsoverthisimplementation.First,weconsiderbothtimeandtime-frequencydomainfeaturesfortrackingthedegradationofthebearing.Inthetimedomain,weusethevariancefeature,asithasbeenestablishedintheliteratureasareliableindicatorofthebear-ingconditionasitapproachesfailure.Inthetime-frequencydomain,weproposetouseanovelentropyfeature,whichcapturesthecomplexityofthesignalinbothdomainssimultaneously.Thisentropyfeaturehasbeenshowntobeagoodindicatorofthesignalcomplexityandrobusttotime-frequencyshiftsinthesignal.Weobservethattheentropyincreasesassoonastherstindicationsoffaultdevelop,whichrelateswelltothebearingPhysicsofFailure,whereaninitiallylocalizedfault(lowentropy)becomesageneralroughnesswithhighentropy.Second,weconsiderdifferentanalyticmodelsformodelingthelifetimeofthebearingandbuildthestatevectorscorrespondingtoeachcase.ThisenablesustofullyunderstandhowdifferentfeaturesevolveoverthelifetimeofthebearingandtheeffectofdifferentmodelassumptionsinthenalRULestimation.Third,weprovideacondenceintervalfortheRULestimatesusingthepredictionerrorscalculatedaspartofEKF.Finally,weillustratehowdifferenttypesoffeaturesmaycarrymoreinformationunderchangingoperatingconditions.Chapter5focusesondeterminingtherelationshipbetweenbearingcurrentandbearingfailure,inordertoexploitthisrelationshipformoreaccurateRULestimation.Sincebearingcurrentsareacauseofbearingfailureandnotaneffect,trackingbearingcurrentsovertimecanprovideinformationaboutanimpendingfailurebeforesignicantchangesoccurinthevibrationdata.Inparticular,itisseenthattheenergyofthevibrationsignalgrowsexponentiallyafteralargeinuxofbearingdischargeevents.Thisphenomenonshowsthatbearingcurrentscanprovideanearly8warningtoanimminentfailurebeforethereisasignicantchangeinthebearingcharacteristicfrequenciesusedinbearingvibrationanalysis.Inthischapter,weproposeanovelapproachwhichrstdetectsthecurrentdischargeeventsfromthecurrentsensorandthenidentiescriticaleventsduringthelifetimeofthebearing.ThesecriticaleventsarethenusedtodeterminethestartingpointofRULestimationfromvibrationdata.Inthismanner,thedependenceofRULestimationfromearlynoisybearingdataiseliminated,thecomputationalcomplexityofestimationisreducedandtheaccuracyofprognosisisincreased.
1.3Background
1.3.1Time-FrequencyDistributions
Time-frequencytransformsareusefulforextractinginformationfromnonstationarysignals,suchasbearingvibrationsignals.WhiletheFouriertransform(FT)cancapturethefrequencycontentofstationarysignals,itdoesnotprovidetime-localizedfrequencyinformation[54,55].Time-frequency(TF)representationsofsignalsareabletoshowhowthespectralpropertiesofasignalchangesovertime.Althoughthereareseveralmethodsinliteraturethatcanbeusedtoobtainatime-frequencyrepresentationofasignal,thereisnospecictransformthathasdistinctadvantagesovertheothersinallcircumstances[55].Somecommontime-frequencytransformmethodsaretheShort-timeFourierTransform,WavelettransformandCohen'sclassoftime-frequencydistri-butions.Theshort-timeFourierTransform(STFT)isalinearTFtransformthatrstdividesthesignalofinterestintomultipletimesegments.Fourieranalysisisconductedoneachtimesegmenttoextractthefrequenciesthatarepresentduringthatspecictimewindow[55].Aftertheanalysisiscompletedoveralltimewindows,thefrequenciesexistinginthesignalisshownchangingovertime.ThemathematicalrepresentationoftheSTFTisgivenby[56]1:1Allintegralsarefrom1to1unlessotherwisestated.9S(t;!)=Zf(˝)g(˝t)ej!td˝;(1.1)wheref(˝)isthesignalandg(t)istheslidingwindowwhichisreal,symmetricandnormalized.Theslidingwindowhasaxedlengthanditslengthintroducesatrade-offbetweenfrequencyandtimeresolution.Longtimewindowsresultingoodfrequencyresolution,whileshorttimewindowsprovidegoodtimeresolution.TheWavelettransform(WT)isanotherlineartransformusedtorepresentasignalinTFdo-main.However,insteadofdecomposingthesignalintosinusoidsatdifferentfrequencies,theWTusesthesuperpositionoftime-shiftedandscaledwaveletfunctions.TheDiscreteWaveletTrans-form(DWT)useslterbankstodecomposeasignalintohighandlowfrequencycomponents[1].First,thesignalispassedthroughalowpasslter,h[n]andsubsequentlydownsampledby2.Themathematicalrepresentationisgivenby[57]:A[k]=Xnx[n]h[2kn](1.2)wherex[n]isthesignal,h[n]isthelowpasslterandA[n]arecalledtherstlevelapproximationcoefcients.Next,thehighfrequencycoefcientsarecomputedusingthesameprocedurewithahighpasslter,givenby:D[k]=Xnx[n]g[2kn](1.3)whereg[n]isthehighpasslterandD[k]arecalledthedetailcoefcients.Thisprocessprovidesonelevelofapproximationanddetailcoefcients.Foreachlevelafterwards,thissameprocedureisiteratedontheapproximationcoefcientsofthepreviouslevel(showninFigure1.3).Cohen'sclassofTFdistributionsarequadraticTFrepresentationsandassuchtheyarecom-putationallymoreexpensivethanSTFTandWT.OneoftheadvantagesofusingCohen'sclassoftime-frequencydistributionsisthattheyprovideuniformresolutionoverbothtimeandfrequency,whilethewavelettransformdoesnot.Cohen'sclassofTFDscomputestheFouriertransformoftheautocorrelationofasignal,whichisthecorrelationofthesignalwithitselfinboththetimeandfrequencydomain.SinceCohen'sclassoftime-frequencydistributions(TFDs)arenotlin-10Figure1.3Diagramofadiscretewaveletdecomposition[1].ear,usingtheseTFdistributionsonsignalscontainingmultiplecomponentsproducesunwantedterms.Sincemostsignalscanbebrokendownintomultiplecomponents,theissueofcross-termsisprevalentinmostcases.However,theeffectofcross-termscanbeminimizedwiththeuseofasmoothingwindow,orkernelfunction.Thekernelfunctionactsasalterinbothtimeandfre-quency[55,58,59].Thekernelfunctionshouldbealow-passlterandmustbedesignedsothatitdecreasesthefartheryoumoveawayfromthe˝axis.Thekernelfunctioncanbedesignedsuchthatitremovesallofthecross-termsbutitcomesatthecostoflossinresolution[60].Cohen'sclassofTFDsisgivenby[59]:C(t;!)=RRR˚(;˝)s(u+˝2)s(u˝2)ej(ut˝!)dudd˝;(1.4)wherethefunction˚(;˝)isthekernelfunctionandsisthesignalofinterest.Inthiswork,theChoi-Williamskernelisusedtolteroutthecross-termsandisgivenby:˚(;˝)=exp((˝)2˙);(1.5)where˙controlsthetrade-offbetweentime-frequencyresolutionandthecross-terms.111.3.2Time-FrequencyFeatureExtraction
Fromthevibrationsignalofbearings,timedomainfeaturesincludingtherootmeansquare(rms),variance,skewness,kurtosisarecommonlyusedinfaultprognosis[36].Inthefrequencydomain,commonlyusedfeaturesincludermsfrequency,frequencycenter,androotvariancefrequency[28,35,36].Inthischapter,wefocusonTFfeaturessincetheyarecapableofjointlycapturingthetimeandfrequencydomaincharacteristics.AsopposedtotheconventionalShannonentropy,R´enyientropyhasbeenselectedduetoitsabilitytohandlepositiveaswellasnon-positivedistributions.R´enyientropyisdenedas[61]:H(C)=11log2XnXkC[n;k]Pn0Pk0C[n0;k0](1.6)where>0istheorder,andnandkarethediscretetimeandfrequencyindices.Entropyiswell-denedfortheTFDaslongasPnPkC[n;k]>0.ConcentrationmeasureshavealsobeenusedtoevaluateTFDs[62].Contrarytotheentropy,concentrationmeasureisastatisticonhowconcentratedasignalisandisdenedas[62]:M[C]=0
@XnXk


C[n;k]Pn0Pk0C[n0;k0]


1p1
Ap(1.7)wherep>1.Furthermore,smallvaluesforp,p<4,arepreferredsincehighpvaluescanempha-sizesmallenergyvaluesdisproportionately.Lastly,commonstatisticalmoments,suchasthemean,varianceandskewness,canalsobeextractedfromtheTFdomain.Onewaytodothisistoconvertthetime-frequencysurfaceintoavectorandcomputethewell-knownmean,variance,andskewnessmeasuresasdenedintheone-dimensionaltimedomain.12CHAPTER2EXPERIMENTALDATA2.1Background
2.1.1ReviewofSomePreviousPlatforms
Morethan50%ofmotorfailuresareduetoballbearings.Assuch,theareaofbearingfaultdiagnosisandprognosishasattractedalotofattentioninrecentyears[63,64].Bearingdegradationcanoccurduetomechanicalstress,resultingfromsourcessuchasradialoraxialloadsplacedonthebearings.Recently,researchhasbeendonetounderstandbearingdegradationduetoelectricalstress,suchastheformationofEDMcurrentstravellingthroughthebearing,causingmechanicaldamagessuchaspitsandracesintheouterring.Inordertoobservethisphenomenon,severaltestrigswereconstructedinrecentyears.In[65],anexperimentalsetupusedtoobservetherelationshipbetweenaxialloadsandthelubricatinglmbetweenthetribologicalsurfacesofdeepgrooveballbearingsisdescribed.Thisexperimentconsistedofamachinewithspeedsof100,500,1000and3000RPM.Axialloadswereplacedonthebearingsusingapiezoelectricactuatortoapplyforcetopreloadeddiscspringsontheouterringofthetestbearingatfrequenciesof2and16Hz.Eachrunwasprecededbya1hourrun-inperiodtoensurethemachinehadreachedasteady-stateoperation.Afterashorttimeintheorderofmilliseconds,theloadwasreleased.Theentireprocedurewasthenrepeated10timesforeachloadlevel,varyingfrom100to800N.Thegoalofthisexperimentwastocalculatethebearingcapacitanceandresultinglubricatinglmthicknessasinputstothesimulationmodelforbearingcurrentprediction.Throughthiswork,theauthorsfoundthatunderastaticloadtothemachine,therewasadecreaseinlmthicknesswhenthebearingswereexposedtolowspeedsandhightemperatures.However,theresultsofthedynamicloadexperimentswerenottrivial.Intheory,onewouldexpecttoseeadecreaseinlmthicknessastheloadincreased,buttheresultsshowedanearconstantvalueofthicknessspanning13theentireloadlevelrange.Itwasreasonedthatthiswasbecausetherewasnotenoughtimebetweenloadchangestoallowthemachinetooperateinsteady-state.Asaresult,itwasshownthatthemaincontributorstothedecreaseoflmthicknessofthebearings,thusleadingtobearingcurrents,aretemperatureandspeed.Asthetemperatureincreasedatanyspeed,thelubricatinglmdecreased,whichtheoreticallyleadstobearingvoltagebreakdownandthusbearingcurrents.In[47]atestrigwasconstructedtoexplorethelinkbetweenbearingvibrationsandinverter-inducedbearingdamage.Thisset-upconsistedofalow-voltage,squirrel-cageinductionmotorwhichwas3-phase,15kWandhad4-poles.Thetestbearingsinthiscasewerealsodeepgrooveballbearingswithoff-the-shelflithiumsoap-basedgrease.Thebearingswereelectricallyinsulatedfromthemotorandasinusoidalvoltageof20Vppand300kHzwasappliedtotheshafttosimulatecommonmodevoltageduetooperationofthemotorbyaninverter.Thesignalwasappliedtotheshaftviaaslipring.Eachbearingtestwasallowedtorunfor1184hoursandthentheexperimentwasstopped.Usingthistestbed,theauthorswereabletomeasurethetemperatureoftheouterraceofthebearing,thebearingvoltage,bearingvibration,bearingcurrent,andthenumberofbearingdischargepulsesviaRFmeasurements.Thevibrationsignalsweresampledat20kHzfor400ksamples/sandthedischargeactivitywasmeasuredasthetotalnumberofdischargesthatoccurredina30secondwindow.Theresultwasaqualitativeanalysisoftherelationshipbetweenvibration,temperatureandbearingdamageduetoEDMcurrents.Aftertheapplicationoftheshaftvoltage,theinnerraceofthetestbearingsappearedbrandnew,whiletheouterraceexhibitedracingstripesandsmallcraters.Althoughenergydissipationwasattributedtotheconstructionofthesecraters,itwasnotedthatthesizeofthecratersshouldhavebeendoubleinsize.Theauthorsattributedthistothefactthatthedischargeactivitymayhaveoccurredbeforetheshaftvoltagereacheditspeak.Quantitatively,thenumberofRFpulsesincreasedovertime,givingatotalnumberofapproximately10milliondischarges.Itwasnotedthatalthoughthiswasalargenumber,theenergydissipatedineachoneoftheseeventswasrelativelylow,around89.15nW,whichisnotenoughtocausesignicantbearingtemperatureincrease.Inconjunctionwiththisfact,thetemperaturereadingsoverthecourseoftheexperimentsprovidedinsignicantinformation14pertainingtodischargeactivity.Thevibrationdataprovidedsignicantresultsforthebearingouterringpassfrequencyandtheballrotationfrequencyandnottheinnerringpassfrequency,whichcorroboratedtheresultsofthevisualinspectionofthetestbearingsaftertheexperiments.Itwasnotedthatthevibrationdatadidnotfollowamonotonictrend,renderingituselessforquantitativeanalysis.Anothertestbedinvolvingbearingcurrentsisdescribedin[66],inwhichthedamageofabearingduetoasinglecurrentpulsewasexamined.Inthistestrig,thrustballbearingswereused,withthenumberofballsbeingmanuallychangedfrom9to3.Outofthe3balls,onlyonewasallowedtobeconductive,thuscurrentonlytraveledthroughitandnottheothers.Theexperimentswereranat60,120and1000RPM.Avoltagewasappliedacrossthebearingracestoinduceasinglecurrentpulseoncommandviaacircuitdesignedforhigh-frequencypulsecurrents.Thecircuitconsistedofcapacitors,aresistanceandpowertransistorsandwasusedtosimulatecurrentpulsesdeliveredbyafrequencyconverterandthedrivingvoltageofthiscircuitvariedfrom0to30Volts.Eachtestbearingwasalsoloadedwithanaxialloadof400N.Eachexperimentwasrunfor5minuteswithloadbutnotappliedvoltageandthen,subsequently,10minuteswithcurrentpulsesapplied.Thegoalofthisworkwastoanalyzethevisualeffectofcurrentpulsesontheouterraceofabearing,whenranatdifferentspeeds.Acomparisonofanimageofthebearingracesunderamicroscopewasmadebetweenabearingwithinducedbearingcurrentsandonewithout.Foreachspeed,therewasasignicantamountofdamagethatcouldbevisiblyseenunderthemicroscopewhenbearingcurrentswereinduced,andrelativelylessdamagewhentheywerenotinduced.Itwasalsonotedthatafter500bearingcurrenteventswererecorded,theaveragepeakcurrentwasaround2.1A.Moreover,keepingthespeedconstantandincreasingthedrivingvoltageresultedingreaterdamagetothebearingraceways.
2.1.2BearingCurrentFormation
Overtheyears,theuseofvariablefrequencydrivestocontrolelectricmachineshasgrownduetotheircapabilityofsavingenergy[66].ModernACdrivesystemscreatethefundamentalvoltage15Figure2.1ThreephasevoltagesofanACdriveandtheaverageofallthree,orthecommonmodevoltage[2].
ofthemotorbyswitchingaDCbusvoltageontothe3phaseterminalsofthemotor.Insine-wavedrivenmachines,thethreephasesofthemachinearebalancedandsymmetric.However,withPulseWidthModulated(PWM)drivenmachines,atanypointintimetheonlyvaluesofeachphasevoltageiseither+VDCor-VDC.Thisimpliesthatwhiletheinverteroutputvoltagesarebalancedandsymmetric,atanyinstanttheaverageofthesephasevoltagesisnonzero[2,3,66,67].Thisnonzerovoltageiscalledthecommonmode(CM)voltageandisbetweenthestatorneutralandframeground.ThefrequencyofthisCMvoltagecanbeinthekHztoMHzscale,asitisequaltoswitchingfrequencyoftheinverter.AnexampleofthethreephaseoutputstothemachineandtheircorrespondingneutralisshowninFigure2.1.CMvoltageaffectsthestator,rotor,shaft,andbearingsthroughthestraycapacitancesofthemotor.Thesecapacitancesarecreatedinherentlythroughtheseparationoftheconductingele-16mentsofinductionmotors.Thus,voltagesbecomepresentinthemotorshaftandstatorduetothecapacitordividersbetweentheCMvoltageandthestraycapacitances.Fromthisinteraction,therearethreetypesofbearingcurrentsthatcanbegenerated:circulatingbearingcurrents,shaftgroundingcurrent,andEDMbearingcurrents[2,3].
2.1.2.1CirculatingBearingCurrents
TheCMvoltagecreatesacapacitivecurrentinthestatorwindingwhenitexcitesthestraycapac-itancebetweenthestatorandframe.Whenthiscurrentasymmetricallyleaksfromthewindingtothestatorframeacrossthestatorcircumference,itcreatesahighfrequencyaxialuxaroundthestator.Thisuxinducesavoltagearoundthemachine,betweentheshaftends,causingcirculatingcurrentstoowintheloopcreatedbytheshaft,bearingsandframe.2.1.2.2ShaftGroundingCurrent
Ifcurrentleaksfromthestatorwindingstotheframe,andtheframeisnotproperlygrounded,thecurrentwillseekaroutetoground.Iftheshaftisgrounded,theimpedanceoftheshaft,bearingandloadissmallerthananyotherpathtoground.Therefore,theleakedcurrentinthestatorframewillchoosetherouteoftravelingthroughthedrive-endbearing,totheshaft,totheloadandnallybacktoground.Thiscurrentonlybecomessignicantifthereisashortinthestatorwinding.2.1.2.3ElectricDischargeMachiningCurrents
Asstatedbefore,ahighfrequencyshaftvoltagebecomespresentduetotheinteractionoftheCMvoltageandthestrayrotorcapacitances.Whenthemotorframeisgrounded,ifthisshaftvoltageexceedsthebearingbreakdownvoltageacurrentdischargeeventwilloccur.Whileasingledis-chargeeventisnotextremelydamagingtothebearing,acollectionoftheseEDMcurrentsoverashortperiodoftimeisextremelydamaging.Duringadischargeevent,thereisalocalizedtempera-tureincreasecausingdeteriorationinthebearinglubricationalongwithpitsinthebearingraceway.17Figure2.2Straycapacitancesofaninductionmotor[3].Thesepitsovertimeleadtocratersanduting(showninFigure1.2)inthebearingraceway,whichsigniesbearingdamage.Undersine-waveoperation,theshaftvoltagenecessarytoexceedthebearingbreakdownvoltagethresholdissignicantlylowerthanunderPWMoperation.Becauseofthis,theresultingEDMcurrentunderPWMinverteroperationarehigherandmoredamagingtobearings[42].Thebearingbreakdownvoltageisdeterminedbythelubricationinthebearing.Thecharacter-isticsofthelubricationaredependentonanumberoffactors,includingthegreaseconductivity,motorspeed,motorload,andbearingvoltage[43,45].First,theconductivegreaseinbearingscanactasasuppressanttoEDMcurrents.Sincethesecurrentsarearesultofthepotentialdifferencebetweentheballsandtheraces,conductinggreaseremovesthatpotentialdifference.However,theauthorsin[45]foundthatconductivegreaseonlyhasthiseffectonbearingsfortherst40hoursofoperation.Afterthistime,conductivegreasebehavessimilarlytononconductinggrease,givingwaytoelectricdischarges.Second,thespeedoftherotatingelementsinthebearinghasaneffect18onthelubricationlmthickness.Atlowspeeds,thelubricationlmisthin,causingmetal-to-metalandquasi-metalsurfacecontact,allowingcirculatinganddischargecurrentstoowfreelythroughthebearing.Becausethelubricationlmisthin,thesedischargeeventsdonotcontainmuchen-ergyandminimallydamagethebearing.Athighspeeds,athickerlubricationlmisbuilt,whichsignicantlyincreasestheresistancebetweenthebearingraceways,leadingtolessmetal-to-metalcontactpoints.Inorderforcurrenttoowthroughthebearing,thebearingvoltagehastobelargeenoughtoexceedthebearingbreakdownvoltage,consequentlyproducingmoredamagingelectricdischargeeventsthanatlowerspeeds.Third,boththerateofchangeofthevoltageamplitudehavesignicantinuenceonthepresenceofelectricdischargeevents.Shaftvoltageswithhighdv=dtplacehighstressonthelubrication,causingbreakdownsandthusdischarges.UnderPWMoperation,thebreakdownthresholdvoltageisbetween8-15Vunder60Hzoperation,whichproduceshighenergydischargeeventscausingseveredamage.Last,theloadassociatedwiththemotorhasminimaleffectonthepresenceofEDMcurrents.However,thepresenceofaloadonbearingsincreasestheirlifeexpectancyasunloadedbearingspresenttheworstcasescenarioforbearingdischargecurrents[42,43,68].Duetotheintrinsicpropertiesofbearingspreviouslydiscussed,therearethreeprobablestatesabearingcanmanifest.ThosestatesareshowninFigure2.3,withinsulated,discharge,andtheohmicinwhichthecurrenthasa180phaseshiftin2.3a,2.3b,and2.3c,respectively[4].Therststate,referredtoasinsulated,iswhenthebearingactspurelycapacitiveandallowsnocurrenttoow.Thisoccurswhenthereisasufcientamountoflubricationseparatingtheballsandtheraces.Thesecondstate,referredtoasdischarge,occurswhenthereisabreakdowninthepreviouslycapacitivestate,causingthebearingcapacitortodischargecreatinganinuxofcurrent.Thethirdstate,orohmicstate,iswhenthecurrentowingthroughthebearingfollowsthetrendoftheshaftvoltage.Thebearingispurelyresistiveinthisstateduetometal-to-metalcontactbetweentheballsandtheracesorconductinglubricationgrease.19a) InsulatedVoltageb) Dischargec) OhmicCurrentFigure2.3Bearingelectricloadstates.Thetoprowshows3specicvoltageprolesandtheircorrespondingcurrentresponsesareshowndirectlybelow.(a)Insulated.(b)Discharge.(c)Ohmic[4].
2.2PRONOSTIAPLATFORM
ThePRONOSTIAplatformwasusedtostudyaccelerateddegradationofballbearingsinordertoproviderealexperimentaldatathatcharacterizethedegradationoftheballbearingsduringtheirlifetime[5].Thisplatformdiffersfromothersinthatthebearingfailureprocessisflnormalflandthebearingsarenottamperedwithduringtheintroductionofdefects.InFigure2.6,anexampleofanormalanddegradedbearingisshown.onewhichincludestheasynchronousmotor,gearboxandtwoshaftsusedtodrivetheexperiment.Themotorpowerwasratedat250Wpowerandratedspeedof2830rpm.Thegearboxallowedthesecondaryshafttorotateataspeedlessthan2000rpmwhilemaintainingthemotor'sratedtorque.Thesecondaryshaftwasthenconnectedtotheinnerraceofthebearingviacompliantandrigidshaftcouplings.Thebearingsupportshaft(seeFigure2.5)isalsoheldbytwopillowblocksontheends.Second,therewasaloadingdevicewithwhichthebearingswereloadedinandtheradialforceswereapplied.Thispartconsistedofapneumatic20Table2.1OperatingConditionSpecicationsforthePRONOSTIAPlatformOperatingConditionCondition1Condition2Condition3RadialLoad(N)400042005000Speed(RPM)180016501500TrainingSetsBearing11Bearing21Bearing31Bearing12Bearing22Bearing32TestingSetsBearing13Bearing23Bearing33Bearing14Bearing24Bearing15Bearing25Bearing16Bearing26Bearing17Bearing27jackandaclampingringofthetestbearing.Forceisappliedtothebearingthroughitsclampingringviatheamplicationoftheforcefromthepneumaticjackthroughaleverarm.Third,therewasameasurementpartinwhichradialforce,bearingshaftspeed,andtorqueweremeasuredatafrequencyof100Hz.Thesethreemeasurementsdeterminedtheoperatingconditionsofthebearing.TheentireoverviewofthePRONOSTIAsetupisshowninFigure2.4.Thecharacterizationofthebearing'sdegradationisbasedontwotypesofsensors:vibrationandtemperature.Thevibrationsensorsconsistoftwominiatureaccelerometersplacedradiallyontheexternalraceofthebearing90toeachother,withonebeingplacedontheverticalaxisandtheotheronthehorizontal.Thetemperaturesensorwasplacedinaholeclosetotheexternalbearing'sring.Thevibrationandtemperaturesignalsweresampledat25.6kHzand10Hz,respectively.Aspartoftheexperiment,threedifferentoperatingconditionswereexplored,inwhichradialloadandspeedwerevaried.Atotalof17run-to-failuredatasetswereproduced.Sixrun-to-failuredatasetswereforalgorithmtrainingandtheremainingmonitoringdataofthe11bearingsweretruncatedfortestingpurposes[5].Table2.1showsacompletebreakdownoftheoperatingconditionsandtheirspecications.21Figure2.4OverviewofPRONOSTIAsetup[5].Figure2.5BearingsupportshaftofPRONOSTIAplatform[5].22Figure2.6Comparisonofnewvs.degradedbearing[5].2.3AcceleratedBearingDegradationPlatformviaElectricalStressTheexperimentalprocedureinthistestbedresultedinaccelerateddegradationofabearingbyinducingacurrentthroughitsraces.Theinducedbearingcurrentproducedanelectricalstressonthebearingcausingabreakdowninthelubricationlmsurroundingtheballsintheballbearing.Thisacceleratedbearingdegradationplatformincludeda3-phaseinductionmachineconnectedtoapillowblockbearingthroughapiece-wiseshaft,includingseveralcouplings.Theinductionmachinewaselectricallyisolatedfromthebearingshaftviaanylonshaftcouplingconnectedtothemotoroutputshaft.Theinsulatedpieceoftheshaftwasthencoupledtoacoppertube,orbearingshaft,viaaHigh-SpeedBellowsFlexibleShaftCoupling.Lastly,thebearingwasthentontothecoppertube,withnoloadattachedtotheshaft.Brusheswereplacedonthebearingshaftinordertoprovideacontactpointtoapplyavoltage.Aplexiglasshieldwasplacedoverthecouplingsasasafetymeasure.AnoverviewofthesetupisshowninFigure2.7.Avoltagewasappliedtotheshaft,fromthebearinginnerracetoouterraceusingavoltagebufferampliercircuit.Thisshaftvoltagewasa20Vpppulsewitha50kHzfrequency.Foreachexperiment,theinductionmachinewasrunat2400RPMwithaninverter.Threetypesofsensors23Figure2.7Overviewofacceleratedbearingdegradationplatformduetoinducedelectricalstress.wereplacedonandaroundthebearingpillowblocktoobtaininformationaboutthebearingfromstarttofailure:temperature,vibration,andcurrent.ThetemperaturewasmeasuredusingaT-Typethermocoupleandwasplacedincloseproximitytothebearingouterrace.Temperaturewassampledonceevery10minutesat1Hzsamplingratefor1second,duetothefacttemperaturehasaslowrateofchange.Thevibrationswerecapturedusingtwoaccelerometers,placedontheverticalandhorizontalaxisofthebearingpillowblock.VibrationmeasurementsweresampledsimilarlytothePRONOSTIAplatform,every10minutes,withasamplingfrequencyof20kHzforadurationof0.1seconds.Thecurrentowingthroughthebearingwasmeasuredbyloopingthebearingcurrent-holdingwirethroughacurrenttransducer.ThecurrenttransducerusedinthisexperimentwasaPearsonCT-4100.Sincetheshaftvoltagewasat50Hzfrequency,thecurrentneededtobesampledat1.2MHzcontinuouslytoseetheentirecurrentwaveform.242.4Conclusions
Inthischapter,wediscussedtheimportanceofstudyingbearingfailureasitcontributestooverhalfofmotorfailures.Wealsodiscussedthecausesofbearingfailure,includingexcessiveradialandaxialloads,temperatureandelectricalstressviabearingcurrentow.Bearingcurrentowcancomein3differentforms:circulatingcurrents,shaftgroundingcurrents,orEDMcurrents.EDMcurrentsaredamagingtobearingsandcreatepitsandcratersontherotatingelementsofbearings.Thesecratersleadtoutingandeventuallybearingfailure.Toobservethis,wepresentedanewtestbedwhichallowstheaccelerateddegradationofbearings.Anelectricalstressisplacedonthebearingsbyapplyingavoltagetothebearingshaft,allowingcurrentowthroughthebearing.Overthecourseoftheexperiment,wecollectedtemperature,vibrationandcurrentdata.Wealsoprovidedinformationaboutaseparatetestbed,thePRONOSTIAplatform,whichusedexcessiveloadstoacceleratetheagingprocessofthebearing.Inthesubsequentchapters,datafromthesetwotestbedsareanalyzedandfeaturesareextractedinordertoperformRULestimation.Infuturework,morebearingsshouldbetestedusingtheplatformweconstructedinordertocollectmoredata.Thiswillhelptoobtainacollectionofhistoricaldataforbearingfaultprognosis.Furthermore,asolutiontomeasurebearingcurrentinreal-worldmotorsetupsshouldbedeveloped.Futureworkshouldalsodevelopmethodologiesfordetectingbearingcurrentsindirectly,usingeitheranRFsystemorbearingcurrentestimationtechniquesfrommeasuredbearingvoltage[47,69].ThiswillgivewaytopracticallyimplementableRULestimationforbearingsthatcanbeusedinindustry,asameanstopreventsystemdowntimeormotorfailureduetobearingfailure.25CHAPTER3DISCOVERINGTHEHIDDENHEALTHSTATESFROMBEARINGVIBRATIONDATA3.1Introduction
Bearingfaultdiagnosisandprognosishasbeenagrowingareaofinterest,sincebearingfailurecausesmorethan50%ofmotorfailurecases.Previousworkshavefoundsuccessintheareaoffaultdiagnosisandprognosisbyutilizinghealthstatestodiscernthestateofasystem'sdegradation[29Œ31].However,bearinghealthstatesarenotdenedbyaphysicalphenomenon.Inthischapter,weexploretheproblemofhealthstateestimationfromvibrationdataforbearingfaultprognosis.Therehasbeenmuchworkdoneinusinghealthstatesasameanstofailureprognosis.In[29],HiddenMarkovModels(HMM)wasusedtoconductprognosisongearfailures.AnHMMwastrainedonthetrainingdataandthelearnedstateswereusedtoclassifytestingdataintoaparticularhealthstate.TheHMMwasthenusedtopredictthenextprobablestateandprovidedawarningifthenextstatewasthefailurestate.However,inthiswork,eachhealthstatehadaphysicalmeaning,aseachstatecorrespondedtoadifferentnumberofbrokenteeth.However,forsystemswhichcontainnodistinctphysicalphenomenacontributingtospecichealthstates,thereisaneedforunsupervisedclustering.In[70],hierarchicalHMMswereusedtoconductbothsupervisedandunsupervisedclusteringonacousticdata.Similarly,HMMswereusedin[71]toclusterecologydataintoclasses.However,therestillremainstheissueofndingthehiddenhealthstatesinbearingvibrationdata.Inthischapter,weproposetwodifferentmethodologiestoaddressthisissue.Therstisanovelunsupervisedclusteringmethodbasedonaneventdetectionframeworkwhichidentiesperiodsofstationarityinthedata.Theseperiodsofstationarityarethenusedtopartitionthedataintoseveralhealthstates.Althoughthisapproachcansuccessfullyandeffectivelyclusterthedataintomeaningfulstates,itispurelyheuristic.Thesecondapproachusesamorestatisticalframework,atemporalhiddenMarkovmodel,topartitionthedataintoclasses.Thesehiddenhealthstatescan2600.020.040.060.080.1-2024Horizontal Raw Vibration DataTime (s)Acceleration00.020.040.060.080.1-2-1012Vertical Raw Vibration DataTime (s)AccelerationFigure3.1RawDataofInitialVibrationSignalforBearing11.beusedtodiagnosetheconditionofabearing,andsubsequentlyestimatethenextprobablestate.3.2HiddenHealthStateIdenticationviaEventDetection
3.2.1FeatureExtraction
ThevibrationdatafromthePRONOSTIAplatform,describedinChapter2,isusedforfeatureextractioninthissection.Anexampleofrawvibrationsignalsfromtheinitial(healthystate)andnal(failure)samplecanbefoundinFigures3.1and3.2.Inthetimedomain,variancewasextractedasafeature.InFigure4.1,wecanseetheprogressionofthevarianceacrosstimeforeachofthetrainingdatasets.Sincethevibrationsareknowntobenonstationary,wealsoconsideredTFdomainfeaturessuchasentropywith=2(seeeqn.1.6).TherawvibrationsignalsweretransformedintotheTFdomainusingtheChoi-Williamsdistributionwith˙=10.TheTFrepresentationsofinitialandfailuresamplesareshowninFigures3.3and3.4,respectively.FromtheTFD,weobservedthattheverticalvibrationTFrepresentationgavelittleusefulinformation.Wenoticedtwophenomenathatwereevidentacrossalltrainingsetsinthehorizontaldataasthefaultprogressed.First,therewasashiftfromasignicantamountofconcentrated2700.020.040.060.080.1-40-2002040Horizontal Raw Vibration DataTime (s)Acceleration00.020.040.060.080.1-40-2002040Vertical Raw Vibration DataTime (s)AccelerationFigure3.2RawDataofFinalVibrationSignalforBearing11.Horizontal Vibrationstime (sec)Frequency (Hz)  5000100001500020000250004080120160200240-0.2-0.15-0.1-0.0500.050.10.150.2Figure3.3Choi-WilliamsTransformationofInitialHorizontalVibrationSignalwith˙=10forBearing11.28Horizontal Vibrationstime (sec)Frequency (Hz)  5000100001500020000250004080120160200240-0.2-0.15-0.1-0.0500.050.10.150.2Figure3.4Choi-WilliamsTransformationofFinalHorizontalVibrationSignalwith˙=10forBearing11.energy,atthestart,toimpulsiveenergydistribution,atfailure,inthe160-200Hzfrequencyband.Second,therewasashiftfrominsignicantenergytoalargeamountofenergyinthe236-256Hzand0-40Hzfrequencybands.Thesethreefrequencybandswereexploredinfeatureextraction,usingentropyandconcentrationmeasures[72Œ74]tocapturethesetrendsandtheresultingfeatureshadcleartrendsacrosstime.
3.2.2EventDetection
Eventdetectionisusedtodetermineachangeinthedatatosignifydifferenthealthstates.Therearemanydifferentwaystodeterminethesechangepoints,butinthisworkweutilizetheZ-scoreasdenedin[32].Firstweconstructedafeaturematrix,2RFT,whereFisthenumberoffeaturesandTisthetotalnumberoftimepoints.Next,weconstructedFF,time-varyingcorrelationmatrices,C(t),from,usingslidingwindowsoflengthWwhere:Cij(t)=


ˆX;Y=E[(XX)(YY)]˙X˙Y


(3.1)whereXandYarelocalizedfeaturematrices(i;tW:t)and(j;tW:t),respectively.Fromthesetime-varyingcorrelationmatricesC(t),wecomputedtheprincipaleigenvector,u(t).29Thisvectoru(t)summarizestheactivityofeachfeatureinthattimeintervalandisgivenbysolvingtheequation:(C(t)maxI)u(t)=0;i;j21;2;:::F;(3.2)whereIisanFFidentitymatrixandmaxisthemaximumsolutionofdet(C(t)I)=0:(3.3)Inordertodeterminethechange-points,thisvectoru(t)iscomparedtoanaverageofallthepreviousW0principaleigenvectors,denotedasr(t1)=1W0W0Pi=1u(ti).TheZ-scoreisgivenasZ(t)=1u(t)Tr(t1).Thus,ifu(t)2RF1isdramaticallydifferentfromr(t1),theirdotproductwillbe0,producingaZ-scoreof1.Ifu(t)andr(t1)aresimilar,theirdotproductwillbecloseto1,producingaZ-scorecloseto0.Sinceu(t)andr(t1)arebothunitvectors,theZ-scoreisalwaysbetween0and1.Finally,changepointscanbedetectedasspikes,orhighscoresintheplotoftheZ-score.
3.2.3Results
Inthiswork,fromtheFEMTOdata,weextractedatotalof6featuresfromtheTFdomain(showninFigure3.5):
1)Entropyfromthe160-200Hzfrequencyband
2)Entropyfromthe0-40Hzfrequencyband
3)Concentrationmeasurefromthe0-40Hzfrequencyband
4)Variancefromthe236-256Hzfrequencyrange
5)Meanfromthe236-256Hzfrequencyrange
6)Skewnessfromthe236-256HzfrequencyrangeIntheeventdetectionstep,awindowsizeofW=W0=100sampleswasusedacrossallsixtrainingsets.Itisassumedthatachange-pointoccursinaparticulartrainingsetiftheZ-score30050010001500200025003000-2-1012345time (samples)SkenwessSkewness of 236-256 Hz Band(a)Skewnessof236-256HzBand050010001500200025003000-5-4-3-2-1012EntropyEntropy of 160-200 Hz Bandtime (samples)(b)Entropyof160-200HzBand050010001500200025003000-5-4-3-2-1012Entropytime (samples)Entropy of 236-256 Hz Band(c)Entropyof236-256HzBand050010001500200025003000-101234567Variance of 236-256 Hz Bandtime (samples)Variance(d)Varianceof236-256HzBand050010001500200025003000-101234567MeanMean of 236-256 Hz Bandtime (samples)(e)Meanof236-256HzBand050010001500200025003000-1.5-1-0.500.511.52Concentration MeasureConcentration Measure of 0-40 Hz Bandtime (samples)(f)ConcentrationMeasureof0-40HzBandFigure3.5FeaturesforBearing11acrosstime.31increasesbeyondathreshold,givenby(Zni)+˙(Zni)whereZni=[Z1Z2:::Zni]andniisthenumberofsamplesintheithtrainingset,isthemeanand˙isthestandarddeviationacrossalltime.
3.2.3.1EstimatingtheHealthStates
Applyingtheproposedchangepointdetectionalgorithmtodifferenttrainingsetscorrespondingtodifferentoperatingconditionsyieldeddifferenttransitionsbetweenhealthstates.Forexample,inFigure3.6wecanseetheZ-scoreresultsforthersttwooperatingconditions.Wenoticedthatacrosstime,thetwotrainingsetswithinthesameoperatingconditionshowedsimilartrendsbuttheyweredistinctlydifferentfromthetrendoftheotheroperatingcondition.Wealsonoticedthatthereweregroupsofchange-pointsincloseproximitytoeachotheraswellasperiodsoflittlechangethroughoutalltrainingdatasets.Wereasonthatmultiplechange-pointswithinawindowcorrespondtoatransitionstagefromonestatetothenext.Inotherwords,timeperiodswherethereweregreatchangesinthedata(i.e.multiplechange-pointsoverashorttimeperiod)werecalledtransitionarystates.Conversely,thetimeperiodswheretherearenochange-pointscorrespondtotheactualhealthstates.AnexampleofthispartitioningofthetimeseriesdataforBearing11,intherstoperatingcondition,canbefoundinFigure3.6a,wheretheshadedareasrepresentthetransitionstagesandtheunshadedrepresenttheestimatedhealthstates.TheothertrainingsetinthisoperatingconditionhadasimilartrendinitsZ-scoreplotovertime,ascanbeseeninFigure3.6b.Giventhesesimilargroupingtrendsfromoperatingcondition1,wecanbuildacompleteoverviewofhealthstatesandtransitionstagesofthebearingfromstarttofailure,asseeninFigure3.7.Inthisoverview,wecansee5distincthealthstatesand4transitionarystagesbetweenthem.Fromthis,wecandeterminethatthereisahealthystate(0),3intermediaryhealthstates(1-3),andthefailurestate(4).TheZ-scoretrendsforthetwotrainingdatasetsinoperatingcondition2canbeseeninFigure3.6cand3.6d.Thesetrainingsetsprovidedadifferenttrendfromtherstoperatingcondition,whichcouldbeattributedtothedifferentloadspeedsandforces.Anoverallviewofthehealth32(a)Bearing11(b)Bearing12(c)Bearing21(d)Bearing22Figure3.6Change-pointgroupingintotransitionstages(shaded)foroperatingconditions1and2.
statesforoperatingcondition2canbeseeninFigure3.8.Inthisoperatingcondition,weseethatthehealthstate(0)isrelativelylongerinrelationtothehealthystatesfoundinoperatingcondition1.
3.2.3.2PerformanceofMultipleFeatures
Asstatedbefore,thetime-varyingtrendsofthefeaturesusedinthischaptercanbeseeninFigure3.5.Fromtheseplots,wecanseethatcertainfeaturesdonotprovidecleartrends.Forinstance,thevarianceandmeanofthe236-256Hzfrequencybandwouldonlyprovideonechange-point,330100200300400500600700800012345time (samples)Health StateHealth State ProgressionFigure3.7HealthstateprogressionforBearing12.0100200300400500600700800900012345time (samples)Health StateHealth State ProgressionFigure3.8HealthstateprogressionforBearing21.whileothers,suchasthetwoentropyfeatures,wouldresultinmultiplechange-pointsduetotheiructuations.Inthissection,weevaluatetheuseofdifferentnumberoffeaturesforchangepointdetec-tion.Forinstance,inFigure3.9,wecanseetheeffectofcomputingtheZ-scoreusingonlytwo(skewnessof236-256Hzbandandentropyof160-200Hzband)ofthese6featuresforBearing12.TheZ-Scoreishighthroughouttheentiretrainingset.Thisisbecauseitishardtodetectchange-pointsinsignalswithhighvolatility,suchasbearingvibrationdata.However,whenwe34010020030040050060070080000.10.20.30.40.50.60.70.80.91time (samples)Z-ScoreZ-Score using two variablesFigure3.9Z-Scorecomputationusing2featuresonBearing12.usemorefeatures,wecanseethatthechange-pointdetectionalgorithmbecomesmorerobusttonoiseandcanidentifymoredistinctchange-points.Thisphenomenoncanbeseenasthefeaturesincreasefrom2to6inincrementsof2.Figure3.10showstheZ-scoreforthealgorithmwhichutilizes4features(skewness,entropyofbothfreq.bands,andthevarianceof236-256Hzband).Inthisplot,wecanseethatthechange-pointsbecomemoreconspicuous.Furthermore,asthenumberoffeaturesisincreasedto6,seeninFigure3.11,thechange-pointsbecomeevenmoreapparent,assomechange-pointstowardsthebeginningoftherunarecombinedintoone,showingthatthereisadirectrelationshipbetweenthenumberoffeaturesusedandtheeaseofidentifyingthechange-points.
3.3HiddenHealthStateIdenticationviaEventDetection
3.3.1TemporalHiddenMarkovModels
TheHMMisastatisticalmethodandhasbeenextensivelyusedinthemodelingandanalyzationofstochasticprocesses.TheHMMhasbeenusedinavarietyofapplicationssuchasspeechrecognition[75],motorfaultdiagnosis[29,76],andbearingfailureprognosis[23].TheHMMisusedtomodelsystemswhichhaveanitenumberofhiddenstates(S1;S2;:::SM)whichareeach35010020030040050060070080000.010.020.030.040.050.060.07time (samples)Z-ScoreZ-Score using four variables  Z-ScoreThresholdFigure3.10Z-Scorecomputationusing4featuresonBearing12.010020030040050060070080000.020.040.060.080.10.12time (samples)Z-ScoreZ-Score using six variables  Z-ScoreThresholdFigure3.11Z-Scorecomputationusing6featuresonBearing12.indirectlyobservableviathesequenceofobservations,(O=o1;o2;:::oN),theyexhibit.TherearethreeparametersthatgoverntheHMM:1)Aninitialprobabilitymatrix,ˇ,whichdeterminestheinitialprobabilityofagivenstate,mathematicallyrepresentedasˇi=P(q0=Si);1iM;(3.4)whereq0istherststateinastatesequence.362)Astatetransitionprobabilitymatrix,A=aij,whichgivestheprobabilityofastateSjgiventhepreviousstatewasSi,whereaij=P(qt=Sjjqt1=Si):(3.5)3)Anobservationprobabilitymatrix,B=bj(ot),whichdetermineshowlikelyanobservationisgiventhestateisSj,wherebj(ot)=P(otjqt=Sj)(3.6)Forsimplicity,anHMMcanbewrittenas=(ˇ;A;B).However,thereisalimitationinusingHMMfortime-seriesdatasinceHMMassumesthateachobservationisstatisticallyindependentoftheprevious.Fortime-seriesdata,thisisusuallynotthecase.ThetemporalHMMwasproposedtoavoidthislimitationbyassumingthateachobservationisstatisticallydependentonitspredecessor.Ifweassumeanitenumberofobservablestates,(W=w1;w2;:::wk),theprobabilityofottakingthevaluewtisgivenbyP(ot=wtjot1=wt1;ot2=wt2;:::o1=w1)=P(ot=wtjot1=wt1):(3.7)ThisissimilartoarstorderMarkovprocessinthattheprobabilityofanobservationattimetonlydependsontheobservationattimet1.Usingthiscondition,theobservationprobabilitymatrixisupdatedtocontaintwoparameters,qandp,whereq(i)=P(o1=wi)andp(i;j)=P(ot=wjjot1=wi).ItisnotedthattheseparametersmustsatisfytheconditionsNXi=1q(i)=1;NXj=1p(i;j)=11i;j;N:(3.8).
Theprobabilitiesqandpcanbecalculatedusing[77]37q(i)=1LLXi=1P(wijo(l)t;)(3.9)andp(i;j)=PL
l=1PTlt=2P(Sijo(l)t1;)P(Sjjo(l)t)PL
l=1PTlt=2P(Sijo(l)t1;);(3.10)where1i;jN;1lL,Listhenumberoftrainingsequences,Tlisthelengthofthelthobservationsequence,andP(Sijo(l)t;)istheposteriorprobabilityofaparticularstate,Si,givenaobservationsequence,o(l)tandHMMmodel,.Oncetheseparameters,whichcharacterizetheHMMmodel,aredened,theprobabilityofanobservationsequence,withlengthT,iscalculatedas[77]P(Oj)=XSP(O;Sj)=XSP(OjS;)P(Sj);(3.11)whereP(OjS;)=q(o1=w1jS;)TYt=2p(ot=wtjot1=wt1;S;)(3.12)andP(Sj)=P(S1j)TYt=2P(StjSt1;)=ˇS1TYt2aSt1St:(3.13)3.3.2FeatureExtraction
Thevibrationdatafromtheacceleratedbearingdegradationplatformviaelectricalstress,describedinChapter2,isusedforfeatureextractioninthissection.Thevibrationsfromthisexperimentfol-lowedasimilartrendasthoseinthePRONOSTIAsetup.Atthebeginningoftherun,theamplitudeofthevibrationsisatitssmallestpoint.Attheendoftherun,whenthebearingmovestowardsfailure,thevibrationsincreaseexponentially(seeFigure3.12).Thevarianceofthevibrationsalsoincreasesexponentiallyoverthedurationofarun(showninFigure3.13).Thus,thevarianceof38(a)HorizontalAccelerometer(b)VerticalAccelerometerFigure3.12AccelerometerresultsforBearing2fromstarttofailure.(a)ShowsthevibrationsfromthehorizontalaccelerometerforBearing2and(b)thevibrationsfromtheverticalaccelerom-eterforBearing2.Figure3.13Varianceofthehorizontalvibrationdata.thevibrationsisagainselectedasafeature.Anexampleofrawvibrationsignalsfromtheinitial(healthystate)andnal(failure)samplecanbefoundinFigures3.14and3.15,respectively.TheircorrespondingTFrepresentations,usingtheChoi-Williamsdistributionwith˙=10,canbefoundinFigures3.16and3.17,respectively.Asthefaultprogresses,thereisasignicantamountofenergyinthe80-90Hzfrequencyband.Thismaybeduetotherotationalspeedofthemotorbeing80Hz.Asthebearingnearsfailure,theenergyacrossthesurfacebecomesmoreimpulsive.39Figure3.14RawDataofInitialVibrationSignalforBearing2.Figure3.15RawDataofFinalVibrationSignalforBearing2.Wealsosawlittleenergydistributioninthe20-80Hzfrequencybanduntilthebearingreachedfailure.InFigure3.18,snapshotsoftheBearing2runfromstarttofailureintheTFdomainisshown.Thesefrequencybands,aswellastheenergyovertheentireTFplane,wereusedforfea-tureextraction.EntropyandconcentrationmeasureswereagainusedtocapturethechangesinthespreadofenergyintheTFdomainacrosstime.Inthetimedomain,therstfourstatisticalmoments,mean,variance,skewnessandkurtosiswereextracted.Inthefrequencydomain,themax,varianceandRMSwerealsochosenasfeatures.Finally,intheTFdomain,theconcentration40Time (seconds)Frequency (Hz)Horizontal Vibrations in TF Domain  00.020.040.060.080.1020406080100120(m/s2)2-0.4-0.3-0.2-0.100.10.20.3Figure3.16Choi-WilliamsTransformationofInitialHorizontalVibrationSignalofBearing2with˙=10.Time (seconds)Frequency (Hz)Horizontal Vibrations in TF Domain  00.020.040.060.080.1020406080100120(m/s2)2-0.4-0.3-0.2-0.100.10.20.3Figure3.17Choi-WilliamsTransformationofFinalHorizontalVibrationSignalofBearing2with˙=10.41measureovertheentireTFplane,andentropyfromthe80-90HzfrequencybandandtheentireTFplanewereextractedasfeatures.
3.3.3CalculatingtheHMMparameters
Inthissection,wewilldescribehoweachparameteroftheHMMiscalculatedfromtheselectedfeatures.
1)Theinitialprobabilityvector,ˇ,andthestatetransitionmatrix,A.First,thenumberofclasses,K,issetarbitrarilyatthebeginning.Inthiswork,weselectedarangeofthreetoveclasses.Next,ak-NNclassicationisrunonthefeaturematrix,classifyingthefeaturevectorateachtimepointintooneofKstates.Theinitialprobabilityvector,ˇ,isinitializedusingtheratioofthenumberoffeaturevectorsineachstatetothetotalnumberoffeaturevectorsgivenasˇi=KiN;(3.14)whereKiisthenumberoffeaturevectorsintheithclassandNisthetotalnumberoffeaturevectors.Theprobabilitiesofthestatetransitionmatrix,A,arethencalculatedinabruteforcemanner,whereaijisgivenbythenumberoftimesafeaturevectorisfoundinstatejattimet,giventhestateattimet1wasi,dividedbythetotalnumberoffeaturevectors.2)Theobservationprobabilitymatrix,B=(q;p).Tocalculatetheparametersoftheobservationprobabilitymatrix,thefeaturesarerstquantizedtothenearesttenthtocreateanitenumberofobservableoutcomes,W=(w1;w2;:::wk).Thetransitionalprobabilities,p(i;j)=P(ot=wtjot1=wt1)areagaincalculatedinabruteforcemannersimilartothecalculationofA,being42Time (seconds)Frequency (Hz)Horizontal Vibrations in TF Domain  00.020.040.060.080.1020406080100120(m/s2)2-0.4-0.3-0.2-0.100.10.20.3(a)InitialSampleTime (seconds)Frequency (Hz)Horizontal Vibrations in TF Domain  00.020.040.060.080.1020406080100120-0.4-0.3-0.2-0.100.10.20.3(b)IntermediateSample(1)Time (seconds)Frequency (Hz)Horizontal Vibrations in TF Domain  00.020.040.060.080.1020406080100120-0.4-0.3-0.2-0.100.10.20.3(c)IntermediateSample(2)Time (seconds)Frequency (Hz)Horizontal Vibrations in TF Domain  00.020.040.060.080.1020406080100120-0.4-0.3-0.2-0.100.10.20.3(d)IntermediateSample(3)Time (seconds)Frequency (Hz)Horizontal Vibrations in TF Domain  00.020.040.060.080.1020406080100120-0.4-0.3-0.2-0.100.10.20.3(e)IntermediateSample(4)Time (seconds)Frequency (Hz)Horizontal Vibrations in TF Domain  00.020.040.060.080.1020406080100120(m/s2)2-0.4-0.3-0.2-0.100.10.20.3(f)FinalSampleFigure3.18ProgressionofthefaultforBearing2intheTFdomain.43p(i;j)=P(ot=wtjot1=wt1)=Numberoftimesobservationwtdirectlyfollowswt1Totalnumberofobservations(3.15)3.3.4UnsupervisedClusteringviaTemporalHMM
Theproposedmethodwasappliedtothedatafromtheacceleratedbearingdegradationviaelectri-calstressplatform.Inthiswork,weextractedatotalof11featuresfromtheTFdomain(showninFigure3.19):
1)VarianceovertheentiresurfaceoftheTFrepresentation
2)Entropyfromthe80-90Hzfrequencyband
3)EntropyovertheentiresurfaceoftheTFrepresentation
4)ConcentrationmeasureovertheentiresurfaceoftheTFrepresentation5)Maxmagnitudefromthefrequencydomain
6)Variancefromthefrequencydomain
7)RMSfromthefrequencydomain
8)Meanfromthetimedomain
9)Variancefromthetimedomain
10)Skewnessfromthetimedomain
11)KurtosisfromthetimedomainTheparametersoftheHMMwerecalculatedasoutlinedinsection3.3.3.Forthiswork,wechosetolookatK=3,4,and5clusters.Afterperformingtheunsupervisedclusteringofthevibrationdata,wecanseeatrendinthetransitionsofthehiddenhealthstatesacrossallbearingsasweincreasethenumberofclasses.InFigure3.20(a),theclusteringresultsforBearing1areshown,withK=3.Asthenumberofclassesincreasestofour(showninFigure3.20(b))andve(showninFigure3.20(c)),theclasses44024681012141618-10Mean (Normalized)time (days)Mean(a)Meanfromthetimedomain024681012141618-1012345Variance (Time Domain)time (days)Variance (Normalized)(b)Variancefromthetimedomain024681012141618-2-1012Skewness (Normalized)Skewnesstime (days)(c)Skewnessfromthetimedomain024681012141618-2-101234Kurtosis (Normalized)time (days)Kurtosis(d)Kurtosisfromthetimedomain024681012141618-1012345Variance (Freq. Domain)time (days)Variance (Normalized)(e)Variancefromthefrequencydomain024681012141618-101234RMS Frequencytime (days)RMS Freq. (Normalized)(f)RMSfromthefrequencydomainFigure3.19FeaturesusedintemporalHMMclusteringforBearing1acrosstime.45Figure3.19cont'd024681012141618-101234Max. Frequencytime (days)Max. Freq. (Normalized)(g)Maxfromthefrequencydomain024681012141618-4-3-2-10123time (days)Concentration Measure over Entire SurfaceConcentration Measure (Normalized)(h)ConcentrationmeasureovertheentiresurfaceoftheTF
representation024681012141618-3-2-10123time (days)Entropy over Entire SurfaceEntropy (Normalized)(i)EntropyovertheentiresurfaceoftheTFrepresentation024681012141618-3-2-1012time (days)Entropy of 80-90Hz bandEntropy (Normalized)(j)Entropyfromthe80-90Hzfrequencyband024681012141618-10123456Variance (TF Domain)time (days)Variance (Normalized)(k)VarianceovertheentiresurfaceoftheTFrepresentation46024681012141618123Bearing 1 - 3 StatesStatestime (days)(a)3States0246810121416181234Bearing 1 - 4 StatesStatestime (days)(b)4States02468101214161812345Bearing 1 - 5 StatesStatestime (days)(c)5StatesFigure3.20TemporalHMMClusteringresultsforBearing1.becomeunevenlydistributed,withthemajorityofclassesattheendoftherun.ThistrendisalsoshowninFigures3.21,3.20forBearings2and4,respectively.InBearing4therstclassdividesintotwostatesasweincreasethenumberofclustersfrom3to4.Althoughthisisslightlycontrarytothetrend,therearestillmoretransitionsbetweenclassesattheendoftherunwhenthenumberofclustersisincreasedto5.Thisanalysisiscongruentwithwhatishappeningphysicallyinthebearinganditsvibrations.Asthebearingnearsfailure,moresignicantinformationisfound,thusrequiringmorestatestocapturethechangingtrend.4702468101214123Bearing 2 - 3 ClassesStatestime (days)(a)3Classes024681012141234Bearing 2 - 4 ClassesStatestime (days)(b)4Classes0246810121412345Bearing 2 - 5 ClassesStatestime (days)(c)5ClassesFigure3.21TemporalHMMClusteringresultsforBearing2.3.4Conclusions
Inthischapter,wehaveintroducedtwocomplementarymethodsforhiddenhealthstateestima-tioninbearingvibrationdata.First,weproposedamultivariateeventdetectionframeworktolearnthehiddenhealthstatesofbearingsfromunlabeledtrainingdata.Wehaveshownthatthesehealthstatesarenotnecessarilydiscreteandevolvecontinuouslyovertime.Moreover,wehaveshownhowincludingmorerelevantfeaturesforhealthstateestimationimprovestheaccuracyandincreasesthereliabilityofstateestimation.Wealsoshowedthatdifferentoperatingconditionsre-4805101520253035123Bearing 4 - 3 ClassesStatestime (days)(a)3Classes051015202530351234Bearing 4 - 4 ClassesStatestime (days)(b)4Classes0510152025303512345Bearing 4 - 5 ClassesStatestime (days)(c)5ClassesFigure3.22TemporalHMMClusteringresultsforBearing4.sultedincontrastingstatetransitionproles,alludingtotheneedforprognosistobeconductedonanoperatingconditionspecicbasis.Second,weintroducedanunsupervisedclusteringmethodbasedontemporalHMMclusteringtoobtainthehiddenhealthstatesofabearingwhendegradedbyelectricalstress.Asthenumberofclassesincrease,thereislessseparationbetweenthelat-terclasses.Thiscorrespondstothephysicalprocessofbearingdegradation,inwhichsignicantchangesinthedataoccurtowardsfailure.Infuturework,discoveringhiddenhealthstatesmayimprovethecurrentstateoftheartdi-agnosisandprognosismethods.Oncethedifferenthealthstatesareidentied,typicalRULsfor49eachhealthstatecanbedeterminedfromtrainingdata.ThisinformationalongwithrepresentativefeaturescorrespondingtoeachstatecanprovideaprobabilisticwayofestimatingtheRULfromnewtestingdata.Furthermore,thetemporalHMMclusteringmethodcanbeusedonothertypesofsensordata,suchascurrent,andfusedacrossdifferentsensors.Thehiddenhealthstatesfoundinthisanalysiscanbecomparedtotheresultsusingonlyvibrationsignalstodetermineanycorre-lations.Lastly,morehistoricaldataisneededtoassurethecertaintyofthismethodacrossalltypesofbearingsandoperatingconditions.50CHAPTER4FAULTPROGNOSISANDRULESTIMATIONONBEARINGSVIAEXTENDEDKALMANFILTER4.1Background
Kalmanlteringisarecursivealgorithmthatestimatesthetruestateofasystembasedonnoisymeasurements.TheKalmanlterhasbeenusedinmanyapplicationsinvolvingnavigation,on-linesystemidenticationandtracking[78,79].Recently,Kalmanlteringhasbeensuccessfullyappliedtofaultprognosis[52].Forexample,in[30],Kalmanlteringisusedtointerpolatethesignalfeaturetrendslearnedfromthelabeleddataandthentoestimatetheevolutionofthefaultsfordiagnosispurposes.However,KalmanlteringassumesalinearsystemdynamicsmodelwithGaussiannoiseinthemeasurementswhichisnotalwaysrealisticinreallifeapplications.Ex-tendedKalmanFiltering(EKF)isanextensionofthisframeworktononlinearsystemdynamicsandhasbeenusedbyLalletal.[52,53]forprognosticsofelectronicinterconnectsandbySahaetal.[80]forbatterylifemanagement.TheEKFequationinthepresenceofprocessnoiseandmeasurementnoiseis:xk=f(xk1;uk1)+wk1(4.1)wherexkisthestatebeingestimated,fisanonlinearfunctionofstates,ukistheinputattimesamplek,wkisrandomzeromeannoisewithcovariancematrixQk.InEKF,therelationshipbetweensystemstates(xk)andmeasurements(zk)canalsobenonlinear:zk=h(xk)+vk;(4.2)wherezkisthemeasurement,hisameasurementfunctionwhichisanonlinearfunctionofstatesandvkiszero-meanrandomprocessdescribedbythemeasurementnoisecovariancematrix,Rk.InordertocarryoutthenormalKalmanFilteroperations,thenonlinearfunctions,fandh,mustbelocallylinearizedaroundtheestimatedstatebycalculatingtheirrespectiveJacobian,producingthematricesFandH,respectively.OneimportantpointintheimplementationofEKFisthechoice51oftheinitialparameters,asthespeedofconvergencedependsontheinitialestimate,^
x0andtheuncertaintymatrix,P0[81].4.1.1EKFParameterLearning
Forthetwodifferenttypesoffeatures,i.e.vibrationandentropy,timedependentdegradationmodelsareobtainedthroughcurvetting.Forexample,forthevariancefeature,anexponentialoftheformaebtwasfoundtobethemostsuitablewhereasforthetime-frequencyentropyfeatureacurveintheformofabectwasmoresuitable(seeFigures4.2aand4.2b).Theparametersofthedegradationfunctionareupdatedwitheachnewmeasurement.Toaccomplishthis,astatevectorxcontainingtheequationforthecurvetaswellastheunknownparametersdescribingthisdegradationmodelateachtimepointaredenedforeachfeature.Forthevariancefeature,theparametersakandbkoftheexponentialcurveareusedtodenethestatevectoras[51,82]:xk=[aebkab]T(4.3)andfortheentropyfeature:xk=[akbkeckkakbkck]T;(4.4)bothwiththemeasurementequationgivenby:zk=h(xk)=xk(1):(4.5)Itisalsonotedthatthereisnoinputtothissystem,soinourcaseukdenedinequation4.1isequaltozero.Witheachtimestep,theparametersofthedegradationmodelareupdatedtoformanewmodel,fk,andanestimateofthenextstate,^xkiscalculated.ThefunctionsfandharethenlocallylinearizedaboutthatestimatetoproduceFkandHkby:Fk=@f@x


^
xk=2
6
6
6
6
40ebkkkakebkk010
0013
7
7
7
7
5;(4.6)5205001000150020002500300002468101214161820Variance - Training Set 1SamplesVariance(a)Bearing11010020030040050060070080090000.20.40.60.811.21.41.61.82Variance - Training Set 2SamplesVariance(b)Bearing120200400600800100000.20.40.60.811.21.4Variance - Training Set 3SamplesVariance(c)Bearing21010020030040050060070080000.10.20.30.40.50.60.70.80.9Variance - Training Set 4SamplesVariance(d)Bearing22010020030040050060000.10.20.30.40.50.60.7Variance - Training Set 5SamplesVariance(e)Bearing3102004006008001000120014001600180000.20.40.60.811.21.4Variance - Training Set 6SamplesVariance(f)Bearing32Figure4.1Medianlteredtime-domainvarianceacrossall6trainingsetsfortheFEMTOdata.5300.511.522.53x 10405101520253035404550time (sec)Variance ((m/s2)2)Curve Fit for Variance Feature  Variance DataCurve Fit(a)VariancecurvetforBearing110100020003000400050006000024681012time (sec)Entropy (bits)Curve Fit for Entropy Feature  Entropy DataCurve Fit(b)EntropycurvetforBearing31Figure4.2Curvettingtovarianceandentropyfeatures.Hk=@h@x


^
xk=100;(4.7)forthevariancetrendandFk=@f@x


^
xk=2
6
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401eckkkbkeckk0100
0010
00013
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5;(4.8)Hk=@h@x


^
xk=1000;(4.9)fortheentropytrend.TheinitialstateestimategiventotheEKF,x0,consistingoftheinitialvalueforthedegradationfeatureaswellastheinitialguessesfortheparameterswasfoundthroughuseofthetrainingdata.Averagesofthesevalueswithinoperatingconditionswerethenusedastheinitialparameterestimatesinx0intheRULpredictionstage.Furthermore,theinitialvaluesforeachtrainingsetwasextractedandtheaverageforeachoperatingconditionwasusedastheinitialvalueforthe54degradationfeatureinx0.Theinitialuncertaintymatrix,P0,waschosenempiricallyas:P0=2
6
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40:10000:10000:13
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5(4.10)4.1.2RULPrediction
Thefailurethresholds,i,whereiistheiththresholdforagivenoperatingcondition,wereextractedasthevalueatthenalsampleofeachtrainingdataset.Thenaltestingthresholdforeachoperatingconditionwascomputedastheaverageoftheseperoperatingcondition,givenby:o=1KKXi=1i;(4.11)whereKisthenumberoftrainingsetsinoperatingconditiono.TrackingofthetestfeaturesbeginsafteracertainamountoftimehaspassedsincethedataatthersttimepointsarenotusuallyreliableforRULestimationandmaynotfollowthetrendlearnedfromthetrainingdata.Anoverallviewofthealgorithmandallitsstepsaregivenbelow:1)Initializex0andP0.2)Predictthenextstate,^xk,anduncertaintymatrix,Mk:^
xk=f(^xk1)+wk1;(4.12)Mk=Fk1Pk1Fk1+Qk1:(4.13)3)Takeinmeasurement,zk.4)UpdatethepredictionsandtheiruncertaintiesusingtheKalmangain,Kk:Kk=MkHT
kHkMkHT
k+Rk1;(4.14)^xk ^xk+Kk(zkHk^
xk);(4.15)55Pk=(1KkHk)Mk:(4.16)5)Thecurrentvalueofthefeaturestateisextrapolatedouttofailurethreshold.Thetotalnumberoftimesteps,n,requiredtoreachthefailurethresholdistakenastheRULattimek.o=^xk+n=fk(^xk+n1)+wk+n(4.17)6)CalculatethecondenceintervalsofRULpredictions(seesectionIII-E).7)Repeatprocessstartingatstep3.
4.1.3RULCondenceIntervals
AnimportantaspectofprognosisisnotjusttopredicttheRUL,buttoaddacondencemeasuretoitaswell.Inthetestingphase,thecondenceintervaloftheRULpredictioniscalculatedbyusingtheerrorcovariancematrix,P.Attheendofeachstep,theerrorcovariancematrixisupdatedgivingtheuncertaintymeasurementbetweenthepredictedstate,^xandthetruemea-surement.Sincetherstvalueinthestatevariableistheactualfeaturevalue,beingeitheren-tropyorvariance,P(1;1)containstheuncertaintyofthepredictedstate.Giventhisuncertainty,a95%condenceintervalcanbeplacedaroundtheestimateandtheupperandlowerbounds,^xubd=^x(1)+2:567P(1;1)and^xlbd=^x(1)2:567P(1;1),areextracted.SimilartooriginalRULestimation,thesevaluescanalsobeextrapolatedtothefailurethresholdtoobtainupperandlowercondenceboundsontheRULpredictions.
4.1.4RULEstimationviaExtendedKalmanFilter
4.1.4.1FeatureExtractionandCurveFitting
ForallofthedatasetsdescribedinTableI,bothtimedomainvarianceandtime-frequencydomainentropyfeaturesareextracted.Adegradationtrendisextractedfromthetrainingdatabasedonthesefeaturesbyanalyticalcurvetting.Theaccuracyofcurvettingwasquantiedbythenormalizedmeansquareerror(NMSE),givenbyk^xxk2kxk2,where^xisthettedcurveandxisthe56Table4.1NMSEBetweenCurveFitsandFeaturesOperatingConditionTrainingsetEntropyVariance110.05100.352520.04340.3326230.05370.714640.05150.5462350.03350.842260.04850.3680Table4.2CurveFittingParametersforTrainingDataOperatingTrainingEntropyVarianceConditionSetabcab119.2961.2860.36611:429e400.0339729.4341.8320.40215:556e150.03916239.9638.920.014130.18620.001592410.27.5090.0079810.13490.002287359.7478.5930.0373:026e80.0336468.8514.6940.0073811:181e650.09228rawdata.Forthevariancefeature,itwasnotedthattherstandthirdoperatingconditionsbehavesimilarly,withthedegradationtrendmodeledbyanexponentialfunctionaebtacrossthelifetimeofthebearings.TheaverageNMSEacrossthefourtrainingsetsinoperatingconditions1and3forthevariancefeatureis0.526.However,forthesecondoperatingcondition,thedegradationtrendforthevariancefeatureinthetestingdatadoesnottthetrendlearnedfromthetrainingdata.Thismaybeduetothefactthatthetestingsamplesprovidedforoperatingcondition2weretruncatedbeforethevariancefeaturecouldcapturethedegradation,i.e.thetestinglifehistoryisshorter,averageof152minutes,comparedtotheotheroperatingconditions.Fortheentropyfeature,wedeterminedthebestcurvettobeoftheformabect,andtheaverageNMSEbetweenthistandtheentropyfromallthetrainingsetswas0.048.Table4.1showsalloftheNMSEvaluesforbothfeaturesandalltrainingsets.Table4.2showsthevaluesoftheparametersforeachtrainingset.WealsonotethattheNMSEisusuallyhigherforthevariancefeaturesinceitsrangeislargercomparedtoentropyasseeninFigure4.2.574.1.4.2RULEstimation
Asnotedabove,weusedtwodifferentfeaturesinordertoestimatetheRULbasedontheoperatingcondition.InTable4.3,theRULestimationresultsusingentropyversusvarianceonallofthetestingdatasetsaregiven.TheresultsarequantiedbythepercentageofRULestimationsthatfallwithin+=20%ofthetrueRULinthelast500seconds.Itisshownthatusingvarianceinthe1stand3rdoperatingconditionswillprovidetheuserwithaccurateRULestimationstowardstheendofthetestsets.However,thevariancefeaturecannotpredictthetrueRULforoperatingcondition2.Thisisduetothefactthatthevariancefeature,asshowninFigure4.2a,staysconstantallthewaythroughtherununtiltheveryend,whereitisabletocapturethefailureofthebearing.Therefore,ifthetestingdatasetistruncatedtooearly,i.e.beforethebearinggetsclosetofailure,thevariancefeatureisnotagoodindicatoroffailureandcannotperformwellinthesubsequentRULestimation.Ontheotherhand,theentropyfeatureshowsaninitialincreaseinvalue,indicativeofthebearing'shealthdeterioration,andthenstaysstablearoundthisvalueasshowninFigure4.2b.Assuch,whenthetestsetisshortcomparedtothelifetimeofthebearing,asinthecaseofoperatingcondition2,theentropyfeatureissuccessfulintrackingtheearlydeteriorationofbearinghealthandinpredictingthetrueRULinmostofthetestcases.Whenthetestdatasetislonger,entropydoesnotyieldaccurateRULestimatessinceduringthelast500secondsofthelifetimeofthebearingtheentropyvaluesdonotchangethatmuch.Therefore,inthenalanalysis,thevariancefeaturewasusedforoperatingconditions1and3,andtheentropyfeaturewasusedforoperatingcondition2.Itisalsoimportanttonotethatforcertaintestsetsfromcondition2,inparticularforBearing23andBearing27,neitherfeaturegetsclosetothetrueRULestimate.Thisisduetothefactthatthetestdatasetisveryshort,e.g.only29minutesinthecaseofBearing27.Inparticular,Figure4.3ashowstheresultsofRULpredictionsonbearing13.PredictionswerestartedatthehalfpointsinceearlyRULpredictionswiththevariancefeaturearenotveryreliableandmeaningful.Asshowninthisgure,theRULpredictionsatthebeginninghavealotoferrorbutastimegoesontheyconvergetothetrueRUL.Figure4.4showstheRULestimations58Table4.3ComparisonofRULusingVariancevs.EntropyOperatingConditionTestsetDuration(min)VarianceEntropy1130096%0%2190100%0%338446%0%438454%0%525040%0%262000%4%71020%70%83340%24%9950%36%10290%0%3115956%0%overtimefortheonlytestingsetinthethirdoperatingcondition,bearing33.Here,too,wecanseethatthetrendofRULpredictionsacrosstimeconvergestothetrueRULline.Itisnotedthattheconvergenceofthissetisslowerthanthatoftheprevious.Thisisduetothelackofsubstantialnumberofsamplesforupdating,i.e.shortertestdataset(59minutes).Figure4.5illustratestheperformanceoftheentropyfeatureinRULpredictionforbearing24fromthesecondoperatingcondition.Althoughthedatasetismoretruncatedcomparedtotheothertwooperatingconditions,entropyisabletotracktheactualRULvalue.WealsonoticedintheRULpredictionalgorithmthatthetimeatwhichEKFtrackingisstartedhasaneffectonhowquicklytheestimatedRULsconvergetothetrueRUL.Ifthetrackingisstartedatthebeginningofthetestingdata,theconvergenceismuchslowerasthereismoreirrel-evantsamplesornoiseinthedatathatlowerstheaccuracyoftheRULprediction.AnexampleofthisphenomenacanbeseeninFigure4.6,whereweshowtheRULestimationsofthesamebearingwithdifferentstarttimesfortracking.Theconvergenceofthealgorithmisfasterwhentheprocedureisstartedmidway,insteadofattheverybeginning.Neitheroneofthesestartingpointsconvergesasfastasstartingpasthalfway,asseeninFigure4.4.5900.511.52x 10401234567x 104time (sec)RUL (sec)RUL predictions for bearing 1_3  Predicted RULTrue RUL(a)RULpredictionovertimeversustheactualRUL1650017000175001800002000400060008000100001200014000160001800020000time (sec)RUL (sec)RUL predictions for bearing 1_3 (closeup)  Predicted RUL95% Confidence BoundsTrue RUL20% Confidence Intervals(b)CondenceintervalfortheestimatedRULnearendof
testingFigure4.3RULEstimationforBearing13withthevariancefeature.050010001500200025003000350040000123456789x 104time (sec)RUL (sec)RUL predictions for bearing 3_3  Predicted RULTrue RULFigure4.4RULEstimationforBearing33withthevariancefeature.60010020030040050060070000.511.522.533.5x 104time (sec)RUL (sec)RUL predictions for bearing 2_4  Predicted RULTrue RUL20% confidence IntervalsFigure4.5RULEstimationforBearing24.3100315032003250330033503400345035000200040006000800010000120001400016000time (sec)RUL (sec)RUL predictions for bearing 3_3  From BeginningFrom MiddleTrue RULFigure4.6RULEstimationforBearing33withdifferentEKFtrackingstarttimes.4.1.4.3ComparisonofEKFvs.KF
Inthissection,wecomparetheperformanceoftheproposedExtendedKalmanFilter(EKF)ap-proachtotheregularKalmanFilter(seeTable4.4),whichiscommonlyusedinliteratureforprognosis,intermsofthepercentageofestimatedRULvalueswithin+=20%ofthetrueRULinthelast500seconds.Forthemajorityofthedatasets,theKalmanFilter(KF)showsnoaccuracytowardstheendofthetestingdatasetwiththeexceptionoftestset6.Overall,EKFbasedRULalgorithmperformsbetterthantheKFforalloperatingconditions.61Table4.4ComparisonofRULestimationsusingEKFvsKFOperatingConditionTestsetEKFKF1196%0%2100%0%346%0%454%0%540%0%264%92%770%8%824%0%936%4%100%24%31156%0%4.1.4.4CondenceIntervalEstimation
InFigure4.3b,wecanseeacloserviewoftheendoftherun,alongwiththecondenceintervalsoftheRULestimations.WealsoshowabandwidtharoundthetrueRUL(+/-20%ofthetrueRULvalue)toseehowwelltheestimatedRULsfallwithinthisband.Atthebeginningofthealgorithm,thetrueRULisnotwithinthecondenceboundsofthepredictions.Thisisduetothefactthatsamplestowardsthebeginningofthetestingdatamaynotprovideenoughinformationtoperformfeaturetracking.However,towardstheendoftherun,thepredictionsaswellastheircondenceintervalsarebothwithin20%oftheactualRUL.4.2Conclusions
Inthischapter,wehaveintroducedanExtendedKalmanFilteringbasedapproachfortrackingtheRULofbearings.First,weintroducedbothtimeandtime-frequencydomainfeaturesofvibrationsignalsandillustratedthatdifferentfeaturesmayworkbetterunderdifferentoperatingconditions.Second,wegaveadetaileddescriptionofRULestimationbasedonEKFalongwithaproceduretoestimatethecondenceintervalalongtheRULestimates.Finally,weappliedtheproposedalgorithmtobearingvibrationdatatoillustratetheconvergenceofthealgorithmalongwithitsbehaviorunderdifferentconditions.62FutureworkshouldconsiderseveralimprovementstoourRULestimationmethodology.First,astheresultsindicatethestartingpointofEKFinthetestingdatahasadirecteffectonthecon-vergencerateoftheRULestimates.FutureworkshouldconsiderslidingwindowbasedEKFimplementationsuchthatateachtimepointonlythemostrecentpasttimepointsareusedtoesti-matetheRUL.Thistypeofanapproachmayhelpwithttingthelearnedtrendofthedegradationproletothetestingfeatures.Sincethetestingdataisatruncatedversionofthefulllifetimeofabearing,thetrendsobtainedfortrainingdatamaynotalwaysbegoodtstothetestingdataresultinginlargeamountofpredictionerror.Second,futureworkshouldconsiderfusionofdiffer-entfeaturesandcorrespondingdegradationprolesforamoreaccuratesystemmodel.Moreover,differentpre-processingmethodstothedatashouldbeexplored.Third,futureworkshouldcon-sideralternativeapproachestolearningthedegradationmodelsfromextracteddatasuchasneuralnetworks.Finally,theproposedframeworkshouldbeextendedtodifferenttypesofsensordata,suchasthecurrent,frombearingstoimprovetheRULestimationaccuracy.63CHAPTER5THEUSEOFBEARINGCURRENTSANDVIBRATIONSINLIFETIMEESTIMATIONOFBEARINGS5.1Background
5.1.1BearingCharacteristicFrequencies
Duetothenatureoftherotatingelementsofbearings,specicfrequencieshavebeenshowntobepresentinfrequencyanalysisofbearingvibrations.ThesebearingcharacteristicfrequenciesarecalculatedfromthegeometryofthebearingandaregiveninTable5.1,wheredisthediameteroftherollingelements,Disthepitchcirclediameter,Zisthenumberofrollingelementsandisthecontactangle.Theseequationsareonlyvalidiftherotatingelementsdonotexhibitanysliding,whichisusuallynotthecaseinmostapplications[83].
5.1.2Data
Thedatainthischaptercomesfromtheaccelerateddegradationplatformviaelectricalstress,describedinChapter2.Table5.1BearingCharacteristicFrequenciesFundamentaltrainfreq.relativetoouterringfr2h1dcosDiFundamentaltrainfreq.relativetoinnerringfr2h1+dcosDiBallpassfreq.ofouterringZfr2h1dcosDiBallpassfreq.ofinnerringZfr2h1+dcosDiRotatingelementspinfreq.Dfr2d1dcosD264024681012141618-50-40-30-20-1001020304050Bearing 1 Raw Horiz. Vib.time (days)Acceleration (m/s2)(a)HorizontalAccelerometer024681012141618-50-40-30-20-1001020304050Bearing 1 Raw Vert. Vib.time (days)Acceleration (m/s2)(b)VerticalAccelerometerFigure5.1AccelerometerrecordingsforBearing1fromstarttofailure.(a)Thevibrationsfromthehorizontalaccelerometerand(b)thevibrationsfromtheverticalaccelerometer.5.2Methodology
5.2.1FeatureExtraction
5.2.1.1BearingVibrationFeatures
Atthebeginningoftherun,theamplitudeofthevibrationsisatitssmallestpoint.Attheendoftherun,whenthebearingstartstofail,theamplitudeofthevibrationsincreasesexponentially(seeFig.5.1).Somecommonfeaturesextractedfrombearingvibrationdatatocapturetrendsarestatisticalmoments,suchasvariance,fromthetimedomainandRMSfrequencyfromthefrequencydomain[28,35,36,84].Inthischapter,theRMSfrequencyfeatureischosen,givenby:RMSFreq:=v
u
u
t1nnXi=1(
Xfft;i
)2;(5.1)wherenisthenumberoffrequencysamplesandXfft;iistheithsampleoftheFouriertransformofthevibrationsignal.655.2.2DetectionandTrackingofEDMCurrents
Intheseexperiments,thebearingsarefoundtobemostlyintheohmicstate,inwhichthebearingcurrentfollowsthesamebehaviorasthebearingshaftvoltage.Fig.5.2ashowsanormalcurrentsamplewithoutanydischargeevents.InFig.5.2c,acloseupplotofthenormalbearingcurrentisshownanditcanbeseenthatthetrendispulse-likeresemblingtheshaftvoltage.Asthebearingdegrades,itbeginsswitchingbetweenthethreeelectricalstates:ohmic,capacitiveanddischarge.InFig.5.2b,acurrentsampleforthesamebearingatalatertimeisshown.Inthistimeperiod,aspikeoccursindicatingadischargeevent.AsshowninFig.5.2d,thecurrentstartsintheohmicstate,transitionsintoacapacitivestate,andnallydischargesandgoesbackintoanohmicstate.Thetrackingofthesedischargeeventsisimportantfordeterminingthehealthstateofabearing.Largeinuxesofbearingdischargeeventsinashortperiodoftimeproducesignicant,irreversibledamagetoabearing,thusfailureisaccelerated.Inthischapter,awaveletdecompositionbasedmethodisusedtodetectthesedischargeeventsfromtherawcurrentsamples.First,Haarwaveletdecompositionisconductedoneachcurrentsamplewith8levelsofdecomposition.TheHaarwaveletischosenbecauseitcloselyresemblesthesquare-wavenatureofthebearingcurrentdataanditisidealfordetectingdiscontinuities.Inthewaveletdomain,eachdischargeevent(showninFig.5.3)canbeobservedinthesignalreconstructedinthesubspacespannedbythelevel8waveletfunctionsgivenby:D8(n)=NXk=1d8;k (28nk);(5.2)whereNisthelengthofthesignal,d8;karethelevel8waveletcoefcientsand isthewaveletfunction,whichinthiscaseistheHaarwavelet.Thisreconstructionorprojectiontothissub-spaceyieldsasignicantpeakatthetimeofthecurrentdischargeevent.Conversely,acur-rentsamplewithnodischargeeventsshowsnosignicantpeaks(showninFig.5.4).Tode-terminewhetherabreakdownoccurred,athresholdT=(D8)+4˙(D8)ischosen,whereD8=[D8(1)D8(2):::D8(N)]isavectorofprojectedsignalsamplesinthissubspace,isthe6601000200030004000500060007000800090000.511.522.533.5Time (m sec)CurrentCurrent Sample(a)NormalBearingCurrentSample010002000300040005000600070008000900000.511.522.533.5Time (msec)CurrentCurrent Sample(b)BearingCurrentSamplew/DischargeEvent4604805005205405605806006206400.511.522.53Time (m sec)CurrentCurrent Sample(c)CloseupofNormalBearingCurrent5300535054005450550055505600565000.511.522.53Time (msec)CurrentCurrent Sample(d)CloseupofBearingDischargeEventFigure5.2BearingcurrentsamplesfromBearing1.(a)Currentsamplefromabearingundernormalconditionand(c)acloseupofthissample.(b)Currentsampleinwhichadischargeeventhasoccurredand(d)acloseupofthisdischargeevent.67meanand˙isthestandarddeviation.Eachinstancewherethereconstructedsignalcrossesthisthreshold,T,iscategorizedasabearingdischargeevent:DischargeEvent(k)=8
>
>
<
>
>
:1;ifD8(k)T0;otherwise:(5.3)Thesedischargeeventsarethentrackedovertime,andacumulativesum(showninthetop3plotsofFig.5.7forBearings1,2,and3,4,and5respectively)isobtainedasTotalDischarges(t)=PktDischargeEvent(k),wheretisthetimevariable.Atthebeginningoftherun,theoccur-renceofdischargeeventsisfew.ForBearing1,fortherst3days,thereareonly0.5dischargeeventsperminute.Atsomepointduringtherun,thenumberofdischargeeventsperminuteincreasessignicantly,whichcausesirreversibledamagetothebearingleadingtofailure.ForBearing1,thisdamagingperiodincurs400dischargeeventsperminute,occurringaroundday10.Directlyafterthispoint,thebearingreachesitsnalstage.Afterthecriticalperiodinthedischargeeventprole,theamplitudeofthevibrationsstarttoexponentiallyincrease(showninthesecondcolumnofFig.5.7).Inordertodeterminewhenacurrentdischargeinuxhasoccurred,ahistoryofmminutesofdischargeeventsisbuilt.mischosensuchthatitislargeenoughtoacquireasignicantamountofhistoryfortraining,yetsmallenoughtocapturechangesinthedata.Next,thecumulativenumberofdischargesaretrackedacrosstime.Foreachtimepointaftertherstmtimepoints,alineoftheforma+btistoverthelastmminutesofthecurrentdischargeeventsandthenormalizedmeansquareerror(NMSE)betweenthettedlineandtheactualdatapointsiscomputed.ThisNMSEistrackedovertime.OncethisNMSEcrossesathreshold,theeventagistriggered,andthecurrentdischargeinuxpointislocated.Thethresholdissetto8e14,tocaptureoccurrencesofhighdischargeinuxes(morethan100dischargeeventsperminute).680100020003000400050006000700080009000-0.2-0.100.10.2Time (msec)Reconstructed Signal  010002000300040005000600070008000900001234CurrentCurrent Sample w/ Discharge EventReconstructed SignalThresholdFigure5.3Currentsamplew/dischargeeventandcorrespondingreconstructedsignalusingthelevel8detailcoefcientsfromaHaarwaveletdecomposition.0100020003000400050006000700080009000-0.0200.020.04Time (msec)Reconstructed Signal  010002000300040005000600070008000900001234CurrentNormal Current SampleReconstructed SignalThresholdFigure5.4Normalcurrentsampleandcorrespondingreconstructedsignalusingthelevel8detailcoefcientsfromaHaarwaveletdecomposition.695.2.3RULPredictionviaEKF
TheRULiscalculatedbasedontheframeworkpresentedin[85]withadifferentrealizationoftheEKFbasedonacontinuous-timestateequation[86].First,curvettingisusedonthetrainingdatatoextractasuitabledegradation,orobservation,modelh.FortheRMSfrequencyfeature,thebesttisanexponentialfunctionoftheformaebt.TheRMSfrequencyfeatureforeachbearingisshowninFig.5.5.Second,thestatevariablesarechosentobex=[ab!]whered!dt=b.Itisalsoassumedthataandbareupdatedlinearly:dadt=waanddbdt=wb,wherewaandwbarewhiteGaussianprocesses.Inthiswork,thecontinuous-timestateequationisgivenbydxdt=Ax+Lw,andA=2
6
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6
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4wawb3
7
5.InordertopredicttheRULofthebearings,afailurethreshold,=1KKPi=1i,whereiisthevalueofeachindividualtrainingsetatfailureandKisthenumberoftrainingdatasets,isdened.Inthischapter,Kisequalto3,anddeterminedusingthedatasetsthatonlycontainvibrationdata.Second,theEKFisinitializedandthepredictionstepisrunrepeatedlyuntilthevalueofaebtreachesthefailurethreshold,.ThetimetoreachthethresholdistakenastheRUL.Finally,theEKFparametersareupdatedwitheachnewmeasurementpointandRULestimationbasedoniscontinued.Inthischapter,anupdatetothisframeworkisproposedbyusinginformationfromthecurrentdischargeevents.Asitisshownin[85],theRULestimationsusingtheEKFprovidedmoreaccurateresultswhenthebearingvibrationsfollowedanexponentialgrowth,whichhappensclosertofailure.Beforethistime,RULestimationscanbehighlyinaccurate.ThisworkproposestousetheeventofasharpincreaseofbearingdischargeeventsinashortperiodoftimeasanindicatortostartpredictingRUL.TherearethreesignicantreasonswhystartingRULestimationsfromthispointisbenecial.First,bearingvibrationsofahealthybearingareusuallynotanaccuratepredictoroffailureatthebeginningofarun.Thus,RULpredictionsbasedonearlyvibrationdatatendtogivehighlyinaccurateRULestimations[85].Second,usingvibrationdataforRUL700510152025303540020406080100120time (days)RMS Frequency of Horiz. Vib  Bearing 1Bearing 2Bearing 3Bearing 4Bearing 5Figure5.5RMSFrequencyfeatureforBearings1,2,3,4and5.estimationsfallsintoa20%condenceintervaltowardstheend,andsometimeswithoutwarning.Afterthecurrentdischargeinuxpoint,signicantinformationstartstobefoundinthevibrationdata,makingallRULestimationsmoreaccurateandmeaningful.Lastly,itiscomputationallyinefcienttorunRULestimationalgorithmsfromthebeginningofarunespeciallyiftheuserisnotcertainoftheaccuracyoftheestimates.Startingafterthecurrentdischargeinuxproposesasolutiontothisasnocomputationisdonebeforetheinux.Theoverviewofthisalgorithmisasfollows.ThetrainingoftheEKFstartsaftermminutes.Next,thedischargeeventsaretrackedacrosstimetodetectthecurrentinuxevent.Oncethiseventisrealized,RULestimationbasedonvibrationsviaEKFisstarted.5.3ExperimentalResults
5.3.1TemperatureAnalysis
Bearingtemperaturedidnotprovideanysignicantinformationaboutthestateofthebearings.Therearenosignicantchangesinthetemperatureforthedurationoftherunforanybearing.71051015202530354025303540455055time (days)temperature (°C)Bearing Temperature  Bearing 1Bearing 2Bearing 3Bearing 4Bearing 5Figure5.6TemperaturesignalforBearings1,2,3,4and5.ThetemperaturesignalexhibitedaslightandsteadydecreaseoverthecourseoftherunasshowninFigure5.6.Sincethistrendisnotclear,temperaturedataisneglectedinfurtheranalysis.Onereasonforthisdecreasemightbeduetonoisegeneratedinthethermocouplereadingsfromthehighfrequencyswitchingacrossthebearingshaft.However,theEDMeventswouldnothavecausedasignicantriseintemperature,astheenergyoftheseeventsisdissipatedintheentirebearing[47].
5.3.2ComparisonwithConventionalVibrationAnalysis
Inthissection,wecompareconventionalvibrationanalysistotheproposedmethodology.Fromtheanalysisofthesensorsignals,aclearrelationshipisseenbetweenthecurrentandthevibrations.Ithasalreadybeenreasonedthatalargeamountofcurrentdischargesinashortperiodoftimecausesirreplaceabledamagetobearings[50].Asstatedbefore,thenumberofdischargesslowlyincreasesduringthebeginningstagesoffailure.Whenthenumberofdischargesincreasesrapidly,thevibrationsbegintheirexponentialgrowthandthebearingisforcedintofailure(showninFig.5.7forall5testBearings).7201020024681012x 104Bearing 1No. Discharge Events05101500.511.522.53x 107Bearing 20102030012345x 106Bearing 30204000.511.52x 107Bearing 402040012345x 106Bearing 50102001020304050607080RMS Frequencytime (days)051015010203040506070time (days)0102030020406080100120time (days)0204001020304050time (days)02040020406080100time (days)Figure5.7Relationshipbetweenbearingcurrentdischargesandvibrationsforthe5testbearings.Therstrowshowsthecumulativebearingdischargesacrosstheentirerun.ThesecondrowshowstheRMSFrequencyofthevibrations,extractedfromthefrequencydomain.Incomparison,Fig.5.8illustratesthebearingcharacteristicfrequenciesacrosstimeforeachbearing.Itcanbeseenthattherearenosignicantchangesinthemagnitudeofthefrequencyspectrumforthemajorityoftheextractedfeaturesuntilthebearingnearsfailure.Thesignalsshowlittlevariationuntilafterthenumberofdischargesrapidlyincreases.InBearings2and3,thereissomeinitialinformationinthefundamentalcagefrequencies(fcoandfci)atthebeginningoftherun,butthisistooearlytouseasanindicator.Afterthisinitialabruptchange,thesesignalsinbothbearingsreachasteadystateuntilnearfailure.Furthermore,thereisnomonotonictrendacrossallbearingsthatcanbeexploitedforawarningindicatororforRULestimationpurposes.Thisshowsagainthattheinuxofcurrentdischargesinashortperiodoftimeprecedesthevibrationgrowth,thusprovidingmoreusefulinformation.Usingthesendings,thisinuxofcurrenteventcanbeusedtoprovideanearlyindicationfortheremainingusefullifeandimminentfailureofabearing.730102000.20.4Bearing 1fco0102000.20.4fci0102000.10.2fbo0102000.20.4fbi0102000.10.2fbtime (days)05101500.10.2Bearing 205101500.10.205101500.20.405101500.505101500.20.4time (days)0102000.51Bearing 30102000.510102000.20.40102000.20.40102000.20.4time (days)0204000.20.4Bearing 40204000.20.40204000.10.20204000.10.20204000.20.4time (days)0204000.10.2Bearing 50204000.10.20204000.20.40204000.20.40204000.10.2time (days)Figure5.8Magnitudeofthefrequencyspectrumateachbearingcharacteristicfrequencytrackedintimeforthe5testbearings.
5.3.3Event-triggeredRULEstimationsusingCurrentDischargeInuxInthealgorithmproposedinSection5.2.3,therststepistodetecttheinuxevent.ItisshowninFig.5.9thatthealgorithmisabletocapturetheinuxofcurrentdischarges,whichsigniesanimpendingfailure.Inthissection,theaccuracyoftheRULestimatesforthetraditionalandproposedEKFtrainingmethodarecomparedtoeachother.ThetraditionalmethodstartsRULestimationatthebeginningoftherunwhereastheproposedmethodusestheinuxeventasacuetostartRULestimation.ThesecomparisonsarequantiedthroughbothMAEoftheRULestimatesandthepercentageofestimatesthatfallwithina20%condenceintervalofthetrueRULvalue.ForbothbearingsshowninFig.5.10,theRULestimatesoftheproposedmethodfallmorefrequentlywithinthe20%condenceintervalscomparedtothetraditionalmethod.Toquantify74Table5.2ComparisonofRULaccuracyfortrainingacrossalltimeversustrainingaftertheinuxeventBearingMAE(days)Within20%C.I.Full*Before*After*After**%*%**11.401.770.900.6627.65%46.89%20.991.180.870.6128.16%57.21%33.514.670.880.5713.78%52.84%410.4617.795.43.9711.53%19.42%535.1243.2216.547.730.81%0.94%*RULestimationsstartfrombeginningofrun**RULestimationsstartfrominuxpointthis,thepercentageofRULestimatesthatfallwithinthe20%condenceintervalsafterthecurrentinuxpointarecalculatedforbothcases.TheseresultsareshowninTable5.2.Foreverybearing,thepercentageofRULestimateswithinthe20%condenceintervalsishigherfortheproposedmethod.Onaverage,ourproposedmethodoffersa200%increaseinthecondenceoftheRULestimatesoverthetraditionalmethod.AlsoshowninthistablearetheMAEbetweentheestimatedandtrueRULs.Forthetradi-tionalmethod,thereismoreerrorbetweentheestimatedandtrueRULsduetothelackofusefulvariationsinvibrationdataatthebeginningofoperation.Sincethereislittleaccuracyduringthistimeperiod,thisapproachiscomputationallyinefcient.However,theRULestimationsstartingfromthebeginningshowgreateraccuracyaftertheinuxpointandtheMAEacrossallbearingsreduces(shownincolumn3ofTable5.2).Thiscorrespondstothepresenceofmoresignicantinformationfoundinthevibrationdataafterthisinuxpoint.Fortheproposedmethod,theMAEbetweenthetrueandestimatedRUListhesmallestacrossallbearings(shownincolumn4ofTable5.2),thusprovidingthehighestaccuracywhiledecreasingcomputationaltime.Onaverage,theMAEbetweenthetrueandestimatedRULsisonly2.7daysfortheproposed,whilebeing10.3daysforthetraditionalmethod.Overall,theproposedmethodbeginsmakingmoreaccurateRULpredictionsonaverageof9.95daysbeforethetraditionalmethod.75024681012141618051015x 104No. of Disch. EventsBearing 1024681012141618-10123x 10-13time (days)NMSE  NMSEthresholdFigure5.9Detectionofthecurrentdischargeinuxevent.Thetopplotshowsthenumberofdischargeeventsacrosstime.ThebottomplotshowstheNMSEbetweenthettedlineandthedatapoints,witheachpointrepresentingtheerroroverthepreviousmminutes.02468101214161802468101214161820time (days)RUL (days)Bearing 1  Start at Current Influx EventStart From BeginningTrue RUL20% Confidence Intervals(a)Bearing1024681012140246810121416time (days)RUL (days)Bearing 2  Start at Current Influx EventStart From BeginningTrue RUL20% Confidence Intervals(b)Bearing2Figure5.10RULEstimationsforBearings1and2.EachplotshowstheresultsofstartingRULestimationsfromthebeginningandfromthecurrentdischargeinuxevent.CondenceintervalsaroundthetrueRULareshowntoevaluatetheaccuracyoftheestimations.765.4Conclusions
Inthischapter,bearingfailureanditsrelationtobearingcurrentowisinvestigatedthroughanoveltestbedandacomputationalapproach.Anewtestbedwhichallowstheaccelerateddegra-dationofbearingsduetoanelectricalstressplacedonthebearingsviashaftvoltageispresented.Overthecourseoftheexperiment,temperature,vibrationandcurrentdataarecollected.Thetemperaturedatadidnotprovideanyusefulinformationrelatedtothestateofbearingfailure.However,adistinctrelationshipisobservedbetweenthenumberofcurrentdischargeeventsandtheenergyofthevibrationsovertime.Thebearingentersfailurestatedirectlyafteraninuxofbearingdischargeevents.Itisshownthattrackingthenumberofdischargeeventsovertimecanprovideanearlywarningdetectionwhichisnotavailablethroughtrackingthebearingcharacteris-ticfrequenciesinvibrationanalysis.Thisisbecausebearingcurrentscauseanincreaseinbearingvibrationandeventuallyfailure.AnimprovedRULestimationalgorithmisproposedusingthissurgeofbearingdischargeeventasacuetostartestimatingtheRULofabearing.ThisnewRULestimationalgorithmprovidesmoreaccurateresultsandrequireslesscomputationtimecomparedtoRULestimationstartingfromthebeginningofabearing'slife.Thischapteralsoprovidessomeinsightintothehiddenhealthstatesofabearingbyrelatingthecumulativedischargeswithbearingvibrations.Tobuildsufcientcollectionofhistoricaldataforbearingfaultprognosis,additionaltestsareneeded.Theadditionalcumulativeeffectsofradialandaxialloadsshouldalsobeinvestigatedtoprovideacompletepictureofallthecomponentsthatattributetobearingfailure.Futureworkalsoentailsdetectingbearingcurrentsindirectly,usingeitheranRFsystemorbearingcurrentestimationtechniquesfrommeasuredbearingvoltage.ThiswouldprovideawayforpracticallyimplementableRULestimationofbearingfaultsinindustry,preventingsystemdowntimeormotorfailureduetobearingfailure.77CHAPTER6CONCLUSIONSMotorsarewidelyusedinavarietyofapplicationsinindustry.Problemsarisewhenthesemotorsfailwithoutwarning.Sincebearingsconstitutealargeportionofthesefailurecases,muchinteresthasbeenshownovertheyearsinstudyingbearingfailure.Bearingsfailduetoanumberoffactorsincludingmechanicalstress,suchasaxialorradialloads,andelectricalstress,suchasEDMcurrents.Currently,themostwidelyusedsolutiontobearingfailureistoperformxedintervalmaintenanceandtherearenowell-knownandacceptedtechniquesforbearingfaultprognosis.Thegoalofthisworkwastoproposemethodologiesforeffectiveprognosissothatroutinemaintenanceonbearingscanbechangedtocondition-basedmaintenance.Thisiscriticalforsystemreliability,safetyandiscostefcientasitdecreasestheamountofdown-timeforthesystem.InChapter2,wereviewedseveraltestrigswhichweredesignedtoacceleratethedegradationprocessofabearing.Wealsopresentedtwoplatforms.TherstwasthePRONOSTIAplatformwhichusedradialloadstodegradebearings.Thesecondwasanexperimentalsetupweconstructedwhichusedelectricalstressviaashaftvoltagetoacceleratebearingdegradation.ThisshaftvoltageinducedEDMcurrentstoowthroughthebearing,causingdamage.Datafrombothoftheseplatformswereusedinthesubsequentchaptersforhiddenhealthstateestimationfrombearingvibrationdataandbearingfaultprognosis.InChapter3,wepresentedtwonovelmethodsforhealthstateestimationfrombearingvibra-tiondata.Therstmethodwasbasedonchange-pointdetectionandndingtransientperiodsinthedata.Thesetransientperiodscorrespondedtothetransitionaryperiodsbetweenthehiddenhealthstates.Wealsoshowedthatdifferentloadingconditionsresultedindifferenthealthstatedurationsandtransitions.Thesecondmethodutilizedastatisticalmodelingtool,thetemporalHMM,toperformunsupervisedclusteringonbearingvibrationdata.Thismethodsuggestedamorestatis-ticalapproachtoestimatingthehiddenhealthstatesandprovidedabetterunderstandingofhowbearingstransitionthroughtheirdegradation.Sincethehealthstatesofabearingarehidden,using78atemporalHMMtoestimatethemonlyseemstting.InChapter4,weusedtheEKFtoperformRULestimationforbearings.First,weintroducedTFfeaturesfrombearingvibrationdataforfaultprognosis,duetothefactthatTFfeatureshavetheabilitytoprovidemoreinformationthanthetimeandfrequencydomain.WeshowedthatRULestimationalgorithmsaremoreaccurateifthealgorithmistrainedinthemiddleofarunratherthanatthebeginning,sincethevibrationdataistoonoisytoprovideusefulfeaturesatthebeginningofarun.Wealsoshowedthatdifferentfeatureswerebettersuitedtocapturetrendsunderdifferentoperatingconditions.Forcertainoperatingconditions,entropywasabletocaptureinformationatthestart.Inthesecases,trackingtheentropyacrosstimewasabletoprovidemoreaccurateRULestimates.Thevariancewasmoresuitabletocaptureinformationattheendforanyoperatingcondition.Thissuggeststheneedforoperatingcondition-specicRULestimationalgorithms.InChapter5,westudiedtheisolatedeffectsofEDMcurrentstogainabetterunderstandingofhowelectricalstressalonecancausedamagetobearings.WealsoprovidedsomeupdatestoourRULestimationworkinChapter4.First,wechangedtheimplementationofourEKFbyusingacontinuous-timedynamicequationtomodeloursystem.Wealsopresentedanovelapproachbasedonwaveletdecompositiontodetectacurrentdischargeinuxeventfrombearingcurrentdata.Thisinuxeventoccurredbeforetherewasanysignicantchangeinvibrationdata.WeproposedtousethiscurrentdischargeinuxeventasacuetostartRULestimation,providingmoreaccuracyandefciency.
6.1FutureWork
6.1.1UsingtheHiddenHealthStatesofBearingsforEffectiveFaultPrognosisInChapter3,weproposedaframeworkwhichgroupedunlabeledbearingvibrationdataintodis-cretehealthstates.Wesawthatthesehealthstateschangedinaccordancetotheoperatingcon-ditionsthebearingwasunder.FutureworkshouldtakeastepfurtherandutilizetheseresultstoobtainaccurateRULestimations.Oneapproachtoachievethisistoperformprognosisintwo79steps.Therststepwouldbetodiagnosewhichhealthstatethebearingisinbasedonthefeaturesatthattimeinstant.Thenextstepwouldbetoestimatethetimeitwilltakeforthebearingtoreachthefollowingstate(s)untilthebearingreachesthefailurestate.Toaccomplishthis,adegradationmodelshouldbebuiltforeachstate,trackingtheevolutionofthefeaturestothenextstate.6.1.2RFDetectionofBearingDischargeEvents
Asstatedbefore,oneofthefundamentalshortcomingsofusingcurrentdataisthatdirectmeasure-mentofbearingcurrentsisnotpossibleinreal-worldapplications.Furthermore,theshaftvoltagewearticiallyinducedisnormallyunknowntotheuser.Futureworkinthisexperimentalsetupwouldbetodetectbearingdischargepulsesindirectly.Onewaytoachievethisisthroughradiofrequency(RF)detection[47,87].Eachbearingdischargeeventcontainsaniteamountofenergygivenby:Ec=12Ctotv2b;(6.1)wherevbisthebearingvoltageatthetimeinstancejustbeforethedischarge,andCtot=Crf+Cb;(6.2)whereCrfistherotor-to-framecapacitance,Cbisthebearingcapacitance.Asstatedin[6],aportionofthisenergyisradiatedoutsideofthemotorandtheseradiationscanbedetectedbyanRFantenna.Thefrequencyofbearingdischargeevents(showninFigure6.1)hasbeenfoundtobeintherangeof100-400MHz,withadurationofapproximately50ns[6,88].In[47],thedischargeeventsdetectedbytheantennawerethencountedusingaeldprogrammablegatearray(FPGA)andthebearingdischargeeventscouldbetrackedthroughoutthedurationoftherun.Infuturework,anadditiontoouracceleratedbearingdegradationplatformcouldbemadeintheformofanRFsystemtodetectthebearingcurrentpulses.Onceaccuratedetectionofbearingcurrentdischargesisobtained,thedetectedpulsescanbeusedtoreplacethemeasuredpulsesinChapter5.ThedetectedbearingpulseswouldthenbeusedasameanstotriggerRULestimation80Figure6.1BearingCurrentDischargeEvent[6].onthebearingsandwillprovideanindustryreadysolutiontotheproblemofunanticipatedbearingfailure.81BIBLIOGRAPHY82BIBLIOGRAPHY[1]G.StrangandT.Nguyen,Waveletsandlterbanks.SIAM,1996.[2]ABB,fiBearingcurrentsinmodernacdrivesystems,flinTech.Guideno.5,1999.[3]D.R.Quintero,W.Mejia,J.Roseroetal.,fiGoodpracticeforelectricdischargemachining(edm)bearingcurrentsmeasurementintheinductionmotoranddrivessystem,flinElectricMachines&DrivesConference(IEMDC),2013IEEEInternational.IEEE,2013,pp.1384Œ1390.[4]H.TischmacherandS.Gattermann,fiBearingcurrentsinconverteroperation,flinElectricalMachines(ICEM),2010XIXInternationalConferenceon.IEEE,2010,pp.1Œ8.[5]P.Nectoux,R.Gouriveau,K.Medjaher,E.Ramasso,B.Chebel-Morello,N.Zerhouni,C.Varnieretal.,fiPronostia:Anexperimentalplatformforbearingsaccelerateddegradationtests.flinConf.onPrognosticsandHealthManagement.,2012,pp.1Œ8.[6]J.Ahola,V.Sarkimaki,A.Muetze,andJ.Tamminen,fiRadio-frequency-baseddetectionofelectricaldischargemachiningbearingcurrents,flElectricPowerApplications,IET,vol.5,no.4,pp.386Œ392,2011.[7]S.Mathew,D.Das,R.Rossenberger,andM.Pecht,fiFailuremechanismsbasedprognostics,flinProc.Int.Conf.PrognosticsHealthManage.IEEE,2008,pp.1Œ6.[8]S.ChengandM.Pecht,fiAfusionprognosticsmethodforremainingusefullifepredictionofelectronicproducts,flinProc.IEEECASE.IEEE,2009,pp.102Œ107.[9]M.Pecht,Prognosticsandhealthmanagementofelectronics.WileyOnlineLibrary,2008.[10]X.Si,W.Wang,C.Hu,andD.Zhou,fiRemainingusefullifeestimationŒareviewonthesta-tisticaldatadrivenapproaches,flEuropeanJournalofOperationalResearch,vol.213,no.1,pp.1Œ14,2011.[11]E.ZioandG.Peloni,fiParticlelteringprognosticestimationoftheremainingusefullifeofnonlinearcomponents,flReliabilityEngineeringandSystemSafety,vol.96,no.3,pp.403Œ409,2011.[12]E.G.Strangas,S.Aviyente,andS.S.H.Zaidi,fiTimeŒfrequencyanalysisforefcientfaultdiagnosisandfailureprognosisforinteriorpermanent-magnetacmotors,flIEEETrans.onInd.Electron.,vol.55,no.12,pp.4191Œ4199,2008.[13]A.LebaroudandG.Clerc,fiClassicationofinductionmachinefaultsbyoptimaltimeŒfrequencyrepresentations,flIEEETrans.Ind.Electron.,vol.55,no.12,pp.4290Œ4298,2008.[14]A.Bouzida,O.Touhami,R.Ibtiouen,A.Belouchrani,M.Fadel,andA.Rezzoug,fiFaultdiagnosisinindustrialinductionmachinesthroughdiscretewavelettransform,flIEEETrans.Ind.Electron.,vol.58,no.9,pp.4385Œ4395,2011.83[15]A.Bellini,F.Filippetti,C.Tassoni,andG.-A.Capolino,fiAdvancesindiagnostictechniquesforinductionmachines,flIEEETrans.Ind.Electron.,vol.55,no.12,pp.4109Œ4126,2008.[16]S.Cheng,M.H.Azarian,andM.G.Pecht,fiSensorsystemsforprognosticsandhealthman-agement,flSensors,vol.10,no.6,pp.5774Œ5797,2010.[17]A.Heng,S.Zhang,A.C.C.Tan,andJ.Mathew,fiRotatingmachineryprognostics:Stateoftheart,challengesandopportunities,flMechanicalSystemsandSignalProcessing,vol.23,no.3,pp.724Œ739,2009.[18]M.E.OrchardandG.J.Vachtsevanos,fiAparticleltering-basedframeworkforreal-timefaultdiagnosisandfailureprognosisinaturbineengine,flinProc.Medit.Conf.onControl&Auto.IEEE,2007,pp.1Œ6.[19]B.SahaandK.Goebel,fiModelingli-ionbatterycapacitydepletioninaparticlelteringframework,flinProc.Conf.PrognosticsandHealthManagement,2009.[20]C.J.LiandH.Lee,fiGearfatiguecrackprognosisusingembeddedmodel,geardynamicmodelandfracturemechanics,flMechanicalsystemsandsignalprocessing,vol.19,no.4,pp.836Œ846,2005.[21]K.W.YuandT.A.Harris,fiNewstress-basedfatiguelifemodelforballbearings,flTribologyTransactions,vol.44,no.1,pp.11Œ18,2001.[22]A.Soualhi,G.Clerc,H.Razik,andF.Rivas,fiLong-termpredictionofbearingconditionbytheneo-fuzzyneuron,flinProc.IEEEInt.SDEMPED.IEEE,2013,pp.586Œ591.[23]A.Soualhi,H.Razik,G.Clerc,andD.D.Doan,fiPrognosisofbearingfailuresusinghid-denMarkovmodelsandtheadaptiven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