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Transcript of 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
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)
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
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
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
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)
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
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.
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
+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
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
I Form ApprovedREPORT DOCUMENTATION PAGE OMB No. 0704-0188
Public reporting bun:lenfor this collectionof infmmaion is estimated to average 1 hourpet response, includingthe time for reviewing instructions,searchingexisting data sources,0=he_ng_ ._,r._.ng,he_, ,.._ed.,_ _ng ._d,._ ,he=_n _ ,r_.._..io_S._ _c_,. ,_=_.ng,._:bu_.° .=_m.._._ =_27s_ _._,hiscollectionof information,includingsuggestionsfor reducingthis burden, to WashingtonHeadqua,-ters_e_,icas, DirectorateTOtin/ormatponoperatm ano Hepons, z :)Davis Highway, Suite 1204, Arlington,VA 22202-4302, and to the Office ol Management and Budget, Papmw0rk ReductionProject(0704-0188), Washington, DC 20503.
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