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外文翻译--使用有限元基神经网络的机器健康检测与寿命管理 英文版【优秀】.pdf

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外文翻译--使用有限元基神经网络的机器健康检测与寿命管理 英文版【优秀】.pdf

M.J.RoemerC.HongS.H.HeslerStressTechnologyInc.,1800BrightonHenriettaTownLineRd.,Rochester,NY14623MachineHealthMonitoringandLifeManagementUsingFiniteElementBasedNeuralNetworksThispaperdemonstratesanovelapproachtoconditionbasedhealthmonitoringforrotatingmachineryusingrecentadvancesinneuralnetworktechnologyandrotordynamic,finiteelementmodeling.Adesktoprotordemonstrationrigwasusedasaproofofconcepttool.Theapproachintegratesmachinerysensormeasurementswithdetailed,rotordynamic,finiteelementmodelsthroughaneuralnetworkthatisspecificallytrainedtorespondtothemachinebeingmonitored.Theadvantageofthisapproachovercurrentmethodsliesintheuseofanadvancedneuralnetwork.Theneuralnetworkistrainedtocontaintheknowledgeofadetailedfiniteelementmodelwhoseresultsareintegratedwithsystemmeasurementstoproduceaccuratemachinefaultdiagnosticsandcomponentstresspredictions.ThistechniquetakesadvantageofrecentadvancesinneuralnetworktechnologythatenablerealtimemachinerydiagnosticsandcomponentstresspredictiontobeperformedonaPCwiththeaccuracyoffiniteelementanalysis.Theavailabilityoftherealtime,finiteelementbasedknowledgeonrotatingelementsallowsforrealtimecomponentlifepredictionaswellasaccurateandfastfaultdiagnosis.IntroductionMaximizingoperatinglifeandavailabilityofallcriticalcomponentsonrotatingmachinery,whileminimizingunplannedmaintenancedowntimeandtheriskofcatastrophicfailure,isacommongoalwithinallindustry.Thispaperdemonstratesafiniteelementbasedneuralsystemforimprovingthepresentstateoftheartinmachineryhealthmonitoringbyincreasingtheeffectivenessofstructuralcomponentdiagnosticsandmonitoring.Inparticular,neuralnetworkclassifiersweredevelopedthatoperateasahubforinformationgatheringandservedtomakeinformeddecisionsonarotorsystemshealthusingexperimentalandanalyticaldata.Thenetworkobservesthebehavioroftherotorsystembeingmonitoredtodiagnosestructuralfaultsandpredictcomponentstressesfromavarietyofpotentialfailuresources.Adesktopdemonstrationrotorwasusedasaproofconcepttool.Sensorsonthedemonstrationrigmeasurevibrationamplitudeandphaseatappropriatelocationsthroughouttherotorsystem.Fromthesemeasurements,theneuralsystemwilldiagnosefaultsandpredictrotatingmembercomponentstressesbywayofaneuralnetworktrainedextensivelyfromadetailed,rotordynamics,finiteelementmodelFEM.Currently,commerciallyavailableexpertsystemsusedforconditionmonitoringuseonlymeasuredsystemdata,withnoknowledgeofrotatingcomponentstresses.Withoutthesestressdata,calculatingremainingcomponentlifedirectlywouldbeverydifficult.Theminiaturizedrotorrigdemonstrateshowtheneuralsystemcanbeusedtoobtainbothrealtime,finiteelementmodelresultsandmachinefaultdiagnostics.Thefiniteelementmodelcapabilityisdemonstratedbyestimatingthedynamicstressesontherotatingshaftandreactionforcesonthebearings.Thediagnosisabilityofthenetworkisillustratedbypredictingthelocation,magnitude,andphaseofdiskunbalances,amountofmisalignment,degreeofrotorrubormechanicallooseness,andbearingclearanceproblems.ThedynamicstressestimationandContributedbytheInternationalGasTurbineInstituteandpresentedatthe40thInternationalGasTurbineandAeroengineCongressandExhibition,Houston,Texas,June58,1995.ManuscriptreceivedbytheInternationalGasTurbineInstituteFebruary27,1995.PaperNo.95GT243.AssociateTechnicalEditorC.J.Russo.structuraldiagnosesarebothperformedfromthevibrationmeasurementstakenfromthebearinglocations.Thispaperalsoshowstheabilityofthenetworktopredictthenonlineardynamicstressesintheshaft,whilesimultaneouslypredictingmechanicalfaults.RotorDemonstrationRigandMeasurementProcessingRotorSystemConfiguration.Adesktoprotorrigwasconstructedtodemonstratetheconceptsproposedinthispaperonactualhardware.Thedemonstrationrigwasdesignedtobeversatileenoughtoduplicatevariousvibrationproducingphenomenafoundinalltypesofrotatingsystems.Manydifferenttypesofvibrationrelatedcharacteristicswerecreatedandmeasuredbychangingrotorspeed,degreeofunbalance,degreeofmisalignment,shaftbow,shaftrub,androtorbearingclearances.Theresultingdynamiccharacteristicsaremeasuredwithproximityprobesand/oraccelerometersandareprocessedwithamultichanneldynamicsignalanalyzer.TherotorconfigurationstudiedinthispaperisshowninFig.1.Therotorsetupconsistsofthefollowingcomponents1JQHPelectricmotor.2Flexiblerubbercoupling.3Rigidsteelcouplingusercontrolledsourceofshaftmisalignment43ballbearingsand3journalbearings.52rotatingdiskswithbalanceweightholes.6gin.diameterand25in.longsteelshaft.7Motorspeedcontrollerwithclosedloopfeedback.8Variousproximityprobesandaccelerometers.9Fixturingtoproviderotorpreloads,rotorrub,andmechanicalloosenessconditions.Tworollerbearingssupportthemotorarmature,whilefouroilimpregnatedbronzesleevebearingsarepositionedbetweenthevariouscouplingsanddisks.Asolid36in.aluminumbasewithadjustablebearingpedestallocationsandrubberisolationfeetprovidesufficientrigiditytotherotorconfiguration.Motorspeedcontrolismaintainedwithaproportionalspeedfeedback830/Vol.118,OCTOBER1996TransactionsoftheASMECopyright©1996byASMEDownloaded19Mar2009to202.198.46.243.RedistributionsubjecttoASMElicenseorcopyrightseehttp//www.asme.org/terms/Terms_Use.cfmMOTORRIGIDCOUPFLEXCOUPBRG3Fig.1Rotordemonstrationrigalgorithm,withspeedsensedbyadedicatedproximityprobesandtoothedwheel.Therotorwasinitiallybalancedwithin0.05milsintwoplanesbeforeanymeasurementsweretaken.Aspeedrunuptestwasperformedtoexperimentallydeterminetherotorscriticalspeeds.Themeasuredrotorresponsefrom0to100HzisgiveninFig.2.Thefirstresonantrotormodewasidentifiedatapproximately80Hzor4800rpm.Therotorwasruncontinuouslyat40Hzinabalancedconditiontodeterminethesensitivityoftherotortochangingconditions.DataAcquisitionandDatabaseDevelopment.VibrationmeasurementsobtainedfromproximityprobesandaccelerometersweresignalconditionedandthenprocessedbyanOnoSokkiCF6400,fourchannel,digitalsignalanalyzer.Themeasuredfrequencyresponseswerethentransferredtoapersonalcomputerwherethepertinent,perrevmagnitudeandphasereadingsweredetermined.Note,theinputparameterstotheneuralnetworkclassifiersweremagnitudemilsandphasedegreesoftheIXperrevrotorspeedatalltransducerlocations.Seededfaultswereintroducedintotherotordemonstrationsystembyapplyingmassunbalancestothedisks,misalignmentacrosstherigidcoupling,looseningthebearingpedestals,andinstallingprewornbearings.Undereachoftheseconditions,measurementswereobtainedfromeachoffourproximityprobestodeterminethemagnitudeandphaseofeachtransducerwithrespecttothereferencekeyphaser.Thespecificmagnitudeandphasemeasurementswerethenloggedintoadatabasewithspecificinputoutputpairsthatareusedintheneuralnetworktrainingprocedure.Alistoftheinputoutputpairsthatareincludedinthedatabaseisgivenbelow.RotordynamicsFiniteElementModelAdetailedmodeloftherotordemonstrationsystemwasdevelopedusingadedicatedfiniteelementrotorprogramdevelopedatSTIcalledRDARotorDynamicsAnalysis.Thiscomputerprogramwasusedtosimulaterotoroperationandtotraintheneuralnetworkclassifiers.RDAisfiniteelementbased,andcontainsanarrayofpreprocessorroutinestofacilitategrid0iiiiiiiiiiiiiiiiiiiiiiiiiiiii61319.527.6344248.664.662.671788886FrequencyHzOptionsHelpT£T1HiiimiiiHiliiitMw3damHxtemw¥i4lttMtrifwpyvfflwvHMinmaHiUlmsRadialOTangentIolOfWalOErantIsometricOBockIsonetrlcOGeneraIRotaterrBadlal||ijjjOlong.Incr,Peg,OBXIOIamnFig.2Rotorresponse0100HzFig.3Calculatedfirstcriticalrotormodegeneration.Thefiniteelementbasedmodelpredictsoverallrotorvibratorycharacteristicsaswellaslocalvibratorystresslevels.Thegeneralgeometryoftherotorisprescribedtothecodeattheoutset,toallowselectionofthepreprocessorandinputinstructionstobemade.Theaddedvalueofhavingafiniteelementmodelbaseddiagnosticsystemisthatitprovidesaveryaccuratepictureoftherotorstressdistributionandreactionforces.Thesestressesandforcesarethecausesofmanyofthecomponentfailuresintherotor,bearings,seals,etc.Withtherotatingshaftcomponentstressespredicted,anautomatedlifeanalysisalgorithmwillbeabletodeterminewhattheexpectedcomponentlifewillbewithanydamagecondition.Thefiniteelementmodelofthedemonstrationrotorconfigurationwasdevelopedandcorrelatedtotheexperimentalresults.Themodelwasusedasanadditionalsourceofinformationforenhancedtrainingoftheneuralnetwork.Inparticular,thenetworkwastrainedfromthemodeltodeterminedynamicstressesandforcesincriticalmechanicalcomponentssothatitwouldbeabletocalculateremainingcomponentlifeasadiagnosticoutput.Figure3illustratesthefirstcriticalmodeassociatedwiththefiniteelementmodel.Notethecloseagreementbetweenthemeasuredandcalculatedfirstcriticalmodes.Thismodelwasusedforcalculatingdynamicstressesintheshaftandbearingreactionforcesundervariousoperatingconditionsincludingunbalancesandmisalignment.NeuralNetworkDescriptionandDevelopmentTheneuralnetworkarchitecturesdevelopedinthispaperservedasahubforinformationgathering/processingandresultedininformeddiagnosesoftheconditionofthedemorotorrigusingacombinationofexperimentalandanalyticaldata.Theinternalinterconnectionsoftheproposedneuralnetworkarchitecturesweredevelopedbasedontheamountofdatatobeprocessedbytheneuralnet.Thisisanalogoustomodelingthenumberofneuronsinthesystemsbraintobeutilizedforaparticularnetwork.Themoreneuronsusedintheentirenetwork,thelargerthesolutionspacewillbeforgeneralizingasystemsbehavior.Severalmultilayer,feedforwardnetworksweredevelopedforthisproject,utilizingthebackpropagationalgorithmforminimizingtheerrorsignals.Twoprincipalneuralnetworkarchitecturesweredevelopedinordertoexaminethesensitivityandaccuracyofdifferentnetworkdesignphilosophies.SingleNetworkArchitecture.ThesinglenetworkconfigurationdevelopedfirstutilizedfourbearingvibrationinputmeasurementsincludingmagnitudeandphaseandfunctionalJournalofEngineeringforGasTurbinesandPowerOCTOBER1996,Vol.118/831Downloaded19Mar2009to202.198.46.243.RedistributionsubjecttoASMElicenseorcopyrightseehttp//www.asme.org/terms/Terms_Use.cfmSBIRNETWORKCONFIGURATIONBRO1MAC.8RGIPHASEBRG2MAG.BRO2PHASEBRO3MAG.8RG3PHASEBRG4MAG.BRG4PHASBUNBALANCEDISKIMOOHUNBALANCEMAOUNBALANCEPHASEdffurttiUNBALANCEDISKI01Wi|UNBAUNCEMAG.grnrnjUNBALANCEPHASEdqjrwiMISALIGNMENTu100OFFSETAMOUNTmitiBENDINGSTRESSDISKIpliBENDINGSTRESSDISK2|puRADIALFORCEBRGIIt|RADIALFORCEBRG2lbIBEARINGWEAR0lOOttMECHANICALLOSSENESS0IQOWIFig.4Singleneuralnetworkarchitectureenhancementsofthesefoursensorinputstoyield24inputnodestothenetwork.Adiscussiononthepracticeofusingfunctionalenhancementstoimprovetrainingaccuraciesandtimingisgivenlater.Onehiddenlayer,consistingof24nodes,isusedtoincreasetheflexibilityofthenetwork.Hiddenlayers,whenusedproperly,canprovidemoreaccuratecorrelationbetweencomplex,linear,andnonlineartrainingpatterns.Theoutputlayerofthenetworkconsistsof14nodes.Figure4isarepresentationofthistypeofsingleneuralnetworkarchitecturewithitscorrespondinginput/outputparameters.Note,duetothespacelimitationassociatedwiththefigure,the24inputandhiddenlayernodeswerereducedtofitonthepage.Thefirstsixnodesoftheoutputlayerarededicatedtodetermining1theprobabilitythatanunbalancemayexist,2themagnitudeoftheidentifiedunbalance,and3thephaselocationoftheunbalanceontheoutofbalancedisk.Thenexttwooutputnodesdetermineifamisalignmentexistsacrosstherigidcoupling.Theprobabilityofhavingamisalignmentisdeterminedalongwiththemagnitudeoftheoffsetinmils.Fouroutputnodesofthenetworkarededicatedtovirtualsensing.Virtualsensingreferstoindirectlymeasuringaparametersuchasshaftstressorbearingforcesbymatchingpatternsofdirectlysenseddatasuchasbearingdisplacementwithafiniteelementmodeltoyieldanaccuratemeasurementoftheunmeasuredparameter.Forthedemonstrationrotorsystem,theshaftbendingstressesandbearingforcesarecalculatedusingadetailedfiniteelementmodeloftherotorforparticularrotorconditions.Theneuralnetworkisthentrainedtorecognizethesensedpatternsandrelatethemtothevaluescalculatedfromthemodel.TheresultisaneuralnetworktrainedfrommeasurementsandFEmodelthatiscapableofvirtuallysensingstressesandreactionforcesonparticularcomponentsinrealtimewithoutactuallyhavinginstalledstraingagesorforcetransducersonboard.Thelasttwonodesoftheoutputlayerdiagnosetheprobabilityofrotorruborbearingclearanceproblemsandstructuralsupportlooseness.DividedNetworkArchitecture.Adivided,multilayernetworkarchitecturewasdevelopedthatusedthesamefourbearingvibrationinputmeasurementsincludingmagnitudeandphaseasthepreviousarchitecture.However,inthiscase,thenewnetworkconfigurationwasbrokenupintosmaller,morespecializedclassifiers.AnillustrationofthisnetworkarchitectureisgiveninFig.5.Thefirstsectionofthisnewnetworkconfigurationdiagnosesthegrossfaultconditionaseither1anunbalanceondiskNo.1,2anunbalanceondiskNo.2,3amisalignmentacrosstherigidcoupling,4abearingwearorclearanceproblem,or5astructural/mechanicalloosenessproblem.Thesecondlayerutilizesthesamebearingvibrationinputstodeterminespecificlevelsofunbalanceand/ormisalignmentabouttheparticularlyidentifiedfaultaswellasgiveimportantvirtualsensinginformationaboutshaftstressesandbearingreactionforces.ThetopnetworkarchitectureinthesecondlayerdeterminesthefaultspecificswithrespecttoadiskNo.1unbalance.Theseverityoftheunbalanceisdiagnosedinthefirstoutputnode.Theseverityoutputvaluesrangefrom0of1,with1representingthemostseverecondition.Thesecondandthirdoutputnodesdeterminethemagnitudeandphaseoftheunbalanceconditionsothatcorrectiveactioncanbetakenatanytime.Theseverityoftheunbalancediagnosisiscontinuouslymonitoredandtrackedtoidentifyaworseningcondition.ThenetworkarchitectureinthesecondlayerdiagnosesanunbalanceconditionondiskNo.2.TheoutputnodespecificsareidenticaltothediagnosisnetworkassociatedwithdiskNo.1.Athirdnetworkinthesecondlayerisusedtodeterminetheseverityandmagnitudeofanymisalignmentacrossthecoupling.Severityvaluesrangebetween0and1,asinthepreviouscases,whilethemisalignmentoffsetamountisreportedinmils.Thefinalnetworkinthesecondlayerisdedicatedtovirtuallysensingmaximumshaftstressesandbearingreactionforcesfromthevibrationpatternsrecognizedatthesensorlocations.NeuralNetworkTrainingandConsultingTrainingofaneuralnetworkinvolvestheprocessofevaluatingtheweightsandthresholdsofthenumerousinterconnectionsbetweentheinputandoutputlayers.Thetrainingoftheneuralnetworkswasconductedutilizingbothunsupervisedandsupervisedprocedures.Theunsupervisedtrainingwasusedtogroupsimilarinputpatternstofacilitateprocessingofthelargenumberoftrainingpatternsused.ThesupervisedtrainingtechniqueisNEURALNETWORKCONFIGURATIONSUNBALANCEROW1SEVERITYMAGNITUDEPHASEVIRTUALSENSORSSHAFTSTRESSDISK1SHAFTSTRESSDISK2BEARING1FORCEBEARINGnFORCEFig.5Dividedneuralnetworkarchitecture832/Vol.118,OCTOBER1996TransactionsoftheASMEDownloaded19Mar2009to202.198.46.243.RedistributionsubjecttoASMElicenseorcopyrightseehttp//www.asme.org/terms/Terms_Use.cfmusedforspecifyingwhattargetoutputsshouldresultfromaninputpattern.Theneuralnetworkvariablesweightsandthresholdsarethenselfadjustedtogeneratethattargetoutput.Thecombinationofthesetwotrainingprocedureswasutilizedduringthisprojectinordertoachievethedesirablenetworkaccuracy.Oncetheinternalstructuresofthenetworkswereconstructed,theyweretrainedbasedonexperimentalcasehistoriesandanalyticallyderivedinput/outputpairsderivedfromtherotordynamicscomputermodel.Developmentofthisdatabasecontainingtheneuralnetworkinput/outputtrainingpatternsrepresentedamajorportionofthispaperseffort.UnsupervisedTraining.Givenasetoftrainingpatterns,anunsupervisedlearningalgorithmwillselforganizetheinputpatternsintogroupsofpatternscalledclusters.BasedonaEuclideandistancesimilaritymeasure,alargenumberofpatternscanbeseparatedintoseveralclusters.Duringthetrainingprocess,networkweightsandthresholdsaremodifiedandclustercentersaredetermined.Thenumberofclustersformediscontrolledbyadjustingtheclustercenterradiusvalue.Afterthetrainingprocessisfinished,thenetworkcanbeconsultedwitheitherknownorunknowninputpatterns.SupervisedTraining.Supervisedlearning,asopposedtounsupervisedlearning,utilizespairsofassociatedinput/outputpatterns.ThistechniqueiscommonlyimplementedusingaGeneralizedDeltaRulenetworkarchitecturewithbackpropagationoferror.Duringthisprocedure,thenetworkarchitectureisspecifiedintermsofthenumberofinputandoutputnodes,aswellashiddenlayernodes.Thetrainingsetisthenusedtospecifywhattargetoutputsshouldresultfromaninputpattern,andthenetworkautomaticallylearnsthesetofparametersweightsandthresholdsthatwillgeneratethisdesiredoutput.Inthislearningprocedure,thenetworklearnsasinglesetofnetworkparametersthatsatisfiesallthetraininginput/outputpairs.Thelearningisnotperfect,butisoptimumonthebasisoftheleastmeansquareerror.Intheconsultingmode,thenetworkisabletogeneralizeandgenerateappropriateoutputpatternsforanyinputpatternappliedtothenetwork.Thisattributeistheprincipaladvantagetoutilizingneuralnetworksinconditionmonitoringapplications.AnadditionalmathematicalenhancementusedinPhaseIthathelpsthenetworkarchitecturereducetheerrorassociatedwiththenumerousinput/outputpairsiscalledtheFunctionalLink.Inthisapproach,theinputpatternsareexpandedtoincludehigherordertermsassociatedwiththeoriginalinputvalues.Althoughthisenhancementisntalwaysnecessary,itoftenreducestheneedforhiddenlayersandresultsindramaticallyreducedtrainingtimes.SpecificNetworkTrainingandConsulting.Bothnetworkarchitecturesweretrainedwiththesame232input/outputtrainingpatternsdevisedfrombothexperimentalmeasurementsandthefiniteelementmodelanalysis.Thetrainingpatternsofthenetworkdatabasefocusedondiagnosingunbalanceconditions,misalignment,bearingreactionforces,andshaftstresses.Asanexample,experimentaldatawerecollectedfromtherigtotraintheneuralnetworktodistinguishthedifferencesbetweenmisalignmentandanunbalancecondition.Bothoftheseconditionsexhibitsimilarone/revvibrationcharacteristics.Phaseanglemeasurementswereobviouslyveryimportantforthenetworktomakethisdistinction.Amajorportionofthetrainingsetswerederivedinordertorecognizethedifferencesbetweensmallchangesinmagnitudeandphaseoftheappliedseededunbalanceforces.Duetothefactthatthekeyphasersignalwasonlyaccuratetowithin±10deg,changesinunbalanceforcesappliedevery22.5degwereusedasthebaseresolutionfromwhichtoidentifythelocationsoftheunbalance.Duetothefactthatunbalancemagnitudechangesof1.2gin.0.0425ozin.onlyproducedaminimalvibrationamplitudechangeof0.2mils,thisvaluewasusedasthebestresolutionpossiblewithinthepracticalconstraintsimposedbytherotorsystem.Therotordynamicsfiniteelementmodelwasexercisedextensivelywithnumerousunbalanceforceandshaftmisalignmentconditions.Theresultsfromeachrunofthefiniteelementmodeltakingapproximately\houreachyieldedsteadystateshaftbendingstressesandbearingreactionforcesforeachoftheseforcingconditions.Theseresultswerethenusedinconjunctionwiththemeasureddatatobuildthetrainingpatterndatabase.ComponentLifeAccumulationAfatiguelifealgorithmwasdevelopedthatutilizedthevirtuallysensedshaftstressesandbearingreactionforcesasabasisforcomputingfatigueinitiationlife.Thealgorithmestimatestheamountoftimetocrackinitiation,withcrackpropagationnotbeingconsidered.Neubersruleisusedtocomputethetruestressandstraininthecrackinitiationregion.Morrowsmethodisusedtoincorporatethemeanstresseffectsinthelifecalculations,whicharebasedonstrainamplitudeandthenumberofreversals.Minerslawcomputesthecumulativefatiguedamage.StrainLifeEquation.Thelocalstrainapproachwasusedtocalculatethetotalstrain,e,includingelasticandplasticcomponents,fromthegivenstressstateandthefatiguepropertiesofmaterialea}a„2Nfb/Eef{2NfYwhereEistheelasticmodulus,a„isthetruemeanstressorthetruesteadystress,andNfisthenumberofcyclesrequiredforcrackinitiation.Intherighthandsideoftheequation,thefirsttermrepresentsanelasticstrainandthesecondtermrepresentsaplasticstrain.Thisequationisthefoundationforthecyclicstrainbasedapproachtofatiguepredictionandisusuallycalledthestrainlifeequation.CycleCountingandCumulativeDamage.Underspectralloading,thedynamicstrainconditionsatcriticallocationsofacomponentmayhaveverycomplexwaveforms.Severalproceduresexisttodealwiththissituation,ofwhichtheRainfiowcyclecountingprocedureiswellknown.Simplystated,thisprocedureconsistsofdividingthecomplexwaveformintoasequenceofsimplecycles,andthencountingthenumberofstraincycleswithinagivenstrainrange.Theresultingnumberisthencomparedwiththetestedfatiguelifeofthematerialatthisstrainleveltodeterminethedegreeofincrementaldamage.ThebestknowncumulativedamageassessmentprocedureisMinerslaw,whichstatesthatthecumulativedamageisequaltothesumoftheincrementaldamageatthevariousstrainrangesThisprocedureisutilizedinthisfatiguelifealgorithm.Thenumberofcyclesn,occurringatagivenstrainlevelisfirstcomputedfromtheRainfiowcyclecountingprocedure.Thenumberofcyclestofailureateachstrainlevel,N,,isbasedontestsampledataandadjustedformeanstresseffects.Thisisobtainedfromthestrainlifeequation.Theportionofdamageatthisstrainlevelbecomesn,/TV,.ThesummationsignintheMinerslawequationindicatesthatthecumulativedamageisthesumofdamageportionduetoallexistingstrainlevels.Accordingly,thecrackinitiationisexpectedtooccurwhenthecumulativedamageisequaltoorgreaterthanunity.ShaftandBearingLifeResults.AnexampleoutputfromthefatiguelifealgorithmusedtoassesstheshaftandbearinglifeisgiveninFig.6.Theshaftfatiguelifesummaryisgivenatthetopofthepage,whilethebearingliferesultsaregivenJournalofEngineeringforGasTurbinesandPowerOCTOBER1996,Vol.118/833Downloaded19Mar2009to202.198.46.243.RedistributionsubjecttoASMElicenseorcopyrightseehttp//www.asme.org/terms/Terms_Use.cfm

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