The design of a gear mesh is treated with the objective of minimizing the gear size for a given gear ratio, pinion torque, pressure angle, and allowable tooth lengths. Tooth strengths considered include scoring, pitting fatigue, and bending fatigue. Kinematic involute interference is avoided. The design variation on standard spur gear teeth called the long and short addendum system, is considered. In this system the mesh center distance and pressure angle are maintained as is the ability to manufacture the teeth with standard tooling. However, the pinion and gear tooth proportions are altered in order to obtain fewer teeth numbers for the same ratio as standard gears without kinematic involute interference. The effect of this nonstandard gearing geometry with on tooth strengths and gear mesh size are studied. For a 2:1 gearing ratio, the optimal nonstandard gear design is compared with the optimal standard gear design

Coy, J. J.

Savage, M.

Townsend, D. P.

English

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https://ntrs.nasa.gov/search.jsp?R=19820020769 2020-03-21T08:01:14+00:00Z
NASA	 AVRADCOM
TecbniM 1 emerandum 82866	 Teftio l Report 92-C-7
The Optimal Design of Involute Gear
Teeth With Unequal Addenda
(NASA-Tm-928b6)
	 THE OPTI!lAL EESIGN U1: 	 \ e2-2F645
INVOLUTE GEAt? TEETH WITH LIE^.UAL ADDErEA
(NASA) 8 p HC A02/MF A01
	 CSCL 131
Unclas
G3/37 28403
Wchael Savage
The Univff3O of Akron
Akron, Ohio
and
John J. Coy
Propulsion Laboratory
AVRADCOM Research and Technology Laboratories
Lowis Research Center
Cleveland, Ohio
and
Dennis P. Townsend
Lewis Research Center 	 X ^^
Cleveland, Ohio rr:	 <C1 {
@ SQ
Prepared for the
Tenth World Congress on System Simulation and Scientific Computation
sponsored by the International Association for Mathematics and
Computers in Simulation (IMACS)
Montreal, Canada, August 8-13, 1982
THE OPTIMAL DESIGN OF INVOLUTE GEAR TEETH
WITH UNEQUAL AUUENUA
Michael Savage
The University of Akron
Akron, Ohio
and
John J. Coy
Propulsion Laboratory
AVRADCOM Research and Technology Laboratories
Lewis Research Center
Cleveland, Ohio
and
Dennis P. Townsend
Lewis Research Center
N	 Cleveland, Ohio
ABSTRACT
The design of a gear mesh is treated with the objective of minimizing the
gear size for a given gear ratio, pinion torque, pressure angle and allowable
tooth strengths. The gear tooth strengths considered are scoring, pitting
fatigue and bending fatigue. Kinematic involute interference is also
avoided. The design variation on standard spur gear teeth called the long and
short addendum system is considered. In this system, the mesh center distance
and pressure angle are maintained as is the ability to manufacture the teeth
with standard tooling. However, the pinion and gear tooth proportions are
altered in order to obtain fewer teeth numbers for the same ratio as standard
gears without kinematic involute interference. The effect of this nonstandard
gearing geometry on tooth strengths and gear mesh size are studied. For a 2:1
gearing ratio, the optimal nonstandard gear design is compared with the
optimal standard gear design.
BACKGROUND
The use of long and short addendum teeth for an involute spur gear mesh
has long been considered an improvement in design over standard involute spur
gear geometry (1). This belief comes from the fact that larger teeth are
stronger in bending fatigue than smaller teeth (2). The first problem
encountered in the minimum tooth design is kinematic involute interference on
the pinion, where the tip of the gear tooth contacts the pinion tooth below
the base circle on the pinion. To take advantage of the increased bending
strength of the larger teeth and avoid the kinematic interference problem
without resorting to special tooling, the long and short addendum system has
been adopted (1). Design procedures for gear meshes utilizing unequal addenda
have been recently presented by Tucker (3) and Estrin (4). Both procedures
are excellent procedures which consider many factors - kinematic, structural,
•	 and manufacturing. However, they do not consider the Hertzian contact
pressure at the tip of the gear tooth which is a prime contributor to gear
scoring. The authors (5) have more recently presented an optimization method
for standard spur gears which includes consideration of this effect. It is
the purpose of the work presented herein to extend the results of (5) to
unequal addenda gears. 	 .
THE DESIGN PROBLEM
A gear set is normally used to transmit an input torque from a shaft
turning at one speed to an output torque on a shaft turning at a lower speed.
The basic parameters describing this situation are the gear ratio,
mg, the input pinion torque, Tp, and the pinion speed, up.
The size of the gear is defined by the gear mesh center distance, C. A
standard gear mesh with 22 teeth on the pinion and 44 teeth on the gear is
shown in figure 1. For a 2:1 gear ratio, these numbers of pinion and ear
teeth result in an optimal design configuration as shown in reference ?5). By
tying the gear face width, f, to the pinion pitch diameter by the ratio,
A,:
fed
I = R	 (1)
p
where Pd is the diametral pitch of the gear set and Np is the number of
teeth on the pinion, the center distance C is made a measure of the gear
blank's volume as well as the gear area. This length to diameter ratio,
X, insures that a uniform load is present across the tooth face. Many
gear meshes have a values in the neighborhood of 0.25, although meshes
with extremely stiff shafts can have values as high as 0.6. In addition to
these parameters, the gear and pinion teeth are defined by the pitch line
pressure angle, v, and the addenda, ap and ag, and the dedenda, d
and d , of the teeth. These teeth are shown in figure 2 for the RGMA
standard values of
ap = ag = 1/Pd	 (2)
and
d  = d g = 1.25/Pd
	(3)
In the long and short addendum system, the addendum of the pinion and the
dedendum of the gear are increased by the tool shift, e, while the addendum
of the gear and the dedendum of the pinion are decreased by the tool shift,
e.
The parameters to be determined in a design are the number of pinion
teeth, NP,, the diametral pitch, Pd, and the tool shift, e, for a given
gear ratio, mg , pinion torque, T p , pressure angle, o, pinion width to
diameter ratio, a, and the material strengths.
The material properties of importance for both the pinion and the gear
are their elastic modulii, Ep and E, their Poisson's ratios, v and
V	 and their bending fatigue strengths and their surface endur nce
sKengths.
2
The optimal solution to the long and short addendum design problem is a
design with an adequate contact ratio (usually greater than 1.4), no kinematic
involute interference, balanced bending and surface endurance strengths,
standard diametral pitch and minimum center distance.
LONG AND SHORT ADDENDUM TEETH
y=
Long and short addendum teeth are shown in figure 3 for a 2:1 gear ratio
with 14 teeth on the pinion and 28 teeth on the gear, a pressure angle of 20
degrees and a tool shift of 0.4/P d . The ad:onua changes in this tooth
system cause the tooth thickness of the pinion tooth to increase and the tooth
thickness of the gear to decrease. This ir-r:ases the bending strength of thef.
pinion tooth but decreases the bending strength of the gear tooth.
In addition, the top land of the pinion tooth is reduced. This reduction
in pinion top land thickness places an upper bound on the amount of tool shift
possible, at which point the pinion tooth becomes pointed. Since kinematic
involute interference produces a lower bound on the tool shift, these two
factors of minimum top land and involute interference place practical limits
on the amount of tool shift possible in a given design.
DESIGN COMPARISON
One can study the effect of various tool shifts on the strength
characteristics of an unequal addendum gear mesh by computer simulation. The
maximum pinion bending stress and the maximum gear bending stress are plotted
versus tool shift in figure 4. For this figure and the next, a face width of
0.5 inches (12.7 mm), a diametral pitch of 10, a pressure angle of 20 degrees,
a gear ratio of 2 and 14 pinion teeth were selected. The gear ratio of 2 was
chosen as a representative reduction. The fewer the teeth number, the larger
the teeth for a given center distance. The number of pinion teeth was
selected as 14 to come down to a number of teeth low enough to produce
kinematic interference with standard teeth anc to display a design
surfficiently different from the standard optimum design to illustrate the
strengths and weaknesses of long and short addendum gearing.
The pinion torque for this figure was set at 1,000 lb-in (113 N-m). The
material chosen was steel with a modulus of 30x106
 psi (200 GPa) and a
Poisson's ratio of 0.25. In this figure, one can see that the gear stress
increases while the pinion stress decreases as the tool shift increases.
Figure 5 is a plot of the Hertzian contact pressure between the teeth at
the tip of the gear tooth and at the highest point of single tooth contact on
the gear. Both pressures are plotted versus tool shift for the same data as
used in figure 4. The intersection of the two curves indicates the tool shift
required to produce equal contact pressure at the gear tip and at the highest
point of single tooth contact on the gear tooth. These are the two locations
of highest contact pressure between the teeth. Since the contact pressures
are much higher than the bending stresses, this amount of tool shift produces
the optimum long and short addendum design fnr 14 pinion teeth and a 2:1 gear
ratio. Dividing this tool shift by the diametral pitch produces a
dimensionless tool shift which produces this balanced design state for all
gear sizes.
3
The Hertzian contact pressure at the tip of the gear tooth is a measure
of how close to the pinion base circle the contact with the gear tooth is
initiated. It is the relative imbalance of this stress (5) which forces the
optimal standard tooth design to have a large number of pinion teeth. This
same factor requires the tool shift to be relatively high for the unequal
addenda design.
It is interesting to note that for equal pinion length to diameter ratios
and equal Hertzian contact pressures, the optimal design indicated by figure 5
wish a tool shift of 0.4/Pd has a center distance similar in size to that of
the optimal standard tooth design with 22 pinion teeth shown in figures 1 and
2. Figure 3 shows the tooth proporitions of the unequal addendum optimal
design.
If one sets 200 ksi (1.34 MP ) as an acceptable contact pressure, then
the 14 tooth pinion design has a giametral pitch of 6, a face width of 0.5
inches (12.7 mm), a center distance of 3.5 inches (88.9 mm), and a maximum
bending stress of 24 ksi (161 kP ). Figure 6 is a plot of the Hertzian
contact pressure versus the involute length on the pinion tooth from the base
circle for this design. In comparison, the standard tooth design from
reference (5) has a diametral pitch of 10, a face width of 0.55 inches (14
mm), a center distance of 3.3 inches (83.8 mm), and a maximum bending stress
of 7.9 ksi (53 kPa) for a maximum contact pressure of 200 ksi (1.34 MPa).
This is due to the fact that the surface curvature is a function of the base
circle size for efficient designs. These two designs have nearly the same
base circle sizes. If the diametral pitches were exactly proportional to the
tooth numbers, the face widths and maximum contact pressures would be equal
for the two designs.
SUMMARY AND CONCLUSIONS
The optimum design for a minimum center distance gear mesh with long and
short addendum teeth was presented in this paper. In considering this design,
the following results were obtained:
(1) by going to the long and short addendum tooth system, no size
improvement can be obtained over an optimally designed standard spur gear mesh,
(2) the optimal design of a long and short addendum gear mesh is
determined by the balance of Hertzian contact stress between the teeth at the
gear tip and at the highest point of single tooth contact on the gear tooth,
and
(3) the optimal design of a long and short addendum gear mesh is defined
by the gear ratio, pressure angle, number of pinion teeth and tool shift ratio
and is valid for any physical size of gear set.
REFERENCES
1. Mabie, H. H.; and Ocvirk, F. W.: Mechanisms and Dynamics of Machinery.
Third ed., John Wiley, 1978.
2. Shigley, J. E.: Mechanical Engineering Uesign. Third ed., McGraw-Hill,
1977.
4
3. Tucker, A. I.: The Gear Design Process. ASME Paper 8U-C2/DET-13, Aug.
1980.
4. Estrin, M.: Optimization of Tooth Proportions for a Gear Mesh. ASME
Paper 80-C2/DET-101, Aug. 1980.
A	 5.	 Savage, M.; Coy, J. J.; and Townsend, D. P.: Optimal Tooth Numbers for
Compact STandard Spur Gear Sets. ASME Paper 81-DET-115, Sept. 1981.
5
too
00
NN
100
VZ
50
0	 20	 40	 60	 w	 100«10-3
TOOL SHIFT, in.
Figure 4. - Tooth bendng stress cem-
wison.
1000
^m 80D LOAD AT GM TIPad
vim+ 600NW
d
LOAD ON SINGLE TOOTH
200
i
0	 20	 40	 60	 Io	 100e10-3
TOOL SHIFT, in.
Figure 5. - Herfcien confect Pressure
congerison.
200
.m
150
4ANN
rU
c 10D
50V
PINION TOM LENGTH, in.
Figure 6. - OpUrA 6d gn Herren con- 	 t
tect Pressure.
ORIGINAL PAGE 13
OF POOR QUALITY
Figure 1. - Stenderd gar mesh.
44 TOOTH GEAR
Figure 2. - Stw4wd Involute mesh ponviry.
14 TOOTH PINION
a TOM GEAR
Figure 3. - Unepuel eddende mesh geometry.


The optimal design of involute gear teeth with unequal addenda

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