A G M A
210 . 0 2
JAN., 1965
AGMA
STANDARD
(
for
r
I
<
Surface Durability
(Pitting)
of Spur Ciear Teeth
MA
.02
5
Put/;s�eJ t�:
AMERICAN
1330 Massachusetts Avenue, N. W.
•
GEAR MANUFACTURERS
Washington, D. C. 20005
ASSOCIATION
(. ) )
FOREWORD
This standard is for rating the surface durability of spur gear teeth. It contains the following:
Basic Rating Formula
This section enumerates the factors known to affect surface durability. Numerical values
are presented for those factors which have been evaluated by analytical means, test results
or field experience. Suggestions are made for the factors which are not now capable of being
expressed accurately. New knowledge and more definite measurement of these parameters
will continually necessitate revisions and improvements,
In addition to the above, it is contemplated to publish design practices such as AGMA 2 1 1.0 2A,
having specific application under the heading of:
Design Practices for Specialized Applications
It is recognized that it is sometimes desirable to provide simplified design practice data
applicable to a specialized field of application. These individual design practices will
enable enclosed speed reducer, mill gear, aircraft or other specialized product designers
to record the modifications and limitations they wish to use.
Information on references, derivations, explanations and examples are contained in AGMA 229.06.
Basic data illustrating the coordination of rating for all types of gears is contained in Tentative
Information Sheet AGMA 215 .01, "Surface Durability ( Pitting) of Spur, Helical, Herringbone and
Bevel Gear Teeth,"
The first draft of this standard was prepared rn October, 196 1.
membership as of June 13, 1964.
It was approved by the AGMA
Tables or other self-supporting sections may be quoted or extracted in their entirety.
lines should read:.
Teeth (AGMA
Credit
"Extracted from AGMA Standard for S urface Durability (Pitting) of Spur Gear
210.02),
with the permission of the publi sher, the American Gea r Manufacturers
Assoc iation 1330 Massachusetts Avenue, N. W., Washington, D. C.
20005."
COPYRIGHT, 1965, BY
AMERICAN GEAR MANUFACTURERS ASSOCIATION
-
2
-
Personnel of
Gear Rating Committee
Technical Division
January, 1965
E. ] Wellauer, Chairman, The Falk Corp., Milwaukee, Wis.
•
D. L. Borden, Jr., The Falk Corp., Milwaukee, Wis.
Wells Coleman, Gleason Works, Rochester, New York
D. W. Dudley, Mechanical Technology, Inc., Latham, New York
] . H. Glover, Ford Motor Co., Dearborn, Michigan
I. Koenig, Hewitt-Robbins, Inc., Chicago, Illinois
C. F. Schwan, Reliance Electric & Engineering Co., Cleveland, Ohio
] . C. Straub, Wheelabrator Corp., Mishawaka, Indiana
F. A. Thoma, Warner & Swa sey Co., Cleveland, Ohio
N. A. Wilson, Morgan Construction Co., Worcester, Mass.
G. L. Scott, AGMA, Washington, D. C.
AGMA Standards are subject to constant improvement, revlSlon or withdrawal as
dictated by experience. Any person who refers to AGMA technical publications should
satisfy himself that he has the latest information available from the Association on
the subject matter.
-
3
-
AGMA STANDARD
SURFACE DURABILITY (PITTING) OF SPUR GEAR TEETH
1. Scope
The symbols used, wherever applicable, con­
1.5
form to Standard AGMA 112.04, "Gear Nomenclature­
Terms,
This standard presents the fundamental fo7mula
1.1
Abbreviations"
and, "Letter Symbols for Mechanics of Solid Bod­
for rating the surface durability (pitting) of spur
gear teeth.
Definitions, Symbols, and
ies," ( ASA Zl0.3-1948) (Rl953).
It contains all of the factors which are
known to affect the resistance of gear teeth to pit­
ting.
This formula is not applicable to other types
of gear tooth surface deterioration such as plastic
yielding, scoring, wel ding, etc.
2.
This standard is
Fundamental
Pitting Durability Formula
based on AGMA Information Sheet 215.0 1, "Informa­
tion Sheet for Surface Durability ( Pitting) of Spur,
The fundamental pitting durability formula for
2.1
Helical, Herringbone and Bevel Gear Teeth" and is
gear teeth is as follows:
coordinated with the durability ratings for helical,
herringbone, and bevel gears.
1.2
Pitting of gear teeth is considered a fatigue
phenomenon.
The two kinds of pitting - 1) initial,
and 2) destructive - are illustrated in Standard
AGMA
110.03,
"Gear-Tooth
Wear and Failure."
Corrective and non-progressive initial pitting is not
deemed serious.
The aim of this standard is to de­
sign the gear teeth so that destructive pitting does
Where:
not occur.
sc
1.3
Surface durability rating practices for a particu­
elastic coefficient (see Section 16)
lar field of gearing may be established by selecting
the proper factors for this general formula.
calculated contact stress number
Proper
evaluation of the various factors in the basic for­
mula will produce suitable ratings for any applica­
transmitted
tion.
tangen tial
load
10
pounds at operating pitch diameter
(See Section 4)
1.4
Load
Where no applicable specific AGMA standard
overload factor
exists, numerical values may be estimated for the
factors in this general formula and an approximate
(See Section 10)
dynamic faccor (See Section 9)
surface durability rating calculated.
-4-
AGMA STANDARD
SURFACE DURABILITY (PITTING) OF SPUR GEAR TEETH
d
=
pinion operating pitch diameter, in.
NOTE:
(or
to account for differences in material properties,
outside
diameter
minus two
addendums when applicable)
Size
Both pinion and gear teeth must be checked
and number of tooth contact cycles under load.
net f�c;e_ width of the narrowest
of the mating gears, in.
Cs
=
size factor (see Section 7)
3.
cm = load
factor
(See
Dis-
The following formula may be used to calculate
3 .1
Section 6)
S tress
the power directly:
geometry factor (See Section 5)
tribution
distribution
cl
surface
Surface Durability Power Formula
condition
factor
np
p ac
(See
F
126,000
Section 8)
Where:
NOTE:
p
ac
The above equation is divided into three
allowable power, hp
groups of terms; the first is concerned with load,
pinion speed, rpm.
the second with gear size, and the third with stress
distribution.
2.2
The relation of calculated contact stress num­
ber to allowable contact stress number is:
4.
4.1
Transmitted Tangential Load
-
W1
The transmitted tangential load is calculated
directly from the power transmitted by the gear set.
(When operating near a critical speed of the drive, a
Where:
careful analysil? of conditions must be made.) When
allowable contact stress number
Section 15)
the transmitted load is not uniform, consideration
(See
should be given not only to the· peak load and its
anticipated number of cycles, but also to intermedi­
ate loads and their number of cycles.
life factor (See Section 12)
4.1.1
hardness ratio factor
(See Section 14)
Frequently a gear set is required to carry a peak
load for a limited number of cycles, and then a much
smaller load for a much larger number of cycles.
temperature factor (See Section 13)
In
such cases the gear rating should be adjusted so
that each load and number of cycles carried by the
factor of safety (See Section 11)
gear set is within allowable stress limits.
- 5-
AGMA 210.02- Jan., 1965
fr.
AGMA STANDARD
SURFACE DURABILITY (PITTING) OF SPUR GEAR TEETH
11:.Z
The transmitted tangential load is :
p x 33,000
D
operating
inches
me
gear ratio
zc
distance (inches) measured along the
line of action from the pitch point to the
lowest point of single tooth contact.
(See Figure A-1) Values for Zc for one
diametral pitch are shown in Figures
A-2 and A-3 for standard operating center
distances. Values for Zc for any dia­
metral pitch can be obtained by dividing
these values by the diametral pitch.
126,000 p
2T
d
Where:
p
transmitted power, hp
T
pinion torque, lb-in
pitch line velocity, fpm
5.
-
Geometry Factor
pitch
diameter
of gear -
Geometry factors for standard center distances
are shown in Figure 1. For non-standard centers or
special tooth geometry, see Appendix B.
5.2
I
5.1 The greatest contact stress oc curs at the lowest
point of single tooth contact of the pinion. The
geometry factor for this position of loading is as
follows. The derivation is presented in Appendix A.
For an external gear :
6,
(; � (- � ( � )
me
2 cot¢
sin¢
Zc
-+ -
me +
2
D
sin¢
Zc
--
· -
Distribution
-
Cm
6.1 The load distribution factor Cm, evaluates the
effects of non-uniform distribution of load, and
depends upon :
d
2
Load
For an internal gear :
2 cot¢
( )(
me
me
) (-- )
sin¢
Zc
----
1
2
D
1. cutting errors
sin¢
-
2
Zc
2. errors m rotating axis m mounting due to
bore tolerances
-
d
3. internal bearing clearance
Where:
d
operating pressure angle - degrees
4. parallelism
operating pitch diameter of prn10n
inches
5. tooth stiffness
of shafts carrying each gear
(includes runout)
- 6-
AGMA STANDARD
SURFACE DURABILITY (PITTING) OF SPUR GEAR TEETH
6. 4 When gears. are hardened after cutting, and the
profiles are not ground or otherwise processed to
insure high accuracy, the tooth distortion will affect
the load distribution. When Cm is selected from
Figure 5 then:
6. blank stiffness
7. shaft stiffness
8. housing stiffness
9. bearing deflection
6. 4. 1 Multiply cm by 1.05 if one element lS hard­
ened after cutting.
10. Hertz deflection
11. thermal expansion and distortion due to
operating temperatures. This is especially
important for wide face gears.
Multiply Cm by 1.10 if both elements are
hardened after cutting.
6. 4.2
6.1. 1 Figures 2 and 3 illustrate misalignment and
its effect on load distribution.
The effect of different rates of misalignment
is shown in Figure 4. Fm represents the face width
having 100 per cent contact for a given tangential
load and misalignment error. Generally, Fm should
exceed F. The amount of misalignment shown in
Figure 4 must consider all factors shown in para­
graph 6. 1.
6.1.2
7.
Size Factor
-
Cs
7 .1 The size factor reflects the effect of dimens1ons on the uniformity of material properties. It
depends primarily on:
1. tooth size
2. gear diameter
Manufacturers of wide-face gears generally find
it necessary to c ontrol misalignment by other means
than by allowed rates of misalignment. To handle
such cases, Table 1 shows appropriate values of
6.2
3. face width
4. ratio of tooth size to gear diameter
cm.
5. area of contact pattern
6. ratio of case depth to tooth size
For spur gears equivalent to those used in
commercial gear units in accuracy and mounting
rigidity, the load distribution factor , cm, shown in
Figure 5 can be used as a guide. The load distri­
bution factor shown inc l udes other factors affecting
gear durability such as size effect, alignment errors,
6.3
7. hardenability and heat treatment of materials.
7. 2 The size factor may be taken as unity for most
gears, provided a proper choice of steel is made for
the size of the parts, hardness pattern or the total
case depth agrees with Figure 6.
F
etc. When-d ratio exceeds 2 , a more detailed analysis is suggested.
-7 -
AGMA 210.0 2
-
] an., 1 96 5
V»
Md
�
V
AGMA STANDARD
SURFACE DURABILITY (PITTING) OF SPUR GEAR TEETH
Table 1
Load Distribution Factor for Precision Wide-Face Spur Gears
-
Cm
-
F
Contact
cm
95% face w idth contact obtained at one-third torque
95% face width contact obtained at full torque
1.4 at 1/3 torque
1.1 at full �0:que
75% face width contact obtained at one-third torque
9 5% face w idth contact obtained at full torque
1 .8 at 1 /3 torque
1 .3 at full torque·
35% face width contact obtained at one-third torque
95% face width contact obtained at full torque
2. 5 at 1/3 torque
1. 9 at full torque
20% face width contact obtained at one-third torque
75% face w idth contact obtained at full torque
4.0 at 1 /3 tcrgue
2.5 at full torque
Ratio ofd
1 .0 or less
---
T eeth are crowned
35% face width contact obtained at one-third torque
85% face w idth contact obtained at full torque
2. 5 at 1 /3 torque
1 .7 at full torque
Calculated combined twist and bending of pinion not
over . 001 in. over entire face
Pinion not over 250 Bhn hardness
7 5% contact obtained at one-third torque
95% contact obtained at full torque
2.0 at 1/3 torque
1 .4 at full torque
Calculated combined twist and binding of pinion not
over .0007 in. over entire face
Over 1
but
Pinion not over 350 Bhn hardness
75% contact obtained at one-third torque
95% contact obtained at full torque
2.0 at l/3 torque
1 .4 at fu]J torque
l ess
than 2
30% contact obtained at one-third torque
75% contact obtained at full torque
4.0 at 1 /3 torque
3.0 a t full torque
Calculate effects of
deflection and in-
Twist and bending exceeds .001 in. over entire face
crease c to allow
m
for misalignment
errors
-8 -
AGMA STANDARD
SURF ACE DURABILITY (PITTING) OF SPUR GEAR TEETH
7 .3
Standard size factors have not yet been estab­
9.2
lished for cases where there is a detrimental size
effect.
Figure 7 shows some of the dynamic factors
that are commonly used:
In such cases a size factor greater than
unity should be used.
Curve # 1 - Should be used for shaved or ground
8, Surface Condition Factor
8.1
spur gears where the factors listed in paragraph
Cf
-
9. 1 result in no appreciable dyna1r.ic load.
The surface condition factor Cf, depends on:
1 . surface
finish
shaving,
as
lapping,
affected
grinding,
by
Curve # 2 - Should be used for shaved or ground
spur gears when the factors listed in paragraph
cutting,
9. 1 can develop a light dynamic load.
shot peening,
etc.
2. residual stress
Curve # 3
3. plasticicy effects (work hardening).
Should be used for shaved or ground
9. 1 can develop a moderate dynamic load. This
curve
8.2
-
spur gears when the factors listed in paragraph
is
recommended
for
commercial spur
gears.
The surface condition factor Cf' may be taken
as unity when a good surface is developed by either
processing or run-in.
9.3
When milling cutters are used to cut the teeth
or when inaccurate teeth are generated, lower dy­
namic factors than shown by these curves are re­
9,
9.1
Dynamic Factor
-
quired.
Cv
The dynamic factor, Cv, depends on:
9.4
1. effects of tooth spacing and profile error
With numerical knowledge of the factors listed
in paragraph 9. 1 and by using methods published in
the literature, other dynamic factor curves can be
2. effects of pitch line speed and rpm
prepared for specific fields of application.
3. inertia and stiffness of all rotating elements
4. transmitted load per inch of face
9.5 When
5. tooth stiffness
actual dynamic loads are computed or
measured and are added to the tangential load, the
6. lubricant properties.
dynamic factor can be unity.
- 9-
AGMA 210.02- Jan., 1 96 5
AGMA STANDARD
SURFACE DURABILITY (PITTING) OF SPUR GEAR TEETH
10.
Overload
Factor
C0
-
10. 4
Service factors have been established where
field
data is available for specific applications.
These service factors include not only the overload
The overload factor makes allowance for the
factor, but also the life factor and factor of safety.
roughness or smoothness of operation of both the
Service factors for many applications are listed in
driving
AGMA
1 0.1
and driven apparatus.
Specific overload
Standards,
and should be
used whenever
factors can only be established after considerable
applicable.
field experience is gained in a particular applica­
place of the overload factor C0, use a value of 1.0
tion.
for CR and CL.
10. 2
If a specific service factor is used in
In determining the overload factor, considera­
tion should be given to the fact that many prime
movers develop momentary overload torques appre­
ciably greater than those determined by the name­
plate ratings of either the prime mover or the driven
apparatus.
10. 3
11.
Factor of Safety
-
CR
In the absence of specific overload factors,
the values in Table 2 should be used.
11.1
The
equation
factor
of safety is introduced in this
to offer the designer an opportunity to
design for high reliability, or in some instances, to
Table 2
Overload Factors
-
design for a calculated risk. Failure does not mean
C 0*
an immediate failure under the applied load, but
rather a shortening of life.
gested list of safety factors.
Character of Load
On Driven Machine
Power
Source
Uniform
Table 3 shows a sug­
Moderate
Heavy
Uniform
Shock
Shock
1.00
1.25
Light Shock
1.25
1.50
Medium Shock
1.50
1.75
1. 75 o r
h igher
Table 3
the factors in Table 2.
-
CR
2.00 or
higher
Requirements of Application
2.25 or
higher
*This table 1s for speed decreasing drives only.
Fo< >pood inma>ing d<ive> add .01
Factor of Safety
t::J
High Reliability
1.25 or higher
Fewer than One Failure in 100
1.00
Fewer than One Failure in Three
0.80**
<o
**At
this value plastic profile deformation might
occur rather than pitting.
- 10 -
AGMA STANDARD
SURFACE DURABILITY (PITTING) OF SPUR GEAR TEETH
12,
Life Factor
-
CL
Table 4
Hardness Combinations
12.1 The life factor, C L, adjusts the allowable
loading for the required number of cycles.
At the present time, there is insufficient data
available to present accurate fatigue curves for any
type of gear. The data available indicates that the
fatigue curves for pitting will have a slope, as in­
dicated in Figure 8.
Temperature Factor
-
Cy
1 3 .1 The temperature factor, Cy, is generally taken
as unity when gears operate with oil or gear blank
temperatures not exceeding 250 degrees F, In some
instances, it is necessary to use a Cy value greater
than unity for carburized gears operating at oil
temperatures above 180 degrees F.
14.
14.1
Hardness Ratio Factor
-
Gear Bhn
Pinion Bhn
180
210
225
265
210
12.2
1 3.
Typical Gear and Pinion
245
245
285
255
300
270
315
285
335
300
350
The hardness ratio factors (CH) given rn
Figure 9 may be used as a guide .
14.3
15,
CH
Allowable Contact Stress Number
-
sac
The allowable contact stress number, sac'
depends on:
15. 1
The hardness ratio factor, CH, depends on:
1. gear ratio
1. material composition
2. hardness of pinion and gear .
2. mechanical properties
3. number of cycles
1 4.2 The gear and pinion hardness combinations
shown in Table 4 have been used successfully.
4. temperature
- 11 -
AGMA 210.0 2
-
] an., 1965
AGMA STANDARD
SURFACE DURABILITY (PITTING) OF SPUR GEAR TEETH
5 . size
1
6. residual stress
7. work hardening.
Where:
15.2
An allowable contact stress number for 10
million cycles of load application is determined by
field experience, for each material and condition of
that material. This stress number is designated -
Poisson's ratio for pinion and gear
respectively
Modulus of elasticity for pinion and
gear
15.3
The allowable contact stress number for gear
materials varies considerably with heat treatment,
forging or casting practice, and material composition.
15 .4
The designer should consult other AGMA
Rating Standards, and use the contact stress num­
bers given there wherever applicable. If the appli­
cation doe s not fit any of these established classes,
then the contact stress numbers" listed in Table 5
may be used as a guide.
In Table 5, the lower values are suggested
for general design purposes. The upper values may
be used when high quality material is used, when
15.5
section size and design allows maximum response
to heat treatment and when proper quality control is
effected by adequate inspection.
16. Elastic Coefficient
-
CP
16.1
The elastic coefficient, CP, is defined by the
following e quation. Its values for various combina­
tions of gear and pinion materials are shown in
Table 6.
- 12 -
AGMA STANDARD
SURFACE DURABILITY (PITTING) OF SPUR GEAR TEETH
Table 5
Allowable Contact Stress Number
Surface Hard ness,
Minimum
Material
Through Hardened
Steel
sac
Cast Iron
50- 60,000
65-75,000
75 - 85,000 .
180 Blm
85 - 95 ,000
240 Bhn
105 - 115 ,000
AGMA Grade 20
AGMA Grade 30
AGMA Grade 40
3 00 Blm�
120- 135,000
Nodular Iron
360 Bhn
145 -160,000
Annealed
165 Bhn
440 Bhn
170- 190,000
Normalized
210 Bhn
Oil Quench
and Temper
255 Bhn
90-1003 of
the sac
value of
steel with
the same
hardness
Tensile Strength
psi (Min.)
Sac
40,000
3 0,000
Case Carburized
(see Note 1)
55 RC
180-200,000
Bronze
60 Rc
200-225,000
Tin Bronze
AGMA 2C (10123 Tin)
Aluminum Bronze
ASTM B 148 -5 2
(Alloy 9C-H.T .)
Flame or Induction
Hardened
so RC
NOTE 1.
sac
Surface Hardness,
Minimum
Material
sac
-
170- 190,000
-
175 Bhn
200 Bhn
....
-
65,000
90,000
For minimum case depths at the pitch diameter as shown rn Figure 6.
Table 6
Elastic Coefficient
-
CP
Gear Material and Modulus of Elasticity - E*
Pinion Material
and Modulus of
Elasticity - E*
Steel
30
x
Cast Iron
106
19
x
106
Aluminum Bronze
17.5
x
106
Tin Bronze
16 x 106
Steel
30
x
106
2300
2000
1950
1900
Cast Iron
19
x
106
2000
1800
1800
1750
Aluminum Bronze
17.5
1950
1800
1750
1700
Tin Bronze
16
1900
1750
1700
1650
Poisson's Ratio
*NOTE:
=
x
x
106
106
0.30
When more exact values of E are obtained from roller contact tests, they can be used.
- 13 -
0
AGMA 210.02 - Jan., 1965
z
� .100
LI:
>­
�
.080
w
�
!il
<!> .060
2
0
3
14t DEGREE
(A)
4
6
5
GEAR RATIO
8
7
9
10
PRESSURE ANGLE FULL DEPTH TEETH STANDARD ADDENDUM = .!..
pd
H
I
a:
�
u
rt
Np= 50 OR MORE
.140
Np= 30
Np= 24
.120
N p= 16
>­
a:
l;j .100
�
!il
(!)
.080
.060
0
2
( B) 20
DEGREE
3
4
5
6
8
7
GEAR RATIO
9
10
PRESSURE ANGLE FULL DEPTH TEETH STAND ARD ADDENDUM::
�
d
�
Np=24
>-
Np= 16
rt .1 00
�
� .080
0
w
(!)
.060
0
2
(C ) 20
3
DEGREE
4
5
GEAR RATIO
6
PRESSURE ANGLE
STANDARD ADDENDUM =
8
7
9
10
STUB TEETH -
�
d
NOTE:
ALL CURVES ARE FOR THE LOWEST POINT OF
SINGLE TOOTH CONTACT ON THE PINION.
FIG. I
EXTERNAL SPUR PINION GEOMETRY FACTOR
FOR STANDARD CENTER DISTANCES
- 14 -
-
I
PINION
GEAR
FIG. 2
EXAMPLE OF A PINION AND GEAR MISALIGNMENT UNDER NO
LOAD. TEETH CONTACT AT LEFT HAND END AND ARE OPEN
AT RIGHT HAND END.
w,
0
<X
F
g
FACE WIDTH- F
F IG. 3
LOAD DIS T RIBU TION ACROSS FACE WIDTH
FOR VARIOUS CONTACT CONDIT IONS
-15 -
ce -
wt
- 1000
e
)
I
Wt
=
e
=
TANGENT IAL
L O AD - LBS
� LIGNME NT E RR O R - INCHES/ INCH
I
'
15,000
14,000
5.0
13,000
12,000
Q)
11,000
Iz
w
10,000
u
u
LL
LL
w
0
u
Cl'.
0
Cl'.
Cl'.
w
w
u
Cl'.
0
LL
4.0
E 3.0
u
9,000
8 ,000
2. 0
7,000
6,000
1.0
0
1.0
Fm
-
5,000
F
4,000
F
=
FAC E WI DTH
(INCHES)
3,000
2,000
1 ,000
0
2
FIG. 4
5
SPUR
GEAR
6
LOAD
- 16 -
DISTRIBUTION
FA CTOR - Cm
2.0
FOR
USE
FOURTH
THIRD
AND
AD DITIONAL REDUCT I 0 NS
VALUES
REDUCTION
e
(.)
I
0::
0
......
(.)
<(
LL
.......
---J
I
z
0
......
::::>
m
0::
......
en
1.8
1.7
1.6
F
Cm=.52F + 2.3
1.5
'
1.4
0
1.3
c
<(
0
-'
1.2
Use above
formulas when
face widths
exceed 18'�
I. I
1.0
F
cm= .55F+ 2.5
0
2
3
4
5
6
8
7
FACE
FIG. 5
SPUR GEAR
9
10
II
12
13
14
15
WIDTH (INCHES)
LOAD
ACRES
DISTRIBUTION
NITOBA LTD.
FACTOR - Cm
16
17
18
20 DP
TO TAL CASE DEPTH IS THE DEPTH
AT WHICH THE CARBON CONTENT
OF THE CASE AND CORE ARE EQUAL.
10 D.P.
::i:::
�
a::
_.
<l
g::
w
�
<l
iS
SOME HEAVY DU TY GEARS
6 D.P.
4 D.P.
-
-
2 D.P.
I D.P.
.5 D.P.
.010
.020
•
.030
.040
.050
.070
.100
.200
APPROXIMATE M INIMUM DEPTH OF CASE (INCHES)
FIG. 6
DEPTH
OF
CASE
- 18 -
AT
PITCH
LINE
.300
.500
.700
'
CURVE I n
. Cy = I
,-1
.80"'
CURVE
2
P Cv =
>
(.)
j 78./Vf
+
78
I
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1-­
(.)
\0
I
,_.
U
�
<X
z
>0
3
CURVE
�
40
.
t:='.�
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t�
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.
r
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'
.
_::::I
'
..
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.
rt
rlt1
-I
T
Cy =
'
'
'
H- ++ -r H-
5+0
v +f
50 ""ry-
'
- '� '
-H
'
'
+ -
:�
'
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II I I ! I I I I I -1-H�-++-++rl--
�
0
1000
,.
2000
3000
4000
5000
PITCH LINE VELO CITY - Vt ( fpl"fl)
FIG. 7
CR��
DYNAMIC
L'
FACTOR - Cv
6000
7000
8000
a>
0
<O
Q
en
w
....J
Q u
>u
,...
z
w
u..
....J
0
I.Li
a::
CD
5
Q 0
w
a::
I
0
iri
. . ,·
0
�
0
rri
<t
0
C\i
IO
10 - �010'1.:t
- 20 -
d
3.:111
Q
a::
0
.....
(.)
<(
u..
w
u..
...J
I
<X>
(!)
LL
K =
BRINELL OF PINION
BRIN ELL OF GEAR
WHEN
K
( 1.2 USE CH= 1.00
l:
0
0:
0
t­
o
<(
�
0
�
0:
(/)
(/)
LU
z
0
0:
<(
:c
'�f
FIG. 9
4
8
SI N GLE
REDUCTION GEAR RATIO
12
HARDNESS - RATIO
- 21 -
16
FACTOR - CH
20
APPENDIX A
DERIVATION OF GEOMETRY FACTOR (I)
FOR A PINION AT THE LOWEST POINT
OF SINGLE TOOTH CONTACT
1.
1.1
STANDARD CENTER DISTANCE
The contact stress for a gear mesh at the lowest point of single tooth contact is:
s·
c 1 cP
s
C1 C P
Where:
C1
R1
WI
+
R2
F cos¢
R1
R2
W
I
R1
-
F cos¢
R1
R2
R2
external gears
internal gears
coeffici ent dependi ng on the type of stress (contact stress, subsurfa
tensile stress, etc.)
ce shear stress,
radii of curvature of bodies 1 and 2 at the point of contact - inches
1.2
The geometry factor is:
Where:
R1
R1
R2
D
=
-
2
D
2
d
2
sin¢
+ Zc
external gears
sin¢
-
zc
internal gears
sin¢
-
zc
- A-1 -
APPENDIX A
Where:
D
operating pitch diameter of gear - inches
d
operating pitch diameter of pinion - inches
Zc
distance (inches) measured along the line of action from the pitch point to t he lowest point of
single tooth contact, (See Figure A-1). Values for Zc for one diametral pitch are shown in
Figures A-2 and A-3. Values for Zc for diametral pitch, Pd canoe obtained by dividing the values
from Figures A-2 and A-3 by the diametral pitch.
( 2 sin¢
d
D
sin¢
2
2
d
(cot ¢)
[
d
± 2
.
(cos¢)
d
sin¢
(
4 sin 2 ¢ - zc
Dd
.
2 sin¢
D
±
d
2
sin¢
)
(D ± d)
�
sin 2 ¢
2D
cl
4
(cot¢)
-
sin2 ¢
zc
2
me ± 1
2 cot ¢
( ) (\
2 cot ¢
( )(
me
sin ¢
me + 1
2
me
me
±
+
1
_
2
±
±
\
J
Zc
_
D
sin¢
( : �)
z
2
c
l
2
Dd
external gears
!...::_) (� �)
D
c
z
2
- A-2 -
_
cl
internal gears
DRIVEN
DRIVER
II
_Q_
2
FIG. A-I
Do
2
DIAG RAM
FOR THE DISTANCE
- A-3 -
-
Zc
-1 .
,,
1 .0
I
14t
I
0-
PRESSURE ANGLE
- DIAMETRAL PITCH
_,_
0.5
,_
'r c
�
0
CJ)
w
:I:
(.)
z
I
"'
I'
-0.5
I"
I"
'
�
"'
.....
"'
"
�
"
..
I'
.
-1.0
...
"'
�..
'
....
.....
"'
I'
"
"
....
"'
"
"
r c
....
�
�
,_
..,,_
'
.......
'
�
"
"
'
......
....
'
�
�
"
�
III
�
�
I.ii:::
�
-1. 5
-0.5
....
I
r"
- 1.0
�
.._
i"o
I ':I
"""
�
...
...
,�..
-1.5
I ,J
....
...
20
10
30
NUMBER
F IG.
....
ll::l
-2.0
A-2
40
50
60
"
1"
-2.0
......
""'
...
70 80 90 100
OF PINI ON TEETH
Zc -DISTANCE ALONG THE LINE OF ACTI O N FROM THE
PITCH PO INT TO THE LOWEST POINT OF SINGLE
TO OTH CONTACT ON THE PINION F O R STA NDARD
CENTER DISTANCES.
- A-4 -
CJ)
w
:I:
(.)
z
0
N
'
�
""
,.......
�
"
"'
"'
"'
"'
0.5
'""lo.�
0
..
�,...
�
0
N
1 .0
n 1n llM
II� '
2.0
1.5
---
-..;
....
I'""'
..
""'
0
""'
....
....
....
""
...
...
""
,.....
""'
....
....
0
�
""'
-0.5
....
...
-0.5
-...._
-........_
20° - PRESS UR'E ANGLE
-1.0
I -DI AMETRAL PITCH
-
20
1.0
��
,...,_
10
30
40
50
60
70
-
80 90 100
NUMBER OF PIN ION TEETH
FIG.
A-3
Zc - DISTANCE ALONG THE LINE OF ACT I ON FROM THE
PITCH POINT TO THE LOWEST POINT OF S INGLE
TO OTH CONTACT ON T HE PINION FOR STANDARD
CENTER DISTANCES.
t .
; .
- A-5 -
CJ)
l.LJ
::I:
u
z
(.)
N
I
....
�
i-..
�
...
....
,.....,
�
1.0
I�
0.5
,....
(.)
N
II
...
'""'
�
I
-...
1.0
0.5
1.5
·�
-
CJ)
l.LJ
::I:
(.)
z
2.0
II�
-
q
APPENDIX B
GEOMETRY FACTOR FOR A SPUR
GEAR AT THE LOWEST POINT OF
SINGLE TOOTH CONTACT
1.
NON-STANDARD CENTER D ISTANCES OR SPECIAL TOOTH GEOMETRY
1 .1
The expression for geometry factor is the same for both standard and non-standard center distanc es, however,
the value of Zc must be calculated when non -standard centers are used , whereas, for standard ce nters , the
value of Zc c an be obtained from Figures A-2 or A-3 .
1.2
The expression for the geometry factor ( I ) 1s :
2 cot¢
2 cot¢
( )(
( 1)J (
me
sin¢
me + 1
2
me
sin ¢
me
+
2
J (�
}
�)
�) (� )
Zc
D
2
_
D
2
e xternal gears
d
�
internal gears
d
Where :
¢
operating pressure angle - degrees.
me
gear ratio
D
operating pitch diameter of gear
d
operating pitch diameter of pinion - inches.
Zc
·-
inches.
distance measured along the line of action from the pitch point to the lowest point of single
tooth contact - inches.
Pb
base pitch - inches.
Z
addendum portion of line of action - inches.
a
- B- 1 -
APPENDIX B
Where :
d0
outside diameter of pinion - inches.
db
base circle diamet er of pinion - inches.
Other terms shown in Figure A- 1 but not used in any formula are :
Where :
-
�]
D0
outside diameter of gear - inches.
Db
base circle diameter of g ear - inches.
D
operating pitch diameter of gear - inches.
.-:::- ·
- B-2 -