外文翻译--其对称性揭示了行星齿轮减速器独立振荡 英文版.pdf

12THIFTOMMWORLDCONGRESS,BESANONFRANCE,JUNE1821,2007REVEALINGOFINDEPENDENTOSCILLATIONSINPLANETARYREDUCERGEAROWINGTOITSSYMMETRYLBANAKHYUFEDOSEEVMECHANICALENGINEERINGRESEARCHINSTITUTEOFRUSSIANACADEMYOFSCIENCESMOSCOW,RUSSIAABSTRACTTHEPLANETARYREDUCER11GEARISASYMMETRICSYSTEMFORITSOSCILLATIONANALYSISTHEREISAPPLIEDTHESYMMETRYGROUPREPRESENTATIONTHEORY,WHICHWASGENERALIZEDFORMECHANICALSYSTEMSITWASFOUNDTHATDUETOREDUCERSYMMETRYTHEOSCILLATIONSDECOMPOSITIONHASARISENTHEREAREINDEPENDENTOSCILLATIONSCLASSES,SUCHASANGULAROSCILLATIONSOFSOLARGEARANDEPICYCLESATELLITESOSCILLATIONSINPHASE；TRANSVERSALOSCILLATIONSOFSOLARGEARANDEPICYCLESATELLITESOSCILLATIONSINANTIPHASESOLARGEARANDEPICYCLEOSCILLATIONSINAPHASEDONOTDEPENDONANGULARSATELLITESOSCILLATIONSKEYWORDSPLANETARYREDUCER,SYMMETRY,GROUPREPRESENTATIONTHEORY,INDEPENDENTOSCILLATIONSIINTRODUCTIONITISWELLKNOWNTHATATTHEOPERATIONOFPLANETARYREDUCERTHEREAREOSCILLATIONSOFITSELEMENTS,SUCHASSOLARGEAR,EPICYCLEANDSATELLITESTHISFACTORESSENTIALLYWORSENSAQUALITYOFREDUCER’OPERATION,ANDINSOMECASESCANRESULTINTHEIRCURVATUREANDBREAKAGEAPLENTYOFPAPERSAREDEVOTEDTOTHEDYNAMICANALYSISOFGEARREDUCERS1BASICALLYTHEREARECOMPUTATIONALRESEARCHESINTHEGIVENPAPERTHEANALYTICALAPPROACHESFORINVESTIGATIONOFREDUCERDYNAMICSISPRESENTEDTHEPLANETARYREDUCERHASAHIGHDEGREEOFSYMMETRYSOTHISPROPERTYWASUSEDANDTHEGROUPREPRESENTATIONTHEORYWASAPPLIEDAPPLICATIONOFTHISTHEORYALLOWSCARRYINGOUTDEEPENOUGHDYNAMICANALYSIS,USINGSYMMETRYPROPERTIESONLYFORTHISPURPOSEITISNECESSARYTOHAVETHEDYNAMICALMODELWHICHISTAKINGINTOACCOUNTSTIFFNESSCHARACTERISTICSINLINKAGESBETWEENREDUCERELEMENTSTHEMATHEMATICALAPPARATUSOFTHESYMMETRYGROUPS’REPRESENTATIONTHEORYISWIDELYUSEDINTHEQUANTUMMECHANICS,CRYSTALLOGRAPHIC,SPECTROSCOPY2,3,4THEADVANTAGESOFTHISAPPROACHAREDIFFICULTFOROVERESTIMATINGWITHITSHELPITISPOSSIBLETODEFINEWITHEXHAUSTIVECOMPLETENESSTHEDYNAMICPROPERTIES,USINGSTRUCTURESYMMETRYOFSYSTEMONLYWITHOUTSOLVINGOFMOTIONEQUATIONSHOWEVERINTHECLASSICALMECHANICSTHISAPPROACHISNOTWIDELYUSEDITISRESULTFROMSOMEPARTICULARFEATURESOFMECHANICALSYSTEMSFIRST,THEREISANEMAILBANLINBOXRUAVAILABILITYOFSOLIDSWITH6THDEGREESOFFREEDOMITISUNCLEARTOWHATSYMMETRYGROUPTORELATEASOLIDINORDERTHATSYSTEMSYMMETRYMAYBERETAINEDSECONDATREALDESIGNSMAYBETECHNOLOGICALERRORSANDMISTAKESATASSEMBLY,SOTHEREISASMALLASYMMETRYANDTHESYSTEMBECOMESQUASISYMMETRICFURTHERTHEMECHANICALSYSTEMSCONSISTFROMVARIOUSSUBSYSTEMSWITHVARIOUSSYMMETRYGROUPSINTHISCONNECTIONITISNECESSARYTOHAVEMETHODSFORTHEANALYSISASSYMMETRICANDQUASISYMMETRICMECHANICALSYSTEMSCONSISTINGOFVARIOUSSUBSYSTEMSANDSOLIDSHAVINGMADESOMEGENERALIZATIONS,THISMATHEMATICALAPPARATUSFORMECHANICALSYSTEMSMAYBEUSEDFORTHISPURPOSEWEPROPOSETOAPPLYTHEGENERALIZEDPROJECTIVEOPERATORS5THESEOPERATORSAREMATRIXESOFTHEAPPROPRIATEORDERINSTEADOFSCALARASINPHYSICSTHEUSEOFGENERALIZEDPROJECTIVEOPERATORSALLOWSTAKINGINTOACCOUNTALLABOVEMENTIONEDFEATURESOFMECHANICALSYSTEMSTHEAPPLICATIONOFTHESEOPERATORSTOINITIALSTIFFNESSMATRIXLEADSTOITSDECOMPOSITIONONINDEPENDENTBLOCKSEACHOFTHEMCORRESPONDSTOOWNOSCILLATIONCLASSININDEPENDENTSUBSPACESTOACCOUNTFORTHESOLIDSSYMMETRYTHEEQUIVALENTPOINTSWEREENTEREDTHESEPOINTSARECHOSENONSOLIDSOTHATTHEIRDISPLACEMENTSWERECOMPATIBLETOCONNECTIONSANDCORRESPONDEDTOGROUPOFSYMMETRYOFALLSYSTEMTHESEOPERATORSENABLEALSOMAYBEAPPLIEDWITHTHEFINITEELEMENTSMODELSFEMIIDYNAMICMODELOFPLANETARYREDUCERSTIFFNESSMATRIXTHEMODELOFAPLANETARYREDUCERSTEPISSUBMITTEDONFIG16THESTEPCONSISTSFROMSOLARGEARS,ITSMASSANDRADIUSAREEQUALTO11,MRITENGAGESINTOMESHWITHTHREESATELLITESSTII1,2,3ITSMASSESANDRADIUSAREIDENTICALANDEQUALTO22,MRSATELLITESINTURNAREENGAGEDINTOMESHWITHEPICYCLEEP33,MRANDTHEYAREFASTENONCARRIERBYELASTICSUPPORTWITHRIGIDITYH6THERIGIDITYOFGEARINGSOLARGEARSATELLITESISEQUALTO1H,THEGEARINGEPICYCLESATELLITESIS3H,ΓISANGLEOFGEARING12THIFTOMMWORLDCONGRESS,BESANONFRANCE,JUNE1821,2007FIG1PLANETARYREDUCERSTEPSSOLARGEAR,ЭEPICYCLE,1,2,3–SATELLITESSTLETSCONSIDERALLOVERAGAINTHEPLANEOSCILLATIONSOFPLANETARYREDUCERSTEPTRANSVERSALX,YANDANGULARΦOSCILLATIONSWITHOUTTHECASINGASTIFFNESSMATRIXMAYBEREPRESENTEDINABLOCKVIEWK123123123SSSTSSTSSTЭЭSTЭSTЭSTSTSTSTKKKKKKKKKKK1HEREONTHEMAINDIAGONALTHEREARETHESTIFFNESSSUBMATRIXES3X3FORAPPROPRIATEELEMENTS,ANDOUTSIDEOFTHEMAINDIAGONALTHEREARESTIFFNESSSUBMATRIXESOFCONNECTIONBETWEENTHESEELEMENTSTHEREARE15GENERALIZEDCOORDINATESX,,；,,,SSSEPEPEPXYXYΦΦ；1113,,STSTSTSTXYΦΦTHECONCRETEVIEWOFTHESEBLOCKSISSUBMITTEDINAPPENDIXTHUS,THETOTALORDEROFMATRIXKIS15X15ANINERTIAMATRIXMISDIAGONALIIIINTRODUCTIONOFEQUIVALENTPOINTSINDYNAMICMODELOPERATORSOFSYMMETRYBYVIRTUEOFSYMMETRYOFSATELLITESFASTENINGTHISSUBSYSTEMHASSYMMETRYSUCHAS3CASTRIANGLETOREVEALSYMMETRY3CATMOVINGOFSOLARGEARSANDEPICYCLEEPWESHALLENTERTHECOORDINATES123,,LLLONSOLARGEARSINPOINTSOFSATELLITESFASTENINGFIG2FIG2EQUIVALENTPOINTSONSOLARGEARS1,2,3,SATELLITESTHEYARE“EQUIVALENTPOINTS”THEIRCOORDINATESARE11112221333121COS3；21SIN31,2,3SSISSISSIRXRУIRIPIΑΒΑΑΒPIΒΑΒΦ−−ℓℓℓ2ORINMATRIXFORMLAXANDANALOGUESRELATIONSFOR“EQUIVALENTPOINTS”ONEPICYCLE,BUTINSTEAD1RIN2MUSTBEWRITTEN3RANDLATERONTHESECOORDINATEOFSOLARGEARANDEPICYCLEWILLBEUSEDINSTEADX,YANDΦAFTERTHATITISPOSSIBLETOCOUNT,THATALLCOORDINATESOFSYSTEMSHOULDVARYACCORDINGTOSYMMETRYGROUP3CAND,HENCE,ITISPOSSIBLETOAPPLYTHEPROJECTIVEOPERATOROFSYMMETRYTOALLSYSTEMELEMENTSS,EP,ANDALSOTOTHREESATELLITESSTII1,2,3FIG3THEORTHONORMALPROJECTIVEOPERATORGOFSYMMETRYFORPOINTGROUP3CISKNOWNAS2ITISG11133312166611022−−−3FORTHEWHOLESYSTEMTHEPROJECTIVEOPERATORMUSTBEREPRESENTEDASBLOCKDIAGONALMATRIX12THIFTOMMWORLDCONGRESS,BESANONFRANCE,JUNE1821,2007GSTGGG4HEREEACHSUBMATRIXCORRESPONDSTOS,EP,ANDALSOTOTHREESATELLITESSTII1,2,3SOBECAUSEWEHAVETHREEIDENTICALSATELLITESANDEACHOFTHEMHAS3DEGREESOFFREEDOM,IISTSTXYANDANGULARIISTSTΦΦ,THEREFOREITISNECESSARYTOENTERGENERALIZEOPERATOR33,4ANDTOCONSIDERSTGASBLOCKMATRIXWHERETHEEACHELEMENTISDIAGONALMATRIX3Х3,THATISITISPOSSIBLETOPRESENTEACHELEMENTASSTG1,11EGEEETHUSTOINITIALCOORDINATES,,,SEPXYΦOFSOLARGEARANDEPICYCLECONSISTENTLYTWOTRANSFORMATIONSAREAPPLIEDAANDGANDRESULTINGTRANSFORMATIONOFANINITIALMATRIXKEQUALSTOPRODUCTOFOPERATORSGATHISORTHOGONALTRANSFORMATIONANDITLOOKSLIKEGАSTGAGAG,WHEREGA223300100ΑΒΒΑBYAPPLYINGOFTHISTRANSFORMATIONTOMATRIXК1,WESHALLRECEIVEКGАКGАTRSOTHECORRESPONDINGTRANSFORMATIONSOFCOORDINATESANDFORCESAREXGАХ,FGАTRF5ASARESULTTHEINITIALMATRIXK15Х15ISDIVIDEDON3INDEPENDENTBLOCKS5X5AND,LOOKINGLIKE,12IIIIIKKKK6THEINERTIAMATRIXMREMAINSDIAGONALBECAUSEMATRIXGAISORTHOGONAL；THEREFORETHEINDEPENDENCEOFOSCILLATIONCLASSESDEFINESMATRIXКONLYIVREVEALINGOFINDEPENDENTMOTIONSCLASSESATFORNATURALANDFORCEDOSCILLATIONSANATURALOSCILLATIONSFROMTHEVIEWOFMATRIX6ITISSEEN,THATOWINGTOSYSTEMSYMMETRYTHEREISADECOMPOSITIONOFINITIALMATRIXK,AND,HENCE,DIVISIONOFOSCILLATIONCLASSESANDASWELLASSPACEOFPARAMETERSTHECONCRETERELATIONSFORSUBMATRIXESIN6SHOWTHATTHEREAREFOLLOWINGINDEPENDENTOSCILLATIONSCLASSESISTCLASSSUBSPACEISUBMATRIXIKANGULAROSCILLATIONOFSOLARGEARANDEPICYCLEOSCILLATIONSOFSATELLITESINAPHASEDIMENSIONOFTHISSUBSPACEISEQUALTO5ITSDETERMININGPARAMETERSARE12313612139,,,,,,,,RRRHHHHRH2NDCLASSSUBSPACEIISUBMATRIXES1IIK2IIKTRANSVERSALOSCILLATIONSOFSOLARGEARANDEPICYCLEOSCILLATIONSOFSATELLITESINANANTIPHASESUBSPACEIIBREAKSUPTOTWOIDENTICALSUBMATRIXES1IIKAND2IIK5Х5ITMEANSTHATINSYSTEMTHEREARE5EQUALFREQUENCIESITSDETERMININGPARAMETERSARE213679,,,,,,RHHHHHΓTHUS,TAKINGINTOACCOUNTONLYPROPERTIESOFSYMMETRYITISPOSSIBLETORECEIVEDEEPENOUGHANALYSISOFDYNAMICPROPERTIESOFSYSTEMOFAPLANETARYREDUCERBESIDESITISPOSSIBLETOSIMPLIFYALSOPROCESSOFSYSTEMOPTIMIZATIONBFORCEDOSCILLATIONSATTHEFORCEDOSCILLATIONSTHEUSEOFTHEINDEPENDENTOSCILLATIONCLASSESISSUITABLEONLYINTWOCASESAIFTHEPOINTSOFAPPLICATIONOFTHEEXTERNALFORCESHAVETHESAMETYPEOFSYMMETRY,ASADESIGNHAS,ORBIFTHEYAREDISPOSEDACCORDINGTOTHEINDEPENDENTCLASSESOFOSCILLATIONSREALLY,THENTRANSFORMATION5BRINGAFORCESVECTORFINTOAFORMCONTAININGZEROELEMENTSORIN1ST,OR2THSUBSPACESTHEANALYSISOFTHEREALLOADINGSFORCESONAREDUCER,SHOWS,THATITISVALIDIFELEMENTSDISBALANCESARETHESAMEАIDENTICALSATELLITESDISBALANCESDISBALANCEOFEPICYCLE；БIDENTICALSATELLITESDISBALANCESDISBALANCEOFSOLARGEARVTHEFURTHERMOTIONSDECOMPOSITIONTHEFURTHERDECOMPOSITIONOFSUBSPACESIANDIIIN6ISPOSSIBLEONLYIFTHEREAREADDITIONALCONDITIONSRAISINGATYPEOFSYSTEMSYMMETRY12THIFTOMMWORLDCONGRESS,BESANONFRANCE,JUNE1821,2007THESECONDITIONS,INPARTICULAR,CANBERECEIVEDFROMSIMILARITYSYMMETRYOFSOLARGEARANDEPICYCLETHEYLOOKLIKE1EQUALITYOFGEARINGSTIFFNESSWITHSANDEP,IE12HH,2EQUALITYOFPARTIALFREQUENCIESFORANGULARMOTIONSSANDEPSEPΦΦΝΝ,WHENCE78HH,OR3EQUALITYOFPARTIALFREQUENCIESATTRANVERSALMOTIONSOFSANDEP,,SEPXYXYΝΝ,WHENCEH72H9SOBYFULFILLMENTOFCONDITIONS1,2OR1,3THEADDITIONALSYMMETRYTYPE2CΝISAPPEAREDREFLECTIONSYMMETRYTOTHISSYMMETRYGROUPTHEOPERATOR2′GOR2′′GISCORRESPONDED2′G11311113；1311RHRH−2′′G111113213112HH−THEAPPLICATIONOFTHESEOPERATORSTOMATRIXKPERMITTOACHIEVETHEFURTHERDECOMPOSITIONOFCORRESPONDINGMATRIXESANDAPPROPRIATEMOTIONSREALLYTHEYMAYHAVESYMMETRICANDANTISYMMETRICOSCILLATIONCLASSESFORSOLARGEARANDEPICYCLETHUSTHECOORDINATETRANSFORMATIONIS11131311113131SEPRRHSEPRRH−XXXXXXAND13121312SEPHSEPH′−XXXXXXBYTHISCOORDINATETRANSFORMATIONTHEFOLLOWINGINDEPENDENTMOTIONTYPESAREARISEN′′⇒′′IIIKKKTHECONCRETERELATIONSFORTHESESUBMATRIXESSHOWTHATTHEREAREFOLLOWINGINDEPENDENTOSCILLATIONSCLASSESISUBSPACEMATRIX′IKANGULAROSCILLATIONSOFSOLARGEARSANDEPICYCLEEPINPHASESATELLITESSTIOSCILLATIONALONGAXISXINPHASE,IISUBSPACEMATRIX′′IKANGULAROSCILLATIONSOFSOLARGEARSANDEPICYCLEEPINANTIPHASESATELLITESSTIOSCILLATIONALONGAXISYANDAROUNDAXISΦINPHASESIMILARLYOCCURSDECOMPOSITIONOFSUBSPACESIIANDMATRIXES1IIK,2IIKBUTINSTEADOFANGULAROSCILLATIONSSANDEPTHEREARETHEIRTRANSVERSALOSCILLATIONALONGANAXISXORY,ANDOSCILLATIONOFSATELLITESINANANTIPHASEASSHOWSTHEANALYSISOFMATRIXI′KTHEOSCILLATIONSOFSANDEPINAPHASEDONOTDEPENDONANGULAROSCILLATIONOFSATELLITESΦSTFROMANALYSISOFI′KANDII′KWENOTICE,THATATH7H8H60AZEROROOTMAYBEARISENTHISOSCILLATIONTYPEMEANSTHEFREEOSCILLATIONSOFSATELLITESNAVIGATIONATANGULARORTRANSLATIONOSCILLATIONSSOLARGEARANDEPICYCLEAFORCEDOSC