AIRCRAFT GEARING

ANALYSIS
of
Test and Service Data

J. O. Almen and J. C. Straub

".: ....

RESEARCH LABORATORIES DIVISION GENERAL MOTORS CORPORATION

Detroit, Michigan
 2

FOREWORD

Since we rarely know the magnitude of loads that are applied to gears
in service or the extent to which the stresses from the applied loads
a.re concentrated by elastic deflections, thermal changes of dimen-
sions, or by dimensional errors; end since we do not know the strength
of our materials in their operating environments, or the effect of
pro.cessing upon metal strength--to mention some of the more important
UDknowns and unknowables--any prediction of performance based upon
assumed knowledge of such variables must necessarily be wide of the
mark.
When gears must be designed in aeoordanoe with textbook procedures,
which procedures are based upon many assumptions and misconceptions,
they will inevi ta.bly be overdesigned and unbalanced with respect to
the several kinds of failures that afflict gears. Most aircraft
gears are greatly overdesigned against tooth breakage, with the re-
sult that overcaution against one kind of failure actually produces
inferior gears because of their proneness to failures from causes not
considered by the designer.

The only known we:y to produce well-balanced designs of machine parts,

where intensive use of material is essential, is by the construction
of empirical formulas from service records of large statistical
samples. To accomplish this objective for aircraft engine gears, the
Research Laboratories Division of the General Motor& Corporation
asked ~nd obtained the cooperation of the major aircraft engine manu-
facturers during the war in the accumulating of suitable data.

The data are taken from service reports of many thousands of engines
u-sed in mili ta.ry end airlir1e operations. They are concerned only
with gears or gear combinations that failed from any cause attrib-
J.1ta.ble to desfg;n, material, or worlananship. It should be understood
that "failure," as here used, does not necessarily mean chronic
failure or catastrophic failure, cut includes also occasione.l damage
occurring to a few gears at some time during the service operation
of +he engine.

Suitable data were obtained on seventy-three gear combinations run

under a variety of loads, speeds, and temperatures. External as
well as internal gears were included. The torque ranged from tvro
pound-feet to 6500 pound-feet; the speeds ranged from 1260 rpm to
28, 000 rpm; and pitch line veloci -cy ranged from 785 feet per minute
to 19,100 feet per minute.

We wish to thank the following aircraft engine manufacturers, whose

contribution of data. made this study possible: Allison Division of
 2e.

the General Motors Corporation, Continental Aviation &: Engineering

Corporation, Packard Motor Car Company, Pratt &: Whitney Aircraft
Division of United Aircraft Corporation, and Wright Aeronautical
Corporation.
 3

TABLE OF CONTENTS

Foreword • • • • • • • • • • • • • • • • • • • • 2

Introduction .• • • • • • • • • • • • • • • 3a

Conclusions • • • • • • • • • • • 4
Discussion • • • • • • • • • • • • • • 5

Figures • • • • • • • • • • • 14
Appendix I • • • • • • • • • • • • • 25
Appendix II • • • • • • • • • • • • 31
 3a.

INTRODUCTION

The loads that ms.y safely be applied to automotive gear teeth exceed
the loads that ms.y be applied to gears made for any other service,
regardless of precision or cost of manufacture. Automotive gears
are not only capable of transmitting greater loads, but they also
meet higher standards of quietness than highly loaded gears in any
other service.

Automotive gears were developed by processes of trial and error over

a period of many years, processes that closely resemble the evolu-
tionary processes in nature. Numerous designers, working independ-
ently, contributed to the development by ma.king various modifications
to meet the dictates of space, tool equipment, or last-minute ratio
changes. Frequently, design modifications were made with the hope
the.t improvement in ce.paci ty or quietness would result. The automo-
tive gears that we use today represent only a small fraction of the
numerous design and material variations that were tried. These gears
survived because actual service in literally millions of automobiles,
used in all kinds of service by all kinds of drivers, proved them to
be the best fitted for +.heir jobs.

Since automotive gears were not developed by the methods that we are
·pleased to consider as rational applications of engineering and
,metallurgical fundamentals, they do not conform in many particulars
to "good" engineering practices. Conventional design fornn1las were
usually used by automotive gear desie71ers, but, having learned by
experience that these forrnul~s were not reliable, each designer applied
numerous modifications based upon his own experience, which, together
with a sixth sense of fitness, enabled him to produce gears of light
'weight and low cost that were usually successful. Sometimes new de-
signs would prove to be regressive, and occasionally, but often with-
out inteni, a new design proved to bee. mutation which enabled a
particular designer to advance, for a time, ahead of his oompetitors.

There is nothing wrong with applying evolutionary processes in machine

design. In fact 1 until we have a much better understanding of the
nature of our structural materials, and until we learn how our machine
parts are loaded in servioe, no other process is available whereby we
may progress toward more intense use of materials of construction.
The process is unsatisfactory, however, in that, being based upon
individual experience, the designers can seldom record their methods
in m.a.thematioal formulas or other precise statements that will enable
their successors to build directly upon past experience.

"Know-how" oan be empirically expressed

This difficulty can be overcome if service data on a sufficient number
of machine parts of a given type are available whereby empirioal
formulas oan be constructed. By empirical methods, we may consolidate
the experiences of all designers, ~.anufacturors, and users and thus
make immediately available that intangible, but essential, ingredient
of modern production called "know-how." By eI:lpirical means, we may
evaluate materials, processes, and design details to the end that any
product for any particular use will give the best customer service
for the least cost.
E)npirical engineering fornn.ilas are in extensive use. They are the
sole guides for determining the capa.ci ty of ball e.nd roller bearings,
and, more recently, empirical formulas have been constructed that
have ta.ken most of the guess work out of the design of bevel gears
for automobile rear axle use and helical and spur gearing for auto-
mobile transmission use.

The construction of empirical formulas for any ma.chine element is

simple in principle but dif'ficul t in practice. It requires extensive
data on the behavior of the type of machine part in question in aotua.l
user service. Laboratory date. a.lone will not suffice, but date. from
laboratory tests that are properly correlated with service performance
may be used to support service date.. The empirical formula that was
developed for automobile spiral bevel gears, now in universal use, is
based upon the service records of many thousands of automobiles in
actual user service. These data are supported by laboratory fatigue
tests on several hundred complete automobile rear axles, in which
test conditions were selected such that the type of failure and the
locations of the failures were the same as failures that occurred in
user service. Since service failures were relatively rare, and since
it was necessary to differentiate between failures that were primarily
the .tault of the gears and failures in whioh the gears were damaged by
some accident or as the result of the failure of related parts, suoh
as failure of the supporting bearings, a very large statistical sample
had to be examined.

Dn.pirioal formulas used in automotive gear design

The manner in which the empirical formulas for rear axle spiral bevel
gears and transmission helical and spur gears were constructed is
described in detail in the papers "Factors Influencing the Durability
of Spiral-Bevel Gee.rs for Automobiles," by J. o. Almen (Automotive
Industries, November 16 and 23, 1935), and "Factors Influencing the
Durability of Automobile Transmission Gears," by J. o. Almen and J. C·
Straub (Automotive Industries, September 25 and October 9, 1937).
Since, for these gears, the service data were supported by laboratory
fatigue tests, the data oould be presented in the form of fatigue
curves. Since the stress was calculated by empirical formulas, the
stress units are not pounds per square inch and the stress units
calculated for spiral bevel gears do not have the same value as the
stress units oaleulated for helical and spur gears. The stress cycles
 3o

shown in the fatigue plots in the papers are, of course, the actual
number of load applications to failure of each gear.

The permissible stress, as calculated for carburized spiral bevel

gears used in passenger automobiles in which the pinions are over-
lmng, as sho'Wil in Fig. 10 and the first fopr illustrations in Fig.
12 of the paper on spiral bevel gears, is 42,000 stress units. If
the pinion is straddle-mounted, as shown in Fig. 11 and the last
illustration of Fig. 12, the permissible stress is increased to 51,000
stress units because of the reduced deflection and consequent reduced
stress concentration.

For truck service, the permissible stress for overhung pinions is

33,000 stress units and 43,000 stress units for straddle-mounted
pinions. For bus service, -the permissible stress for overhung
pinions is 30,000 stress units and for straddle-mounted pinions the
permissible stress is 40,000 stress units. The permissible stress
ma.y be further increased by approximately ten per cent by shot peen-
ing as a final operation.

By "permi$Sible stress," is meant that spiral bevel gears stressed to

not more than the amotmts indicated above would not fail by breakage
during the life of the vehicle, but at greater stresses, occasional
tooth breakage might be expected.

Clutches influence strength of gears

At the time the spiral bevel gear fornru.la was oonstructed, all automo-
biles were equipped with mechanical friction clutches, and because of
harsh action of many clutches, the gear loads were often grea.ter than
the maximum engine torque multiplied by the transmission ratio. such
overloads would necessarily appear in the empirical formula and would
help to establish the permissible stress. It is to be expected that
automobiles equipped with hydraulic clutches may safely operate at
greater calculated stress because harsh action is completely avoided,
but the form of the empirical formula would probably not be altered.

Dnpirioe.l formulas for predicting the fatigue strength by breakage or

automotive hypoid gears have not yet been constructed for lack of
sufficient authenticated service failures. The few service failures
by breakage of teeth that have been recorded, together with a limited
number of laboratory fatigue tests on hypoid rear axles; provide
sufficient data to justify the use, at least temporarily, of the
spiral bevel gear fornrula. However, for gear sets having ring gears
of the same diameters, the permissible stress may be ten per cent
greater for hypoids than for spiral bevels.
Machine parts subject to several kinds of failure

Many machine parts, inoluding gears, are subject to several kinds of

failures. As discussed above, gear teeth may fail by breakage, which
 3d

is bending .fatigue of the teeth loaded as cantilever beams. They may

also fail by pitting, which is fatigue failure of the surfaoes of the
teeth by tensile stresses induced by the compressive loads; or they
may fail by abrasion, one form of which results from welting of small
.areas of contaoting teeth because of the heat of friction and intense
pressure.

Since these three forms of failure are controllable by design, it is

necessary, to assure the success of highly loaded, high-speed gears,
that adequate preoautions are taken against each form of failure. It
is also necessary to avoid overdesign against one kind of failure at
the expense of another. This requires that empirical formulas be
constructed to establish design limits for eaoh form of failure.

Given sufficient data, empirical formulas can be constructed to

establish design limits for each form of failure for any particular
type of service. It is probable that any adequate collection of data
can be successfully expressed by more than one empirical fornn.2la, but
no formula can be successful, except by rare chance, until adequate
statistical data have been aooumule.ted. It is desirable that an
empirical formula should be rational; that is, that the ·formula should
be based upon the conventions used in ordinary design calculations.
This is not essential, however, since a formula may serve its purpose
even though the te:nns used appear to be co~pletely irrational.

statistical data on aircraft gears

Prior to our entry int.o World War II, there were not enough airplane
engines in use to provide sound bases for statistical studies of
service failures that might lead to the construction of empirical
formulas. The great increase in production and the extensive use of
these engines in severe military service indicated that studies of
service records might be profitable in supplying useful data for
future designs.

The Research Laboratories Division of the General Motors Corporation

initiated such a study in 1943 by asking the cooperation of sevetal
of the le.rge lllB.Ilufac'b.lrers of airplane engines in supplying data on
gear tooth failures from service records in airline end military
operations. Data were requested on any gear used in production that
had developed one or more of the several forms of distress enumerated
above; that is, pitting, breakage, or severe wee.r. De.ta. were also
requested on the type of service, the time that the gears had been in
service, and design deta.ils of both gears of a pair. The requested
cooperation was freely given in every case, end useful data were
supplied on one hundred and forty-siz designs, ranging from main re-
duction gee.rs to high-speed supercharger gears.

Service failures by breakage and by pitting were not numerous enough

to be significant for the purpose of establishing definite design
 3e

limits, but comparisons of bending stress and surface compressive

stress of the one hundred and forty-Gix gear designs that had been
submitted, with the bending and compressive stresses in automobile
gears previously analyzed, showed that the aircraft engine gears
should be relatively immune to breakage and to pitting.

"scoring" of gear teeth

The wear-damaged gears were not available for examination, but since
severe wear is almost always caused by local friction welding, here-
after called "scoring," this was assumed to be the cause of the wear
distress in most cf the one hundred and forty-six gears that were
reported. This assumption was later supported by the high values of
sliding and unit loading that were found in the course of the analy-
sis. Scoring of automobile rear axle gears, lubricated with ordinary
mineral oil, had previously been found to ocour in gear sets in which
the product of the Hertz compressive stress, in psi, and the tooth
sliding velocity, in feet per second, exceeded 1,500,000. This simple
method of predicting scoring in rear axle gears was successful, prob-
ably because the service, when scoring occurred, was approximately
the same as to load, temperature, and sliding velocity, since the
axle gears covered a relatively narrow range of size.

It was not to be expected that this simple method would suffice for
predicting scoring in aircraft engine gears, for which the tempera-
ture, unit loading, and velocity of sliding varied over a much great-
er range. Actual trials had shown that for certain aircraft gears,
PV values greater than 4,000,000 did not soore (see "Dimensional
Value of Lubricants in Gear Design, "J. o. Almen, SAE Journal, Vol,
50, 1942). It was hoped that the new and more extensive data sup-
plied by the cooperating aircraft engine manufacturers would aid in
clearing up this discrepancy.

As in all empirical formula constructions, a considerable number of

methods were tried to find numerical values for borderline scoring
conditions in service. These included several trials of the general
fonn pmvn, none of which were successful, as will be seen in the
discussion of this report. .Another geometric factor., T, was then
added, as is shown in Fig. 1, which resulted in establishing a very
good empirical limit, PVT, applicable to all gear sets (except as
noted in the discussion) for scoring of aircraft gears lubricated
with normal mineral oil.

Design conventions not aPPlicable

It is not too difficult to accept the product PV as a measure of

friction heat for conditions of borderline lubrication, but the new
factor T is foreign to our normal concepts. The real and only reason
for the factors used in any formula, empirical or otherwise, is to
 3f

enable the designers to accurately predict the behavior of their

designs in actual service. Whenever possible, the factors used should
conform to established conventions, but the success of the formula
should never be jeopardized for the sake of custom. The factor T
may be said to imperfectly represent the accumulated heat--the ambient
temperature--of the gear teeth. It is better, however, to accept the
factor T, as well as P and V, for what they are; that is., parts of a
measuring system--a go and no-go gauge.

It should be remembered that we cannot express incipient welding in

rational terms because we do not know the coefficient of friction of
the rubbing teeth, nor do we know the tempera uire required to form a
weld under the conditions of pressure and surface contamination that
prevail in the welded areas. Unless these things are known., none of
the factors used to express scoring can have individual, quantitative
meaning. Until measurements have been made of the acuial temperatures,
the actual pressure., the actual service loads, the distribution of
the load, and the manner in which these and many other factors influ-
ence weldability, the methods that we usually consider to be rational
must be classed as mere conventions.
 4

CONCLUSIONS

A method of calculating the scoring resistance of spur gears was

developed which checks quite well with the test data.. The method
is based on the total heat generated per cycle by the sliding action
under pressure of the mating teeth. This method is given herein.
The results of this method as applied to the test data. are shown in
Fig. 1 which shows good correlation. The sate limit of the scoring
· factor is 1.,500.,000 with mineral oil.

The G.M.R. method of calculation of bending stress on spur and helical

gears was applied to the data but it was found that the data on bend-
ing failures were insu.tfioient to appraise the validity of the method.

A method of calculating oompressiTe stress as & factor ·or pitting

resistance was also applied to the test data. but here again the in-
formation on pitting failures was insufficient to appraise the method.
 5

DISCUSSION

Scoring Factor for Spur and Helioe.l Gee.rs

'!'he method of calculation of the scoring factor, PVT, for spur and
helical gee.rs which is presented here he.s been established by test
date.. The date. were obtained from manufacturers of aircraft engines
end the scoring factor was oaloulated for each gear. The results of
these calculations in conjunction with the test data are illustrated
in Fig. 1. While the method is set up for spur and helical gears,
the test date. e.ve.ile.ble are confined to spur gee.rs. The points on
this cha.rt a.re plotted in the following me.nner:

Each vertical line designe. tes a gear design and is numbered for ref-
erence. To the left of the line is plotted the ce.loule.ted scoring
factor at the tip of the pinion tooth and to the right the scoring
factor at the tip of the gear tooth. Scoring failures are repre-
sented by open circles and gee.rs showing no scoring failure are rep-
resented by closed circles. !his cha.rt includes gear tests on 73
designs run under a wide variety of conditions.

Reduction gears at moderate speeds and high torques, as well as super-

charger gears at speeds up to 28,000 RPM and lower torques, are rep-
resented in the chart. Both external end internal gears are included.
Several thousands of tests are represented on the chart. Note that
gears with a scoring factor in excess of approximately 1,500,000 show
failure whereas those with less than 1,500,000 are free from scoring.

It should be mentioned here that a limiting factor (1,500,000) cen be

used as a scoring criterion because this type of failure is not
fatigue in character but rather a "yes or no" -cyrpe. That is, if
gears do not score on the first service application of max. speed
and torque, they are not likely to score at all.

With the exception of the pinion of design #6, which shows a high
factor wi thou.t scoring, there is 1i ttle deviation from the line marked
"safe limit. 11 Design :/If, is one in which the tooth profile was modi-
fied considerably and the end of action is, therefo~e, in doubt. As
will be seen from the method of oaloulation presented later, this
would have a marked effect on the calculated scoring factor.

Fig. 2 shows the results of the same group of tests using a conven-
tional method of calculating a scoring factor. The test results are
plotted in the same manner as in Fig. 1. The factor in this case is
the coDmlonly used value of PV, in which the unit pressure is simply
multiplied by the sliding velocity. The assumptions of load distribu-
tion are the seme as in the PVT method here presented. Such a calcu-
lation is evidently inadequate for a wide range of conditions of
speed and torque, even though it apparently shows some correlation in
 6

limited ranges of torque and speed. It should be noted that this

chart is plotted to 1/3 of the soale in Fig • ..L•

Method of Calculation - Nomenolature

The scoring factor used in the cha.rt of Fig. l is based on the total
heat generated by the oontacting surfaces during the cyole of sliding
from the pitch line to the beginning or end of action. This is in
contrast to the PV factor often used, whioh gives the instantaneous
rate of generation of heat at the area of contact.
The method involves the calculation of the unit pressure based on
the Hertz equations. The load is assumed to be distributed uniformly
on the average total length of contact lines.

The nomenclature used in the oaloulation is as follows:

Tp • pinion torque, pound inches

RPMp • pinion speed, revs. per min. with respect to its own axis

Np, Ng• number of teeth on pinion and gear respectively

F = face length, inches (of pinion or gear, whichever is
smaller)_

o< • normal pressure angle, degrees

..6.. helix angle at pitch line, degrees

1• pressure angle in plane of rotation, degrees
tan cj .. tan d sec 4
CD~ operating 0'8?lter distance, inches

P~, PRg • operating pitch radii of pinion and gear respectiv~ly, inohee

ORp, ORg • outside radii of pinion and gear respectively, inches

IRg • inside radius of internal gear, inches

Calculation of PVT - external gears

Referring to Fig. 3

Ag • ~ 0Rg 2 - PRg 2 oos 2 ~

 7

Na • Ap + Ag • CD sin 1
The unit pressure for external gears is calculated by the expressions

CD sin d.., at the tip of the

Pp • 2290V F2rrNaTp Np Ap (CD sin f - Ap) pinion tooth

and

\} 2TT Tp CD sin d at the tip of the

Pg • 2290F Na Np Ag (CD sin? - Ag) gear tooth

Having determined the unit pressure, the scoring factor, PVT, is cal-
culated by the expressions

at the tip of the

pinion tooth

and

at the tip ot the

PVTg • Tr RPMp (
360 Ng ) (Ag - PRg sinr,1.)a Pg
1 +~ gear tooth

In the case of spur gears the helix angle is zero and the pressure
angle Jn the plane of rotation is the normal pressure angle and there-
fore Cf • o( for spur gears.

With mineral oil as a lubricant, the safe limit of PVT is 1,soo.000

as shown by the chart Fig • ..!..•

The derivation of this method of calculation is given in the Appendix.

Calculation of PVT - Internal Helical Gears

Referring to Fig • .!_

A' p • Vo~ 2
- PRp 2 oos 2 f

N'a • A'p - A'g + CD sin <f

 8

The unit pressure for internal gears is calculated by the expressions

pt
p = 2290 v 2 'IT Tp
F N'a Np A'
. p (A'p
C D sin cl
+ C D sin,)
at O.D. of pinion

a.nd

P' g = 2290F2 v T~
'IT
N'a Np
CD sin cf..
A'g (A' g - C D sin~) at I.D. of gear

Having determined the unit pressure, the scor~ng factor, PVT, is

calculated by the expressions:

PVT'
P
= 'IT360
RPM;g (
1
~ ) (A'
- Ng P - PRp sin 1)
2
P' p at o.D. 01· pinion

and

at I.D. of gear

For spur gears 9

=o(, the same as for external gears. The safe limit
is 1,500,000 'With mineral oil as a lubricant .as is the case with
external gears. The derivation of these equations is given in the
Appendix.
Factors Influencing Scoring Resistance

In many gear designs in use today, scoring difficulties could be

reduced or even eliminated without additional cost if proper atten-
tion were paid to the tooth design. This applies also to pitting
difficulties, and in most eases there is no conflict between remedies
for pitting and scoring.
One factor that is often overlooked is- the proportion of tooth height
with respect to the pitch line, in .relation to the tooth ratio. In
a gear design of a ratio nearly one to one, no special scoring diffi-
culties are liable to result from the ' use of tooth proportions with
equal adaenda on gear and pinion.

But as the tooth ratio increases it becomes increasingly important

to proportion the addenda in order to avoid high unit pressures ·at
the tip of the gear.. Further, this oondi tion becomes aooentuated
in coarse pitoh gears. The reason for the necessity of such consider-
ations is quite simple. The unit pressure varies roughly as the
reciprocal of the smaller radius of curvature of the teeth, and that
radius is theoretically ;ero at the base circle. Therefore, to avoid
 9

high unit pressures and high scoring factors, it is necessary to avoid

action near the base circle. Fig. 1 shows the unit pressure a.nd
scoring factor for a pair of gears "of 3.8 ratio. The variation in
unit pressure along the line of action ts shown immediately above the
diagram of the gears. Note that where the line of action approaches
the base circle of the pinion, at the left, the unit pressure rises
asymptotically. The gears shown have equal addenda on gear and pinion
with the result that at the tip of the gear tooth the unit pressure is
more than twice as great as that at the tip of the pinion tooth. The
sliding velocity along the line of action varies directly as the
distance from the pitch line as shown above the unit pressure chart.
Next above the sliding velocity is shown the variation of PV or the
instantaneous rate of heat generated. It is an intermediate step in
determining the scoring factor PVT which is illustrated in the top
curve of Fig. 7 • The values of PVT plotted vertically in this
curve are obta'Iiied as the area under the PV ourve, starting from the
pitch line. For instance, at the tip of the gear the PVT value
represents the approximate area of the PV curve to the left of the
pitch line. It is approximate in that the PV curve is considered
as a straight line from the pitch line to the tip of the gear.

Note again that at the tip of the gear the scoring factor is very
much higher than at the tip of the pinion for the same reason that
the unit pressure is excessive there. It should be emphasized that
the ratio of the gears illustrated here is rather moderate in com-
parison to many designs which require ratios of 8 to l or higher.
In such cases, the situation becomes even worse, because with increas-
ing ratios the pitch line moves more and more to the left, or towards
the pinion base circle. The ratio chosen for the illustration was
selecied because the factors involved can be pointed out more clearly.
Designs such as that shown in Fig. 1 are not uncommon even in higher
ratios. Such a design can be substantially improved at little, if
any, additional cost by simply changing the tooth proportions.

Fig. 8 shows the same design as shown in Fig. 7 except that the
gear addendum is decreased and the pinion addendum increased. Note
that this change results in a marked decrease in the maximum unit
pressure and also a decrease in the maximum. scoring factor.
A, striking example of such an improvement in actual practice is shown
by comparing designs 38 and 39 in the sooring factor chart of Fig. 1.
Design 38 is one in which the ratio is approximately 2 to 1. The
pinion addendum in this pair is slightly greater than that of the
gear but the gears are of coarse pitch, end consequently of long
addenda • . The result is that action talces place rather olose to the
base oircle. As shown in Fig. 1, the scoring factor at the tip of
the gear in this design is 2,650,000. Approximately 400 gear sets
0£ this design were tested, of whioh 33% scored at the tip of the
gear.

Design 39 is identical to 38 just mentioned except that the gear

addendum is .070" shorter than that in 38. Referring again to Fig. 1,
this change reduced the PVT to 1,750,000 in design 39, ·and the scoring
 10

dropped to less than 2% in 3000 tests. This comparison is even more

striking in the fact that the pinion addendum is not increased to
:maintain the same working depth. In other words, this improvement was
made in the sooring resistance in spite of the fact that the contact
ratio was materially reduced •

.Another example is shown by comparing designs 35 and 36 in Fig. 1 •

In this case the scoring failures were reduced from almost 5% of~ears
tested in 35 to 1/3 of 1% in 36. This improvement is also the result
of reducing the gear addendum, thereby reducing the PVT factor from
1,780,000 to 1,550,000.
In the design of a pair of gears the selection of the diametral pitch
is of great importance unless weight of the gear box is of no conse-
quence. The diametral pitch should be a compromise between bending
strength on the one hand and resistance to pitting and scoring on
the other. Low bending stresses are obtained by the use of a coarse
pitch because the tooth thickness is large. However, as the pitch
becomes coarse, it is necessary to increase the tooth height in order
to maintain a reasonable contact ratio. This means that action ap~
proaches closer to the base circles of the gears, which in turn means
high compressive stresses which may cause pitting. Further, the in-
creased tooth height increases the sliding velocity which in combina-
tion with the high compressive stress may cause scoring. From the
standpoint of scoring and pitting this increase of tooth height to
accomodate a coarse pitch becomes increasingly important as the
tooth ratio increases. A good design, · then, is one in which these
factors are balanced.

The pressure angle is also important in the strength of gear teeth,

but in this case there is no oonfliot between bending stress and
unit pressure or PVT faotor. Higher pressure angles tend. to decrease
the bending stress as well as the unit pressure and PVT factor. To
oite an example of such improvement in scoring resistance, we can
refer again to Fig. 1, designs 35, 36 and 37. It will be recalled
that the scoring failures were reduced in this design from 5% in
design 36 to 1/3 of 1% in design 36 by decreasing the gear addendum.

A second change was made by increasing the pressure angle from 20°
to 25° in design 37. As shown in Fig. 1 , design 37 has a reduced
PVT factor as compared to 36. Over 2000-gear sets were tested after
this ohange was made and failure was oompletely eliminated.

Some designers place great emphasis on the contact ratio as a factor

influencing the strength of gears. But if a high contact ratio is
accompanied by high unit pressures, its advantages may be more than
offset by pitting and scoring difficulties. The examples cited above
bear out the faot that a high contact ratio of itself is not necessarily
beneficial. Design 37, just mentioned, is one which has a contact
ratio of less than 1.2. In spite of the fact that this appears very
low, no failures have occurred, whereas designs 35 and 36, both o_f
which had definitely higher contact ratios, failed by scoring.
 11

Pitting Fa.c tor, P

Pitting, in contrast to scoring, is a fatigue type of failure and

therefore more detailed data are required to appraise a method of
oaloulation. In the appraisal of a scoring faotor it is sufficient
to know the torque and speed of the gears tested and whether or not
scoring occurred under these conditions. The appraisal of a pitting
factor on the other hand requires data on the actual life of the
gears before such failure, if any, occurred. Such data could then
be plotted on a log-log fatigue chart with life plotted against the
calculated factor. The factor thus calculated would be appraised on
the basis of whether . the results fall reasonably close to a fatigue
line.

The available data on pitting is incomplete from this standpoint.

However, a "yes or no" chart is plotted in Fig. 5 to determine
whether such a chart would give any limiting values of a pitting
factor. As can be seen from the chart, no such limiting value is
indicated.
The factor used in calculating the pitting resistance of spur and
helical gears is the unit pressure, P, as calculated in the PVT
faotor given above. That is, for external gears, the unit pressure
is

CD sin d. at the tip of the

Ap (CD sin~ - Ap) pinion tooth

and

CD sin cl.. at the tip of the

gear tooth

For internal gears the unit pressure

pt . .. 2290 , I 2 1T T~ CD sin O(' at o.D. of pinion

P \J F Na p A' p (A' p + CD sin ¢')

and

, /2 1T T CD sino<' at 1.D. of gear

pt g • 2290 \J F Na ip A' g (A' g - CD sin~)
 12

Bending Stress Factor

The third type of gear failure, bending or tooth breakage, is similar

to pitting failure in that it is fatigue in oharacter. Therefore,
data on the life of gears tested are necessary for appraisal of a
method. This information is lacking in the available data as was
the case in pitting failures and,for the same reason as pointed out
tmder pitting, the data are not sufficiently complete to appraise a
bending stress formula. As was the case in pitting, the bending
stress factor is plotted in Fig. 6 in an effort to show limiting
values if they exist. As oan be seen from the ohart, no limiting
value is apparent. The bending stresses were calculated by the
G.M.R. method described in Report D-314.
Factors Other Than Tooth Design

Aside from the design of the gear teeth themselves there are other
factors which must be considered as part of the overall design of
a set of gears. The housing, shafts, bearings, etc. should be
designed in such a way as to allow the teeth to mesh uniformly for
the full length of the teeth as nearly as possible when the load is
applied. A heavy tooth bearing on either end of the teeth inevitably
results in load ooncentrations at that end. This load concentration
is undesirable from every standpoint. It cannot be entirely eliminated
because spur or helical gears have straight lines of contact and it
is impossible to maintain parallelism of the potential lines of contaot
when high loads are applied to the gears. Suoh load concentration oan
be reduced, however, by proper design of the gear mounting. It is,
of course, desirable to support each gear between two bearings when-
ever possible, in order to avoid more deflection at one end of the
gears than at the other. The case and shafts should be sufficiently
rigid to avoid excessive deflections. However, this should not be
interpreted as a statement that all gear mountings should be as rigid
as possible. In some oases it is more practical to balance deflections
in one sense with others of the opposite sense. This is a matter of
study of the indivi~ual case.

The material used in gears is important with respect to its hardness.

The data presented in Fig. 1 and 2 deal entirely with hardened .
steel gears. That is, gear~with ~file hard surface. It is very
important in gears which are highly loaded to maintain a hard sur-
face. This is true for all three types of failure. Where the sur-
face hardness is low, the bending stress, unit pressure and scoring
factor nrust be correspondingly low, if failure is to be avoided.

Another factor which must be consi~erad in a gear design is the cool-

ing system. This varies with the job that the gears are expected to
do. Where the power to be transmitted is high, more cooling oapaoi ty
must be provided than if the gears are to run at light load and
:moderate speed. In some cases it is diffioult to provide adequate
cooling for the gear box. In suoh instances it may be necessary to
operate at higher temperatures. Here again a lower scoring faotor
 13

can be tolerated than if cooling were sufficient to maintain a

reasonably low oil temperature. Fig. JL is a qualitative chart
showing the influence of surface hardness and temperature upon the
unit pressure and sliding velocity which can be tolerated. This is
a three-dimensional chart in which the curved surfaces within the
cube represent the influence of pressure, sliding velocity and oil
temperature on the scoring characteristics. Note that for hard
material, higher values of load, speed and oil temperature can be
tolerated than for soft material. As the oil temperature increases,
using a given hardness of material, the pressure or velocity or
both must decrease in order to be sa.fe. Finally, assuming a given
oil temperature, lower pressure will tolerate a higher sliding
velocity and vice versa.

Some gear designs are required to carry extremely high loads and
speeds under adverse conditions. In such cases, the use of extreme
preseure lubricants may be necessar-.r• With the use of such lubri-
cants the scoring resistance of gears can be definitely increased.
The use of extreme pressure lubricants, however, is usually limited
to isolated gear boxes because such lubricants are likely to attack
non-ferrous bearing materials• The effect of an extreIT1e pressure
lubricant on the scoring resistance is illustrated in Fig. 10, which
is another three-dimensional chart similar to Fig. 9 • In this case
the chart shows that with an extreme pressure lubricant, higher loads
and speeds can be used than with mineral oil. These characteristics
are summarized in Fig. 11, which shows the influence of the factors
mentioned above. The higher range of pressure-velocity combinations
is for E.P. oils, while the lower range is for mineral oils. These
tv,,o groups are in turn influenced by surface hardness and oil
temperature.
The method of lubrication is another factor influencing the scoring
resistance of gears. It is important to remember that the function
of a lubricant is one of cooling as well as lubrication. The oil
carries the heat from the gears to the housing or heat exchanger as
the case may be, and then to the outside. Where the gears are lubri-
cated by passing through a sump of oil, exo~ssive oil must be avoided
as well as an inadequate supply. With excessive oil, where the gears
are submerged in a large volume of oil, the churning losses may be
so great as to heat the oil, and excessive temperatures may develop.

The method of lubrication to be preferred, if possible, is that of

using an oil pump and a jet of oil on the teeth as they come out of
mesh. The reason for the jet being preferred on the out-of-mesh
side rather than on the entering side is because the teeth are at
their maximum temperature after meshing and, therefore, the cooling
is more effeotive. The oil remaining on the teeth when sprayed on
the leaving side is ample for lubrication, because only a thin film
of oil is required for that purpose. A further advantage of this
type of lubrication is that dirt or any other foreign material is
thrown off and thus excluded from the gear teeth in action.
 •
II

c p

:)
p

•
11 • 0
Pc
c
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h
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c

1234567891() 15 20 25 30 35 40 45 50 55 60 65 70
DESIGN NUMBER
(PINION TIP, LEFT OF LINE; . GEAR TIP 1 RIGHT OF LINE)

o- TEETH SCORED
• - NO SCORING

CORRELATION OF SCORING FACTOR, PVT,

WITH ACTUAL TEST DATA ON 73 GEAR DESIGNS
FIG. I
 15.4
:
9 •
• p

II • p
II
(/) 6
z •
Q
_J
• t)
•
I
-
_J

~
p
II
• I I
I . p p D

• •• c
II
• " •
II II
>3
Q..
•• I
•• I ,11
•
0
I
••
la

••
• ••• • • ,11 •• • •
• 0 • •• • II II
• •• •
0
• •• •
I 2. 3 4 5 6 7 8 9 IO 15 20 25 JO .35
DESIGN NUMBER
(PINION TIP. LEFT OF LINE ; GEAR TIP, RIGHT OF LINE)
o-TEETH SCORED
•- NO SCORlNG
FAILURE OF "PV'' TO CORRELATE WITH
ACTUAL TEST DATA OF 39 AIRCRAFT
ENGINE SPUR GEAR DESIGNS
FIG. 2
 I~

RADIUS OF CURVATURE
EXTERNAL ~GEARS

PLANE OF ACTION

FIG. 3
 17

RADIUS OF CURVATURE
INTERNAL GEARS

FIG. 4
 (1)4
0 - )

I 3 !'\

II

I
• •• la ... :>
~ • - ~

"
•
19' Ill

• •••• • It •• • • b
•••
•• • • • •
I b 0
m10 Ill •
.. . ••
••• c 19
c c
b
b. c c c
• c
II
II
• 19
f • • It 41
•• ••• •
0
; O I 2 3 4 5 6 7 6 9 JO JS 20 25 JO 35
DESIGN NUMBER
(PINION TIP LEFT or UN£ GEAR TIP RIGHT or LINE)
o-TE£TH PITTED
•-NO PITTING

LACK OF CORRELATION OF UNIT PRESSURE WITH

ACTUAL TEST DATA ON 39 AIRCRAFT
ENGINE SPUR GEAR DESIGNS
.....
FIG. 5 (X)
 7

5n I
•
• c
p

II
•
cp
••• .
• • • I

•• t '"'
p
••
b p

U)
cD
• p
( I
U)
w 2 • •
• Kii l'I
., c
~ •
K I

C)
z
0
I
I I
It•
••-
.•.• .
•~
It "
~.• I I

• b It •
• •o c
p

~ 0 I 2 3 4 56 7 8 9 to 15 20
• 25 1
•• .D 35
m DESIGN NUMBER
(~INION TIP, LEFT OF LINE~ GEAR TIP, RIGHT OF LINE)
o-TOOT~~REAKtGE
1
::~ ~ BREU~~
BENDING FAILURES IN ACTUAL TEST DATA .
AIRCRAFT ENGINE SPUR GEAR DESIGNS ......
co

FIG. 6
 20

!----
>
SCORING .
FACTOR-PVT
a.1-+--+----+-+---------l
o---~11&-~---~~----

--~--t-- SLIDING
VELOCITY-V
==i
- NO. OF TEETH ON
PINION 13
RATIO 4.23: I OR 55: 13
U I PRESS ANGLE 25°
r ----r-- PRESSURE-P

a. ~___:;:===~-----==-=-=---~
--
TGEIPAORF ol--"----~---------4 TIP OF
---...--r-,-r- --------PINION

-'I'----
PITCH
LINE

SCORING FACTOR WITH EC.UAL ADD.

FIG. 7
 + PV _ _ ____..
&:--~....__----f
0----¥----------------

SCORING FACTOR WITH SHORT GEAR ADD.

AND LONG PINION ADD.
FIG. 8
 SOFT
MATERIAL

>-
r HARD
-
( /) MATERIAL
z
w
t-
z
w
a::
::)
(/)
(/)
w
a::
Q..
II
Q..
116..------------
V=SLIDE VELOCITY-.

EFFECT OF PRESSURE, TEMP., SLIDING

VEL. & HARDNESS ON SCORING LIMIT
FIG. 9 N
N
MINERAL OIL

>-
r E.P.
-(/) LUBRICANT
z
w
r
z
-

FIG. 10
 BASE CURVE EP OIL
t....
>-
t - - ~ _ , . _ _ ~ ~ - + - - - i BASE

MINE.R L
CURVE, - - - -
IL
~ LOWER TEMP.
Z
w H
~~a&-4--~~~

....z
-
~
:::>
(/)
~t-----1r-----i~~~~~~=-------+-~~
w
a= HIGHER TEMP. OR
~ SOFTER MATERIAL
~----------------__..-------------------~
V=SLIDING · VELOCITY •

EFFECT OF HARDNESS TEMP, AND

LUBRICANT ON SCORING LIMIT
FIG. 11
 25

APPENDIX I

Derivation of Method ot Caloulation of PVT (External Gears)

Unit Pressure

As mentioned pr~viously, the unit pressure as calculated is based on

the Hertz equations. Of necessity certain assumptions must be made
as to the distribution of the load on the teeth in contact. This
method assumes that the load is distributed uniformly on the average
total length of contact line. The term average is used rather loosely
here for want of a better term. The actual length over which the load
is assumed to be distributed is calculated as Lin the following
expression:

(see Fie;. 3) (1) .

in which $NP is the basic normal pitch and the other symbols
correspond with the nomenclature given on page 6.

The basic normal pitch is equal to

f N
p . 2 TT PRp cos
Np
f cos..6o.~

in which ~(3 • helix angle at the base circle.

Sub!tituting ~NP in the expression for L:

F Na Np
L • 2 TT PRp cos 1 cos A(3 (2)

The total load normal to the teeth is

TJ;2 (3)
p a PRp cosf> cos~

The load per inch of contact line then becomes

p
P ,,,. L (4)

( 5)
 26

Having thus determined the load per inoh on the contact lines, the
unit pressure is calculated, assuming the tooth surfaces as elements
of cylinders contacting eaoh other.
In general, the maximum compressive stress occurs at the tip of the
gear tooth or that of the pinion tooth, depending upon the radii
of curvature of the tooth flanks.

When the contact is at the tip of the gear tooth, the normal radius
of curvature of the gear tooth at that point is

~- .. A~
·o cos.A.~

and that of the pinion tooth in the oorrespon~ing position is

R ,. CD sinf - A~
p COS A~

The Hertz equation for unit pressure at the tip of the gear tooth
using steel gears is

Sg • ~ 5,250,000 p R~ +RR' ( 6)
p g

The expression can be simplified as follows,

R;p + R, .. A, + CD sin [ - A, oos 2 ~€3

Rp Rg cos Ll(3 • Ag (CD sin f - Ag)

CD sin .......~----J-.-...-
.. ~--~
Ag CD
But
cos .6 cos c( .
cos_.1 ~ • cos f'
 27

Further

tan j • tanc(
i oos ..6.
Finally
tano(
CD oos 2.. oos A cos <:/.._
Ag (CD sin~ - Ag)
which simplifies to

Rp + Rg • ~-,.C~D~s~i~n__.ioo-~.._. ( 7)
Rp Rg Ag CD sin - Ag

Substituting (5) and (7) in (6)

Sg • 5,250,000 p Rp + R~ ( 6)
Rp Rg

• 2290 \ I 2 TT Tp CD sin (7a)

~ F Na Np Ag CD sin • Ag

This gives the unit pressure at the tip ot• the gear tooth. It is
apparent that at the tip of the pinion tooth the unit pressure is

\ ~;p
Sp• 2290 ~~ Np Ap
CD sin
CD sin - Ap (7b)

Sp and Sg are used to determine the unit pressure on external gears

as shown on the chart, Fig. 7. Sp and Sg as calculated by these
expressions are plotted as "P" at the tip of gear and tip or pinion
in Fig. 7.

Sliding Velocity ( external gears)

The sliding velocity of the tooth surfaces in contact is determined

as follows:

The relative angular velocity of one gear with respect to the other
is
.. ( 8)
 28

2
in whioh W'p • 6~ RPMp ~ angular velooity ot pinion, radians/sec.

211 RPMg
and Wg • ~ • angular velooity of gear, radians/ sec • .

RPMp and RPMg • speed ot pinion and gear respeotively, revs./min.

Substituting tor wp and "'g in (8)

1f
w • '!O" ( RPMp + RPMg)

' ' w• ~ RPMp (1 + t) rad,/seo'

The distance from the instantaneous center of rotation (pitch point)

is

PRt sin
ag • At - 12
p at tip or gear tooth, feet

ap •
A;g - PR;g sin
12
f at tip of pinion tooth, teat

The linear sliding velocity is then

V • ,.. a
g g
• 1T
~
RPM
P
(i + No)
°ffg At - PR, sin
12
<j at tip of
gear tooth, (Sa)
tt./sec.
or

V. • w
P
a_ •
!-'
1T
10"
RPMp (.1 + !J2.)
W-g
:'p - PRp sin
12
;i at tip of
pinion (Sb)
tooth, tt./seo.
The 'V'&lues of Vat the tip of the gear and the tip of the pinion in
Fig. 7 are calculated by these expressions.
 29

It has been more or less comm.on practice to use a "PV" factor for
a measure of scoring resistance. That is, the product of unit
pressure P and sliding velocity Vis. referred to as the "PV" factor.

Such a PV factor is obtained by the product of equations (7a) and

(Sa) or (7b) and (Sb):
at tip
of' gear (9a)

at tip of
sp pinion (9b)

The values of PV at the tip of the gear and tip of the pinion in
Fig. 7 are calculated by these expressions. The PV factor as cal-
culated by this method is not a tr-ue measure of the scoring resistance,
as demonstrated in Fig. 2.

Scoring Factor PVT (external gears)

The PVT factor which shows correlation with test data and does serve
as a measure of scoring resistance is obtained as the approximate
area under the PV curve from the pitch line to the tip of gear or tip
of pinion along the line of action.

The area under the PV curve is approximate in that the PV ourve is

considered a straight line, from zero at the pitch line to the value
of PV at the tip of gear or pinion.

Considering the tip of the gear, the area under the PV curve to the
left of the pitch line in Fig. 7 is simply the area of the triangle
in which the vertical side is PVg and the horizontal aide is ag·
The area of this triangle is

Area • PVg i'Lg

2

But ag • -~--__,,,P.,,.R.,[_s_i_n....,f1--
12

Area= PV~ (.Ai - PRir sin(:)

24

Since the calculated values are only relative, the denominator of

the above expression is eliminated for ease of calculation, and the
scoring factor is expressed as
 30

Substituting for PVg, the scoring factor becomes

PVTg .. ~ (1 + t) (Ag - PRg sinf ) 2 Sg at tip

of' gear
(lOa)

By similar procedure

PVTp ,. TT RPM,, (' 1 + ~ ) (Ap - PRp sin). )2 S -..I at tip of' (lOb)
~ Ng; T P pinion

These are the expressions used in the oalculation of the scoring

factors f'or external gears at the gear and pinion tips as shown in
Fig. 7. The intermediate points of the PVT ourve as sho19Il a.re
plotted only for illustration. A word ot explanation should be given, .
however, as to the derivation of' the PVT curve. At any point along
the line of action, the value of PVT is the area under the PV ourve
(considered as a straight line) from the pitch point to the point in
question. The question might be asked as to why the PVT factor is
not plotted as zero at the beginning of action, inoreasing to a maximum
at the pitoh line for the approach action. The reason for the curve
being plotted as shown is due to the fa.ct that the designs which had
scoring failures were scored, in general, at the tip of the gear or
pinion regardless of whether it happened to be the beginning or end
of action. The actual value of the scoring factor is not affected by
the cha.rt because the area under the total PV curve is the same in
either case.
 31

APPENDIX II

Derivation of Method of Calculation of PVT (Internal Gears)

The factors involved in the PVT factor for internal helical gear
are similar to those for external helical gears.

It is apparent that the load per inch of contact line can be calcu-
lated in the same ws:y as for external gears. That is

(11)

Unit Pressure (tip of gear)

The determination of the radii of ourvature, however, requires some-

what different expressions. Referring to Fig. 4, when the oontaot
is at the tip of the gear tooth (I.D.), the radius of curvature of
the gear tooth at that point is

A'·g
R' g • - --
oos -
.A'3

and that of the pinion tooth is

R,
p
• A' g -
cos
CD
~?
sin i

For steel internal gears the Hertz equation for unit pressure at the
tip of the gear tooth is

S'g • ~ 5,250,000 p R' - R'

Rf p R' gp (12)

The tenn (R'g - R'p) is the difference between the radii of curvature
rather than their sum because the ourvature is in the same sense
for internal gears.
 32

Simplifying the expression R'e; - R'p

Rp Rg

R' - R' A' - A' + CD sin i cos 2 Ll.(3

ff•g R'~ • g ool6.~ A'g (A'g - CD sin¢)

• CD sin
A'g Ag -

But COSA~ •
cos A oosci.
cos</'

R' - R' CD sin cosA cosd CD tan cos A cos o(

Rf p R' g p "" . ,.A......
g_o_o_s_,~-A.,..'_ _C__D_s"""in-,1-.- ""
g - . A'g A'g - CD sin

Further tan~ • tand.

r cos.A

(13)

Substituting (11) and (13) in (12)

S'g • 2290 2 'TT T CD sin o(

F Na.~ A'g {A'g - CD sin#)

This expression gives the unit pressure at the tip ( I .D.) of the
internal gea.r and is the maximum value of unit pressure on the basis
of the foregoing assumptions.

Unit Pressure (tip of pinion)

When the contact is at the tip of the pinion tooth, the radius o~ the
pinion tooth is (see Fig. 4)

A'p
R' p • - --
cos -
4~
 33

and that of' the gear tooth is

R'g • A'p + OD sinf

OOSA~

R'f - R';g, • A'p + CD sin<P - A'p

R g R'p cos ..C1.13r A'p (A'p + C
cos 2 ~g sinf)

But cos ..6. oosc<

cos 4~ -=
cos~

R'v - R'p • cos ..6. cos c( ::

CD tan cos Li oos o(
R g R'p cos A' p + CD sin A' p A' p + CD sin

Further

tan f • tan o( sec A

CD sin o<
• A'~ (A'p + CD sin,) (14)

Substituting (11) and (14) in the expression for unit pressure, as

before

2 ,r T CD sin d..
S' p 2 2290
F Na ~p A'p (A'p + OD sin/)

This is the unit pressure at the tip of' the pinion tooth.
 34

Sliding Velocity
In the case of internal gears the relative angular velocity is

W'"" Wp - Wg
in which ll'p and Wg are the angular velooi ties of pinion and gear.

Using the same nomenolature as for external gears:

W.p =2 TT RPMp
60

rad./sec.

The distance from the instantaneous oenter of rotation (pitch point)

, PR~ sin <j; - A'~

a g • 12 at tip of gear tooth (I.D.) feet

and

A' - PlL sin<}

a'p = p 12-P at tip of pinion feet

The sliding velocity is

Vg =- w a' g .. 30 RP1:1p
TT (
l - ~
Nn.J PR
_g
sin
l2
p- A'e:"'
at tip of gear
tooth (I.D.),
ft./seo.
or

Vp ~ w e.'p ~ ~
3 0
ll1'Mp (1 - t) {A'p - PRp sin fl .at tip of
pinion
tooth,
f't./seo.
 35

Scoring Factor PVT

Finally, the sooring faotor PVT is obtained by

PVTg • ~
3 0
RPMp (1 - t) (PRg sin f- A' g)2 S' g at tip of gear
tooth (I.D.)

or

at tip of pinion
tooth