Dynamics Of Planetary Gear Trains - NASA
NASA Contractor Report 3793iNASA-CR-3793 19840017959Dynamics of Planetary Gear TrainsrR. August, R. Kasuba,v- : t-,J. L. Frater, and A. Pintz",GRANT NAG3-186JUNE 1984--L. ',/#":. ./
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NASA ContractorDynamicsReport 3793of PlanetaryR. August, R. Kasuba,J. L. Frater, and A. PintzCleveland State UniversityCleveland, OhioPrepared forLewis Research Centerunder Grant NAG3o186N/LSANationalAeronauticsand Space AdministrationSclentlllc and TechnicalInformationBranch1984Gear Trains
hase .59IncorporatingPositioning66Synchronousof PGT'sStaticPlanetPlanetErrors.Positioning.75Load .4.2.1IV.Remarks1
iv4. i. IXABAnalysisPlanet4.1.2.2DisplacementofSun Gear eedAPPENDIX D104.ii0.113.Operation.of116118113138Developmentof the GearVariableMesh rEpicyclic GearingAPPENDIX Types140of.148Calculation of Total IntegrationTime and Step Size .153Computer Program Package .166
1CHAPTER IINTRODUCTIONI.iGeneral RemarksEpicyclic gearing arrangementsare comprisedof fourdifferent elements that produce a wide range of speed ratiosin a compact layout.externallyThese elements are:(i) Sun gear, antoothed ring gear co-axial with the gear train;(2) Annulus, an internally toothed ring gear co-axial withthe gear train;(3) Planets, externally toothed gears whichmesh with the sun and annulus;and (4) Planet Carrier,asupport structure for the planets, co-axial with the train.The name "epicyclic" is derived from the curve traced by apoint on the circumferenceof a circle as it rolls on thecircumference of a second fixed circle.By fixing one of the co-axialremainingmembersand usingthetwo for input and output, three types of simplesingle-stageepicyclicgearingthese are called planetary,are possible.Generally,star, and solar arrangements.This investigation is primarily concerned with planetary geardrives(Fig. I) which have a fixed annulus with the planetcarrier rotating in the same direction as the sun gear.The principaladvantagesof epicyclicgearsoverparallel shaft gears are considerable savings in weight andspace (see Table i).These advantages stem from the fact
////'///////Ring gearm gearPlanetcarrier(Ou tput)IPlanet gearFigure 1 - Planetary Gear Train
ApplicationPowerTurbo-pump4360 h.p.Speeds, rpmRatio6160/11205.5 : 1Turbo-generator1060 kw.6000/600i0 : 1Drive train typeWeight, lb.Heaviest component, lb.Parallel shaft34001730Epicyclic25001600Parallel shaft38002100Epicyclic22001500Circular pitch, in.Pitch circle dia., in.0.907Pinion 5.774Gear31.7540.725Pinion 3.695Gear36.95096.50.518Sun2.474Planet 9.897Ring 22.26958.5Pitch line velocity, fpsHorsepower losses1550.605Sun4.619Planet 8.083Ring 20.785102BearingToothGearcase size, LengthWidthHeightTable 1 - Comparison of Parallel Shaft -vs- Epicyclic Gear Trains
eesixplanets,toeachcarryoneofdoofnota similarto ttorque.isthe bearingsgearingequivalentbeco-axialof a hip.bethetwiceradialmusthavea gearingmorethewouldthatadvantageorthreeof the piniontransmitcommerciallyA secondwithConsequently,preferred,aregearloada nstobegivehavingthetheandshaftsarrangementbe tandoftotalwouldgear-traintootha planetarythesun sedbyexternalgearingis thatconditions.Anotheradvantageof epicyclicthesmaller gears used can be made more accurately and with lessdifficultyTheythan the larger ones of parallelare easierto handleandto harden,shaft gearing.and distortionduring hardening is not such a serious problem.
5Whilethe ring gear is not carburizedbecauseof thedifficulty of precision grinding of internal teeth, it can behardened by nitriding.gearscanNormally,be runThis causes so little distortion thatwithouthardeningany post-nitridingprocessing.of the ring teeth is not that criticalsince the surface stress between internal and external teethis less than that betweensurfaceof the internaltwo externa!teeth.tooth in contactThe concavewiththe convexsurface of the planet tooth results in a larger contact areathan for two externalwear load.teeth, thus increasingAlso, for a given size and number,the limitingteeth cut onan internal ring are stronger than those cut on an equivalentexternal wheel.Use of smaller components giyes lower pitch line velocities.This accountsfor epicyclicgear trains being morequiet than parallel shaft gear trains.mesh,not shiftingHaving more teeth inthe load so abruptly,also reducesthenoise level.Generally,single-stage epicyclic gear trains are moreefficient than equivalent parallel shaft gear trains becausepowerlosses occurringlosses are reduced.through tooth frictionand bearingTooth friction losses are approximatelyproportional to the tooth load and the pitch line velocities.With smallertooth loads and slower pitch line ve!ocities,the frictionloss in epicyclicgears is less than parallelshaft gears running at the same rotationalsame load.Bearinglosses are dependentspeed with theon bearingsize,
6which are smaller on epicyclic gears since no tooth reactionloads are carried.Epicyclic gear systems have a long history of industrialuse.As early as 1781, James Watt patented a sun and planetgear arrangement used in one of his early engines.advancesin internalgear manufacturingHowever,did not parallelthose in external gears, limiting the development of edtransmissions with higher power ratings, the performanceepicyclicofgear systems became poorer at higher loads sinceload equalization among the planet gears was not realized dueto poor manufacturing and assembly techniques.However, the compact layout and inline arrangement foundfavor with the early automobile swereno longersince thepowerstationaryintegral part of the moving machine.generallygearscreditedin automotiveWeight and sizeobjects,butanDr. F. W. Lanchester iswith being the first to useapplicationssource andwithepicyclicthe annulusof thefirst stage used as the planet carrier of the second stage toform a compound planetary transmission.W. G. Stoeckicht adaptedand marine1700applications.RPM gearsaircraftepicyclicHe designedgears for aircraftthe 3300 HP, 3200 tofor the Jumo 222, the largest pistonengine developedin Germany.typeHe also designeda5000 HP, 3770 to 550 RPM marine main propulsion unit, whichhad a gear case three feet in diameter,2 1/2 feet long, and
whose totalweight was less than one ton.Presently, planetary gears are frequently used as mainreductionships.gearsin propulsionThey are widely usedhelicopter aircraft.gasturbinesfor merchantin rotor drive gearboxesforIn lower horsepower applications, highratio planetary systems are combined with hydrostatic drivesto producewheeldrivesfor agricultura!and off-highwayequ ipment.In an attemptto save weightand materialand obtainbetter load sharing, current gear train designs are increasingly incorporating lighter, more flexible structures.result,at highertooth-meshingfrequencies,As athe dynamicbehavior of the gears becomes of increasing interest becauseof its affect on gearbox noise and life and powerrating ofthe transmission.Efforts have been made to determinethe instantaneousloads on the gear teeth, but few have examined the influenceof the entire gear train on the dynamicloads.Most modelsused to date did not account for non-conjugategear actioncaused by the deflectionof teeth and other elementsunderload, or by inherent errors caused by gear manufacturing andassembly.Also, most models did notconsiderload sharing among planetsteethat eachplanetchanges in thewhen the number of engaged gearchange.Accuratemodelingof thesun/planet and planet/ring tooth engagements directly affectsthe determination of the instantaneous load.In additionto the tooth engagementvariations,the
8dynamicloadingon the gearteethis dependentupontheinteraction of the components that make up the gear train andthe load transmittedgators,fortheby the gear advariations which normally occur because of the elastic natureof the gear train elements.consideringconstantUsuallytorques appliedconstant velocity ratios maintainedthe analysisis madeto the input member,betweenmeshinggears,and a constant torque withdrawn from the output member.More recently,have madeto permittrain.large-scaledigitalcomputerprogramsit possible to investigate gear tooth interactionsa morerealisticThis wouldallowmodelfor the epicyclicthe study of dynamicgearloadingbyconsidering tooth engagement which is affected by characteristics of the gears in mesh, such as errors, tooth stiffnesses, massesof the gears and contactratios.Also to beconsidered would be the influence of the gear train composedof connecting shafts, gears, bearing supports, couplings, andtorque inputs.It is the purposeof this investigationto developamore comprehensive mode! which considers the affect of toothengagement and system parameters.This model will be used toimprove the current analytical methods for determininginstantaneous loads to which gear teeth are subjected.theSinceplanetary gear systems are most commonly used, and techniquesexistto determinecomprehensivemethodequivalentplanetaryfor analyzinggeartrains,athe static and dynamic
loading in a planetaryvariable-variablegear train will be developed using amesh stiffnessand internal gear teeth.linearsharing,toothnormeshhaveNo currentstiffnessany(VVMS) model for externaltechnique uses a non-in xamining the effect of the phase relationships of the VVMSon the dynamicbehaviorof the gears.Consequently,thiswork extends the scope of engineering analysis of epicyclicgearing.The analysis is applicable toward either involuteor modified spur gearing.
l0CHAPTER IILITERATURE REVIEWSince the planetarygear train (PGT) is an assemblyofboth external and internal spur gears, this literature reviewrelieson informationcomponentsepicyclicfor PGT's as well as its individualin showinggearing.the progressInvestigationand currentstatusinto the actionofof theseparate elements must be made in order to better understandthe behavior of the entire system.Consequently, informationis presented chronologically, and in separate sections, forexternal/internal spur gears and PGT's, respectively.2.1 Spur GearsA major factor in the design of spur gears is the powerthat must be transmittedfrom the primemoverto the load.The force that is transmitted becomes important in designingfor gear tooth beamfactor.strength,contactand scoringDynamic effects must be considered since the load isbeing transmitted by an elastic medium,teeth.stressConsequently,the nstantaneousi.e., deflected gearload to which theengaged gear teeth are subjected is generally higher than thenominal statically calculated load.Duringthelate1920's and earlySociety of Mechanica!1930's,the AmericanEngineers Research Committeeinvesti-
iigated dynamic loading of gear teeth.Lewis and Buckinghamconducted tests to determine the effects of operating speedand manufacturingerrorsin theinvolutetooth profile.Their report presented a method to evaluate the dynamic loadincrement due to gear train dynamics and tooth error.Thesestudies presented a dynamic load solution more commonly knownas Buckingham's Equation's [i]*Tuplin[2] usedan equivalentspring-masssystemrepresenting the gears in mesh to determine the dynamic loadsin gear teeth.The masswas determinedfrom equivalentmasses of the gears concentrated at the pitch circles.springstiffnessdeflectionof twowasdeterminedcontactingfromteeth.thestaticIt wasTheload-consideredconstant and linear. He felt that dynamic loads occured fromthe passage of "thick" teeth through the meshing zone.Theexcess width impressed a displacement upon the mesh spring,introducing dynamic effects.shapes of excitation,thatthe shapeTuplin investigated differenti.e., profileof excitationerrors, and concludedhad littleeffecton hisresuits.His calculatedaverage transmittedload increment was independent of theload and was inversely proportionalthe pitch-line velocity.toAlso, his equations did not accountfor system damping nor multiple tooth contact.In general,Tuplin's analysisa constantcan be consideredas usingequivalent mesh stiffness.*Numbers in brackets refer to bibliography
12Bollinger[3] was one of the first to consider the toothstiffnessas a periodiceffectivestiffnesschange fromsingleback from doublestudy correlatedfunction.betweengear pairsto doubleto singleDiscontinuitiesresultedtooth contacttooth contact.the experimentalin thefromtheand the changeResultsand analyticalof hiswork verywell.This techniquemesh stiffness modelcan be defined as a fixed-variable(FVMS).gearAlthough it is an improvementover the constant mesh stiffness model,the simplificationsused in the model can be generalized as:a.It does not account for irregular sharing of theload between simultaneously engaged tooth pairs.Therefore, analysis is limited to contact ratiosbelow 2.0.b.Gear tooth errors have negligible effect or none onmesh stiffness.This implies for a given load, agear with errors will have an equivalent meshstiffness as the same gear without errors.c.Contactaction.d.The contact ratio and/or mesh stiffness is notaffected by pre-matureor post-maturecontactcaused by deflection of the gear teeth under load.e.The mesh stiffness has a fixed functional relationto the displacement of the gear.is assumedIn an effortto occur only on the line ofto determinetoothmeshstiffnesses,investigations have been made to determine gear tooth deflections as a functionWeberactualof the point[4] used the strain energyshapeof thetoothof load of application.techniqueprofileinalong with thehisdeflection
13analysis.Normal, shear, and bending energy in the tooth andits an deformationwas calculated using the tooth profileradiias equivalentof curvaturecylinders.Attia[5]expanded Weber's model by including circumferential deformation of the gear rim and the deflection of a tooth under theeffect of loaded neighboring teeth.KasubaandEvans[6] useda largescaledigitizedapproach to calculate directly the gear mesh stiffness of twoexternal spur gears as a functionprofileerrors,of transmittedgear tooth deflections,load, geargear hub deforma-tions, and position of tooth contact and the number of toothpairsin r mesh stiffness (Fig. 2) was defined as beinga variable-variablemesh stiffness(VVMS).The VVMS modelhas the following properties:a.It simulatesthe gear system by including theelastic effects of the entire system and not justthe gears.b.The stiffness of the gear teeth is consideredfunction of the position of contact.c.It allows for multiple tooth contact by examiningeach pair of teeth that are in and close to thetheoretical contact zone.d.The gear tooth profiles are defined with errors tosimulate tooth profile errors found in manufacture.e.It allows for backlash in the gears.The actual load sharing and deflectionaare calculatedfor discrete positions within an established mesh arc.For
14-4ZGear mesh stiffness cycleTooth pair mesh stiffness cycleANGLE OF ROTATION(degrees)Point APoint B-Reference tooth pair enters contact zonePreceding tooth pair exits contact zonePoint CPoint D-Following tooth pair enters contact zoneReference tooth pair exits contact zoneFigure 2 - Variable-Variable Mesh Stiffness
15any positionKP(k)i,i, the calculated kth gear tooth pair tiffnesseffectsdueKGi,andto profileand tooth deflections.loadsharingerrors,profileThe individualgearpair stiffness then isKP(k) i Q(k)i/ (k) iwhere:(2.1)Q(k)i gear pair static tooth load(k)i tooth deflection due to bending,compression, and shear forces; andtotal hub deformationIf the effective errors prevent contact, KP(k)i O.The mesh stiffness at the ith positiongeartooth pairstiffnessesforis the sum of theall pairsin contactatposition i,KG i KP(k)i Theconceptof theVVMSwas(2.2)furtherexpandedbyintroducing an iterative procedure to calculate the VVMS bysolving the staticallyindeterminate problemof multi-paircontacts, changes in the contact ratio, and mesh deflections.The results of the VVMS model showed that:a.The maximum instantaneous load occurred immediatelyafter a change in the number of teeth in contact.b.Dynamic load factors decrease with increasingaverage transmitted load between gears.c.Dynamicload factorsvary with the speed ofoperationof the system and closenessto theresonances of the system.
16d.The gear system can be tuned through the use oftorsionallyflexible gear bodies/hubs/rimstoreduce dynamic load factors.e.Dynamic load factors are reduced when the contactratio is increased.f.Dynamic load factors are reduced when the systemdamping is increased.Pintz[7] used a similarVVMS for an internal-externaltechniquefor developinggear mesh.theThis was used todetermine the dynamic loads experienced in an interna! geardrive.Pintz's model used an internalradiallysupportedring gear which wasand driven by an externallyHe stated that the VVMS model was appropriate"very high contactdrives.uniqueratios" encounteredwithtooth gear.in light of theinternalgearUse of the VVMS allowed investigation into problemsto internal gears such as ring gear deflections.Heexamined the effect of ring deflections on the gear mesh, andfound considerable missing and backhitting of the gear teethsimilar to the performanceerrors.of gears with sinusoidal profileAlso, larger dynamic load factors were found with anincrease in radial deflection.Theeffectiveworkwayof KasubaandPintzto date of determiningrepresentsthemostthe discontinuous,non-constant, gear mesh stiffness.2.2Ep icyclic GearsEpicyclicgearshave been used as early as the 170O's.However,the firstrecordedattemptto apply engineeringanalysiswas made in the very late 18OO's.Lanchester[8]
17recognized the need for "precision in workmanship" to achieveoptimum performancefrom the planetarygear train by equalload distribution among the planet gears.workingclearancefor a three planetHe found that thesystemallowedtheplanets to find an equilibrium postion such that the load wasevenly distributed among the planets.combinationassembledBut for a four pinionwith the highest degreeof accuracypossible, it was found that the load was not quite so evenlydistributed among the four planets.Wilson[9] wasableto showthathisthrees
Generally, single-stage epicyclic gear trains are more efficient than equivalent parallel shaft gear trains because power losses occurring through tooth friction and bearing losses are reduced. Tooth friction losses are approximately proportional to the tooth load and the pitch line velociti
This sun gear drives three planetary wheels, which roll off on an internal gear with internal gear teeth, as a single-stage gearbox and four planetary wheels as a two-stage gearbox. The planetary gears are fixed in the planet carrier, which acts as the output. By dividing the load among the planetary gear on three or four planetary wheels, a .
Work out the gear ratios of these gears: (numbers inside the circles are the numbers of teeth on each gear) Blue drive gear, green driven gear, pink idler gear, yellow compound gear 1: Simple meshed gear 2: Simple gear train with idler gear 15 3: Compound gear train 4: Choose the correct words to make these statements true for these gear trains.
70% of the gearbox business are of planetary type with a torque range up to 5,000 kNm. Helical planetary gear units This design was originally developed for roller crushers in the cement industry. It consists of a double planetary stage and a helical stage on the input. The planetary stages are standard while
3. Explain what a gear train does? 4. Explain what a simple gear train is? 5. Explain torque? 6. What does a larger gear driving a smaller gear achieve? 7. What does a smaller gear driving a larger gear achieve? 8. What is the driven gear? 9. What is the driver gear? 10. What is the idler gear? 11. What is the calculation for working out gear .
form a gear set or a gear train . In a gear set, the input gear is called _ gear. The output gear is called the _ gear. Note: An idler gear or intermediate gear changes direction of rotation without change in gear set speed or torque Spinner &
element model of a planetary gear set to determine the effect of internal gear flexibility on deflections and bending stresses. Bo-das and Kahraman [10] examined the influence of carrier and gear manufacturing errors on planetary gear sets using a two di-mensional finite elem
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