Impulse vibration transmissibility characteristics in the presence of localized surface defects

ProcIMechEPartK:JMulti-bodyDynamics2014,Vol.228(1)62–81!IMechE2013

Reprintsandpermissions:

sagepub.co.uk/journalsPermissions.navDOI:10.1177/1464419313514572pik.sagepub.com

Impulsevibrationtransmissibilitycharacteristicsinthepresence

oflocalizedsurfacedefectsindeepgrooveballbearingsystems

JingLiu1,YiminShao1andTeikC.Lim2Abstract

Thepresentstudyanalyzestheimpulsevibrationtransmissionthroughthedeepgrooveballbearingsystemscausedbyalocalizedsurfacedefectonthebearingraceways.Theanalysisisperformedtounderstandtheuseofvibrationsignalinrotatingmachinesforconditionmonitoringanddiagnostics.Anewdynamicmodelofadeepgrooveballbearingsystemthataccountsforthecontactstiffnessbetweentheouterraceandhousing,andincludestheeffectofalocalizeddefectonracewaysisformulatedinthisstudy.Thismodelisappliedtoexaminetheeffectofhousingmaterialsonthecontactstiffnessbetweentheouterraceandhousing,andthevibrationtransmissioncharacteristicsthroughtheballbearingsystems.Aseriesofparametricstudiesisalsoperformedtounderstandtherelationshipsbetweenthematerialproper-ties,theouterrace–housingcontactstiffness,andballbearingvibrationresponse,andaswellasthevibrationtransmis-sioncharacteristicsoftheimpulsecausedbythedefectswithdifferentsizes.Thenumericalresultsdemonstratethattheproposedmodelprovidesawayforsimulatingvibrationtransmissibilitycharacteristicsthroughtheballbearingsystems,whichwaspreviouslynotpossible.Anexperimentalinvestigationisalsopresentedtovalidatetheproposedmodel.Keywords

Ballbearingdynamics,bearingdefects,bearingsupportcompliance,vibrationtransmissioncharacteristics

Datereceived:27July2013;accepted:1October2013

Introduction

Ballbearingsystemsareoneofthemaincomponentsofrotatingmachines.Thevibrationofthesemachineswillbedependentontheload,speed,andthefitbetweenouterraceandhousing.Thus,havinganin-depthunderstandingoftheeffectoftheinterfacebetweenouterraceandhousingontheresponsesofthemachinesisdesirableinconditionmonitoringanddiagnosisapplications.Inpractice,thebearingusedinthesemachinesmaygeneratelocalizeddefectssuchascracks,pits,andspallsduringitsoperationallife.1Apulseisgeneratedwhenadefectononesurfaceofthebearingcomponentsinteractswithitsmatingsurface,whichexcitesvibrationandnoiseoftherotor-bearingsystem.2,3Intheory,thevibrationandnoiseresponsescanbemonitoredtodetectanddiagnosethepresenceofthefaultsanddefectsinthebearings.3,4However,thefaultsordefectsareordinarilyinsidethemechan-icalsystemsandthesensorsaremostlymountedontheoutercasing.5Fortunately,forarotor-bearingsystem,themeasuredexternalsignalscontaininfor-mationabouttheoriginalfaultsordefectssincetheexcitationsaretransmittedthroughthebearingstruc-tures,theinterfacebetweentheouterraceand

housing,andthehousingstructure.Thus,developinganaccuratemodeltostudythevibrationtransmissioncharacteristicsofthebearing–housingsystemsisthemotivationforthepresentwork.Inthisanalysis,thefocusistoinvestigatetheeffectsofouterrace–housinginterfacesupportstiffnessandtheflexibilityofhous-ingonthevibrationtransmissionthroughtherotor-bearingsystemconsideringalocalizedsurfacedefectonthebearingraceways.Therefore,theproposedfor-mulationpresentedinthispaperwillincludethecon-tactstiffnessbetweentheouterraceandhousingtoensurethatthevibrationtransmissioncharacteristicsofthedefectsignalsthroughtherotor-bearingsystemmaybepredictedaccurately.

StateKeyLaboratoryofMechanicalTransmission,ChongqingUniversity,Chongqing,P.R.China2CollegeofEngineeringandAppliedScience,Cincinnati,UniversityofCincinnati,USA

Correspondingauthor:

YiminShao,StateKeyLaboratoryofMechanicalTransmission,ChongqingUniversity,No.174,Sha-Zheng-Jie,Sha-Ping-Ba-Qu,Chongqing400030,P.R.China.Email:ymshao@cqu.edu.cn

1Liuetal.

Manystudies6–13havereportedsuccessindetectinganddiagnosingthefaultsordefectsinrollingelementbearings.Wardle6andMeyeretal.7haveshownthatbearing-inducedfrequenciesandthoseduetodefectscanbeeasilyisolatedusingaverticalspindlerigwithsomelightpreloading.Lynaghetal.8presentedthewholerangeofprimaryandsecondarybearing-inducedfrequenciesforaspindleconsideringfreerun-ning.Thecagefrequency,ballpassfrequency,balltoinnerfrequency,andthecorrespondingside-bandfre-quencyarestudiedintheirnumericalandexperimen-talworks.JackandNandi,9OcakandLoparo,10Purushothametal.,11andShaoandNezu12alsoproposedvariousmethodsbasedontime-orfre-quency-domainsignalsfordetectinganddiagnosingthebearingdefects.Althoughmanydifferentmethodshavebeenusedfordetectionanddiagnosisofbearingdefects,modelingandsimulationmethodsprovideanincreasinglyaccurateapproachforstudyingthedynamicperformanceofsystemsthatincludeballbearings.3McFaddenandSmith14,15developedasimplemodeltostudythevibrationofrollingelementbearinghavingasingle-andmultiple-pointdefectsunderradialload.Theymodeledthepointdefectasaseriesofrepeatedimpulseswhentherollingelementstrikesthepointofdefect.SopanenandMikola16,17proposedadynamicmodelforadeepgrooveballbearingincludinglocalizedanddistributeddefects.Thedefectisdescribedusingtangentandstepfunc-tionsthatapproximateaHeavisidestepfunctionwithcubicpolynomials.However,theshaftandhousingwerenotconsideredintheirmodel.Rafsanjanietal.3developedananalyticalmodelforrollingelem-entbearinghavinglocaldefectsontheinnerrace,outerrace,androllingelement.Intheirmodel,thedefectisalsodefinedasaseriesofrepeatingimpulses.Moreover,theshaftandhousingeffectsareneglected.Sassietal.18studiedthedynamicbehaviorofaballbearingconsideringalocalizedsurfacedefect.Theyappliedtheimpactassumptionforthedefectdynam-icsanddefinedthedefectmodelasrectangularorsineimpulses.Boththeshaftandthehousingwerenottakenintoaccount.Ashtekaretal.19,20studiedtheeffectsofthesurfacedentsonHertziancontactsusingadrycontactmodelandthesuperpositionprin-ciple,respectively.NakhaeinejadandBryant21pro-posedamultibodydynamicsmodeltoinvestigatetheinfluencesofdentsandpitsoninnerrace,outerrace,andballsonthevibrationsofrollingelementbearingusingvectorbondgraphs.Thedefectsaredefinedintheirmodelbasedonthesurfaceprofilechange.TheyalsodidnotconsiderthemassoftheshaftandhousingsimilartothestudiesreportedinRefs.22,23.

TandonandChoudhury2,24presentedavibrationmodeltopredictthevibrationfrequenciesofrollingelementbearing.Intheirmodel,thedefectsarerepre-sentedasfinitewidthtriangular,rectangular,andhalf-sinefunctions.Theyconsideredtheeffectsof

63

thelumpedmasselementsofshaftandhousing.KiralandKaragulle25developedaforcemodeltosimulatetheeffectsofthenumberofdefectsanddefectlocationofracesonvibrationoftherollingelementbearing.Thehousingisconsideredbuttheshaftisignored.CaoandXiao26presentedadynamicmodelfordouble-rowsphericalrollerbearinghavinglocalizedanddistributeddefectsonracesconsideringtheinfluenceoftheshaftmass.Thepointdefectisdefinedbytheangularposition,thetangentialsize,andtheradialdepthofthedefectivefeature.Patiletal.27proposedananalyticalmodelthatdescribesthedefectasacircumferentialhalf-sinefunctiontoinvestigatetheinfluenceoflocalizeddefectonthevibrationoftheballbearing.Inthismodel,theouterraceisassumedfixedtoarigidsupportandtheinnerraceisrigidlyfixedtothemotorshaft.ArslanandAkturk28presentedashaft-bearingmodeltostudythevibrationofanangularcontactballbearingwithasingledefectontheinnerracesurface,theouterracesurface,andtheballsurface,respectively.Inthisanalysis,thedefectismodeledasseriesofrepeatingrectangularimpulses.Theycon-sideredthemassoftheshaftbuthaveneglectedthemassofthehousing.Pateletal.1developedadynamicmodeltostudyvibrationsofadeepgrooveballbear-ingsincludingsingleandmultipledefectsonsurfaceofinnerandouterraces.Thedefectsmodelisalsoconsideredasrectangularfunctionintheirmodel.Inaddition,themassesofshaftandthehousingarecon-sidered.Liuetal.29,30proposedatime-varyingdeflec-tionexcitationmodeltodescribetheimpulsecausedbyalocalizedsurfacedefectontheracesofaballbearingbasedontheshapeandsizesofthedefect;however,theeffectoftheshaftandthehousingisignoredintheirmodel.

Theliteraturereviewearlierindicatesthatthemajorityofthepreviousstudieshaveemployedvari-ousvibrationmodelsforlocallydefectiverollingelementbearingsconsideringacombinationofthemassoftheshaft,themassofthehousing,orthemasselementsoftheshaftandthehousing.However,inthesemodels,thejointinterfacebetweentheshaftandtheinnerraceisneglected,andaswellastheinterfacebetweentheouterraceandhousing.Sincetheseinterfacesdoexistinrotor-bearingsysteminpracticeandtheproducedattenuationatthejointinterfacescansignificantlyaffectthestrengthandcharacteristicofthemeasuredsignal,5neglectingthemisnotdesirable.Hence,investigatingthevibra-tiontransmissioncharacteristicsoftherotor-bearingsystemduetoimpulseexcitationgeneratedbyaloca-lizeddefectiscriticalformonitoringlocalizeddefectsinthissystem.

Ontheotherhand,avastamountofresearchstu-dieshavebeenpublishedonstudyingthevibrationtransmissioncharacteristicsoftheballbearingsystemasseeninRefs.31–34.Forabearingsys-tem,aninherentbearingvibrationproblemisits

64

variablecompliance.6,35Moreover,thereareothertwosourcesofcompliances.AccordingtotheexperimentalresultspresentedbyAinietal.,36itisshownthatotherspectralcontributions(besidesthebearingelementcontributions)appeartobethenaturalfrequenciesoftheshaft-bearingsystem,andintheircase,duetoacombinationofthehousingcomplianceandshaftflexi-bility.Fortheaboveanalysis,thebearingsupportcompliancesincludethebearing–housinganditsinter-facestiffnesswiththeouterbearingring,andtheflexi-bilityoftheshaft-bearingassembly.Therefore,theinherentbearingvariablecompliance,thecomplianceofthebearing–housinganditsinterfacestiffnesswiththeouterbearingring,andtheshaft-bearingassem-bly’sflexibilityshouldbeconsideredintheformulationofthevibrationproblemsofabearingsystem.Althoughsomeoftheanalysesconsideredtheinfluenceofthehousingstiffness,thecontactstiffnessbetweentheouterraceandhousingisusuallyignoredandtheouterraceisconsideredtoberigidlymountedinthehousing.Forinstance,White31proposedatwo-degree-of-freedom(DOF)dynamicmodeltostudythevibra-tionoftherollingelementbearingtransfercharacter-istics.Inthismodel,thehousingisconsideredasflexibleandthesupportstiffnessofthehousingismod-eled.Inanotherstudy,LimandSingh32developedasix-DOFkinematic-basedbearingstiffnessmodeltoinvestigatethevibrationtransmissionthroughrollingelementbearings.Eventhoughnumerousotherstu-dies33,34alsoconsideredtheeffectofthehousingstiff-nessonthevibrationresponseoftherollingelementbearings,theiranalysisdidnotconsiderthetransmis-sioncharacteristicsofvibrationgeneratedbylocalizeddefectinsidethebearings,andalso,thecontactstiff-nessbetweentheouterraceandhousingisoftenneg-lected.Gaoetal.37proposedavibrationmodelforacylinderrollerbearing-rotorsystem.Intheirmodel,thebearing–housingdeflectionsareconsideredasaseriesofdeflectionangles.Theydidnotprovidethecalculationmethodofthebearing–housingdefectionsorstiffness.

Theliteraturereviewshowsthatmoststudiesonthevibrationresponseofrollingelementbearingsarelimitedtoconsideringtheinfluenceofthehousingstiffness.31–34Inaddition,theinterfacebetweentheouterraceandhousingisconsideredtobeafixedconnectionasshowninFigure1(a).However,inprac-ticetheflexibilityoftheinterfaceisafunctionofthematerialsoftheouterraceandthehousingasshowninFigure1(b).Iftheinterfaceofthesetwojoiningpartsisconsideredsimplyasfixedconnection,theeffectofraceway–housinginterfacesupportstiffnesscannotbeaccountedforwhencomputingthevibra-tionresponsesofrollingelementbearings.Moreover,itcannotexplainthedifferencebetweenthevibrationresponseoftheouterraceandhousing,andalsocannotrepresentthevibratorymotiontransmittedfromtheouterraceintothehousingandothercon-nectingstructuresofthebearingsystem,especially

ProcIMechEPartK:JMulti-bodyDynamics228(1)

Figure1.Schematicmodelofinterfacebetweenouterraceandhousingwithdissimilarmaterials:(a)rigidconnection,and(b)elasticconnection.

whentheelasticmodulibetweentheouterraceandhousingareverydifferent.Itiscommonlyknownthattheseinterfacesexistinrotor-bearingsysteminprac-ticeandtheattenuationatthejointinterfacescansignificantlyaffectthestrengthandcharacteristicofthemeasuredsignal.5Thus,toinvestigateaccuratelythetransmissioncharacteristicsofpulsesignalgener-atedbylocalizeddefectsthroughtherotor-bearingsystem,theflexibilityoftheinterfacebetweentheouterraceandthehousingshouldbeconsidered.Inthisstudy,asix-DOFdynamicmodelincludingtheeffectsofthelocalizedsurfacedefectonitsraceisproposedtostudythetransmissioncharacteristicsofimpulseexcitationcausedbylocalizeddefectsinballbearingsystems.Insteadofassumingrigidconnectionbetweentheouterracewayandhousing,thecontactstiffnessbetweentheouterraceandhousingistakenintoaccountinthismodel.Themodelmakesitpos-sibletosimulatethevibrationtransmissioncharacter-isticsthroughtherotor-bearingsystemmorerealistically.Inaddition,thelocalizedsurfacedefectmodelisdescribedasahalf-sinefunctionthatdependsonthedefectsizeandtheratiooftheballsizetothedefectsize.Finally,thedevelopedmodelisemployedtoinvestigatethevibrationtransmissioncharacteris-ticsofimpulseexcitationcausedbylocalizeddefectsinballbearingsystems.Anexperimentalinvestigationisalsopresentedtovalidatetheproposedmodel.Thispaperisorganizedasfollows.Thedynamicmodelisdevelopedin‘‘Dynamicmodel’’section.Thehousingstiffnessandthecontactstiffnessbetweentheouterraceandhousingarepresentedin‘‘Supportstiffness’’section.Thenumericalresultsforadeepgrooveballbearingwithdifferentlocalizedsurfacedefectsconsideringtheeffectsofthehousingwithdif-ferentmaterialsarestudiedin‘‘Numericalresults’’section.Anexperimentalvalidationisproposedin‘‘Experimentalvalidation’’section.Finally,conclud-ingremarksaregivenin‘‘Conclusions’’section.

Liuetal.Dynamicmodel

Inthiswork,adeepgrooveballbearing(designation:

6308)isthestudybearingwhichismountedinahous-ingasshowninFigure2(a).Theformulationofthebearingsystemiscarriedoutusinglumpedmassesandsprings.AschematicdiagramofthemodeledbearingsystemunderinvestigationisillustratedinFigure2(b).TheformulationtodevelopthedynamicmodelispartlybasedontheassumptionsandconsiderationspresentedinRefs.35,38–42.Inthisstudy,thefollow-ingassumptionsandconsiderationsaremade:(a)therigidconnectionoftheinterfacebetweentheouterraceandhousingissubstitutedbytheelasticconnec-tion;(b)themassandstiffnessofthehousingstruc-tureisconsidered;(c)waviness,ballspin,andoff-sizedballsarenottakenintoaccount;(d)theshaftandinnerraceofthebearingisrigidlycon-nected;(e)theball–racecontactsareconsideredasdryelastostaticHertzian,andthus,thedeformationbetweentheballandracewayofthebearingfollowsHertzcontacttheory;(f)themassoftheballis

Figure2.Schematicdiagramofalumpedspring-masssystemandthehousing:(a)housing,and(b)lumpedspring-masssystem.

65

neglected;(g)theshaftisconsideredasrigidbody,sothedeformationoftheshaftisneglected;(h)thethrustloadingisnottakenintoaccount;and(i)thelocalizedsurfacedefectonracewaysofthebearingisincorporatedintheformulation.

Thegoverningequationsofmotionforeachcom-ponentcanbederivedinthefollowingmanner.Fortheinnerrace

minX€inþCbÀX_inÀX_outÁþXNb

Ki󰀂1i:5cos󰀃i¼Frx

i¼1

ð1Þ

minY€inþCbÀY_inÀY_outÁþXNb

Ki󰀂1i:5sin󰀃i¼Fry

i¼1

ð2Þ

Fortheouterrace

moutX

€outþKhoðXoutÀXhÞþChoÀX_outÀX_hÁÀCð3Þ

bÀX_inÀX_outÁÀXNb

Ki󰀂1i:5cos󰀃i¼0

i¼1

moutY

€outþKhoðYoutÀYhÞþChoÀY_outÀY_hÁÀCbÀNb

Y_inÀY_outÁÀXKi󰀂1:5ð4Þ

isin󰀃i¼0

i¼1

Forthehousing

mhX

€hþKhxXhþChX_hÀChoÀX_outÀX_hÁÀKð5Þ

hoðXoutÀXhÞ¼0

mhY

€hþKhyYhþChY_hÀChoÀY_outÀY_hÁÀKhoðYoutÀYhÞ¼0

ð6Þ

wheremin,mout,andmhare,respectively,themasselementsofinnerraceandshaft,outerrace,andhous-ing,andCb,Cho,andChare,respectively,thebearinginternaldamping,dampingoftheinterfacebetweentheouterraceofthebearingandthehousing,anddampingofthehousing.Thesymbols,Ki,Kho,andKhdenotethecontactstiffnessbetweentherollingelementsandraces,contactstiffnessbetweentheouterraceandhousing,andthehousingstiffness,respectively.Thedisplacementoftheinnerrace,outerrace,andhousingisdenotedbyXin,Xout,Xh,Yin,Yout,andYh.Also,FrxandFryrefertotheappliedloadsintwoorthogonalaxesontheinnerraceofthebearing.

Theexpressionsfor󰀃iand󰀂i,asappearedinequa-tions(1)to(4)aregivenasfollows

󰀃󰀄i

i¼

2Nþ!ctþ󰀃0ð7Þ

b

66

󰀂i¼ðXinÀXoutÞcos󰀃iþðYinÀYoutÞsin󰀃iÀðcþH0Þ

ð8Þwhereidescribestheithrollingelement,!cdenotestherotationalspeedofthecage,󰀃0istheinitialangularofthefirstrollingelement,andc¼cd=2ð1Àcos󰀃iÞ.1Moreover,H0istheadditionaldeflectionatthedefectoftheballwhenitpassesoverthedefectanditisdefinedbythediameteroftheballandthewidthofthedefectasfollows

H¼À

󰀄0:5DbÀðð0:5DbÞ2

Àð0:5BÞ2Þ

0:5

Á

sin0:5󰀄󰀅󰀅

ð󰀆iÀ󰀆0Þ

ð9Þ

andwhere󰀅isgivenby󰀅¼

&

arcsinðL=DoÞdefectontheouterracearcsinðL=DiÞ

defectontheinnerrace

ð10Þ

Theracewaycontactangle󰀆iisrepresentedby

8

>><2󰀄NðiÀ1Þþ!ctouterracecontactangle󰀆¼

b

i>>:2󰀄NðiÀ1Þþð!cÀ!sÞtinnerracecontactangleb

ð11Þ

Theinitialangularoffsetofthedefectofthejthballis

8>2󰀄defectontheouterrace󰀆<ðiÀ1Þþ’o0¼N>:2󰀄bNðiÀ1Þþ’idefectontheinnerrace

b

ð12Þ

where’iand’oaretheinitialangularoffsetsofthedefectofthefirstball,whichareassumedtobezerointhisstudy.

WhentheinterfacebetweentheouterraceandthehousingisconsideredasrigidconnectioninRefs.27,28,31,32,thegoverningequationsofmotionsfortheinnerraceandtheouterracecanbegivenasshowninthefollowingderivations.FortheinnerraceminX

€inþCbÀ

X_inÀX_outÁ

þXNbKi󰀂1i

:5

cos󰀃i¼Frx

i¼1

ð13Þ

minY

€inþCbÀY_inÀY_outÁþXNbKi󰀂1i:5

sin󰀃i¼Fry

i¼1

ð14Þ

Fortheouterrace

m0outX€outþKhxXoutþChX_outÀCbÀX_inÀX_outÁ

À

XNbKi󰀂1i:5

cos󰀃i¼0

ð15Þ

i¼1

ProcIMechEPartK:JMulti-bodyDynamics228(1)m0outY€outþKhyYoutþChY_outÀCbÀY_inÀY_outÁ

XNbÀ

Ki󰀂1i:5sin󰀃i¼0

ð16Þ

i¼1

wherem0outisthesumoftheouterraceandhousing

mass.

SupportstiffnessFiniteelement(FE)model

ThevariousmaterialsanalyzedinthisnumericalstudyaretabulatedinTable1.TheouterraceofthebearingismadeofbearingsteelGCr15,andthehous-ingismadefromeither42CrMoA,QT400-18AL,orZL102.43–45ThephysicalpropertiesofthesematerialsusedineachcomponentaregiveninTable2.

Inthisstudy,a6308-typedeepgrooveballbearingisstudied.TheparametersofthebearingarelistedinTable3andtheschematicmodelisgiveninFigure2(a).Inaddition,thesizeofthehousingisshowninFigure2(b)anditsthicknessisequalto23mmalongtheZ-direction.

Inordertoignoretheeffectofthehousingstiffnessonthecontactdeformationbetweentheouterraceandhousing,asubstructureFEmodelincludingtheouterraceandthehousingwithapartialpart(thethicknessis4mm)isstudied,asshowninFigure3(a).Theclearancebetweentheouterraceandthehousingischosentobezerointhisstudy.TheFEmodelisdevelopedbyacommercialsoftwareprogram.46Theouterraceandthehousingaremeshedusingthree-dimensionalsolidelements,whichisnamelyaneight-nodetetrahedronsolidelem-enttypehavingthreeDOFateachnode,whichistypicallyusedfor3Dmodelingofsolidstructures.46TheCoulombfrictionlawwithaconstantfriction

Table1.Materialsemployedinthenumericalstudy.Typeofthe

materialsOuterraceHousing1GCr1542CrMoA2GCr15QT400-18AL3

GCr15

ZL102

Table2.PhysicalpropertiesofmaterialsgiveninTable1.DensityElastic

Poisson’sMaterials(kg/m3)modulus(GPa)ratioGCr15

78302190.342CrMoA78502060.28QT400-18AL70501470.25ZL102

2670

68

0.35

Liuetal.

Table3.Bearingspecificationsandsimulationparameters.

Innerdiameter,d(mm)40Outerdiameter,D(mm)90Racewaywidth,Br(mm)23Pitchdiameter,dm(mm)65Balldiameter,Db(mm)15.081Numberofballs,Nb8Outergrooveradius,ro(mm)8.01Innergrooveradius,ri(mm)7.665Outerracewayradius,Do(mm)80.088Innerracewayradius,Di(mm)49.912Contactangle,󰀇(o)

0Materialdensity,󰀈(g/cm3)7.85Elasticmodulus,E(Gpa)210Poisson’sratio,󰀉

0.3

Figure3.FEmodel:(a)outerraceandpartialhousing,and(b)housing.

67

coefficient0.1isusedfordescribingthecontactbetweentheouterraceandthehousing.ThepartialFEmodelincludingtheouterraceandthepartialhousinghas81,792elementsand99,401nodes.Moreover,inordertocalculatethehousingstiffness,thewholehousingstructureisalsomeshedusing3Dsolidelement.AsshowninFigure3(b),thewholeFEmodelofthehousinghas76,560elementsand83,150nodes.

FEresults

InordertoverifytheFEmethodusedinthisstudy,theHertziancontacttheory47isutilized.Thesizeofthetwocylindersisasfollows:R1¼10mm,R2¼20mm,L1¼L2¼40mm.Thecenterlineofthebiggercylinderisrestricted.Theradialloadisappliedtothecenterlineofthesmallerone.Theloadisequalto500,1000,and1500N,respectively.ThecontactdeformationofthetwocylindersiscalculatedusingFEmethodandtheresultsareplottedinTable4.Moreover,theresultsarecomparedtotheresultsusingHertziancontacttheory.AspresentedinTable4,theerrorsbetweenthesetwomethodsarelessthan4.5%,whichisconsideredtobequitereasonable.

Inthissection,thehousingstiffnessandthecontactstiffnessbetweentheouterraceofthebearingandhousingareinvestigatedusingthestaticFEmethod.ForthepartialFEmodeloftheouterraceandthehousing,displacementintheX-andZ-directionsoftheinnersurfaceisrestricted,andfixeddisplacementconstraintsareappliedtotheoutsidesurfaceofthepartialhousingmodel.Thedifferentradialloadsof1000,2000,3000,4000,and5000Nareappliedtothecenteroftheouterraceofbearing,respectively.Figure4showstherelationshipbetweenthecontactdeformationandradialloads.Asshowninthefigure,thecontactdeformationvarieslinearlywiththeradialloads.AccordingtotheHooke’slaw,thecontactstiff-nessisequalto2.72Â107N/mmforthecase1,thevalueofcase2is2.04Â107N/mm,anditis1.30Â107N/mmforthecase3.

Inaddition,fortheFEmodelofthehousing,thedisplacementsatthebottomofthehousingarecon-strainedandthevariousloads(1000,2000,3000,4000,and5000N)areappliedtothecenterofthe

Table4.ComparisonofnumericalresultsbetweenHertziancontacttheoryandFEmethod.

Radialload(N)

Numericalmethod

50010001500Hertzcontacttheory(10–3mm)0.480.901.31Finiteelementmethod(10–3mm)0.460.911.35Errors,%

4.17

1.11

3.05

68Figure4.Contactdeformationbetweentheouterraceandthehousingunderdifferentradialloads(-Á-,materialtype3;-*-,materialtype2;-«-,materialtype1).

Figure5.Thedeformationofthehousingunderdifferentloads:(a)stiffnessinX-direction,and(b)stiffnessinY-direction(-Á-,materialtype3;-*-,materialtype2;-«-,materialtype1).

mountinghole.Therelationshipbetweenthedeform-ationandtheloadsalongtheX-directionisshowninFigure5(a).Thefigureshowsthattheirrelationshipsarealsolinear.Usingthesamemethod,therelation-shipalongtheY-directionisshowninFigure5(b)andthedeformationislinearwiththeloadstoo.UsingtheHooke’slaw,thehousingstiffnessiscalculated.Forthecase1,thestiffnessinX-direction(Khx1)is3.16Â106N/mmandit(Khy1)is3.70Â105N/mminY-direction.Thestiffnessofhousingwiththematerial2isgivenasfollows:Khx2¼2.24Â106N/mm,Khy2¼2.64Â105N/mm.Forcase3,thevalue

ProcIMechEPartK:JMulti-bodyDynamics228(1)

Table5.Sizeanddepthofdefects(inmm)studiedinthenumericalanalysis.TypeofthedefectsLength(L)Width(B)Depth(H)10.20.20.220.30.30.33

0.4

0.4

0.4

Figure6.Vibrationresponseforinnerraceofanormalballbearingsystemformaterialtype3:(a)X-directionacceleration,and(b)spectrumofX-directionacceleration.

isequalto1.06Â106N/mm(Khx3)and

1.22Â105N/mm(Khy3).

Numericalresults

Thedynamicresponsesofthebearingsystemareana-lyzedbyusingthefourth-orderRunge–Kuttamethod48(seeAppendix2formoredetails)withafixedtimestep.Thetimestepforthesimulationischosentobe5Â10–6s.Thenumberofstepsischosentobe1Â105.Forfastconvergence,theinitialdisplacementsaresettobex0¼10À6mand

y0¼10À6mandtheinitialvelocitiesarex

_0¼0m=sandy

_0¼0m=s.AcomputerwithtwoCPUand12GBRAMisusedtocalculatethesimulationresults.TheCPUtimeisequalto30sforeachcase.ThedampingcoefficientsCb,Cho,andCharechosento

Liuetal.69

Figure7.FrequencyspectraofaccelerationresponseintheX-directionofthenormalbearingsystem:(a)innerraceformaterialtype1,(b)innerraceformaterialtype2,(c)innerraceformaterialtype3,(d)outerraceformaterialtype1,(e)outerraceformaterialtype2,and(f)outerraceformaterialtype3(----,rigidconnectcase;—,elasticconnectcase).

be200,200,and100Nm/s,respectively.1,3,49TheexampleballbearingsystemusedinthispaperisplottedinTable3.Themassesoftheshaftandtheinnerraceare1.5and0.6kg,respectively.Themassoftheouterraceandthehousingare0.314and4.16kg,respectively.Therotorspeedis2000r/min,andtheradialloadsappliedontotheinnerraceofbearingaregivenbyFrx¼20NandFry¼0N.Also,thesizeanddepthofthethreedifferentdefectcasesemployedinthisstudyarepresentedinTable5.

Vibrationresponseofnormalballbearingsystem

Inordertoverifytheeffectivenessoftheproposeddynamicmodel,thenumericalresultsofthemodel

arecomparedtotheresultsusingtheHarris’smethod.50Thevibrationresponsefortheinnerraceoftheballbearingsystemforthematerialtype3usingthedevelopeddynamicmodelisshowninFigure6.ThefrequencyspectrumoftheaccelerationsignalisobtainedusingtheconventionalfastFouriertrans-form.Thepeakvalueisat102.2Hz.AccordingtotheHarris’smethod,whentheshaftrotatesunderthespeedof2000r/min,theballpassingfrequencyofouterrace(BPFO)is102.4Hz(BPFO¼(Nb/2)(Ns/60)(1ÀDb/dm)).ThefrequencyappearstomatchthoseinFigure6.ThesimilarresultsarealsopresentedinRefs.1–3,14,17,18,21,24,27,29,30.Hencetheresultcanprovideapartialvalidationofthepro-poseddynamicmodel.

70ProcIMechEPartK:JMulti-bodyDynamics228(1)

Figure8.Spectraofaccelerationsofnormalrotor-bearingsystemalongX-direction:(a)spectrumofaccelerationsofinnerrace,(b)spectrumofaccelerationsofouterrace,(c)spectrumofaccelerationsofhousing(—,materialtype1;-󰀄-󰀄-,materialtype2;----,materialtype3).

Time-andfrequency-domainvibrationresponsecomparison

Figure7plotsthefrequencyspectraoftheacceler-ationresponsesoftheinnerandouterracewaysalongtheX-directionforbothoftherigidandelasticconnectcases.Thefrequencyspectraoftheacceler-ationresponsesalongtheX-directionoftheinnerraceforthematerialtype1,2,and3areshowninFigure7(a)to(c),respectively.Figure7(d)and(f)displaysthefrequencyspectraoftheaccelerationresponsesalongtheX-directionoftheouterraceforthemater-ialtypes1,2,and3,respectively.AsshowninFigure7,forthematerialtypes1and2,theamplitudesofthepeaksatthe102.2Hzanditsharmonicsinthefre-quencyspectraoftheaccelerationsignalsoftheinnerraceareverysimilarbetweentherigidconnectcaseandtheelasticconnectcaseforthethreematerialtypes.Buttheamplitudesofthepeaksatthe102.2Hzanditsharmonicsinthefrequencyspectraoftheaccelerationsignalsoftheouterraceshowsomedif-ferencesforthethreematerialtypes,andthepeakvalueoftheelasticconnectcaseishigherthantherigidconnectcase.Ontheotherhand,theamplitudeoftheaccelerationresponseoftheinnerraceisslightlyinfluencedbythedifferentmaterialtypes,butfortheouterrace,theresponseincreasesasthe

housingstiffnessandthecontactstiffnessbetweentheouterraceandhousingdecreaseduetothedecreasingvalueofelasticmodulusgoingfrommaterialtypes1to3.Theseresultsimplythatthevibrationresponsesofthebearingsystemareaffectedbythehousingstiff-nessandthecontactstiffnessbetweentheouterraceandhousing.

Inordertohaveanin-depthunderstandingoftheinfluencesoftheelasticconnectcaseonthevibrationresponseofthebearingsystem,theeffectsofthehous-ingmaterialpropertiesonthevibrationresponsesofthenormalrotor-bearingsystemarediscussednext.Figure8illustratesthecomparisonofthefrequencyspectraoftheaccelerationresponsesalongtheX-directionoftheinnerrace,theouterrace,andthehousingforthethreecases.Figure8showsthatthevibrationresponsesaresignificantlyinfluencedbythematerialcharacteristicsofthehousing.Fortheinnerrace,theouterrace,andthehousing,theamplitudesofthepeaksatthe102.2Hzinthefrequencyspectraoftheaccelerationresponsearesignificantlydifferentbetweenthesethreecases.Moreover,fortheinnerandtheouterraces,theamplitudeforthecase2isthemaximumwhilethevalueformaterialtype3isminimum.Forthehousing,theresultsshowthattheamplitudesofthepeaksat102.2Hzinthefre-quencyspectraoftheaccelerationresponseofthe

Liuetal.71

Figure9.X-directionaccelerationresponseofdefectivebearingsystemwithdefectcase1onitsouterrace:(a1)innerraceformaterialstype1,(b1)outerraceformaterialstype1,(c1)housingformaterialstype1,(a2)innerraceformaterialstype2,(b2)outerraceformaterialstype2,(c2)housingformaterialstype2,(a3)innerraceformaterialstype3,(b3)outerraceformaterialstype3,and(c3)housingformaterialstype3(----,rigidconnectcase;—,elasticconnectcase).

72ProcIMechEPartK:JMulti-bodyDynamics228(1)

Figure10.FrequencyspectracorrespondingtotheX-directionaccelerationresponseinFigure9:(a1)innerraceformaterialstype1,(b1)outerraceformaterialstype1,(c1)housingformaterialstype1,(a2)innerraceformaterialstype2,(b2)outerraceformaterialstype2,(c2)housingformaterialstype2,(a3)innerraceformaterialstype3,(b3)outerraceformaterialstype3,and(c3)housingformaterialstype3(----,rigidconnectcase;—,elasticconnectcase).

Liuetal.73

Figure11.RMSvalueofX-directionaccelerationresponseofnormalbearingsystemformaterialtypes1,2,and3:(a)innerrace,(b)outerrace,and(c)housing(-*-,rigidconnectcase;-4-,elasticconnectcase).

rotor-bearingsystemincreaseowingtotheelasticmodulusdecreases.Itisbecausethesupportstiffnessofthehousingandthecontactstiffnessbetweentheouterraceandhousingdecreaseastheelasticmodulusofthehousingdecreases.

Figure9depictsthevibrationresponsesalongtheX-directionofbearingsystemwithdefectcase2(0.2mmÂ0.2mmÂ0.2mm)fortherigidandelasticconnectcasesinthetimerangefrom0.458to0.47s.TheX-directionaccelerationresponsesoftheinnerrace,outerrace,andhousingforthematerialtype1areshowninFigure9(a1toc1),respectively.Figure9(a2toc2)plotstheX-directionaccelerationresponsesofthebearingsystemforthematerialtype2.TheX-directionaccelerationresponsesoftheinnerrace,outerrace,andhousingforthematerialtype3areshowninFigure9(a3toc3).

Fortheinnerrace,thedifferencebetweentherigidandelasticconnectcasesisverysmall.However,theconnectcaseshaveasignificantinfluenceonthetimedomainwaveformofthevibrationresponsefortheouterraceandhousingofthebearingsystem.AsshowninFigure9,thelargestdifferencebetweentherigidandelasticconnectcasesisthattheelasticcon-nectcasecanbetterabletodescribetheattenuationtimedomainwaveformcausedbytheimpulseexcita-tiontransmissionthroughtheinterfacebetweentheouterraceandhousing,ascomparedtotherigidcon-nectcase.ThesimilarresultsarealsoreportedinSmith’sexperimentalresults,51whichcanprovideapartialvalidationoftheproposeddynamicmodel.Moreover,theamplitudeofaccelerationresponse

Figure12.Vibrationtransmissionratioofnormalbearingsystemformaterialtypes1,2,and3:(a)betweenouterraceandinnerrace,and(b)betweenhousingandinnerrace(-*-,rigidconnectcase;-«-,elasticconnectcase).

74ProcIMechEPartK:JMulti-bodyDynamics228(1)

Figure13.RMSvalueofX-directionaccelerationresponseofdefectivebearingsystemversusdefectsizeformaterialtypes1,2,and3:(a1)innerraceformaterialtype1,(b1)outerraceformaterialtype2,(c1)housingformaterialtype3,(a2)innerraceformaterialtype1,(b2)outerraceformaterialtype2,(c2)housingformaterialtype3,(a3)innerraceformaterialtype1,(b3)outerraceformaterialtype2,and(c3)housingformaterialtype3(-*-,rigidconnectcase;-«-,elasticconnectcase).

Liuetal.

fortheelasticconnectcaseispredictedmorethantherigidconnectcasefortheouterraceandhousing,especiallyforthematerialtype3.Onereasonisbecausethesupportstiffnessisthecontactstiffnessbetweentheouterraceandhousingfortheelasticconnectcase,whileitisthehousingstiffnessfortherigidconnectcaseandlessthanthecontactstiffness.Anotherreasonisthatthemassoftheouterraceorhousingfortheelasticcaseislessthantherigidcon-nectcasewhichfixedtheouterraceandhousing.Inaddition,thefrequencyspectracorrespondingtothetimehistoriesinFigure9areshowninFigure10.Therearealsosomedifferencesbetweenthefrequencyspectrafromtherigidconnectcaseandthosefromtheelasticcase.Theamplitudedifferencesinthefre-quencydomainaresimilartothoseinthetimedomain.Fortheinnerraceandhousing,theampli-tudeshaveslightdifferencesbetweentherigidandelasticconnectioncases.Fortheouterrace,theamp-litudeshavesignificantdifferencesforthetwoconnectcases,andtheamplitudefromtheelasticconnectcaseislargerthanthatfromtherigidconnectcase.Theaboveresultsshowthattheelasticconnectcasefortheinterfacebetweentheouterraceandthehousingshouldbeconsideredinthebearing–housingsystemmodel.

Vibrationtransmissibilitycharacteristicscomparison

Therootmeansquare(RMS)valuesofthenumericalaccelerationresponsesareextractedandappliedtocharacterizethetransientresponseandthevibrationtransmissionthroughtheinterfacesinbearingsystemwithdefectcases1,2,and3.Thefundamentalexpres-sionforRMSisexpressedasvuRMS¼uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit1X

NffiNa2

j

ð17Þ

j¼1whereNisthetotalnumberofdatapointsofsamples,anditiscalculatedusingthenumericaldataintimedurationof0.2s(from0.2to0.4s)inthisstudy.TheRMSvaluesversusthematerialpropertiesofhousingforthenormalbearingsystemareplottedinFigure11.ItshouldbenotedthatRMSvaluesoftheouterraceandhousingdecreaseasthestiffnessofhousingandstiffnessbetweentheouterraceandhousingincreasefortherigidconnectcaseandelasticconnect11,theerrors(err¼󰀈󰀈case.AsshowninFigureRMSelasticÀRMSrigid󰀈󰀈=󰀈󰀈RMSrigid󰀈

󰀈Â

100%)betweentheelasticconnectcaseandtherigidconnectcaseare2.84,0.19,and1.69%forthethreematerialtypes,respectively.TheerrorsoftheRMSvalueofouterraceare38.23,12.59,and1.38%,andfurthermore,37.36,13.18,and1.64%canbeobtainedfortheRMSvaluesofthehousing.Therefore,itcanbeseenthatthevibrationresponseofthebearing

75

Table6.ComparisonofRMSvaluesforelasticandrigidconnectcases.

Errors,%

MaterialDefectDefectDefecttype

case1case2case3Innerrace

10.010.010.4620.120.350.5032.432.331.95Outerrace

13.871.352.2425.566.747.31362.5139.0446.73Housing

11.870.620.6124.515.685.863

1.07

0.03

0.54

systemissignificantlyaffectedbytheinterfacebetweentheouterraceandhousinginbearingsystem.Thetransmissionofvibrationthroughtheinter-facesinthebearingsystemischaracterizedbythetransmissionratio󰀊,whichisdefinedastheratioofRMSvaluesoftheaccelerationresponsebetweentheinterfacesinthebearingsystemandwrittenasfollows󰀊n¼RMSn=RMSin

ð18Þ

whereRMSinistheRMSvalueoftheaccelerationresponseoftheinnerrace,andRMSndenotestheRMSvalueoftheaccelerationresponseoftheouterraceorthehousing(n¼oorharefortheouterraceorthehousing).

Figure12showsthevibrationtransmissionratioversusthehousingmaterialproperties.Itshowsthatvibrationtransmissionratiobetweentheouterraceandinnerracedecreaseswithdecreaseinthehousingstiffnessandcontactstiffnessbetweentheouterraceandhousing.Thesameeffectcanbeseeninthevibra-tiontransmissionratiobetweentheouterraceandhousing.Forthevibrationtransmissionratiobetweentheouterrace(err¼󰀈󰀈󰀊elasticÀ󰀊rigid󰀈󰀈and=󰀈󰀈inner󰀊rigid󰀈

󰀈race,theerrorsÂ100%)betweentheelasticconnectcaseandtherigidconnectcaseare34.42,12.42,and3.13%forthethreetypesofmater-ialsstudied,respectively.Forthevibrationtransmis-sionratiobetweenthehousingandinnerrace,theerrorsbetweentheelasticconnectcaseandtherigidconnectcaseare33.57,13.02,and0.05%forthethreematerialtypes.Althoughthevibrationtransmissionratioissimilarfortherigidconnectcaseandelasticconnectcaseformaterialtype3,itissignificantlyaffectedbyinterfacebetweentheouterraceandhous-inginbearingsystemforthematerialtypes1and2asshowninFigure12.

Figure13illustratestheRMSvaluesoftheaccelera-tionresponseofthedefectivebearingsystemalongtheX-directionversusthedefectsizeformaterialstypes1,2,and3.ItisobviousthattheRMSvaluesfortheinner

76ProcIMechEPartK:JMulti-bodyDynamics228(1)

Figure14.Vibrationtransmissionratioofdefectivebearingsystemwithdifferentdefectcases1,2,and3formaterialtypes1,2,and3:(a1)betweenouterraceandinnerraceformaterialtype1,(b1)betweenouterraceandinnerraceformaterialtype2,(c1)betweenouterraceandinnerraceformaterialtype3,(a2)housingandinnerraceformaterialtype1,(b2)betweenhousingandinnerraceformaterialtype2,and(c2)betweenhousingandinnerraceformaterialtype3(-*-,rigidconnectcase;-«-,elasticconnectcase).

race,outerrace,andhousingincreaseasthesizeofdefectincreasesforbothoftherigidconnectcaseandelasticconnectcase.TheerrorsoftheRMSvaluesbetweentheelasticconnectcaseandrigidconnectcasefordifferenttypesofmaterialsanddefectsarelistedinTable6.AspresentedinTable6,theerrorfortheinnerraceisminimum,butthevaluefortheouterraceismaximum.Thus,itmaybenotedthattheamplitudeoftheRMSresponsefortheouterraceandhousingissignificantlyinfluencedbytheconnectcasesandthematerialstypes.Furthermore,theamplitudeofRMSvaluesfortheinnerraceisslightlyaffectedbytheconnectcasesbetweentheouterraceandhousing,andaswellasthematerialstypeofhousing.

Table7.Comparisonofvibrationtransmissionratioforelasticandrigidconnectcases.

Errors,%

Materialtype

󰀊o

123123

Defectcase14.026.01160.241.954.900.36

Defectcase21.367.6167.850.646.472.38

Defectcase32.768.4291.390.166.862.98

󰀊h

Liuetal.77

Figure15.Experimentalsetup:(a)anoverview,(b)sensorlocation,and(c)shaftdimensions.

Figure14showsthevibrationtransmissionratioversusthesizesofthedefectforthethreetypesofmaterialsstudied.Table7givestheerrorsofthevibrationtransmissionratiobetweentheelasticandrigidconnectcases.Itmaybenotedthatthevibrationtransmissionratiobetweentheouterraceandinnerracedecreasesasthesizeofdefectincreasesforbothoftherigidconnectandelasticconnectcases.Similarinfluenceisseenforthevibrationtransmissionratioofthethreetypesofmaterials.However,asshowninFigure14,thevibrationtransmissionratioissignifi-cantlyaffectedbytheconnectcasesontheinterfacebetweentheouterraceandhousing.Inaddition,thematerialtypesofthehousingalsohaveasignificanteffectonthevibrationtransmissionratioofthebearingsystem.

Table8.Angularcontactballbearing(7008C)specifications.Innerdiameter,(mm)Outerdiameter,(mm)Racewaywidth,(mm)Pitchdiameter,(mm)Balldiameter,(mm)NumberofballsContactangle,(o)

406815547.1441825

Table9.Bearinginducedfrequencies49inHzforshaftspeedof2000r/min.fs33.33

BPFO102.40

BPFI164.27

Experimentalvalidation

Tovalidatetheproposedmodel,anexperimentissetupasshowninFigure15.TheexperimentalsystemconsistsofashaftsupportedontwoangularcontactballbearingsasshowninTable8anddrivenbyavariablespeedmotor.ThedimensionsoftheshaftareshowninFigure15(c).Twopulleysandabeltareutilizedtoconnecttheshaftandthemotor,whichcanisolatethevibrationsfromthemotor.Thetestballbearingwiththehousingisplacedonthenondrivingendoftheshaft.Theexperimentisperformedatsshaftspeedof2000r/min(33.33Hz)witha300Nradialloadontheouterrace.Theshaftspeedfrequency(fs),BPFO,andballpassing

frequencyinnerrace(BPFI)arelistedinTable9.ThedatacollectionsystemconsistsofaKistleraccel-erometerwithasensitivityof100mV/g,aLMSdatacollectionsystem(LMSInternational),andacom-puter.Theaccelerometerismountedonthehousingtomeasurethevibrationsfromthetestbearing.Thesignalsaresampledat10.24kHzwithasamplingtimeof60s.BasedonthesignalprocessingmethodinRef.30,alow-passfilterof2kHzisusedtoremoveunde-siredhighfrequencynoise,andthecut-offandalias-ingfrequenciesarechosentobe4kHz.

78Figure16.ComparisonofthefrequencyspectraoftheaccelerationsintheX-directionfromthesimulationandexperiment:(a)simulationresultsofthenormalballbearing,and(b)experimentalresultsofthenormalballbearing.

Becausetheamplitudeofvibrationissomewhatdampedbythevibrationsfromthetestbearingsystem,theamplitudeofthevibrationfromtheexperimentalresultcanbedifferentfromthesimula-tionresult.Therefore,onlytheBPFOinthefrequencyspectrumfromtheexperimentalresultiscomparedwiththatfromthesimulationresult.Figure16showsthefrequencyspectraoftheaccelerationsfromtheexperimentandsimulationforthehealthybearing,whichareobtainedfromtheaccelerationsusingthesignalprocessingmethodinRef.30.ThedifferencebetweentheBPFOofthehealthybearingfromthesimulationandexperimentis–0.1%,asshowninFigure16,whichcanvalidatetheproposedmodel.Inaddition,fortheexperimentalresults,theshaftfrequencyandtheBPFIarealsoshowninFigure16(b).ThesimilarexperimentalresultsaregivenbyLynaghetal.8Thus,itseemsthattheaboveresultsarevalidinthisstudy.

Conclusions

ThisstudyemploystheFEapproachtoinvestigatetheeffectsofhousingstructurematerialsonitsstiff-nessandonthecontactstiffnessbetweentheouterraceandhousing.Inaddition,adynamicmodelof

ProcIMechEPartK:JMulti-bodyDynamics228(1)

thebearingsystemisdevelopedtoanalyzetheeffectofcontactstiffnessbetweentheouterraceandhousingonthevibrationresponseofnormalanddefectivebearingsystems.ExperimentalinvestigationvalidatedthemodelpredictionoftheBPFO.Thefol-lowingspecificconclusionsofthepresentstudyareobtained:

1.Thetime-domainwaveformofthepulseisinflu-encedgreatlybythecontactstiffnessbetweentheouterraceandhousing.Also,thefluctuationofthetime-domainpulsewaveformfortheelasticconnectcaseismorethantherigidconnectcase.Moreover,thepulsewaveformfromtheinnerracetohousingwillbechangedduetotheeffectsoftheinterfacebetweentheouterraceandhousing.Theamplitudeofthevibrationspectrumisalsoinflu-encedbytheinterfacebetweentheouterraceandhousing.However,thereisnodriftinbearing-inducedfrequencycontent.

2.Forthethreetypesofmaterialsstudied,themax-imumdifferencesofRMSvaluesoftheacceler-ationresponsesbetweentherigidconnectcaseandelasticconnectcaseare2.84,38.23,and37.36%fortheinnerrace,outerrace,andhousingofthenormalbearingsystem,respectively;themaximumdifferenceofthevibrationtransmissionratiosbetweentheouterraceandinnerraceis34.42%forthenormalbearingsystem,thatbetweenthehousingandinnerraceis33.57%;themaximumdifferencesofRMSvaluesoftheaccelerationresponsesbetweentherigidconnectcaseandelasticconnectcaseare2.43,62.51,and5.86%fortheinnerrace,outerrace,andhousingofthedefectivebearingsystemwiththethreedefectcases,respectively;themaximumdifferenceofthevibrationtransmissionratiosbetweentheouterraceandinnerraceis160.24%forthedefectivebearingsystemwiththethreedefectcases,thatbetweenthehousingandinnerraceis6.86%.

3.Thecontactstiffnessbetweentheouterraceand

housinghassignificantinfluenceonthevibrationresponseandthevibrationtransmissionratioontheinterfacesofthebearingsystem.Furthermore,theeffectofthecontactstiffnessbetweentheouterraceandhousingonthevibrationresponseoftheouterraceandhousingismoresignificantthantheinnerraceofthebearing.Therefore,itismoreaccuratetoconsidertheelasticconnectionbetweentheouterraceandhousingindynamicmodelofthebearingsystemforsimulationofthevibrationresponse.

4.Theamplitudeofthevibrationresponseofthe

bearingsystemisaffectedbythehousingmaterialproperties.Thus,thedeliberateuseofspecifichousingmaterialsisconstructiveinreducingthevibrationresponseofthebearingsupportsystemjustlikegearbox.

Liuetal.

Ourfutureworkwillextendthismodeltosimulatetheenergytransmissionandlosscharacteristicsthroughrollingelementbearingsandincludetheeffectsofinterfacefactorssuchasroughness,materialcharacteristics,contactstates,etc.Funding

Thisresearchreceivednospecificgrantfromanyfundingagencyinthepublic,commercial,ornot-for-profitsectors.

Acknowledgement

TheauthorsaregratefulforthefinancialsupportprovidedbytheNationalNaturalScienceKeyFoundationofChinaunderContractNo.51035008.

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Notation

aiBBrcdCbChoChddmDDbDoDiE

Frx,FryHH0ii0jj0

Khx,KhyKhoKiL

L1,L2minmoutm0outmhNbNroriR1,R2RMSelasticRMSrigidt

Xin,YinXout,YoutXh,Yhaccelerationdata(m/s2)widthofthedefect(m)racewaywidth(mm)

internalradialclearance(mm)

dampingfactorofbearing(Ns/m)dampingfactorofinterfacebetweenouterraceandhousing(Ns/m)dampingfactorofhousing(Ns/m)innerdiameter(mm)pitchdiameter(mm)outerdiameter(mm)balldiameter(mm)

outerracewayradius(mm)innerracewayradius(mm)elasticmodulus(GPa)radialload(N)defectdepth(mm)

additionaldeflection(m)ithball

i’thfunction

jthdatapointofthevibrationsignalj’thtimestep

stiffnessofhousing(N/mm)contactstiffness(N/mm)Hertziancoefficient(N/m3/2)lengthofthedefect(m)cylinderlength(mm)massofinnerrace(kg)massofouterrace(kg)

totalmassofouterraceandhousing(kg)

massofhousing(kg)numberofballs

numberofsignaldataoutergrooveradius(mm)innergrooveradius(mm)cylinderradius(mm)

RMSvalueofelasticconnectcase(g)RMSvalueofrigidconnectcase(g)Time(s)

DisplacementofinnerraceinX-andY-direction(m)

DisplacementofoutraceinX-andY-direction(m)

DisplacementofhousinginX-andY-direction(m)

Liuetal.yi’Differentialequations󰀇Contactangle(degree)

󰀂iContactdeformationofithball(m)󰀊nVibrationtransmissionratioofRMSvalvesbetweeninterfacesinbearingsystem

󰀊elasticVibrationtransmissionratioofelasticconnectcase

󰀊rigidVibrationtransmissionratioofelasticconnectcase

󰀆iContactangleoftheithball(rad)

󰀆0Initialangularoffsetofthedefectoftheithball(rad)󰀉Poisson’sratio

󰀃0Initialangularoffirstballelement(degree)

󰀃iAngularpositionofithball(rad)’iInitialangularoffsetsofinnerracedefectoffirstball(rad)

’oInitialangularoffsetsofouterracedefectoffirstball(rad)!cSpeedofthecage(rad/s)!sSpeedoftheshaft(rad/s)

Appendix2

Fourth-orderRunge–Kuttamethod

Fordifferentialequations,whicharegivenby

y

_1¼f1ðt,y1ðtÞ,y2ðtÞ,...,ynðtÞÞy_2¼f2ðt,y1ðtÞ,y2ðtÞ,...,ynðtÞÞÁÁÁÁÁÁ

ð19Þ

y

_n¼fnðt,y1ðtÞ,y2ðtÞ,...,ynðtÞÞy1ðt0Þ¼y10,y2ðt0Þ¼y20,...,ynðt0Þ¼yn0

81

Thetimestepisassumedtobeh.Whenthevalueofyi0,j0ði0¼1,2,...,nÞatt¼tj0isagivenvalue,accord-ingtofourth-orderRunge–Kuttamethod,thevalueofyi0,j0þ1att¼tj0þ1canbegivenby

y1

i0,j0þ1¼yi0,j0þ6ðki01þki02þki03þki04Þ

ki01¼hfi0À󰀄tj0,y1j0,y2j0,...,ynj0

Á

khki02¼hfi0tj0þ,y1j0þ

11󰀅

,yþk2j021,...,yknj0þn1

󰀄2222khki03¼hfi0tj0þ,y1j0þ122,yk2j0þ22,...,yþknj0n2

󰀅

ki04¼hfi0À2

22tj0þh,y1j0þk13,y2j0þk23,...,ynj0þkn3

Á

ð20Þ

wherej0isthej0thtimestep.Theconvergencecriterionis

max󰀈

󰀈󰀈yh=2󰀈i0,j0þ1Àyh

i0,j0þ1󰀈󰀈5\"

ð21Þ

where\"isasmallparameterforerrortolerance.