1、10FTM16AGMA Technical PaperAnalysis of LoadDistribution inPlanet-Gear BearingsBy L. Mignot, L. Bonnard andV. Abousleiman, Hispano-SuizaAnalysis of Load Distribution in Planet-Gear BearingsLouis Mignot, Loic Bonnard and Vincent Abousleiman, Hispano-SuizaThe statements and opinions contained herein ar
2、e those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractIn epicyclic gear sets aimed at aeronautical applications, planet-gears are generally supported by sphericalroller bearings with bearing outer race being integral
3、to the gear hub. This paper presents a new method tocompute roller load distribution in such bearings where the outer ring cant be considered rigid. Based on wellknown Harris method, a modified formulation enables to account for centrifugal effects due to planet-carrierrotation and to assess roller
4、loads at any position throughout the rotation cycle. New model load distributionpredictions show discrepancies with results presented by Harris, but are well correlated with 1D and 3D FiniteElement Models. Several results validate the use of simplified analytical models to assess the roller loaddist
5、ribution instead of more time consuming Finite element Models. The effects of centrifugal effects due toplanet-carrier rotation on roller loads are also analyzed. Finally the impact of the positions of rollers relative tothe gear mesh forces on the load distribution is shown.Copyright 2010American G
6、ear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October 2010ISBN: 978-1-55589-991-23Analysis of Load Distribution in Planet-Gear BearingsLouis Mignot, Loc Bonnard, and Vincent Abousleiman, Hispano-SuizaIntroductionEpicyclic gear sets are known to be powertrans
7、mission systems that provide high capacity,power density and efficiency. For these reasons,they are widely used in various aeronauticalapplications such as helicopter main gearboxes andturboprop power gearboxes where the weight is acritical performance criterion. In planetary andepicyclic gearboxes,
8、 high loads are transmitted viathe planet-carrier which could result in misalignedcontacts on gear meshes or planet bearings.Conventional gearbox designs thus include spher-ical roller bearings to support the planets on theplanet-carrier axles. These bearings can cope withmisalignment angles up to 1
9、.5 degrees 1 whileproviding a good radial load carrying capacity. Apast study 2 has shown that bearings are a majorsource of failure in epicyclic gear sets. The authorsdemonstrated also that the optimization ofplanet-bearings design can provide significantweight reduction since the saving obtained o
10、n oneplanet is multiplied by the number of planets whichis generally greater than four. At the early gearboxdesign phase, it is thus essential to performparametric studies in order to find the mostoptimized design for the planet bearings.Spherical roller bearings in aeronautical epicyclicgear sets a
11、re characterized by two main features:S For weight saving reasons, the outer-ring ofthese spherical roller bearings are usually madeintegral to the planet-gear hub which is, inaddition, made as thin as possible. Gear meshforces induced by the sun-planet and thering-planet meshes are thus applied dir
12、ectly tothe outer-ring at localized points. The conven-tional assumption of rigid bearing outer-ringsubmitted to a concentrated load is here not val-id. The outer-ring must be considered deform-able to determine the roller load distribution.S In epicyclic gear sets, the carrier is rotating whilethe
13、ring is stationary (Figure 1). This makes theplanet-bearings kinematics rather complex withthe inner-ring rotating around the gearbox mainaxis while the outer-ring is rotating around theinner-ring (planet-carrier axle). The effect of thecentrifugal loads induced by the outer-ringweight could influen
14、ce the roller load distributionsince the ratio of centrifugal loads to accumu-lated radial gear loads can be as high as 20% fortypical turboprop applications.In this regard, several studies have been conductedto determine the influence of a deformableouter-ring on the bearing loading. An analyticala
15、pproach has been proposed in 1963 by Jones andHarris 3 and also described in Harris 4. The mainresults showed that the outer race distortion modi-fies significantly the roller load distributioncompared with rigid outer race assumption: thenumber of loaded rollers increases and the mostloaded roller
16、is no longer located along gear tangen-tial direction but close to mesh force application.Effects of bearings diametral and out-of-roundclearance were also obtained with the same modelby Harris et al. 5. For this model, though, no finiteelement (FE) validation exists which could give anidea of the p
17、erformances of the model.Liu and Chiu 6 proposed a model that accounts forinertial effects induced by planet-carrier rotationand roller centrifugal forces. The main resultsshowed the influence of the bearing diametralclearance on roller load distribution and fatigue life.Some discrepancies were also
18、 observed by theauthors compared to Jones and Harris study 3.Other authors have proposed an FEM approach.Drago et al. 7 studied the effect of planet bearingouter race deformation on gear stresses using a 3Dfinite element model. The authors demonstratedthat the optimization of roller loads can detrim
19、entallyaffect the gear stresses and that the planet bearingcant be designed without accounting for gearstresses.The model presented in this paper is based onJones and Harris 3 approach. In a first part, arecall of Jones and Harris model will be given. Acomparison of predicted loads and deformationsw
20、ith 1D and 3D Finite Elements models will showdiscrepancies that can be explained by theassumptions made in Jones and Harris equations.4In a second part, a new model is proposed that cansolve non-symmetric problems to account forcentrifugal effects due to planet-carrier rotation.The results analyze
21、the effects of centrifugaleffects. The influence of rollers positions withrespect to the mesh loads is also studied.Analysis of state of the art modelJones and Harris analytical model descriptionIn Jones and Harris approach, the outer-ringflexibility is modeled as a thin elastic ring with amean radi
22、us R and a section moment of inertia I(Figure 1). The effect of gear teeth on ring stiffnessis neglected. The loads acting on the bearing outerring are simplified as two equal and diametrically op-posed loads representing respectively sun-planetand ring-gear-planet meshes (Figure 2). The meshloads a
23、re assumed to act along the line of action andon the pitch radius RP.Figure 1. Loads acting on a planetThese loads can be decomposed in elementaryradial (Fs) and tangential (Ft) forces and a moment(M) acting on the elastic ring mean radius R.Therollers are assumed to be equally spaced aroundthe oute
24、r-ring with the first roller being located alongthe Ox axis defined in Figure 3. The position of theroller number j is characterized by an angle iandthe reaction force of this roller on the outer-ring isnoted Qi. In summary, the planet-gear outer-ring issubmitted to mesh forces Fs, Ft, M and roller
25、contactloads Qi.Figure 2. Simplification of mesh loadsFigure 3. Forces acting on the planet-gearouter-ringThe system studied is thus symmetric around theOx axis. This makes it impossible to study theinfluence of centrifugal effects which introduce anasymmetric force acting along the Oy axis, or tost
26、udy the load distribution with arbitrary rollerpositions. The radial elastic ring deflection at point idue to a load P is expressed by means of influence5coefficients CPiwhich detailed expressions are notgiven in this paper but can be found in 3 and 4.uPi= CPiP (1)It is worth noting that the elastic
27、 ring deflection takesonly into account the bending in the ring but not thetension or shear. The validity of this approximationfor rings such as the planet-gears will be illustratedlater in this paper. The total radial displacement atany location i is thus obtained by combining theeffects of all ele
28、mentary loads which yields:+jCQjijQjui= u0cosi+ CFsiFs+ CMiM(2)Using Lundberg and Palmgren relationship, acontact condition can be defined at any roller j:Qj= K ujPd2bif ujPd2 0Qj= 0ifujPd2 0(3)wherePdis the diametral clearance,b = 3/2 for point contacts and b = 10/9 for linecontactsCombing equation
29、s 2 and 3 and writing the forceequilibrium along the Ox axis yields the followingsystem of N + 1 equations, where N is the number ofrollers: KjCQjijujPd2b= 0:iFt KjjujPd2bcosj= 0:0ui u0cosi CFsiFs CMiM(4)With,j= 0.5 for j=0 or j= 180j= 1 in all other casesAs suggested by the authors 3 and 4, thenon-
30、linear system can suitably be solved by usingthe iterative Newton Raphson method.Remarks on the model formulaeIn Jones and Harris model, the symmetric system issolved on a half ring. Therefore, the jcoefficientswere introduced in the force equilibrium equation(o) to take into account half of the loa
31、ds at rollerpositions j=0 and j= 180 , i.e., solving:Ft Q12 Q2cos2 Qjcosj= 0(5)In displacements equations (i), the displacementat any point i of the ring is calculated by consideringa pair of roller loads Qjsymmetric with respect to theOx axis (Figure 4a). It follows that, when the effectof the forc
32、e of roller #1 is taken into account, itshould be divided by two in order not to consider apair of loads Q1as shown in Figure 4b.a) b)Figure 4. Jones and Harris model assumption of symmetric roller loads6A modified system of equations is proposed to solvethis problem by introducing the coefficient j
33、in theterm:KjCQjijujPd2b(6)It yields: KjjCQjijujPd2b= 0:iFt KjjujPd2bcosj= 0:0ui u0cosi CFsiFs CMiM(7)With,j= 0.5 for j=0 or j= 180j= 1 in all other casesComparison of analytical models to FEMTwo different Finite Element Models (FEM) havebeen built in order to validate the results obtainedwith the H
34、arris initial model and the proposedmodified analytical model.The first FEM uses one-dimensional (1D) FiniteElements (Figure 5):S The ring is modeled as an assembly of beamelements which account for tension, bendingand shear effects.S The rollers contacts are modeled as non-linearsprings with force-
35、deflection relationship intro-duced via tabulated data following the Lundbergand Palmgren contact deflection law. Thisforce-deflection law accounts for the diametralclearance of the bearing as in equation 3.The second FEM uses three-dimensional (3) FiniteElements (Figure 6):S The ring is meshed with
36、 3D linear hexahedricand pentahedric elements.S The roller contact force-deflection is describedin the same way as in the previous 1D FEM. Inaddition, rigid body elements are connecting theouter race nodes to each roller spring in order todistribute the contact load along the race width.S The gear m
37、esh loads are assumed to beuniformly distributed across the gear width.Figure 5. Example of 1D Beam FEMFigure 6. Example of 3D FEMThe example used in this paper is based on the datafound in Table 1.7Table 1. Sample dataNumber of rollers per row 12Number of rows 1Roller diameter 12,5 mmRoller length
38、40 mmBearing clearance 0mmRoller contact angle 0Outer ring section moment ofinertia3081 mm4Radius of outer ring neutralaxis70,3 mmGear pitch radius 79,5 mmGear mesh tangential force 27096 NGear mesh separation force 9862 NGear moment 249372 N mmFigure 7 shows the roller loads for the initial Harrism
39、odel, the 1D and 3D FEM and the modified Harrismodel according to equation 5. The abscissarepresents the roller number according to conven-tion given in Figure 2. The plots show a goodcorrelation between the 1D FEM and the modifiedHarris model for all rollers, whereas the Harris initialmodel predict
40、s significantly lower contact load forthe roller number #1 and higher loads for the otherrollers. This result confirms that the missing term inJones and Harris model has a strong importance onthe roller load distribution and must be taken intoaccount.The good agreement between the 1D FEM and themodi
41、fied analytical model confirms also that neg-lecting the tension and shear effects in the elasticring deflection formula of the Harris analytical modelis a valid assumption.Finally, the plot in Figure 7 shows that the roller loaddistribution given by the 3D FEM is similar to that ofthe 1D FEM and th
42、e modified Harris model. In thiscase, the contact loads are greater but distributedon a smaller number of rollers since rollers #5 and#9 are not loaded for the 3D FEM. These small dif-ferences demonstrate that the impact of teethstiffness and 3D effects dont play a significant rolein the load distri
43、bution.Figure 7. Roller load distribution for severalmodelsFigure 8 illustrates the ring radial displacement forthe modified analytical model and the 1D and 3DFEM. The Finite Element Model results appear tobe in a good agreement while the predicteddisplacements differ significantly with the analytic
44、almodel. The area of highest differences corres-ponds to the unloaded zone of the bearing (rollers#6 to #8) which explains why there is no impact onthe load distribution as observed on the Figure 7.These differences are believed to be a con-sequence of the effects of tension and shear whichis not ta
45、ken into account in the analytical model.Figure 8. Ring radial displacement for severalmodelsThe effect of the way the gear mesh force is appliedto the model has been investigated. The 3D FEMhas been run for two configurations:S The reference model considers the gear meshforce as a unique force appl
46、ied at the gear pitchdiameter and along the line of action.8S The second model accounts for the actual loadapplication points on the different teeth incontact. In this case (Figure 9), the contact isassumed to be distributed on three teeth (HCRgears).Figure 9. Influence of the gear mesh forceapplica
47、tionFigure 10 shows that the roller load distributionobtained with both models are very similar. A pre-cise description of the tooth load distribution on theplanet-gear is thus not necessary as long as theassessment of the roller load distribution is con-cerned.Figure 10. Roller load distribution fo
48、r severalmodelsThe results from Figure 7 through Figure 10validates the use of a simplified analytical model inthe prediction of roller load distribution. This is animportant result since the analytical models can befast to run and enable to perform parametric studiesin a short time which is less ea
49、sy with 1D or 3D FEM.Improved analytical modelModel descriptionIn the preceding sections, the Jones and Harrismodel was modified to include a missing term in theequations that appeared to be crucial in the analysisof load distribution in planet-gear bearings. But thismodel is based on the assumption that the problemstudied is symmetric about the Ox axis. Thisprevents, in particular:S Account for the centrifugal force Fc ind
