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AN ELEMENTARY TEXT-BOOK

MACHINE DESIGN

A STUDY OF METHOD WITH
NUMEROUS ILLUSTRATIONS

For Students, Mechanics, Draftsmen, and Others
Desiring an Introduction to the Sub-
ject With or Without the Aid
OF AN Instructor

CfiARtrES LEWIS griffin S. B.

Mbchanical Enoinbbr, Sbmbt-Solvay Company
Ambrxcan Socibtt op Mechanical Enoinbbrs

AMERICAN SCHOOL, OF CORRESPONDENCE

ARMOUR institute OF TECHNOLOGY

CHICAGO, IL,L,INOIS

1904

coftkxght 1904 bt
Chari«es Lbwis Grippin

PREFACE

THIS little book is intended to be written so plainly as to be
interpreted readily without the aid of an instructor, and
yet it strenuously avoids the use of ''rules of thumb" so
prevalent in works of this character. It is written on a purely
scientific, technical basis, not to compete with the many excellent
works on Machine Design already available, but rather as prepar-
atory and auxiliary to them.

The author's experience has been that the student's chief dif-
ficulty in his elementary study of Machine Design is lack of
ability to attack a practical problem. However much valuable
illustration and data are laid before the student in engineering
books, he cannot avail himself of them until he has developed a
method or plan of action by which he can systematically interpret
and measure their value for his particular problem.

This method or plan of action it is hoped to make clear in the
pages of Part I ; and, to properly enforce the same, the designing
of a simple machine is undertaken, and is followed from the orig-
inal data, through the preliminary sketches, to the finished detail
drawings.

Part II is auxiliary to Part I, and illustrates the method of
design applied to specific cases of the simplest machine elements.
Elaborate designs are carefully avoided, simplicity being the con-
trolling element in all cases. Hence Part II must not be con-
sidered as a compendium of design, but rather as an introduction
and a basis for further study.

In fact, the whole purpose of this book will be abundantly
fulfilled if the student grasps and gets under his control the
method of Machine Design, and finds himself incited to study the
more elaborate works. It is confidently believed that much, if not
all, of his discouragement at undertaking the study of standard
works on Machine Design will be relieved, when he stands on his

own footing of Analysis, Theory, and Practical riodification as

taught in these pages.

The author desires to express his appreciation of the services
of Mr. H. Edw. Dunkle, of the Pennsylvania State College, in the
preparation of Part II, a large amount of matter and drawings
being due to his interested labor.

CHARLES LEWIS 6RIFFIK
Sybaouse, N. Y., April, 1904.

CONTENTS

PART I.

PAOB

Definition 3

Method of Design .

Constructive Mechanics

Friction and Lubrication .

Working Stresses and Strains

Application to Practical Case

Preliminary Sketch and Calculations

Preliminary Layout to Scale .

. 39

Pulleys ......

Gears ......

Brackets and Caps

Drum and Brake . .

Base, Brake Strap Bracket, and Foot-lever

Gear Guard and Brake Relief Spring

General Drawing .....

General Comments on Problem

Classification of Machinery

■

Machine Tools .

Motive Power Machinery

Structural Machinery .

Mill and Plant Machinery

Original Design

PART IL

Introduction ........ 75

Belts . . . . .

» i

1 i

Pulleys

» i

> i

» <

SpUt PuUeys

Special Forms of Pulleys .

Shafts ....

100

Spur Gears .

» i

. 114

Bevel Gears .

» 1

. 126

Worm and Worm Gears

. 132

Friction Clutches .

. 139

Couplings ....

■

. 145

Bolts, Studs, Nuts, and Screws

1 4

. 149

Keys, Pins, and Cotters

1 «

. 160

Bearings, Brackets, and Stands

. 170

MACHINE DESIGN,

PART 1.

Definition, Machine Design is the art of mechanical thought,
development, and specification.

It is an art, in that its routine processes can be analyzed and
systematically applied. Proficiency in the art positively cannot
be attained by any " short cut " method. There is nothing of a
spectacular nature in the methods of Machine Design. Large
results cannot be accomplished at a single bound, and success is
possible only by a patient, step-by-step advance in accordance
with well-established principles.

" Mechanical thought " means the thinking of things strictly
from their mechanical side; a study of their mechanical theory,
structure, production, and use; a consideration of their mechanical
fitness as parts of a machine.

" Mechanical development " signifies the taking of an idea in
the rough, in the crude form, for example, in which it comes from
the inventor, working it out in detail, and refining and fixing it in
shape by the designing process. Ideas in this way may become
commercially practicable designs.

" Mechanical specification " implies the detailed description
of designs, in such exact form that the shop workmen are enabled
to construct completely and put in operation the machines repre-
sented in the designs.

The object of Machine Design is the creation of machinery
for specific purposes. Every department of a manufacturing
plant is a controlling factor in the design and production of the
machines built there. A successful design cannot be out of
harmony with the organized methods of production. Hence in
the high development of the art of Machine Design is involved a
knowledge of the operations in all the departments of a manu-
facturing plant. Tlie student is therefore urged not only to
familiarize himself with the direct production of machinery, but to
study the relation thereto of the allied commercial departments.

MACHINE DESIGN

He sliouW get into the spirit of business at the start, get into the
shop atmosphere, execute his work just as though the resulting
design were to be built and sold in competition. He should visit
shops, work in them if possible, and observe details of design and
methods of finishing machine parts. In this way he will begin
to srore up bits of information, practical and commercial, which
will have valuable bearing on his engineering stuay.

The labor involved in the design of a complicated automatic
machine is evidenced by the designer's wonderful familiarity with
its every detail as he stands before the completed machine in
operation and explains its movements to an observer. The intri-
cate mass of levers, shafts, pulleys, gears, cams, clutches, etc., etc.,
packed into a small space, and confusing even to a mechanical
mind, seems like a printed book to the designer of them.

This is so because it is a familiar journey for the designer's
mind to run over a path which it has already traversed so many
times that he can see every inch of it with his eyes shut. Every
detail of that machine has been picked from a score or more of
possible ideas. One by one, ideas have been worked out, laid
aside, and others taken up. Little by little, the special fitness of
certain devices has become established, but only by patient, care-
ful consideration of others, which at first seemed equally good.

Every line, and corner, and surface of each piece, however
small that piece may be, has been through the refining process of
theoretical, practical, and commercial design. Every piece has
been followed in the mind's eye of its designer from tite crude
material of which it is made, through the various^processes of fin-
ishing, to its final location in the completed machine; thus its
bodily existence there is but the realization of an old and familiar
picture.

What wonder that the machine seems simple to the designer
of it! As he looks back to the multitude of ideas invented,
worked out, considered and discarded, the machine in its final
form is but a trifle. It merely represents a survival of the fittest.

No successful machine, however simple, was ever designed
that did not go through this slow process of evolution. No
machine ever just simply happened by accident to do the work
for which it is valued. No other principle upon which the sue-

MACHINE DESIGN

ce88ful design of machinery depnds is so important as this careful,
patient consideration of detail. A machine is seldom unsuccessful
because some main point of construction is wrong. The principal
features of a machine are usually the easiest to determine. It is
a failure because some little detail was overlooked, or hastily con-
sidered, or allowed to be neglected, because of the irksome labor
necessary to work it out properly.

There is no task so tedious, for example, as the devising of
the method of lubricating the parts of a complicated machine.
Yet there is no point of design so vital to its life and operation as
an absolute assurance of an adequate supply of oil for the moving
parts at all times and under all circumstances. Suitable means
often cannot be found, after the parts are together, hence the
machine goes into service on a risky basis, with the result, per-
haps, of early failure, due to "running dry." Good designers
will not permit a design to leave their hands which does not pro-
vide practically automatic oiling, or at least such means of lubri-
cation that the operator can offer no excuse for neglecting to oil
his machine. This is but a single illustration of many which
might be presented to impress the definite and detail character
necessary in- work in Machine Design.

Relation. The relation which Machine Design should cor-
rectly bear to the problems that it seeks to solve, is twofold; and
there are, likewise, two points of view corresponding to this two-
fold relation, from which a study of the subject should be traced.
Keither of these can be discarded and an efficient mastery of the
art attained. These points are —

I. Theory.
n. Production.

I. Theory. From this point of view. Machine Design is
merely a skeleton or framework process, resulting in a repre-
sentation of ideas of pure motion, fundamental shape, and ideal
proportion. It implies a working knowledge of physical and
mathematical laws. It is a strictly scientific solution of the
problem at hand, and may be based purely on theory which has
been reasoned out by calculation or deduced from experiment.
This is the only sure foundation for intelligent design of any sort.

But it is not enough to view the subject from the standpoint

6 MACHINE DESIGN

of theory alone. If we stopped here we should have nothing but
mechanisms, mere laboratory machines, simply structures of
ingenuity and examples of fine mechanical skill. A machine may
be correct in the theory of its motions; it may be correct in the
theoretical proportions of its parts; it may even be correct in its
operation for the time being; and yet its complication, its mis-
directed and wasteful effort, its lack of adjustment, its expensive
and irregular construction, its lack of compactness, its difficulty
of ready repair, its inability to hold its own in competition — any
of these may throw the balance to the side of failure. Such a
machine, commercially considered, is of little value. No shop
will build it, no machinery house will sell it, nobody will buy it
if it is put on the market.

Thus we see that, aside from the theoretical correctness of
principle, the design of a machine must satisfy certain other
exacting requirements of a distinctly business nature.

II. Production. From this point of view. Machine Design
is the practical, marketable development of mechanical ideas.
Viewed thus, the theoretical, skeleton design must be so clothed
and shaped that its production may be cheap, involving simple
and efficient processes of manufacture. It must be judged by the
latest shop methods for exact and maximum output. It must
possess all the good points of its competitor, and, withal, some
novel and valuable ones of its own. In these days of keen com-
petition it is only by carefully studied, well-directed effort toward
rapid, efficient, and, therefore, cheap production that any machine
can be brought to a commercial basis, no matter what its other
merits may be. All this must be thought of and planned for in
the design, and the final shapes arrived at are quite as much a
result of this second point of view as of the first.

As a good illustration of this, may be cited the effect of the
present somewhat remarkable development of the so-called " high
speed " steels. The speeds and feeds possible with tools made of
these steels are such that the driving power, gearing, and feed
mechanism of the ordinary lathe are wholly inadequate to the
demands made upon them when working the tool to its limit.
This means that the basis of design as used for the ordinary tool
steel will not do, if the machine is expected to stand up to the

MACHINE DESIGN

cuts possible with the new steels. Hence, while the old designs
were right for the old standard, a new one has been set, and it
thorough revision on a high-speed basis is imminent, else the
market for them as machines of maximum output will be lost.

From these definitions it is evident that the designer must
not only use all the theory at his command, bujt must continually
inform himself on all processes and conditions of manufacture,
and keep an eye on the tendency of the sales markets, both
of raw material and the finished machinery product. This is
what in the broadest sense is meant by the term "Mechanical
Thought,'' thought which is directed and controlled, not only by
theoretical principle but by closely observed practice. From the
feeblest pretenders of design to those engineers who consummate
the boldest feats and control the largest enterprises, the process
which produces results is always the same. Although experience
is necessary for the best mechanical judgment, yet the student
must at least begin to cultivate good mechanical sense very early
in his study of design.

Invention. Invention is closely related to Machine Design,
but is not design itself. Whatever is invented has yet to be
designed. An invention is of little value until it has been refined
by the process of design.

Original design is of an inventive nature, but is not strictly
invention. Invention is usually considered as the result of genius,
and is announced in a flash of brilliancy. We see only the flash,
but behind the flash is a long course of the most concentrated
brain effort. Inventions are not spontaneous, are not thrown off
like sparks from the blacksmith's anvil, but are the result of hard
and applied thinking. This is worth noting carefully, for the
same effort which produces original design may develop a valuable
invention. But there is little possibility of inventing anything
except through exhaustive analysis and a clear interpretation of
such analysis.

Handbooks and Empirical Data. The subject matter in
these ia often contradictory in its nature, but valuable nevertheless.
Empirical data are data for certain fixed conditions and are not
general. Hence, when handbook data are applied to some specific
case of design, while the information should be used in the freest

8 MACHINE DESIGN

manner, yet it must not be forgotten that the case at hand is prob-
ably different, in some degree, from that upon which the data were
based, and unlike any other case which ever existed or will ever
again exist. Therefore the data should be applied with the greatest
discretion, and when so applied will contribute to the success of
the design at least as a check, if not as a positive factor.

The student -should at the outset purchase one good handbook,
and acquire the habit of consulting it on all occasions, checking
and comparing his own calculations and designs therefrom. Care
must be taken not to become tied to a handbook to such an extent
that one's own results are wholly subordinated to it. Independence
in design must be cultivated, and the student should not sacrifice
his calculated results until they can be shown to be false or based
on false assumption. Originality and confidence in design will be
the result if this course be honestly pursued.

Calculations, Notes, and Records. Accurate calculations are
the basis of correct proportions of machine parts. There is a right
way to make calculations and a wrong way, and the student will
usually take the wrong way unless he is cautioned at the start.

The wrong way of making calculations is the loose and shift-
less fashion of scratching upon a scrap of detached paper marks
and figures, arranged in haphazard form, and disconnected and
incomplete. These calculations are in a few moments' time totally
meaningless, even to the author of them himself, and are so easily
lost or mislaid that when wanted they usually cannot be found.

Engineering calculations should always be made systemati-
cally, neatly, and in perfectly legible form, in some permanently
bound blank book, so that reference may always be had to them at
any future time for the purpose of checking or reviewing. Put
all the data down. Do not leave in doubt the exact conditions
under which the calculations were made. Note the date of calcu-
lation.

If a mistake in figures is made, or a change is found neces-
sary, never rub out the figures or tear out the leaf, or in any way
obliterate the figures. Simply draw a bold cross through the wrong
part and begin again. Often a calculation which is supposed to
be wrong is later shown to be right, or the facts which caused the
error may be needed for investigation and comparison. Time which

MACHINE DESIGN 9

is spent in making figures is always valuable time, time too pre-
cious to be thrown away by destroying the record.

The recording of calculations in a permanent form, as just
described, is the general practice in all modern engineering offices.
This plan has been established purely as a business policy. In
case of error it locates responsibility and settles dispute. Con.
sistent designing is made possible through the records of past
designs. Proposals, estimates, and bids may often be made
instantly, on tiie basis of what these record books show of sizes
and weights. This bookkeeping of calculations is as important a
Victor of systematic engineering as bookkeeping of business
accounts is of financial success.

The student should procure for this purpose a good blank book
with a firm binding, size of page not smaller than 6 by 8 inches
(perhaps 8 by 11 inches may be better), and every calculation, how-
ever small and apparently unimportant, should be made in it.

Sample pages of engineering calculations are reproduced in.
Figs. 8 to 9. Note the sketch showing the forces. Note the clear
statement of data. Note the systematic writing of the equations,
and the definite substitutions therein. Noto the heavy double
underscoring of the result, when obtained. There is nothing in
the whole process of the calculation that cannot be reviewed at
any moment by anybody, and in the briefest time.

The development of a personal note-book is of great value to
ihe designer of machinery. The facts of observation and experi-
ence recorded in proper form, bearing the imprint of intimate
personal contact with the points recorded, cannot be equalled
in value by those of any hand or reference book made by another.
There is always a flavor about a personal note-book, a sort of
guarantee, which makes the use of it by its author definite and
sure.

The habit of taking and recording notes, or even knowing
what notes to take, is an art in itself, and the student should
begin early to make his note-book. Aside from the value of the
notes themselves as a part of his personal equipment, the facility
with which his eye will be trained to see and record mechanical
things will be of great value in all of his study and work. How
many men go through a shop and really see nothing of the opera-

10 MACHINE DESIGN

tioDB going on therein, or, seeing them, remember nothing I An
engineer, trained in this respect, will to a surprising degree be
able to retain and sketch little details which fall under his eye for
a brief moment only, while he is passing through a crowded shop.

Some draftsmen have the habit of copying all the standard
tables of the various oiBces in which they work. While these are
of some value in a few cases, yet this is not what is meant by a
good note-book in the best sense. Ideas make a good note-book,
not a mere tabulation of figures. If the basis upon which stan-
dards are founded can be transferred to permanent personal record,
or novel methods of calculation, or simple features of construc-
tion, or data of mechanical tests, or efficient arrangement of
machinery — if these can be preserved for reference, the note-book
will be of greatest value.

Whatever is noted down, make clear and intelligible, illus-
trating by a sketch if possible. Make the note so clear that
reference to it after a long space of years would bring the whole
subject before the mind in an instant. If this is not done the
author of the note himself will not have patience to dig out the
meaning when it is needed; and the note will be of no value.

METHOD OF DESIGN.

The fundamental lines of thought and action which every
designer follows in the solution of any problem in any class of
work whatsoever, are four in number. The expert may carry all
these in mind at the same time, without definite separation into a
a step-by- step process; but the student must master them in their
proper sequence, and thoroughly understand their application.
In these four are concentrated the entire art of Machine Design.
When they have become so familiar as to be instinctively applied
on any and all occasions, good design is the result. The only
other quality which will facilitate still further the design of good
machinery is experience; and that cannot be taught, it must be
acquired by actual work.

I. Analysis of Conditions and Forces. First, take a good
square look at the problem to be solved. Study it from all sides,
view it in all lights, note the worst conditions which can possibly
exist, note the average conditions of service, note any special or
irregular service likely to be called for.

MAGHIKE DESION U

WIA Aese eooditions well in mind, make a caroful analyiilii
of all tbe faneSj maximimi as well as ayeraf2;e, wliioli rnuy Ih^
bron^it into play. Make a rough sketch of the picnui luicldr con
M i iffialinnj and put in these forces. Be sure that th(*ri(t forcu*^ aro
at least a^iraxiiiiately right. Oo over the analytiiN imn^fiilly
j^gun and again. Bemember that time savtMl at tlui l)o^hnilnf^
hybaetjand poor analysis will actually be time loHt at thn immI;
and if tlie madiine actually fails from this reuHoii, lumvy Ihiaiicliil
loBB in material and labor will occur. Any hiiHlii towiinl ooni
pletion of die structure beyond the roughent ontlinn, without thin
eaiefol study of forces, is a blind leap in the diirlc, nntlrnly nn.
scientific, and almost certain to result in ultiiniito fiiilurM.

On the other hand this principle may be ciirrifMl Um far. Iti
trying to make the analysis thorough and tlio Utrroa fU'cMimtf^, it in
quite possible to consume more than a reaHonahle iirnount of titnn.
Again, it is not always easy, and fre(jU(Milly irnpoMHibli^, to doinr.
mine exactly the forces acting on a glvon piecn. )tut ibf^r vafurp^
whether sudden or slowly applied, rapid in H(*Mon or only oc-
curring at intervals, and their ^//7>/v>j?////^//^' dlro<!tlc»n and rnHgni-
tude at least, are always capable of analyHlpt. 1'hnre are f(>\v, if
any, cases where close assumptions canttot bo niaclo on tho above
basis and the design proceeded with m^oonllngly, Ilcnre the
danger of too great refinement of analysiH in simply to bo avoided
by the designer's plain business Hense.

The first tendency of the student is to pass ovor the study of
the forces as dull and dry, and attempt the design at once. IIo
soon finds himself facing problems of which he boob no poBsiblo
solution, and he bases his dciHign on pure guuHHwork. This is
the only solution possible from such a j)oint of viow, and is really
no solution at all. A guess whicli has sorno rational backing is
often successful; but in that case some analyniH is rocjuired, and It
is not a pure guess, but falls under the v<jry principle wo are
considering.

There is no short cut to the design of maelilnci partM whioli
avoids this full understanding of tliu foreos that thtty niuAt
sustain. The size of a belt de])t3n(U u])on the maximum pull
upon it, and the designing of belts is nothing but providing
sufficient cross-section of leather to prevent the belt tearing under

12 MACHINE DESIGN

the pull. Again, if pulley arms are not to break, or shafts twist
off, or bolts be torn apart, or the teeth of gears fail, or keys and
pins shear off, we must first, of course, find out what forces exist
which are likely to produce stress that may lead to such
breakage. We should not guess at the sizes, and then run the
machine to see if breakage results, and then guess again. Ma-
chines are sometimes built in this way, but it is an unreasonable
and uncertain method. We must use every effort to foresee the
stress which a piece is liable to receive, before we' decide its size.
We must know all the forces approximately, if not positively.
The analysis must be thorough enough to permit of reasonable
assumption, if not positive assertion. It is manifestly impossible
to solve any problem until we know exactly what the problem is;
and a full analysis is the statement of the problem.

a. Theoretical I>esis:n. After we know by careful analysis
what stress the machine part has to sustain, the next step is so to
design it that it will theoretically resist the applied forces with
the least expenditure of material.

We often see machinery with the metal of which it is made
distributed in the worst possible manner. In places where the
stress is heavy and a rigid member is needed, we find a weak,
springy part; while in other parts, where there are no forces to be
resisted, or vibration to be absorbed, there seems to be a waste of
good material. Whether in such case the analysis of the forces
was poor, or perhaps not made at all, or whether a knowledge of
how to design so as to resist the given forces was wholly absent,
cannot be told. At any rate, lack of either or both is clearly
shown in the result.

Any member of a machine may vary in form from a solid
block or chunk of material to an open ribbed structure. The solid
chunk fills the requirement as far as strength is concerned, unless
it is 00 heavy as to fail from its own weight. But such construc-
tion is poor design, except in cases where the concentration of
heavy mass is necessary to absorb repeated blows like those of a
hammer. The possibility of these blows should, however, have
been determined in the analysis; and the solid, anvil construction
then becomes theoretical design for that analysis.

For steadily applied loads an open, ribbed, or hollow box

MACHINE DESIGN 13

structure can be made wbicli will distribute the metal where it is
theoretically needed, and each fiber will then sustain its proper
share of the load. In this way weight, cost, and appearance are
heeded; and the service of the piece is as good as, and probably
better than, it would be with the clumsy, solid form.

There is no such thing as putting too much theory into the
design of machinery. The strongest trait which an engineer can
have is absolute faith in his analysis and calculations, and their
reproduction in his theoretical design. Theoretical design is^an
indication of scientific advance in the art, and some of the greatest
steps of progress which have been made in recent years have been
accomplished through a purely theoretical study of machine
structure.

It will never do, however, to be satisfied with theoretical
ilebign when it is not in accord with modern commercial and manu-
facturing considerations. Hence the next step after the determina-
tion of the theoretical design is the study of it from the producing
standpoint.

3. Practical Modification. All theoretical design viewed from
the business standpoint is worthless, unless it has been subjected
to the test of cheap and efficient production. Each machine detail,
though correct in theory, may yet be improperly shaped and unfit
for the part it is to play in the general scheme of manufacture.

The conditions here involved are changeable. What is good
design in this decade may be bad in the next. In this light the
designer must be a close student of the signs of the times; he must
follow the march of progi'ess, closely applying existing resources,
conditions, and facilities, otherwise he cannot produce up-to-date
designs. The introduction of new raw materials, the cheapening
of production of others, the changing of shop methods, the use of
special machinery, the opening of new markets, the development
of new motive agents, — all these and many others are constantly
demanding some modification in design to meet competition.

Illustrative of this, note the change which has been wrought
by the development of electric power, the rise and decline of the
bicycle business, the present manufacture of automobiles, the last
named especially with reference to the development of the small
motive unit, the gasolene engine, the steam engine, etc. The

14 MACHINE DESIGN

design of much machinery has been materiallj changed to meet
the exacting demands of these new enterprises.

Practical modifications of design necessary to meet the limi.
tations of construction in the pattern shop, foundry, and machine
shop are of daily application in the designer's work. He must
keep in his mind's eye at all times the workmen and the processes
they use to create his designs in metal in the shop.

"How can this be made?" "Can it be made at all?"
" Can it be made cheaply ? " " Will it be simple in operation
after it is made ? " " Can it be readily removed for repair ? "
" Can it be lubricated ? " " How can it be put in place ? " " How
can it be gotten out?" "Will it be made in small quantities
or large?" "Will it sell as a special or standard machine?"
etc., etc.

The consideration of such questions as these is a practical
necessity as a business matter. No other feature affects the
design of machinery more, perhaps; for designs which cannot be
built as business propositions are no designs at all.

The student, it is true, may not have the extended shop
knowledge which is essential to this; but he can do much for
himself by visiting shops whenever possible, getting hold of shop
ways of doing things, and invariably treating his work as a
business matter. Though a man may not be a pattern maker,
molder, blacksmith, or machinist, yet he can soon gain ideas of the
processes in each of these branches which will be of immense
advantage to him in his designing work.

4* Delineation and Specification. This means the clear and
concise representation of the design by mechanical drawings.

This is as much a part of the routine method of Machine De-
sign as the other three points which have been discussed. The
mere act of putting the results of mechanical thinking on paper is
one of the greatest helps to force thinking machinery to system-
atic and definite action. A designer never thinks very long
without drawing something, and the student must bring himself to
feel that a drawing in its first sense is a means of helping his own
thought, and must freely use it as such.

In its second and final sense, the drawing is an order and
specification sheet from the designer to the workman. Design

MACHINE DESIGN 15

whicli stops short of exact, finished delineation in the form of
working shop drawings is only half done. In fact the possibility
of a piece being thus exactly drawn is often the crucial test of its
feasibility as a part of a machine. It is easy to make general out-
lines, but it is not so easy to get down to finished detail. It is
safe to say that there is no one thing productive of more trouble,
delay and embarrassment, and waste of time and money in the
shop, when there need be none from this cause, than a poor detail
drawing. The efficiency of the process of design is not fully real-
ized, and failures are often recorded where there sliould be success,
merely because the indefiniteness permitted by the designer in the
drawings naturally transmitted itself to the workman, and he in
turn produced a part indefinite in form and operation.

The actual process of drawing in the development of a design
may be outlined as follows :

Rough sketches merely representing ideas, not drawn to scale,
are first made. These are of use only so far as the choice of me-
chanical ideas is concerned, and to carry preliminary dimensions.

Following these sketches, comes a layout to scale, of the
favored sketch, a working out of the relative sizes and location of
the parts. This drawing may be of a sketchy nature, carrying a
principal dimension here and there to fix and control the detailed
design. In this drawing the design is developed and general detail
worked out. The minute detail of the individual parts is, however,
left to the subsequent working drawing.

This layout drawing may now be turned over to an expert
draftsman or detail designer, who picks out each part, makes an
exact drawing of it, studying every little detail of its shape, and
finally adds complete dimensions and specifications so that the
workman is positively informed as to every point of its construction.

General drawings and cross sections constitute the last step
in the process of complete delineation. These show the parts
assembled in the complete machine. They also serve a valuable
purpose to the draftsman in checking up the dimensions of the
detail drawings. Errors which have escaped previous notice are
often discovered in this way. The layout, mentioned above, is
sometimes finished up into a general drawing; but it is safer to
make an entirely new drawing, as changes in detail are often
necessary after the layout is made.

16 MACHINE DESIGN

The four fundamental lines of thought and action noted
above may be summarized thus — ** analy^^e and theorize, modify
and delineate/* This is a maxim easy to remember, applicable
to every problem in Machine Design, and always provides the
answer to the question ^< What shall I do, how shall I proceed ? "
by pointing out the proper sequence in the course to be followed.

CONSTRUCTIVE MECHANICS.

Mechanics is a constructive science, its principles lying at the
root of the design and operation of all machinery. It is usually
taught, however, as an advanced mathematical subject; and the
student gets his original conceptions of forces, moments, and
beams in the abstract, before he realizes the constructive value of
such conceptions. By "Constructive Mechanics" is meant the
study of a machine purely from its constructive side, the viewing
of the parts with respect to their " mechanics," and satisfying the
requirements of the same in form and arrangement.

The student may cultivate this habit of clear, mechanical per-
ception by constantly noting the "mechanics" of the simple
structures which he sees in his daily routine of work. Aside
from machinery, in which the "mechanics" is often obscure,
the world is full of simple examples of natural strength and
symmetry, explainable by application of the principles of pure
" mechanics."

Posts and pillars are largest at their bases; overhanging
brackets or arms are spread out at the fastening to the wall;
heavy swinging gates are counter-balanced by a ponderous weight;
the old-fashioned well sweep carries its tray of stones at the end,
adjusting the balance to a nicety; these are examples of things
depending for their form and operation upon the principles of
"mechanics." The building of them involved "constructive
mechanics," and yet their constructor perhaps never heard of the
science, using merely his natural sense of mechanical fitness.
Such simple reasoning is, however. Constructive Mechanics.

Forces, Moments, and Beams. Machines are nothing but a
collection of (1) parts taking direct stress, or (2) parts acting as
loaded beams. Forces acting without leverage produce direct
stress on the sustaining part. Forces acting with leverage pro-

MACHIKE DESIGN 17

dnce a moment; the sustaining member is a beam, and the stress
therein depends on the theory of beams, as explained in ^^Me-
chanics/'

An example of the first is the load on a rope, the force acting
without leverage, and the rope therefore having a direct stress put
upon it.

An example of the second is a push of the hand on the crank
of a grindstone. A moment is produced about the hub of the
crank; the arm of the crank is a beam, and the stress at any point
of it may be found by the method of theory of beams.

Tension, Compression, and Torsion. The stress induced in
the sustaining part, whether tensile, compressive, or torsional, is
caused by the application of forces, either acting directly without
leverage, o^ with leverage in the production of moments.

Tlie forces applied from external sources are at constant war
with the resisting forces due to the strength of the fibres of the
piaterial composing the machine members. The moments of the
external forces are constantly exerted against and balanced by the
moments of the internal resistance of the material. Hence,
design, from a strength standpoint, is merely a balancing of
internal strength against external force. In other words, we may
in all cases write a sign of equality, place the applied effort on
one side, the effective resistance on the other, and we shall have
an equation, which, if capable of solution, will pve the proper
proportions of the parts considered.

External Force = Internal Kesistance.
External Moment = Internal Moment of Kesistance.
Expressed in terms of the " Mechanics:"

P=A8 (i)

B or T=^ (a)
c ^ ^

In these formulas, which are perfectly general,

P= direct load in pounds.

A = area of effective material, in square inches.

S=working fibre stress of the material (teieile, compressive, or shear-
ing), in pounds per square inch.

B or T=external moment (bending or torsional), in inch-pounds.

I = moment of inertia (direct or polar), of the resisting section.

c=distance of the most remote fibre of the resisting section from the
neutral axis.

18 MACHINE DESIGN

P may produce direct tensile, compressive, or shearing stress.

B may produce tensile or compressive stress, and requires use of direct
moment of inertia in either case.

T produces shearing stress, and requires use of polar moment of
inertia.

The origin of formula (1) is obvious, the assninption being

that the fibre stress is equally distributed to every partide in the
area "A."

The development of formula (2) is given in any text-book in
Mechanics. It requires the aid of the Calculus, however. Any
good handbook gives values for both the direct moment of inertia
and the polar moment of inertia for quite a large variety of sections,
so that further reference is an easy matter for the student. These
values are also obtained through the methods of the Calculus.

The reason for introducing these formulas at this time is to
call the attention of the student especially to the fact of their
universal and fundamental use in all problems concerning the
strength of machine parts. Nearly every computation may be
reduced to or expanded from these two simple equations. Many
complex combinations occur, of course, which will not permit sim-
ple and direct application of these formulas, but the student will
do well to place himself in perfect command of these two. Assuming
that he is able to analyze forces, and compute the simple moment
at the point where he wishes to find the strength of section, the
rest is the mere insertion of the assumed working fibre stress of
the material in the formula (2) above, and solution for the quantity
desired.

When the case is one of combined stress, the relation becomes
more complicated and difficult of analysis and solution. The most
common case is where bending is combined with torsion, as in the
case of a shaft transmitting power, and at the same time loaded
transversely between bearings. In fact there are very few cases of
shafts in machines, which, at some part of their length, do not
have this combined stress. In this case the method of procedure
is to find the simple bending moment and the simple torsional
moment separately, in the ordinary way. Then the theory of
elasticity furnishes us with a formula for an equivalent bending
or an equivalent torsional moment which is supposed to produce
the same effect upon the fibres of the material as the combined

MACHINE DESIGN 19

action of the two simple moments acting together. In other
words, the separate moments combined in action, being impossible
of solution in that form, are reduced to an equivalent simple
moment and the solution then becomes the same as for the prev-
ious case.

These equivalent equations are given below, the subscript "e"
being added to express separation from the simple moment:

B~+ii/W+V (3)

T,=B+y/W+T (4)

Bq and Tg, found from these equations, are the external mo-
ments, and are to be equated to the internal moments of resistance
of the section precisely as if they were simple bending or torsional
moments. Either may be used. For shafts (4) is generally used,
being the simpler of the two in form.

FRICTION AND LUBRICATION.

The parts of a machine which have no relative motion with
regard to each other are not dependent upon lubrication of their
surfaces for the proper performance of their functions. In cases
where relative motion does occur, as between a planer bed and its
ways, a shaft and its bearing, or a driving screw and its nut,
friction, and consequent resistance to motion, will inevitably
occur. Heat will be generated, and cutting or scoring of the
surfaces will take place if the surfaces are allowed to run together
dry.

This difficulty, which exists with all materials, cannot be
overcome, for it is a result of roughness of surface, characteristic
of the material even when highly finished. The problem of the
designer, then, is to take conditions as he finds them, and, as he
cannot change the physical characteristics, of materials, so choose
those which are to rub together in the operation of the machine
that friction will be reduced to the lowest possible limit. Now it
fortunately happens that there are certain agents like oil and
graphite, which seem to fill up the hollows in the surface of a
solid material, and which themselves have very little friction on
other substances. Hence, if a machine permits by its design an
automatic supply of these lubricating agents to all surfaces having

20 MACHINE DESIGN

motion between them, friction may be reduced to the lowest limit.

If this full supply of lubricant be secured, and the parts still
heat and cut, then the fault may be traced to other causes, such as
springy surfaces, localization of pressure, or insufficient radiating
surface to carry away the heat of friction as &st as it is generated.

Lubricating agents are of a nature running from the solid
graphite form to a thick grease, then to a heavy dark oil, and
finsdly to a thin, fluid oil flowing as freely as water. The solid and
heavy lubricants are applicable to heavily loaded places where the
pressure would squeeze out the lighter oils. Grease, forced be-
tween the surfaces by compression grease cups, is an admirable
lubricator for heavy machinery under severe service. High-speed
and accurate machinery, lightly loaded, requires a thin oil, as the
fits would not allow room for the heavier lubricants to find their
way to the desired spot. The ideal condition in any case is to
have a film of lubricant always between the surfaces in contact,
and it is this condition at whidi the designer is always aiming in
his lubricating devices.

Oil ways and channels should be direct, ample in siee,
readily accessible for cleaning, and distributing the oil by natural
flow over the full extent of the surface. Hidden and remote
bearings must be reached by pipes, the mouths of which should
be clearly indicated and accessible to the operator of the machine.
Such pipes must be straight, if possible, and readily cleaned.

There is one practical principle affecting the design of
methods of lubrication of a machine which should be borne in
mind. This is, " Neglect and carelessness by the operator w/ust
be provided for." It is of no use to say that the ruination of a
surface or hidden bearing is due to neglect by the operator, if the
means for such lubrication are not perfectly obvious. This is
" locking the door after the horse is stolen." The designer has
not done his duty until he has made the scheme of lubrication so
plain that every part must receive its proper supply of oil, except
by gross and willful negligence, for which there can be no
possible just excuse.

WORKING STRESSES AND STRAINS.

Some persons object to the use of these terms, as one is
frequently used for the other, and misunderstanding results. This

MACHINE DESIGN 21

is doubtless true; T)nt the student may as well learn the true
relation of the terms once for all, because he will frequently run
across them in his reading and reference work, and should inter*
pret them rightly. The strict relation of the two is as follows:

Stress is the internal force in a piece resisting the external
force applied to it. A weight of ten pounds hanging on a rope
produces a stress of ten pounds in the rope.

Strain is the change of shape, or deformation, in a piece
resisting an external force applied to it. If the above weight of
ten pounds stretches the rope J inch, the strain is J inch.

Unit stress is stress per unit area, e. g., per square inch.

Unit strain is strain per unit length, e. g., per inch length.

In the above case, if the rope Were J square inch in area
and 80 inches long, the unit stress, or intensity of stress, is
10^^=20 pounds per square inch; the unit strain is J-5-30=-j^
inch per inch.

When stress is induced in a piece, the strain is practically
proportional to the stress for all values of the stress below the
elastic limit of the material; and when the external load is re-
moved the strain will entirely disappear, or the recovering power
of the material will restore the piece to the original length.

Illustrating by the case above, on the supposition that the
elastic limit has not been reached by the stress of 20 pounds per
square inch, if the load of 10 pounds were taken oflP, the J-inch
strain would disappear and the rope return to its original length;
if the load were changed to J of 10 pounds, or 5 pounds, the
strain would be J of J inch, or J inch.

Now it is found that if we wish a piece to last in service for
a long time without danger of breakage, we must not permit it
to be stressed anywhere near the elastic limit value. If we do,
although it will probably not break at once, it is in a dangerous
condition, and not well suited to its requirements as a machine
member. The technical name for this weakening effect is " fa-
tigue." It is further found that the fatigue due to this repeated
stress is reached at a lower limit when the stress is alternating in
character than when it is not. In other words, if we first pull on
a piece and then push on it, we shall first have the piece in tension
and then in compression; this alternation of stress repeated to

22 MACHINE DESIGN

near the elastic limit of the material will fatigue it, or wear out
the fibres, and it will finally fail. If, however, we first pull on
the piece with the same force as before, and then let. go, we shall
first have the piece in tension and then entirely relieved; such
repetition of stress will finally " fatigue " the material, but not so
quickly as in the first case. Experiments indicate that it may
take twice as many applications in the latter case as in the former.

The working stress of materials permissible in machines is
based on the above facts. The breaking strength divided by a
liberal factor of safety will not necessarily give a desirable work-
ing stress. The question to be answered is, " Will the assumed
working fibre stress permit an indefinite number of applications
of the load without fatiguing the material ? '^

Hence we see that the same material may be safely used under
diflferent assumptions of working stress. For example, a rotating
shaft, heavily loaded between bearings, acts as a beam which in
each revolution is having its particles subjected, first to a maxi-
mum tensile stress, and then to a maximum compressive stress.
This is obviously a very diflEerent stress from that which the same
piece would receive if it were a pin in a bridge truss. In the
former we have a case where the stress on each particle reverses at
each revolution, while in the latter we have merely the same stress
recurring at intervals, but never becoming of the opposite char-
acter. For ordinary steel, a value of 8,000 would be reasonable in
the former case, while in the latter it may be much higher with
safety, perhaps nearly double.

From the facts stated above, it is evident that exact values for
working fibre stress cannot be assumed with certainty and applied
broadly in all cases. If the elastic limit of the material is defi*
nitely known we can base our working value quite surely on that.

With but a general knowledge of the elastic limit, ordinary
steel is good for from 12,000 to 15,000 pounds per square inch
non -reversing stress, and 8,000 to 10,000 reversing stress. Cast
iron is such an uncertain metal on account of its variable structure
that stresses are always kept low, say from 3,000 to 4,000 for non-
reversing stress, and 1,500 to 2,500 for reversing stress.

With these values as a guide, and the special conditions con-
trolling each case carefully studied, reasonable limits may be

• I

KACmNE DESIGN 88

assigned for working stress, not only of steels, various grades of
cast iron, and mixtures of the same, but of other alloys, brass,
bronze, etc. Gun metal, semi-steel, and bronze are intermediate
in strength between cast iron and steel. Data on the strength of
materials are available in any of the handbooks, and should be con-
sulted freely by the student. They will be found somewhat con«
flicting, but will assist the judgment in coming to a conclusion.

Application to Practical Case. In actual practice the only
information which the designer has, upon which to base his design,
Is the object to be accomplished. He must choose or originate
suitable devices, develop the arrangement of the parts, make his
own assumptions regarding the operation of the machine, then
Analyze and Theorize^ Modify and Delineate each detail as
he meets it.

This, it will be found, is a very different matter from taking
some familiar piece of machinery, such as a pulley, or a shaft, or
a gear, as an isolated case, the load being definitely given, and
proceeding with the design. This is easily done, but is only half
the problem, for machine parts, such as pulleys, gears, and shafts,
do not confront the designer tagged or labeled with the conditions
they are to meet. He is to provide parts to meet the specific con-
ditions, and it is as much a part of his designing method to know
how to attack the design of a machine as it is to know how to
design the parts in detail after the attack has reduced the members
to definitely loaded structures. The whole process must be gone
through, the preliminary sketches, calculations, and layout, all of
which precede the detail design and working drawings; and no step
of the process can be omitted.

It is for this reason that the present case used for illustration
is carried out quite thoroughly, ^e student should make himself
familiar with every step of the designing method as applied to this'
simple case of design. More complex problems, handled in the
same way, will simplify themselves; and when the point is reached
where confidence exists to take hold of the design of any machine,
however unfamiliar its object may be, or however involved its
probable detail appears, the student has become the true designer.
It is the knowing how to attack a problem, to start definite work
on it, to go ahead boldly, confident that the method applied will

MACHINE DESIGN

produce reenlts, that gives command of the design of mftdmieiy
and wioB engiueenQg success.

The special case which Las been chosen to illustrate the
application of the principles stated in the foregoing pages is ideal,

MACHINE DESIGN 25

in that it does not represent any actual machine at present in
operation. Probably builders of hoisting machinery have devices
which would improve the machine as shown. In detail, as well
as arrangement, they could doubtless make criticism as manufac-
turers. The arrangement as shown is merely intended to bring
out in simplest form the common elements of transmission ma«
chinery as parts of some definite machine, instead of as isolated
details. The design is one entirely possible, practical, and me-
chanical, but special attention has been paid to simplicity in order
to enable the student to follow the method closely, for the method
is the chief thing for him to acquire.

The student is expected to refer constantly to Part II for a
more formal and general discussion of the simple machine ele-
ments involved in the case considered. Part II is intended to be
a simplified and condensed reference book, carried out in accord-
ance with the method of machine design as specified in Part I.
The student should not wait until he has completed the study of
this part before taking up Part II, for the latter is intended for
use with the former in the solution of the problems.

In the case of power transmission about to be studied, the
running, conversational method employed assumes that the student
is in possession of the matter in Part II on the subject considered.
Thus, in the design of the pulley, reference to the subject of
" Pulleys " in Part II is necessary to follow the train of calcula-
tion; in designing the gear, consulIT" Gears;" in calculating size
of shafts, see " Shafts," etc., etc.

Problem. A machine is to be designed to be set on the floor

.of a building to drive a wire rope falling from the overhead

sheaves of an elevator or hoist. "Without regard to details of this

overhead arrangement, for its design would be a separate problem,

suppose that the data for the rope are as follows:

Load on rope 5,000 pounds.

Speed of rope 150 feet per minute.

Ijength of rope to be reeled in 200 feet.

We shall further assume that the driving power is to be an
electric motor belted to the machine, that the required speed
reduction can be satisfactorily obtained by a single pair of pulleys
and one pair of gears, and that a plain band brake is to be applied
to the drum.

MACHINE DESIGN

With this data we shall proceed to work out the detail design
of the machine.

PFellmlnary Sketch. The first thing to do is to sketch
roughly the proposed arraQgement of the machioe.

This might appear like Fig. 1 except that it would have no
dimensions in addition to the data given above. If the scheme
seems suitable, the next step is to make such preliminary calcula>
tions as will give further data, exact or closely approximate Bizea,
to be put at once on the sketch, to outline the future design.

Rope and Drum. Beferring to tables of strength of wire rope
(Kent's Pocket Book gives the manufacturers' list), we find that
a g-inch cast-steel rope will carry 5,000 pounds safely, and that the
proper size of dmta to avoid
excessive bending of the rope
around it is 27 inches diameter. J^^.^^ J^ V ^l-jf^o^C

Allowing ^ inch between the
coils &s the rope winds on the
drum, the pitch of coil will be
9 inch as shown in sketch, Fig.
2. The length of one complete
. „ 27X3.U16
coil IS, practically, 7g Pig. 2.

200
=7.07 feet. To provide for 200feet will require ;^=28+coilB.

To be safe, let us provide for 30 coils, for which a length of drum
(30X|)+|=23J inches is required.

l^e space for brake strap may be assumed at 5 inches, and the
thickness to provide necessary strength determined later in the
design. The frictional surface of the strap may be of basswood
blocks, say IJ inches thick, screwed to the metal band. The
diameter of brake surface may be 28 inches.

Driving Oears. The size of drum gear evidently depends
upon the method of fastening to the drum, and, other things being
equal, should be kept as small as possible. One way would be to
key the gear on the outside of the drum, another to bolt the gear
to the end of the drum. The latter has the advantage that a
standard gear pattern can be used with the slight change of

MACHINE, DESIGN

addition of bolt flange on the anns. 'Ibis makes a Bimple, direct,
and strong drive, the bolts being in shear.

Sketching this arrangement as the preferred one (Fig. 2A), it
iaevident that the diameter of the gear shoald be at least as large
as the dmm in order to keep the tooth load down to a reasonable
figure. On the other hand, if made too large, it spreads out the
machine and destroys its compactness. As a diameter of 36
inches is not excessive, let us assume this, and see if a desirable
proportion of gear tooth can be found to carry the load.

For a pitch diameter of 36 inches there will be a theoretical

load of ^'^y^''' =3,750 pounds at the pitch line. . Bat the load

FiR. 2A.

on the tooth most not only impart a pull of 5,000 pounds to the
rope, but mu«t overcome friction between the gear teeth in action,
also between the drum shaft and Its bearings. Assuming the
efficiency between the rope and tooth load to be 95 per cent, the

net load, therefore, which the tooth must take is ' ' — = 3,947, say

4,000 pounds.

28 MACHINE DESIGN

Assuming involute teeth, and applying the "Lewis" formula,
(Part II, "Gears"):

W=sxpx/Xy W=4,000

«=6,000
4,000=6,000 XpXfX .116 y =.116 (number of teeth

assumed at 75)

P ^/= am^^ 11^ =^-'^ inches p=circular pitch

0,UUU X 'iio

/=face of gear
Let/=3 p (a reasonable proportion for machine-cut teeth).
Then3Xi>2=5.7

i)2=1.9

p=\/ 1.9=1.378 inches
The diametral pitch corresponding to this is

which is just between the regular standard pitches, 2 and 2^, for
which stock cutters are made. To be safe, let us take the coarser
pitch, which is 2. The circular pitch corresponding to this is

«j = 1.57, and making the face about three times the circular

pitch gives

3 X 1.57 = 4.71, say 4J inches.

The number of teeth in the gear is then 36 X 2 = 72.
Referring to the value assumed for the tooth factor in calculation
above, it is seen that y was based on 75 as the number of teeth,
which is near enough to 72 to avoid the necessity of further check-
ing the result.

The pinion to mesh with this gear should be as small as possi-
ble in order to get a high-speed ratio between pinion shaft and
drum, otherwise an excessive ratio will be required in the pulleys,
making the large one of inconvenient size. Small pinions have
the teeth badly undercut and' therefore weak, 13 teeth being the
lowest limit usually considered desirable, for that reason. Choos-

ing that number, we have a pitch diameter of ■9"= 6.5 in., which

is probably ample to take the shaft and key, and still leave suf-
ficient stock under the tooth for strength. If made of cast iron,
however, the pinion teeth, on account of the low number, will be
narrower at the root than those of the gear of 72 teeth. Yet it

MACHINE DESIGN 29

was upon the basis of tlie latter that the pitch was chosen, for it
will be remembered that the value of y in the formula was
taken at .116. Hence the pinion will be weaker than the gear
unless we make it of stronger material than cast iron, of which
the large gear is supposed to be made. Steel lends itself very
readily to this requirement; and in practice, pinions of less than 20
teeth are usually made of this material, hence we shall specify the
pinion to be of steel.

Pulleys* The question now is whether or not we can get a
suitable ratio in the pulleys without making the large one of incon-
venient size, or giving the motor too slow speed for an economical
proportion.

Suppose we limit ourselves to a diameter of 42 inches for the
large pulley, and try a ratio of 4 to 1; this will give a diameter
for the small pulley of i^=10^ inches. We shall then have

72 288

Total ratio between drum and motor "Tq"^ 4=— -=22.2

Rev. per min. of drum to give 150 f . p. m. of

^'^ ^=^-^

Rev. per min. of motor 22.2 X 21.2=470

Horse-power of motor at 80 per cent efficiency ^^^ ^ ^*^ =3Q
^ ^ "^ 33,000 X .80

A 30 H. P. motor running 470 r. p. m. would be classed as a
slow speed motor and would be a heavier machine and cost more
than one of higher speed. It will be noticed, however, that the
diameter of the small pulley is already quite reduced, and it is
hardly 'desirable to decrease it still further. Neither can we
increase the large pulley, as we have already set the limit at 42
inches. Hence, for our present problem we cannot improve mat-
ters much without increasing the size of the large gear, which is
undesirable, or putting in another pair of gears, which is contrary
to the conditions of the problem. As such a motor is perfectly
reasonable, we shall assume it to be chosen for the purpose.

In commercial practice it would be well to pick out some
standard make of motor of the required horse-power, note the speed
as specified by the makers, and then, if possible, suit the ratio in
the machine to this speed. It is always best to use standard ma-
chinery, if possible, both from the standpoint of first cost^ as well

30 MACHINE DESIGN

t(ridt^ o^^S^JLPf '. 9Kou ^,ao,/fa3

-^ ':p^' 2.-7-2. f/CcO-^J'f^ yC<,^,3

• •

2.^6

-7J= ^^*>"

Fig. a

MACHINE DESIGN

as ease of replacing worn parts. Machinery ordered special is
expensive in first cost of designing, patterns, and tools, and extra
spare parts for emergency orders are not often kept on hand.

Tabulation of Torsional Moments. For future reference, it is
desirable at this point to tabulate the torsional moment, or torque,
about each of the three shaft axes, assuming reasonable efficiencies
for the various parts, as follows:

Efficiency between drum and gear tooth 95 per cent

Efficiency between drum and pinion shaft 90 per cent

Efficiency between drum and motor shaft 80 per cent

TABLE OF TORSIONAL MOMENTS.

Axis.

Inch Lbs. Torque
at 100 Per Cent Efficiency.

Inch Lbs. Torque,
Efficiency as Above.

Drum

Pinion

Motor

5,000X^ =67,500

27 1^
5,000X^X-i| =12,187

5,000X^X^X^= 3,047

.80 ^'®®

This means that the motor develops a torque of 3,809 inch*
pounds delivering to pinion shaft 13,541 inch-pounds, and to drum
71,052 inch-pounds.

Width of Belt. The page of calculation for belt width is repro-
duced in Fig. 3.

The calculation as given is strictly scientific, based on the
working strength of a cemented joint (^=400 lbs. per square inch).
This is a favorable situation for the use of a cemented joint, be-
cause it is easy to provide means of adjusting the belt tension by
placing the motor on a sliding base. Otherwise a laced joint could
be used, requiring relacing when the belt slackens through its
stretch in service. Under the assumption that a double laced belt
is used, the empirical formula below is one often applied:

p ^XV ^XMOO
540 "" 540 ""

This gives w==^ 1 qoo ^^^^'^ inches (say 12 inches).

It should be remembered that this value is purely empirical;
it applies to a laced joint, and could not be expected to check the

32 MACHINE DESIGN

value of 9 inches obtained by the first computation for a cemented
joint. It is fairly in proportion. For the quite definite service
required of the belt in the present case, the width of 9 inches h
doubtless sufficient, considering the cemented joint.

Length of Bearings. C!onsiderable latitude in choice of length
of bearings is permissible, especially in such slow-speed machinery.
There is probably little danger from heating, and the question then
becomes one of wear. It is better in such cases as the one in ques-
tion, to choose boldly a length which seems to be reasonable and
proceed with the design on that basis, even if the length be later
found out of proportion to the shaft diameter, than to waste too
much time in the preliminary calculation over the exact determina-
tion of this question. Probably in most cases of commercial piuc-
tice the existence of patterns, or some other practical consideration,
will decide the limits of length.

In the present instance it seems reasonable that a length of
6 inches would fill the requirement for the worst case, that of the
drum shaft, and it is obvious that the bearings for the pinion shaft
would naturally be of the same length on account of being cast on
the same bracket, and faced at the same setting of the planer tool.

Height of Centers. The large pulley should naturally swing
clear of the floor. This will require, say, a total height of 23
inches, out which we may take 4 inches for the base, leaving 19
inches as the height, center of bearing to base of bracket.

Data on 5lcetch. The data as found above should now be put
on the sketch previously made; it will then have the appearance
shown in Fig. 1.

This sketch is now in form to control all the subsequent detail
design, and it is expected that the figured dimensions as shown can
be maintained. It is impossible to predict this with positiveness,
however, as in the working out of the minor details certain changes
may be found desirable, when, of course, tl^y should be made.

The shaft sizes do not appear on this sketch, hence before
proceeding further the several shaft diameters must be calculated.

Sizes of Shafts. The calculations of the shaft diameters are
good instances of systematic engineering computations, hence they
are reproduced in the exact form in which they were made. The
student ehoold learn a valuable lesson in making and recording

MACHINE DESIGN

calcalationa by following these carefully. Note that each set of
figures is independeut, both in the statement of given data, as well
as in the actual computation. Observe how easy it would be for
the anther of these figures or anyone else to check them even after

SyuAxJX ■^l.<.MlM -^vr t,. ?Ha^S^/,

TU-= lost

B=/4'73«^=7365"
7"» (i¥^*S-3S'33f&

= 73iSy t/3.0 = /SV-tS'

/S¥-SS= ^^f-^

^.^fc/rf- 80Vr>

■3

Fig. 1.
ft loDg lapse of time. If the machine should unexpectedly fail in
service the figures are always available to prove or disprove theor-
etical weaknesB. The right triangles merely indicate that the
value of i/U'+'i" was found by the graphical method Boggeated in

MACHINE DESIGN

Part II, " ShaftB," the figures being pnt on the triangle as a eim-
pla and direct way oF recording both proceBS and resnJt.

Attention is especially called to the fact that in the pinion
shaft the size is changed for each piece upon the shaft. This ib

7iXeuf9-l^03

X^'/^si

Ti^BrifBr-t-T-

'13 c s-f-hiz^t /ss*^

'73£ST/^4f:iS' 2,1790

^■X19e - ££r2jr
S.I

done partly because it is desired to show the student that the shaft
at each of these points shonld be theoretically of different size.
It is also done because as a practical feature of construction it ia a
good plan to change the size when the fit changes, partlyfor rea-
sons of production in the shop, partly for ease in slipping pieces

MACHINE DESIGN

freely endwise on the shaft until they reach their proper fit and
location in the assembling of the machine.

This should not be taken as an absolute requirement in any
sense. A straight shaft would be satisfactory in the present case;
but the shouldered shaft is a little better construction, in a mechan-
ical sense, and does not cost much more. Hence it is used. For
the drum the straight shaft seems to answer the requirenxent well
enough.

I&CaaJuuj^ ^hJL^'ti Aa^f y uz^ ^yJJbuf . Tijcu u j SLi^

ts^atj

75- //7^^

-r^ (>^¥.»Ji ^/ sr/5 5'4^/

^s /3-tyy^'^tT^

iniH-

« //7S<^^///7^'^f/^S^/^
^//7»¥ '^/7fSo « ^973^

Fig. 6.

Small Pulley Bore. Fig 4.

Large Pulley Bore. Fig. 5.

Bearing Next to Large Pulley. Fig. 6.

The diameter, 2f^J, as calculated, is based on the supposition
tnat the greatest bending moment is caused by the belt pull on the
overhanging pulley, that is, by the forces existing at the left-hand
side of the center of the bearing.

MACHINE DESIGN

But the pinion tooth load produeea a heavy bending on the
shaft in the bearing, the ehaft in this case acting as a beam sap-

Tj' 3i-7S ^ ^ov-o X..33.S- (^r^UC^j aj^^ q)
7^' -i^^g-f 33-J~^ a^s-£
^= V^rw - 3 v^ J-^ - ^* a.

3 = 3¥S8 A 9-S- ■- /2 / <? 3 0)Uu. TeT ^

^^m Plt^fli -2.fZ. K>^^^

Fig. 7.
ported at the two bearings and having the tooth load applied as
lihown. If tbiB latter effect be greater than the former, that is, if

= *»■

MACHINE DESIGN 37

the bending moment produced by tlie pinion tooth load be greater
than the bending moment produced by the belt pull, then the diam-
eter must be increased to satisfy the latter case. As is seen by
the second calculation of Fig. 6, this is not the case, and the diam-
' eter stands at 2^ as made.

Pinion Bore. Fig. 7. The pinion being a driving fit upon
flie shaft, reinforces the shaft to such an extent that it is hardly
possible for the shaft to break off very far inside the face of the
pinion; but it is quite possible that the metal of the pinion may
give enough, or be a little free at the ends of the hole, so that the
shaft may be broken off, say \ inch inside the face. In this case,
it may fail from the moment of the force at the left-hand bearing
or of that at the right. It may fail then at (a) or (b), depending
on which section has the greater bending moment. Trying both,
it is seen by the calculation that the right-hand moment is the
controlling one, and it, therefore, is used.

Sliaft Outside of Pinion. Fig. 8. As there is no power
transmitted through this portion of the shaft, there is no torsional
moment in it, and the bending moment remains practically the
same as inside the pinion.

The size figures about 2i|, but since there is no use in turn-
ing off material just to reduce the size to this, it is well to make
it 2J, or just smaller than the fit in the pinion.

Pinion Shaft Outer Bearing. Fig. 8. This diameter, of
course, figures small, as there is no torsion in it, and the bending
moment is not heavy. The practical question comes in, however,
whether it is advisable to make the outer bracket different from
the inner one just on account of this bearing. The commercial
answer to this would probably be " No," hence the size as figured
next to the pinion will be maintained (2{^J).

Drum Shaft. Fig. 9. In this case, as previously inferred,
the simplest thing to do is to use a piece of straight cold-rolled
steel, and make both bearings alike, the size being determined
according to the worst case of loading which can occur as the
rope travels from end to end of the drum. This case is evi-
dently when the rope is at the end of its travel close to the brake,
for at that time both the load on the rope and the load on the pinion
tooth which is driving it are exerted upward, and produce the

38 MACHINE DESIGN

greatest reaction at the bearing next to tlie gear. The analysis of
the forces for this condition is shown in Fig. 9.

Other conditions of loading would be when the brake is on
and the tooth load relieved, but then the resultant of the brake
strap tensions would be diagonally downward and would reduce

« •

X^' c3 •

^ /fi.

/O.2.

V*» /6937^ /a-3 . . ^ ,

J ^ - - «, ^ A

^^^^

^^s /^ i 6 X Z^?.

22- ^?3

Fig. 8.

rather than add to the rope load. Again, when the rope is at
the end of the drum farthest from the gear, the load on it and
the load on the pinion tooth are both exerted upward as before, but
the reaction cannot be as great as. in the case of Fig. 9, because the
tooth load is still concentrated at the other end of the shaft and
produces a relatively small reaction at the rope end.

MACHINE DESIGN

Preliminary Layout. Fig. 10. Proceeding now with the lay-
oat to Bcale, the detail of the parts may be worked out as com-
pletely as the scale of the drawing will permit. The work on this
drawing may he of an unfinished, sketchy nature, hnt the meaenre-
mentfl must be exact as far as they go, for this drawing is to serve
as the reference sheet, from which all futnre detail is to be worked np.

In this layout may be worked out the sizes of the arms and
hubs of pulleys and gears, the proportions of the drum and brake

^■m

s&t

31-7 S

■t- A-ooo »

"■ /O.2.

/O.2.

/e.T.

Pig. 9.
Strap, and the general dimensions of the side brackets and the
base. "When the detail becomes too fine to work out to advan-
tage on this drawing it may be worked out full size by a separate
sketch, or left to be finished when it is regularly detailed. The
preliminary layout, it should be remembered, is a service sheet
only, a means of carrying along the design, and not intended for

MACHINE DESIGN 41

a finished drawing. The moment that the free use of the layout
is impaired by trying to make too much of a drawing of it, its
value is largely lost. A designer must have some place to try out
his schemes and devices, and the layout drawing is the place to do
it. This drawing may be recurred to at intervals in the progress
of the design, details being filled in as they are worked out, as
they may control the design of adjacent parts.

As the discussion of the design of each of the members
involved in the present problem can be better taken up in con-
nection with the detail drawing of each, it will be given there,
rather than in connection with the layout, although many of the
proportions thus discussed could be worked out directly from the
latter.

Pulleys. Fig. 11. The analysis of the forces in the belt
gives, according to the calculation of Fig. 3, a tension in the tight
side of 1,059 pounds, and in the slack side 414 pounds. The
difference of these, or 1,059 — 414=645 pounds, is transmitted to
the pulley and produces the torque in the shaft. Of course in
the small pulley the torque is transmitted from the motor through
the pulley to the belt, but both cases are the same as far as the
loading of the pulleys is concerned.

The only other force theoretically acting is the centrifugal
force due to the speed of the pulley. This produces tension in
the rim and arms, but for the low value of 1,300 feet per minute
peripheral velocity in this case may be disregarded.

Considering the arms as beams loaded at the ends, and that
one-half the whole number of arms take the load, and for con-
venience, figuring the size of the arms at the center of the pulley,
gives the following calculation for the large pulley:

^X21=^^=.0393X2,500X/1» Let S=2,500

.^ •• ^=breadth of oval

fc»=.^^=46 •« .4/i=thickn©ss of oval

fc= v^'46=3.6 (say 3.5)

.4fc=.4x3 5=1.4 (say 17-16)

This is about all the theoretical figuring necessary on this

pulley. The rim is made as thin as experience judges it capable

of being cast; the arms are tapered to suit the eye, thus giving

ample fastening to the rim to provide against shearing off the rim

MACHINE DESIGN 43

from the arms; generous fillets join the arms to both rim and hub;

and the hub is given thickness to carry the key, and length

enough to prevent tendency to rock on the shaft. Uncertaia

strains due to unequal cooling in the foundry mold may be set up

in the arms and rim, but with careful pouring of the metal they

should not be serious, and the low value chosen for the fibre stress

allows considerable margin for strength.

The small pulley has the same forces to Withstand as the

large pulley, but on account of its small diameter there is not

room enough for arms between the rim and the hub, hence it is

made with a web. The web cannot be given any bending by the

belt pull, the only tendency which exists in this case being a

shearing where the web joins the hub. This shearing also exists

throughout the web as well, but at other points farther from the

center it is of less magnitude, and moreover, there is more area of

metal to take it. The natural way to proportion the thickness of

the web is to give it an intermediate thickness between that of the

hub and rim, thus securing uniform cooling, and then figure the

stress as a check. Making this value ^ inch gives a shearing area

of J multiplied by the circumference of the hub, which is 3.1416

645 X 5 25
X 4 = 12.56. The shearing force at the hub is g-^— =1,693

pounds. Equating the external force to the internal resistance

1,693 = JX 12.56 XS

^ 1,693X8 ... , . , , ,

= 7 wf o gr> = 15^ pounds per square inch (approx.).

This is a very low figure, even for cast iron, hence the web is
amply strong. The rim and hub are proportioned as for the large
pullfey.

The keys are taken from the standard list. They may be
checked for shear, crushing in the hub, and crushing in the shaft,
but the hubs are so long that it is at once evident without figuring
that the stress would run very low in both cases.

Qears. Fig. 12. The analysis of the forces acting on the
gears has been given on page 28, 4,000 pounds being taken at the
pitch line. Using this same value, and choosing a T-shaped
arm as a good form for a heavily loaded gear like the present one,
let us consider that the rim is stiff enough to distribute the load

MACHINE DESIGN 45

equally between all the arms, and that each acts as a beam loaded
at the end with its proportion of the tooth load. Before we can
determine the length of these arms, however, we must fix upon the
size of the flange which is to carry the driving bolts. This is taken
at 13 inches. It could be smaller if desired, but drawing the bolts
in toward the center increases the load on them, and 13 inches
seems reasonable until it is proved otherwise. This makes the

.1-1. 4,000X11.5 ^^^^

maximum moment which can come on an arm ^ =7,666

6 '

inch-pounds.

Now it is evident that the base of the T arm section, which
lies in the plane of rotation, is most effective for driving, and
that the center leg of the T does not add much to the driving
. capacity of the arm, although it increases the lateral stiffness of
the arm, as well as providing in casting a free flow of metal between
the rim and the hub. Hence the simplest way of treating the sec
tion of the arm for strength is to consider the base of the T
only, of rectangular section, breadth S, and depth A, for which

the internal moment of resistance is -^ '

Also, it is simplest to assume one dimension, say the breadth,
and the allowable fibre stress, and figure for the depth. Taking
the breadth at 1 J inches, which looks about right, and the fibre
stress at 2,500, and equating the external moment to the internal,
we have

^ ggg _ 2,500 X 1.125 X A'
' 6

6X7,666
^ "" 2,500 X 1.125 ""■^^•*

h = i/IS:i = 4.05 (say 4J)
Drawing in this size, and tapering the arm to the rim as in
the case of the pulleys, making the depth of the rim according to
the suggested proportions given in Part H, " Gears," giving the
center leg of the T a thickness of J inch tapering to 1 inch, and
heavily filleting the arms to the rim and center flange, we have a
fairly well proportioned gear.

The next thing to determine is the size of the driving bolts.
The circle upon which their centers lie may be 11 inches in diam-

MACHINE DESIGN 47

eter, and there will naturally be six bolts, one between each arm.
These bolts are in pure shear, and the material of which they are
to be made ought to be good for at least 8,000 pounds per square
inch fibre stress. The force acting at the circumference of an

11-inch circle would be -^ — r-r =13,091 pounds.

Equating the load on each bolt to the resisting shear giyes

1 3,091 onnAv.A 8,000x3.1416Xcg Let A =area resisting shear,
"e ~ ' ^ ^^ ^ Let d=dia. of bolt.

Then A=— ^

^ 4X13,091 «-

"^"6X8,000X3.1416""

d="|/.35 (say .6) ^-inch bolts would do.

But g-inch bolts are pretty small to use in connection with such
heavy machinery. They look out of proportion to the adjacent
parts. Hence ^-inch bolts have been substituted as being better
suited to the place in spite of the fact that theoretically they are
larger than necessary. The extra cost is a small matter. These
bolts may crush in the flange as well as shear off, but as there is

' 13 091
an area of J X If = 1.422 square inches to take — ^ — =2,182

pounds, the pressure per square inch of projected area is only

2 182

Y^Qp= 1,534 pounds, which is very low.

This gear needs no key to the shaft because all the power
comes down the arms and passes off to the drum through the bolts,
thus putting no torsional stress in the shaft. The face of the
flange is counterbored so as to center the gear upon the drum,
without relying upon the fit of the gear upon the shaft to do this.

The pinion is solid and needs no discussion for its design.

Brackets and Caps. Fig. 13. As the size of the drum shaft
was determined by considering the rope wound close up to the
brake, thus giving in combination with the load on the gear tooth
the maximum reaction at the bearing as 6,748 pounds, the cap and
bolts should be designed to carry the same load.

For a bearing but 6 inches long, two bolts are sufficient under
ordinary conditions and might perhaps do for this case. The load
is pretty heavy, however, and it is deemed wise to provide four
boliS| thus securing extra rigidity, and permitting the use of bolts

48 MACHINE DESIGN .

of comparatively small size. If the load were distributed equally
over all the bolts each would take one-fourth of the whole load,
but it is not usually safe to figure them on this basis, because it
is difficult to guarantee that each bolt will receive its exact share
of stress. Assuming that the two bolts on one side take |^ the
whole load instead of J, which provides for this uncertain extra
stress, each bolt must take care of -J- of 6,748, or 2,249, pounds.
Allowing 8,000 pounds per square inch fibre stress calls for an

2 249
area at the root of the thread of ^j^ = .281 square inch. Con-
sulting a table of bolts we find that the next standard size of bolt
greater than this is |, which gives an area of .302 square inch.

Choosing this size as satisfactory, the bolts should be located
as close to the shaft as will permit the hole to be drilled and tapped
witnout breaking out. A center distance of 5 J inches accomplishes
this result. The distance between centers in the other direction
is somewhat arbitrary, although the theoretical distance between
the bolt and the end of the bearing to give equal bending moment
at the center of the cap and at the line of the bolts is about -^^ of
the length, or -^^ of 6 = 1 J inches. This proportion answers
well for the present case, although for long caps it brings the
bolts too far in to look well.

The thickness of the cap may be determined by assuming it
to be a beam supported at the bolts and loaded at the middle.
This is not strictly true, for the load is distributed over at least a
portion of the shaft diameter; moreover, the bolts to some extent
make the beam fixed at the ends. It being impossible to determine
the exact nature of the loading, we may take it as stated, supported
at the ends and loaded in the middle, and allow a higher fibre
stress than usual, say 3,500. The longitudinal section at the
middle of the cap is rectangular, of breadth 6 inches, and deryfh
unknown, say A. The equation of moments is

WXI _ SXI _ SX^XA'
4 ~ c "" 6

6,748X5.5 _ 3,500X 6 XA^
4 6

64X6,748X5.5
^ - 4X3,500X6 -"^'^^
h = V^2.65=1.02 (IJ will probably answer)

MACHINE DESIGN 49

For the other bearing next to the pinion, the load on the tooth
acts downward, and the resultant pull of the belt is nearly hori-
zontal, hence the cap and bolts must stand but little load, and
calculation would give minute values. In a case like this it is
well to make the size the same as for the larger bearing, unless
the construction becomes very clumsy thereby. This saves chang-
ing drills and taps in making the holes, and preserves the symmetry
of the bracket. The |-inch bolts are good proportion for the
smaller bearing, hence that size will be maintained throughout.

The body of the bracket is conveniently made with the web at
the side and horizontal ribs extendincr to the outside. The load due
to the rope is carried directly down the side ribs and web into
the bottom flanges and to the bolts. The analysis of the forces
on these bolts is shown in Fig. 14. It is evident from the figure
that the resultant belt pull tends to hold the bracket down, while
the load on the rope tends to pull it up, the point about which it
tends to rotate being the corner furthest from the drum. It is also
evident that the bolts nearest this corner can have little effect on
the holding down, because their leverage is so small about the cor-
ner, hence we shall assume that the pair of bolts at the right-hand
end of the bracket takes all the load. The belt pull, being hori-
zontal, tends to slide the bracket along the base, but this tendency
is small, and at any rate is easily taken care of by the two dowel
pins, which are thus put in shear.

The load on the bolts being 4,954 pounds, a heavy bending
moment is thrown on the flange of the bracket, tending to break
it off at the root of the fillet. The distance to the root of the fillet
is 2 inch; the section tending to break is rectangular, of breadth
5 J inches, and unknown depth A. The equation of moments is

c o

4,954X3 2,600 X 5.5 X A '
4 ~" 6

6X4,954X3
'^ "4X2,500X5.5 -■^•*'''

A=l/r62 = 1.3(saylJ).
The thickness of the web and ribs of this bracket is hardly
capable of calculation. The figure ^ inch has been chosen in pro-

MACHINE DESIGN

portion to the size of the large dram bearing, giving ample stiff,
ness and rigidity, and permitting uniform flow and cooling of the
metal in the mold. The opening in the center is made merely to
save material, as in that part little stress would exist, the two sides

^^•^/^ '4<AJi^ "^•CfS

•^

.3L^-/?J?^

zj&5?t

37.&7S-

"""■^ Fig. 14.

carrying the load down to the base bolts, and the top serving as a
tie between the bearings.

This bracket might be made with the web in the center of the
bearings instead of at the side, in which case the expense of the

MACHINE DESIGN 51

pattern would be slightly greater. It could also be made of closed
box form, but would in liiat case probably weigh more than as
shown.

Drum and Brake. Fig. 15. The analysis of the forces acting
on the drum is simple, but its theoretical design is more compli-
cated. It is evident that the drum acts as a beam of hollow circular
cross section, and that its worst case of loading is when the rope is
at or near the middle of the drum length. At the same time the
metal of this circular cross section is in a stiate of torsion between
the free end of the rope and the driving gear, due to the load on
the gear tooth and the reaction of th^ rope. Also the wrapping of
the rope around the drum tends to crush the metal of the
section beneath it, the maximum effect of this action being near
the free end of the rope where its tension has not been reduced by
friction on the drum surface.

Now the " mechanics " to solve the problem of these three
combined actions is rather complicated. It can be at least approx-
imately solved, however, for it satisfies fairly well the case of
combined compression and shear. But on a further study of this
particular case, it is seen at once that the diameter of the drum is
relatively large with respect to its length, which means that the
thickness of the metal may be very small and yet give a large
resisting area, or value of " I,'' both in direct bending as well as
torsion; also it is so short that the external bending moment will
be small. The practical condition now comes in, that the drum
can be safely cast only when the thickness of the metal is at a
minimum limit, for the core may be out of round, not set centrally,
or by some other variation produce thin spots or even develop holes
reaching out into the rope groove, discovered only when the latter
is turned in the lathe.

Hence it seems reasonable and safe in this case to make the
thickness of the drum depend simply upon the crushing caused by
the wrapping of the rope around it, and we shall take the coil
nearest the free end of the rope, assuming that it carries the full
load of 5.000 pounds throughout one complete wrap around the
drum.

The area resisting the crushing action may be considered to
be that of the cross section of a ring, of width equal to the pitch

MACHINE DESIGN 53

of the groove doubled. Assuming that | inch is the least thick-
ness which can safely be allowed under the groove for casting pur-
poses, let us figure th^ crushing fibre stress to see if this is suf-
ficiently strong. Disregarding the small amount of metal existing
above the bottom of the groove, this gives the areat to resist the
crushing f X f X 2, or .94 inch. Since there are fwo of these
sections and the rope acts on both sides, the equation of forces is:
5,000.x 2 = SX .94 X 2

5,000 X 2 .o.Q . . ,

S = — ^ . ,. = 5,319 pounds per square inch.

This, for cast iron, in pure crushing, allows plenty of mar-
gin for the extra bending and torsional stress, which for such a
considerable thickness would be slight.

The above case indicates a method of reasoning much used in
designing machinery, which while following out the specified
routine of thought as previously given in these pages, stops short
of elaborate and minute theoretical calculation when such is obvi-
ously unnecessary. If a drum of great length were to be designed,
and of small diameter, the same method of reasoning would deduce
the fact that the design should be based on the bending and the
torsional moments, the thickness in such a case being so great to
withstand these that the intensity of the crushing due to wrap of
the rope becomes of inappreciable value.

The remaining points of design of the drum are determined
from practical considerations and judgment of appearance. The
ribs behind the arms are put in to give lateral stiffness and guard
against endwise collapse. The arms are subject to the same bend-
ing as those of the gear, but as they are equally heavy it is not
necessary to calculate them. The flange at the driving end is of
course matched to that already designed for the gear. The rope
is intended to be brought through the right-hand end with an
easy bend and the standard form of button wedged on to prevent
its pulling through.

This drum would probably be cast with its axis vertical, and
the driving flange down to secure sound metal at that point.
Heavy risers would be left at the other end to pscure soundness
where the rope is fastened. Drums are often cast with the axis
horizontal, but the vertical method is more certain to produce
a sound casting. The grooves should be turned from the

54 MACHINE DESIGN

solid metal, partly because it is a difficult matter to cast them, but
principally because the rope should run on as smooth surface as
possible to avoid undue wear. On drums which carry chain instead
of wire rope the grooves are sometimes cast with success, although
even in this case the turned groove is generally preferable.

The brake consists of a wrought-iron band to which are fast-
ened wooden blocks, the iron band giving the requisite strength
while the blocks give frictional grip on the drum surface and can
be easily replaced when worn. As in the designing of a belt the
object in view is the grip on the pulley surface by the leather to
enable power to be transmitted from the belt to the pulley, so in
the case of the brake if we put the proper tension in the strap it
can be made to grip the brake drum so tightly that motion between
it and the drum cannot occur. The latter case is really the reverse
of the first, if the driven pulley be considered, but is identical with
the case of the driving pulley, in which the power is transmitted
from the pulley to the belt. Of course in the case of the brake
no power is transmitted, as when the brake holds no motion occurs,
but the principle of the relative tensions in the strap is the same
as for the belt.

Since the brake drum surface is 28 inches in diameter, the load
at that surface which the brake must hold is

_ 5,000X27 . ^^^ ,

P= 14X2 = ^^821 pounds.

We have then the following calculation corresponding exactly
to that of the belt given in Fig. 3.

log.^=2.729x^X^ ^*!:i:?5

T„— T° = P = 4,821

Tn ft _nn fti- ,P,^ rve^ft (f OF which the natural Dumber
log. Y-= 2.729 X .25 X .75 = 0.512 ^ j^ g 35).

T T

Then ^=3.25 T„ =

To " 3.25

Tn-To = 4,821 T„_3^g=^3^ = 4,821

T„ = ^^^^^^= 6,963 pounds (say 7,000)
To = 6,963—4,821 = 2,142 pounds (say 2,200)

MACHINE DESIGN 55

The tight end of the strap must then be eapable of carrying a
load of 7,000 pounds, and since the width has already been taken at
4^ inches, the problem is to find the necessary thickness. Equating
the external load to the internal resistance we have

7,000 == A X S Let ^ = thickness

« S = fibre stress = 12,000
7,000 = 4.5X^X12,000

. 7,000 ^_ .

^ = 4.5X12.000 ^'^^^^"^

This, however, can be but a preliminary figure, for the riveting
of the strap will take out some of the effective area, and the thick-
ness will have to be increased to allow for this. Suppose on the
basis of this figure we assume the thickness at a slightly increased
value, say ^ inch, and proceed to calculate the rivets.

A group of five rivets will work in well for this case, which

gives -^ — = 1,400 pounds per rivet. A safe shearing fibre stress
is 6,000, hence the area necessary per rivet is n\r\rx = .23 square

inch. This comes nearest to the area -^^ diameter, but for the
sake of using the more general size of rivet (f inch) the latter is
chosen, for which the area is .30.

We must now try these rivets in a y^^-inch plate for their safe
bearing value. The projected area of a |-inch hole in a y3g..inch plate

is |X-A=-11''' square inch. -yy=- = 11,965 (15,000 would be safe)

Taking out two §-inch rivets from the full width of 4 J inches
leaves 4^ — (2Xf)= 3.25, and makes the net area of strap to take
stress 3.25 X ^=-61 square inches. Ee-calculating the fibre stress
for this area gives

7,000 = .61 X S

j^ 7,000 = 11,475 (which approximates the previous value

^ ~ "TeT of 12,000).

The slack end of the strap has to take but 2,200 pounds, hence
a different calculation might be made for this end giving smaller
rivets; but as it is impractical to change the thickness of the strap
to meet this reduced load, it is well to maintain the same propor-
tion of joint as at the tight end. The spacing of the rivets in both

t^~" — r

[^A

i ^ J]

>■ i it

il ■

MACHINE DESIGN 57

cases follows the ordinary rule allowing at least three times the
diameter of the rivet as center distance, and one-half this value to
the edge of the plate.

The threaded end of the forging on the strap also has to carry
the load of 2,200 pounds, for which a size smaller than 1 inch
would suflSce. It is natural, however, for the sake of general pro-
portion to make the bolt as strong as the strap, and a l-inch bolt
gives an area of .52 square inch, nearly equalling the value of
.61 net area of strap noted above.

Base, Brake-5trap Bracket and Foot Lever. Fig 16. The
base cannot be definitely calculated, and can best be proportioned
6y judgment. It must not distort, twist, or spring in any way to
throw the shaf ts^ out of line. The area in contact with the founda-
tion upon which it rests must be ample to carry the weight of the
whole machine with a low unit pressure. Although the form
shown is perfectly practicable to cast and machine, and is simple
and rigid, yet it is questionable if a bolted-up construction, say of
four pieces, might not be equally rigid and yet involve greater
facility of production in both the foundry and machine shop on
account of the reduced sizes of parts to be handled. This is a
question which depends on the equipment and methods of the
individual shop, and is an illustration of the practical control of
design by manufacturing conditions.

The brake-strap bracket and foot lever, also shown in this
figure, are examples of machine parts which are quite definitely
loaded, and the designing of which is a simple matter. Further
discussion of their design is not made, the student being given
opportunity for some original thought in determining the forces
and moments that control their design.

Qear Quard and Brake-Relief Spring. In exposed machin-
ery of this character it is desirable to cover over the gears with a
guard to prevent anything accidentally dropping between the
teeth and perhaps wrecking the whole machine.. This guard is not
shown, as it involves little of an engineering nature to interest
the student. It could readily be made of sheet metal or light
boiler plate, bent to follow the contour of the gears and fastened
to the top flange of the main bracket.

If the brake be not automatically supported at its top it will

58 MACHmE DESIGN

lie with considerable pressure, due to its own weight, on the brake
surface when it is supposed to be free from it, and by the friction
thereby created will produce a heavy drag and waste of power. ^
A spring connection fastened to an overhead beam is a simple way
of accomplishing the desired result. A flat supporting strap car-
ried out from the gear guard, having some degree of spring in it,
is a neater method of solving the problem. The spring should
be just strong enough to counterbalance the weight of the strap
and yet not resist to an appreciable degree the force applied to
throw the brake on.

GENERAL DRAWING.

The last step in the process of design of a machine is the
making of the assembled or general drawing. This should be
built up piece by piece from the detail drawings, thereby serving
as a last check on the parts going together. This drawing may
be a cross section or an outside view. In any case it is not wise
to try to show too much of the inside construction by dotted lines,
for if this be attempted, the drawing soon loses its character of
clearness, and becomes practically useless. A general drawing
should clearly hint at, but not specify, detailed design. It is
just as valuable a part of the design as the detail drawing, but
it cannot be made to answer for both with any degree of success.
A good general drawing has plenty of views, and an abundance of
cross sections, but few dotted lines.

The general drawing of the machine under consideration is
left for the student to work up from the complete details shown.
It would look something like the preliminary layout of Fig. 10, if
the same were carefully carried out to finished form. A plain out-
side view would probably be more satisfactory in this case than a
cross section, as the latter would show little more of value than the
former. The functions which the general drawing may serve are
many and varied. Its principal usefulness is, perhaps, in showing
to the workman how the various parts go together, enabling him to
sort out readily the finished detail parts and assemble them, finally
producing the complete structure. Otherwise the making of a
machine, even with the parts all at hand, would be like the putting
together of the many parts of an intricate puzzle, and much time

MACHINE DESIGN 69

would be wasted in trying to make the several parts fit, with per-
haps never complete success in giving each its absolutely correct
location.

The general drawing also gives valuable information as to the
total space occupied by the completed machine, enabling its loca-
tion in a crowded manufacturing plant to be planned for, its con-
nection to the main driving element arranged, and its convenience
of operation studied.

In some classes of work it is a convenient practice to letter
each part on the general drawings and to note the same letters on
the specification or order sheet, thus enabling the whole machine
to be ordered from the general drawings. This is a very excel-
lent service performed by the general drawing in certain lines of
work, but for such a purpose the drawing is quite inapplicable
in others.

Merely as a basis for judgment of design, the general drawing
fulfils an important function in any class of work, for it approaches
the nearest possible to the actual appearance that the machine will
have when finished. A good general drawing is, for critical pur-
poses, of as much value to the expert eye of the mechanical
engineer as the elaborate and colored sketch of the architect is to
the house builder or landscape designer.

From the above it is readily understood that the general
drawing, although a mere putting together of parts in illustration,
is yet of great assistance in producing finished and exact machine
design.

GENERAL COMMENTS ON PRECEDING PROBLEM.

After following through the detail of work as given in the
preceding pages, it is worth while to stop for a moment and
take a brief survey or review of the subject as illustrated therein.

If the text be carefully studied it will be seen that in every
part to be designed the same routine method has been followed,
regardless of the final outcome. In some cases it may seem a
roundabout procedure to follow a train of thought that finally
ends in a design apparently based on purely practical judgment,
the theory having had but very little if any influence. The ques-
tion at once arises — Why not use the empirical rule or formula in

60 MACHINE DESIGN

the first place ? Why not make a good guess at once ? Why not
save all the time and energy devoted to a careful analysis and
theory, if we are finally to throw them away and not base our
design on them ?

The principle to be noted in this connection is, that it is just
as fatal to good design to rely upon bare experience and upon
judgment alone, as it is to construct solely according to what pure
theory tells us. There are many things in the operation of
machinery that are totally inexplicable from the purely practical
point of view, and will forever remain so until we analyze them
and theorize on them. Many good things in machinery have
been the result of what might be called "reversed" machine
design. When a new machine is started, it frequently, or we
might almost say always, fails to do its work just as it is expected
to do it. This is because some little point of design is bad, owing
to the inability of drawings, however good they may be, to show
all that the machine itself in bodily form and in motion shows.

Now, if our analysis and theory have been good in the
designing process, it is almost sure that we can very readily
analyze and theorize on the trouble that exists when the machine
is finished, can detect the weakness, and can correct it with com-
paratively small change in the general design. This is " reversed "
machine design.

If, on the contrary, we have based our design purely on guess-
work, allowing our fancy full and free play to work out the details
without further basis, we may consider ourselves lucky if the
machine runs at all. This, however, is not the worst of the
situation. If the machine does actually operate, even as well as
it might reasonably be expected to, but still has the usual diffi-
culty of some little kink or hitch that was not expected, then, as a
result of the method upon which the whole thing has been con-
structed, we have no definite plan of action to proceed upon. We
must try first this, then that scheme to obviate the trouble. We
may be fortunate enough to " strike it " the first time ; we may
never strike it. It is doubtful if the machine ever can be made to
work at highest efficiency ; and if fairly good results be finally
obtained we never know the reason why, and have nothing on
which to base any future action or design.

MACHINE DESIGK 61

This haphazard process is not machine design at all, either
in name or in result.

As has previously been stated in these pages, there is no
such thing as too much analysis or theory in the designing of
machinery. Even if we carefully analyze, theorize with rigorous
exactness, and then practically modify our construction to such a
point that the original theoretical shape is almost or entirely lost,
the apparently roundabout process is not in vain, for we are in per-
fect control of our design. We know exactly what it has to take in
the way of forces, blows and vibrations. Wa know what its ideal
shape should be. We know where we can practically modify its
form without weakening it excessively or adding excess of material.
In other words we know all about it, and therefore know exactly
what we can do with it ; and whether it follows in its shape the
outline that pure theory gives it or some other outline, it is never-
theless well designed.

"Reversed" machine design, as described above, based on
observation and experiment with regard to machines already in
operation, is just as impossible witltioot exact analysis and theory
as is original design based merely on mechanical ideas in the
abstract. The method once learned and made a habit of mind
will produce results with equal facility in either case, and results
are what the mechanical world is seeking.

Another point worth noting in the progress of the problem
88 given is the absolute necessity of possessing some knowlege of
Mechanics. The more of this subject the designer can have at
his finger ends, the more ready and successful will he be in all
problems of Machine Design. However, the principles of forces
and moments clearly understood, and the application of the same
in the all-important subject, "Strength of Beams," constitute a
fund of information that will give a splendid start and a good
working basis for simple designs. It should always be remem-
bered that a complicated design is little more than a combination
of simple designs, and if one has the ability to dissect and analyze
what seems at first like a bewildering maze of parts, complication
is speedily changed to simplicity.

Common sense goes a long way in good designing. There is
nothing mysterious about the process If the beginner will only

62 MACHINE DESIGN

avoid doing things that are foolish and ridiculous on their very
face, if he will exercise the same judgment that he uses in the
daily affairs of his life and will mix in something of mechanics and
mechanical method, he will be on the direct road to success in the
art.

Good drawing is an essential element of good design, and it
is especially urged that the sketches and drawings as reproduced
in the preceding text be studied with this in mind. By a good
drawing is meant not a showy piece of work, finely shaded or
artistically lettered, but an exact layout, definite and measurable,
correctly dimensioned if in detail, and meaning exactly what it
says. Machine design is an exact science, and the designer can-
not shirk responsibility by permitting his work to be shiftless and
loose. If he cannot delineate, clearly and in definite form what he
determines in his mind the structure should be, then it is purely
good luck if he achieves success, and it may safely be asserted that
the success is due to some subsequent care and finished design
added to his feeble effort, rather than to any expertness of his own.
Such success is of a very doubtful nature, and if not bordering on
financial loss it is at least secured only at a low working efiBciency.

As examples of good drawings the plates shown are not
claimed to be anything extraordinary, but it will be noted that they
are clean-cut and definite, and that even the sketches are unmis-
takable as to that which they are intended to illustrate. The
information as to the design is all there; nothing is left to the
imagination.

Classification of flachinery. It is intended to be made clear
in all that has preceded, that the same method of attack and pro-
cedure may be applied to the designing of machinery, whatever
may be the class or kind. This is a fundamental principle.
When it is logically carried cut, however, it produces very differ-
ent results, as is evidenced by the characteristics of style peculiar
to each of the classes of machinery to one or another of which
all machines belong.

For example, an engine lathe has a style similar to a drill
press, or a boring mill, or a screw machine, or a milling machine.
It is very different, however, from the style of a steam engine, or
a pump, or an air compressor, or a locomotive; it is still more dif-

MACHINE DESIGN 63

ferent from the style of a rolling mill, or a link belt conveyor, or
a coal crusher, or a stamp mill.

These classes of machinery are so distinctly marked that the
novice is easily able to perceive that there is some controlling
influence in each which marks its peculiar style. He should at
the same time see that the very analysis that has been so strongly
insisted upon in these pages is the direct cause of the marked
characteristic in design. Each class of machinery must satisfy
certain exacting conditions different from those of any other, and
it is the careful study of these conditions, as fundamentally
enforced, which leads to the strictly logical design.

A few of the most common classes are enumerated below,
and their prominent features noted. It is hoped that a study of
them will familiarize the student in a general way with the
requirements of each, and serve as a guide to a more comprehen-
sive study of their detail design than is possible in these pages.

Machine Tools. Examples:: — lathe, planer, milling machine,
drill press, screw machine, boring mill, grinding machine, etc., etc.

The machines of this class are all utilized for the finishing of
metal surfaces. They are really at the root of the production of
machinery of all other classes. Accuracy is their prime character-
istic — accuracy of construction, accuracy of operation, accuracy of
adjustment. Any inaccuracy that exists primarily in a machine
tool is reproduced in every piece upon which it produces a finished
surface ; and since the mere act of finishing a surface upon any-
thing implies that a rough and inaccurate surface will not answer,^
the tool then fails of its purpose if it cannot produce a true sur.
face: it does not accomplish that for which it was designed.

The effect that this element of accuracy has upon the design
of a machine tool is to require long bearings, convenient and exact
methods of adjustment, stiffness, excess of material to absorb
vibration, special shapes to facilitate application of jigs, fixtures,
and exact manufacturing devices insuring interchangeability of
parts, dust guards, and automatic lubrication.

Machine tools are essentially machines of maximum output,
and depend for their success, not only upon their accuracy as
noted, but also upon their ability to do the greatest amount of work
per square foot of space occupied, with the least amount of manual

64 MACHINE DESIGN

labor and attention on the part of the operator. ' This is especially
true of automatic machinery, which perhaps might be classed by
itself in this respect, but which is nevertheless included under the
broad term of a machine for producing finished surfaces, being
merely the highest and most refined form of same. For machines
of this class the designer has to study every detail with the most
minute attention, packing away the operating parts into the
smallest space and yet providing ready means for access, removal,
and repair. Clearances that would be too little for other kinds of
machinery are permitted and provided for; material of high grade,
strength, and wearing quality, though expensive in first cost, and
requiring the most expert skill to finish and to fit into place, must
be used in order to keep the machine compact and yet of large
capacity, to make it reasonably light in weight and yet amply
strong.

Another point which has a great influence on the design of a
machine tool is that we can never tell in advance just what it will
have to stand in work, for the variation in the material that it fin-
ishes, the uncertain skill of the operator who runs it, the crowding
to its limit of capacity and even beyond in times of press of business,
and the many other stresses that may suddenly and without warn-
ing be thrown upon it, must all be thought of and provided for.

The points above mentioned are but a few of those which the
designer of machine tools has to meet, and are presented merely
as illustrations to show the special skill required in this class of
machinery. It is readily seen that while the machine tool
designer has great latitude in choice of material and in expendi-
ture of money for refinement of structure — perhaps greater lati-
tude than in any other class, yet he is held down as in no other
to the final productive results, a small percentage of failure entirely
throwing out the machine as a marketable product.

The style and external appearance of machine tools have a
character of their own resulting from this extreme detailed care in
design. Corners and fillets are carefully rounded; surfaces and
intersections are definitely made; in short, the mechanical beauty
of a machine tool is seen only from a near view and close inspec-
tion, and it is to this end that the design is constantly directed
Appearance is a large factor in the sale of a fine tool, and the

MACHINE DESIGN 65

prestige of the American trade abroad in this respect is very
noticeable.

Motive-Power flachinery. Examples: — Steam engine, gas
engine, air compressor, steam pump, hydraulic machinery, etc., etc.

The element of heat enters into the design of all machinery
in this class. The natural agents, air, gas, and water, in their
various forms, are taken into the machine in the most efiScient
form in which it is possible to obtain them, are robbed of their
energy to provide power, and are discharged in a form as weak
and inert as the eflSciency of the machine will determine.

In contrast to the class of machinery just studied, it should
be noted that these machines do not produce any material thing;
that is, they do not produce finished surfaces on metals, make
screws or bolts, bore holes in castings, or tnm line shafting.
They merely take the energy of the natural agent, which is not in
a form available for use, and transform it into motive power for
general use.

Hence the element of accuracy as entering into the design of
these machines is necessary only for their own eflScient operation,
and not for the quality of the thing which they produce, as in the
case of machine tools. For example, the power furnished by one
steam engine to drive a line shaft is as good as that of another as
far as the rotating of the shaft is concerned, provided, of course,
that both are equipped with the same quality of governing mechan-
ism. The fact that one of the engines has a good adjusting device
on the main bearing while the other has not is of no consequence
from the standpoint of the line shaft, but it is, of course, of con-
sequence respecting the efiBcient operation of the engines.

The design of steam engines and similar machines is of a
rough nature compared with that of machine tools, as far as the
detail of surface is concerned. General accuracy is nevertheless
essential for the machine's own sake, but while in the machine
tool we deal with thousandths of an inch, in the steam engine
hundredths of an inch indicates fine work.

These machines are subject to extremes of temperature that
have to be provided for in the design and arrangement of the parts.
Being prime movers, controlling the operation of many machines,
they must be certain to run during their period of work; hence

66 MACHINE DESIGN

design and adjustment mnst be positive, and when the latter can-
not be made while running, it must be quickly and definitely accom-
plished when a stop is made. Simplicity of construction is essential,
facilitating cheap and quick repairs. The design should be such
that constant attention while running is avoided, the usual atten-
tion of the engineer being a safeguard rather than an implied fac-
tor of the original design. General rigidity and stiffness are
important, also good balancing of the moving parts, and weight for
absorption of vibration ; otherwise under the constant daily run the
machines will tear to pieces not only themselves but their founda-
tions.

As far as external appearance goes in this and subsequent
classes to be mentioned we are on a very different basis from that
of machine tools. General mechanical symmetry of form is aimed
at in the design, and the several smaller parts depend for their out-
line (aside from considerations of strength, which are, of course,
always in order) upon the harmonious relation which they bear to
the main and fundamental elements of the machine. Such
machinery as air compressors, steam engines, pumps, and the like
are viewed as a whole, and criticised, not detail by detail, as is the
machine tool, but as to general effect of outline observed from
some distance. To convey the desired effect to the eye the design
must be bold and massive, connections simple and direct, and the
smaller parts must not be so dwarfed in size as to appear like deli-
cate ornaments instead of integral parts of the machine. The lines
of connected parts must be continuous from one part to the other;
and when interrupted by flanges, bosses, or lugs, the latter, which
are merely incidental to the former must not be allowed to obscure
wholly the main lines of the fundamental pieces.

It is attention to such points as these that marks the difference
between well-designed motive-power machinery and that of the
opposite character. Even though the little details of fillets and
corners and surfaces may have their effect from a close point of
view, the design will stand or fall in excellence on its bolder
features, as noted above.

Structural Machinery. Examples : — Hoists, cranes, elevators,
transfer tables, locomotives, cars, conveyors, cable-ways, etc., etc.

In the two preceding classes that have been noted, cast iron

MACHINE DESIGN 67

in the form of foundry castings enters as the principal material.
Steel is utilized for shafts, studs, pins, and keys. Also special
forgings, malleable iron and steel castings enter as factors in the
production of the machinery discussed. Foundry castings, how-
ever, compose the great body of the material used, and the chief
problems involved are those of the expert moulding of cast iron,
and the handling and finishing of the same. For the operating
parts, steel of fine grade is used in highly finished form, expens-
ive because of its fineness, and yet a necessity to the extent it is
used. Brass and bronze are used in the same way, generally in
connection with the bearings for the shafts.

Structural machinery, on the contrary, uses steel as the basis
of its construction. The fundamental structure is built up of
plates, channels, beams, and angles; castings, though numerous,
are relatively small, being riveted or bolted to the main structure
and controlled in their design by its requirements.

Steel is used in this manner partly because the exclusive use
of castings is prohibited on account of the excessive weight, and
therefore expense, and partly because castings could not be made
which would possess the necessary toughness and strength. In
many cases the size of the machinery is such that castings, even
if they could be made, would not support their own weight.
Moreover, machinery of this class is subjected to rough service,
and yet must be practically infallible under all conditions, neither
being uncertain in operation at critical moments nor entirely fail-
ing under an extraordinary load.

The design of structural machinery is tied up to con-
ditions existing largely outside of the locality in which the ma-
chinery is built. The steel plates and structural shapes required,
being products of the rolling mill, have to conform to the latter's
standards. The rivets, bolts and other fastenings have to be in
accordance with the established practice of the structural iron
worker, in order to permit punching, shearing and bending ma-
chinery of regular form to be utilized. Shipment on standard
railway cars has to be considered, the design often requiring to be
modified to permit this apd nevertheless insure positive and
accurate assembling in the field.

Steel castings, both large and small, find ready application in

68 MACHINE DESIGN

this class of work; also steel forgings, requiring to be worked
under a heavy hammer and in many cases by specially devised
processes.

In structural design less of the actual process of manu&ctnre
is under the eye of the designer than in the former classes of
machinery which have been considered, and hence more allowance
has to be made for things not coming exactly right to the fraction
of an inch. It would be bad design to plan any structural piece
of work with the same closeness of detail permitted, and in fact
required, in the case of machine tools, or even in the case of motive-
power machinery. In planning structural work the idea must be
carried out, of certainty of operation in spite of roughness of detail
and variations of construction. This does not necessarily imply
inaccuracy, or shiftless, loosely constructed machinery; on the con-
trary, quite the reverse. The locomotive, for example, ia one of
the most refined pieces of mechanism that exists today; and yet
the methods applied to the construction of machine tools would
prove a failure on the locomotive. The design of a car axle box
has to be just right else it will heat and destroy itself; the same is
true of the spindle of a fine engine lathe; and yet how rough the
former is compared with the latter, and how unsuited either would
be for use on the service of the other.

As a general rule structural machinery can be more closely
proportioned to theoretically calculated size than can the preceding
types. The rolled material of which it is made is of a uniform
and homogeneous nature owing to its process of manufacture,
hence its every fibre may be counted on to sustain its share of the
total load imposed upon it. This is in sharp contrast to the case
of cast iron, which is of such a porous and irregular structure that
we have to use a large factor of safety to cover this inherent
defect.

Steel castings of both small and large size (which are quite
apt to be utilized in this class of machinery for parts that can with
difficulty be made out of rolled material), if properly designed of
uniform thickness, with all corners well filleted and with the
channels for the flow of the molten metal direct and ample, are
nearly as reliable as rolled steel. In parts subject to excessive
vibration, shocks, and sudden wrenchings, as, for example, the

MACHINE DESIGN 69

Bide frames or the connecting rod of a locomotive, the forged and
hammered material is practically a necessity. This is especially
the case when the possible breakage of the part would cause
eerions consequences involving heavy loss of lif e^ and property.

From the several points of view as above considered, it can
be readily appreciated that, while structural work is in one sense
rough and unpolished, yet it requires, from an engineering stand-
point, quite as much breadth of experience and judgment as any
of the other types. The fine-tool designer, least of all, perhaps,
requires book theory, but does require an extended machine-shop
experience. The designer of motive-power machinery needs pure
physical theory and shop experience of a large and broad scope.
The structural designer is least of all concerned with refined and
minute finishing processes, but utilizes his theory absolutely, even
though roughly.

Mill and Plant Machinery, Examples: — Rolling mills,
mining machinery, crushers, stamps, rock drills, coal cutters, the
machinery of blast furnaces and steel mills, tube mills, etc., etc.

This machinery constitutes a class which in the roughness
of its operation exceeds all others. Moreover, it is machinery
which for the most part is in continuous operation— 24 hours per
day and 365 days in the year. Hence refinement, even such as
might be permitted in the preceding class of Structural Machinery,
would be fatal here. The conditions that surround plant machinery
are unfavorable in the extreme to the life of any material or metal,
and it is not possible to change these conditions or give more
than partial protection to the operating parts. Hence the design
of such machinery must proceed primarily on the assumption
that abuse and neglect, grinding away of surfaces, chemical eating
away of metal, flooding of parts with water gritty and corrosive,
subjection to sudden bursts of flame and intense heat, etc., will
in a relatively short time totally destroy, perhaps, the entire
structure.

In view of the continuous nature of the working process,
which must be kept up in spite of these almost insurmountable
conditions, the problem in each case becomes one of expediency;
and the designs and arrangement of machinery must be so worked
out that operation, repair, construction, and installation can all go

70 MACHINE DESIGN

on simnltaneonslj without stopping the continuous process, and
with but a small degree of inconvenience to the operation of the
plant.

This problem, difficult though it may seem, can be worked
out successfully, as is evidenced by the great number of plants of
the continuous character operating at high efficiency throughout
the world. The engineering and designing skill required to ac-
complish this, is perhaps of the highest degree met with in mod-
ern practice, for in it is involved a working knowledge of the
possibilities, if not the detailed designs of machinery included in
all classes. And yet, as in the most elementary case of simple
design that can be conceived, the result is accomplished in the
same way, namely, by studying the conditions (analysis), devel-
oping an ideal application to those conditions (theory), and then
reducing the ideal design to a practical basis (modification).

A Few Pointed 5ug:g:estions on Orig^inal Desig^n. Original
design deals with the development of original mechanical ideas.
The prime requisite for the development of an idea is to under-
stand thoroughly the idea in the rough. See distinctly the mark
aimed at, and never lose sight of it. If a method of reaching it
is already outlined, understand that also thoroughly and the prin-
ciples involved. It is impossible to go ahead blindly and hope to
come out right. No good machine was ever built that does not
stand for hours of concentrated thought on the part of its designer.
Good machines never happen^ they always grow.

Just as soon as the object to be accomplished is clearly under-
stood, begin to produce some visible work on the problem. Sketch
something. Get some ideas on paper. Ideas on paper suggest
other ideas. If the problem, for example, is one of lathe
design, sketch a rectangle, and call it the headstock; another rec-
tangle, and call it the footstock; a couple of scratches for the
centers; some steps for the cone pulley; three or four lines for the
bed; and as many more for the supports. There is now something
on paper to look at; the design is begun.

It is much better to stare at this sketch, than into blank
space trying to imagine the finished design. No matter how rough
the sketch may be, a short study of it will develop some linaiting
conditions that before were not apparent. Guess at a few rough

MACHINE DESIGN 71

dimensions; put them on the sketch; develop another view — a plan
or a side elevation — all still in the roughest style, without any
regard to finished detail. Information will be growing all the
while, and the problem will be opening up. At this stage it is
probable that the sketch can easily be seen to be wrong in many
respects. Perhaps the arrangement will not do at all.

This is a good sign. It shows that the design is progressing.
It is a valuable thing to know that certain plans cannot be fol-
lowed. Do not rub out part of the sketch already made and try
to remedy it. Begin again. Make another sketch. Sketch paper
is cheap. By and by it may prove to be very desirable to have
that first rough outline available for comparison ; or it may be that
some of its ideas can be applied on other sketches. The second
sketch may "show up" little or no better than the first. Make
another, and another, and another, until the subject is thoroughly
digested. It is wonderful how helpful it is to have some marks
on paper relative to a design, even though they be of the utmost
crudeness. They save imaginative power tremendously; and, even
with them, all available powers of imagination will be needed
before the design is perfected.

A careful comparison of one's sketches, rejecting here, and
approving there, will, little by little, bring about a definite opinion,
and the scale drawing can be begun.

As in the case of the first sketch, so in the case of the first
scale drawing, get some lines on paper as quickly as possible.
Draw something, even if it is nothing more than a straight hori-
zontal line. Do not stare at blank paper for an hour trying to
imagine how the tenth or eleventh line is going to be drawn in
relation to the first line. Do not worry about the later lines
until it is time for them. Draw the first line at once; and, when
the second line is drawn, if the first line proves to be wrong,
make it right. As in the rough sketch, that first horizontal line
is an immense relief from the great waste of blank paper of a
fresh sheet. It is something to look at. It is the beginning of a
detailed design. If it happens not to be the absolutely correct
foundation to build upon, it at least is something to tear down.
The main purpose of these preliminary drawings is to keep the
mind active on the problem; and advance toward the final acdom-

72 MACHINE DESIGN

plishment of the design is often made quite as rapidly by discover-
ing what to tear down as by consistently building up.

When a detail draftsman who has been used to having all his
work laid out for him by an expert designer attempts to take up
original work for himself, he encounters the drawing of that first
line in a way he never did before, lie is apt to worry for some
time over the possible or impossible results of drawing that first
line. If he continue this, he will be sure to fail. The second line
is much easier to draw than the first, and the third than the second;
and the next hundred will follow on in comparatively smooth
sequence, all because of bold action on the first few lines.

And yet, just as the design appears to be progressing smoothly,
and the advanced progress of the drawing seems cause for congratu-
lation, careful consideration may disclose a "snag" not previously
known to exist in the problem. Further study pursued along
the line of this new discovery may show that the whole layout
thus far has been radically wrong, and that a fresh start will have
to be made. At such a time the young designer is apt to feel
that his labor has all been thrown awav, and he becomes discour-
aged. There is, however, no cause for discouragement. Machine
Design might almost be defined to be the "successful elimination
of snags." It takes some ability to discover an obstacle of this
sort; to know a "snag" when an opportunity to see it is given.
It takes a good designer to eliminate such a difiiculty after it has
been found. If there were no "snags" it would not require great
ability to design machines. Many machines fail because in them
there are a lot of undiscovered "snags." Others fail because the
"snags," although discovered, were not eliminated by careful design.

Do not be afraid to make a lot of "first" drawings. It is
just as important to digest the design thoroughly by means of
scale drawings, as it was to digest it originally by means of
the rough sketches. An attempt to make the first drawing of an
original design absolutely right would, it is safe to say, produce a
poor design, one that could be much improved by further trial.
Let the drawings multiply, one after another, until the final one
is reached, in which the perfection of detail will eliminate all tbe
bad points of the preceding drafts and incorporate good ones of its
own based on the study of the others.

MACHINE DESIGN 73

And yet it is often true that the first design laid out, even
after many others have been developed, may be found to possess
features that render a return to it desirable. This is why it is
always better to produce a collection of designs than to attempt
to rub out and work over the first one. The best designers usually
have a great number of sketches showing how to accomplish a
single result. Likewise, they also have a series of layouts to scale,
showing in detailed form the development of their various ideas.
This is because, without a careful consideration of many methods,
they themselves feel incompetent to judge of the best design pos-
sible for accomplishing a given result.

Sketches and origin^ designs should always be dated and
signed. Different designers may be working on the same prob-
lem, and priority of design will never be allowed except upon
signed and witnessed papers. It is embarrassing to find, after
months and perhaps years have passed since an original drawing
was made, that one's rights have been preempted merely because
there was no date or signature to define them.

In redesigning or modifying an existing machine, never make
a change merely for the sake of doing so. Give the good points of
the machine a chance, and devote attention in the new design to
correcting the bad points. It is in bad taste, if it be not actually
childish, to "look wise and suggest a change" in details which
happen to have been designed by another party, but which, never-
theless, are by common engineering judgment pronounced good
for the special work intended. This element of unfair and selfish
criticism has more than a moral bearing. When it is carried into
the superintendence of designing work, it extinguishes. the person-
ality of the subordinate draftsman; his efficiency as an original
thinker is lowered; and narrow designs are produced.

**The best way for a subordinate to dispose of what
appears to be a poor suggestion from a superior, is to work it out
to the best degree possible.'* If it turns out to be good the
credit of working it out belongs to the man who did it. If it is
actually bad, a careful working out will usually develop the fact
beyond dispute, and save unprofitable argument. For the success
or failure of a machine there is only one argument better than
the detail drawings, and that is the machine itself in operation.

76 MACHINE DESIGN

the pnlleys than exists in the middle of the span. This increase
of tension due to the weight of the belt would make but little dif-
ference in the unit-stress in the material of which the belt is made;
hence it may safely be assumed that the tension in the belt when
at rest is uniform throughout its entire length.

When we start to transmit power through the belt by turning
one of the pulleys, thereby driving the other pulley the condition
of stress in the belt is at once materially changed. As the belt is
a flexible member, we can transmit only a pull to the other pulley,
thereby turning it around, the push which is at the same time
given to the other side of the belt merely acting to make the belt
sag or become slack. Hence the immediate effect of starting mo-
tion in a belt is to change the condition of equal tension through-
out its length, to that of unequal tension in the two sides. The
driving side is tight, while the other is loose, the former having
gained as much tension as the latter has lost, and the sum of the
two being practically equal to the sum of the tensions in the two
sides of the belt when at rest. This is not strictly true, as will be
shown later; but it is sufficiently accurate to form a good basis
for the practical design, at least of slow-speed belts.

This condition of tight and slack sides is made possible by
the fact that the belt, in being WTapped around the pulleys under
tension, has friction on their surfaces. Thus, we can pull hard on
one side without slipping the belt around the pulleys, but could
not do this if the pulleys were perfectly smooth or frictionless, for
in that case the slightest pull on one side would slip the belt
around the pulleys. In fact, it would be impossible to produce
any pull by means of the driving pulley, for the pulley would
merely slip around inside the belt.

The amount of pull we can apply to the belt is therefore lim-
ited by the tension at which the belt slips around the pulley.
Moreover, since the force of friction between the belt and pulley
is dependent upon the normal force with which the belt is pressed
against the pulley, and the coefficient of friction between the two,
it is evident that the tighter the belt is laced up, and the rougher
the surfaces of the pulley and belt, the greater is the force that
can be transmitted through the belt. This leads to the conclusion
that it would be possible to transmit any amount of power through

MACHINE DESIGN 77

any belt however small, if the belt were only laced up tight
enough.

This conclusion is literally true; but the important fact now
comes in, that the strength of the material of which the belt iff
made is limited, and while theoretically we might be able to ac-
complish the above, it would be impossible to do so in practice,
for at a certain point the belt would break under the strain. Other
practical considerations also come in, which fix this limit of power
transmission at a point far below the breaking strength of the ma-
terial.

The complete analysis is not quite as simple as the above, es-
pecially for high-speed belts. When the driving side of the belt
becomes tight, it stretches and grows longer; and at the same
time the other side of the belt becomes slack and grows shorter.
But it is not true that the increase in the one side is the same as
the decrease in the other, and this fact produces the condition that
the sum of the tensions in motion is not quite the same as the sum
of the tensions at rest.

Again, when the belt, as it passes around the pulley, changes
its straight-line direction to circular motion, each particle of the
belt — like a body whirling at the end of a cord about a ceLterof
rotation — tends by centrifugal force to fly away from the surface
of the pulley, thereby decreasing the normal pressure, and hence
the friction. This centrifugal force also changes somewhat the
tensions in the belt between the pulleys. As the centrifugal force
increases in proportion to the square of the linear velocity, it is
evident that the effect is greater at high speeds than at moderate
or low speeds.

A further circumstance that affects the driving power of a
belt is the stiffness of the leather or other material of which the
belt is made. As it passes around the pulley, the belt is bent to
conform to the circumference of the pulley, and is again straight-
ened out as it leaves the pulley. Hence the theoretically perfect
action is modified somewhat according to the sharpness of the
bending and the thickness or flexibility of the belt; in other words,
a small pulley carrying a thick belt would be the worst case for
successful calculation on a theoretical basis. '

THEORY. The condition of the tight and loose sides of a

MACHINE DESIGN

belt transmitting power, is similar to that of the weighted strap
and fixed pulley shown in Fig. 17. If motion is desired of the
strap around the pulley, it is necessary to make the weight W, of
such a magnitude that it will overcome not only the weight W„
but also the, friction between the strap and the pulley. The strap
tension T^ is, of course, equal to Wj, and T^ to W,. The equation
showing the balance of forces for the condition when motion is
about to occur, is:

T„ - To = F = P (driving force). (5)

If the pulley be free to turn on its axis, instead of being fixed

as in Fig. 17, the strap by its
friction on the pulley will turn
the pulley, and the force of
friction F becomes the driving
force for the pulley as noted
in equation 5 above.

In Fig. 18, let us sup-
pose that W is a weight repre-
senting the resistance to be
overcome. The tensions T^
and Tq, equal at first owing to
stretching the belt tightly
over the pulleys at rest, change
when an attempt is made to
raise the weight by turning
the larger pulley; and just as
the weight leaves the floor, the
equality of moments about
the axis of the driven pulley
gives the following equation:

Fig. 17.

(T^ -TJ r = F X r = P X r = W X r,. (6)

This equality of moments remains as long as the motion of
the weight is uniform, and represents closely the conditions under
which belt pulleys work.

Although we know from the above what the difference of the
belt tensions is, and what this difference will do when applied to

MACHINE DESIGN

the surface of a given pnlley, we do not yet know what either
T„ or T„ actually is; and until we do know, we cannot correctly
proportion the belt. Hence we must find another relation between
T„ and Tj, which we can combine with equations 5 and 6. This
relation is deduced by a process of higher mathematics, which re-
enlts aa follows:

Common logarithm ^= 2.729 ^ {1 - e)n. (7)

Treating equations 5 and 7 as simultaneous, values of both
Tq and T(, can be found by the regular algebraic solution. As T„
is the larger, the actual area of belt to provide the necessary strength
must be made to depend upon it.

The factors in equation 7 depends upon the centrifugal force

developed by the weight of the belt paasingaround the pulley. Its
value, found from mechanics, is:

«) X V

9,660 Xi:
Having found the maximum pull on the belt, it now remains
to write the equation :

External force ^= Internal resistance;

or, T„ ^ fi X A X «. (8)

Usually the most convenient way to handle this equation is ■
to assume A and t, and then solve for *.

80 MACHINE DESIGN

Summing up the theoretical treatment of belt design, we
simply combine equations 5, 6, 7, and 8, and solve for the quantity
desired. Discussion of the constants involved in these equations,
and of the practical factors controlling them, is given in the fol-
lowing :

PRACTICAL MODIFICATION. The force of friction F, which
is the same as driving force P, depends on:

Coefficient of friction (fi) between belt and pulley;

Tightness of the belt;

Centrifugal force of the belt;

Angle of contact of belt with pulley.

The coefficient of friction (/ut), according to experiments and
observed operation of belts transmitting power, varies from .15 to
.56 for leather on cast iron. An average value consistent with a
reasonable amount of slip, the belt being in good running order,
is .30. If the belt is oily, or likely to become so in use, a lower
value should be taken.

The tighter the belt is drawn up, the greater is the pressure
against the pulley, and hence the greater is the force of friction.
But if we pull the belt up too tightly, when we begin to drive,
Tjj becomes too great, and the belt breaks or is under such stress
that it wears out quickly. Moreover, the great side pressure on
the bearings carrying the shaft produces excessive friction, and the
drive is inefficient. This is why a narrow belt driven at high
speed is more efficient than a wide belt at slow speed, for we can-
not pull up the former as tightly as the latter without overstraining
it, and yet it is possible to get the required power out of the nar-
row belt by running it at high speed.

The centrifugal force is of small importance for low speeds,
say of 3,000 feet per minute and less; and it therefore may usu-
ally be neglected. The factor b then becomes zero in the expres-
sion 1 - 2 in equation 7, and the second member of the equation
stands simply 2.729 X fiX n.

The angle of contact of belt with pulley is important, as a
large value gives a great difference between T^ and T^; and it is
desirable to make this difference as great as possible, because there-
by the driving force is increased. The loose side of a horizontal
belt should always be above, as then the natural sag of the loose

MACHINE DESIGN 81

side due to its slackness tends to increase the angle of contact with
the pulley, while the tightening up of the lower side acts against
its sag to make the loss of wrap as little as possible. Vertical belts
which have the driving pulley uppermost, utilize the weight of the *
belt to increase the pressure against the surface of the pulley, slightly
increasing its capacity for driving. The angle of contact may
be artificially increased by a tightening pulley which presses the
belt further around the pulley than it would naturally lie. It
adds however, the friction of its own bearing, and impairs the effi-
ciency of the drive. For ordinary horizontal belts, the angle of
contact is but little more than 180°, and the value of n in equation
7 may be safely assumed at ^ unless the pulleys are of relatively
great difference of diameter and very close together.

Strength of Leather Belong;. The breaking tensile strength
of leather belting varies from 3,000 to 5,000 pounds per square
inch. Joints are made by lacing, by metal fasteners, or by cement-
ing. The strength of a laced joint may be about -j^, of a metal-
fastened joint, about ^, and of a cemented joint, about equal to
the full strength of the belt cross -sectional area. The proper
working strength of belting depends on the use to which the belt
is put. A continuously running belt should have a low tension
in order to have long life and a minimum loss of time for repairs.
For double leather belting it has been shown that a working ten-
sion of 240 pounds per square inch of sectional area gives an an-
nual cost — for repairs, maintenance, and renewals — of 14 per
cent of first cost. At 400 pounds working tension, the annual ex-
pense becomes 37 per cent of first cost. These results apply to
belts running continuously; larger values may be used where the
full load comes on but a short time, as in the case of dynamos.

Good average values for working tensions of leather belts are:

Cemented joints, 400 pounds per square inch.
Laced joints, 300 " " " "

Metal joints, 250 " " " "

Horse-Power Transmitted by Belting* If P is the driving
force in pounds at the rim of the pulley, and V is the velocity of
the belt in feet per minute, the theoretical horse-power transmitted
is evidently :

82 MACHINE DESIGN

^- ^' ^ apoo • (^)

It is evident from the above that the horse-power of a belt de-
pends upon two things, the driving force P and the velocity V. If
either of these factors is increased, the horse-power is increased.
Increasing P means a tight belt. Hence a tight belt and high
speed together give maximum horse-power. But a tight belt
means more side strain on shaft and journal. Therefore, from the
standpoint of efficiency, use a narrow belt under low tension at as
high a speed as possible.

Empirical rules for horse-power of belting, if used with judg-
ment, give safe results when applied to very general cases. A
common rule used by American engineers is:

^- ^' = i;ooo • ('®>

For a double belt, assuming double strength, this%becomes:

With large pulleys and moderate velocities, this may hold
good. With small pulleys and high velocities, however, the un-
certain stresses induced by the bending of the fibers of the belt
around the pulley, and the relatively great loss due to centrifugal
force, modify this relation* and a safer value for a double belt of
the ordinary kind is:

^- ■^- "" "liO" ' ('^)

or, still safer, H. P. = -t^qq— (13)

If we compare the theoretical value of equation 9 with the
empirical value of equation 10 by putting them equal to each
other, thus:

H P _ FXV bxY
' ' 38,000 " 1,000 •

and solve for P, we get :

MACHINE DESIGN 83

P = 33i. (i4)

This develops the fact that the empirical rule of equation 10 as-
Slimes a driving force of 33 pounds per inch of width of single
belt.

Another way of expressing equation 10 is: A single belt
will transmit one horse-power for every inch of width at a belt
speed of 1,000 feet per minute.

5peed of Belting^. The most economical speed is somewhere
between 4,000 and 5,000 feet per minute. Above these values
the life of the belt is shortened; also "flapping," "chasing," and
centrifugal force cause considerable loss of power. The limit of
speed with cast-iron pulleys is fixed at the safe limit for bursting
of the rim, which may be taken at one mile per minute.

Material of Belting^. Oak-tanned leather, made from the
part of the hide which covers the back of the ox, gives the best re-
sults for leather belting. The thickness of the leather varies
from .18 to .25 inch. It weighs from .03 to .04 pound per cubic
inch. The average thickness of double leather belts may be taken
as .33 inch, although a variation in thickness from -J- inch to -^
inch is not uncommon. Double leather belts may be ordered
light, medium, or heavy.

In a single-thickness belt the grain or hair side should be
next to the pulley, for the flesh side is the stronger and is there-
fore better able to resist the tensile stress due to bending set up
where the belt makes and leaves contact with the pulley face.
Double leather belts are made by cementing the flesh sides of
two thicknesses of belt together, leaving the grain side exposed
to surface wear.

Kaw hide and semi-raw hide belts have a slightly higher co-
efficient of friction than ordinary tanned belts. They are useful in
damp places. The strength of these belts is about one and one-
half times that of tanned leather.

Cotton, cotton -leather, rubber, and leather link belting are
some of the forms on the market, each of which is especially
adapted to certain uses. For their weights and their tensile and
working strengths consult the manufacturers' catalogues.

A prominent manufacturer's practice in regard to the sizes of

MACHINE DESIGN

leather belting will be found useful for comparison, and is indicated
in the table on page 12.

Initial Tension in Belt. On the assumption that the sum of
the tensions is unchanged, whether the belt be at rest or driving,
we should have the following relation :

whence.

T =

(15)

This is not strictly true, however, as is stated in the " Analysis
of " Belts." It has been found that in a horizontal belt working at
about 400 lbs. tension per square inch on the tight side, and hav-
ing 2 per cent slip on cast-iron pulleys ( i. «., the surface of the

Sizes of Leather Belting.

THICKNESS.

DTH.

Single.

Double.

1 inch.

inch.

■

7
16

driven pulley moving 2 per cent slower than that of the driver ),
the increase of the sum of the tensions when in motion over the
sum of the tensions at rest, may be taken at about ^- the value of
the tensions at rest. Expressing this in the form of an equation

A fi V T

T =

(T„ + T„).

(i6)

MACHINE DESIGN 85

The value of T thus found would be the pounds initial tension to
which the belt should be pulled up when being laced, in order to
produce T^ and T^ when driving.

This value is not of very great practical importance, as the
proper tightness of belt is usually secured by trial, by tightening
pulleys, by pulley adjustment (as in motor drives), or by shorten-
ing the belt from time to time as needed. It is worth noting,
however, that for the most economical life of the belt it would be
very desirable in every case to weigh the tension by a spring bal-
ance when giving the belt its initial tension. This, however, is
not always easy or even feasible; hence it is a refinement with
which good practice usually dispenses, except in the case of large
and heavy belts.

PROBLEMS ON BELTS.

1. Determine the belt tensions in a laced belt transmitting 50
horse-power at a velocity of 3,500 feet per minute. Suppose that
the arc of contact is 180^; weight of belt = .035 pound per cub.
in.; and coefficient of friction 25 per cent.

2. What is the width of above belt if it is -^^ inch in thick-
ness ?

3. What initial tension must be placed on above belt ?

4. The main drive pulley of a 120-horse-power water wheel
is 6 feet in diameter. A cemented leather belt is to- connect the
main pulley to a 3-foot pulley on the line shafting in a mill. The
horizontal distance between centers of shafting is 24 feet; coeffi-
cient of friction, 30 per cent; re volutions, per minute of line shaft-
ing, 180. Design the belt for this drive.

5. An 8 -inch double belt | inch thick connects 2 pulleys of
30-inch and 20-inch diameter respectively. The horizontal dis-
tance between the centers is 12.5 feet. The coefficient of friction
is 0.3, and the weight of belt per cubic inch is 0.035 pound.
Working tension, 300 pounds per square inch. Speed of belt
5,000 feet per minute. Lower face of 30-inch pulley is the driv-
ing face. Required the H. P. which may be transmitted (theo-
retically).

6. Compare the theoretical horse-power in problem 5 with
that obtained by the use of empirical formula.

86 MACHINE DESIGN

PULLEYS.

NOTATION— The following notation is used throughout the chapter onPollayi:

A =Area of rim (sq. in.). I =Length of hub (inches).

a = *» **arm(" "). N = Number of arms.

b = Center of pulley to center of belt n = ** " rim bolts, each side.

(inches; practically equal to R). P =DriTing force of belt (lbs.).

Ci=Totalcentrifugalforceof rim (lbs.). Pi = Force at circumference of shaft
c = Distance from neutral axis to outer (lbs.).'

fiber (inches) . P2=/Force at circumference of hub (lbs.) .

D = Diameter of pulley (inches) . p = Stress in rim due to centrifugal force
Di= ** "hub ( '• ). (lbs. per sq. in.).

di = " " bolt at root of thread R =Radius of pulley (inches).

(inches). S = Fiber stress (lbs. per sq. in.).

d =Diameter of bolt holes (inches). « =Fiber stress in flange (lbs. per sq. in.).

g = Acceleration diie to gravity (ft. T = Thickness of web (inches).

per sec.). t = *' ** rim ( *• ).

h = Width of arm at any section (inches). fs = ** ** ** bolt flange (inches).

I = Moment of inertia. T n= Tension of belt on tight side (lbs.) .

L =Length of arm, center of belt to hub To= " " " "loose " ( " ).

(inches). v =Yelocity of rim (ft. per sec.).

Li= Length of rim flange of split pulley to =Weightof material Qbs. per cub. in.).

(inches).

ANALYSIS. If a flexible band be wrapped completely about
a pulley, and a heavy stress be put upon each end of the band, the
rim of the pulley will tend to collapse just like a boiler tube with
steam pressure on the outside of it. A compressive stress is in-
duced which is very nearly evenly distributed over the cross-sec-
tion of the rim, except at points where the arms are connected
thereto. At these points the arms, acting like rigid posts, take
this compressive stress. Now, a pulley never has a belt wrapped
completely round it, the fraction of the circumference embraced by
the belt being usually about ^, and seldom, even with a tightener
pulley, reaching £. Assuming the wrap to be ^ the circumference,
and that all the side pull of the belt comes on the rim, none being
transmitted through the arms to the hub, we then have one-half of
the rim pressed hard against the other half by a force equal to the
resultant of the belt tensions, which, in this case, would be the
sum of them. Dividing the pulley by a plane through its center
and perpendicular to the belt, the cross-section of the rim cut by
this plane has to take this compressive stress- >

This analysis is satisfactory from an ideal standpoint only, for
the intensity of stress due to the direct pull of the belt, with the
usual practical proportions of rim, would be very small. More-
over, the element of speed has not been considered.

When the pulley is under speed, a set of conditions which

MACHINE DESIGN 87

complicates matters is introduced. The centrifugal force due to
the 'Weight of the rim and arms is no longer negligible, but has
an important influence upon the design and material used. This
centrifugal force acts against the effect of the belt wrap, tending
to reduce the compressive stress, or, overcoming the latter entirely,
sets up a tensional stress both in the rim and in the arms. It also
tends to distort the rim from a true circle by bowing out the rim
between the arms, thus producing a bending moment in the rim,
maximum at the points where the rim joins each arm.

It can readily be imagined that the analysis in detail of these
various stresses in the rim acting in conjunction with each other
is quite complicated — far too much so in fact, to be introduced
here. As in most cases of such design, however, one controll-
ing inijuence can be separated out from the others, and the de-
sign based thereon with suflBcient margin of strength to satisfy
the more obscure conditions. This is rational treatment, and the
" theory '' will be studied accordingly.

The rim, being fastened to the ends of the arms, tends, when
driving, to be sheared off, the resisting area being the areas of the
cross-sections of the arms at their point of joining the rim. The
force that produces this shearing tendency is the driving force of
the belt, or the difference between the tensions of the tight and
loose sides.

Again, at the point of connection of the arms to the hub, a
shearing action takes place, so that, if this shearing tendency were
carried to rupture, the hub would literally be torn out of the arms.
Now, viewing the arms as beams loaded at the end with the driv-
ing force of the belt, and fixed at the hub, a heavy bending stress
is set up, which is maximum at the point of connection to the
hub. If the rim were stiff enough to distribute this driving force
equally between the arms, each arm would take its proportional
share of the load. The rim, however, is quite thin and flexible;
and it is not safe to assume this perfect distribution. It is usual
to consider that one-half the whole number of arms take the full
driving force.

THEORY — Pulley Rim. Evidently it is practically impossible
to make so thin a rim that it will collapse under the pull of a belt.
As far as the theory of the rim is concerned, its proportion prob-

MACHINE DESIGN

ably depends more upon the calculation for centrifugal force than
upon anything else. •

In order to separate this action from that of any other forces,
let us suppose that the rim is entirely free from the arms and hub,
and is rotating about its center. Every particle, by centrifugal
force, tends to fly radially outward from the center. This condi-
tion is represented in Fig. 19. The tendency with which one-half
of the rim tends to fly apart from the other is indicated by the
force Cj', and the relation between C, and the small radial force c
for each unit-length of rim can readily be found from the prin-
ciples of mechanics. The case is exactly like that of a boiler or a
thin pipe subjected to uniform internal pressure, which, if carried
to rupture, would split the rim along a longitudinal seam.

Fig. 19. Fig. 20.

The tensile stress thus induced per square inch can be found
by simple mechanics to be:

P =

(i7)

or, since w = 0.26 pound, and g = 32.2 feet per second,

J) ^ 0.097 v' ( say-ryr) ;

(i8)

and. Up be taken equal to 1,000 pounds per square inch, which is
as high as it is safe to work cast iron in this place,

V = 100 feet per second. ('9)

This shows the curious fact that the intensity of stress in the rim

MACHINE DESIGN

IB directly proportional to the square of the linear velocity, and
wholly independent of the area of crosa- section. It is also to be
noted that 100 feet per second is abont the limit of speed for cast-
iron pulleys to be safe against bursting.

If we wish to consider theoretically the rim together with the
anne as actually connected to it, we get a much more complicated
relation. This condition is shown in Fig. 20, where the rim, ex-
panding more than the arms, bulges oat between them. This
makes the rim act something like a contiauons beam uniformly
loaded; but even then the resulting stress is not clearly defined on
account of the variable stretch in the arms. Investigation on this
basis is not needed further than to note that it is theoretically
better, in the case of a split pulley, to make the joint close to the
arms, rather than in the middle of a span.

Pulley ArinS' The centrifugal force developed by the rim
and arms tends to pull the arms from the hub. On the belt side,
this is balanced to some extent by the belt wrap, which tends to
compress the arm and relieve the tension. On the side away
from the belt, the centrifugal action has
full play, but the arm is usually of such
cross-section that the intensity of this stress
is very low. It may safely be neglected.

The rim being very thin in most cases,
its distributing effect cannot be depended
on, hence the driving force of the belt may
be taken entirely by the arms immediately
under the portion of the belt in contact with
the pulley face. For a wrap of 180° this
means that only one-half of the pulley arms can be considered as
effective in transmitting the turning effort to the hub. Each of
these arms is a lever iixed at one end to the hub and loaded at the
other. A lever of this description is called a " cantilever " beam,
its maximum moment existing at its fixed end. The load that each

. P

of these beanis may be subjected to is-j^, and therefore the maxi-

2~
2FL

Pig. 21.

mnm external moment at the hub is -

From mechanics we

90 MACHINE DESIGN

know that the internal moment of resistance of any beam section
is — , and that equilibrium of the beam can be satisfied only

when the external moment is equal to the internal moment of re-
sistance of the beam section. Equating these two, we have:

2PL SI , .

■TT =V (^^)

The arms of a pulley are usually of the elliptical or segmental
cross-section, and may be of the proportions shown in Fig. 21.

For either of these sections the fraction — is approximately equal

to 0.0893A'. For convenience (the error caused being on the safe

side), L may be taken as equal to the full radius of the pulley B,

whence

2PR_ 2(Tn-T„)R _ Q ogggg^.^ ^^,^

in which S may be from 2,000 to 2,250 for cast iron

Taking moments about the center of the pulley, and solving
for P„ the force acting at the circumference of the hub, we have :

2PE _ P^,

~F~~ 2 '

The area of an elliptical section is ^ times the product of the
half axes. With the proportions of Fig. 21, this becomes:

a = n X 0.2A X 0.5A = nh\ (23)

Equating the external force to the internal shearing resistance, we
have :

4PR , ..

MACHINE DESIGN

in which the shearing stress Sg may run from 1,500 to 1,800 for
cast iron.

Although both bending and shearing stresses as calculated
above exist at the base of the arms, the bending is, in practically
every case, the controlling factor in the design of the arms. An
arm-section large enough to resist bending would have a very low
intensity of shear.

If the number of arms be increased indefinitely, we come to
a continuous arm or web, in which the bending action is elimi-
nated. It may still shear off at the hub, where the area of metal
is the least, at minimum circumference. In this case the area
under shearing stress is ^DjT; and the force at the circumference
of the hub, as before, is :

P.=

4PK
ND,

Fig. 22.

Equating external force to in-
ternal shearing resistance, we
have :

4PK _ __
jfp =7rDjT8g;

Pulley Hub. As in the
case of the arms, centrifugal
force does not play much part in the design of the hub of a pulley.
The hub is designed principally to carry the key, and through it
transmit the turning moment to the shaft. Considered thus, the
hub may tear along the line of the key or crush in front of the key.

For example, in Fig. 22, if the connection with the lower
arms be neglected, and the upper arms be held fast while a turning
force P„ at the surface of the shaft, is transmitted to the hub
through the key, then the metal of the hub directly in front of the
key is under crushing stress; and the metal along the line eb^ from
the comer to the outside, is under tensile stress. This condition ia
the worst that could possibly happen, because the bracing effect of
the lower arms has been neglected, and the key is located between
the arms.

92 MACHINE DESIGN

Taking moments about the center of the shaft, the value of the
force at the shaft circumference, or the "key pull," is:

P k
Now-p^ = — , k being the distance from the center of shaft to

center of eb, and the area of metal which is subjected to the tearing
action P, is Z X eb. Equating the external force to the internal
resistance, and assuming that the stress is equally distributed over
the area I X eb^ we have:

r ?• PR

P, = -^P, = -^ X -;r- = S X Z X « J/

The intensity of crushing on the metal in front of the key, due
to force Pj, depends upon the thickness of the key, and is properly
discussed later under "Keys."

PRACTICAL MODIFICATION— Pulley Rim. The theoretical
calculation for the thickness of the rim may give a thickness that
could not be cast in the foundry, and the section in that case will
have to be increased. As light a section as can be readily cast will
usually be found abundantly strong for the forces it has to resist.
A minimum thickness at the edge of the rim is about -j^^ inch;
and as the pulleys increase in size, the rim also must be made
thicker; otherwise the rim will cool so much more quickly than
the arms, that the latter, on cooling, will develop shrinkage cracks
at the point of junction.

For a velocity of 6,000 feet per minute, we find from equation
18 that the tension in pounds per square inch, in the rim, due to
centrifugal force, is 970. Though this in itself is a low value, yet
the uncertain nature of cast iron, its condition of internal stress,
due to casting, and the likely existence of hidden flaws and pockets,
have established the usage of this figure as the highest safe limit
for the peripheral speed of cast-iron pulleys. It is easily remem-
bered that cast-iron pulleys a7*e safe for a linear velocity of about
one rmle per minute.

MACHINE DESIGN

To prevent the belt from running off the pulley, a "crown"
or rounding surface is given the rim. A tapered face, which is
more easily produced in the ordinary shop, may be used instead.
This taper should be as little as possible, consistent with the belt
staying on the pulley; ^ inch per foot each way from the center
is not too much for faces 4 inches wide and less; while above this
width J inch per foot is enough. As little as J inch total crown
has been found to be sufficient on a 24-inch face, but this is
probably too little for general service.

Instead of being "crowned," the pulley may be flanged at the
edges; but flanged pulley rims chafe and wear the edge of the belt.

The inside of the rim of a cast-iron pulley should have a taper
of J inch per foot to permit easy withdrawal from the foundry

y////////////////////j^//7^^//^^^^

wzzzsm^

Fig. 23.

mould. This is known as "draft." If the pattern be of metal, or
if the pulley be machine-moulded, the greater truth of the casting
does not require that the inside of the rim be turned, as the pulley,
at low speeds, will be in sufficiently good balance to run smoothly.
For roughly moulded pulleys, and for use at high speeds, however,
it is necessary that the rim be turned on the inside to give the
pulley a running balance.

Fig. 23 shows a plain rim a also one stiffened by a rib h.
Where heavy arms are used this rib is essential so that there will
not be too sudden change of section at the junction of rim and arm,
and consequent cracks or spongy metal.

Pulley Arms. The arms should be well fiUetted at both rim
and hub, to render the flow of metal free and uniform in the mould.
The general proportions of arms and connections to both hub and
rim may perhaps be best developed by trial to scale on the draw-
ing board. The base of the arm being determined^ it may gradu-

94 MACHINE DESIGN

ally taper to the rim, where it takes about the relation of § to J
the dimensions chosen at the hub. The taper may be modified
until it looks right, and then the sizes checked for strength.

Six arms are used in the great majority of pulleys. This
number not only looks well, but is adapted to the standard three-
jawed chucks and common clamping devices found in most shops.
Elliptical arms look better than the segmental style. The flat,
rectangular arm gives a very clumsy and heavy appearance, and is
seldom found except on the very cheapest work.

A double set of arms may be used on an excessively wide
face, but it complicates the casting to some extent.

Although a web pulley may be calculated for shear at the
hub, yet it will usually be found that with a thickness of web in-
termediate between the thickness of the rim and that of the hub,
which will satisfy the casting requirements, the requirements as to
strength will be fully met.

Pulley Hub. The hub should have a taper of ^ inch per foot
draft, similar to that of the inside of the rim. The length of the
hub is arbitrary, but should be ample to prevent rocking on the
shaft. A common rule is to make it about | the face width of
the pulley.

The diameter of the hub, aside from the theoretical consider,
ation given above, must be sufficient to take the wedging action of a
taper key without splitting. This relation cannot well be calcu-
lated. Probably the best rule that exists is the familiar one that
the hub should be twice the diameter of the shaft. This rule,
however, cannot be literally adhered to, as it gives too small hubs
for small shafts and too large ones for large shafts. It is always
well to locate the key, if possible, underneath an arm instead of
between the arms, thus gaining the additional strength due to the
backing of the arm.

SPLIT PULLEYS.

ANALYSIS and THEORY. The split pulley is made in
halves and provided with bolts through flanges and bosses on the
hub for holding the two halves together. When the pulley is in
place on the shaft, bolted up as one piece, it is subjected to the
same forces as the simple pulley. Hence its general design fol-

MACHINE DESIGN

lov/s the same principles, and we need only study the fastening of
the two halves, and the effect of this fastening on the detail of rim
and hnb.

The simplest stress we have to consider on the rim bolts is
one of pure tension, due to the centrifugal force of the halves
of the pulley, A safe assumption to make is that the rim is free

Pig. 24.

from the arms and hub, as in the simple pulley, and that the oen-
trifogsl force developed by it has to be taken by the rim bolta
alone. In other words, consider the rim bolta as belonging en-
tirely to the rim, and make them as strong as the rim, leaving the
hub bolts to take the centrifugal force of the arms and hub, and
the spreading tendency due to the key.

Another tensile stress is induced in the rim bolts by the fact,
that, having made an open joint in the rim, and in addition placed
the extra weight of lugs there, the centrifugal action at this point
is increased, and at the same time a point of weakness in the rim

MACHINE DESIGN

iotrodcced. Beferring to Fig. 24, the rim flanges EJ tend to fly
out due to the centrifngal force Cp. This tends to open the joiot
J at the outside of the rim ; to throw a bending stress od the rim,
niaxinium at the point F ; and to "heel" the rim flanges about
the point £. The rim bolts acting on the leverage e about the
point E must resist these tendencies, and are thereby put in
tension.

Beferring to equation 18, we find the intensity of etrese due to
the centrifngal force of the rim in lbs. per square inch to be :

If A is the sectional area of the rim in square inches, this i

that the total strength of

the rim is represented by

Aw"
-r^. The strength of a

bolt is represented by the

there are n bolts in the

flange, the total resisting

. , , , , , n^vd^

force of the bolts is — T"'

and the equation represent-
ing equality of strength be-
tween rim and bolts is :

To' ^"T"'

from which, by a proper
assumption of the fiber
stress S, which should be

(28)

Fig. ^

low, the opening-up tendency of the joint being neglected, the diam-
eter at the root of the thread rf, may be calculated, and the nom-
inal bolt diameter chosen. Eeference to the table for strength
of bolts, given in the chapter on Bolts, Studs, etc., will be found
convenient.

MACHINE DESIGN 97

It is very doubtful if the tension on the flange bolts, due to
the " heeling" about E can be calculated with sufficient accuracy to
be of much value. It is probably better to assume S at a low value,
say 4,000, and, in addition, for large and high-speed pulleys, stif-
fen the rim by running a rib between the flange and the adjacent
arm. It is evident that if we make the rim so stiff that it cannot
deflect, there will be no " heeling " about E ; and the bolts will
be well proportioned by the preceding calculation, giving them
equal strength to that of the rim section.

For the bolt flange itself, any tendency to open at the joint J
would cause it to act like a beam loaded at some point near its
middle with the bolt load, and supported at J and E. This
condition is shown in Fig. 25. Probably the weakest section
would be along the line of the bolt centers. We have just noted

that the carrying capacity of the bolts is . ^ . Hence, assum-
ing that e = ^fj which is about the worst case which could hap-
pen, we have a beam of length/* loaded at the middle with — j— i-

and supported at the ends. Equating the external moment to
the internal moment, we have :

—4- ^ x = 6 ' ^^^>

from which the fiber stress % in the flange may be calculated and
judged for its allowable value.

Lj maybe assumed a little narrower than the pulley face; and
t^ from 1 inch to 2 inches or more, depending on the thickness of
the rim.

The hub bolts doubtless assist the rim bolts in preventing
the halves of the pulley, from flying apart. They also clamp the
hub tightly to the shaft, preventing any looseness on the key.
Their function is a rather general one; and the specific stress
which they receive is practically impossible to calculate. As a
matter of fact, if the hub bolts were left out entirely, the pulley
would still drive fairly well, but general rigidity and steadiness
would be impaired. Hence the size of the hub bolts is more a
practical question than one involving calculation. The rim bolts

98 MACHINE DESIGN

should be figured first, and their size determined on ; then the hub
bolts can be judged in proportion to the rim bolts, the diameter of
shaft, the thickness and length of the hub, and the general form
of the pulley. Often appearance is the deciding factor, it being
manifestly inconsistent to associate small fastenings with large
shafts or hubs, even though the load be actually small.

PRACTICAL MODIFICATION. Practical considerations are
chiefly responsible for the location o^ the joint in a split pulley
between the arms instead of directly at the end of an arm, where
theoretically it would seem to be required. It is usually more
convenient in the foundry and machine shop to have the joint be-
tween the arms; so we generally find it placed there, and strength
provided to permit this. It is possible, however, to provide a
double arm, or a single split arm, in which case the joint of the
pulley comes at the arm, and the " heeling " action of the rim
flanges is prevented.

The rim bolts should be crowded as close as possible to the
rim in order to reduce the stress on them, and also the stress in
the flange itself. The practical point must not be forgotten, how-
ever that the bolts must have suflicient clearance to be put into
place beneath the rim.

While it is evident that the rim bolts are most effective in
taking care of the centrifugal action of the halves, yet in small
split pulleys it is quite common to omit the rim bolts and to
use the hub bolts for the double purpose of clamping the shaft
and holding the two halves together. The pulley is cast with its
rim continuous throughout the full circle, and it is machined in
this form. It is then cracked in two by a well-directed blow of a
cold chisel, the casting being especially arranged for this along the
division line by cores so set that but a narrow fin of metal holds
the two parts together. This provides suflScient strength for cast-
ing and turning, but permits the cold chisel to break the conneo-
tion easily.

SPECIAL FORilS OF PULLEYS.

The plain cast-iron pulley has been used in the foregoing
discussion as a basis of design. A pulley is, however, such a
common commercial article, and finds such universal use, that

MACHINE DESIGN 99

special forms, which can be bought in the open market, are not
only cheaper but better than the plain cast-iron pulley, at least for
regular line-shaft work.

Cast iron is a treacherous and uncertain material for rims of
pulleys. It is not well suited to high fiber stresses; hence the range
of speed permissible for pulley rims of cast iron is limited. Steel
and wrought iron, having several times the tensional strength of
cast iron, and being, moreover, much more nearly homogeneous
in texture, are well suited for this work; one of the best pulleys on
the market consists of a steel rim riveted to a cast-iron spider.
Such an arrangement combines strength and lightness, without
increasing complication or expense.

The all-steel pulley is a step further in this direction. Here
the rim, arms, and hub are each pressed into shape by specially
devised machinery, then riveted and bolted together. This pulley
is strictly a manufactured article, which could not compete with the
simpler forms unless built in large quantities, enabling automatic
machinery to be used. Large numbers of pulleys are built in this
way, and are put on the market at reasonable prices.

Wood -rim pulleys have been made for many years, and,
except for their clumsy appearance, are excellent in many respects.
The rim is built up of segments in much the same way as an ordi-
nary pattern is made, the segments being so arranged that they
will not shrink or twist out of shape from moisture. The hubs
may be of cast iron, bolted to wooden webs, and carrying hard-
wood split bushings, which may be varied in bore vdthin certain
limits 80 as to fit di£Ferent sizes of shafting. The wooden pulley
is readily and most often used in the split form, thus enabling it
to be put in position easily at any point of a crowded shaft. It is
often merely clamped in place, thus avoiding the use of keys or
set screws, and not burring or roughening the shaft in any way.

PROBLEMS ON PULLEYS.

1. Calculate the tensile stress due to centrifugal force in
the rim of a cast-iron pulley 30 inches in diameter, at 500 revolu-
tions per minute.

2. The driving force of a belt on a 36.inch pulley is 800
lbs., and the belt wrap about 180°. Calculate proportions of el-

100

MACHINE DESIGN

liptical arms to resist bending, the allowable fiber stress being
2,000.

3. A pulley 12 inches in diameter, |-inch web, 4-inch diam-
eter hub, transmits 25 horse-power at a belt speed of 3,000 ft.
per minute. Calculate the maximum shearing stress in the web.

4. In Fig. 24 assume the following data: L, = 7 inches;
^a= 1 inch; e = l^ inches; /= 3 inches; area of rim = 8 sq.
in.; allowable tensile stress in rim 1,000 lbs. per sq. in. Calculate
the diameter of the rim bolts.

5. Calculate the fiber stress in the rim bolt flange along the
line of the bolts.

SHAFTS.

NOTATION— The foUowing notation is used throoffhout the chapter on Shafts :

AA=An8ralar deflection (degrees).

B =Simple bending moment (inch-lbs.).

B«= Equivalent bending moment (inch-
lbs.).

e = Distance from neutral axis to outer
fiber (inches).

d, do, d2t dst d4= Diameters of shaft
(inches).

di= Internal diameter of shaft (inches).

E= Direct modulus of elasticity (a
ratio).

e = Transverse deflection (inches).

0=Transyerse modulus of elasticity (a

ratio).

H=Horse-power (33,000 ft.-lbs. per min-
ute).

I = Moment of inertia.

K=: Distance between bearings (inches).

L =Liength along shaft (inches).
Li, L2=Liength of bearings (inches.)
M = Distance between bearings (feet.)
N = Number of revolutions per minnte.
P ^Driving force of belt (lbs.).
Pi = Load applied as stated (lbs.).

R = Radius at which load as stated acts

(inches).
S = Fiber stress, tension, oompressioii»

or shearing (lbs. per sq. in.).
T = Simple twisting moment (inch-lbs.).
Te=Equivalent twisting moment (ineh-

Ibs.).
Tn=Tension in tight side of belt (lbs.)«
To=Tension in loose side of belt (lbs.).
W=Load applied as stated (lbs.).

ANALYSIS. The simplest csLse of shaft loading is shown in
Fig. 26. The equal toTcea W, similarly applied to the disc at the
distance B from its center, tend to twist the shaft o£F, the tendency
being equal at all points of the length L between the disc and the
post, to which the shaft is rigidly fastened. The fastening to the
post, of course, in this ideal case, takes the place of a resisting
member of a machine. A state of pure torsion is induced in the
shaft; and any element, such as ea^ is distorted to the position ob^
acib being the angular deflection for the distance L.

The case of Fig. 27 is illustrative of what occurs when a belt
pulley is substituted for the simple disc. Here the twisting action
is caused by the driving force of the belt, which is T^ - T^ = P,

MACHINE DESIGN

101

acting at the radius R. Torsion and angular deflection exist in
the shaft, as in Fig. 26. In addition, however, another stress of
a different kind has been introduced; for not only does the shaft
tend to be twisted off, but the forces T^ and T^ , acting together,
tend to bend the shaft, the bending moment varying with every
section of the shaft, being nothing at the point o, and maximam
at the point c. This combined action is the most common of any
that we find in ordinary machinery, occurring in nearly every case
with which we have to deal.

In Fig. 27, if the forces T„ and T^ be made equal, there will
be no tendency at all to twist off the shaft, but the bending will
remain, being maximum at the point e. This condition is iUastra-
tive of the case of all ordinary pins and studs in machines. In
this sense, a pin or a stud is sim-
ply a abaft which is fixed to the
frame of the machine, there be*
ing no tendency to turning of the
pin or stud itself. The same
condition would be realized if
the disc in Fig. 27 were loose
upon the shaft. In that case,
the bending moment would be
caused by Tq + T^ acting with
the leverage L, Of course there
would have to be some resistance
for T^-T^ to work against, in
order that torsion should not be
transmitted through the shaft.
This condition might be intro-
duced by having a similar disc
lock with the first one by means
of lugs on its face, thus receiving and transmitting the torsion.

If the distance L becomes very great, both the angular defleo.
tion due to twisting, and the sidewise deflection due to bending,
become excessive, and not permissible in good design. This
trouble is remedied by placing a bearing at some point closer to
tlie disc, which, as it decreases L, of course, decreases the bending
moment and therefore the transverse deflection. The angular de-

Pig. 26.

102

MACHINE DESIGK

flection can be decreased only by bringing the resiatance and load
nearer together.

The above implies, of course, that the diameter. of the ehftft ia DOt
eluuiged. It being obvious that increase of diameter means increase of itreiig^th
and corresponding decrease of both angular and tranaverse deflection.

If the speed of the shaft be very high, and the distance be-
tween bearings, represented by L, be very great, the shaft will take
a shape like a bow string when it is vibrated, and smooth action
cannot be maintained.

It is necessary to carry the cases of Figs. 26 and 27 bnt a

Pig. 27.

single step farther to illustrate the actual working conditions of
shafting in machines. Suppose the rigid post to hare the shaft
passing clear through it, and to act as a bearing, so that the shaft
can freely rotate in it, the resistance being exerted somewhere be-
yond. The twisting moment will be unchanged, also the bending
moment; but the effect of the bending moment will be on each
particle of the shaft in Bncceesion, now putting compression on a
given particle, and then tension, then compresBion ^ain, and so
on, a complete cycle being performed for each revolution. This

" MACHINE DESIGN 103

brings out a very important difference between the bending stress
in pins and the bending stress in rotating shafts. In the one case
the bending stress is non -reversing; in the other, reversing; and
a much higher fiber stress is permissible in the former than in the
latter.

THEORY — Simple Torsion. In the case of simple torsion
the stress induced in the shaft is a shearing one. The external
moment acts about the axis of the shaft, or is a polar moment;
hence in the expression for the moment of the internal forces, the
polar moment of inertia must be used. Now, from mechanics we
have:

I d?
and ' — "^"kT" (^^^ circular section of diameter d)\

therefore, T = -^-=-5 (3^)

from which the diameter for any given twisting moment and fiber
stress can readily be found.

For a hollow shaft this expression becomes:

T=?a-4a. (3.)

Simple Bending. The stresses induced in a pin or shaft under
simple bending are compression and tension. The external moment
in this case is transverse, or about an axis across the shaft; hence
the direct moment of inertia is applicable to the equation of forces.

X. SI

B= — ;

I cP
and — = tt™ (for circular section of diameter rf);

therefore, ^^102' (3^)

For a hollow shaft or pin this expression becomes:

Combined Stresses. In the greater number of cases met with

104

MACHINE DESIGN

in practice, we find two or more simple stresses acting at the same
timie, and, although the shaft may be strong enough for any one of
them alone, it may fail under their combined action. The most
common cases are discussed below.

Tension or Pressure Combined witli Bending:. In Fig. 28,
the load W produces a tension acting over the whole area of dy due
to its direct pull. It also produces a bending action due to the
leverage R, which puts the fibers at B in tension and those at the
opposite side in compression. It is evident, therefore, that by
taking the algebraic sum of the stresses at either side we shall
obtain the net stress. It is also evident that the greatest and

Fig. 28.

Fig. 29

controlling stress will occur on the side where the stresses add, or
on the tension side. Hence, from mechanics.

or.

S =

Also,

WR =

(due to direct tension). (3^)

or.

S =

10.2'
10.2 WE

(due to bending). (35)

Hence the combined tensional stress acting at the point B, or, in

MACHINE DESIGN

fact, at any point on the extreme ontside of the vertioal shaft to-
ward the force W, is:

4W , 10^ WB

(36)

If Wftcted in the opposite direction, the greatest stress would
BtiU be at the side B, but weald be a compressioti instead of a too*
sioD, of the same magnitude as before.

Tension or Compression Combined with Torsion, In Fig. 29,
V might be the end load on a vertical ehaft; and the two forces W
might act in conjunction with it ae in the case of Fig. 26, at the
radins K. This case is not very often met with. It is nsaally
possible to combine the moments, find an equivalent moment of a
simple kind, and use the corresponding simple fiber stress. Id the
case in qnestion we have a direct stress to be combined with a
shearing stress, and mechanics gives us the following solution:

Fig. 30.

Let 83 = simple shearing stress (lbs. per sg in.).
Let Be = simple compressive stress (lbs. per aq. in.).
Let S„= resultant shearing stress (lbs. per sq. in.).
Let &„=resultaDt compressive stress (Ibe. per sq. in.).
We then have ;

^ 5.1 '

2TVTl =

S,=

6.1(2WR)

(37)

106 MACHINE DESIGN

8c = ^V. (38)

Now, from a solution given in simplest form in " Merriman's
Mechanics" — which the student may consult, if desired — values
for the resultant stresses can be found. Whichever of these is
the critical one for the material used, should form the basis for its
diameter:

= \|Ss'+T- (39)

Also, s„ =-1 + ^83'+^ (40)

Benditis: Combined with Torsion. In Fig. 30, the load W
acts not only to twist the shaft off, but also presses it sidewise
against the bearing. As it is usually customary to figure the
maximum moment as taking place at the center of the bearing,
the length L, which determines the bending moment, is taken to
that point. The theory of the stress induced in this case is com-
plicated. In order to make the magnitude of the moments clearer,
let us introduce the two equal and opposite forces F and P, each
equal to W, at the point C. We can evidently do this without
changing the equilibrium of the shaft in any way. We now see
that W and F^ act as a couple giving a twisting moment WR ;
and that F acts with a leverage L, producing a bending moment
FL = WL, at the middle of the bearing.

If, now, we find an equivalent twisting moment, or an equiv-
alent bending moment, which would produce the same effect on
the fibers of the shaft as the two combined, we can treat the cal-
culation of the diameter as a simple case, and proceed as in the
cases of simple torsion and simple bending considered above. This
relation is given us in mechanics:

=4+T>l^'+'^- (41)

Te = B + ^B» + T'. (43)

These expressions are true in relation to each other, on the assunxp-
tion that the allowable fiber stress S is the same for tension, com-

MACHINE DESIGN 107

pression, and shearing. For the material of which shafts are usu-
ally made, this is near enough to the truth to give safe and practi-
cal results. Using the expressions for internal moments of resist-
ance as previously noted for circular sections, we then have :

Be = i^- (43)

Also, ^«""5l" ^^^^

Either equation may be used ; the diameter d will result the same
whichever equation is taken. For the sake of simplicity, equation
42 is generally preferred, equation 44 being taken in conjunction
with it.

The expression l/B^ + P is one that would be a long and
tedious task to calculate. By inspection it is readily seen that
this quantity can be graphically represented by means of a right-
angled triangle having B and T as the sides. We may then lay
down on a piece of paper, to some convenient scale, the moments
B and T as the sides of a right-angled triangle, when, upon
measuring the hypothenuse, we can easily read off to the same

scale i/B*^ + P. Even if the drawing is made to a small scale,
the accuracy of the reading will be suflScient to enable the value
for d to he solved very closely. This graphical method is illus-
trated in Part I.

Deflection. For a shaft subjected to pure torsion, as in Fig.
26, the angular deflection due to the load may be carried to a cer-
tain point before the limit of working fiber stress is exceeded.
The equation worked out from mechanics for this condition, is:

Ao 584 TL .. .

which at once gives the number of degrees of angular deflection

for a shaft whose modulus of elasticity, torsional moment, and

length are known.

The jshearing modulus of elasticity of ordinary shaft steel runs from
10,000,000 to 13,000,000, giving as an average about 12,000,000.

By the well-known relation of " Hooke's law " (stresses pro-
portional to strains within the elastic limit of the material), we have:

108 MACHINE DESIGN

A° SL .
360° ~ iriid'

_ AirQd f ^v

°' S =-360L- (46)

A twist of one degree in a length of twenty diameters is a
usual allowance. Substituting A = 1, L =^ 20d, and G = 12,000,
000, we have:

S = 5,240 (nearly). (47)

This is a safe value for shearing fiber stress in steel. In fact, in
calculations for strength, even for reversing stresses, the usual
figure is 8,000 (lbs. per square inch), thus indicating that the re-
lation of one degree to twenty diameters is well within the limit
of strength.

For a hollow shaft the above formula becomes :

_ 584 TL .

^ -G(d„*-d*)' ^^^^ ■

Transverse deflection occurs when the shaft is subjected to a
bending moment. It may therefore exist alone or in conjunction
with angular deflection. Transverse deflection of shafts, however,
rarely exists up to the point of limiting fiber stress, because before
that point is reached the alignment of the shaft is so disturbed
that it is not practicable as a device for transmitting power. A
transverse deflection of .01 inch per foot of length is a common
allowance ; but it is impossible to fix any general limit, as in many
cases this figure, if exceeded, would do no harm, while in others —
such as heavily loaded or high-speed bearings — even the figure
given might be fatal to good operation.

The formula for transverse deflection, deduced from mechan-
ics, varies with the system of loading. The three most common
conditions only are given below, reference to the handbook being
necessary if other conditions must be satisfied:

Fixed at one end, loaded at the other,

WL' . .

' = SET- <49)

MACHINE DESIGN 100

Supported at ends, loaded in middle,

_ WL*
*~48EI'

Supported at ends, loaded unif ormly,

5WL'
384 EI"

(SO)

e =

(51)

For transverse deflection the direct modulus of elasticity must
be used, for the fibers are stretched or compressed, instead of being
subjected to a shearing action. The most usual value of the di-
rect modulus of elasticity for ordinary steel is 80,000,000, and is
denoted in most books by the symbol £. Both the shearing and
direct moduli of elasticity are really nothing but the ratio of the
stress to the strain produced by that stress, it being assumed that
the given material is perfectly elastic. A material is supposed to
be perfectly elastic up to a certain limit of stress, and it is within
this limit that the relation as above holds good.

Expressed in the form of an equation this would be :

T. S SL , .

E = T- = -i- (52)

Centrifusfal Whirlitis:. If a line shaft deflect but slightly,
due to its own weight, or the weight or pressure of other bodies
upon it, and then be run at a high speed, the centrifugal force set
up increases the deflection, and the shaft whirls about the geomet-
rical line through the centers of the bearings, causing vibration
and wear in the adjoining members. It is evident that the prac-
tical remedy for this tendency in a shaft of given diameter and
speed is to locate the bearings sufiiciently close to render the action
of small effect.

Many formulsB might be given for this relation, each being
based on different assumptions. Perhaps as widely applied and
as simple as any, is the ^' Bankine " formula, which sets the limit
of length between bearings for shafts not greatly loaded by inter-
mediate pulleys or side strains :

H = 175 J|: (53)

110

MACHINE DESIGN

Horse-Power of Shafting. Horse-power is a certain specific

rate of doing work, viz,^ 33,000 foot-pounds per minute. Hence,
to find the horse-power that a shaft will transmit, we must first
find the work done, and then relate it to the speed. Take, for ex-
ample, the case of a pulley, the symbols being the same as before
— namely, P = driving force at rim of pulley (lbs.); B = radius
of pulley (inches); N = number of revolutions per minute; and
H = horse- power. Then,

Work = force X distance = P X (2 tt KN) = H X 33,000 X 12;

or.

PR =

63,026H

(54)

This is one of the most useful equations for calculations involving
horse-power. By it the number of inch-pounds torsion for any
horse-power can be at once ascertained.

It should be clearly noted, however, that in this equation the
bending moment does not enter at all. Hence any shaft based in
size on horse-power alone^ is based on torsional moment aloney
bending moment being entirely neglected. In many cases the
bending moment is the controlling one as to limiting fiber stress.
Hence empirical shafting formulae depending upon the horse-
power relation are unsafe, unless it is definitely known just what
torsional and bending moments have been assumed.

The only safe way to figure the size of a shaft is to find
accurately what torsional moment and bending moment it has to
sustain, and then combine them according to equation 41 or 42

MACHINE DESIGN 111

introducing the element of speed as basis for assumption of a high
or low working fiber stress.

PRACTICAL MODIFICATION. The practical methods of
handling the theoretical shaft equations have reference to the fit of
the shaft within the several pieces upon it. The running fit of a
shaft in a bearing is usually considered to be so loose that the shaft
could freely deflect to the center of the bearing. This is doubtless
an extreme view of the case, but it is the only safe assumption.
Hence a shaft running in bearings (see Fig. 31) is supposed to be
supported at the centers of those bearings, and its theoretical
strength is based on this supposition.

For a tight or driving fit upon the shaft, a safe assumption to
make is that there is looseness enough at the ends of the fit to per-
mit the shaft to be stressed by the load a short distance within the
faces of the hub, say from -J inch to 1 inch. For example, refer-
ring to Fig. 31, suppose P, to be the transverse load, exerted
through a hub fast upon the part of the shaft d^. Taking mo-
ments about the center of one bearing, and solving for the reaction
at the center of the other, we have :

F^u=R, K;

^^^ Ri = -IT"* (55)

Also, P, ^ = B, K;

R, = tJ^. (56)

Now, as far as the part of shaft d^ is concerned, it may depend for
its size on the bending moment R, J, or on Rj a. The reason the
lever arm is not taken to the point directly under the load P„ is
because it is not practically possible to break the shaft at that
point, on account of the reinforcement of the hub, which is tightly
fitted upon it. Trying these moments to see which is the greater,
we shall find that the greater moment always occurs in connection
with the longer lever arm. Hence Rj b will be greater than Rj a.
We then write the equation of external moment = internal mo-
ment:

112 MACHINE DESIGN

* ''""^

. = -jm^. (57)

For the size of bearing A we have the mazimnm bending mo-
ment:

,10.2 R,L,
or, a

.= ;/2^S5: (58)

For the size of bearing B we have the maziinam moment:

^» 2 ~ 10.2 ■'

or,

,. = -1^^: (5,)

The above calculations are, of course, on the assumption that no tonion
is transmitted either waj through this axle. We should in that case have
combined torsion and bending. This has been made sufficiently clear in pre-
ceding paragraphs and in Part I, to require no further illustration.

The dotted line in Fig. 31 shows the theoretical shape the
axle should take under the assumed conditions. The practical
modification of this shape is obvious. At the shoulders of the
shaft the corners should not be sharp, but carefully filleted, to
avoid the possible starting of a crack at those points.

Often the diameter of certain parts of a shaft may be larger
than strength actually calls for. For example, in Fig. 31, the
part d^ need only be as large as the dotted line; but it is obvious
that unless the key is sunk in the body of the shaft, the hub could
not be slipped into place over the part d^. If, however, the diam-
eter d^ be made large enough so that the bottom of the key will
clear d^y the rotary cutter which forms the key way in d^ will also
clear d^y and the key way can be more easily produced.

In cases where fits are not required to be snug, a straight
shaft of cold-rolled steel is commonly used. Here any parts fast-
ened on the middle of the shaft have to be driven over a consider-
able length of the shaft before they reach their final position.
Moreover, there is no definite shoulder to stop against, and meas*
urement has to be resorted to in locating them.

MACHINE DESIGN 113

It does not pay to turn any portion of a cold-rolled shaft, un-
less it be the very ends, for relieving the " skin tension " in such
material is sure to throw the shaft out of line and necessitate
subsequent straightening.

Turned -steel shafts for machines may with advantage be
slightly varied in diameter wherever the fit changes; and although
the production of shoulders costs something, yet it assists greatly
in bringing the parts to their exact location, and enables the work-
man to concentrate his best skill on the fine bearing fits, and to
save time by rough-turning the parts that have no fits.

Hollow shafts are practicable only for large sizes. The advan-
tages of removing the inner core of metal, aside from some specific
requirement of the machine, are that it eliminates all possibility of
cracks starting from the checks that may exist at the center, per-
mits inspection of the material of a shaft, and, . in case of hollow-
forged shafts, gives an opening for the forging mandrel. In the
last case, the material is improved by a rolling process.

The material most common for use in machine shafting is the
ordinary " Machinery Steel," made by the Bessemer process. This
steel is apt to be '^ seamy," and often contains checks and flaws
that are detected only upon sudden and unexpected breakage of a
part apparently sound. This characteristic is a result of the proc-
ess employed in the manufacture of the steel, and thus far has
never been wholly eliminated. Bessemer steel is, nevertheless, a
very useful material, and the above weakness is not so serious but
that this kind of steel can be used with success in the great majority
of cases.

When a more homogeneous shaft is desired, open-hearth steel
is available. This is a more reliable material to use than the Bes-
semer, and costs somewhat more. It makes a stiff, true, fine-sur-
faced shaft, high-grade in every respect. It is usually specified
for armature shafts of dynamos and motors.

Steels of special strength, toughness, and elasticity are made
under numerous processes. Nickel steel is perhaps the most con-
spicuous example. While for this steel a high price has to be
paid, yet its great strength, in connection with other valuable qual-
ities, makes it a material extremely valuable for service where light
weight is essential, or where contracted space demands small size.

114 MACHINE DESIGN

The range of strength of these various steels is so great that it is well-
nigh useless to go into a discussion of it here. Reference should be had to
the extended discussions of the handbooks, and to special trade pamphlets.
A study of the possibilities of steel in its various forms for use in shafting,
is very valuable as a basis for design, as it can almost be said that a machine
consists chiefly of a ''collection of shafts with a structure built round them."
The shafts are like a core, and evidently the size of the core determines the
shell about it.

PROBLEilS ON SHAFTS.

1. Kequired the twisting moment on a shaft that transmits
30 horse-power at 120 revolutions per minute.

2. Find the diameter of a steel shaft designed to transmit 50
horse-power at 150 revolutions per minute.

3. Assuming same data as in Problem 1, find the diameters
of a hollow shaft for a value of S = 8,000.

4. A belt on an idler pulley embraces an angle of 120
degrees. Assuming tension of belt 1,000 pounds on each side,
and pulley located midway between bearings, which are 30 inches
from center to center, what is the diameter of shaft required ?

5. Calculate the diameter of a steel shaft designed to transmit
a twisting moment of 400,000 inch-pounds and also to take a
bending moment of 300,000 inch-pounds.

6. Find the angular deflection in a 4-inch shaft 20 feet long
when subjected to a load of 5,500 pounds applied to an arm of
30-inch radius. Assume transverse modulus of elasticity equal to
12,000,000.

7. The overhung crank of a steam engine has a force of
32,000 lbs. at the center of the crank pin, which is 12 inches from
the center of the shaft bearing, measured parallel to the shaft.
The radius of crank arm is 10 inches. Assume S equal to 10,000.
Calculate the diameter of the crank shaft.

8. On a short, vertical steel shaft the load is 5,000 pounds.
A gear, 36 teeth, IJ diametral pitch, at top of shaft, transmits a
load of 4,000 pounds at the pitch line. Safe shear = 7,500. What
is the diameter of the shaft ?

SPUR GEARS.

NOTATION— The foUowlng notation is used throughout the chapter on Spur Oears:

6 = Breadth of rectangular section of M, Mi =Revolutionsi)er minute.

arm (inches) . U=Ooefficient of friction between teeth.

MACHINE DESIGN

115

C = Width of arm extended to pitch

line (inches).
c =Distance from neutral axis to outer

fiber (inches).
D=: Pitch diameter of gear (inches).
P=Face of gear (inches).
f = Clearance of tooth at bottom

(inches).
Q= Thickness of arm extended to pitch

line (inches).
H=Thickness of tooth at any section

(inches).
h = Depth of rectangular section of arm

(inches).
I =Moment of inertia.
K=Thickness of rim (inches).
L= Distance from top of tooth to any

section (inches).

N=Number of teeth,
n = Number of arms.
P= Diametral pitch (teeth i)er inch of

diameter).
Pi= Circular pitch (inches).
Qi Qi= Normal pressure between teeth

(lbs.).
R, Ri= Resultant pressure between

teeth (lbs.),
r, n= Radius of pitch circles (inches).
S =Fiber stress of material (lbs. per

sq. in.).
a = Addendum of tooth (inches) sDe-

dendum of tooth.
t = Thickness of tooth at pitch line

(inches).
W=Load at pitch line (lbs.).
y =Coefficlent for " Lewis *' formula.

ANALYSIS. If a cylinder be placed on a plane surface, with
its axis parallel to the plane, an attempt to rotate the cylinder
about its axis would cause it to roll on the plane.

Again, if two cylinders be provided with axial bearings, and
be slightly pressed together, motion of one about its axis will
cause a similar motion of the other, the two surfaces rolling one
on the other at their common tangent line. If moved with care,
there will be no slipping in either of the above cases — which is
explained by the fact that no matter how smooth the surfaces may
appear to be, there is still suflScient roughness to make the little
irregularities interlock and act like minute teeth.

The magnitude of the force possible to be transmitted de-
pends not only on the roughness of the surfaces, but on the
amount of pressure between them. Suppose that one cylinder is
a part of a hoisting drum, on which is wound a rope with a weight
attached. We can readily make the weight so great that, no mat-
ter how hard we press the two cylinders together, the driving
cylinder will not turn the hoisting cylinder, but will slip past it.
If now, instead of increasing the pressure, which is detrimental
both to cylinders and bearings of same, we increase the coarseness
of the surfaces, or, in other words, put teeth of appreciable size
on these surfaces, we attain the desired result of positively driving
without excessive side pressure.

These artificial projections, or t<3eth, must fit into one another;
hence the surfaces of the original cylinders, having been broken
up into alternate projections and hollows, have entirely disap-

llfi

MACHINE DESIGN

peared to the eye; they nevertheless exist as ideal or imaginary
surfaces, which roll together with the same Barface velocities as if
in bodily form, provided that the carves of the teeth are correctly
formed. Several mathematical curves are available for Tise as
tooth ontlines, but in practice the Involute and cycloldal curve*
are the only ones used for this purpose.

The ideal surfaces are known as pitch cylinders or pitch
circles. In Fig. 32 is shown an end view of such a pair of cylin-
ders in contact at their pitch point F. lu gear calculations we
assume that there is no slip between the pitch circles, acting aa
driving cylinders; hence the speeds of the two pitch circles at the

. Fig. 32.

pitch point are equal. If M and HL^ be the revolutions per minute
of the cylinders respectively, r and r^ their radii, then
27rrM = 2ffr,Mi;

Mr, ,- V

That is, the number of revolutions varies inversely as the radii. Q
The simple calculation as above is the key to all calcolations
involving gear trains in reference to their speed ratio.

Fig. 33 represents cycloidal teeth in the two extreme positions
of beginaing and ending contact. The normal pressure Q or Qj
between the teeth in each position acts through the pitch point O,
as it must always do in order to insure the condition of ideal roll-

MACHINE DESIGN

117

ing of the pitch circles, and the velocity ratio proportional to — -

As the surfaces of the teeth slide together, frictional resistance is

produced at their point of contact. This force is widely variable,

depending on the material and condition of the tooth surfaces,

whether smooth and well lubricated, or rough and gritty. As this

resistance acts in conjunction with the normal force between the

teeth, we may construct a parallelogram of forces on these two as

a base, the resultant pressure between the teeth being slightly

changed thereby, as shown in Fig 33.

Assuming a coefficient of friction fl, the force of friction is fi Q or fi Qi
and the resultant pressure R or Ri.

Tooth B of the follower is therefore under a heavy bending moment
measured by the product RL, L
being the perpendicular distance
from the 6enter of the tooth at
its base to the line of the force.
This tooth also has a relatively
small compressive stress due to
the resolved part of R along the
radius, and a relatively small
shearing stress due to the re-
solved part of R along a tangent
to the pitch circle.

Tooth D of the driven wheel
or FOLLOWER has a relatively
large shearing stress, a small
bending moment, and practi-
cally no direct compressive
stress.

Tooth A of the driving wheel
or DRIVER has a relatively large
shearing stress, a small bending moment, and small compressive stress.

Tooth C of the driver has a large bending moment, but small com-
pressive and shearing stresses.

The conditions as noted above are not those of every pair of

gears, in fact they vary with every difference of pitch circle, or of

detail and position of tooth. It is true, however, that in nearly

all cases in practice the bending stress is the controlling one from

a theoretical standpoint. Moreover, the designer must consider

the form and strength of the tooth when it is under the condition

of maximum moment. This evidently, from the above, occurs at

the beginning of contact, for the follower teeth; and at the end of

contact, for the driver teeth. In the particular case illustrated in

118

MACHINE DESIGN

Fig. 33, if the material in both gears were the same, tooth C,
being the weaker at the root, wonld probably break before B; but
if C were of steel, and B of cast iron, B might break first.

It will be noticed that It is nearly parallel to the top of the
tooth; and it may easily happen that the friction may become of
saeh a ralne that it will turn the direction of R until it lies along
the top of the tooth exactly, which is the condition for maximum
moment. For strength calculations it is usual to consider this
condition as existing in all cases.

At the beginning of contact there is more or less shock when
the teeth strike together, and this effect is much more evident at
high speeds. There is also at the beginning of contact a sort of
chattering action as the driving tooth rubs along the driven tooth.

Uniform distribution of pressure along the face of the tooth is
often impaired by uneven wear of the bearings supporting the gear
shafts, the pressure being localized on one corner of the tooth. The
same effect is caused by the accidental presence of foreign material
between the teeth. Again, in cast gearing, the spacing may be
irregular, or, on account of draft on the pattern, the teeth may bear
at the high points only. While it is
usual to consider that the load is evenly
distributed along the face of the tooth,
yet the above considerations show that
an ample margin of strength must al-
waya he allowed on account of these
uncertainties.

When the number of teeth in the
mating gears is high, the load will be
distributed between several teeth ; but,
as it is almost certain that at some time
the proper distribution of load will not

exist, and that one tooth will receive the full load, it is considered
that practically the only safe method is so to design the teeth that
a single tooth may be relied upon to withstand the full load without
failure.

THEORY. Based on the Analysis as given, the theory of gear
teeth assumes that one tooth takes the whole load, and that this load
is evenly distributed along the top of the tooth and acts parallel with

Fig. 34.

MACHINE DESIGN 119

its base, thus reducing the condition of the tooth to that of a
cantilever beam. The magnitude of this load at the top of the
tooth is taken for convenience the same as the force transmitted at
the pitch circle. This condition is shown in Fig. 34. Equating
the external moment to the internal moment, we then have, from
mechanics:

^, SI SFH^ ,. .

WL = - = -g— (6l)

The thickness H is usually taken either at the pitch line or at
the root of the tooth just before the fillet begins; and L, of
course, is dependent on the tooth dimensions. The formula is
most readily used when the outline of the tooth is either assumed
or known, a trial calculation being made to see if it will stand the
load, and a series of subsequent calculations followed out in the
same way until a suitable tooth is found. This method is pursued
because there are certain even pitches which it is desirable to use;
and it is safe to say that any calculation figured the reverse way
would result in fractional pitches. The latter course may be used,
however, and the nearest even pitch chosen as the proper one.

As stated under "Analysis," there are a great many circum-
stances attending the operation of gears which make impossible
the purely theoretical application of the beam formulae. For this
reason there is no one element of machinery which depends so
much on experience and judgment for correct proportion as the
tooth of a gear. Hence it is true that a rational formula based on
the theoretical one is really of the greater practical value in tooth
design.

If we examine formula 61, we find that in a form solved for
W, we have:

,^ SFH^ ,^ V

W = -6L- (<^*)

Of these quantities, H and L are the only variables, for we can
make the others what we choose. H and L depend upon the
circular pitch P* and the curvature and outline of the tooth. If
now we could settle on a standard system of teeth, we could estab-
lish a coeflScient to be used to take the place of the variable part

120

MACHINE DESIGN

of H and L, which depends on the outline of tooth, and we should
thas have an empirical formula which would be on a theoretical
basis.

This, Mr. Wilfred Lewis has done; and it is safe to say
that this formula is more universally used and with more satis-

2
•^10

t =

% =.

Fig. 35.

factory practical results than any other formula, theoretical or
practical, that has ever been devised. His coeflScient is known as
y, and was determined from many actual drawings of different
forms of teeth showing the weakest section. This coefScient is
worked out for the three most common systems as follows:

For 20" involute, y = 0.154 - ^^^ (63)

For 15° involute
and cycloidal.

y = 0.124 -

For radial flanks, y = 0.075 -

0.684

0.276

(64)
(65)

The tooth upon which the above is based is the American standard or
Brown & Sharpe tooth, for which the proportions are shown in Fig. 35.

The " Lewis " formula* is:

W = SF Fy. (66)

A table indicating the value of S for different speeds follows:

Safe Working Stresses for Different Speeds.

Speed of teeth,
ft. per min.

100

200

300

600

900

1200

1800

2400

Cast iron

8000

6000

4800

4000

3000

2400

2000

1700

Steel

20000

15000

12000

10000

7500

6000

5000

4300

*NoTE. A full and convenient statement of the Lewis formula will
be found in «*Kent's Pocket Book. "

MACHINE DESIGN

121

A usual relation of F to PMs:

For cast teeth, P = 2Pi to SPK
For cut teeth, F = 3Pi to 4Pi.

(66)
(67)

The usual method of handling these formulae is as follows:

The pitch circles of the proposed gears are known or can be assumed;
hence W can readily be figured, also the speed of the teeth, whence S can
be read from the table. The desired relation of F to Pi can be arbitrarily
chosen, when Pi and y become the only unknown quantities in the equation.
A shrewd guess can be made for the number of teeth, and y calculated there-
from. Then solve the equation for P'' which will undoubtedly be fractional.
Choose the nearest even pitch, or, if it is desired to keep an even diametral
pitch, the fractional pitch that will bring an even diametral pitch. Now,
from this final and corrected pitch, and the diameter of the pitch circle,
calculate the number of teeth N in the gear. Check the assumed value of y
by this positive value of N.

Another good way of using this formula is to start with the
pitch and face desired, and the diameter of the pitch circle. In

Fig. 37.

this case W is the only unknown quantity, and when found can be
compared with the load required to be carried. If too small,
make another and successive calculations until the result approxi-
mates the required load.

SPUR GEAR Rin, ARilS, AND HUB.

ANALYSIS and THEORY. The rim of a gear has to transmit
the load on the teeth to the arms. It is thus in tension on one side
of the teeth in action, and in compression on the other. The sec-
tion of the rim, however, is so dependent on other practical con-
siderations which call for an excess of strength in this respect, that

122

MACHINE DESIGN

it is not conBidered worth while to attempt a calculation on this
basiB.

Gears seldom run faet enongh to make necessary a calculation
for centrifogal force ; and in geaeral it can be said that the design
of the rim is entirely dependent on practical considerations. These
will appear later nnder " Practical Modification. "

The arms of a gear are stressed the same as pnlley arms, the
same theory answering for both, except that a gear rim always be-
ing much heavier than a pulley rim, the distribation of load
amongst the arms is better in the case of a gear than of a pulley,
and it is asnally safe to assume that each arm of a gear takes ita fall
proportion of load ; or, for an oval section, equating the external
moment to the internal moment aa in the case of pnlleys, we have :

= 0.0393 SA'.

(68)

Heavy spar gears have the arms of a cross or T section (Fig.

Pig. 38.

37), the latter being especially applicable to the case of bevel gears
where there is considerable side thrust. The simpleBt way of
treating such sectioiiB is to consider that the whole bending mooient
is taken by the rectangular section whose greater dimension is in
the direction of the load. The rest of the section, being close to
the neutral axis of the Bection, is of little value in resisting the
direct load, its function being to give sidewise stiffneBS. The
equation f.oi the cross or T style of arm, then is :

W D S5A* ,, ,

MACHINE DESIGN 123

Either h or h may be assnmed, and the other determined. As a
guide to the section, h may be taken at about the thickness of the
tooth.

Gear hubs are in no wise diflferent from the hubs of pulleys or
other rotating pieces. The depth necessary for providing suflS-
cient strength over the key to avoid splitting is the guiding ele-
ment, and can usually be best determined by careful judgment.

PRACTICAL MODIFICATION. The practical requirements,
which no theory will satisfy, are many and varied. Sudden and
severe shock, excessive wear due to an atmosphere of grit and corros-
ive elements, abrupt reversal of the mechanism, the throwing-in of
clutches and pawls, the action of brakes — these and many other
influences have an important bearing on gear design, but not one
that can be calculated. The only method of procedure in such
cases is to base the design on analysis and theory as previously
given, and then add to the face of gear, thickness of tooth, or pitch
an amount which judgment and experience dictate as suflScient.

Excessive noise and vibration are diflScult to prevent at high
speeds. At 1,000 feet per minute, gears are apt to run with an
unpleasant amount of noise. At speeds beyond this, it is often
necessary to provide mortise teeth, or teeth of hard wood set into
a cast-iron rim (see Fig. 38). Kawhide pinions are useful in this
regard. Fine pitches with a long face of tooth run much more
smoothly at high speeds than a coarse pitch and narrow-faced tooth
of equal strength. Greater care in alignment of shafts, however,
is necessary, also stiffer supports.

Should it be impracticable to use a standard tooth of suflScient
strength, there are several ways in which we can increase the
carrying capacity without increasing the pitch. These are:

1. Use a stronger material, such as steel.

2. Shroud the teeth.

3. Use a hook tooth.

4. Use a stub tooth.

Shrouding a tooth consists in connecting the ends of the teeth
with a rim of metal. When this rim is extended to the top of the
tooth, the process is called " full- shrouding" (Fig. 39); and when
carried only to the pitch line, it is termed "half -shrouding"
(Fig. 40). The theoretical effect of shrouding is to make the tooth

124

MACHINE DESIGN

act like a short beam built in at the sides; and the tooth will
practically have to be sheared out in order to fail. This modifica-
tion of gear design requires the teeth to be cast, as the cutter
cannot pass through the shrouding. The strength of the shrouded
gear is estimated to be from 25 to 50 per cent above that of the
plain-tooth type.

Fig. 39.

Fig. 40.

The hook-tooth gear (Fig. 41) is applicable only to cases
where the load on the tooth does not reverse. The working side
of the tooth is made of the usual standard curve, while the back is
made of a curve of greater obliquity, resulting in a considerable
increase of thickness at the root of the tooth. A comparison of
strength between this form and the standard may be made by
drawing the two teeth for a given pitch, measuring their thickness
just at top of the fillet, and finding the relation of the squares
of these dimensions. The truth of this relation is readily seen from
an inspection of formula 61.

l^e stub tooth merely involves the shortening of the height

MACHINE DESIGN 125

of the tooth in order to reducse the lever arm on which the load
acts, thus reducing the moment, and thereby permitting a greater
load to be carried for the same stress.

The rim of a gear is dependent for its proportions chiefly on
questions of practical moulding and machining. It must bear a
certain relation to the teeth and arms, so that, when it is cooling in
the mould, serious shrinkage stresses will not be set up, forming
pockets and cracks. Moreover, when under pressure of the cutter
in the producing of the teeth, it must not chatter or spring. This
condition is quite well attained in ordinary gears when the thick-
ness of the rim below the base of the tooth is made about the same
as the thickness of the tooth.

l-IGHT PRESSURC
ON BACK or TOOTH.
35*ir

UOADCD aioc

ViNvourre:.

Fig. 41.

The stiffening ribs and arms must all be joined to the rim by
ample fillets, and the cross-section must be as uniform as possible,
to prevent unequal cooling and consequent pulling-away of the
arms from the rim or hub. Often the calculated size of the arms
at both rim and hub has to be modified considerably to meet this
requirement.

The arms are usually tapered to suit the designer's eye, a
small gear requiring more taper per foot than a large one. Both
rim and hub should be tapered ^ inch per foot to permit easy
drawing-out from the mould.

The proportions given in the following table have been used
with success as a basis of gear design in manufacturing practice.
The table will serve as an excellent guide in laying out, and can be
closely followed, in most cases with but slight modification.
Web gears are introduced for small diameters where the arms begin
to look awkward and clumsy.

12(1

KACHINE DESIGN

Ocar D«*lEii Data.

Diametral pitch ..

1»

«i

TliioknesB of arm
when extended
to pitch line

1«

Width of arm when
extended to
pitch line

a»

1«

Thickneseofrim..

2»

"Depth of rib

1»

Thickneea of web.

Number of arme, 6.

dive inside of rims and hub a draft of i inch per foot.

BEVEL GEARS.

NOTATIO^T— The following notation ia used tbroushoQt the chapter OD

4 cApei distance at pitch elemeot of

i=Apei dl9ti

h (inches:
B =Angleof botlom of tooth (degrees).
C = Pitch angle (degrees).
D =Pitch diameter (inches).
E =RadiasiDcreDieDtofge&T(inches).
F =Face of gear (inches).
' =ClearBnce at bottom (inches).
3 =Aiigle of (ace (degrees).
H =Oittiag angle (degree
K =RadiQ3 iacremeat

(in,

ea).

pinion

}f teeth.

lI=Nnml»t.

Hi=ForaiatiTe nnmbor of teeth, or
the number corresponding to the
spur gear on which the outline of
tooth is made.

) D=OQtEide diameter (inches).

' =Dianietral pitch related to pitch

diameter (teeth per inch).
pi =CircolBr pitch measured on the

circumference of D (inches).
S = Working strength of material (lbs.

per sq. in.).
t i^Addendum, or beifiht of tooth

above pitch line (inches).
• +/=Depth of tooth below pitch line

(inches).
T =Angle of top of tooth (degrose}.
( =Thickness of tooth at pitch Una

(inches).
W =Worl[ing load at pitch line (lba.>.
V ^Factor in "I>ei«ia" fonunla.

ANALYSIS. It is possible to consider bevel gears as the
general case of which spur gears are a special form. The pitch

MACHINE DESIGN

127

surfaces of spur gears described above as cylinders, mathematically
considered, are cones whose vertices are infinitely distant, while
bevel gears likewise are based on pitch cones, but with a vertex at
some finite point, common to the mating pair. Hence, as we
might expect, the laws of tooth action are similar in bevel gears
to those in the case of spur gears. The profile of the tooth in the
former case, however, is based, not on the real radius of the pitch
cone, but on the radius of the normal cone ; and in the develop-
ment of the outline the latter is treated just as though it were the
radius of a spur gear. The tooth thus formed is wrapped back up-
on the normal cone face, and becomes the large end of the taper-
ing bevel-gear tooth (see Fig. 44).

Pig. 42.

The teeth of bevel gears, being simply projections with bases on
the pitch cones, have a varying cross-section decreasing toward the
vertex ; also a trapezoidal section of root, the latter section acting
as a beam section to resist the cantilever moment due to the tooth
load.

The arms must, as in the case of spur gears, transmit the load
from the tooth to the shaft; in addition, the arms of a bevel gear
are subjected to a side thrust due to the wedging action of the
cones. Hence sidewise stiffness of the arms is more essential in
this type of gear than in the case of the spur gear.

THEORY. It is evident that the calculation of tooth strength
based on a trapezoidal section of root would be somewhat compli-

MACHINE DESIGN

cated ; also that the trapezoid in moBt cases wonld be but little
different from a true rectangle. Hence the error will be but
slight if the average croBs-section of the tooth be taken to repre-
sent its strength, and the calculation made accordingly.

130 MACHINE DESIGN

Fig. 45 showfl a bevel-gear tooth with the average cross-sec-
tion in dotted lines. For the purpose of calculation, the assump-
tion is made that the section A is carried the full length of the
face of the gear, and that the load which this average tooth must
carry is the calculated load at the pitch line of section A. This
is equivalent to saying that the strength of a bevel-gear tooth is
equal to that of a spur-gear tooth which has the same face, and a
section identical with that cut out by a plane at the middle of the
bevel tooth. The load, as in the case of the spur gear, should be
taken at the top of the tooth; and its magnitude can be con-
veniently calculated at the mean pitch radius of the bevel face,
without appreciable error.

This similarity to spur gears being borne
in mind, the calculation for strength needs no
further treatment. Once the average tooth is
assumed or found by layout, a strict following-
out of the methods pursued for spur-gear
teeth will bring consistent results.

The detail design of a pair of bevel gears
involves some trigonometrical computations
in order properly to dimension the drawing
for use in finishing the blanks and subse-
Fig. 45. quently in Cutting the teeth, or, in the case

of cast gears, in making the pattern. These
calculations, although simple, are yet apt to be tedious; and inac-
curacies are likely to creep in if a definite system of relations
be not maintained. Hence the results of these calculations are
given below in condensed and reduced form. The deduction of
these formulae is a simple and interesting exercise in trigonometry;
and it is urged that they be worked out by the student from the
figure, in which case he will feel greater confidence in their use.

Axes of Gears at 90 Degrees.

Use subscript 1 for gear; P for pinion. Letters refer to Fig. 44.

(70)

(71)
(72)

N
D

TT
Pl'

8 =

71 •

t =

y.p-

MACHINE DESIGN 131

y-^-ZL-X. ^ (73)

tan Cp = -j^; tan Ci = jp- (7^)

«, s 2 sin C /.7.V

tanT ='X = — N ^ ^

. „ «+/ 2.3UsinC /-^.

tan B = —^ = jj (76)

« +/ = A tan B = ^ = 0^68Pi. (^7)

^= 2P8mC =^l/W + Np» =4"^Pi' + Pp'' ^^®^

A i_ ^ 5 l79\

-^-cosB -2Pcos BsinC ' ^ '

Gi = 90°-(Ci + T); Gp = 00° - (Cp + T). (80)

E = S cos Ci = S Bin dp . (81)

K = S cos Cp = 8 sin Ci. (82)

PRACTICAL MODIFICATION. The practical requirements
to be met in transmission of power by bevel gears are the same as
for spur gears; but in the case of bevel gears even greater care is
necessary to provide stiffness, strength, true alignment, and rigid
supports. As far as the gears themselves are concerned, a long
face is desirable; but it is much more difficult to gain the ad-
vantage of its strength than in the case of spur gears, because full
bearing along the length of the tooth is hard to guarantee.

The rim usually requires a series of ribs running to the hub
to give required stiffness and strength against the side thrust which
is always present in a pair of bevel gears. Instead of arms, the
tendency of bevel-gear design, except for very large gears, is toward
a web on account of the better and more uniform connection
thereby secured between rim and hub. This web may be lightened
by a number of holes, so that the resultant effect is that of a num-
ber of wide and flat arms.

The hubs naturally have to be fully as long as those of spur
gears, because there is greater tendency to rock on the shaft, due
to the side thrust from the teeth, mentioned above.

The teeth on small gears are cut with rotary cutters, at least
two finishing cuts being necessary, one for each side of the taper-
ing tooth. The more accurate method is to plane the teeth on a
special gear planer, and this method is followed on all gears of
any considerable size. The practical requirement here is that no
portion of the hub shall project so as to interfere with the stroke

132

MACHINE DESIGN

of the planer tool. The requirements of gear planers vary some-
what in this regard.

Finally, after all that is possible has been done in the design
of the gear itself to render it suitable to withstand the varied
stresses, especial attention must be paid to the rigidity of the
supporting shafts and bearings. Bearings should always be close
up to the hubs of the gears, and, if possible the bearing for both
pinion and gear should be cast in the same piece. If this is not
done, the tendency of the separate bearings to get out of line and
destroy the full bearing of the teeth is difficult to control. Thrust
washers are desirable against the hubs of both pinion and gear;
also proper means of well lubricating the same.

With these considerations carefully met, bevel gears are not
the bugbear of machine design that they are sometimes claimed
to be. The common reason why bevel gears cut and fail to work
smoothly, is that the gears and supports are not designed carefully
enough in relation to each other. This is also true of spur gears,
but the bevel gear will reveal imperfections in its design far the
more quickly of the two.

WORM AND WORM GEAR.

NOTATION— The following notation is used thronghont the chapter on Worm and
WormOear:

D = Pitch diameter of gear (inches).
E = Efficiency between worm shaft and

gear shaft (per cent).
f =Clearance of tooth at bottom

(inches),
t = Index of worm thread (1 for single*

2 for double, etc.).
L =Lead of worm thread (inches).
M = Revolutions of gear shaft

minute.
Mw= Revolutions of worm shaft

minute.
N = Number of teeth in gear.

per

Pi = Circular pitch = Pitch of worm

thread (inches).
R = Radios of pitch circle of worm gear

(inches).
« = Addendum of tooth (inches).
T =Twisting moment on gear shaft

(inch-lbs.).
Tw=Twisting moment on worm shaft

(inch-lbs.).
t = Thickness of tooth at pitch line

(inches).
W =Load at pitch line (lbs.).

ANALYSIS. The simplest way of analyzing the case of the
worm and worm gear is to base it upon an ordinary screw
and nut. Take, for example, the lead screw of a common lathe.
The carriage carries a nut, through which the lead screw passes.
By the rotation of the screw, the carriage, being constrained by the
guides to travel lengthwise of the ways, is moved. This motion

MACHINE DESIGN 133

is, for a single -threaded screw, a distance per revolution equal to
the lead of the screw.

Now, suppose that the carriage, instead of sliding along the
ways, is compelled to turn about an axis at some point below the
ways. Also, suppose the top of the nut to be cut off, and its length
made endless by wrapping it around a circle struck from the center
about which the carriage rotates. This reduces the nut to a
peculiar kind of spur gear, the partial threads of the nut now
having the appearance of twisted teeth.

This special form of spur gear, based on the idea of a threaded
nut, is known as a worm gear, and the screw is termed a worm.
The teeth are loaded similarly to those of a spur gear, but with the
additional feature of a large amount of sliding along the tooth
surfaces. This, of course, means considerable friction; and it is in
fact possible to utilize the worm and worm gear as an eflScient
device, only by running the teeth constantly in a bath of oil.
Even then the pressures have to be kept well down to insure the
required term of life of the tooth surfaces.

It is evident that for one revolution of a single-threaded worm,
one tooth of the gear will be passed. The speed ratio between the
worm gear and worm shaft will then be equal to the number of
teeth in the gear, which is relatively great. Hence the worm and
worm gear are principally useful in giving large speed reduction
in a small amount of space.

THEORY. The theory of worm-wheel teeth is complicated
and obscure. The production of the teeth is simple, a dummy worm
with cutting edges, called a "hob," being allowed to carve its way
into the worm-gear blank, thus producing the teeth and at the
same time driving the worm gear about its axis.

It is clear that if we know the torsional moment on the worm-
gear shaft, and the pitch radius of the worm gear, we can find the
load on the teeth at the pitch line by dividing the former by the
latter. Expressed as an equation:

WR = T;orW=-^. (83)

How we shall consider this value of W as distributed on the
teeth, is a question difScult to answer. The teeth not only are

134 MACHINE DESIGN

curved to embrace the worm, but are twisted across the face of the
gear, so that it would be practically impossible to devise a purely
theoretical method of exact calculation. The most reasonable thing
to do is to assume the teeth as being equally as strong as spur-gear
teeth of the same circular pitch, and to figure them accordingly.
It is probably true, however, that the load is carried by more than
one tooth, especially in a hobbed wheel; so we shall be safe in
assuming that two — and, in case of large wheels, three — teeth
divide the load between them. With these considerations borne
in mind, the case reduces itself to that of a simple spur-gear
tooth calculation, which has already been explained under the
heading "Spur Gears."

The worm teeth, or threads, are probably always stronger than
the worm-gear teeth; so no calculation for their strength need be
made.

The twisting moment on the worm shaft is not determined so
directly as in the case of spur gears. The relative number of
revolutions of the two shaiFts depends upon the " lead " of the
worm thread and the number of teeth in the gear.

Lead (L) is the distance parallel to the axis of the worm which
any point in the thread advances in one revolution of the worm.
Pitch (P^) is the distance parallel to the axis of the worm between
corresponding points on adjacent threads. The distinction between
lead and pitch should be carefully observed, as the two are often
confounded, one with the other.

The thread may be single, double, triple, etc., the index of the
thread t, being 1, 2, 3, etc., in accordance therewith. The relation
between lead and pitch may then be expressed by an equation, thus:

L = iP\ (84)

When the index of the thread is changed the speed ratio is
changed, the relation being shown by the equation:

^ = 4 (85)

If the eflSciency were 100 per cent ^ between the two shafts,
the twisting moments would be inversely as the ratio of the speeds
thus:

MACHINE DESIGN

135

or.

T_ =

(86)

but for an efficiency £ the equation would be:

or.

T_ =

_T£

(S7)

The diameter of the worm is arbitrary. Change of this
diameter has no effect on the speed ratio. It has a slight effect on
the eflSciency, the smaller worm giving a little higher efficiency.
The diameter of the worm runs ordinarily from 3 to 10 times the
circular pitch, an average value being 4P^ or 5P\

•A longitudinal cross-section through the axis of the worm
cuts out a rack tooth, and this tooth section is usually made of the
standard 14r|° involute form shown in Fig. 46 for a rack.

The end thrust, of a mag-
nitude practically equal to the r— T'
pressure between the teeth,
has to be taken by the hub of
the worm against the face of
the Bhaft bearing. A serious
loss of efficiency from friction
is likely to occur here. This

is often reduced, however, by roller or ball bearings. With two
worms on the same shaft, each driving into a separate worm gear,
it is possible to make one of the worms right-hand thread, and
the other left-hand, in which case the thrust is self-contained in
the shaft itself, and there is absolutely no end thrust against the
face of the bearing. This involves a double outfit throughout, and
is not always practicable.

There are few mathematical equations necessary for the dimen-
sioning of a worm and worm gear. The formulae for the tooth
parts as given on page 120 apply equally well in this case.

PRACTICAL MODIFICATION. The discussion of the effi-
ciency E of the worm and worm gear is more of a practical than

Fig. 46.

136 MACHINE DESIGN

of a theoretical nature. It seems to be true from actual operation,
as well as theory, that the steeper the threads the higher the effi-
ciency. In actual practice we seldom have opportunity to change
the slope of the thread to get increased efficiency. The slope
is usually settled from considerations of speed ratio, or available
space, or some other condition. The usual practical problem is to
take a given worm and worm gear, and to make out of it as efficient
a device as possible. With hobbed gears running in oil baths, and
with rfioderate pressures and speeds, the efficiency will range between
40 per cent and 70 per cent. The latter figure is higher than is
usually attained.

To avoid cutting and to secure high efficiency, it seems es-
sential to make the worm and the gear of different materials.
The worm-thread surfaces being in contact a greater number of
times than the gear teeth, should evidently be of the Harder material.
Hence we usually find the worm of steel, and the gear of cast iron,
brass, or bronze. To save the expense of a large and heavy bronze
gear, it is common to make a cast-iron center and bolt a bronze
rim to it.

The worm, being the most liable to replacement from wear,
it is desirable so to arrange its shaft fastening and general acces-
sibility that it may be readily removed without disturbing the
worm gear.

The circular pitch of the gear and the pitch of the worm
thread must be the same, and the practical question comes in as to
the threads per inch possible to be cut in the lathe in the pro-
duction of the worm thread. The pitch must satisfy this require-
ment; hence the pitch will usually be fractional, and the diameter
of the worm gear, to give the necessary number of teeth, must be
brought to it. While it would perhaps be desirable to keep an
even diametral pitch for the worm gear, yet it would be poor de-
sign to specify a worm thread which could not be cut in a lathe.

The standard involute of 14|°, and the standard proportions
of teeth as given on page 120, are usually used for worm threads.
This system requires the gear to have at least 30 teeth, for if fewer
teeth are used the thread of the worm will interfere with the
flanks of the gear teeth. This is a mathematical relation, and
there are methods of preventing it by change of tooth proportions

MACHINE DESIGN 137

or of angle of worm thread ; but there are few instances in which
less than 30 teeth are required, and it is not deemed worth while
to go into a lengthy discussion of this point.

The angle of the worm embraced by the worm-gear teeth
varies from 60° to 90°, and the general dimensions of rim are made
about the' same as for spur gears. The arms, or the web, have the
same reasons for their size and shape. Probably web gears and
cross-shaped arms are more common than oval or elliptical sections.

Worm gears sometimes have cast teeth, but they are for the
roughest service only, and give but a point bearing at the middle
of the tooth. An accurately hobbed worm gear will give a bearing
clear across the face of the tooth, and, if properly set up and cared
for, makes a good mechanical device although admittedly of some-
what low efficiency.

Fig. 47 shows a detail drawing of a standard worm and worm
gear. It should serve as a suggestion in design, and an illustration
of the shop dimensions required for its production.

PROBLEMS ON SPUR, BEVEL, AND WORM QEAR5.

1. Calculate proportions of a standard Brown & Sharpe
gear tooth of 1^ diametral pitch, making a rough sketch and put-
ting the dimensions on it.

2. Suppose the above tooth to be loaded at the top with
5,000 lbs. If the face be 6 inches, calculate the fiber stress at the
pitch line, due to bending.

3. A tooth load of 1,200 lbs. is transmitted between two
spur gears of 12-inch and 30-inch diameter, the latter gear making
100 revolutions per minute. Calculate a suitable pitch and face
of tooth by the " Lewis " formula.

4. Assuming a ^-inch web on the 12-inch gear, calculate the
shearing fiber stress at the outside of a hub 4 inches in diameter

5. Design elliptical arms for the 30-inch gear, allowing
8 = 2,200.

6. Design cross-shaped arms for 30-inch gear.

7. Calculate the dimensions shown in formulae 70 to 82 in-
clusive for a pair of bevel gears of 20 and 60 teeth respectively, 2
diametral pitch, and 4-inch face. (The use of logarithmic tables
makes the calculation much easier than with the natural functions.)

MACHINE DESIGN

139

8. A worm wheel has 40 teeth, 3 diametral pitch, and double
thread. Calculate (a) its lead; (b) its pitch diameter.

FRICTION CLUTCHES.

NOTATION— The following notation is used throughout the chapter on Friction Clutohet:.

a sAngle between dutch face
and axis of shaft (degrees )

H aHorse-power (JttfiOO ft.4bB.
per minute).

fl s Coefficient of friction (per
cent).

N ^Number of reTolutions per
minute.

P aPoroe to hold clutch in gear
to produce W (lbs.).

R aMean radius of friction sur-
face (inches).

T ^Twisting moment about
shaft axis (inch-lbs.).

V B3 Force normal to dutch face
(lbs.).

WaLoad at mean radius of
friction surface (lbs.).

ANALYSIS. The

friction clutch is a de-
vice for connecting at
will two separate pieces
of shaft, transmitting an
amount of power be-
tween them to the capac-
ity of the clutch. The
connection is usually ac-
complished while the
driving shaft is under
full speed, the slipping
between the surfaces
which occurs during the
throwing-in of the
clutch, permitting the
driven shaft to pick up
the speed of the other
gradually, without ap-
preciable shock. The
disconnection it made in
the same manner, the

140

MACHINE DESIGN

amoant of slipping which occurs depending on the saddennees with

which the clatch is thrown out.

The force of friction ie the sole driving element, hence the
problem is to secure as
large a force of friction
as possible. But friction
cannot be secured with-
out a heavy normal pres-
sure between surfaces
having a high coefficient
of friction between them.
The many varieties of
friction clutches which
are on the market or de-
signed for some special
purpose, are all devices
for accompliBhing one
and the same effect, viz.,
tbeprodnctionofaheavy
normal force or pressure
between surfaces at such
a radius from the. driven
axis, that the product of
. the force of friction

S-:tw thereby created and the
**" radius shall equal the
deeired twisting moment
about that axis.

Three typical metb-
odsof accomplishing this
are shown in Figs. 48,
and 50. None of
these drawings is worked
out in operative detail.
They are merely illas-

trations of principle, and are drawn in the simplest form for that

purpose.

In Fig, 48 the normal pressure is created in the simplest poa-

MACHINE DESIGN

141

sibid way, an absolutely direct pasb being exerted betveen the
discB, dne to the thrust P of the clatch fork.

Id Fig. 49 advantage U taken of the wedge action of the in-
clined faces, the result be-
ing that it takes less thrust
P to produce the required
normal pressure at the ra-
dius K.

In Fig. 50 the inclin-
atioQ of the faces is carried
fio far that the angle a of
Fig. 49 has become zero;
and by the tt^gle- joint ac>
tion of the link piroted to
the clutch collar, the nor-
mal force produced may be
verygreatfor a slight thrust
P. By careful adjustment
of the length of the link so
that the jaw Cakes hold of
the clutch surface, when
the link stands nearly rer-
tical, a very easy operating
device ib secured, and the
. thmst P is made a mini-

THEORY. Referring
to Fig. 48 in order to cal-
cnlate the twisting mo-
ment, we must remember
that the force of friction
between two surfaces is
eqnat to the normal pres-
flare times the coefficient of
friction. This, in the form
of an eqnatioD, nsing the symboU of the figure, is :

W. ,.I-. (88)

142 MACHINE DESIGN

Hence we may consider that we have a force of magnitude /xP
acting at the mean radius R of the clutch surface. The twisting
moment will then be :

T = WR = /LtPR. (89)

Referring to equation 54, which gives twisting moment in terms
of horse-power, and putting the two expressions equal to each other,
we have :

_ 63,025H
T = ij = fiFR;

H = -6W- (90)

This expression gives at once the horse-power that the clutch will
transmit with a given end thrust P.

In Fig. 49 the equilibrium of the forces is shown in the little
sketch at the left of the figure. The clutch faces are supposed
to be in gear, and the extra force necessary to slide the two to-
gether is not considered, as it is of small importance. The static
equations then are :

P = 2 -5- sin a;

or, V = P cosec a. (pi)

W = fiY = /jiP cosec a. (9^)

T = WR = ^PR cosec a. (93)

63,025H
T = ^ = firK cosec a;

or.

^NPR cosec a ^ ^

S — aT(^9K V94;

63,025

In Fig. 50, P would of course be variable, depending on the
inclination of the little link. The amount of horse-power which
this clutch would transmit would be the same as in the case of the
device illustrated in Fig. 49, for an equal normal force V produced.

The further theoretical design of such clutches should be in
accordance with the same principles as for arms and webs of
pulleys, gears, etc. The length of the hubs must be liberal in

MACHINE DESIGN 143

order to prevent tipping on the shaft as a result of uneven wear.
The end thrust is apt to be considerable; and extra side stiffness
must be provided, as well as a rim that will not spring under the
radial pressure.

PRACTICAL nODIFICATION. It is desirable to make the
most complicated part of a friction clutch the driven part, for then
the mechanism requiring the closest attention and adjustment may
be brought to and kept at rest when no transmission of power is
desired.

Simplicity is an important practical requirement in clutches.
The wearing surfaces are subjected to severe usage; and it is
essential that they be made not only strong in the first place, but
also capable of being readily replaced when worn out, as they are
sure to be after some service.

The form of clutch shown in Fig. 50 is the most eflBcient
form of the three shown, although its commercial design is consid-
erably different from that indicated. Usually the jaws grip both
sides of the rim, pinching it between them. This relieves the
clutch rim of the radial unbalanced thrust.

Adjusting screws must be provided for taking up the wear,
and lock nuts for maintaining their position.

Theoretically, the rubbing surfaces should be of those materials
whose coeflBcient of friction is the highest; but the practical ques-
tion of wear comes in, and hence we usually find both surfaces of
metal, cast iron being most common. For metal on metal the
coefficient of friction /i cannot be safely assumed at more than 15
per cent, because the surfaces are sure to get oily.

A leather facing on one of the surfaces gives good results as
to coefficient of friction, fi having a value, even for oily leather, of
20 per cent. Much slipping, however, is apt to burn the leather;
and this is most likely to occur at high speeds.

Wood on cast iron gives a little higher coefficient of friction
for an oily surface than metal on metal. Wood blocks can be so
set into the face of the jaws as to be readily replaced when worn,
and in such case make an excellent facing.

The angle a of a cone friction clutch of the type shown in
Fig. 49, may evidently be made so small that the two parts will
wedge together tightly with a very slight pressure P; or it may

MACHINE DESIGN

MACHINE DESIGN 145

be so large as to have little wedging action, and approach the oon*
dition illustrated in Fig. 48. Between tliese liiuita there la a
practical value which neither gives a wedging action so great aa to
make the surfaces difficult to pull apart, nor, on the otlier hand,
requires an objectionable end thrust along the shaft in order to
make the clutch drive properly.

For a = about 15^^, the surfaces will free themsolvea when P ia rolioved.
" a = " 12°, " " require slight pull to bo freed.
" a = " 10°, " " cannot be freed by direct pull of the
hand, but require some leverage to produce the necessary force P.

PROBLEMS ON FRICTION CLUTCHES.

1. With what, force must we hold a friction dutch in to
transmit 30 horse-power at 200 revolutions per minute, assuming
working radius of clutch to be 12 inches; coefficient of friction 16
per cent ; angle a = 10° ?

2. How much horse-power could be transmitted, other con-
ditions remaining the same, if the working radius were increased
to 18 inches ?

3. What force would be necessary in problem 1, if the angU
a were 15°, other conditions remaining the same ?

COUPLINGS.

NOTATION.— The foUowing notation is UHed throughout tb« cb*pter on OoupUofti

D = Diameter of shaft (inches). Se a Safe crtmblng tlmr nivmm (itm, ptr
d =Diameter of bolt body (inches). sq. In.)*

n ^Number of bolts. T ^Twitting mtmumi iinoh'\\m»),
R =Raditi8 of bolt circle (inches). ^aThickoiNM of flange Oaehm),

S =Safe shearing fiber stress (lbs. per W iLoad on bolM (lbs.),
sq. in.).

ANALY5IS. Rigid couplings are intended to make ib#
shafts which they connect act as a solid^ continuous shaft In
order that the shaft may be worked up to its full strength capaC'
ity, the coupling must be as strong in all resfjects as the stiaft,
or, in other words, it must transmit the same torsional tmnufint.
In the analysis of the forces which r^;me ii\Hm i\umt iU)n\)\\U{fn^ it
is not considered that they are U) take any si^li; Umd^ but that itii^
are to act purely as torsional elements. It is douhiUfnn true thai fa
many cases they do have to provide s^>rrie nuUi strength and Biift*
ness, but this is not their natural function, nor tim one upon whU^
their design is bailed.

146 MACHINE DESIGN

Beferring to Fig. 51, which is the type most convenient for
analysis, we have an example of the simplest form of flange coup-
ling. It consists merely of hubs keyed to the two portions, with
flanges driving through shear on a series of bolts arranged con-
centrically about the shaft. The hubs, keys, and flanges are Bub-
ject to the same conditions of design as the hubs, keys, and web of
a gear or pulley, the key tending to shear and be crushed in the
hub and shaft, and the hub tending to be torn or sheared from the
flange. The driving bolts, which must be carefully fitted In
reamed holes, are subject to a purely shearing stress over their full
area at the joint, and at the same time tend to crush the metal in
the flange, against which they bear, over their projected area.
This latter stress is seldom of importance, the thickness of the
flange, for practical reasons, being sufficient to make the crushing
stress very low.

THEORY. The theory of hubs, keys, and flanges, being like
that already given for pulleys and gears, need not be repeated for
couplings. The shearing stress on the bolts is the only new point
to be studied.

In Fig. 51, for a twisting moment on the shaft of T, the load

at the bolt circle is W = ■^. If the number of bolts be n^

equating the external force to the internal strength, we have:

W = -^ = -^n. (95)

Although the crushing will seldom be of importance, yet for
the sake of completeness its equation is given, thus:

T
W = -w = Scdtn. (96)

The internal moment of resistance of the shaft is -f-t- 5

5.1

hence the equation representing full equality of strength between
the shaft and the coupling, depending upon the shearing strength
of the bolts, is :

MACHINE DESIGN

The theory of the other types of couplings is obscnre, except
ae regards the proportions of the key, which are the same in all
cases. The shell of the damp coupling, Fig. 52, should be thick
enough to give eqoal torsional strength with the shaft; but the
exact function which the bolts perform is diiEcult to determine.
In general the bolts clamp the coupling tightly on the abaft and
provide rigidity, but the key does the principal amoant of the
driving. The bolt sizes, in these couplings, are based on judgment
and relation to surrounding parts, rather than on theory.

PRACTICAL nODIFICATION. All couplings must be made
witih care and nicely fitted, for their tendency, otherwise, is to

spring the shafts oat of line. In the case of the flange coupling,
the two halves may be keyed in place on the shafts, the latter then
swung on centers in the lathe, and the joint ^ed off. Thus the
joint will be true to the axis of the shaft; and, when it is clamped
in position by the bolts, no springing out of line can take place.

A flange F (see Fig. 51) is sometimes made on this form of
coupling, in order to guard the bolts. It may be used, also, to take
a light belt for driving machinery; but a side load is thereby thrown
on the shaft at the joint, which is at the very point whore it is desir-
able to avoid it.

The simplest form of rigid coupling consists of a plain sleere
slipped over from one shaft to the other, when the second is butted
up against the first. This is known as a muff coupling. When
once in place, this is a very excellent coupling, as it is perfectly
smooth on the outside, and consists of the fewest possible parts,
merely a sleeve and a key. It is, however, expensive to fit,

148

MACHINE DESIGN"

difficult to remove, aod requires an extra space of half its length '
on the shaft over which to be slipped back.

The clamp coupling is a good coupling for moderate- sized
shafts, where the flange type of Fig. 51 wonid be anneceBsarily
expeasire. Theclamp coupling, Fig. 52, iBsimply a muff coupling
split in halves, and recessed for bolts. It is cheap and is easily
applied and removed, even with a crowded shaft. If bored with a
piece of paper in the joint, when it is clamped in position it will
pinch the shaft tightly and make a rigid connection. It is desir-
able to have the bolt-heads protected as much as possible, and this
may be accomplished by making the outside diameter large enough
so that the bolts will not project. Often an additional shell is
provided to encase the coupling completely after it is located.

Fig. 53.

There are many other special forms of couplings, some of
them adjustable. Most of them depend upon a wedging action
exerted by- taper cones, screws, or keys. Trade catalogues are to
be sought for their description.

The claw couplitiK, Fig. 53, is nothing but a heavy flange
coupling with interlocking claws or jaws on the faces of the flanges,
to take the place of the driving bolts. This coupling can he thrown
in or out as desired, although it usjially performs the service of a
rigid coupling, as it is not suited to clutching-in during rapid mo-
tion, like a friction clutch.

Flexible couplings, which allow slight lack of alignment, are
made by introducing between the flanges of a coupling a flexible
disc, the one flange being fastened to the inner circle of the disc,
the other to the outer circle. This is also accomplished by pro-
viding the faces of the flange coupling with pins that drive hy

MACHINE DESIGN

149

pressure together or through leather straps wrapped round the
pins. These devices are mostly of a special and often uncertain
nature, lacking the positiveness which is one essential feature
of a good coupling.

PROBLEHS ON COUPLINGS.

1. A flange coupling of the type of Fig. 51 is used on a shaft
2 inches in diameter. The hub is 3 inches long, and carries a
standard key, of proportions indicated below in the table of " Pro-
portions for Gib Keys " ( page 166 ). The bolt circle is 7 inches
in diameter, and it is desired to use |-inch bolts. How many
bolts are needed to transmit 60,000 inch-lbs., for a fiber stress in
the bolt of 6,000 ?

2. Using 6 bolts, what diameter of bolt would be required ?

8. If four |-inch bolts were used on a circle of 8 inches di-
ameter, what diameter of shaft would be used in the coupling to
give equal strength with the bolts ?

BOLTS, STUDS, NUTS, AND SCREWS.

NOTATION— The following notation is used throughout the chapter on Bolts, Studs,

Nuts, and Screws:

d =
di =

H =
I =
k =

L =

Diameter of bolt (inches).

Diameter at root of thread (in-
ches).

Height of nut (inches).

Initial axial tension (lbs.).

Allowable bearing pressure on sur-
face of thread (lbs. per sq. in.).

Lead, or distance nut advances
along axis in one revolution
(inches).

I = Length of wrench handle (inches)

n = Number of threads in nut=: — .

P = Axial load (lbs.).

p = Pitch of thread, or distance be-
tween similar points on adjacent
threads, measured parallel to
axis-(inche8).

S = Fiber stress (lbs. per sq. in.).

W = Load on bolt (lbs.).

Fig. 54.

ANALYSI5. A bolt is simply a cylindrical bar of metal
upset at one end to form a head, and having a thread at the other
end, Fig. 54. A stud is a bolt in which the head is replaced by
a thread; or it is a cylindrical bar threaded at both ends, usually

150

MACHIKE DESIGN

having a Bmall plain portion in the middle, Fig. 55. The object
of boltB and studs is to clamp machine parts together, and yet
permit these same parts to be readily disconnected. The bolt
passes through the pieces to be connected, and, when tightened,
causes surface compression between the parts, while the reactions
on the bead and nut produce tension in the bolt. Studs and tap
bolts pass through one of the connected parts and are screwed
into the other, the stud remaining in position when the parts are
disconnected.

As all materials are elastic within certain limits, the action of

. ^^ -,

Fig 55

a bolt in clamping two machine parts together, more especiaUy if
there Is an elastic packing between them, may be represented
diagram matically by Fig. 56, in which a spring has been introdnced
to take the compression due to screwing up the nut. Evidently
the tension in the bolt is equal to the force necessary to compress
the spring. Now, suppose that two weights, each equal to ^ W,
are placed symmetrically on either side of the bolt, then the tension
in the bolt will be increased by the added weights if the bolt is
perfectly rigid. The bolt, however, stretches; hence some of the
compression on the spring is relieved and the total tension io the

MACHINE DESIGN

151

bolt is less than W + I, by an amount depending on the relative
elasticity of the bolt and spring. •

Suppose that the stud in Fig. 55 is one of the studs connect-
ing the cover to the cylinder of a steam engine, and that the studs
have a small initial tension ; then the pressure of the steam loads
each stud, and, if the studs stretch enough to relieve the initial
pressure between the two surfaces, then their stress is due to the
steam pressure only; or, from Fig. 56, when I = W ; the initial
pressure due to the elasticity of the joint is entirely relieved by the
assumed stretch of the studs. Except to prevent leakage, it is
seldom necessary to consider the initial tension, for the stretch
of the bolt may be counted on to relieve
this force, and the working tension on the
bolt is simply the load applied.

For shocks or blows, as in the case
of the bolts found on the marine type of
connecting-rod end, the stretch of the
bolts acts like a spring to reduce the re-
sulting tensions. So important is this
feature that the body of the bolt is fre-
quently turned down to the diameter of
the bottom of the thread, thus uniformly
distributing the stretch through the full
length of the bolt, instead of localizing it
at the threaded parts.

In tightening up a bolt, the friction
at the surface of the thread produces a twisting moment, which
increases the stress in the bolts, just as in the case of shafting
under combined tension and torsion; but the increase is small in
amount, and may readily be takeu care of by permitting low values
only for the fiber stress.

In a flange coupling, bolts are acted upon by forces perpen-
dicular to the axis, and hence are under pure shearing stress. If
the torque on the shaft becomes too great, failure will occur by
the bolts shearing oflf at the joint of the coupling.

A bolt under tension communicates its load to the nut through
the locking of the threads together. If the nut is thin, and the
number of threads to take the load few, the threads may break or

Fig. 56.

152 MACHINE DESIGN

shear off at the root. With a V thread there is produced a com-
ponent force, perpendicular to the axis of the bolt, which tends to
split the nut.

In screws for continuous transmission of motion and power,
the thread may be compared to a rough inclined plane, on which a
small block, the nut, is being pushed upward by a force parallel to
the base of the plane. The angle at the bottom of the plane is the
angle of the helix, or an angle whose tangent is the lead divided
by the circumference of the screw. The horizontal force corre-
sponds to the tangential force on the screw. The friction at the
surface of the thread produces a twisting moment about the axis of
the screw, which, combined with the axial load, subjects the screw
to combined tension and torsion. Screws with square threads are
generally used for this service, the sides of the thread exerting no
bursting pressure on the nut. The proportions of screw thread
for transmission of power depend more on the bearing pressure
than on strength. If the bearing surface be too small and lubrica-
tion poor, the screw will cut and wear rapidly.

THEORY. A direct tensile stress is induced in a bolt when
it carries a load exerted along its axis. This load must be taken
by the section of the bolt at the bottom of the thread. If the area

at the root of the thread is — ^, and if S is the allowable stress

per square inch, then the internal resistance of the bolt is
— — L. Equating the external load to the internal strength we have :

W = ^. (98)

For bolts which are used to clamp two machine parts together
so that they will not separate under the action of an applied load,
the initial tension of the bolt must be at least equal to the applied
load. If the applied load is W, then the parts are just about to
separate when I = W. Therefore the above relation for strength
is applicable. As the initial tension to prevent separation should
be a little greater than W, a value of S should be chosen so that
there will be a margin of safety. For ordinary wrought iron and
steel, S may be taken at 6,000 to 8,000.

MACHINE DESIGN

If, however, the joints must be each that there is do leakage
between the earfoces, as in the case of a steam cylinder head, and
supposing that elastic packings are placed in the joints, then a
mach larger margin should be made, for the mazimQm load which
may com© on the bolt ie I + "W, where W is the proportional
shue of the internal pressare carried by the bolt. In such cases
8 = 3,000 to 5,000, asing the lower value for bolts of less than
|-inch diameter.

The table given on page 154 will be found very useful in pro-
portioning bolts with U. S, standard thread for any desired fiber
stress.

To find the initial tension due to screwing up the nut, we

Fig. 560.
may assume the length of the handle of an ordinary wrench, meas-
ured from the center of the bolt, as about 16 times the diameter
of the bolt. For one turn of the wrench a force F at the handle
would pass over a distance 2Trl, and the work done is equal to the
product of the force and apace, or F X Zvl. At the same time
the axial load P would be moved a distance p along the axis.
Assuming that there is no friction, the equation for the equality
of the work at the handle and at the screw is;

F2irl = Pp.

(99)

FrictioD, however, is always present; hence the ratio of the osefol
work (P^) to the work applied (FStt^) is not unity as above re.

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""■""°'"°'"2S(I3tSESSSS£gg|gg2|ggg|Kg^

aSHSlL.,

^ism..

^^''555'HSSH5HH|'|S|

i- = s i« a * 3 2 J J » ^ 5 f J jj

^'5=!=:::!::SHS5S!5H:-

,.S.,K,.

MACHINE DESIGN

1(6

From numerous experiments on the friotion of
screws uid nots, it lias been found Uist the efGcienry may be u
low S8 10 per cent. Introdacing the efficiency in kbove eqoatioa,
it imy be written :

P» 1 ,

Assuming tliat 50 ponnda is exerted by a vorkman in

Fig. 68.
tightening ap the net on a 1-inch bolt, the equation above ihovi
UiBtF:= 4,021 pounds; or the initial tension ia somewhat 1ms
than the tabular safe load shown for a 1-inch bolt, with S assumed
at 10,000 pounds per sq. inch.

For shearing stresBeB the bolt should be fitted so that the body
of the bolt, not the threads, resists the force tending to shear off
the bolt perpendicular to its axis. The internal strength of the
bolt to resist shear is the allowable stress S times the area of the

bolt in shear, or— ^ — If W represents the external force tending

to shear the bolt the equality of the external force to the internal
strength is :

W = ^. (,o.)

156 • MACHINE DESIGN

Reference to the table on page 154 for the shearing strength of
bolts, may be made to save the labor of calculations.

Let Fig. 58 represent a square thread screw for the transmis-
sion of motion. The sui'face on which the axial pressure bears, if

n is the number of threads in the nut, is --- (e^- di) n. Suppose

that a pressure of k pounds per square inch is allowed on the
surface of the thread. Then the greatest permissible axial load P
must not exceed the allowable pressure; or, equating,

V = h^{cP- rfi») n. (I02)

The value of h varies with the service required. If the motion be
slow and the lubrication very good, Jc may be as high as 900, For
rapid motion and doubtful lubrication, k may not be over 200.
Between these two extremes the designer must use his judgment,
remembering that the higher the speed the lower is the allowable
bearing pressure.

PRACTICAL MODIFICATION. It will be noted in the
formulsB for bolt strengths that different values for S are assumed.
This is necessary on account of the uncertain initial stresses which
are produced in setting up the nuts. For cases of mere fastening,
the safe tension is high, as just before the joint opens the tension
is about equal to the load and yet the fastening is secure. On
the other hand, bolts or studs fastening joints subjected to internal
fluid pressure must be stressed initially to a greater amount than
the working pressure which is to come on the bolt. As-this initial
stress is a matter of judgment on the part of the workman, the
designer, in order to be on the safe side, should specify not less
than |-inch or £-inch bolts for ordinary work, so that the bolts
may not be broken off by a careless workman accidentally putting
a greater force than necessary on the wrench handle. In making
a steam-tight joint, the spacing of the bolts will generally deter-
mine their number; hence we often find an excess of bolt strength
in joints of this character.

Through bolts are preferred to studs, and studs to tap bolts
or cap screws. If possible, the design should be such that through
bolts may be used. They are cheapest, are always in standard

MACHINE DESIGN 1B7

stock, and well resist rough usage in connecting and disconnecting.
The threads in cast iron are wesik and have a tendency to crumble;
and if a through bolt cannot be used in such a case, a stud, which
can be placed in position once for al), should be employed — not a
tap bolt, which injures the thread in the casting every time it ii
removed.

The plain portion of a stud should be screwed up tight
against the shoulder, and the tapped hole should be deep enough
to prevent bottoming. To avoid breaking oft the itud at the
shoulder, a neck, or groove, may be made at the lower end of the
thread entering the nut.

To withstand shearing forces the bolts must be fitted so that
no lost motion may occur, otherwise pare shearing will not be
secured.

Nuts are generally made hexagonal, but for rough work are
often made square. The hexagonal nut allows the wrench to turn
through a smaller angle in tightening up, and is preferred to the
square nut. Experiments and calculations show that the height
of the nut with standard threads may be about ^ the diameter of
the bolt and still have the shearing strength of the thread equal to
the tensile strength of the bolt at the root of the thread. Pra((ti-
cally, however, it is difficult to apply such a thin wrench aH this
proportion would call for on ordinary bolts. More commonly the
height of the nut is made equal to the diameter of the bolt no that
the length of thread will guide the nut on the bolt, give a low
bearing pressure on the threads, and enable a suitable wrench to
be easily applied. The standard proportions for bolts and niitii
may be found in any handbook. Not all manufacturers conform
to the United States standard; nor do manufacturers in all cases
conform to one another in practice.

If the bolt is subject to vibration, the nuts have a t<;ndency to
loosen. A common methcxl of preventing this is to urn) doable
nuts, or lock nut^f as they are (sailed (mh^ Fig. C5 A;. Ttie under
nut is screwed tightly against the nurt$U!49, and held \fj a wrench
while the second nut is screwed down tightly against the first.
The effect is to cause the thn^arls of the tipj>er nat to l>ear against
the under sides of the threa^ls of the UAU Tlie hfH4l on the UAt is
sustained therefore by the upper nut, which should be the thiekid'

168

MACHINE DESIGN

of the two ; but for conveDience in applying wreDcbeB the positicoi
of the nutB is often reversed.

The form of thread adapted to transmitting power is the
square thread, which, although giving less bursting preesnre
on the nut, is not as strong as the V thread for a given length,
since the total section of thread at the bottom is only ^ as great.
If the pressure is to be transmitted in but one direction, the two

J\.f^

ty^ J P, _, id

of the proportions shown in Fig. 59. Often, as in the carrii^ of
a lathe, to allow the split nut to be opened and closed over the lead
screw, the sides of the thread are placed at a small angle, say 15°,
to each other, as illustrated in Fig, 60,

The practical commercial forms in which we find screwed
fastenings are included in five classes, as follows:

Fig. 60.

1. Through bolts (Fig. 61), usually rough stock, with sqiiare
upset heads, and square or hexagonal nuts.

2. Tap bolts (Fig. 62), also called cap screws. These osu-
ally have hexagonal heads, and are found both in the rough form,
and finished from the rolled hexagonal bar in the screw machine.

3. Studs (Fig, 63), rough or finished stock, threaded in the
screw machine.

4. Set screws (Fig. 64), usually with square beada, and
case-hardened points. Many varieties of Bet screws are made, the

MACHIJSE DESIGN

159

priDcip&l distinguishing feature of each beiog io the shape of the
point. Thus, in addition to the plain beveled point, we find
the " capped," roanded, conical, and " teat " points.

Fig. 81. Pigr. 62. PJb. 69.

6. Machine screws (Fig. 04u), usually round, '■buttOD/'
or countersunk head. Common proportions are indicated relative
to diameter of body of screw.

Kin. 6*.
PROBLEnS ON BULTA, STVUH, NUTS, AND SCREWS.

1. CalcuUtu tlm diatuvtor of a bolt to suataiu a load of
6,000 Iba.

160

MACHINE DESIGN

2. The shearing force to be resisted by each of the bolts of m
flange coupling is 1,200 lbs. What commercial size of bolt ie
reqaired ?

3. With a wrench 16 times the diameter of the bolt, and an
efficiency of 10 per cent, what axial load can a man exert on a
standard |-inch bolt, if he palls 10 Ibe. at the end of (he wrench
handle ?

4. A single, sqnare-threaded screw of diameter 2 inchea,
lead ^ inch, depth of thread | inch, length of nat 3 inches, is to
be allowed a baring pressnre of 300 lbs. per square inch. What
axial load can be carried 2

5. Calculate the shearing stress at the root of the thread in
problem 4.

KEYS, PINS, AND COTTERS.

= AverBge diameter ol rod (Incbes).
= Outslda diameter of aocket (In.

= Diameter ol shaft (laches).
= Length ofkey (Inches). -
= Driving force (lbs.).
= Axial load on rod (lbs.).
= Radius atwblcb P acts (Inches).
m Sate crushing fiber atresa (lbs.
persq. In.).

IB chapter on Keys. Ptns. and

ST = Sate shearing flher; Btress:{lba.: :

per sq. In.}.
Si = Safe tensile fiber stress (Itn. po^

T = Thickness of key (Inches).

W = Width ot key (Inchea).

w = Average width of cotter (lnches>.

toi = End o[ slot to end ot rod (Inchea).

v!i = End ot slat to end ^>t socket (In^

KEYS AND PINS.
ANALYSIS. Keys and pins are nsed to prevent relative

MACHINE DESIGN 161

rotary motion between machine parts intended to aot together as
one piece. If we drill completely through a hub and across the
shaft, and insert a tightly fitted pin, any rotary motion of the one
will be transmitted to the other, provided the pin does not fail by
shearing off at the joint between the shaft and the hub. The
shearing area is the sum of the cross-sections of the pin at the
joint.

We may drill a hole in the joint, the axis of the bole being
parallel to the axis of the shaft, and drive in a pin, in which case
we introduce a shearing area as before, but the area is now equal
to the diameter of the pin multiplied by its length, and the pin is
stressed sidewise, instead of across. It is evident in the sidewise
case that we may increase the shearing area to anything we please,
without changing the diameter of the pin, merely by increasing
the length of the pin.

As there are some manufacturing reasons why a round pin
placed lengthwise in the joint is not always applicable, we may
make the pin a rectangular one, in which case it is called a key.

When pins are driven across the shaft as in the first instance,
they are usually made taper. This is because it is easier to ream
a taper hole to size than a straight hole, and a taper pin will drive
more easily than a straight pin, it not being necessary to match the
hole in hub and shaft so exactly in order that the pin may enter.
The taper pin will draw the holes into line as it is driven, and can
be backed out readily in removal.

Keys of the rectangular form are either straight or tapered,
but for different reasons from those just stated for pins. Straight
keys have working bearing only at the sides, driving purely by
shear, crushing being exerted by the side of the key in both shaft
and hub, over the area against the key. The key itself does not
prevent end motion along the shaft; and if end motion is not
desired, auxiliary means of some sort must be resorted to, as, for
example, set screws through the hub jamming hard against the top
of the key.

If end motion along the shaft is desired, the key is called a
spline, and, while not jammed against the shaft, is yet prevented
from changing its relation to the hub by some means such as
illustrated in Fig. 65.

162

MACHINE DESIGN

Taper keys not only drive through sidewise shearing strength^
but prevent endwise motion by the wedging action exerted between
the shaft and hub. These keys drive more like a stmt from corner to
corner; but this action is incidental rather than intentional, and the
proportions of a taper key should be such that it will give itS' full
reeisting area in shearing and crushing, the same as a strai^t li^y.

Fig. 65.

THEORY. Suppose that the pin illustrated in Fig. 66 passes
through hub and shaft, and the driving force P acts at the radius
K; then the force which is exerted at the surface of the shaft to

2 PR

shear oflf the pin at the points A and B is — ^ — . If D^ is the

average diameter of the pin, its shearing strength is -: — ^ ^ ,

Equating the external force to the internal strength, we have :

2PR 2^Dj^ S.

or.

D, = \|

4PII

TT^S.

(I03>

In Fig. 67 a rectangular key is sunk half way in hub and
shaft according to usual practice. Here the force at the* surfaoo

MACHINE D^ltiJS

ti^

of dw shaft, aleitlsted tbe b»q» u befoTi\ not oaly Iwttvlii to ^mat
tiff die Iray ftlong the tioe AB. bat ti&uis to crash huth th« pw^
tion in the shaft and in thehab. The aheuing itrMgih kloog \il»

line AB is LWS,, Gqaatiag external forue to iDternal atrangth,
we have;

""srs;:"

(104)

The craahing atrength is, of ouurae, that due tu tlw weaker

metal, whether in shaft or hub. T^et 8^ Ihi this ItiMt ufe nruahlng

LT
fiber etrsBB. The crushing strength then ia -'^r- t),, and, eqnating

ezternaLforcetoioternal strength, we bavo:

Cos)

2PR LT

T - *™

^ - -djjs:-

The proportiooB of the key mast be such that the afjiutioaa M
above, both for shearing and for crasbiog, aball \hs mtiwAui.

PRACTICAL MODIFICATfON. Pins across t\u> riuih mn b*
wed to drive lif^t work only, fw the abeariog area cannot \m w*rj
hx^ A laige pio cnt* awsjr too much nrm ttt tbe ab»ft, Amrimi-
iogthe latter's Btrengtb, FinsM-euaeful in ffl-«r«nt!ngMi4n*«tloa,
b^ is this ease are expected to take no sbew, aed nwy b««f souU

164 MACHINE DESIGN

diameter. The common split pin is especially adapted to this
service, and is a standard commercial article.

Taper pins are nsuallj listed according to the Morse standard
taper, proportions of which may be found in any handbook. It
is desirable to use standard taper pins in machine construction, as
the reamers are a commercial article of accepted value, and readily
obtainable in the machine-tool market.

With properly fitted keys, the shearing strength is usually
the controlling element. For shafts of ordinary size, the standard
proportions as given in tables like that below are safe enough
without calculation, up to the limit of torsional strength of the
shaft. For special cases of short hubs or heavy loads, a calcula-
tion is needed to check the size, and perhaps modify it.

Splines, also known as '< feather keys," require thickness
greater than regular keys, on account of the
sliding at the sides. A table suggesting
proportions for splines is given on page 166.

Though the spline maybe either in the
shaft or hub, it is the more usual thing to
find the spline dovetailed (Fig. 67a),
"gibbed," or otherwise fastened in the
hub; and a long spline way made in the
shaft, in which it slides.

The straight key, accurately fitted, is Fig. 67a.

the most desirable fastening device for ac-
curate machines, such as machine tools, on account of the fact that
there is absolutely no radial force exerted to throw the parts out of
true. It, however, requires a tight fit of hub to shaft, as the key
cannot be relied upon to take up any looseness.

The taper key (Fig. 68), by its wedging action, will take up
some looseness, but in so doing throws the parts out slightly.
Or, even if the bored fit be good, if the taper key be not driven
home with care, it will spring the hub, and make the parts run
untrue. The great advantage, however, that the taper key has of
holding the hub from endwise motion, renders it a very useful
and practical article. It is usually provided with a head, or " gib,"
which permits a draw hook to be used to wedge between the face
of the hub and the key to facilitate starting the key from its seat.

MACHINE DESIGN

166

Two keys at 90° from each other may be used io casee where
one key will not sufijce. The doe workmanship inTolved in
spacing these keys so that they will drire equally makes this plan
ioadrisable except in case of positive and unavoidable necessity.

The " Woodruff " key (Fig. 69) is a useful patented article
for certain locations. This key is a half-disc, sunk in the shaft

r*'i

*.T

^TAPER il PER FT.

and the hub is slipped over it. A simple rotary cutter is dropped
into the shaft to produce the key seat; and on account of the
depth in the shaft, the tendency to rock sidewise is eliminated,
and the drive is purely by shear.

Keys may be milled out of solid stock, or drop.forged to
within a small fraction of finished size. The drop-forged key is
an excellent modem production and requires but a i

amount of fitting. Any key, no matter how produced, requires
some hand fitting and draw filing to bring it properly to its seat
and give it full bearing.

It is good mechanical policy to avoid keyed fastenings
whenever possible. This does not mean that keys may never be
used, but that a key is not an ideal way to produce an absolutely
positive drive, partly because it is an expensive device, and partly
because the tendency of any key is to work itself loose, even if
carefully fitted.

The following tables are suggested as a guide to proportions

166

MACHINE DESIGK

of gib keys and feather keys, and will be found aseful in the
abBBDce of any manafactarer'B atandard list:

Ff«. 70. PROPORTIONS FOR OIB KEYS.

Diametar of shaft {d), inchee.
Width (W), inchee.
Thickness (T), inches.

3*3

H
H

It".
1

Tig. n. PRCK>ORTIONS FOR PBATHBR KEYS.

Diameter of Shaft (d), inches,
Width (W), inches.

Thickneaa (T), inches.

3 m

I I

COTTERS.

ANALYSIS. Cotters are used to fasten hubs to rods rather
than shafts, the distinction between a rod and a shaft being that
a rod takes its load in the direction of its length, and does not
drive by rotation. A cotter, therefore, is nothing but a cross-pin
of modified form, to take shearing and crnshing stress in the
direction of the azis of the rod, instead of perpendicular to it.

fieferring to Fig. 73, one will see that the cotter is made
long and thin — long, in order to get sufficient shearing area to resist
shearing along lines A and B; thin, in order to cut as little cross-
sectional area out of the body of the shaft as possible. The cotter
itself tends to shear along the lines A and B, and crush along the
surfaces K, G, and J. The socket tends to crush along the surfaces
K and G. The rod end D tends to be sheared out along the lines
C H and Q E, and also to be crushed along the surface J. The
socket tends to be sheared along the lines V IT and X Y.

The cotter is made taper on one side, thus enabling it to draw
up the flange of the rod tightly against the head of the socket.
This taper must not be great enough to permit easy " backing oat "
and loosening of the cotter under load or vibration in the rod. In
responsible situations this cannot be safely guarded against except
through some auxiliary locking device, such as lock nuts on the
end of the cotter (Fig. 73).

THEORY. Referring to Fig, 72, assume an axial load of F„
as shown. The successive equations of external force to internal

MACUIXE DESIGN

iin

Btrength are enomented belov, fw the diiliannt Mctioua that t«k»
place:

For ehearing along lines A and fi, to being the av«rage
width of cotter, and S, safe shearing stress of cotter,

P, = 2TwS,.

(I06)

For crushing along nurlmPAUt K and H, H, t^rii^ Imm( Mf«
emshing ctreM, wh«tb«r tif i-zdU^ or mielu/tf

p, ^r(t>, v^^

(«»7)

F«jr cnubing »Utng nttriai-M A , Vt, lKf)«(( ImmI mU vnuhtu^

MACHINE DESIGN

For sheariDg along surfaces CII and QE, S, beiog safe shear'
ing stress of rod end, and v, end of slot to end of rod,

P, = 2«.,DS.. (109)

For teoeion id rod end at section across slot, S, being eafe
tensile stress in rod end,

p.=(^-

TD)8,.

(110)

For tension in socket at section across slot, S^ being safe 1
eile stress in socket,

P,=

-T(D,-D)]St. (ill)

For Bhearing in socket along the lines VTJ and XY, 8, being
safe shearing stress in the socket, and w^ end of slot to end of
socket,

Pi = 2«',(D,- D)S,. (113)

The proportions of cotter and socket may be fixed to some
extent by practical or as
Bumed conditions. The di-
mensions may then he tested
by the above equations, that
the safe working stresses may
not be exceeded, the dimen-
sions being then modified ac-
cordingly.

The Bteel of which both
cotter and rod would ordina-
rily be made has range of
— working fiber stress as
follows :
Tension, 8,000 to 12,000 (lbs. per

aq. in.)
Compression, 10,000 to 16,000 (lbs.

■^ Shear, 6,000 to 10,000 (Iba. per

sq. in.)

^«- '^ The socket, if made of

cast iron, will be weak as regards tension, tendency to shear out at

MACHINE DESIGK

169

the end, and tendency to split. The uncertainty of cast iron to
resist these is so great that the hob or socket mnst be very clumsy
in order to have enough surplus strength. This is always a notice-
able feature of the cotter type of fastening, and cannot well bo
avoided.

PRACTICAL MODIFICATION. The driving faces of the cot-
ter are often made semicircular. This not only gives more shear-
ing area at the sides of the slots, but makes the production of the
slots easier in the shop. It also avoids the general objection to
sharp corners — namely, a tendency to start cracks.

A practicable taper for
cotters is ^ inch per foot.
This will under ordinary
circumstances prevent the
cotter from backing out
under the action of the load.
When Bet screws against
the side of the cotter, or
lock nnts are used, as in
Fig. 73, the taper may be
greater than this, perhaps
as much as 1^ inches per
foot.

In the common use of
the cotter for holding the
strap at the ends of con-
necting rods, the strap acts like a modified form of socket. This is
shown in Figs. 73 and 74. Here, in addition to holding the strap
and rod together lengthwise, it may be necessary to prevent their
spreading, and for this purpose an auxiliary piece G with gib ends
is used. The tendency without this extra piece is shown by the
dotted lines in Fig. 74.

The general mechanical fault with cottered joints is that the
action of the load, especially when it constantly reverses, as in
pump piston rods, always tends to work the cotter loose. Vibra-
tion also tends to produce the same effect. Once this looseness is
started in the joint, the cotter loses its pure crushing and shearing
action, and begins to partake of the nature of a hammer, and

Pig. 74.

170

MACHINE DESIGN

pounds itself and its bearing surfaces out of their true shape.
Instead of a collar on the rod, we often find a taper fit of the rod
in the socket; and any looseness in this case is still worse, for the
rod then has end play in the socket, and by its ^< shucking " baek
and forth tends to split open the socket.

The only answer to these objections is to provide a positive
locking device, and take up any looseness the instant it appears.

PROBLEMS ON KEYS, PINS, AND COTTERS.

1. Calculate the safe load in shear which can be carried on a key
^ inch wide, § inch thick, and 6 inches long. Assume S, = 6,000.

2. Assuming the above key to be -^ inch in hub and ^ inch
in shaft, test its proportions for crushing, at S^ «= 16,000.

3. A gear 60 inches in diameter has a load of 3,000 lbs. at
the pitch line. The shaft is 4 inches in diameter, in a hub,
6 inches long; and the key is a standard gib key as given in the
table. Test its proportions for shearing.

4. A piston rod 2 inches in diameter carries a cotter § inch
thick, and has an axial load of 20,000 lbs. Calculate the average
width of the cotter. S^ = 9,000.

5. Calculate fiber stress in rod in preceding problem at
section through slot.

6. How far from the end of rod must the end of slot be ?

7. Calculate the crushing fiber stresses on cotter, rod, and
socket.

8. How far from the end of socket must the end of slot be,
assuming the socket to be of steel ?

BEARINGS, BRACKETS, AND STANDS.

NOTATION— The following notation Is used tliroughout the chapters on
Brackets, and Stands.

A = Area (square Inches).

a = Distance between bolt centers

(Inches).
b = width of bracket base (Inches) .
e = Distance of neutral axis from outer

fiber (Inches).
D = Diameter of shaft (Inches).
d = Diameter of bolt body (Inches).
d\= Diameter at root of thread (Inches).
H = Horse-power.

h = Thickness of cap at center (Inches).
I E Moment of Inertia.
L = Length of bearing (Inches).
fl^ Ooeffloient of friction (per oMit).

N ^Number of revolutions per minute.

n = Number of bolts In cap.

ni= Number of bolts In bracket base.

P = Total pressure on bearing (lbs.).

p = Pressure per square inch of pro-
jected area (lbs).

S = Safe tensile fiber stress (lbs.).

Si= •• shearing •• (LXm,},

T ='Totalload on bolts at top of
bracket (lbs.).

t = Thickness of bracket base (Inches).

X = Distance from line of action of load
to any section of bracket (inches).

MACIUNK 1>K81UK ITl

ANALYSIS. Machine surfact>^ taking xvt^ight i^uii pt>(MiMA\uv
of oth^ parts in motion upon thorn im\ in );^(Mu>rHK knt»^'n M
beuiiqrs. If the motion is rectilint>Ar, Uio Inmrin^ U tt^rn^i^i n
slide, sn><l^ or way, 6uch as the cross slide of a lathe^ tlie eixuit^
head goide of a steam engine, or the Vfhy% of a lathe IhhK

If the motion is a rotary one, like that of Uio tplndle of a lalhf^i
the simple word ** bearing '* is generally nainl.

In any bearing, sliding or roUiry, theiH) tnuHt be utrength lo
carry the load, stiffness to distribute the pressure evenly over thu
fall bearing surface, low intensity of such proHsure to pn^vont thi^
lubricant from being squeezed out and to uunimlKO the weari and
sufficient radiating surface to carry away the hoat generated by
friction of the surfaces as fast as it is generated. Bllding bearln^N
are of such varied nature, and exist under conditions ho peculiar
to each case, that a general analysis is praoiically InipoMidble
beyond that given in the sentence above.

Kotary bearings can be more definitely studied, tin there ar«
but two variable dimensions, diameter and length, and it Im th(i
proper relation between these two that determines a^ood bearing.
The size of the shaft, as noted under " Bhafts," Im calculaU^d by
taking the bending moment at the center of the Ixjiirlng, e^ornbin
ing it with the twisting moment, and solving for tlie diameter
consistent with the assumed fiber stress. But this si//e inuMt thmi
be tried for deflection due to the bending loa^l, hi order that thu
requirement for stiffness may be fulfilled. When this \n iU'AUim
plished, the friction at the bearing surfaci5 may Ntill fHMwrtiU$ MO
much heat that the exposed surface of the bearing will not ra^liaUf
it as fast as generated, in which case the U^rum ftfiiM h^Ht^^r and
hotter, until it finally burns out the lubricant and uuMn th« lining
of the bearing, and ruin results.

The heat condition is usually the eniuthl mtt^^ a# ft \n fwy
easy to make a short bearing which is strong tfuoufsh and amply
stiff for the load it carries, but which nHrefi\ihi^n in a ihWnm a#
a bearing, because it has so small a rtuHaiiufi^ nurlH/fjf ifiai it mn^
not run cool.

The side load which causes the trifciurtt and ti$M fUft$mf\nsfi$i
development of heat, is due to the pnll of i\u9 U'.li iu i>#^ €«tm hi
puUeys, the load on the teeth of g^fi^ i\i^ poll ffh HTMukM *o4

172 MACHINE DESIGN

levers, the weight of parts, etc. If we could exert pure torsion on
shafts without any side pressure, and counteract all the weight
that comes on the shaft, we should not have any trouble with the
development of heat in bearings; in fact, there would theoretically
be no need of bearings, as the shafts would naturally spin about
their axes, and would not need support.

It can be shown, theoretically, that the radiating sur&ce of a
bearing increases relatively to the heat generated by a given side
load, only when the length of the hearing is increased. In oth^r
words, increasing the diameter and not the length, theoretically
increases the heat generated per unit of time just as much as it
increases the radiating surface; hence nothing is gained, and heat
accumulates in the bearing as before. This important fact is veri-
fied by the design of high-speed bearings, which, it is always
noted, are very long in proportion to their diameter, thus giving
relatively high radiating power.

Bearings must be rigidly fastened to the body of the machine
in some way, and the immediate support is termed a bracket*
frame, or housing. ^' Bracket " is a very general term, and ap-
plies to the supports of other machine parts besides " bearings."
It is especially applicable to the more familiar types of bearing
supports, and is here introduced to make the analysis complete.

The bracket must be strong enough as a beam to take the
side load, the bending moment being figured at such points as are
necessary to determine its outline. It may be of solid, box, or
ribbed form, the latter being the most economical of material, and
usually permitting the simplest pattern. The fastening of the
bracket to the main body of the machine must be broad to give
stability; the bolts act partly in shear to keep the bracket from
sliding along its base, and partly in tension to resist its tendency
to rotate about some one of its edges, due to the side pull of the
belt, gear tooth, or lever load, as the case may be. The weight of
the bracket itself and of the parts it sustains through the bearing,
has likewise to be considered; and this acts, in conjunction widi
the working load on the bearing, to modify the direction and
magnitude of the resultant load on the bracket and its fastening.

Stands are forms of brackets, and are subject to the same
analysis. The distinction is by no means well defined, although

MACHINE DESIGN

173

we usnally think more readily of a stand as having an upright or
inverted position with reference to the ground. The ordinary
" hanger " is a good example of an inverted stand; and the regular
" floor stand," found on jack shafts in some power houses, is an
example of the general class.

THEORY, As the method of calculation of the diameter of
the shaft, as well as its deflection, has been coAsidered under
" Shafts," we may assume that the theoretical study of bearings
starts on a given basis of shaft diameter D. The main problem
then being one of heat control, let us first calculate the amount of
heat developed in a bearing by a given side load. The force of
friction acts at the circumference of the shaft, and is equal to the
coefficient of friction times the normal force; or, for a given side load
P, Fig. 75, the force of friction

would be [jlP. The peripheral

speed of the shaft for N revolu-

. ttDN ,
tions per minute is — ^o" '^^^

per minute. As work is " force
times distance," the work wasted

in friction is then h^ — foot-
pounds per minute. One horse-
power being equal to 33,000 foot-
pounds per minute, we have the
equation.

Fig. 75.

H =

/iPttDN
12 X 33,000

(113)

The value of jjl for ordinary, well-lubricated bearings, may run as
low as 5 per cent; but as the lubrication is often impaired, it
quite commonly rises to 10 or 12 per cent. A value of 8 per cent
is a fair average. This amount of horse-power is dissipated
through the bearing in the form of heat. If we could exactly
determine the ability that each particle of the metal around the
shaft had to transmit the heat, or to pass it along to the outside
of the casting, and if we could then determine the ability of the
particles of air surrounding the casting to receive and carry away

174 MACHINE DESIGN

this heat, we could calculate just such proportions of the bearing
and its casing as would never choke or retard this free transfer of
heat away from the running surface.

Such refined theory is not practical, owing to the complicated
shapes and conditions surrounding the bearing. The best that we
can do is to say that for the usual proportions of bearings the side
load may exist up to a certain intensity of " pressure per square
inch of projected area " of bearing, or, in form of an equation,

F=j>U). (114)

The constant ^ is of a variable nature, depending on lubrication,
speed, air contact, and other special conditions. For ordinary
bearings having continuous pressure in one direction, and only
fair lubrication, 400 to 500 is an average value. When the pres-
sure changes direction at every half-revolution, the lubricant has
a better chance to work fully over the bearing surface, and a
higher value is permissible, say, 500 to 800. In locations where
mere oscillation takes place, not continuous rotation, and reversal
of pressure occurs, as on the cross-head pin of a steam engine, p
may run as high as 900 to 1,200. On the crank pins of locomo-
tives, which have the reversal of pressure, and the benefit of high
velocity through the air to facilitate cooling, the pressures may run
equally high. On the eccentric crank pins of punching and shear-
ing machines, where the pressure acts only for a brief instant and
at intervals, the pressure ranges still higher without any dangerous
heating action.

When a bearing, for practical reasons, is provided VTith a cap
held in place by bolts or studs, the theory of the cap and bolts is
of little importance, unless the load comes directly against the cap
and bolts. Except in the latter case, the proportions of the cap and
the size of the bolts are dependent upon general appearance
and utility, it being manifestly desirable to provide a substantial
design, even though some excess of strength is thereby introduced.

For the worst case of loading, however, which is when the
cap is acted upon by the direct load, such as P in Fig. 76, we have
the condition of a centrally loaded beam supported at the bolts.
It is probable that the beam is partially fixed at the ends by the
clamping of the nut; also that the load P, instead of being con-

MACHINE DESIGN

175

centrated at the center, is to some extent distribnted. It is hardly

Va Fa ,
ndr to assume the external moment equal to -^ or -7—' ^^^ ^°®

being too small, perhaps, and the other too large. It will be .rea-

Fa
sonable to take the external moment at — ^, in which case, equat-

I
I

I .
I

■J

•iu.

4,^'

Pig. 76.
ing the external moment to the internal moment of resistance,

(115)

~6~'

from which, the length of bearing being known, we may calculate
the thickness A.

One bolt on each side is sufficient for bearings not more
than 6 inches long, but for longer bearings we usually find two
bolts on a side. The theoretical location for two bolts on a side,
in order that the bearing may be equally strong at the bolts and
at the center of the length, may be shown by the principles of

mechanics to be -^ L from each end, as indicated in Fig. 76.
The bolts are evidently in direct tension, and if equally loaded

176 MACHINE DESIGN

would each take their fractional share of the whole load P. This

2
is difficult to guarantee, and it is safer to consider that -x- P may

be taken by the bolts on one side. On this basis, for total number
of bolts Tij equating the external force to the internal resistance of
the bolts, we have :

from which the proper commercial diameter may be readily found.
The bracket may have the shape shown in Fig. 77. The
portion at B is under direct shearing stress; and if A be the area
at this point, and S, the safe shearing stress, then, equating the
external force to the internal shearing resistance,

P=AS, (117)

The same shear comes on all parts of the bracket to the left of the
load, but there is an excess of shearing strength at these points.

At the point of fastening, the bolts are in shear, due to the
same load, for which the equation is

P = -^n,S, (ii8)

For the upper bolts, the case is that of direct tension, assum-

ing that the whole bracket tends to rotate about the lower edge E.

To find the load T on these bolts, we should take moments about

the point E, as follows:

PT
PLi=TZ;orT=±p. (up)

Then, equating the external force to the internal resistance,

i - -^ = -^X-^^. (I20)

The upper flange is loaded with the bolt load T, and tends to
break off at the point of connection to the main body of the
bracket, the external moment, therefore, being Tr. The section
of the flange is rectangular; hence the equation of external and in-
ternal moments is:

MACHINE DESIGN

Tr =

SJC

(121)

It may be noted that the lower bolts act on such a small leverage
about E, that they would stretch and thus permit all the load to
be thrown on the upper bolts; this is the reason why they are not
subject to calculation for tension.

Fig. 77.

The section of the bracket to the left of the load P is depend-
ent upon the bending moment, for, if this section is Urge enough
to take the bending moment properly, the shear may be disregard-
ed. It should be calculated at several points, to make sure that
the fiber stress is within allowable limits. The general expression
for the equation of moments is, for any section at leverage x,

from which, by the proper substitution of the moment of in-

178

MACHINE DESIGN

ertia of the section, the fiber Btrese' can be calcnlated. The mo-
ment of inertia for simple ribbed sections can be found in most
handbooks. The process of solution of the above equation, though
simple, is apt to be tedious,
and is not considered neces-
sary to illustrate here.

PRACTICAL MODIFI-
CATION. Adjustment is an
important practical feature of
bearings. Unless the propor-
tions are so ample that wear
is inappreciable, simple and
ready adjustment must be
provided. The taper bush-
ing. Fig. 79, is neat and sat-
isfactory for machinery in
which expense and refinement
are permissible. This is true
of some machine tools, but is
not true of the general " run " of bearings. The most common
form of adjustment is secured by the plain cap (which may or may

not be tongued into the bracket), with liners placed in the joint
when new, which may subsequently be removed or reduced so as
to allow the cap to close down upon the shaft. Several forms of
cap bearings are illustrated in Figs. 80, 81, and 82,

MACHINE DESIGN

179

Large engine shaft bearings have special forms of adjustment
by means of wedges and screws, which take up the wear in all
directions, at the same time accurately preserving the alignment
of the shafts; but this refinement is seldom required for shafts of
ordinary machinery.

In cases where the cap bearing is not applicable, a simple
bushing may be used. This may be removed when worn, and a

Pig. 80. Fig. 81.

new one inserted, the exact alignment being maintained, as the
outside will be concentric with the original axis of shaft, regard-
less of the wear which has taken place in the bore.

The lubrication of bearings is a part of the design, in that
the lubricant should be intro-
duced at the proper point, and
pains taken to guarantee its dis-
tribution to all points of the run-
ning surface. The method of
lubrication should be so certain
that no excuse for its failure
would be possible. Grease is a
successful lubricator for heavy
loads and slow speeds, oil for
light loads and high speeds.

In order to insure the lubri- pjg, g2.

cant reaching the sliding sur-
faces and entering between them, it must be introduced at a point

180 MACHINE DESIGN

where the pressure is moderate, and where the motion of the parts
will naturally lead it to all points of the bearing. Grooves and
channels of ample size assist in this regard. A special form of
bearing uses a ring riding on the shaft to carry the oil constantly
from a small reservoir beneath the shaft up to the top, where it is
distributed along the bearing and finally flows back to the reser-
voir and is used again.

The materials of which bearings are made vary with the
service required and with the refinement of the bearing. Cast iron
makes an excellent bearing for light loads and slow speeds, but
it is very apt to ^' seize " the shaft in case the lubrication is in the
least degree impaired. Bronze, in its many forms of density and
hardness, is extensively used for high-grade bearings, but it also
has little natural lubricating power, and requires careful attention
to keep it in good condition.

Babbitt, a composition metal, of varying degrees of hardness,
is the most universal and satisfactory material for ordinary bear-
ings. It affords a cheap method of production, being poured in
molten form around a mandrel, and firmly retained in its casing or
shell through dovetailed pockets into which the metal flows and
hardens. It requires no boring or extensive fitting. Some
scraping to uniform bearing is necessary in most cases, but this is
easily and cheaply done. Babbitt is a durable material, and has
some natural lubricating power, so that it has less tendency to
heat with scanty lubrication than any of the materials previously
mentioned. Almost any grade of bearing may be produced with
babbitt. In its finest form the babbitt is hammered, or pened,
into the shell of the bearing, and then bored out nearly to size, a
slightly tapered mandrel being subsequently drawn through, com-
pressing the babbitt and giving a polished surface.

A combination bearing of babbitt and bronze is sometimes
used. In this the bronze lies in strips from end to end of the
bearing, and the babbitt fills in between the strips. The shell,
being of bronze, gives the required stiffness, and the babbitt the
favorable running quality.

PROBLEMS ON BEARINGS, BRACKETS, AND STANDS.

1. The allowable pressure on a bearing is 300 pounds per

MACHINE DESIGN 181

square inch of projected area. What is the required length of
the bearing if the total load is 4,500 pounds and the diameter is
3 inches ?

2. The cross -head pin of a steam engine must be 2.5 inches
in diameter to withstand the shearing strain. If the maximum
pressure is 10,000 pounds, what length should be given to the pin ?

3. The journals on the tender of a locomotive are 8 J X 7
inches. The total weight of the tender and load is 60,000 pounds.
If there are 8 journals, what is the pressure per square inch of
projected area ?

4. What horse-power is lost in friction at the circumference
of a 3-inch bearing carrying a load of 6,000 pounds, if the number
of revolutions per minute is 150 and the coefficient of friction is
assumed to be 5 per cent ?

5. The cast-iron bracket in Fig. 77 has a load P of 1,000
pounds. Determine the fiber stress in the web section at the base
of the bracket if the thickness is taken at ^ inch, and L^ = 12
inches; Z := 20 inches; k= 11 inches; ^ = 1 inch.

6. Calculate the diameter of the bolts at the top of the
bracket.

7. Assuming r equal to 6 inches, what is the fiber stress at
the root of flange ?

PROBLEMS ON PART L

The drawings made in accordance with the problems below
should be traced in ink on tracing cloth 18 by 24 inches in size,
and having a border line J inch inside the edge of the paper.

PROBLEMS.

1. Suppose a 30-inch pulley is substituted for the 42 -inch
in the problem given, and that the pulley on the motor remains
10^ inches as before,' how fast must the motor run to give the rope
the same speed, 150 feet per minute ?

2. Will the horse-power of the motor be changed with this
new condition ? Explain fully.

3. Calculate the width of double belt for above condition.

4. What is the torque on the motor shaft for above condition?

5. Calculate the size of shaft in the small pulley for above
condition.

6. Calculate the size of shaft in the 30-inch pulley for above
condition.

7. Design and draw both pulleys for above condition, mak-
ing complete working drawings, and giving all calculations in full.

8. Taking the original problem as given in the text, suppose
it is desired to increase the large gear to 45 inches diameter, cal-
culate the load on the tooth, and a suitable pitch and face to take
this load.

9. How many teeth must the pinion have to give the same
speed of rope, 150 feet per minute, assuming that the motor
runs 470 revolutions per minute, for condition in Problem 8 ?

10. Calculate the bore of pinion for this case.

11. Design and draw the gears for the conditions of Prob.
lems 8 and 9, giving all calculations in full.

MACHINE DESIGN

12. When there is but 3,000 pounds on the rope, what are
the tensions in each end of the brake strap^ assuming that the size
of drum and other conditions remain the same ?

13. How much pressure on the foot lever would it take to
hold this load of 3,000 pounds on the rope ?

14. Suppose we put a bearing 9 inches long on the drum
shaft; the distance, center to center of bearings, would then be 3
feet 8| inches, gears, drum, brake, and load being same as in the
original problem of the text. Calculate the diameter of the drum
shaft.

15. Suppose the height of bracket, center to base, to be 15
inches; length and diameter of bearing, as in Problem 14; and that
we use a separate bracket for the drum bearings, not connected
with the pinion -shaft bearings. Design and draw such a bracket.

16. Calculate, design, and draw all the parts for a machine
similar to that of the text, from the following data:

Load on rope 4,000 pounds.

Speed of rope 175 feet per minute.

Length of rope to be reeled in ... . 250 feet.

Problem 16, is supposed to be worked out on the same lines as
the text, but is wholly original in its nature, being based on entirely new data.
It is not expected that this problem will be attempted except by well-advanced
students who can give considerable time to working it out completely. It will
be found, however, an excellent exercise in original and yet simple design.

^^^■in^iilP 1

^^^^^^^H

STANFORD UNIVERSITY LIBRARIES B
STANFORD AUXILIARY LIBRARY T
STANFORD, CALIFORNIA 94305-6004 '
(650) 723-9201
5olcirc@sulmail.slanford.edu
All books ore subject to recoil.
DATE DUE

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