Table Of ContentANALYTICAL
MECHANICS OF GEARS
By
EARLE BUCKINGHAM
Professor of Mechanical Engineering, Emeritus
Massachusetts Institute of Technology
New York
DOVER PUBLICATIONS, IN(t
.
.
IIA Lib.,
©
Copyright 194:9 by Earle Buckingham
All rights reserved under Pan American a.n.d
International Copyright Conventions.
Published in. the United Kingdom. by
Constable and Coxn.pa.n.y, Limited,
10 Orange Street, Lon.don. -W-. C. 2
This Dover edition, first published in 1 963, is an un
abridged and corrected republication of the 'W'ork
first published by the McGra-w-Hill Publishing
Company in 1949.
Standard Book Nu-rnber: 486-61073-X
Library of Congress Catalog Card Num,ber:: 63-2.1.682
Manufactured in t:he United States of America
Dover Publications. Inc.
180 Varick Street
New York, N.Y. 10014
PREFACE
This text is the gratification, to some degree at least, of a suppressed
desire of some twenty-five years' standing: this has been the development
of a. reasonably complete mathematical analysis of the mechanics of
geanng.
The subtitle of this treatise could well be "Final Report of the ASME
Special Research Committees on Worm Gears and the Strength of Gears."
The information and the inspiration from the work of these committees
and from personal contact with their individual members has been the
major motive force behind this work.
To quote from a note to one of my colleagues: "If one understands
the significance of the calculus, then the subject of conjugate gear-tooth
action should be mastered with ease. On the other hand, if one has a
visual imagination and can grasp the significance of this conjugate gear
tooth action, then this presentation should be of material aid to help
him to master the significance of the calculus. If however, as with
myself, he has neither, then continued application to the problem may
give him some slight appreciation of both."
This is not a text on gear design. It should form, however, a sound
foundation upon which logical design practices and design data can be
erected. These design structures, however, must be erected by the spe
cialized engineering groups that have need of them. In the final analy
sis, there are so many unknown and uncertain factors that in critical
cases our only answer is to "try and see .. " An analysis such as this
should be of material aid in the interpretation and in the application of
the information that is made available by definite tests and by actual
. .
service experience.
The trends of today indicate a growing demand for the seemingly
opposed requirements of higher speeds and greater loads with more reli
ability and quietness of operation. These demands are met in part by
improved materials, better balancing, more nearly perfect machined sur
f aces, and more intensive attention to many details of design. This last
should also include a rigorous mathematical analysis of both the kine
matic and dynamic conditions of operation.
The major objection to the thorough analysis of any mechanism is
the amount of time required to make it, and also the dearth of individuals
v
vi PREFACE
who have both the ability and the inherent urge to do such work. If
each new project must be started from the beginning, the time required
to carry it through might be excessive for any single application. If,
however, we can build up a foundation of general solutions of different
basic types of mechanisms, then the specific task for any given project is
to arrange the general solution for the particular case and solve. Even
so, considerable time must be spent in computing the results, and a cal
culating machine is one of the essentials.
Among the machine elements that have received much less analytical
attention than they deserve are linkages, cams, and gears. The book
attempts to give in usable form such a general analysis of gears.
As noted before, this is not a text on gear design. The major effort
has been to make apparent the nature of the action and contact between
the contacting teeth of the different types of gears. A clear understand
ing of these features is essential if the designs are to exploit the full pos
sibilities of these mechanical elements.
The :first chapters give an analysis of conjugate gear-tooth action,
nature of the contact, and resulting gear-tooth profiles of the several
types of gears. These include spur gears, internal gears, helical gears,
spiral gears, worm. gears, bevel gears, and hypoid or skew bevel gears.
Spur, internal, and helical gears are used to drive parallel axes. Bevel
gears are used to drive intersecting axes. Spiral, worm, and hypoid
gears are used to drive nonparallel, nonintersecting axes.
The last chapters give an analysis of gear teeth in action. These
include frictional heat of operation and its dissipation, friction losses and
efficiencies, dynamic loads in operation, beam strength or resistance of
the teeth to breakage and fatigue, surface-endurance limits of materials,
and the limiting wear loads or the potential resistance to surface disin
tegration and excessive wear.
No claim is made that this analysis is complete. In the first place,
space limitations prevent the full development of many interesting and
pertinent factors of this general problem. While in the second and more
important place, the writer has much to learn about the subject. It is
probably much larger than the capacity of any individual to master com
pletely. Even this incomplete analysis is the result of the original work
of many individuals.
The effort has been made to give due credit in the text to the many
sources of information that have made possible this analysis. In addi
tion to this, much help both direct and indirect has been obtained from
close personal contact with Wilfred Lewis, Carl G. Barth, Charles H.
Logue, Ernest Wildhaber, and all the other members of the ASME Spe-
PREFACE vii
cial Research Committees on Worm Gears and the Strength of Gear
Teeth, as well as from many other individuals who are struggling with
various types of gear problems. And last, but not least, a debt of grat
itude is owed to Guy L. Talbourdet, who has studied this manuscript crit
ically and who has checked carefully all the derivations of the equations.
EARLE BUCKINGHAM
CAMBRIDGE, MASS.
March, 1949
