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ORIGINALARTICLEOptimalmachinetoolspindledrivegearboxdesignD.R.SalgadoF.J.AlonsoReceived4August2006/Accepted23March2007/Publishedonline4May2007SpringerVerlagLondonLimited2007AbstractManymachinetoolsareequippedwithamotorgearboxtoextendtheconstantpowerrangeofthemachinetoolspindledrivemotoratlowspeeds.Currently,inthelatestspindledrivemotortechnology,thegearboxesareintegratedinlinebetweenthewatercooledmotorandthespindleinsidethemachinetoolsram.Thefunctionalityofaspindlegearboxdependsdirectlyuponitsconstructionalsolution,andonthekineticenergycorrespondingtothissolution.Inthiswork,spindlegearboxesareoptimizedtakingthisdesignfactorintoaccount.Intheauthorsopinion,theresultscouldbeofgreatinterestforspindledrivegearboxmanufacturers.KeywordsSpindlegearboxdesign.Optimaldesign.Minimumkineticenergy.OptimizationNomenclatureσHHertzcontactstressσFBendingstressσHPAllowableHertzcontactstressσFPAllowablebendingstressσHONominalHertzcontactstressσFONominalbendingstressσHlimMaximumallowedHertzcontactstressσFlimMaximumallowedbendingstressαPressureangleFtTangentialgearforcebFacewidthβHelixangledPitchdiametermModuleKAApplicationfactorKHatrans.loadsharingfactorforpittingresistanceKHblong.loadsharingfactorforpittingresistanceKFatrans.loadsharingfactorforbendingstrengthKFblong.loadsharingfactorforbendingstrengthYFaFormfactorforbendingstrengthYNTLifefactorforbendingstrengthYRrelTRelativerugosityfactorYSaStressconcentrationfactorYSTStressconcentrationfactorYXSizefactorforbendingstrengthYdrelTNotchrelativesensitivityfactorYContactratiofactorforbendingstrengthYbHelixanglefactorforbendingstrengthZEMaterialfactorZHGeometryfactorforpittingresistanceZLViscosityfactorZNLifefactorforpittingresistanceZRRugosityfactorforpittingresistanceZVVelocityfactorZWHardnessratiofactorZXSizefactorforpittingresistanceZbHelixanglefactorforpittingresistanceZContactratiofactorforpittingresistanceKVDynamicfactorNpNumberofplanetgearsKEKineticenergyofplanetarysystemωiAngularspeedofgeariIntJAdvManufTechnol200837851–860DOI10.1007/s0017000710286D.R.SalgadoDepartmentofElectronicsandElectromechanicalEngineering,UniversityofExtremadura,Sta.TeresadeJornet38,06800Mérida,Spainemaildrsunex.esF.J.AlonsoDepartmentofElectronicsandElectromechanicalEngineering,UniversityofExtremadura,Avda.Elvass/n,06071Badajoz,Spainv4SpeedoftheplanetgearmiMassofgeariIiMomentofinertiaofgeariZiNumberofteethongearZnlToothratioofthegearpairformedbylinksnandlηEfficiencyofplanetarygeartrainspindlespeederηoOrdinaryorstationaryefficiencyofthegearpair1IntroductionThespindleisoneofthemainmechanicalcomponentsinmachiningcenters,sinceitsdesigndirectlyaffectsthefinishedqualityofworkpiecesandmachiningproductivity.Consequently,spindledesignhasbeenstudiedindepthinseveralworks1–4.Machinetoolscannotprovidehightorquesovertheirentirespeedrangeunlessthemotorisoversized.Nevertheless,therequirementsofsuchanoversizedmotortoprovideconsistenttorqueacrossthemachinetoolsspeedrangeincreasenotonlythecostofthemotorbutalsotheoperatingcostsduetohigherpowerconsumption.Additionally,themotorsweightwouldexceedtheweightofamotorgearboxcombination.Forthesereasons,spindledrivegearboxesareused.Theuseofmachinetoolspindledrivegearboxesallowsonetoextendtheconstantpowerrangeofthemachinetoolspindledrivemotors.ThisrangeextensionofpowerandtorqueisillustratedinFig.1,inwhichthedashedlinerepresentstherangeextensionwhenthereducergearboxisintroducedbetweenthemotorandthetoolaxis.Figure1arepresentsthepowerextensionandFig.1bthetorqueextensionversustheoutputspeeddatafromanindustrialmachinetool.Insum,theuseofthereducergearboxincreasestheflexibilityofmanufacturing,enablingmachinetoolstooperateat–hightorqueforhoggingoutsteelorcastironreducerspeedratioR1,R1–highspeedforfinishingcutsdirectspeedratio11Traditionally,thesegearboxesareusedinconventionalapplicationswherethemotorandgearboxaremountedoutsidetheheadstock,andthespindleisdrivenbybelts,chains,orgears.Nowadays,thegearboxesareintegratedinlinebetweenthewatercooledmotorandthespindleinsidethemachinetoolsram.Thisintegrationenablestheunittofitintotheboreofthemachinetoolramandshareinthecoolantsystem.Thiscompact,lightweightconfigurationmaximizesefficiencywhileminimizingvibrationsandnoise.Itisthedesignofthisconfigurationthatisstudiedinthepresentwork.Notethatintheinlinegearboxintegrateddesign,theoptimaldesigndependsmainlyonthedesignofthetransmissionfortherequiredspeedratioandpower.Hence,andforthesamereasonsmentionedaboveforthespindledesign,spindledrivegearboxesalsoneedtobestudiedindepthfortheirdesigntobeoptimized.Inthiscontext,themostimportantdesignfactorforspindledrivegearboxesisthatthekineticenergyofthetransmissionisminimal,foranoptimalfunctionalityduringmachiningandensuringalongworkinglife.PlanetarygeartrainsPGTsareusedforthegearboxdesignsincetheyofferaverycompactreducedweightandsizeandefficientsolutionhighspeedratiosandhighefficiencyincomparisontoordinarygeartrains.Theobjectiveofthispaperistogiveasetofoptimaldesignsofspindlegearboxesfordifferentpowersandspeed101001000100000200400600800100012001400OutputspeedrpmTorqueNm1010010001000001020304050OutputspeedrpmPowerkWRangeextensionRangeextensionabFig.1Rangeextensionofpowerandtorqueusingaspindlegearbox852IntJAdvManufTechnol200837851–860ratios.Inparticular,thetwospindlegearboxconfigurationsusedbymanufacturersarestudiedforallthemarketedrangeofpowersandspeedratios,andtheoptimaldesignsoftheseconfigurationsaregivenandcomparedforallthatrange.2ConsiderationsonthedesignofspindledrivegearboxesInthissectionweexplainsomeimportantconsiderationsthatmustbetakenintoaccountforspindlegearboxdesign.ThemembersofPGTsareofthreetypes,dependingontheirmovementsandlinkswithothermembers.Inthepresentwork,theywillbecalledsuns,arms,andplanets.TwodifferentPGTconfigurationsareusedbyspindlegearboxmanufacturers.TheyareshowninFig.2aandb.Inthesefigures,members1and2arethesuns,3isthearm,and4and4′aretheplanets.2.1EconomicandoperatingconsiderationsThespindlegearboxconfigurationofFig.2bhastheadvantageofbeingmoreinterestingeconomically,sinceitdoesnotincludearinggear.Thereasonisthatspindlegearboxgearsmustbehardened,tempered,andgroundtoavoidhighheating,andagroundringgearismoreexpensivethananongroundringgear.Also,iftheringgearisnotground,heatbuildupwilloccurmorequickly,andthisheatinglimitsandreducestheinputspeedandtorque.2.2EfficiencyconsiderationsAnotherinterestingconsiderationinspindlegearboxdesignisthatitispossibletoprovethattheefficiencyofthereducersbasedonthesetwoPGTconfigurationsisgreateriftheyaredesignedwiththeinputbeingthesunmember.ThisiswhyallspindlegearboxesaredesignedasreducerPGTswiththesunmember1asinputandthearmmember3asoutput,asshowninFig.2aandb.2.3PlanetmemberconsiderationsInspindlegearboxdesign,itisquiteimportanttochooseanoptimalnumberofplanetsfortherequiredpowerandspeedratio.Inthiscontext,thenumberofplanetmembersinaPGTNpisthenumberofthesemembersthatarearrangedaroundthePGTsprincipalaxis.Forexample,thecommercialspindlegearboxshowninFig.2chastwoplanetmembers,i.e.,Np2.Thisnumbermustbeassmallaspossibletoreducetheweightandthekineticenergyofthetransmission,whileensuringagooddistributionoftheloadtoeachoftheplanetgears.Thisnumbercanbetwo,three,four,orevenmore,dependingontheapplication.Whicheverthecase,theplanetsmustalwaysbearrangedconcentricallyaroundthePGTsprincipalaxistobalancethemassdistribution.3TheformulationofconstraintsinspindlegearboxdesignThissectiondescribestheconstraintsinspindlegearboxdesign.Theyaregroupedintothreesets,accordingtothetypeofconstraint.Theseare–Constraintsinvolvinggearsizeandgeometry–Planetarygeartrainmeshingrequirements–Contactandbendingstresses3.1ConstraintsinvolvinggearsizeandgeometryThefirstconstraintisapracticallimitationintherangeofacceptablefacewidthsb.Thisconstraintisasfollows9mC20bC20mð1Þwheremisthemodule.InputmemberOutputmemberInputmemberOutputmemberabcFig.2a,b,ConstructionalsolutionsofthePGTsusedtoextendtheconstantpowerrange.cAnexampleofspindlegearboxbasedonthePGTofFig.2bIntJAdvManufTechnol200837851–860853AllthekinematicanddynamicparametersofthetransmissiondependonthevaluesofthetoothratiosZnl,,whereZnlisthetoothratioofthegearpairformedbythelinkingmembersnandl.Inparticular,ZnlisdefinedasZnl¼ZnZlð2ÞForthedefinitionofthetoothratiostosatisfytheWillisequations,Znlmustbepositiveifthegearisexternalandnegativeifitisinternal5.ForthetrainofFig.2a,onewouldhavetotakeZ140andZ24′0.Intheory,toothratioscantakeanyvalue,butinpracticetheyarelimitedmainlyfortechnicalreasonsbecauseofthedifficultyofassemblinggearsbeyondacertainrangeoftoothratios.Inthiswork,thetoothratiosconsideredforthedesignofspindlegearboxesarequiteclosetotherecommendationofMüller6andtheAGMAnorm7.Theyare02Znl5ð3ÞC07ZnlC022ð4ÞwheretheconstraintofEq.3isforexternalgears,andthatofEq.4isforinternalgears.Inthiswayonealsoensuresthatthereexistsnointerferencebetweenthegears.Anotherconstraintthatwillbeimposedisontheratioofthediametersofthegearsconstitutingtheplanetsmembers4and4′13D4D043ð5ÞOtherrelationshipsmustbesatisfiedasaconsequenceofthegeometryofthesePGTs.Forexample,inthePGTofFig.2athetoothratiosZ14andZ24arerelatedtotheradiiofthegearsconstitutingtheplanets.Inparticular,thefollowinggeometricrelationshipmustbesatisfiedR1þR4¼R2C0R04ð6ÞExpressingtheaboveequationintermsofthemoduleofthegears,itisstraightforwardtofindthattheratioofthediametersofgears4and4′conditionsthevalueofZ14andZ24′.ThisratioisR04R4¼Z14þ1Z240jjC01ð7ÞLikewise,forthecaseofthePGTofFig.2boneobtainstheexpressionR04R4¼Z14þ1Z240þ1ð8ÞLastly,oneassumesaminimumpiniontoothnumberZminC2118ð9Þ3.2PlanetarygeartrainmeshingrequirementsThemeshingrequirementforequallyspacedplanetswiththeconfigurationsofFig.2aandbisgivenbytheAGMAnorm7,andisZ2P2C6Z1P1Np¼anintegerð10ÞwhereZ1andZ2arethenumberofteethonmembers1and2,respectively,andP1andP2arethenumeratoranddenominatoroftheirreduciblefractionequivalenttothefractionZ04C14Z4,whereZ04andZ4arethenumberofteethontheplanetgearsseeFig.2Z04Z4¼P1P23.3ContactandbendingstressesThetorquesoneachgearoftheproposedspindlegearboxdesignswerecalculatedtakingpowerlossesintoaccount.Thisaspectallowsonetoreallyoptimizethespindlegearboxdesign,unlikeoptimizationstudiesinwhichtheselossesarenotconsidered8,9.Theprocedurefordeterminingthetorquesandtheoverallefficiencyofthespindlegearboxisdescribedin10.Foreachofthegears,thefollowingconstraintsrelativetotheHertzcontactandbendingstressesmustbesatisfiedsHsHPð11ÞsFsFPð12ÞForthecalculationofthegears,theISOnormwasfollowed.ThevaluesofthestressesofEq.11andEq.12aredefinedbythisnormassH¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKAC1KVC1KHbC1KHapC1ZHC1ZEC1ZeC1ZbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFtbC1dC1uþ1urð13ÞsF¼KAC1KVC1KFbC1KFaC1FtbC1mC1YFaC1YSaC1YeC1Ybð14ÞThevaluesofσHPandσFParegivenbysHP¼sHlimC1ZNC1ZLC1ZRC1ZVC1ZWC1ZXð15ÞsFP¼sFlimC1YSTC1YNTC1YdrelTC1YRrelTC1YXð16Þ854IntJAdvManufTechnol200837851–860
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