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ORIGINALARTICLEOptimalmachinetoolspindledrivegearboxdesignD.R.Salgado&F.J.AlonsoReceived:4August2006/Accepted:23March2007/Publishedonline:4May2007#Springer-VerlagLondonLimited2007AbstractManymachinetoolsareequippedwithamotor-gearboxtoextendtheconstantpowerrangeofthemachinetoolspindledrivemotoratlowspeeds.Currently,inthelatestspindledrivemotortechnology,thegearboxesareintegratedin-linebetweenthewater-cooledmotorandthespindleinsidethemachinetoolsram.Thefunctionalityofaspindlegearboxdependsdirectlyuponitsconstructionalsolution,andonthekineticenergycorrespondingtothissolution.Inthiswork,spindlegearboxesareoptimizedtakingthisdesignfactorintoaccount.Intheauthorsopinion,theresultscouldbeofgreatinterestforspindledrivegearboxmanufacturers.KeywordsSpindlegearboxdesign.Optimaldesign.Minimumkineticenergy.OptimizationNomenclatureHHertzcontactstressFBendingstressHPAllowableHertzcontactstressFPAllowablebendingstressHONominalHertzcontactstressFONominalbendingstressHlimMaximumallowedHertzcontactstressFlimMaximumallowedbendingstressPressureangleFtTangentialgearforcebFacewidthHelixangledPitchdiametermModuleKAApplicationfactorKHatrans.loadsharingfactorforpittingresistanceKHblong.loadsharingfactorforpittingresistanceKFatrans.loadsharingfactorforbendingstrengthKFblong.loadsharingfactorforbendingstrengthYFaFormfactorforbendingstrengthYNTLifefactorforbendingstrengthYRrelTRelativerugosityfactorYSaStressconcentrationfactorYSTStressconcentrationfactorYXSizefactorforbendingstrengthYdrelTNotchrelativesensitivityfactorYContactratiofactorforbendingstrengthYbHelixanglefactorforbendingstrengthZEMaterialfactorZHGeometryfactorforpittingresistanceZLViscosityfactorZNLifefactorforpittingresistanceZRRugosityfactorforpittingresistanceZVVelocityfactorZWHardnessratiofactorZXSizefactorforpittingresistanceZbHelixanglefactorforpittingresistanceZContactratiofactorforpittingresistanceKVDynamicfactorNpNumberofplanetgearsKEKineticenergyofplanetarysystemiAngularspeedofgeariIntJAdvManufTechnol(2008)37:851860DOI10.1007/s00170-007-1028-6D.R.Salgado(*)DepartmentofElectronicsandElectromechanicalEngineering,UniversityofExtremadura,Sta.TeresadeJornet38,06800Mrida,Spaine-mail:drsunex.esF.J.AlonsoDepartmentofElectronicsandElectromechanicalEngineering,UniversityofExtremadura,Avda.Elvass/n,06071Badajoz,Spainv4SpeedoftheplanetgearmiMassofgeariIiMomentofinertiaofgeariZiNumberofteethongearZnlToothratioofthegearpairformedbylinksnandlEfficiencyofplanetarygeartrain(spindlespeeder)oOrdinaryorstationaryefficiencyofthegearpair1IntroductionThespindleisoneofthemainmechanicalcomponentsinmachiningcenters,sinceitsdesigndirectlyaffectsthefinishedqualityofworkpiecesandmachiningproductivity.Consequently,spindledesignhasbeenstudiedindepthinseveralworks14.Machinetoolscannotprovidehightorquesovertheirentirespeedrangeunlessthemotorisoversized.Nevertheless,therequirementsofsuchanoversizedmotortoprovideconsistenttorqueacrossthemachinetoolsspeedrangeincreasenotonlythecostofthemotorbutalsotheoperatingcostsduetohigherpowerconsumption.Additionally,themotorsweightwouldexceedtheweightofamotor-gearboxcombination.Forthesereasons,spindledrivegearboxesareused.Theuseofmachinetoolspindledrivegearboxesallowsonetoextendtheconstantpowerrangeofthemachinetoolspindledrivemotors.ThisrangeextensionofpowerandtorqueisillustratedinFig.1,inwhichthedashedlinerepresentstherangeextensionwhenthereducergearboxisintroducedbetweenthemotorandthetoolaxis.Figure1arepresentsthepowerextensionandFig.1bthetorqueextensionversustheoutputspeed(datafromanindustrialmachinetool).Insum,theuseofthereducergearboxincreasestheflexibilityofmanufacturing,enablingmachinetoolstooperateat:hightorqueforhoggingoutsteelorcastiron(reducerspeedratioR:1,R1)highspeedforfinishingcuts(directspeedratio1:1)Traditionally,thesegearboxesareusedinconventionalapplicationswherethemotorandgearboxaremountedoutsidetheheadstock,andthespindleisdrivenbybelts,chains,orgears.Nowadays,thegearboxesareintegratedin-linebetweenthewater-cooledmotorandthespindleinsidethemachinetoolsram.Thisintegrationenablestheunittofitintotheboreofthemachinetoolramandshareinthecoolantsystem.Thiscompact,light-weightconfigurationmaximizesefficiencywhileminimizingvibrationsandnoise.Itisthedesignofthisconfigurationthatisstudiedinthepresentwork.Notethatinthein-linegearboxintegrateddesign,theoptimaldesigndependsmainlyonthedesignofthetransmissionfortherequiredspeedratioandpower.Hence,andforthesamereasonsmentionedaboveforthespindledesign,spindledrivegearboxesalsoneedtobestudiedindepthfortheirdesigntobeoptimized.Inthiscontext,themostimportantdesignfactorforspindledrivegearboxesisthatthekineticenergyofthetransmissionisminimal,foranoptimalfunctionalityduringmachiningandensuringalongworkinglife.Planetarygeartrains(PGTs)areusedforthegearboxdesignsincetheyofferaverycompact(reducedweightandsize)andefficientsolution(highspeedratiosandhighefficiency)incomparisontoordinarygeartrains.Theobjectiveofthispaperistogiveasetofoptimaldesignsofspindlegearboxesfordifferentpowersandspeed101001000100000200400600800100012001400Outputspeed(rpm)Torque(Nm)1010010001000001020304050Outputspeed(rpm)Power(kW)RangeextensionRangeextensionabFig.1Rangeextensionofpow-erandtorqueusingaspindlegearbox852IntJAdvManufTechnol(2008)37:851860ratios.Inparticular,thetwospindlegearboxconfigurationsusedbymanufacturersarestudiedforallthemarketedrangeofpowersandspeedratios,andtheoptimaldesignsoftheseconfigurationsaregivenandcomparedforallthatrange.2ConsiderationsonthedesignofspindledrivegearboxesInthissectionweexplainsomeimportantconsiderationsthatmustbetakenintoaccountforspindlegearboxdesign.ThemembersofPGTsareofthreetypes,dependingontheirmovementsandlinkswithothermembers.Inthepresentwork,theywillbecalledsuns,arms,andplanets.TwodifferentPGTconfigurationsareusedbyspindlegearboxmanufacturers.TheyareshowninFig.2aandb.Inthesefigures,members1and2arethesuns,3isthearm,and4and4aretheplanets.2.1EconomicandoperatingconsiderationsThespindlegearboxconfigurationofFig.2bhastheadvantageofbeingmoreinterestingeconomically,sinceitdoesnotincludearinggear.Thereasonisthatspindlegearboxgearsmustbehardened,tempered,andgroundtoavoidhighheating,andagroundringgearismoreexpensivethananon-groundringgear.Also,iftheringgearisnotground,heatbuildupwilloccurmorequickly,andthisheatinglimitsandreducestheinputspeedandtorque.2.2EfficiencyconsiderationsAnotherinterestingconsiderationinspindlegearboxdesignisthatitispossibletoprovethattheefficiencyofthereducersbasedonthesetwoPGTconfigurationsisgreateriftheyaredesignedwiththeinputbeingthesunmember.ThisiswhyallspindlegearboxesaredesignedasreducerPGTswiththesun(member1)asinputandthearm(member3)asoutput,asshowninFig.2aandb.2.3PlanetmemberconsiderationsInspindlegearboxdesign,itisquiteimportanttochooseanoptimalnumberofplanetsfortherequiredpowerandspeedratio.Inthiscontext,thenumberofplanetmembersinaPGT(Np)isthenumberofthesemembersthatarearrangedaroundthePGTsprincipalaxis.Forexample,thecom-mercialspindlegearboxshowninFig.2chastwoplanetmembers,i.e.,Np=2.Thisnumbermustbeassmallaspossibletoreducetheweightandthekineticenergyofthetransmission,whileensuringagooddistributionoftheloadtoeachoftheplanetgears.Thisnumbercanbetwo,three,four,orevenmore,dependingontheapplication.Which-everthecase,theplanetsmustalwaysbearrangedconcentricallyaroundthePGTsprincipalaxistobalancethemassdistribution.3TheformulationofconstraintsinspindlegearboxdesignThissectiondescribestheconstraintsinspindlegearboxdesign.Theyaregroupedintothreesets,accordingtothetypeofconstraint.Theseare:ConstraintsinvolvinggearsizeandgeometryPlanetarygeartrainmeshingrequirementsContactandbendingstresses3.1ConstraintsinvolvinggearsizeandgeometryThefirstconstraintisapracticallimitationintherangeofacceptablefacewidthsb.Thisconstraintisasfollows:9mC20bC20m1wheremisthemodule.InputmemberOutputmemberInputmemberOutputmemberabcFig.2a,b,Constructionalso-lutionsofthePGTsusedtoextendtheconstantpowerrange.cAnexampleofspindlegearboxbasedonthePGTofFig.2bIntJAdvManufTechnol(2008)37:851860853AllthekinematicanddynamicparametersofthetransmissiondependonthevaluesofthetoothratiosZnl,whereZnlisthetoothratioofthegearpairformedbythelinkingmembersnandl.Inparticular,Znlisdefinedas:ZnlZnZl2ForthedefinitionofthetoothratiostosatisfytheWillisequations,Znlmustbepositiveifthegearisexternalandnegativeifitisinternal5.ForthetrainofFig.2a,onewouldhavetotakeZ140andZ240.Intheory,toothratioscantakeanyvalue,butinpracticetheyarelimitedmainlyfortechnicalreasonsbecauseofthedifficultyofassemblinggearsbeyondacertainrangeoftoothratios.Inthiswork,thetoothratiosconsideredforthedesignofspindlegearboxesarequiteclosetotherecommendationofMller6andtheAGMAnorm7.Theyare:0:2Znl53C07ZnlC02:24wheretheconstraintofEq.(3)isforexternalgears,andthatofEq.(4)isforinternalgears.Inthiswayonealsoensuresthatthereexistsnointerferencebetweenthegears.Anotherconstraintthatwillbeimposedisontheratioofthediametersofthegearsconstitutingtheplanets(members4and4):13D4D0435OtherrelationshipsmustbesatisfiedasaconsequenceofthegeometryofthesePGTs.Forexample,inthePGTofFig.2athetoothratiosZ14andZ24arerelatedtotheradiiofthegearsconstitutingtheplanets.Inparticular,thefollowinggeometricrelationshipmustbesatisfied:R1R4R2C0R046Expressingtheaboveequationintermsofthemoduleofthegears,itisstraightforwardtofindthattheratioofthediametersofgears4and4conditionsthevalueofZ14andZ24.Thisratiois:R04R4Z141Z240jjC017Likewise,forthecaseofthePGTofFig.2boneobtainstheexpression:R04R4Z141Z24018Lastly,oneassumesaminimumpiniontoothnumber:ZminC211893.2PlanetarygeartrainmeshingrequirementsThemeshingrequirementforequallyspacedplanetswiththeconfigurationsofFig.2aandbisgivenbytheAGM