Ahmet Kahraman 1Director, Center for Gear Research

The University of Toledo

Nitschke Hall 4045,

Toledo, OH 43606-3390

e-mail: akahrama@eng.utoledo.edu

Sandeep VijayakarAdvanced Numerical Solutions Inc.

3956 Brown Park Dr. Suite B,

Hilliard, OH 43026

Effect offlexibility of an internal gear on the quasi-static behavior of a planetary gear setis investigated. A state-of-the-art finite elements/semi-analytical nonlinear contact me-chanics formulation is employed to model a typical automotive automatic transmissionplanetary unit. The model considers each gear as deformable bodies and meshes them topredict loads, stresses and deformations of the gears. Actual support and spline condi-tions are included in the model. The rim thidkness of the internal gear is varied relativeto the tooth height and gear deflections and bending stresses are quantified as a functionof rim thickness. Influence afrim thickness on the load sharing amongst the planets is alsoinvestigated with and without floating sun gear condition. The results are discussed indetail and guidelines regarding the design cj>f a planetary internal gear are presented.[DOl: 10.1115/1.1371477]

A segment of the internal gear was modeled using conventionalFinite Element Method (FEM) with same boundary conditionsapplied in order to simulate the actual suppon conditions. Thesemodels do not include the other gears in the planetary gear set asthe forces acting on the planet-internal gear mesh are representedby a single point force applied mostly on a single tooth along theline of action. These models were instrumental in qualitativelydescribing the influence of the rim thickness on the bendingstresses of an internal gear, but they were not fully capable ofdescribing the behavior observed in a number of experiments onthis subject matter [11,14-17]. The accuracy of the stress predic-tions were strongly dependent on the suitability of the conven-tional FEM meshes to simulate the tooth, the boundary conditionsimposed to represent the actual suppon conditions, and the as-sumption that a point load can fully describe the actual loads onplanet mesh. Since a large ponion of the internal gear and theother gears (planets and the sun gear) are left out of these models,it was not possible to investigate the effect of internal gear rimthickness on the overall quasi-static (negligible dynamic effects)behavior of the planetary gear set including its influence on thestresses of planets and the sun gear and the load sharing amongstthe planets. Similarly, an accurate prediction of the shape and theamount of the internal gear deflections was also not possible forthe same reasons. Interactions of the internal gear flexibility witha large number of other factors including floating condition sungear and carrier, rim deflections of planets and the sun gear, andmanufacturing, assembly and piloting conditions and errors of thegears and the carrier cannot be studied by using these models.

In this study the influence of the flexibility of an internal gearon the quasi-static behavior of a planetary gear set is investigated.A state-of-the-art nonlinear contact mechanics formulation is usedto model a typical automotive automatic transmission planetaryunit. The model includes all of the gears of planetary gear set intheir deformable form, addressing the shoncomings of the previ-ous simplified models cited above. The model is used to quantifythe impact of the internal gear flexibility on the gear stresses anddeflections. The influence of rim thickness on the load sharingamongst the planets is also investigated with and without thefloating sun gear condition. At the end, guidelines regarding thedesign of a planetary internal gear are presented.

1 IntroductionPlanetary gear sets are used commonly in a number of automo-

tive and aerospace applications. Compact, coaxial design of plan-etary gear sets provides significant packaging advantages overcounter-shaft gear units. Different speed ratios can be achievedusing the same planetary gear set by simply changing input, out-put or reaction (fixed) members. In addition, planetary gear setsalso have a reputation of being quieter than counter-shaft systems.

Given their kinematic and geometric configuration, planetarygear sets require unique design knowledge [1]. As one of the keyparameters, the rim thickness of the internal gear must be definedcarefully by the designer in order the meet certain design objec-tives regarding power density, load sharing, noise and durability.The rim of the internal gear must be as thin as possible in order tomaximize the power density and minimize mass. For a typicalautomotive automatic transmission planetary gear set, a 1 mmreduction on rim thickness of an internal gear may result in up to20 percent reduction in mass of the internal gear, which is rathersignificant. Besides reducing mass, added internal gear flexibilitythrough reduced rim thickness is believed to offer a number offunctional improvements to the planetary gear set. Amongst them,reduced rim thickness is believed to reduce stresses on the othergears. It also potentially reduces the influence of a number ofinternal gear and carrier errors, and piloting inaccuracies. In addi-tion, it was reported that a flexible internal gear could help im-prove the load sharing amongst the planets when a number ofmanufacturing and assembly related gear and carrier errors arepresent [2-4]. Many of these benefits are based on hypotheses oractual field experience as a math-based set of guidelines on howthin an internal gear should be in order to achieve such improve-ments without failing itself [5] are yet to be established. This isthe main objective of this study.

The effect of rim thickness on gear stresses attracted significantattention in the past although the majority of it focused on theexternal spur gears with a thin rim. A number of theoretical stud-ies [6-9] modeled mostly a segment of an external spur gear witha thin rim. The gear segment was constrained using certain bound-ary conditions typically at the cut ends of the gear and a point loadalong the line of action was applied to simulate the forces im-posed on the gear by the mating pinion. The same approach wasapplied to planetary internal gears by various researchers [10-13].

lCorresponding author. Phone: (419) 530-8224, Fax: (419) 530-8206.Contributed by the Power Transmission & Gearing Committee for publication in

the JOURNAL OF MECHANICAL DESIGN. Manuscript received May 2000. AssociateEditor R. F. Handschuh.

2 Physical ModelResearchers have been attempting to model planetary gear sys-

tems numerically for over a decade. The work by Valco [18} isprobably the most recent attempt at a computational model for a

408 I Vol. 123, SEPTEMBER 2001 Transactions of the ASMECopyright @ 2001 by ASME

Table 1 Example planetary gear set design data. All dimen-sions are in mm unless specified

Sun InternalPlanet

341.5

21.31.8950.2

Number of teethModulePressure angle, deg.Circular tooth thicknessHob tip radiusFillet radiusOuter diameterRoot diameterMinor diameterInner (bore) diameterLinear tip modificationStarting roll of relief, deg.

52.7446.00

18 701.5 1.521.3 21.32.585 1.8950.2 ----0.5

30.50 116.55-129.6523.75 110.00

--103.4515.00 --0.010 --27 --

26.300.010

21

solve a problem with contact constraints. Convergence is notguaranteed, and if convergence does occur, it is usually very slow.

The constraints imposed by the contact between mating sur-faces are essentially linear inequality constraints [20]. A RevisedSimplex solver [21-23] is used in this formulation that provides aguarantee of convergence within a predetermined number of itera-tions. Furthermore, this solver can detect ill-posed contact prob-lems even before the solution process is started. The solver isspecifically designed for the linear inequality type constraintsfound in contact problems.

In the planetary gear system, many rigid body type degrees offreedom or mechanisms are constrained only by the contact con-ditions. This means that if a non-linear finite element code with'gap elements' is used, then the incremental stiffness matrices canbecome singular. Most commercial codes cannot proceed whenthis happens. Some workarounds are commonly used, such asadding imaginary linear and torsional springs to make the systemstiffness matrices non-singular. The spring stiffness can be madesmall, but the accuracy of results computed by such almost sin-gular stiffness matrices is questionable. The approach used herehas been to attach a reference frame to each individual compo-nent, and to carry out the finite element computations for eachindividual component separately in its own reference frame. Aslong as each finite element mesh is sufficiently well constrained toits reference frame, the stiffness matrices are well behaved. TheRevised Simplex based contact solver can take into account anyfree mechanisms in the system while computing the contact loadsin a very natural manner.

3 Parametric Study

3.1 Example System. A four-planet planetary gear set de-fined in Table 1 will be used as the example system to study theinfluence of the internal gear rim thickness. Here, this design re-flects a spur gear version of a typical automatic transmission plan-etary gear set. As shown in the finite element model of Fig. I, fourplanets are equally spaced around the sun gear 90 degrees apartfrom each other. The carrier, which is not shown in Fig. 1 forclarity purposes, is chosen to be the input member, and the sungear is chosen to be the output member. The internal gear is keptstationary allowing a 1:3.06 speed ratio. Eight equally spacedstraight splines at the outside surface of the internal gear prevent itfrom rotating. Each spline has a width of 6 mm and a height of 2rom. As suppon conditions can be very influential [24], specialcare is given to the modeling of the actual contacts at the splines.The model also comprehends any potential loss of contact be-tween the spline and housing surfaces due to excessive internalgear deformations. Planets, the carrier and the sun gear are allsupponed radially by isotropic bearings of stiffnesses 106 N/mm

multi-mesh gear system. He attempted to build a static two-dimensional finite element model of a planetary gear s:-;stem witha flexible ring gear. Because of the non-linearity introduced bytooth contact, a commercial non-linear finite element package wasused. However, there is a mismatch between the capabilities ofsuch general-purpose commercial programs and the spc'~';;,J needsof gear contact analysis. Consequently, run times for :i.,,: :"",1delran into hundreds of hours on a contemporary supercomplii.'.:r

In this study, the deformation of the planetary system underload is modeled using a combination of finite element and semi-analytical techniques. The gears of the planetary system havecomplex shapes that are best modeled by the finite elementmethod. However, the contact zones are extremely small, and aretypically two orders or magnitude smaller than the working depthsof the gear teeth. In order for contact equations to be well condi-tioned, it is necessary to have a large degree of freedom concen-trated inside the contact zones. Additionally, the contact zone isnot stationary, and traverses the surface of a gear tooth. Using apure finite element model under these circumstances would re-quire that one either uses an extremely refined mesh, or refines themesh only in the contact zone and re-meshes for every contactposition. Neither of these options is attractive.

Semi-analytical deformation models based on the half-space so-lution for a concentrated load, and numerically integrated over thecontact zone do not suffer from the problem of ill conditioning.They can accurately capture the steep displacement gradients in-side the contact zone. However, they cannot easily model theshape of the gear teeth.

The approach used in this study [19] is based on the use of thefinite element models only to compute relative deformation andstresses for points that are away from the contact zones. For pointswithin the contact zone, we use semi-analytical techniques tocompute the relative deformations and stresses. The 'near field'semi-analytical solution and the 'far field' finite element solutionsare matched at a 'matching surface'. Such a model is significantlymore difficult to program, but once implemented, can providemuch better resolution without using a highly refined finite ele-ment mesh.

The finite element model implemented here uses separate inter-polation schemes for the displacements and coordinates. The toothsurfaces are modeled by elements that have a very large numberof 'coordinate nodes', and can therefore accurately represent theinvolute shape and surface modifications. In the fillet region, theelements have a large number of 'displacement nodes' to correctlycapture the steep stress gradients.

The total number of finite element degrees of freedom in aplanetary model can be extremely large. This is so even with thefinite element model refined only as much as is necessary for thefar field solution. The total number of finite element degrees offreedom in the typical 2D planetary model is around 50,000. Theamount of CPU time and memory needed to run an analysis withsuch a large degree of freedom would make it impractical. Ahierarchical representation of the system was used, in which thesystem is built from many substructures, with each substructure inturn being composed of many substructures. The processes ofstiffness decomposition and load vector back-substitution now be-come very complex, and involve multiple recursive traversals ofthe substructure hierarchy. Nevertheless, it is now possible tokeep CPU requirements down to about 8 seconds per time stepand memory needs down to 128 MB. This has been made possibleby accepting programming complexity in exchange for an in-crease in speed.

Most commercial finite element codes treat contact using "gapelements." These gap elements are non-linear springs with zerostiffness when open and a very large stiffness when closed. Ageneral purpose non-linear equation solver is used to solve thesystem. Poor convergence of contact conditions is one of the big-gest problems caused by using such a general non-linear solver to

SEPTEMBER 2001, Vol. 123 / 409Journal of Mechanical Design

Table 2 Internal gear rim thickness parameter A as a functionof outer diameter (in mm)

O.D.,mm A

116.118.119.121.123.126.129.

1.001.251.501.752.002.503.00

Fig. 1 FE model of the example system

3.2 Effect of Rim Thickness on Gear Stresses and Deflec-tions. This study will first focus on the influence of the internalgear flexibility on the bending stress levels of the all three types ofgears within the planetary gear set. The flexibility of the internalgear is varied by essentially changing dour according to Table 2 toachieve different A values ranging from A = 1 to 3 as the rest ofthe planetary system parameters are kept the same. This rangecovers most automotive and aerospace applications in production.The analysis is also performed for the limiting case of a fixedinternal gear outer surface where the internal gear rim deflectionsare prevented completely (case of A = 00). First, stresses at fixedroot locations are predicted and compared as illustrated in Fig. 3.Here, stresses are predicted at 200 discrete angular positions,which span a 90-degree rotation of the carrier. This ensures thatany tooth on the internal gear goes through a complete loadingcycle since there are four planets in the system. In Fig. 3(a),principal stresses recorded at a fixed point near the root of aninternal gear tooth are plotted against the carrier rotation angle forthree different internal gear rim thickness values, A= 1,2 and 00.When A = 1, maximum and minimum values of the stresses arerather large as a direct result of rim deflections. These stress levelsdiminish as A is increased. As expected. for A = 00, the stressvalues are very low since the point at which the stresses are pre-dicted is at the root and the rim is not allowed to deflect. Such astrong dependence on A does not exist as far as the sun gear andplanet stresses are concerned as shown in Figs. 3(b) and 3(c).Overall planet stresses at this location near the root are influencedby both the gear bending and planet rim deformations whereas thesun gear is too thick for any rim effect.

The critical point where the maximum principal stress occurswithin the region that covers the entire fillet and the root is de-pendent on the value of A. In Fig. 4(a), the internal gear stressesare compared at four different locations along the fillet for A= 1. A surface parameter s is defined to start at the root (s=O)and increases linearly reaching a maximum value of s = 15 at the

per mm gear face width. In cases when the sun gear is allowed tofloat (move freely in radial direction), the bearing support is re-moved from the sun gear.

A dimensionless internal gear rim thickness parameter A is de-fined as the ratio of the rim thickness to the tooth height as

(1)dour-droolA=

droo,-dminor

~~~

where dou!' drool and dminor are the outside, root and minor di-ameters of the internal gear, respectively. Table 2 lists the internalgear outside diameters and the corresponding A values used inthis study. Segments of three internal gears with different A areillustrated in Fig. 2. Similar rim thickness parameters can be de-fined for the sun gear and planets again as the ratio of rim thick-ness to tooth heights. This parameter is also known as the back-upratio. The example system defined in Table 1 and illustrated inFig. 1 has values of 2.9 and 1.3 as the rim thickness parametersfor the sun gear and the planets, respectively. Given their smallradii relative to the internal gear, the rim deflections of these gearsare expected to be significantly less than those of the internal gear.The same input torque value of 75,000 N-mm per mm face widthis applied to the carrier throughout this study. This corresponds toan input torque value of 1500 N-m for a 20 mrn face width gearset.

~

A = 1.0 A = 1.5 A = 2.0

Fig. 2 Internal gears of the example system with different A

Transactions of the ASME410 I Vol. 123, SEPTEMBER 2001

55198246103765

1000 1000(8) Internal Gear

(a)A=1& 500~ f"I!I.- ,./"--~~ 500

I""0

"CO,Cog;t -500

toCl.~~ 0~ r

8 S=2 -s=6 --s= 10

---s= 14

~

~ -500-1000

400

(b) Planet.1000

400~ 200~.,-"'.,.= 0(I)

d

~

(b) /\=2.5 AI"'g~~~

~

--~

~:~~:::::::::::

r ~-200 '"

a.:?:.; 0'"~

US

-200

-400 ~

0 30 60

Carrier Rotation Angle. degrees

locate the critical principal stresses accurately. Even more teethmight be required to be included in the search when splines arenot equally spaced on the internal gears.

In Fig. 6, critical maximum and minimum principal stresses ofthe internal gear, planets and the sun gear are plotted as a functionof A. Here, ten consecutive teeth on the internal gear are consid-ered in order to properly include spline effects mentioned above.Also the entire root and fillet regions of all gears are consideredinstead of a particular fixed point. The critical principal stressvalues predicted for A=oo, umax,A=~ and umin.A=~, are used tonormalize the stress levels predicted for each A value, that isamax=umax/Umax.A=~ and amin=Umin/Umin.A=~. Here, at A=oo,umax,A=~ values for the internal, planet and sun gears are predictedto be 356, 437 and 594 MPa, respectively. Similarly, umin.A=~values for the internal, planet and sun gears are -147, -727 and-155 MPa, respectively. In Fig. 6, both amax and arnjn exhibit anexponential increase as A is reduced. At A= I, amax is greaterthan 2 and a miD is almost 10 which indicates that the internal gear

1500

I--Tooth#1 Ica 1000 I-Tooth#5 Ia.~

-500II)II)Q)J= 0(/)

"iij

.§" -500

.s

It -1000

-1500

0 60 120

Carrier Rotation Angle, degrees

~

-400

200"--.

(c) Sun Gear

~

(II

~ 100.;'"Q)

iT> 0

~~~

90

-100

Fig.

4 C:omparison of internal gear stresses at different rootand fillet locations-200

0 30 60 90Carrier Rotation Angle, degrees

Fig. 3 Comparison of stresses at fixed root locations as afunction of A; (a) internal gear, (b) planet (c) sun gear

end of tooth fillet. In Fig. 4, the sttesses are plotted at s = 2 (nearthe root), s=6, s= 10 and s= 14 (near the end of the fillet). In Fig.4(a), the sttess at s=2 is the highest amongst all four points ofinterest since the internal gear is very flexible (A = I) indicatingthat the rim effects are dominating. The critical maximum tensilesttess value occurs at around 75 degrees carrier rotation anglewhen the internal gear tooth is away from the gear mesh. Sttesslevels decrease as s is increased (moving away from the root)since the rim bending deflections do not influence sttesses at thefillet region significantly. Hence, if one is interested in the criticalsttess value, a point at the root would be the most appropriate toconsider for A = I. On the other hand, the situation is quite dif-ferent for A=2.5 as shown in Fig. 4(b). Now, the point at s=6experiences the highest tensile stress levels mostly due to thetooth bending effects while the tooth is still in mesh. This suggeststhat the location and magnitude of the critical sttess is sttonglydependent on A, and the entire root and fillet regions must besearched when the critical sttesses are sought.

In addition, internal gear stresses are influenced by the locationof the internal gear tooth of interest relative to the outside splines.This is illusttated in Fig. 5 for two internal gear teeth, tooth #1 ata spline location and tooth #5 at a location roughly midway be-tween two adjacent splines. Here, at the point where the sttessesrecorded, maximum sttess is 2 percent higher and minimum sttessis 24 percent lower for tooth #5 than those for tooth #1. Thisillusttates the influence of the splines on ring gear stresses asplanets pass through them. Therefore, in addition to consideringentire fillet and root regions (s = 0 to 15), one must also considerall internal gear teeth between two adjacent splines in order to

~~~

180

Fig.

5 Comparison of internal gear root str.~sses at (tooth #1)and away from (tooth #5) an outer spline loc:ation

Journal of Mechanical Design SEPTEMBER 2001, Vol. 123 I 411

~

10(a) Internal Gear

'\

8 '\r~~\:1~

4 --,~ Ic !

,,

..(/)OJ(/)(/)

~U5

""8-'u.EO

Q:"0OJ

.~"(ij

E0z

~ compressivf6

"""' ,l4

~ Fig. 7 Deflection shapes of internal gears having different A.Deflections are exaggerated 25 times.

0' max tensile2

0

1.5 2 2.5

1.10

;. 1.05(/)

1.00

~C-

'0

.1:It"CQ)

.~~E0z

tion in both inward and outward direction is plotted as a functionof A in Fig. 8. When A = 1, maximum inward and outward radialdeflections are predicted to be 0.128 and 0.156 mm, respectively.If an internal gear has the flexibility, which allows it to deflect thismuch, then one can expect that a number of manufacturing errorsassociated with the internal gear such as the roundness error andthe run-out error can be tolerated as long as their magnitudes areless than the amount of deflection.

3.3 Effect of Rim Thickness on Planet Load Sharing. Alarge number of inaccuracies associated with the manufacturing,assembly and piloting of each component of the gear set areknown to influence the load levels on each planet. Readers arereferred to [15,25] for a detailed classification of such inaccura-cies. Since the total torque transmitted has to remain the same,any unequal load sharing amongst the planets will overload cer-tain planets as the others carry lower levels of load than they aresupposed to. Internal gear flexibility was cited by a number ofinvestigators [2-4] as one of the methods to improve the loadsharing amongst the planets in the presence of such inaccuracies.Here, a tangential planet pin position error on the carrier will beintroduced as an example of such inaccuracies, and the influenceof A on the planet load sharing will be quantified. Tangentialplanet pin position errors (resulting in an error in center positionof a planet gear in circumferential direction) were reported to beone of the most significant types of inaccuracies affecting planetload sharing [25]. A 0.05 mm tangential position error is intro-duced to the pin holding planet-l such that this planet is pushedahead of the other three planets in its tangential position relativeto the sun and internal gears. The tangential loads on each planetbearing F i are compared to calculate the load sharing factors,LSF, of each planet as the percentage of the total load:

0.95

0.901.5 2 2.5 3

Fig. 6 Critical princi!pal stresses as a function of A; (a) theinternal gear, and (b) planets and the sun gear. The stress val-ues are normalized with respect to the stress values when A= (x;

200

~c0

--eQ)

~(:)

~:gCI:E:JE'x(IS~

100

stress levels are i creased significantly as the rim thickness of theinternal gear is re uced to the levels of the tooth height. From theexponential natur of the relationship, it can be said for this par-ticular example c e that any A below A = 1 is not feasible as theinternal gear stre ses values at the root become overwhelminglylarge beyond any ypical design stJ;ess limit unless the applicationrequires a very li ht load. Meanwhile, the u max values of planetsand the sun gear e influenced slightly by increasing the flexibil-ity of the intern gear, as shown in Fig. 6(b). The amounts ofreduction of U m at A = I are around 7 and 2 percent for theplanets and the su gear, respectively, when compared to those atA = 3. The U min V lues of the same gears are relatively insensitiveto the internal ge flexibility except in the region A < 1.5 whereU min of planets in rease slightly. From Fig. 6, it can be concludedthat reducing of i ternal gear rim thickness has a negative influ-ence on the dur ility of the internal gear itself. Also, addingflexibility to the nternal gear is not a very effective method toreduce the stress evels of planets and the sun gear.

For small A, th internal gear deflects significantly as a result ofthe loading as ill ted in Figs. 3: to 6. This is of course possibleonly if the splin s of the intern~ gear allow radial deflectionswithin a housing, as is the case in this study. The reason for theradial deflections of the internal gear is simply the fact that eachinvolute planet-in mal gear mesh creates a radial separating forcecombined with be ding moment reacted by the spline support. Forthe four-planet s stem, this results in a "square-like" deflectionshape of the inte al gear as illustrated in Fig. 7(a) for A = 1. Onlythe deflected inte al gear is shown in Fig. 7 for clarity purposesalthough the mod I considers the entire gear set. Some amount ofbase rotation is so evident from Fig. 7(a) as a result of thebending moment mentioned above. The gear deflections for A= 1.5 and 2.0 are shown in Fig. 7(b, c) using the same deflectionmagnification fac or as Figure 7(4). The amount of radialdeflec-

-200

1 1.5 2 2.5 3

A

Fig. 8 Maximum internal gear deflections as a function of A

Transactions of the ASIVIE412 I Vol. 123~ SEPTEMBER 2001

~I .' s g ' A-,. "

f~ ffI

1!~~~!iI~~'I~~~~J~'/

0

.100

(a)A=3

J, """""""","-..rC 50Q)eQ)c.u::C/)-J 25

--Planet 1

--Planet 2

--Planet 31-o-~anet4

"o;'~~~~A..?~A"';:p"~ "'"\

0

75

(b)A=1 --Planet 1

--Planet 2--Planet 3-Planet 4

:

+"---"""""'---""--50~Q)

~Q)0-

u:cn..J 25

~..

0

0 15 30Carrier Rotation Angle, degrees

45

~p

Fig. 9 Lc,ad sharing factors of each planet as a function ofcarrier rotiation angle for the case of a 0.05 mm tangential pinposition elrror on planet-1 and non-floating sun gear; (a) A=3,and (b) A'=1 ~~

J~ ~F.

(LSF).o:~, io: 1 to 4.

~F..)1=1

(2)

i

g§ ~

In Fig. 9, (LS F) i, it 1 to 4, are p1@tted as a function of thecarrier rotation angle ~ r A = 3 and A ~ 1. Here neither the sungear nor the carrier is al owed to float ra~al1y. For A = 3, planet-1carries on an average 54.5 percent of th~ total torque transmitted.This amounts to gear mesh force of 523i N per mm face width onplanet-1 compared to ollly 240 N per mtn face width in case of aperfect load sharing. .s is a significant increase that is likely tolead to durability pr blems. Similar~y, the bearing holdingplanet-1 carries a load f 1050 N/mm fuce width instead of car-rying only 480 N per face width when the load sharing isperfect. Such an increa e could have severe consequences on theplanet bearing as well. While p1anet-1 ~s severely overloaded inFig. 9(a), other three pi ets carry less titan 20 percent of the totalload each. When this re atively rigid intdrnal gear is replaced by aflexible one having A = 1, as shown in Ifig. 9(b), the load sharingimprove significantly, I wering the m*imum planet load from54.5 percent to 38.8 pe cent. The differ~nce is distributed to theother three planets as th largest share g~ing to planet-3. Here thesmall amplitude, small period fluctuatipns are due to counter-phasing condition of th planet meshes! [26]. The average valuearound which these flu tUations take pl~ce also varies as a func-tion of carrier rotation gle depending dn the position of a planetrelative to the outside s lines. Correspopding stress contours areshown in Fig. 10 toge er with deflect~d state of the planetarygear set. This figure ill strates the dom~ant influence of the pinposition error on the ar deflections ~d stress levels. For A= I, the internal gear is effected into ani asymmetric shape as themaximum deformation takes place in tj1e vicinity of the planetwith the pin position e or. The resulta~t tooth bending stressesand local internal gear s sses are also h1gher in the same area. In

Fig. 10 Maximum principal stress contours and deflectionshapes for the same cases as Figure 9

Fig. 10, some of the meshed teeth look like they are disengagedsimply due to the fact that the deflections are exaggerated graphi-cally for visualization.

The average (LSF)i predictions are plotted against the rimthickness parameter A in Fig. 11 for the same case as Fig. 9. Herethe improvement of load sharing obtained by reducing A is quiteevident. The load on planet-l is reduced steadily as the oth~rplanets experience moderate increases. It can be concluded fromFig. 10 that increasing internal gear flexibility is a viable methodto improve the load sharing amongst the planets provided that therim has the fatigue strength required.

Another more common method of improving planet load shar-ing is to allow the carrier or sun gear to float [15,25]. In order tocompare the effectiveness of this method with the method of aflexible internal gear, the same case shown in Figure 9 is analyzedunder floating sun conditions. Fig. 12 shows (LSF); as a functionof the carrier rotation angle for A = 3 and 1. Now the load iscarried by pairs formed by diagonally opposed planets. The plan-ets forming each pair carry the same amount of load, which wasreported to be due to the load distribution capability of the floatingsun gear in four-planet systems [15,25]. The maximum load on

Journal of Mechanical Design SEPTEMBER 2001, Vol. 123 I 413

75A=3

50CQ)~Q)Co

u.-rn-J 25

~ "-"! ,

~~ ~~0

2 2.5 31.5

Fig. 11 Influence of A on average load sharing factors of eachplanet for the case of a 0.05 mm tangential pin position error onplanet-1 and non-floating sun gear

planets 1 and 3 is at 32.3 percent for A = 3 when the sun gear isfloating compared to 54.5 percent in Fig. 9 corresponding to thecase of non-floating sun. The maximum planet load goes down to28.9 percent in Fig. 12(b) for A= 1 when the sun gear is floating.This is a direct result of the combined effect of the floating sunand the internal gear flexibility on the planet load sharing. This isalso evident from the maximum stress contours shown in Fig. 13.Since floating the sun gear is capable of equating loads diago-nally, the internal gear takes a diamond-like deflection shape. Insummary, a flexible ring gear is less effective in distributing theexcess load to other planets than the floating sun gear. In case ofa four-planet system, a flexible ring gear can help distribute dif-ferent amounts of the excess load to all other planets whereas thefloating sun gear can only distribute the excess load on a planet to

A=l

50

Fig. 13 Maximum principal stress contours and deflectionshapes for the same cases as Figure 12

25

c:'"~'"0-

u:(/)-J

the one that is positioned diametrically from it. Use of both meth-ods in combination results in the best load sharing characteristics.

0

50

25

c~&u.(I)-J

00 15 30 45

Carrier Rotation Angle, degrees

Fig. 12 Load sharing factors of each planet as a function ofcarrier rotation angle for the case of a 0.05 mm tangential pinposition error on planet-1 and floating sun gear; (a) A=3, and

(b)A=1

4 Conclusions and Design GuidelinesIn this study, the effect of flexibility of an internal gear on the

quasi-static behavior of a planetary gear set is investigated. Astate-of-the-art contact mechanics formulation is employed tomodel a typical automotive automatic transmission final driveplanetary unit. The model considers each gear as a deformablebody and meshes them to predict loads, stresses and deformations.Based on the results presented, the following conclusions can bemade:

.Rim thickness influences the stresses of the internal gear sig-nificantly. For internal gears with thick rims, the tooth bendingeffects tend to dominate while the rim bending effects tend todominate over for cases of thin rims. The location of the criticalstress point moves from the fillet to the root of the internal geartooth as the rim thickness is reduced.

.Added internal gear flexibility has limited effect in reducingthe stress levels of planets and the sun gear.

.A flexible internal gear deflects significantly under load mak-ing internal gear piloting, roundness and run-out errors less im-

Transactions of th,e ASME414 I Vol. 123, SEPTEMBER 2001

portant in the quasi-static behavior of the planetary gear set.Larger manufacturing tolerance bands can potentially be used tolimit such errors provided that the internal gear is flexible enough.

.A flexible internal gear helps improve the load sharingamongst the planets. However, it is not as effective as the methodof floating the sun gear.

.The quasi-static behavior of an internal gear rim thickness isdependent on the amount of input torque, the gear tooth designand the internal gear material used. Therefore it is not possible toarrive at a generalized guideline on how thin the internal gear rimshould be. The results of this study indicate that the objectivemust be to use the most flexible internal gear possible given thematerial fatigue strength. The model presented in this study isshown to be suitable for this purpose.

There are a number of potential consequences of using flexibleinternal gears that are not addressed in this study. This studyconsidered only one type of internal gear support condition, astraight splined joint with clearances in both major and minordiameters. Changing certain parameters such as number, spacingand geometry of splines or employing other means of supportsuch as press-fit internal gears should influence the behavior pre-sented in this study. Also, as the internal gear happens to be at theforce transmission path from the gear set to the housing, the flex-ibility of the internal gear together with its support conditions canbe expected to influence the forces going into the housing whichact as excitations for structure-borne gear noise as is the case forthe final drive planetary gear units of the front wheel drive auto-motive automatic transaxles. In addition, the modal characteristicsof a planetary gear set were observed to change as the flexibilityof the internal gear is varied [27]. Interactions of ring gear bend-ing natural modes with the overall planetary modes have not beenfully investigated. Such issues are the focus of our ongoing re-search on this subject matter.

[6] Chang, S. H.. Huston, R. L., and Coy, J. J., 1982. "A Finite Element StressAnalysis of Spur Gears including Fillet Radii and Rim Thickness Effects."ASME J. Mech. Des.. 82-WA/DE-35.

[7] Bibel, G. D., Reddy, S. K., Savage, M., and Handschuh, R. F., 1991, "Effectof Rim Thickness on Spur Gear Bending Stress." NASA Technical Memoran-dum 104388, AVSCOM Technical Report 91-C-O15.

[8] Clerici, P., Girotti, A.. and Perazzolo, A., 1992, "Comparison of Web StressConcentration Factors and Safety Margins for a Thin Webbed Spur Gear Sub-jected to Static and Cyclic Loading Conditions," DE-Vol. 43-1, ASME Inter-national Power Transmission and Gearing Conference, pp. 53-58.

[9] Brazakis. C. A., and Houser. D. R., 1994. "Finite Element and ExperimentalAnalysis of the Effects of Thin-Rimmed Gear Geometry on Spur Gear FilletStresses," International Gearing Conference. Newcastle upon Tyrne, UK, pp.41-46.

[10] Hidaka, T., Terauchi, Y., and Nohara, M., and Oshita, J., 1977, "DynamicBehavior of Planetary Gear (3rd Report, Displacement of Ring Gear in theDirection of Line of Action," Bull. JSME, 20, pp. 1663-1672.

[II] Chong, T. H., and Kubo. A.. 1985. "Simple Stress Formulas for a Thin-Rimmed Spur Gear. Part 3: Examination of the Calculation Method and StressState of Internal Spur Gear Pair," ASME J. Mech., Transm., Autom. Des.,107, pp. 418-423.

[12] Ods, S.. and Miyachika, K.. 1987. "Root Stress of Thin-Rimmed Internal SpurGear Supported with Pins," JSME Int. J., 30, pp. 646-652.

[13] Vaujany, J. P., Kim, H. C.. Guingand, M., and Play. D., 1996. "Effects of Rimand Web on Stresses of Internal Cylindrical Gears," Proceedings pfthe ASMEInternational Power Transmission and Gearing Conference. DE-Vol, 88, pp.73-80.

[14] Linke, H. and Jahn, C., 1997, "Bending Load on Internal Gears of PlanetaryGear Sets," AGMA Technical Paper, 97FTM7.

[15] Kahraman, A., 1999, "Static Load Sharing Characteristic of TransmissionPlanetary Gear Sets: Model and Experiment," SAE Paper 1999-01-1050.

[16] Krantz, T. L., 1992, "Gear Tooth Stress Measurements of Two HelicopterPlanetary Stages," NASA Technical Memorandum 105651, A VSCOM Techni-cal Report 91-C-038.

[17] Botman, M., 1980, "Vibration Measurements on Planetary Gears if AircraftTurbine Engines," J. Aircr. 17, pp. 351-357.

[18] Valco, M.. 1992, "Planetary Gear Train Ring Gear and Support StructureInvestigation," Ph.D. Dissertation, Cleveland State University.

[19] Vijayakar, S., 1991, "A Combined Surface Integral and Finite Element Solu-tion for a Three-dimensional Contact Problem," Int. J. Numer. Methods Eng.,31, pp. 525-545.

[20] Vijayakar, S., Busby, S., and Houser, D: R., 1987, "Linearization of Multi-body Frictional Contact Problems," Comput. Struct., 29, pp. 569-576.

[21] Hadley, G., 1964, Nonlinear and Dynamic Programming, Addison WesleyPublishing Company.

[22] Hadley, G., 1962, Linear Programming, Addison Wesley.[23] Murty, K. G., 1976, Linear & Combinatorial Programming. John Wiley &

Sons.[24] Ehrlenspiel, K., 1967, "Planetengetriebe-Lastausgleich und konstructive En-

twicklung," VDI-Berichte, Nr. 105, pp. 57-67.[25] Kahraman, A., 1994, "Load Sharing Characteristics of Planetary Transmis-

sions," Mech. Mach. Theory, 29, pp. 1151-1165.[26] Kahraman, A., 1994. "Planetary Gear Train Dynamics," ASME J. Mech.

Des., 116, pp. 713-720.[27] Hidaks, T., Terauchi. Y., and Ishioka, K., 1976, "Dynamic Behavior of Plan-

etary Gear (2nd Report, Displacement of Sun and Ring Gear)," Bull. JSME.19, pp. 1563-1570.

References[I] Colbourne, J. R., 1987, "The Geometric Design of Internal Gear Pairs,"

AGMA Technical Paper, 87 FTM 2.[2] Cunliff, F., Smith, J. D., and Welbourne, D. B., 1974, "Dynamic Tooth Loads

in Epicyclic Gears," ASME J. Eng. Ind. May, pp. 578-584.[3] Hidaka, T., Terauchi, Y., and Nagamura, K., 1979, "Dynamic Behavior of

Planetary Gear (7th Report, Influence of the Thickness of the Ring Gear),"Bull. JSME, 22, pp. 1142-1149.

[4] Balasubramanian, B., 1993, "Dynamische Lastferteilung in Planetensatzen,"Ph.D. Dissertation, University of Karlsruhe, Germany.

[5] Lewicki, D. G., and Ballarini, R., 1996, "Effect of Rim Thickness on GearCrack Propagation Path," Proceedings of the AMSE International PowerTransmission and Gearing Conference, DE-Vol-88, pp. 53-64.

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