Internal Gear

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NASA Technical Memorandum 107012 Bending Strength Model for Internal Spur Gear Teeth Army Research Laboratory Technical Report ARL-TR-838 / M. Savage and K.L Rubadeux The University of Akron Akron, Ohio and H.H. Coe Lewis Research Center Cleveland, Ohio (NASA-TM-IO7012) BENDING STRENGTH MO_EL FOR INTERNAL SPUR GEAR TEETH (NASA. Lewis Research Center) 12 p G3/37 N95-33908 Unclas 0060532 Prepared for the 31st Joint Propulsion Conference and Exhibit cosponsored by AIAA, ASME, SAE and ASEE San Diego, California, July 10-12, 1995 National Aeronautics and Space Administration U.S. ARMY RESEARCH LABORATORY

Transcript of Internal Gear

Page 1: Internal Gear

NASA

Technical Memorandum 107012

Bending Strength Model forInternal Spur Gear Teeth

Army Research Laboratory

Technical Report ARL-TR-838

/

M. Savage and K.L RubadeuxThe University of AkronAkron, Ohio

and

H.H. Coe

Lewis Research Center

Cleveland, Ohio

(NASA-TM-IO7012) BENDING STRENGTH

MO_EL FOR INTERNAL SPUR GEAR TEETH

(NASA. Lewis Research Center) 12 p

G3/37

N95-33908

Unclas

0060532

Prepared for the

31st Joint Propulsion Conference and Exhibit

cosponsored by AIAA, ASME, SAE and ASEESan Diego, California, July 10-12, 1995

National Aeronautics and

Space Administration

U.S. ARMY

RESEARCH LABORATORY

Page 2: Internal Gear
Page 3: Internal Gear

BENDING STRENGTHMODEL FOR INTERNAL SPUR GEAR TEETH

M. SavageK. L.Rubadeux

The University of AkronAkron, Ohio 44325phone (2161 972- 7737fax (216) 972- 6027

H. H. CoeNASA Lewis Research CenterCleveland, Ohio 44135

Abstract

Intemalspurgear teethare normally

strongerthan pinionteeth ofthe same pitchand face width sinceexternalteethare

smalleratthe base. However, ringgears

which are narrower,have an unequaladdendum or are made ofa materialwitha

lower strengththan thatof the meshing pinion

may be loaded more criticallyinbending. In

thisstudy,a model forthe bending strengthof

an internalgear toothas a functionofthe

applied load pressureangle ispresentedwhich isbased on the inscribedLewisconstant

strengthparabolicbeam. The bending modelincludesa stressconcentration factorand an

axialcompression term which are extensionsof

the model foran externalgear tooth.The

geometry ofthe Lewisfactordeterminationis

presented, the iterationto determine the

factorisdescribed and the bending slrengthJ

factoriscompared tothatof an externalgear

tooth.Thisstrengthmodel willassistoptimal

design effortsforunequal addendum gears

and gears of mixed materials.

Nomenclature

Symbols

B gear dedendum {ram,in)

c center distance (mm,in)

Copyright @ 1995 by Michael Savage.

Published by the American Institute ofAeronautics and Astronautics, Inc. with

permission.

f face width (mm,in)

h heightof Lewisparabola (mm,in)

H height,y distance (mm,in)orstress

concentration constant

J AGMA bending strengthfactor

Kf stressconcentration factorm module (mm)N number of teeth

Pd diametral pitch (1 .O/inch)R pitch radius (ram,in)

RC radial distance to the parabola apex(mm,in)

RF cutter tip radius {ram,in)

RFF radius to trochoid point {mm,in)

R0 cutter radius to center of cutter tip(mm,in)

RT gear radius to the center of the cutter tipfillet (ram,in)

RA distance from the cutting pitch point to

the cut point (mm,in)RD distance from the cutting pitch point to

the center of the cutter tip fillet (ram,in)

t Lewis parabola tooth thickness {mm,in}ct pitch circle tooth thickness (ram,in)

_J_t tangential (kN,Ibs)load

x Lewis form factor distance (mm,in)X abscissa coordinate (mm,in)

Y ordinate coordinate (ram,in) or Lewis form

factor

a angle between the tangent to thetrochoid and the tooth centerline

(radians)

13 central gear angle {radians)

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V angle on the gear from the cutting pitch

point to the trochoid cutting point

(radians)

61 angle on the cutter from the pitch pointon the tooth surface to the center of the

tip circle (radians)

5 2 angle on the gear from the pitch point onthe tooth surface to the center of the tip

circle (radians)

A internal tooth half bottom land angle

(radians)

rl angle at the cutting point between the

line of centers and the normal through the

tip fillet center (radians)

B roll angle (radians)

ef central angle on the gear from the centerof the tip of the trochoid to the trochoid

point (radians)

eft central angle on the gear from the centerof the gear tooth to the trochoid point

{radians)

A supplement of the angle a (radians)

p radius of curvature (ram,in)

o bending stress (Pa, psi)

pressure angle (radians)

slope of the trochoid at the contact point

Subscripts

A point of contact

b base cErcle

C apex of parabola

E involute point

f . fillet

F trochoid point

z of the involute at the cutter tip fillet center

I cutter

2 internal gear

Suoerscriots

L stress concentration equation constant

M stress concentration equation constant

lnfroduc:tiorl

In the design of spur gear teeth, bending

strength is a significant concern ''z_. Gearteeth which break off at the root become free

debris in a gear box to cause secondary

failures. In a very short time, a tooth bending

fatigue failure will cause a completebreakdown of the transmission in which it

occurs. So tooth bending fatigue limits in a

transmission are a primary concern in all stages

of design.

For equal addenda gears made of thesame material and the same width, the pinion

teeth have the lowest bending fatigue limitsince their bases are smaller and the loads on

the pinion and gear teeth are equal. Thus

most of the gear tooth bending stress models

are for external gear teeth 4"s. However, there

are situations in which an internal gear tooth

may have a higher chance of failure than its

meshing external pinion tooth. It may bemade of a weaker material or its tooth

thickness may be reduced to enlarge the

pinion's tooth thickness and balance the

bending strengths in the mesh.

Present models for the bending strength

of internal gear teeth _'7use the straight line

tangent model of Hofer with a slope relative to

the tooth centerline of forty-two to fifty-nine

degrees. Both studies recommend forty-five

degrees for thick rimmed gears with larger

angles for thin rimmed gears. The AGMA

Aerospace Gearing Design Guideline Annex 8,

gives a procedure for finding the inscribed

parabola which yields the highest stress

estimate for a given tooth and loading. This

procedure assumes a solid body gear and a

circular fillet tangent to the tooth involute.

In this work, the classic method of

inscribing a constant strength parabola inside

the tooth is used to estimate the tooth

strength 9. This is the method of Wilfred Lewis I°

which has been used for many years by the

AGMA as the basis of the external gear tooth

strength model s. A stress concentration factorhas been added to the calculation as an

extension of the Dolan and Broghamer

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factor". The model also includes axial

compression to match the AGMA bending

strength J factorS. Required information to

specify the internalgear J factor are: I)the

dedendum ratioof the tooth, 2} the nominal

pressure angle, 3) the pitch circletooth

thickness,4) the number of teeth on the

internalgear, 5J the number of teeth on the

meshing pinion to find the highest point of

single tooth loading, 6)the number of teeth on

the pinion shaper cutter,and 7} the tipradius

of the cutter.

Tooth Slrenqth Model

Wilfred Lewis developed the basic model

for bending stress in gear teeth in 1892 _°. In his

analysis, Lewis considered a gear tooth to be aloaded cantilever beam with a force applied

to the tip of the gear. He made the following

assumptions:

I. the load is applied to the tip of a gear

tooth;

2. only the tangential component of the

force will be a factor {the radial

component is neglected);3. the load is distributed uniformly across the

entire face width of the gear;,

4. forces due to tooth sliding friction are

negligible; and

5. no sJTessconcentration is present in thetooth fillet.

Lewis took into account the geomeJry of the

gear tooth by inscribing a constant strength

parabola within the tooth form. The vertex of

the parabola is located at the intersection of

the tooth centerline and the applied load's

line of action. At the location on the profile of

the tooth where the inscribed parabola is

tangent, the Lewis equation for the tooth

bending stress is expressed as:

Wt" Pdo - (1)

f-Y

where W t is the tangential load on the tooth,

Pd is the diameJTal pitch, f is the gear facewl_th, and Y is the Lewis form factor based on

the geomeJry of the tooth. The point of

application of the load is described by the

pressure angle of the applied load at the

tooth surface, (I)A.

For the stress analysis of internal gears,

both involute and trochoid geometry are used

in checking for the smallest inscribed parabola

in the tooth.

Involute Geometry

The involute is the locus of a point on a

line unrolling from its base circle. The involute

profile is described in terms of a coordinateframe with its center at the gear center and

the y axis through the center of the tooth. The

coordinates of the profile are obtained as

projections of the base radius R. _, and the• oradius of curvature, p_, of the involute point

onto this x,y coordinate frame.

The involute function of a pressure angle,

(I), depicted in Figure I, is the difference

between the roll angle and the pressure angle

at that point. Mathematically, the involute of

an angle is expressed as:

INV(_) = e-_ = tan(_) -_ (2)

The pitch radius of the internal gear is

expressed by the equation:

N 2

R2 - (3}

2"P d

or

m

R2 = N2.-- (4}2

for metric units, where N_ is the number of

internal gear teeth, Pt.l isZ'the diamefral pitchand m is the module.-The base radius is:

Rb2 = R2 - cos((1)) (5)

Delta, A, isone half of the bottom land angle

of the tooth involute. InFigure I,A can be

seen as the angle from the center of the tooth

to the involute at the base circle,which is:

3

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t

- P INV(¢) (6}

2- R2

where tcircle, p

is the tooth thickness at the pitch

As shown inFigure 2,the radius to the

loaded lineof action at the centerline of the

loaded tooth isR_. Thisisalso the radial

distance to the parabola apex. The pressure

angle at RC isequal to the sum of the tangent

of _)A'the pressure angle at the tooth surface,

and A. So RC can be expressed as:

Rb2 Rb2

RC - - (7}

cos((I)c} cos(tan((I)A) + A)

In Figure 3, 8 is the roll angle to the point

on the involute which is tangent to the

inscribed parabola with its apex at RE. Sincewe must iterate to find e, an initial estimate for

8 can be expressed as:

8 = 1.5- tan(cl)A) {8)

X_ and Y, are the coordinates of the involute

point which is cut at the roll angle e. Thesecoordinates are measured with respect to the

center of the loaded tooth. From the

geometry of Figure 3, XE and YE are:

XE = PE" Cos(A + 8) - Rb2" sin{A + 8) {9)

YE = PE" sin(A + 8) + Rb2- cos{Z_ + e) (10)

In Figure 3, H 1 is the y distance from thetangent point on the involute to the

intersection of the involute's tangent with thetooth centerline.

X E

H I = (11)tan(A + 8)

H 2 isthe distance from that same point on theparabola to the intersectionof the parabola's

tangent with the tooth centedlne. Since the y

distance to the apex of a parabola is one half

the distance to the intersection of the tangent

with the centedine, H2 can be expressed as:

H 2 = 2- (YE- RC) (12)

An interval halving iterafive process isused to find the location on the tooth surface

at which the largest parabola is tangent to theinvolute. 8 X_ Y_ H., and HA are calculated

• ' I-' I Zeach time in _is process. The angle 8 is

increased by a r0(ed step• Ae, in each iteration,

with A8 set to 0.01 radians initially. When the

difference in H. and H_ changes sign AE} is setI

to -A8/2 to close in on _e solution. When the

values of H I and H_ ore equal• the location onthe tooth surface a_ which the largest

parabola is tangent to the involute has beendetermined.

Trochoid Geometry

For an internal tooth, the largest inscribed

parabola may be tangent to the involute or it

may be tangent to the trochoid at the base of

the tooth. Therefore, Jrochoid geometry is also

used to find the point of maximum sITess. In

the following analysis, the cutter is gear I and

the internal gear is gear 2 with R I being the

pitch radius of the cutter and R2"being thepitch radius of the internal gear.

As shown in Figure 4, R0 is the cutterradius to the center of the cutter tip fillet:

R0 = R I+B-R F (13)

where B is the dedendum of the internal gear

and RF is the cutter tip radius. The pressureangle on the cutter to the involute of the

cutter tip fillet center is denoted by (l)Z and is:

{14)

The radius of curvature of the involute at the

cutter tip fillet center is P7 which can be

determined from R0 ancT(J)Z by:

4

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PZ = RO" sin((I)zJ (15)

81 is the angle on the cutter from thepitch I_oint on the tooth surface to the center

of the cutter tip fillet and 69 is the conjugaterotation of the gear from the pitch point on thetooth surface to the center of the curler tip

fillet on the cutting frochoid. The angles 61and 62 can be calculated as:

PZ + RF61 - q_Z - INV(_) (16)

Rbl

R1(52 = (31._

R2

{17)

Figure 5 shows the paths of the frochoidson the internal gear tooth and also displaysthe locations of point C on the trochoid of thetip center and its corresponding pitch point, D.The inner frochoid is for the point at the centerof the cutter tip fillet. The outer trochoid is forthe envelope of the cutter tip positions whichis the cut shape on the tooth root. The line

O2C L locates the tooth centerline in thesefigures.

In Figure 5 8. is the rotation of the cutter• /

and egis the corresponding rotation of thegear. -While the cutter rotates the center ofthe tip fillet from point F to point C, the gearrotates the apex of the trochoid, which is afillet radius above point F, to point G. The lineO^G then is the centerline of the lrochoid onthe gear. The angular rotation of the gear canby expressed in terms of the rotation of thecu_er as:

R1e2 = e 1 -_ (181

R2

R_ isthe radius from the gear center to thecentet[ of the cutter tip fillet. From triangle ABC

in Figure 6 and the law of cosines, RT can bedefined as:

RT = [ c2 + R02 + 2.C.Ro.COS(8 I) ] I 12 (19)

where c is the gear to cutter center distance

which isequal to R2 minus RI . The angle 13isthe central angle on the gear from the cuttingpitch point, D, to the center of the cutter tipfillet. In Figure 6, the perpendicular distancefrom point C to the gear-to-cutter line ofcenters is:

RT •sin(13) : R0 • sin(e1)

therefore,

13= sin-l( Ro'sin{el)

Rr

(2o)

(21)

RD is the distance from the cuffing pitch pointto the center of the cutter tip fillet. Applyingthe law of cosines to triangle ADC, yields:

RD = [ R22 + RT2 - 2"R2"RT'COS(13)]1/2 {22)

The angle at the cuffing pitch point betweenthe line of centers and the cuffing normalthrough the tip fillet center• rl, isfound from thelaw of cosines in triangle ADC:

2

-I ) (23)rl = cos ( R22+RD2-RT

2-R2-RD

R__is the radius from the gear center to thet-_.

actual trochoid point:

RFF =_ [ R22.÷ RA2 _2.R2.RA.cos{rl } ] 1/2 (24)

where RA equals RD plus RF. Gamma, y, is theangle on the gear from the cuffing pitch pointto the trochoid cuffing point:

-1 R22 + RFF2 - RA2y=cos ( ) (25)

2"R2"RFF

An expression for el, the central angle on thegear from the cenfer of the tip of the frochoidto the trochoid paint, is:

ef = y- e2 (26)

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InFigure6,Offisthe centralangle on the gearfrom the ceriferofthe gear toothto the

trochoidpoint.Itcan be expressed as the arc

from the tooth centerlinetothe pitchpointon

the involuteplusthe arc from the pitchpoint

to the center ofthe tiptrochoidminus the arc

to the frochoidpoint:

t

eft - P + 62- of (27)

2-R2

The coordinatesofthe filletdeveloped on

the internal gear are XF and YF:

XF = RFF.sin(Oft) (28I

and

YF = RFF"cos(eft} (29)

Psi,_,isthe slopeof the trochoidat the

contact pointmeasured relativetoa line

perpendicular tothe centedine of the tooth.The frochoidsurfaceisnormal tothe lineDE in

Figure5 sinceD isthe instantcenterforthe

relativemotion of the cutterwithrespectto

the gear. InRgure 7,the angle at E between

the tangent to the trochoidand the radialline

to O_ isn12 -(n-y-rl}ory+rl-nl2.Thismakes theangl'@between the tangent to the Irochoidand the toothcenterline:

a = n - Off - (y + rl - n/2}

or

(3o)

a = 3n/2 - eft - y - rl

its supplement is

[31)

X = n-a = Ofl+y+rl-n/2 (32)

This angle is the complement of u_,therefore:

= n/2 - (-n/2 + Off + y + rl)

of

(33)

= n - n - y - Oft (34)

InFigure7,H. isthe radialdistance on theI

toothcenterlinefrom the pointofinterestto

where the trochoidtangent crossesthe centerofthe tooth:

H I = XF •tan(_) (35)

H^ isdefined as the radialdistance on thez

toothcenterlinefrom the pointofinterestto

where the parabola tangent crossesthe

centerof the tooth.Since one-halfH2 equals

YF minus RC:

H2 = 2- {YF-Rc) {361

A similar interval halving iterative processisused to findthe locationofthe fillet

developed on the internalgear which is

tangent to a pointon the inscribedparabola.

When the valuesof H. and H_ are equal theI _' '

locationof the tangent pointon the internal

gear isdetermined.

The resultsobtained from the involuteand

trochoidgeometries are then compared. Thesmallerx coordinate identifiesthe weaker

inscribedparabola forthe internaltooth.Thisx

coordinate and itscorresponding ycoordinateare used to calculatethe Lewisform factor,

and the AGMA bending strengthJfactorwhich includesa stressconcentration factor

and a term foraxialcompression inthe tooth.

Bendina StrenathFactor

The Lewisform factor,Y,originallydefinedforextemal teeth,is:

2

y = ---Pd-X3

(37)

where

XE2 tc 2x - - (38)

Rc-Y E 4"(Rc-Y E}

One of the most importantfactorswhich

Lewisoverlooked inhisanalysiswas the effect

ofstressconcentrations.Large localized

Page 9: Internal Gear

stresses occur in the fillets of gear teeth due to

the sudden change in the cross-section of the

tooth. By examining these factors and

determining their exact effect on the bending

stress in a gear tooth, Lewis' work wasextended.

In 1940, professors TJ. Dolan and E.L.

Broghamer of the University of Illinois used the

photoelastic method of stress analysis to do

this 11, They examined various types of gear

teeth and determined the location and the

magnitude of the maximum stresses whichoccur in the tooth fillets. Their research

showed that the maximum stress is located

closer to the root circle than Lewis had

predicted. However, the distance between

Lewis' location and Dolan's and BroghameKs

location of the maximum stress is relatively

small. Thus, the use of Lewis' model to

determine the bending stress location in gear

teeth was confirmed by Dolan and

Broghamer. They also determined that the

primary factors affecting the stressconcentration at the tooth fillet are the fillet

radius, the tooth thickness, the height of the

load position on the tooth, and the tooth

pressure angle. They developed the followingstress concentration factor curve fit relationZn:

L M

(tc tc

Pf

where, t is the tooth thickness at the critical. C

sechon, pf is the minimum radius of curvatureof the fillet curve, and h is the height of the

Lewis parabola.

From a curve fit of the experimental data

of Dolan and Broghamer, AGMA s gives the

following values for the constants H, L, and M

in terms of the pitch circle pressure angle, (J):

H = 0.331 -0.436-(J) {40)

L = 0.324 - 0.492-_b 141)

M = 0.261 + 0.545-(I) {42)

The modified Lewis model for determining the

bending stress in gear teeth, which includesthis stress concentration factor and a term for

the axial compression in the tooth from the

radial component of the tooth load, is:

Wt'P da- {43)

f.J

where the AGMA J factor, in terms of e_, the

pressure angle at the apex of the paraJSola on

the tooth centerline, is:

I

J =

Kfc°SI*c)( 6-2h tanl,c))cos(C) tc tc

Since the bending strength factor isa

function of the tooth shape, itisdependent on

the number of teeth on the gear. Thisisshown

inFigure 8,which isa plot of the J factor versus

the number of gear teeth for both an external

gear and for an internalgear. As the number

of external gear teeth increases, the Lewis

form factor increases at a decreasing rate;

while as the number of internalgear teeth

increases, the Lewis form factor decreases at a

decreasing rate. Since the tooth shape on the

two gears approach each other as the

number of teeth increase, the form factor

values for the internaland external gears

approach each other as well.

Conclusions

An estimate for the bending strength of

an internal spur gear tooth has been

developed. This model uses the inscribed

parabola approach of Wilfred Lewis in

combination with an extrapolation of the

Dolan and Broghamer stress concentration

factor and the addition of an axial

compression term.

The estimate is obtained considering boththe involute surface of the tooth and the

trochoid fillet at the base of the tooth as

produced by a pinion shaper cutter.

Page 10: Internal Gear

produced bya pinionshapercutter.Generationequationsarederivedfor both theinvoluteand the trochoid. Dueto the general

nature of the model, the bending strength

prediction is valid for a load applied at any

point on the tooth. The load location is

identified by the tooth surface pressure angle

at the point of application of the load.

A direct and stable iteration procedure is

used to determine the size of the largest

inscribed parabola in the internal gear tooth.

Based on the size of this parabola, the Lewisform factor is established.

To complement the base stress estimate,a stress concentration factor and an axial

compression component are added to the

strength model. This stress concentration

factor is an extrapolation of the Dolan and

Broghamer factor and is consistent with the

AGMA J factor for external gears. A

comparison of the bending strength model for

an external gear and for an internal gear is

given for gears of increasing size meshing with

a twenty-five tooth pinion. Both gears have

twenh/-degree pressure angles and are cut

with a twenty-tooth pinion shaper.

By improving the estimate of the bending

strength of an internal gear tooth, this model

will allow designers to vary the material of a

Hng gear from that of its meshing external

gear. A long and shorf paddendum design

system may also be evaluated to balance

the bending strengths of the external and

internal gears.

References

I. Buckingham, E. K., Analytical Mechanicsof Gears. McGraw Hill, New York, 1949.

. Drago, R. J., Fundamentals of Gear

s_Qff=_jgD,Butterworths, Stoneham,

Massachusetts, 1988.

. AGMA STANDARD, "Fundamental RatingFactors and Calculation Methods for

Involute Spur and Helical Gear Teeth,"

=

o

.

.

.

.

10.

11.

ANSI/AGMA 200 I-B88, Alexandria,

Virginia, September 1988.

Mitchner, R. G. & Mabie, H. H., "The

Undercutting of Hobbed Spur Gear

Teeth", ASME Journal of Meghanical

Vol. 104, No. I, 1982, pp. 148-158.

AGMA STANDARD, "Geometry Factors for

Determining the Pitting Resistance and

Bending Strength of Spur, Helical and

Herringbone Gear Teeth," AGMA 908-1389,

Alexandria, Virginia, April 1989.

von Eiff, H., Hirschmann, K. H. & Lechner,

G., "Influence of Gear Tooth Geometry onTooth Stress of External and Internal

Gears," ASME Journal of Mechanical

Vol. 112, No. 4, 1990, pp. 575-583.

Miyachika, K. & Oda, S., "Bending Strength

of Internal Spur Gears," Proceedinas of the

Intemotional _onference on Motion and

Power Transmissions. November 23-26,

1991, Hiroshima, Japan, pp. 781-786.

AGMA Information Sheet, "Spur Gear

Geometry Factor Including Internal

Meshes," Annex A, AGMA 91 I-A94, Design

Guidelines for Aerospace Gearing,

Alexandria, Virginia, 1994.

Rubadeux, K. L. "PLANOPT - a Fortran

Optimization Program for Planetary

Transmission Design," M.S. Thesis, The

University of Akron, May, 1995.

Lewis, W., "Investigation of the Strength of

Gear Teeth," Proceedings of the Engineers

Club, Philadelphia, Pennsylvania, 1892.

Dolan, T. J. & Broghamer, E. L., "A

Photoelastic Study of Stresses in Gear

Tooth Fillets," University of Illinois,

Engineering Experiment Station, BulletinNo. 335, 1942.

8

Page 11: Internal Gear

+0 2

PE

Rb2

X= YE-Rc

RCYE

0 2 x

Figure 1 Involute and Top Land Angles for anInternal Gear Tooth

Figure 3 Involute Coordinate Geometry for anInternal Gear Tooth

OOTH\ SURFACE

B

Rb2 RC

PZ RO R1

Rbl

¢Z

01

Figure 2 Location of the Parabola Apex onthe Internal Gear Tooth

Figure 4 Cutter Tip Geometry

9

Page 12: Internal Gear

Dedendum G

R o

Figure 5

A

02

TrochoidLocation on the InternalGear Tooth

RF

FF

R2

Figure 6

O2

Internal Gear Tooth Trochoid CuffingPoint Geometry

10

i Hll , H2/2

"7

Oft

0 2

R C

(F

I,Figure 7 Trochoid Coordinate Angles on the

Internal Gear Tooth

J FACTOR

.6

.5

.4

.3

.2

.1

00

Figure 8

INTERNAL

/ EXTERNAL

25 TOOTH PINION

20 TOOTH CUTI'ER

O.3/P, CUTTER TIP RADIUS_0

I I I ! I i

50 1 O0 150 200 250 300

NUMBER OF GEAR TEETH

Bending StrengthJ FactorValues for

Externaland InternalGears withthe

Load atthe HighestPointof SingleTooth Contact

Page 13: Internal Gear

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. _EPORT TYPE AND DATES COVERED

July 1995 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Bending Strength Model for Internal Spur Gear Teeth

6. AUTHOR(S)

M. Savage, K.L. Rubadeux and H.H. Coe

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Lewis ResP..,tux_ Center

Cleveland, Ohio 44135-3191

and

Vehicle Propulsion Directorate

U.S. Amay Research Laboratory

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, D.C. 20546-0001

and

U.S. Army Research Laboratory

Addphi, Maryland 20783-1145

W'U-505--62-361L162211A47A

8. PERFORMING ORGANIZATIONREPORT NUMBER

E-9797

10. SPONSO_NG/MONITORING

AGENCY REPORT NUMBER

NASA TM-107012ARJ..-TR-838

11. SUPPLEMENTARYNOTESM. Savage and K.L. Rubadeux, The University of Akron, Akron, Ohio 44325; H.H. Coe, NASA Lewis Research Center.

Responsible person, H.H. Coe, organization code 2730, (216) 433-3971.

12a. DISTRIBUTIOI_AVAILABILrI'Y STATEMENT 12b. DISTRIBUTION CODE

Unclassified -Unlimited

Subject Category 37

This publication is available from the NASA Center for Aerospace Information, (301) 621-0390.

13. ABSTRACT (Maximum 200 words)

Internal spur gear teeth are normally stronger than pinion teeth of the same pitch and face width since external teeth aresmaller at the base. However, ring gears which are narrower, have an unequal addendum or are made of a material with a

lower strength than that of the meshing pinion may be loaded more critically in bending. In this study, a model for thebending strength of an internal gear tooth as a function of the applied load pressure angle is presented which is based onthe inscribed Lewis constant strength parabolic beam. The bending model includes a stress concentration factor and an

axial compression term which are extensions or the model for an external gear tooth. The geometry of the Lewis factordetermination is presented, the iteration to determine the factor is described and the bending strength J factor is comparedto that of an external gear tooth. This strength model will assist optimal design efforts for unequal addendum gears and

gears of mixed materials.

14. SUBJECT TERMS

Gears; Design; Transmissions; Spur gears; Internal teeth

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

NSN 7540.-01-280-5500

18. SECURITY CLASSIRCATIONOF THIS PAGE

Unclassified

19. SECURITY CLASSIFICATIONOF ABSTRACT

Unclassified

15. NUMBER OF PAGES

1216. PRICE CODE

A0320. LIMITATION OF ABu I HACT

Standard Form 298 (Rev. 2-89)

Prescribed by ANSI SId. Z39-18298-102

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