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12thIFToMMWorldCongress,BesançonFrance,June1821,2007RevealingofIndependentOscillationsinPlanetaryReducerGearowingtoitssymmetryL.BanakhYu.FedoseevMechanicalEngineeringResearchInstituteofRussianAcademyofSciencesMoscow,RussiaAbstractTheplanetaryreducer11gearisasymmetricsystem.Foritsoscillationanalysisthereisappliedthesymmetrygrouprepresentationtheory,whichwasgeneralizedformechanicalsystems.Itwasfoundthatduetoreducersymmetrytheoscillationsdecompositionhasarisen.Thereareindependentoscillationsclasses,suchasangularoscillationsofsolargearandepicyclesatellitesoscillationsinphasetransversaloscillationsofsolargearandepicyclesatellitesoscillationsinantiphase.Solargearandepicycleoscillationsinaphasedonotdependonangularsatellitesoscillations.Keywordsplanetaryreducer,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,thereisanEmailbanlinbox.ruavailabilityofsolidswith6thdegreesoffreedom.Itisuncleartowhatsymmetrygrouptorelateasolidinorderthatsystemsymmetrymayberetained.Secondatrealdesignsmaybetechnologicalerrorsandmistakesatassembly,sothereisasmallasymmetryandthesystembecomesquasisymmetricFurtherthemechanicalsystemsconsistfromvarioussubsystemswithvarioussymmetrygroups.Inthisconnectionitisnecessarytohavemethodsfortheanalysisassymmetricandquasisymmetricmechanicalsystemsconsistingofvarioussubsystemsandsolids.Havingmadesomegeneralizations,thismathematicalapparatusformechanicalsystemsmaybeused.Forthispurposeweproposetoapplythegeneralizedprojectiveoperators5.Theseoperatorsarematrixesoftheappropriateorderinsteadofscalarasinphysics.Theuseofgeneralizedprojectiveoperatorsallowstakingintoaccountallabovementionedfeaturesofmechanicalsystems.Theapplicationoftheseoperatorstoinitialstiffnessmatrixleadstoitsdecompositiononindependentblockseachofthemcorrespondstoownoscillationclassinindependentsubspaces.Toaccountforthesolidssymmetry«theequivalentpoints»wereenteredthesepointsarechosenonsolidsothattheirdisplacementswerecompatibletoconnectionsandcorrespondedtogroupofsymmetryofallsystem.TheseoperatorsenablealsomaybeappliedwiththefiniteelementsmodelsFEM.II.Dynamicmodelofplanetaryreducer.Stiffnessmatrix.Themodelofaplanetaryreducerstepissubmittedonfig.16.ThestepconsistsfromsolargearS,itsmassandradiusareequalto11,mr.ItengagesintomeshwiththreesatellitesStii1,2,3itsmassesandradiusareidenticalandequalto22,mr.SatellitesinturnareengagedintomeshwithepicycleEp33,mrandtheyarefastenoncarrierbyelasticsupportwithrigidityh6.Therigidityofgearingsolargearsatellitesisequalto1h,thegearingepicyclesatellitesis3h,γisangleofgearing.12thIFToMMWorldCongress,BesançonFrance,June1821,2007Fig.1Planetaryreducerstep.Ssolargear,Эepicycle,1,2,3–satellitesSt.Letsconsideralloveragaintheplaneoscillationsofplanetaryreducersteptransversalx,yandangularϕoscillationswithoutthecasing.AstiffnessmatrixmayberepresentedinablockviewK123123123SSStSStSStЭЭStЭStЭStStStStKKKKKKKKKKK1Hereonthemaindiagonaltherearethestiffnesssubmatrixes3x3forappropriateelements,andoutsideofthemaindiagonaltherearestiffnesssubmatrixesofconnectionbetweentheseelements.Thereare15generalizedcoordinatesX,,,,,SSSEpEpEpxyxyϕϕ1113,,...StStStStxyϕϕTheconcreteviewoftheseblocksissubmittedinAppendix.Thus,thetotalorderofmatrixKis15x15.AninertiamatrixMisdiagonal.III.Introductionofequivalentpointsindynamicmodel.Operatorsofsymmetry.Byvirtueofsymmetryofsatellitesfasteningthissubsystemhassymmetrysuchas3Castriangle.Torevealsymmetry3CatmovingofsolargearSandepicycleEpweshallenterthecoordinates123,,lllonsolargearSinpointsofsatellitesfasteningfig.2..Fig.2EquivalentpointsonsolargearS.1,2,3,satellitesTheyareequivalentpoints.Theircoordinatesare11112221333121cos321sin31,2,3.SSiSSiSSirXrУiripiαβααβpiβαβϕ−−ℓℓℓ2orinmatrixformLAXAndanaloguesrelationsforequivalentpointsonepicycle,butinstead1rin2mustbewritten3r.Andlateronthesecoordinateofsolargearandepicyclewillbeusedinsteadx,yandϕ.Afterthatitispossibletocount,thatallcoordinatesofsystemshouldvaryaccordingtosymmetrygroup3Cand,hence,itispossibletoapplytheprojectiveoperatorofsymmetrytoallsystemelementsS,Ep,andalsotothreesatellitesStii1,2,3.fig.3Theorthonormalprojectiveoperatorgofsymmetryforpointgroup3Cisknownas2.Itisg11133312166611022−−−3Forthewholesystemtheprojectiveoperatormustberepresentedasblockdiagonalmatrix12thIFToMMWorldCongress,BesançonFrance,June1821,2007GStggg4HereeachsubmatrixcorrespondstoS,Ep,andalsotothreesatellitesStii1,2,3.SoBecausewehavethreeidenticalsatellitesandeachofthemhas3degreesoffreedom,iiStStxyandangular...iiStStϕϕ,thereforeitisnecessarytoentergeneralizeoperator33,4andtoconsiderStgasblockmatrixwheretheeachelementisdiagonalmatrix3х3,thatisitispossibletopresenteachelementasStg1,11EgEEEThustoinitialcoordinates,,,SEpxyϕofsolargearandepicycleconsistentlytwotransformationsareappliedAandG.AndresultingtransformationofaninitialmatrixKequalstoproductofoperatorsGA.ThisorthogonaltransformationanditlookslikeGАStgAgAg,wheregA223300100αββαByapplyingofthistransformationtomatrixК1,weshallreceiveКGАКGАtrSothecorrespondingtransformationsofcoordinatesandforcesareXGАХ,FGАtrF5AsaresulttheinitialmatrixK15х15isdividedon3independentblocks5x5and,lookinglike,12IIIIIKKKK6TheinertiamatrixMremainsdiagonalbecausematrixGAisorthogonalthereforetheindependenceofoscillationclassesdefinesmatrixКonly.IV.RevealingofindependentmotionsclassesatfornaturalandforcedoscillationsA.NaturaloscillationsFromtheviewofmatrix6itisseen,thatowingtosystemsymmetrythereisadecompositionofinitialmatrixK,and,hence,divisionofoscillationclassesandaswellasspaceofparameters.Theconcreterelationsforsubmatrixesin6showthattherearefollowingindependentoscillationsclassesIstclasssubspaceIsubmatrixIKangularoscillationofsolargearandepicycleoscillationsofsatellitesinaphase.Dimensionofthissubspaceisequalto5.Itsdeterminingparametersare12313612139,,,,,,,,.rrrhhhhrh2ndclasssubspaceIIsubmatrixes1IIK2IIKtransversaloscillationsofsolargearandepicycleoscillationsofsatellitesinanantiphase.SubspaceIIbreaksuptotwoidenticalsubmatrixes1IIKand2IIK5х5.Itmeansthatinsystemthereare5equalfrequencies.Itsdeterminingparametersare213679,,,,,,.rhhhhhγThus,takingintoaccountonlypropertiesofsymmetryitispossibletoreceivedeepenoughanalysisofdynamicpropertiesofsystemofaplanetaryreducer.Besidesitispossibletosimplifyalsoprocessofsystemoptimization.B.ForcedoscillationsAttheforcedoscillationstheuseoftheindependentoscillationclassesissuitableonlyintwocasesaifthepointsofapplicationoftheexternalforceshavethesametypeofsymmetry,asadesignhas,orbiftheyaredisposedaccordingtotheindependentclassesofoscillations.Really,thentransformation5bringaforcesvectorFintoaformcontainingzeroelementsorin1st,or2thsubspaces.Theanalysisoftherealloadingsforcesonareducer,shows,thatitisvalidifelementsdisbalancesarethesameаidenticalsatellitesdisbalancesdisbalanceofepicycleбidenticalsatellitesdisbalancesdisbalanceofsolargear.V.Thefurthermotionsdecomposition.ThefurtherdecompositionofsubspacesIandIIin6ispossibleonlyifthereareadditionalconditionsraisingatypeofsystemsymmetry.12thIFToMMWorldCongress,BesançonFrance,June1821,2007Theseconditions,inparticular,canbereceivedfromsimilaritysymmetryofsolargearandepicycle.Theylooklike1.EqualityofgearingstiffnesswithSandEp,i.e.12hh,2.EqualityofpartialfrequenciesforangularmotionsSandEpSEpϕϕνν,whence78hh,or3.EqualityofpartialfrequenciesattranversalmotionsofSandEp,,SEpxyxyνν,whenceh72h9.Sobyfulfillmentofconditions1,2or1,3theadditionalsymmetrytype2Cνisappearedreflectionsymmetry.Tothissymmetrygrouptheoperator2′Gor2′′Giscorresponded2′G113111131311rhrh−2′′G111113213112hh−TheapplicationoftheseoperatorstomatrixKpermittoachievethefurtherdecompositionofcorrespondingmatrixesandappropriatemotions.Reallytheymayhavesymmetricandantisymmetricoscillationclassesforsolargearandepicycle.Thusthecoordinatetransformationis11131311113131SEprrhSEprrh−XXXXXXAnd13121312SEphSEph′−XXXXXXBythiscoordinatetransformationthefollowingindependentmotiontypesarearisen′′⇒′′IIIKKKTheconcreterelationsforthesesubmatrixesshowthattherearefollowingindependentoscillationsclassesIsubspacematrix′IKangularoscillationsofsolargearSandepicycleEpinphasesatellitesStioscillationalongaxisxinphase,IIsubspacematrix′′IKangularoscillationsofsolargearSandepicycleEpinantiphasesatellitesStioscillationalongaxisyandaroundaxisφinphase.SimilarlyoccursdecompositionofsubspacesIIandmatrixes1IIK,2IIKbutinsteadofangularoscillationsSandEptherearetheirtransversaloscillationalonganaxisxory,andoscillationofsatellitesinanantiphase.AsshowstheanalysisofmatrixI′KtheoscillationsofSandEpinaphasedonotdependonangularoscillationofsatellitesφSt.FromanalysisofI′KandII′Kwenotice,thatath7h8h60azerorootmaybearisen.Thisoscillationtypemeansthefreeoscillationsofsatellitesnavigationatangularortranslationoscillationssolargearandepicycle.A.Forcedoscillations.Thechoiceofexitedforcesaccordingoneoftheseoscillationtypesdonotinducetheotheroscillationtypesbecausetheyareorthogonaleachother.TheexcitingforcesactinginsubspaceIIprovideanindependenceofsymmetricandantisymmetricoscillationsSandEpiftheyareappliedsimultaneouslytoSandEpandareequalonvalue.ThentransformationofexternalforcesFlookslike,submittedintable.