油電混合動(dòng)力汽車(chē)行星齒輪箱設(shè)計(jì),油電混合動(dòng)力汽車(chē)行星齒輪箱設(shè)計(jì),混合,動(dòng)力,汽車(chē),行星,齒輪箱,設(shè)計(jì) Received19April2002;receivedinrevisedform12March2003;accepted16May2003 dynamics of planetary transmissions has drawn much attention. However, almost all the pub- lishedstudiesontheplanetarytransmissionsfocusedonlyontheirlinearvibration1,2. Intheadvancedmechanicalsystemsrunningathighspeed,suchasexactantennasandauto- maticweaponsystems,whichusuallycontainanumberofplanetarygearsets,thegearsystems * Correspondingauthor.Fax:+86-21-56334458. 1.Introduction Planetarygearsystemshavebeenwidelyusedinengineeringowingtotheiradvantagessuchas little space required, large ratio of transmission and high eciency. One of the most popular applicationsistotheautomatictransmissionsinautomobiles. Becauseofmachinerycomplexity,mostofearlierstudiesontheplanetarygearsystemswere conned to their static behaviors and sharing characteristics. Over the past two decades, the Abstract Presentedinthispaperisonthenonlineardynamicsofaplanetarygearsystemwithmultipleclearances takenintoaccount.Alateraltorsionalcoupledmodelisestablishedwithmultiplebacklashes,time-varying meshstiness,errorexcitationandsun-gearshaftcomplianceconsidered.Thesolutionsaredeterminedby usingharmonicbalancemethodfromtheequationsinmatrixform.ThetheoreticalresultsfromHBMare veriedbyusingthenumericalintegration.Finally,eectsofparametersarediscussed. C211 2003ElsevierLtd.Allrightsreserved. Keywords:Planetarygeartransmission;Nonlinearvibration;Fouriertransform;Harmonicbalancemethod;Dynamic Nonlineardynamicsofaplanetarygearsystem withmultipleclearances TaoSun a,* ,HaiYanHu b a DepartmentofPrecisionMechanicalEngineering,ShanghaiUniversity,Shanghai200072,PRChina b InstituteofVibrationEngineering,NanjingUniversityofAeronauticsandAstronautics,Nanjing210016,PRChina MechanismandMachineTheory38(2003)13711390 E-mailaddress:(T.Sun). 0094-114X/$-seefrontmatter C211 2003ElsevierLtd.Allrightsreserved. doi:10.1016/S0094-114X(03)00093-4 1372 T.Sun,H.Y.Hu/MechanismandMachineTheory38(2003)13711390 Nomenclature b backlash often undergo startup and brake interactively or run at high speed and under light load. The currentstudieshaveshownthatagearpairwouldlikelylosecontactandthetoothseparations C, c damping e statictransmissionerror f nonlineardisplacementfunction I rotaryinertia K, k stiness M, m mass N describingfunction P, p force q displacement r radius t time T torque X, x displacement n, g transversedisplacement U angle a pressureangle s dimensionlesstime X, x frequency u phaseangle Subscripts a alternatingcomponents b basecircle c carrier d relativeto h high-spedpart i ordinalnumber l low-speedpart m meancomponent p planetgear r ringgear s sungear n horizontal g vertical occurduetotheunavoidablebacklash.Accordingly,thebacklash,namelytheclearance,tendsto T.Sun,H.Y.Hu/MechanismandMachineTheory38(2003)13711390 1373 bringgearsystemstoexhibittypicalnonlineardynamicalbehaviors. Agearpairisboundtohavesomebacklash,whichmaybeeitherdesignedtoprovidebetter lubrication and to eliminate interference or due to manufacturing errors and wear. Backlash- inducednonlinearvibrationsmaycausetoothseparationandimpactsinunloadorlightlyloaded geared drives. Such impacts result in intense vibration and noise problems and large dynamic loads,whichmayaectreliabilityandlifeofthegeardrive. Experimentalstudiesonthedynamicbehaviorofaspurgearpairwithbacklashstartedalmost 40yearsagoandstillcontinue35.Forinstance,Kuboetal.4observedajumpinthefre- quencyresponseofagearpairwithbacklasheventhoughthetestset-upwasheavilydamped. Suchexperimentalstudies,albeitlimitedinscope,haveclearlyshownthatthedynamicsofa gear pair can hardly be predicted on the basis of a linear model. Consequently, the nonlinear dynamicsandmathematicalmodelsofagearpairwithbacklashhavebeenintensivelystudiedin thepastdecade.Althoughmostofthenonlinearmathematicalmodelsusedtodescribethedy- namic behavior of a gear pair are somewhat similar to each other, they dier in terms of the excitationmechanismsconsideredandespeciallythesolutiontechniqueused. Thenonlinearityofagearbacklashhastobemodeledbyadiscontinuousandnondierentiable function,whichrepresentsastrongnonlinearinteractioninthedynamicequationofwholesys- tem. Comparin and Singh addressed this problem in 6 and pointed out that most techniques availableintheliteraturescannotbedirectlyappliedtosolvingthisproblem.Manyresearchers haverecognizedthisproblemimplicitlyandthereforeemployedeitherdigitaloranalogsimula- tion techniques in their studies. Kahraman and Singh 7 made a detailed review of nonlinear geardynamicsavailableincurrentpublications.Theirtheoreticalstudyalsomadecontributions to the nonlinear dynamics of a spur gear pair with backlash subject to the static transmission error. Althoughthereisavastbodyofliteratureconcernedwithnonlineardynamicsofageneralgear pairwithclearance,thestudiesonnonlineardynamicsofaplanetarytransmissionsystemwith multipleclearancesarestillverylimited.Kahramantookthepossibilityoftoothseparationina planetarygearsystemintoaccount2.Inhisstudy,however,themodelwasnotconsideredasa nonlineardynamicsystem.Instead,astepfunctioninlinearmodelwasusedtodistinguishthe toothcontactandthetoothseparationroughly.Asaresult,thebackcollisionsofteethwerenot includedinhisstudy. Comparedwithabove-mentionedworks,thenonlineardynamicsofaplanetarygearsystemis muchmorecomplicated.Suchasystemisinherentlynonlinearowingtothemultipleclearances, andincludesthetemporallyandspatiallyvaryingsystemparameters.Moreover,therequestfor predictionandexaminationofdynamicbehaviorsisprogressivelyurgentinthedesignofmore quietandreliableplanetarytransmissions.Thepublishedstudiesonthenonlineardynamicsof geareddrivewithbacklash,however,focusononlyasinglegearpair,ratherthananyplanetary gearsystems.Accordingly,thefocusofthispaperisonthenonlineardynamicsofplanetarygear systems.Inthispaper,somecontributionswillbemadetoanumberofkeyissuessuchasnon- lineardynamicsmodeling,solutiontechniquestothenonlineardierentialequationsanddynamic behaviorsofplanetarygearsystems. 2.Dynamicmodelof system 2.1.Modelandassumptions Theplanetarygearsystemofconcerninthisstudyisasingle-stage2K-Htypeplanetarygearset asshowninFig.1.Thesystemconsistsofahigh-speedparth,alow-speedpartl,asungears,a ringgear r, n planetgears p andacarrier c.Here, n,thenumberofplanetgears,istakenas3 throughout the paper. All the gears are mounted on their exible shafts supported by rolling elementbearings. Toestablishthemathematicalmodelofsystem,afewassumptionsareintroducedinthecaseof speedreductionintheplanetarygearshowninFig.1asfollowing. (1) The inertial eects of prime mover (high-speed part) and load inertia (low-speed part) are takenasthoseoflumpedmass.Hence,theplanetarygearsystemhas4 nrotationaldegrees offreedom(DOF),includingtherotationaldisplacementsoflow-speedpart,high-speedpart, sungear,carrierand n planetarygears,respectively. 1374 T.Sun,H.Y.Hu/MechanismandMachineTheory38(2003)13711390 (2) Consideringthebendingstinessoftheshaftofsungear,thecomplianceofbearingsandpo- tentialdisplacementasarigidbodycausedbyoating,thehorizontalandverticaltransverse DOFofthesungearareincluded. (3) Asthebendingstinessoftheplanetgearshaftisverylarge,thedeectionofthisshaftcanbe neglected.Thus,thetransversedisplacementofplanetgearisnotconsidered. (4) Becausetheringgearitselfisapartofgearbox,thedisplacementofringgearasarigidbodyis insignicantandthecenterofringgearisassumednottomove. Basedontheaboveassumptions,thedynamicmodeloftheplanetarygearsystemisestablished asshowninFig.2.AllsymbolsinFig.2canbefoundinthenomenclaturepresentedandfurther explanationsaregiveninsubsequentsections.ThetotalnumberofDOFinthemodelisn 42, r s c l h p Fig. 1.Aplanetarygearset. T.Sun,H.Y.Hu/MechanismandMachineTheory38(2003)13711390 1375 including n 4 rotational DOF and 2 transverse DOF, respectively. Obviously, this is a com- plicated lateraltorsional coupled nonlinear system with multiple clearances and time-varying parameters. 2.2.Equivalentdisplacements Inthecaseofspeedreduction,themeshingrelationoftheplanetarygearsisshowninFig.3. Regardingtheangulardisplacementofeachgearindriving,thedirectionoftherevolutioncaused bythedrivingtorqueisassumedtobepositive.Namely,theangulardisplacements h s and h c of thesungearandthecarrierareinthesamedirection.Andtheangulardisplacementsh r andh p of theringgearandtheplanetgeararereversedinthedirectionsof h s and h c . In order to establish the equations of motion easily, both torsional and transverse displace- mentsareuniedonthepressurelineintermsofequivalentdisplacements. (b)(a) Fig. 2.Thedynamicmodelofaplanetarygearset. (b) line of action Ring gear Planet gear Sun gear (a) Fig. 3.Meshrelationinaplanetarygearset. where r b (subscripts C212sC213, C212rC213, C212pC213)arethebasecircleradiusofgears,while r bc isthenominalbase ring gear Todescribe carrier, system 3.Equations 1376 T.Sun,H.Y.Hu/MechanismandMachineTheory38(2003)13711390 InthedynamicalmodelshowninFig.2,thegearmeshisdescribedbyanonlineardisplacement functionf.Here,f isdenedasanonlinearfunctionintherelativegearmeshdisplacementC22qand C22 with ofmotion where n representsthenumbersofplanetgears. / i 2pi C01=n; i 1;2;.;n; 5 withthecarrier.Therefore,n s andg s representthehorizontalandverticaltransversedisplacements ofsungearwithregardtothemovablereferenceframerespectively.Asthevariationofpressure anglecausedbythetinytranslationofsungearcanbeneglected,theequivalentdisplacementinthe pressurelinedirectionderivedfromthetransversedisplacements n s and g s isrepresentedas x sdi C0n s sin/ i C0 ag s cos/ i C0 a: 4 Here a isthepressureangleofgearpair,and / i meansthatthe ithplanetgearismountedata theoreticalpositionangle/ i onthecarrierwithrespecttopositivedirectionofaxis-n.Ifthecenter oftherstplanetgearisassignedtopositionedatangle0, / i willbe X cl x c C0 x l i 1;2;3: ; theplanemotionofthesungear,akineticCartesiancoordinatesystemattachedtothe insteadofgeneralxedcoordinatesystem,isintroduced.Theorigin-o ofthiscoordinate coincideswiththecenterofcarrier.Andthecoordinateaxesmarkedasn;grotatealong whencalculatingthemeshstiness. By using X to represent the relative displacements in the direction of press line, the relative displacementsareobtainedasfollowsaccordingtothemeshingrelationshowninFig.3andthe equivalentdisplacements X hs x h C0 x s ; X spi x s C0 x pi C0 x c ; X rpi x pi C0 x c C0 x r x pi C0 x c ; 9 = 3 gear r issettobezerosincetheringgearisxedongearboxinthe2K-Htypeplanetary setshowninFig.1.Thatis,h r 0.However,thetoothdeectionoftheringgearisincluded Withregardtothedirectionoftheequivalentdisplacements,thedirectionofdeectioncausedby drivingtorqueisassumedtobepositive. Itshouldbepointedoutthatinthecaseofspeedreduction,therotationaldisplacementofthe r bc r bs r bp r br C0 r bp : 2 circleradiusofthecarrierdenedasfollowing: The equivalent transverse displacements in the pressure line direction caused by rotational displacementsarewrittenasfollows: x h r bs h h ; x s r bs h s ; x r r br h r ; x pi r bp h pi ; x c r bc h c ; x l r bc h l ; i 1;2;3; 1 backlash2b asparameter fC22q C22q C0 C22 b; C22q C22 b; 0; C0 C22 b6C22q6 C22 b; C22q C22 b; C22q C0 C22 b: 8 inthedirectionoflineofaction, k n and k g thebearingstiness. Eq. system, The tothe equation,itisnotpossibletowriteoutthegoverningequationinmatrixform,whileageneral matrixform. Therefore, The also variable isobtained 1378 T.Sun,H.Y.Hu/MechanismandMachineTheory38(2003)13711390 Eq.(9)issimpliedfurtherbyusingasetofnewvariables X hs x h C0 x s ; X sdi x s C0 x pi C0 x c C0 n s sin/ i C0 ag s cos/ i C0 aC0C22e spi ; X n n s ; X g g s ; X rdi x pi C0 x c C0 C22e rpi ; X cl x c C0 x l i 1;2;3: 9 = ; 10 newcoordinatevariablesdenedinabovenotonlyhaveintuitionalphysicalmeaning,but eliminatetherigidbodymotions.Furthermore,f canbewrittenasasetoffunctionsinsingle intermsofvariablesgiveninEq.(10).Hence,thesetofsimpliedgoverningequations solution technique applicable to the systems of multiple degrees of freedom must be based on (9)featuresconsiderabledicultiesinitssolvingprocessasfollows.(1)Itisasemi-denite which predicatesprospective trivialsolutions correspondingto rigid body motions. (2) function f isnonlinearmultivariate,andthenumberofvariablesisevendierentaccording externalandinternalgearpairs.(3)Asbothlinearandnonlinearrestoringforcesexistinthe M l x l C0 C cl _ X cl C0 k cl X cl C0P l i 1;2;3; ; where M h I h r 2 bs ; M s I s r 2 bs ; M pi I pi r 2 bp ; M l I l r 2 bc ; M c I c r 2 bc 3 m p cos 2 a ; k hs K hs r 2 bs ; k cl K cl r 2 bc ; P h T h r bs ; P l T l r bc : Here,thesubscripts C212hC213, C212sC213,pi, C212cC213 and C212lC213 representthehigh-speedpart,thesungear,theplanet gear,thecarrierandthelow-speedpart,respectively.I representstheinertia,mtheactualmass,M theequivalentmass,Cthedampingcoecient,T h andT l theinputandoutputtorquerespectively, P h andP l theequivalentforce,K hs andK cl thetorsionalstiness,k hs andk cl theequivalentstiness m s n s C n _ n s C0 P i1 D spi C1sin/ i C0 ak n n s C0 P i1 W spi C1sin/ i C0 a0; m s g s C g _g s P 3 i1 D spi C1cos/ i C0 ak g g s P 3 i1 W spi C1cos/ i C0 a0; M pi x pi C0 D spi D rpi C0 W spi W rpi 0; M c x c C0 P 3 D spi C0 P 3 D rpi C cl _ X cl C0 P 3 W spi C0 P 3 W rpi k cl X cl 0; = 9 M h x h C hs _ X hs k hs X hs P h ; M s x s C0 C hs _ X hs P 3 i1 D spi C0 k hs X hs P 3 i1 W spi 0; 9 bycombiningEqs.(9)and(10) M X C0 C X C X C X C0 C X where T.Sun,H.Y.Hu/MechanismandMachineTheory38(2003)13711390 1379 C0 sin/ i C0 aM sdi m s C n _ n s cos/ i C0 aM sdi m s C g _g s C0 M sdi M p C rpi _ X rdi M sdi M c X 3 i1 C rpi _ X rdi C0 M sdi M s k hs X hs M sdi M p k spi C1 fX sdi X 3 i1 M sdi M s C18 M sdi M c M sdi m s C19 k spi C1 fX sdi C0 M sdi M c k cl X cl C0 sin/ i C0 aM sdi m s k n _ n s cos/ i C0 aM sdi m s k g g s C0 M sdi M p k rpi fX rdi M sdi M c X 3 i1 k rpi fX rdi C0M sdi C1 C22e spi i 1;2;3; m s X n C n _ X n C0 X 3 i1 C spi _ X sdi sin/ i C0 ak n X n C0 X 3 i1 k spi C1 fX sdi C1sin/ i C0 a0; m s X g C g _ X g X 3 i1 C spi _ X sdi cos/ i C0 ak g X g X 3 i1 k spi C1 fX sdi C1cos/ i C0 a0; M rdi X rdi C0 M rdi M p C spi _ X sdi X 3 i1 M rdi M c C spi _ X sdi M rdi M p C rpi _ X rdi X 3 i1 M rdi M c C rpi _ X rdi C0 M rdi M c C cl _ X cl C0 M rdi M p k spi fX sdi X 3 i1 M rdi M c k spi C1 fX sdi M rdi M p k rpi fX rdi C0 M rdi M c k cl X cl X 3 i1 M rdi M c k rpi fX rdi C0M rdi C1 C22e rpi i 1;2;3; M cl X cl C0 M cl M c X 3 i1 C spi _ X sdi C cl _ X cl C0 M cl M c X 3 i1 C rpi _ X rdi C0 M cl M c X 3 i1 k spi C1 fX sdi k cl X cl C0 M cl M c X 3 i1 k rpi fX rdi M cl M l P; 11 M hs M s M h M s M h ; M rdi M p M c M p M c ; M cl M c M l M c M l ; M sdi M s M pi M c m s M M m M M m M M m M M M : sdi sdi M s hs hs M p spi sdi i1 M s M c m s spi sdi M c cl cl M hs X hs C hs _ X hs C0 M hs M s X 3 i1 C spi _ X sdi k hs X hs C0 M hs M s X 3 i1 k spi C1 fX sdi M hs M h P h ; M sdi _ M sdi _ X 3 M sdi C18 M sdi M sdi C19 _ M sdi _ pi c s s c s s pi s pi c s M diag M hs ; M sd1 ; M sd2 ; M sd3 ; m s ; m s ; M rd1 ; M rd2 ; M rd3 ; M cl C138; 14 coordinates C22q C0 b ; C22q b ; where The Eq. i i c i i i i i where dynamic Kahraman balance clearances 1380 T.Sun,H.Y.Hu/MechanismandMachineTheory38(2003)13711390 toreveal the nonlineardynamicsof multi-degree-of-freedomplanetarygear systemsexcited by transmission method(HBM).SuccessfulexamplesofHBMforsolvingnonlinearsystemswithmultiple havenotbeenavailableinarchivalpublications.Inthissection,theHBMispresented pairwithbacklashexcitedbyexternaltorqueandinternalexcitationsbyemployingtheharmonic andSingh5andComparinandSingh6locatedtheperiodicvibrationofagear 4.Solutionsby harmonicbalancing MqsC _qsKf qs ps; 18 where s t C1 x n isdimensionlesstime. b c isacharacterlengthand x istheexcitationfrequency.Thus,wehavethedimensionless equations q i b i ; q i b i ; 0; C0b 6q 6b ; 8 17 purpose,let x n k sp1 =M sd1 p ,wherek sp1 isthemeanvalueofmeshstinessbetweenthesungear andplanetgear-1.Otherdimensionlessparametersaredenedas q C22q=b c ; M I; X x=x n ; c ij C22c ij =C22m ii x n ; k ij C22 k ij =C22m ii x 2 n ; 2 C22 b i 0; for i 1;5;6;10; 16a C22 b i1 C22 b spi ; C22 b i6 C22 b rpi ; for i 1;2;3: 16b stinessmatrix K,thedampingmatrix C andtheforcevector C22p aregiveninAppendixA. (13) can be further simplied by using a number of dimensionless parameters. For this f i fC22q i i i i i 0; C0 C22 b i 6C22q i 6 C22 b i ; C22q i C22 b i ; C22q i 1; C26 20c Hl C01; l 1; 8 : 20d l C6 C6b i C0 q mi =q ai : 20e N mi 1 2q mi Gl C0Gl C0 C138; 20a N ai 1C0 1 2 Hl C0l C0 C138; 20b where ThesubstitutionofEqs.(17)and(19b)intoEqs.(19d)and(19e)gives q ai N mi q mi ;q ai 2pq mi 0 fq i dh i ; 19d N ai q mi ;q ai 1 pq ai Z 2p 0 fq i cosh i dh i ; h i Xs u i : 19e fq i N mi q mi N ai q ai cosXs u i ; 19c where N mi and N ai arethedescribingfunctionsdenedas 1 Z 2p Harmonicbalancemethod is paid to the periodic vibrations of system under the harmonic excitation. The ofsolvingEq.(18)byHBMincludesfouraspectsasfollowing. Formofexcitation:Accordingtotheassumptionofharmonicexcitations,excitationsgivenin Eq.(17)canberepresentedapproximatelyas p i p mi p ai cosXs u pi ; 19a wheresubscripts C212mC213 and C212aC213 representthemeanandalternatingcomponentsofforce,and u pi isphaseangleofforce. Formofresponse:FortheharmonicexcitationsgivenbyEq.(19a),theentriesinapproximate solutionvector q areassumedintheform q i q mi q ai cosXs u i ; 19b where q mi and q ai arethemeanandalternatingcomponentsofthesteadystateresponse,and u i isphaseangle. Nonlinearfunction:Thesteadystatesolution q i sq i s T inEq.(19b)isassumedtobe periodicwithperiod T 2p=X.Accordingly, fq i s fq i s T mustalsobeperiodic. Akey step of HBM is to represent the nonlinear function f in Eq. (18) in the following form: This (4) Algebraic To rithm where algebraicequationsofsystem Eq. As numerical 1382 T.Sun,H.Y.Hu/MechanismandMachineTheory38(2003)13711390 y 4 fN ai q ai sinu i g nC21 ; y m fN mi q mi g nC21 ; p 1 fp ai cosu pi g nC21 ; p 2 fp ai sinu pi g nC21 ; K 1 fk aij cos/ ij g nC2n ; K 2 k aij sin/ ij C138 nC2n : 24 (23)includesfollowing3n unknownvariablestobesolved q ai ; q mi ; u i ; i 1;2;.;n: 25 it isimpossibletosolvethenonlinearalgebraicequation(23)byanyanalyticalmethods,a where y 1 fq ai cosu i g nC21 ; y 2 fq ai sinu i g nC21 ; y 3 fN ai q ai cosu i g nC21 ; K m y m 1 2 K 1 y 3 K 2 y 4 C0p m f0g nC21 ; K m y 3 K 1 y m C0 X 2 My 1 C0 XCy 2 C0 p 1 f0g nC21 ; K m y 4 K 2 y m C0 X 2 My 2 C0 XCy 1 C0 p 2 f0g nC21 ; 8 : 23 BysubstitutingEqs.(19a)(19c)intoEq.(18)andbalancingthesameharmonics,oneobtainsthe f qfN mi q mi g nC21 fN ai q ai cosXs u i g nC21 : 22c q fq mi g nC21 fq ai cosXs u i g nC21 ; 22b p fp mi g nC21 fp ai cosXs u pi g nC21 ; 22a Eqs.(19a)(19c)canbewritteninthefollowingmatrixform K K m DK; 21b K m k mij C138 nC2n ; DK k aij cosXs / ij C138 nC2n : 21c Hence,K iswrittenintermsoftwoseparatematricesformeanstinessandalternatingstiness k ij k mij k aij cosXs / kij : 21a getthemeanvalueandtheharmonicsofthetime-varyingmeshstiness,theFFTalgo- isintroducedtotheperiodicfunctionofmeshstiness. varyingperiodicmeshstinessinFig.4,theentriesinstinessmatrix K inEq.(18)canbe writtenas N mi 1; N ai 1: C26 20g resultagreeswiththephysicalfactthatdescribingfunctionoflinearfunctionisreally1. equations: Considering the mean value and the fundamental harmonics of time- SubstitutingEq.(20f)intoEqs.(20a)(20e)yields b i 0: 20f For i 1,5,6,10,wehave routine (DNEQBJ of IMSL 8) is used to determine the solution. The routine DNEQBJ uses a secant method with BroydenC213s update to nd the zeros of a set of nonlinear algebraicequations. 4.2.Solutionsandvalidation Astherstexample,a2K-HtypeplanetarygearsystemshowninFig.1isstudied.Thegeo- metric parameters of system are listed in Table 1 and other parameters are given in Table 2. Furthermore,thecharacteristiclengthissettob c 0:01mm.Theexcitationsareassociatedwith themeshstinessuctuationanderrors.Thefrequencyresponsesofsystemaredeterminedby usingtheHBMintheprevioussubsection.Asacomparison,thefrequencyresponsesareshownin Fig.5(a)(e),togetherwiththeresponsesofcorrespondinglinearsystemswithoutanybacklash. These gures demonstrate that a planetary gear system with clearances taken into