1、11FTM05AGMA Technical PaperEpicyclic Load SharingMap Application as aDesign ToolBy A. Singh, General MotorsCompanyEpicyclic Load Sharing Map Application as a Design ToolDr. Avinash Singh, General Motors CompanyThe statements and opinions contained herein are those of the author and should not be con
2、strued as anofficial action or opinion of the American Gear Manufacturers Association.AbstractOneofthemainadvantagesofplanetarytransmissionsisthattheinputtorqueissplitintoanumberofparallelpaths. Therefore, in an n planet planetary system, each sun-pinion-ring path is designed to carry 1/n of theinpu
3、t torque. However, equal load sharing between the planets is possible only in the ideal case. In thepresence of positional type manufacturing errors, equal load sharing is not realized, and the degree ofinequality in load sharing has major implications for gear system sizing, tolerancing schemes, an
4、d torqueratings.Inthispaper,theconceptofanEpicyclicLoadSharingMap(ELSM)willbeexplained. TheELSMisaphysicsbasedtoolthatisderivedfromaphysicalexplanationoftheloadsharingphenomenon. ItisaplotoftheLoadratio (or % of input torque) versus a non dimensional parameter Xe. The non-dimensional parameter is af
5、unction of combined system stiffness, tolerance level, and operating torque. The ELSM maps out theoperatingspaceofanyepicyclicgearset,andagivengearsetatagivenoperatingconditionmapstoapointontheELSM. OnceagearsetislocatedontheELSM,itsbehaviorunderanyloadanderrorconditioncanbequickly predicted. Also,
6、the advantages of adding extra planets can be accurately estimated.Inthispaper,theapplicationoftheELSMasadesigntoolwillbediscussed. Thegeneralcasewhenthereareerrorsonthepositionofeverycarrierpin-holewillbeconsidered. Statisticalsimulationswillbeperformedforagiven manufacturing error distribution for
7、 3 to 7 planet systems.Copyright 2011American Gear Manufacturers Association1001 N. Fairfax Street, 5thFloorAlexandria, Virginia 22314October 2011ISBN: 978-1-61481-004-93 11FTM05Epicyclic Load Sharing Map Application as a Design ToolDr. Avinash Singh, General Motors CompanyIntroductionEpicyclictrans
8、missionsarecompactastheinputtorqueissplitintoanumberofparallelsun-pinion-ringpaths,and each path is designed to transmit a fraction of the input torque. In the absence of manufacturingvariations, perfect load sharing between the different parallel paths is possible. The power density of suchepicycli
9、c gearsets can be improved by simply adding additional planets (up to the maximumnumber thatcanfit).However, in reality due to the presence of various manufacturing variations that cause positional differencesin the location of the individual planets, such equal load sharing cannot be achieved 1-14.
10、 Some of theplanets will transmit higher than nominal loads, while others transmit lower than nominal loads. Previousexperimental 1-6 and computational 6-13 studies of varying complexities have demonstrated the signific-antloadsharinginequalitiesthatresult fromthese errors. Thisload sharinginequalit
11、y needsto beaccuratelyestimated in order to properly size epicyclic gearsets and reliably estimate their torque capacity 14.Recent research work has shown that the load sharing behavior is associated withpositional deviationsfromideal location that causes one or more planets to lead or lag the other
12、 planets. These deviations from ideallocationareduetomanufacturingvariations,andwillbereferredtoas“positionalerror”orsimply“error”inthispaper. BodasandKahraman11classifythemanufacturingerrorsintotimeinvariant,assemblyindependenterrors(pinholepositionerror,pinholediametererror),timeinvariant,assembly
13、dependenterrors(planettooththickness, planet pin and bore eccentricities), and time variant, assembly dependent errors (run-outs of thegears). Theyalsoofferawaytocombinealltheseerrorsintoacumulativepositionalerror. Inthiswork,“error”will refer to this cumulative positional error that includes the co
14、ntributions from all sources.There are several key factors that influence the load sharing behavior. Some of these factors are thetransmittedtorque,errorlevel,directionalityoferror,systemflexibility,number of planets, andamount offloatin the system. The sensitivity of load sharing inequality to many
15、 of these variables has been studied 1-5,10-14. These factors are also recognized in ANSI/AGMA 6123-B06 15.While the research activities have revealed much about the factors influencing load sharing, and providedcomputationalmeansofquantifyingthe loadinequalities, abasic physicalunderstanding ofthe
16、truemechan-ism that leads to the load sharing behavior was lacking. In recent papers 16-18, the author has proposed aphysical mechanism that explains all known load sharing behavior. Both floating and non-floating (fixedcenters) systems were treated. The physical explanation leads to simple expressi
17、ons that seem to com-pletelydescribethecomplexloadsharingbehavior. Theseexpressionsareinnon-dimensionaltermsandcanbeappliedtoanyepicyclicgearsetunderanyoperatingcondition. Comparisonstocomputationalmodelsandexperimental results have shown excellent correlation.The proposed physical explanation also
18、leads to the concept of an epicyclic load sharing map (ELSM). TheELSM is a plot of the Load ratio (or % of input torque) versus a non dimensional parameter Xe.Thenon-dimensionalparameterisafunctionofcombinedsystemstiffness,tolerance level,and operatingtorque.The ELSM maps out the operating space of
19、any epicyclic gear set, and a given gear set at a given operatingcondition maps to a point on the ELSM. The ELSM contains curves for 3, 4, 5, 6, and 7 (and more) planetsystems. Once a gear set is located on the ELSM, its behavior under any load and error condition can bequickly predicted. Also, the
20、advantages of adding extra planets can be accurately estimated.The load ratio term used in the ELSM is defined similar to the mesh load factor, K, defined in the AGMAstandards 15. AGMA recommends estimating Kby measurement, or using a table provided in 15. The4 11FTM05ELSM provides an alternate meth
21、od of defining the load sharing inequality which is based on anunderstanding of the physical behavior, and implicitly includes the influence of the key variables like error,stiffness, number of planets, transmitted torque, etc.Inthispaper, wewill firstbriefly reviewthe physicalexplanation ofthe load
22、sharing phenomenonfor fixedandfloatingsystems. Wewillalsosummarizethepreviouslypublishedfindingsonthedetailedmechanismofloadsharingin37planet systems. A detailed derivation of a five planet system will be provided for the sake ofcompleteness. Next, the concept of the Epicyclic Load Sharing Map will
23、be discussed. An equivalent errormetric that captures the cumulative effect of errors on the position of each planet in the system will also bediscussed. A comparison between the values predicted by the ELSM and those found in 15 will also bediscussed.Finallyastatisticalsimulationwillbeperformedtode
24、monstratetheapplicationoftheELSMtoactualgearsetswith varying levels of manufacturing accuracy.Key elements of the proposed frameworkThe following are the key elements of the framework that will be used to describe the planetary load sharingbehavior:S Tangential position error is the root causeS Syst
25、em float partially neutralizes the errorsS Elastic deformation under load neutralizes the remaining portion of the errorsS Non-dimensional neutralizing ratioS Equal load sharing in the absence of errorsTangential position error as root causePositionerrorisdefinedasthedeviationinthelocationofthecente
26、roftheplanetsfromtheirideallocations. Ithas been widely reported that the presence of positional error results in the phenomenon of unequal loadsharing between the planets. Several recent publications 5,12 havealso shownthat theepicyclic systemissensitive to errors in the tangential direction and in
27、sensitive to errors in the radial direction.Consider anepicyclic system withanerror eon the location of one of the planets, while all the other planetsareattheirideallocation. Figure 1showsaschematicoftheplanetwiththeerror. Underunloadedconditions,the error will cause the planet contacting surfaces
28、to come closer to, or move farther away from, their matingsurfaces. If the error causes the planet to come in contact earlier than the other planets, then the error isconsidered to be positive (planet leads all the other planets) and the planet with the error will carry more loadthanalltheotherplane
29、ts. Ontheotherhand,iftheerrorisnegative,theplanetwilllagalltheotherplanetsandcarryalighterloadthantheotherplanets. Themagnitudeofinequalityintheloadsharingwilldependuponthemagnitudebywhichtheplanet errorcauses themating sun-planetand ring-planetsurfaces tocome closerto(or move away from) each other.
30、Let OS and OR be lines parallel to the sun-pin and pin-ring planes of action; be the operating pressureangle; ebethepinholepositionerror; er, es, eTbetheerrorcomponentsalongthesunLOA,ringLOAandtangential direction; and be the orientation of the error with respect to the tangential direction OX.Then,
31、 the component of error along the planet-ring plane of action is:(1)er= ecos ( )The component of error along the planet-sun plane of action is:(2)es= ecos ( + )5 11FTM05Figure 1. Pinhole position errorThese are the amounts by which the planetsurfaces comecloser to(or movefarther awayfrom) theirmatin
32、gsurfaces. When er es, the firstsurface pair(say sunmesh) thatcomes incontact cannotcarry loaduntilthe planet rotates about its axis and the other surface pair (say ring mesh) also comes in contact. Ingeneral,the planet comes closer to its mating surfaces by an amount:(3)= eTcosep=er+ es2= ecoscosEq
33、uation 3showsthatwhentheerrorisinthe radialdirection, =90and ep=0. This explainswhy anerrorintheradialdirectionhasno influenceon epicyclicload sharing. Also,the magnitudeof theerror ismaximumwhen =0 or 180. When =0, epis positive and the planet will lead all the other planets, and when = 180, epis n
34、egative and the planet will lag all the other planets. For any arbitrary error direction, themagnitude of error in the tangential direction is the only relevant parameter.System floatInnon-floatingsystems,nomovementispossiblebetweenthecentersofthecoaxialmembers(sun,ring,andcarrier). Inthesesystems,a
35、lltheerror hasto beneutralized bythe elasticdeformation inthesystem. Ontheother hand, in floatingsystems thecenter of at least one of the coaxial members is free to move radially, andthusrelativemotionbetweenthecoaxialmembersispossible. Themajoradvantageoffloatingsystemsovernon-floatingsystemsisthat
36、aportionofthepositionalerrorisneutralizedbysystemfloat. Theremainingerroris neutralized by system deflections. The portion of the error that is neutralized by system deflection is thecause of the load sharing inequality.Elastic deformation under loadIn a rigid system (rigid gear tooth surfaces and r
37、igid bearing supports), the presence of a positive error willcause the entire load to be carried by the planet with the error, and all other planets will remain unloaded.However, in elastic systems, as the planet with the error gets loaded, the tooth flanks in mesh and the planetonthe needlebearing
38、supportsundergo elasticdeformation, andthis causesthe errorto beneutralized. Theforcerequiredtoneutralizeagivenerrorwillbecalledtheneutralizingforceandthecorrespondingtorquewillbe called the neutralizing torque. Since the only relevant error is the component in the tangential direction,6 11FTM05and
39、the net resultant of forces acting on the planet center is in the tangential direction, all computations ofstiffnesses and deflections will be performed in the tangential direction and at the center of the planet.Let Kb, Ksand Krbe the bearing stiffness (includes needlesand planetarypin), thesun-pla
40、net meshstiffness(duetodeformationofboththesunandpinionmembers),andtheplanet-ringmeshstiffness(duetodeforma-tion of the pinion and ring members), respectively. The effective stiffness of the sun-planet-ringbearingsystem in the tangential direction is:(4)1Keff=1Kb+1Ks+ KrKeffis the cumulative stiffne
41、ss due to Hertzian contact at the sunplanet and planetring meshes, the toothbending deflections, the tooth base rotation, and the planet bearing and pin stiffnesses. Keffcan be con-sideredtobelumpedatthecenteroftheplanet,andtherestofthesystemcanbeconsideredtoberigid. Keffisa property of the sun-plan
42、et-ring-needle bearing system and isgenerally invariantwith thenumber ofplanetsin the system.Intherestofthispaper,wewillfocusonthetangentialerror eT(orsimplyerrore)andthetangentialneutraliz-ingforceortorquerequiredtoneutralizethiserror. Also,all thestiffnesses willbe lumpedin theKeffterm,andthe rest
43、 of the system will be assumed to be rigid.Non-dimensional neutralizing ratioIf an n planet epicyclic gear set has an error e on the position of one of its planets, and all other planets are attheir ideal location, then the neutralizing force is given by:(5)Fe= KeffeThe corresponding non-dimensional
44、 neutralizing ratio is the ratio of the neutralizing force to the total inputforce, or the neutralizing torque to the total input torque:(6)Xe=KeffeFl=FeFl=TeTlThe neutralizing ratio captures the influence of the system flexibility, the amount of error, and the loading onthe gear set. In this paper,
45、 we will express the load sharing behavior in terms of the neutralizing ratio. Thedevelopedexpressionswillbeapplicableforanyepicyclicgearset,regardlessoferrororloadinglevel,systemstiffness or number of planets.Equal load sharing in the absence of errorsConsider an n planet epicyclic gear set and ass
46、ume the meshes to be in-phase (phasing has a transientsecondary effect on mesh load sharing). Under ideal conditions, all n mesh paths are simultaneously in con-tact. Under this condition, all meshes will share the load equally, and 1/n of the load will pass through eachsun-planet-ringpath. Equallys
47、pacedplanets,intheabsenceoflocationerrors,alwayssharetheloadequallyamongstalltheplanets. Evenslightlyunequallyspacedplanets(smalldeviationsfromequallyspaced duetoassembly considerations) share the load approximately equally.Schematic representationWewilluseaschematicrepresentationtodescribetheloadsh
48、aring behavior. Figure 2(a)shows anexampleof a 5 planet epicyclic gear set with no position errors and all planets in contact, and Figure 2(b) shows itsequivalentschematic. Intheschematic,thecentersoftheplanetsareconnectedtothecenterofthecarrierbyrigid arms and the sun-ring bodies are represented by
49、 another rigid body. The combined stiffness, Keff,ofallthe elastic bodies (sun, planet, ring, bearings, planet pin, and carrier) is lumped at the interface between theplanetandthesun-ringbodies. Thoughthisstiffnessisnotexplicitlyshownintheschematics,itisassumedtoalwaysbepresentattheplanetcenters. Theregionsofcontactarerepresentedby as shownin thefigure,andallloadedelasticdeformations ( )takeplaceintheseregions. Interferencewillberepresentedasshownin Figure 2(c), and such material penetration will
