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12thIFToMMWorldCongress,Besançon(France),June18-21,2007RevealingofIndependentOscillationsinPlanetaryReducerGearowingtoitssymmetryL.Banakh*Yu.Fedoseev+MechanicalEngineeringResearchInstituteofRussianAcademyofSciencesMoscow,RussiaAbstract-Theplanetaryreducer11gearisasymmetricsystem.Foritsoscillationanalysisthereisappliedthesymmetrygrouprepresentationtheory,whichwasgeneralizedformechanicalsystems.Itwasfoundthatduetoreducersymmetrytheoscillationsdecompositionhasarisen.Thereareindependentoscillationsclasses,suchas:angularoscillationsofsolargearandepicycle-satellitesoscillationsinphase；transversaloscillationsofsolargearandepicycle-satellitesoscillationsinantiphase.Solargearandepicycleoscillationsinaphasedonotdependonangularsatellitesoscillations.Keywords:planetaryreducer,symmetry,grouprepresentationtheory,independentoscillationsI.IntroductionItiswellknownthatattheoperationofplanetaryreducerthereareoscillationsofitselements,suchassolargear,epicycleandsatellites.Thisfactoressentiallyworsensaqualityofreduceroperation,andinsomecasescanresultintheircurvatureandbreakage.Aplentyofpapersaredevotedtothedynamicanalysisofgearreducers1.Basicallytherearecomputationalresearches.Inthegivenpapertheanalyticalapproachesforinvestigationofreducerdynamicsispresented.Theplanetaryreducerhasahighdegreeofsymmetry.Sothispropertywasusedandthegrouprepresentationtheorywasapplied.Applicationofthistheoryallowscarryingoutdeepenoughdynamicanalysis,usingsymmetrypropertiesonly.Forthispurposeitisnecessarytohavethedynamicalmodelwhichistakingintoaccountstiffnesscharacteristicsinlinkagesbetweenreducerelements.Themathematicalapparatusofthesymmetrygroupsrepresentationtheoryiswidelyusedinthequantummechanics,crystallographic,spectroscopy2,3,4.Theadvantagesofthisapproacharedifficultforoverestimating.Withitshelpitispossibletodefinewithexhaustivecompletenessthedynamicproperties,usingstructuresymmetryofsystemonlywithoutsolvingofmotionequations.Howeverintheclassicalmechanicsthisapproachisnotwidelyused.Itisresultfromsomeparticularfeaturesofmechanicalsystems.First,thereisan*E-mail:banlinbox.ruavailabilityofsolidswith6-thdegreesoffreedom.Itisuncleartowhatsymmetrygrouptorelateasolidinorderthatsystemsymmetrymayberetained.Secondatrealdesignsmaybetechnologicalerrorsandmistakesatassembly,sothereisasmallasymmetryandthesystembecomesquasisymmetricFurtherthemechanicalsystemsconsistfromvarioussubsystemswithvarioussymmetrygroups.Inthisconnectionitisnecessarytohavemethodsfortheanalysisassymmetricandquasisymmetricmechanicalsystemsconsistingofvarioussubsystemsandsolids.Havingmadesomegeneralizations,thismathematicalapparatusformechanicalsystemsmaybeused.Forthispurposeweproposetoapplythegeneralizedprojectiveoperators5.Theseoperatorsarematrixesoftheappropriateorderinsteadofscalarasinphysics.Theuseofgeneralizedprojectiveoperatorsallowstakingintoaccountallabovementionedfeaturesofmechanicalsystems.Theapplicationoftheseoperatorstoinitialstiffnessmatrixleadstoitsdecompositiononindependentblockseachofthemcorrespondstoownoscillationclassinindependentsubspaces.Toaccountforthesolidssymmetry«theequivalentpoints»wereentered:thesepointsarechosenonsolidsothattheirdisplacementswerecompatibletoconnectionsandcorrespondedtogroupofsymmetryofallsystem.Theseoperatorsenablealsomaybeappliedwiththefiniteelementsmodels(FEM).II.Dynamicmodelofplanetaryreducer.Stiffnessmatrix.Themodelofaplanetaryreducerstepissubmittedonfig.16.ThestepconsistsfromsolargearS,itsmassandradiusareequalto11,mr.ItengagesintomeshwiththreesatellitesSti(i=1,2,3)(itsmassesandradiusareidenticalandequalto22,mr).SatellitesinturnareengagedintomeshwithepicycleEp(33,mr)andtheyarefastenoncarrierbyelasticsupportwithrigidityh6.Therigidityofgearingsolargear-satellitesisequalto1h,thegearingepicycle-satellitesis3h,isangleofgearing.12thIFToMMWorldCongress,Besançon(France),June18-21,2007Fig.1Planetaryreducerstep.S-solargear,-epicycle,1,2,3satellites(St).Letsconsideralloveragaintheplaneoscillationsofplanetaryreducerstep:transversal(x,y)andangular()oscillations(withoutthecasing).AstiffnessmatrixmayberepresentedinablockviewK=123123123SSStSStSStStStStStStStKKKKKKKKKKK(1)Hereonthemaindiagonaltherearethestiffnesssubmatrixes(3x3)forappropriateelements,andoutsideofthemaindiagonaltherearestiffnesssubmatrixesofconnectionbetweentheseelements.Thereare15generalizedcoordinates:X=(*,；,SSSEpEpEpxyxy；1113,.StStStStxy)TheconcreteviewoftheseblocksissubmittedinAppendix.Thus,thetotalorderofmatrixKis(15x15).AninertiamatrixMisdiagonal.III.Introductionofequivalentpointsindynamicmodel.Operatorsofsymmetry.Byvirtueofsymmetryofsatellitesfasteningthissubsystemhassymmetrysuchas3C(astriangle).Torevealsymmetry3CatmovingofsolargearSandepicycleEpweshallenterthecoordinates123,lllonsolargearSinpointsofsatellitesfastening(fig.2.).Fig.2EquivalentpointsonsolargearS.1,2,3,-satellitesTheyare“equivalentpoints”.Theircoordinatesare:1111222133312(1)cos3；2(1)sin31,2,3.SSiSSiSSirXriripipi=(2)orinmatrixformL=AXAndanaloguesrelationsfor“equivalentpoints”onepicycle,butinstead1rin(2)mustbewritten3r.Andlateronthesecoordinateofsolargearandepicyclewillbeusedinstead(x,y)and().Afterthatitispossibletocount,thatallcoordinatesofsystemshouldvaryaccordingtosymmetrygroup3Cand,hence,itispossibletoapplytheprojectiveoperatorofsymmetrytoallsystemelements:S,Ep,andalsotothreesatellitesSti(i=1,2,3).(fig.3)Theortho-normalprojectiveoperatorgofsymmetryforpointgroup3Cisknownas2.Itisg=11133312166611022(3)Forthewholesystemtheprojectiveoperatormustberepresentedasblock-diagonalmatrix12thIFToMMWorldCongress,Besançon(France),June18-21,2007G=Stggg(4)HereeachsubmatrixcorrespondstoS,Ep,andalsotothreesatellitesSti(i=1,2,3).SoBecausewehavethreeidenticalsatellitesandeachofthemhas3degreesoffreedom(,iiStStxyandangular.iiStSt),thereforeitisnecessarytoentergeneralizeoperator(3)3,4andtoconsiderStgasblockmatrixwheretheeachelementisdiagonalmatrix(33),thatisitispossibletopresenteachelementasStg=1,11=EgEEEThustoinitialcoordinates,(,)SEpxyofsolargearandepicycleconsistentlytwotransformationsareapplied:AandG.AndresultingtransformationofaninitialmatrixKequalstoproductofoperatorsGA.ThisorthogonaltransformationanditlookslikeG=StgAgAg,wheregA=223300100+Byapplyingofthistransformationtomatrix(1),weshallreceive*=(G)()(G)trSothecorrespondingtransformationsofcoordinatesandforcesareX*=(G),F*=(G)trF(5)AsaresulttheinitialmatrixK(1515)isdividedon3independentblocks(5x5)and,lookinglike,*(1)*(2)=IIIIIKKKK(6)TheinertiamatrixMremainsdiagonalbecausematrixGAisorthogonal；thereforetheindependenceofoscillationclassesdefinesmatrix*only.IV.RevealingofindependentmotionsclassesatfornaturalandforcedoscillationsA.NaturaloscillationsFromtheviewofmatrix(6)itisseen,thatowingtosystemsymmetrythereisadecompositionofinitialmatrixK,and,hence,divisionofoscillationclassesandaswellasspaceofparameters.Theconcreterelationsforsubmatrixesin(6)showthattherearefollowingindependentoscillationsclasses:I-stclass(subspaceI-submatrix*IK):angularoscillationofsolargearandepicycle+oscillationsofsatellitesinaphase.Dimensionofthissubspaceisequalto5.Itsdeterminingparametersare:12313612139,.rrrhhhhrh2-ndclass(subspaceII-submatrixes*(1)IIK(2)*IIK):transversaloscillationsofsolargearandepicycle+oscillationsofsatellitesinanantiphase.SubspaceIIbreaksuptotwoidenticalsubmatrixes*(1)IIKand(2)*IIK(55).Itmeansthatinsystemthereare5equalfrequencies.Itsdeterminingparametersare:213679,.rhhhhhThus,takingintoaccountonlypropertiesofsymmetryitispossibletoreceivedeepenoughanalysisofdynamicpropertiesofsystemofaplanetaryreducer.Besidesitispossibletosimplifyalsoprocessofsystemoptimization.B.ForcedoscillationsAttheforcedoscillationstheuseoftheindependentoscillationclassesissuitableonlyintwocases:a)ifthepointsofapplicationoftheexternalforceshavethesametypeofsymmetry,asadesignhas,orb)iftheyaredisposedaccordingtotheindependentclassesofoscillations.Really,thentransformation(5)bringaforcesvectorF*intoaformcontainingzeroelementsorin1-st,or2-thsubspaces.Theanalysisoftherealloadingsforcesonareducer,shows,thatitisvalidifelementsdisbalancesarethesame:)identicalsatellitesdisbalances+disbalanceofepicycle；)identicalsatellitesdisbalances+disbalanceofsolargear.V.Thefurthermotionsdecomposition.ThefurtherdecompositionofsubspacesIandIIin(6)ispossibleonlyifthereareadditionalconditionsraisingatypeofsystemsymmetry.12thIFToMMWorldCongress,Besançon(France),June18-21,2007Theseconditions,inparticular,canbereceivedfromsimilaritysymmetryofsolargearandepicycle.Theylooklike:1.EqualityofgearingstiffnesswithSandEp,i.e.12hh=,2.EqualityofpartialfrequenciesforangularmotionsSandEp()()SEp=,whence:78hh=,or3.EqualityofpartialfrequenciesattranversalmotionsofSandEp(,)(,)SEpxyxy=,whence:h7=2h9.Sobyfulfillmentofconditions1,2(or1,3)theadditionalsymmetrytype2Cisappeared(reflectionsymmetry).Tothissymmetrygrouptheoperator2G(or2G)iscorresponded2G=11311113；1311rhrh2G=111113213112hhTheapplicationoftheseoperatorstomatrixK*permittoachievethefurtherdecompositionofcorrespondingmatrixesandappropriatemotions.Reallytheymayhavesymmetricandantisymmetricoscillationclassesforsolargearandepicycle.Thusthecoordinatetransformationis:111*3131111*3131SEprrhSEprrh=+=XXXXXXAnd1*3121*312SEphSEph=+=XXXXXXBythiscoordinatetransformationthefollowingindependentmotiontypesarearisen()*()IIIKKKTheconcreterelationsforthesesubmatrixesshowthattherearefollowingindependentoscillationsclasses:Isubspace(matrix*IK):-angularoscillationsofsolargearSandepicycleEpinphase+satellitesStioscillationalongaxis*xinphase),IIsubspace(matrix*IK):-angularoscillationsofsolargearSandepicycleEpinantiphase+satellitesStioscillationalongaxis*yandaroundaxisinphase.SimilarlyoccursdecompositionofsubspacesIIandmatrixes*(1)IIK,(2)*IIKbutinsteadofangularoscillationsSandEptherearetheirtransversaloscillationalonganaxisx*(ory*),andoscillationofsatellitesinanantiphase.Asshowstheanalysisofmatrix*IKtheoscillationsofSandEpinaphasedonotdependonangularoscillationofsatellites(St).Fromanalysisof*IKand*IIKwenotice,thatath7=h8=h6=0azerorootmaybearisen.Thisoscillationtypemeansthefreeoscillationsofsatellites("navigation")atangular(ortranslation)oscillationssolargearandepicycle.A.Forcedoscillations.Thechoiceofexitedforcesaccordingoneoftheseoscillationtypesdonotinducetheotheroscillationtypesbecausetheyareorthogonaleachother.TheexcitingforcesactinginsubspaceIIprovideanindependenceofsymmetricandantisymmetricoscillationsSandEpiftheyareappliedsimultaneouslytoSandEpandareequalonvalue.ThentransformationofexternalforcesF*lookslike,submittedintable.