LIBRARY
OF THE
UNIVERSITY OF CALIFORNIA.
Class
SELF-TAUGHT
MECHANICAL DRAWING
AND ELEMENTARY
MACHINE DESIGN
A Treatise
Comprising the First Principles of Geometric and Mechanical
Drawing, Workshop Mathematics, Mechanics, Strength
of Materials, and the Design of Machine Details,
including Cams, Sprockets, Gearing, Shafts,
Pulleys, Belting, Couplings, Screws and
Bolts, Clutches, Flywheels, etc. Pre-
pared for the Use of Practical
Mechanics and Young
Draftsmen.
By F. L. .SYLVESTER, M.E.
With Additions
By ERIK OBERG
Associate Editor of "Machinery," Author of "Hand-Book of Small
Tools," "Shop Arithmetic for the Machinist," "Advanced Sho^
Arithmetic for the Machinist " "The Use of Logarithms,"
''Solution of Triangles," etc.
FULLY ILLUSTRATED
NEW YORK
THE NORMAN W. HENLEY PUBLISHING CO.
132 NASSAU STREET
1910
•< eA
Copyrighted, 1910, by
The Norman W. Henley Publishing Co.
PREFACE
THE demand for an elementary treatise on
mechanical drawing, including the first principles
of machine design, and presented in such a way
as to meet, in particular, the needs of the student
whose previous theoretical knowledge is limited,
has caused the author to prepare the present vol-
ume. It has been the author's aim to adapt this
treatise to the requirements of the practical me-
chanic and young draftsman, and to present the
matter in as clear and concise a manner as possible,
so as to make " self -study " easy. In order to meet
the demands of this class of students, practically
all the important elements of machine design have
been dealt with, and, besides, algebraic formulas
have been explained and the elements of trigo-
nometry have been treated in a manner suited to
the needs of the practical man.
In arranging the material, the author has first
devoted himself to mechanical drawing, pure and
simple, because a thorough understanding of the
principles of representing objects greatly facilitates
further study of mechanical subjects ; then, atten-
tion has been given to the mathematics necessary
iii
01 9.J.4.1
IV PREFACE
for the solution of the problems in machine design
presented later, and to a practical introduction to
theoretical mechanics and strength of materials;
and, finally, the various elements entering in ma-
chine design, such as cams, gears, sprocket wheels,
cone pulleys, bolts, screws, couplings, clutches,
shafting, fly-wheels, etc., have been treated. This
arrangement makes it possible to present a con-
tinuous course of study which is easily compre-
hended and assimilated even by students of limited
previous training.
Portions of the section on mechanical drawing
was published by the author in The Patternmaker
several years ago. These articles have, however,
been carefully revised to harmonize with the pres-
ent treatise, and in some sections amplified. In
the preparation of the material, the author has
also consulted the works of various authors on
machine design, and credit has been- given in the
text wherever use has been made of material from
such sources.
Several important additions have been made by
Mr. Erik Oberg, Associate Editor of Machinery.
In the preparation of these additions, use has partly
been made of material published from time to time
in Machinery.
THE PUBLISHER.
APRIL, 1910.
CONTENTS
PREFACE Page iii
CHAPTER I
INSTRUMENTS AND MATERIALS
General Remarks on the Study of Drawing — Drawing
Instruments — Pencils — Use of the Instruments —
Paper — Ink Page 1
CHAPTER II
DEFINITIONS OF TERMS USED IN GEOMETRICAL AND
MECHANICAL DRAWING
Point — Line — Surface — Solid — Plane— Angle— Circle
— Parallelogram — Polygon — Ellipse — Involute —
Cycloid — Parabola Page 10
CHAPTER III
GEOMETRICAL PROBLEMS
Bisecting of Lines and Angles — Perpendicular Lines —
Tangents — Regular Polygons — Inscribed and Cir-
cumscribed Circles — Ellipses — Spirals — Involutes
— Cycloids — Parabolas Page 17
Vi CONTENTS
CHAPTER IV
PROJECTION
Mode of Representing Objects — Projections of Inclined
Prisms — Surface Developments of Cones and Pyra-
mids— Intersecting Cylinders, and Cylinder and
Cone— Projection of a Helix — Isometric Projec-
tion Page 32
CHAPTER V
WORKING DRAWINGS
Object of Working Drawings — Assembly Drawings —
Detail Drawings — Dimensions — Finish Marks —
Sectional Views — Cross-section Chart — Screw
Threads— Shade Lines — Tracing and Blue-print-
ing Page 50
CHAPTER VI
ALGEBRAIC FORMULAS
The Meaning of Formulas — Square and S'quare Root —
Cube and Cube Roots — Exponents — Areas and
Volumes of Plane Figures and Solids Page 79
CHAPTER VII
ELEMENTS OF TRIGONOMETRY
Angles — Right-angled Triangles — Trigonometrical
Functions — Tables of Natural Functions — Solution
of Right-angled Triangles — Solution of Oblique-
angled Triangles — Laying Out Angles by Means
of Trigonometric Functions Page 96
CONTENTS Vii
CHAPTER VIII
ELEMENTS OF MECHANICS
Resolution of Forces—Levers — Fixed and Movable
Pulleys — Inclined Planes — The Screw — Differential
Screw — Newton's Laws of Motion — Pendulum —
Falling Bodies — Energy and Work — Horse-power
of Steam Engines Page 120
CHAPTER IX
FIRST PRINCIPLES OF STRENGTH OF MATERIALS
Factor of Safety— Shape of Machine Parts— Strength
of Materials as Given by Kirkaldy's Tests — Stresses
in Castings Page 151
CHAPTER X
CAMS
General Principles — Design of Cams Imparting Uniform
Motion — Reciprocating Cams — Cams Providing
Uniform Return — Uniformly Accelerated Motion
Cams — Gravity Cam Curve — Harmonic Action
Cams — Approximate Gravity Cam Curve . .Page 164
CHAPTER XI
SPROCKET WHEELS
Object of Sprocket Wheels — Drafting of Sprocket
Wheels for Different Classes of Chain— Speed
Ratio Page 185
Viii CONTENTS
CHAPTER XII
GENERAL PRINCIPLES OF GEARING
Friction and Knuckle Gearing — Epicycloidal Gearing
— Gears with Strengthened Flanks — Gears with
Radial Flanks — Involute Gears — Interference in
Involute Gears — The Two Systems Compared —
Twenty-degree Involute Gears — Shrouded Gears —
Bevel Gears — Worm Gearing — Circular Pitch —
Proportions of Teeth — Diametral Pitch — The
Hunting Tooth — Approximate Shapes for Cycloidal
Gear Teeth — Involute Teeth — Proportions of Gears
—Strength of Gear Teeth— Thurston's Rule for
Gear Shafts— Speed Ratio of Gearing. . .Page 190
CHAPTER XIII
CALCULATING THE DIMENSIONS OF GEARS
Spur Gearing — Bevel Gears — Worm Gearing. . Page 222
CHAPTER XIV
CONE PULLEYS
Conical Drums — Influence of Crossed Belt — Cone
Pulleys — Smith's Rule for Laying Out Cone
Pulleys Page 239
CHAPTER XV
BOLTS, STUDS AND SCREWS
Kinds of Screws — United States Standard Screw Thread
—Check or Lock Nuts— Bolts to Withstand Shock
— Wrench Action — Screws for Power Transmission
— Efficiency of Screws — Acme Standard Thread-
Miscellaneous Screw Thread Systems — Other Com-
mercial Forms of Screws Page 243
CONTENTS ix
CHAPTER XVI
COUPLINGS AND CLUTCHES
Simple Forms of Couplings — Calculation of Flange
Coupling Bolts — Oldham's Coupling — Hooke's
Coupling or Universal Joint — Toothed Clutches —
Friction Clutches— Cone Clutches Page 259
CHAPTER XVII
SHAFTS, BELTS AND PULLEYS
Calculation of Shafting— Horse-power of Belting —
Speed of Belting — Pulley Sizes and Speed Ratios
—Twisted and Unusual Cases of Belting . .Page 272
CHAPTER XVIII
FLY-WHEELS FOR PRESSES, PUNCHES, ETC.
Object of Fly-wheels — Formulas for Fly-wheel Calcu-
lations— Example of Fly-wheel Calculation for
Shears Page 289
CHAPTER XIX
TRAINS OF MECHANISM
To Secure Increase of Speed — To Secure Reversal of
Direction — The Compound Idler — The Screw Cut-
ting Train — Simplified Rules for Calculating Lathe
Change Gears — Back-Gears Page 297
X CONTENTS
CHAPTER XX
QUICK RETURN MOTIONS
Object of Quick Return Motions — Examples of Simple
Designs of Quick Return Motions — The Whitworth
Quick Return Device — The Elliptic Gear Quick
Return Page 313
SELF-TAUGHT
MECHANICAL DRAWING
CHAPTER I
INSTRUMENTS AND MATERIALS
ONE who is to study the subject of drawing
should not merely read a book on the subject, but
should prepare sheets of exercises. This will fix
the principles which he learns in his mind in a way
as reading alone will not do, and will give him
practical experience in the use of the tools. The
geometrical problems given in this book make
perhaps the best of subjects for a beginning, as
their proper execution will require careful work.
Later, the student may make dimensioned free-
hand sketches of some machine with which he is
familiar, and from these sketches he may make up
a set of finished working drawings. In all of this
work, care should be taken to have it so laid out,
with proper margins and spaces between different
parts, that the drawing when finished shall pre-
sent an appearance of neatness and methodical
arrangement.
For the purposes of the student, a drawing board
about 15 by 18 inches will be large enough. With
this should be an 18-inch T-square, a pair of 6-inch
triangles, and a set of three or four irregular curves.
1
SELF-TAUGHT MECHANICAL DRAWING
For drawing full-size work, a good flat beveled-
edge rule will answer ordinary requirements, but
for making half- or quarter-size drawings some
kind of a "scale" ' will be found desirable. The tri-
angular scale shown in Fig. 1 is perhaps the one
mostly used, and it has the advantage of possess-
\ Ytt \V\\ \ Y\\ \\\\ \\\\ \X\\ \\Tvl\\ \Y\\ \
FIG. 1. -The Triangular Scale.
ing six surfaces for graduations, giving variety
enough for all sorts of conditions, but it has the
disadvantage of persistently presenting the wrong
edge, and putting one to the trouble of turning it
over and over to get the desired edge. This trouble
may, of course, be overcome by using a scale guard
such as is shown in Fig. 2, but the guard is itself
often in the way. As
but two or three differ-
ent scales, aside from
full size, will be likely
to be required, it will be
found much more con-
FIG. 2. -Scale Guard or Holder venient to have a sep-
used on Triangular Scale. arate flat scale for each
graduation. Such scales
may be purchased, or, if one is satisfied with the
open graduation system shown in Fig. 3, he may
make them without much trouble himself. In this
system, only one inch is divided, this inch being
numbered 0; and measurements which include a
INSTRUMENTS AND MATERIALS 3
fractional part of an inch are reckoned from the
required whole number to the proper place on the
divided inch.
The drawing instruments themselves, while not
necessarily of the highest price, should be of a
good serviceable quality of German silver. The
cheap brass or nickel plated school sets should not
be considered, as they will prove unsatisfactory
for regular work. It is not necessary to have a
large number of instruments. A very good set,
sufficient for all ordinary requirements, might be
as follows: First a pair of about 4J- or 5-inch com-
11 12 13 14 15 16 17 "18
FIG. 3.— Inexpensive Type of Scale.
passes with fixed needle points (bayonet points are
useless) and interchangeable pin and pencil points,
with lengthening bar. Then, a pair of hair-
spring spacers of about the same size. These re-
semble ordinary plain compasses, but the steel end
of one leg is made adjustable by means of a
thumb screw. Next, a pair of ruling pens, one
large and one small, and, lastly, a set of three
spring instruments, pen, pencil and spacers, for
small work. Rather than to get cheap instru-
ments, it would be advisable to obtain a set gradu-
ally by getting the large instruments and one pen
first, and adding the second pen and the spring
instruments later. The large compasses can, if
necessary, be used to make circles of from about
i inch to about 18 or 20 inches in diameter, so
4 SELF-TAUGHT MECHANICAL DRAWING
that they will do very well for a beginning. For
making larger circles, beam compasses, in which
separate heads for the needle point and for the pen
or pencil point are attached to a wooden bar, after
the manner of workmen's trammels, are used.
A convenient case for the instruments, when
they are bought separately, is shown in Fig. 4,
and is made as follows: Take two pieces of
chamois skin or thin broadcloth, one of them about
one-half longer than the longest instrument, and
somewhat wider than all of them when they are
FIG. 4.— Home-made Instrument Case.
laid out side by side, and the second one of the
same width as the first, but somewhat shorter than
the longest instrument. This second piece is
sewed onto the large piece at one end by the outer
edges. Pockets for the reception of the instru-
ments are then made as shown, and when the free
end of the large piece is folded over, the instru-
ments are rolled up together.
The pencils, which to avoid scratching particles,
should be of best quality, should not be sharpened
to a round point, but to a flat oval point, as such
a shape will wear longer than a round point ; the
leads used in the compasses, however, should be
INSTRUMENTS AND MATERIALS 5
only slightly flattened. It will be found desirable
to have two grades of pencils, one quite hard,
about "4H," to be used for laying out work, and
a softer one, about "2H," to be used for going
over the lines of work which is not to be inked in.
In laying-out work where the hard pencil is used,
only a moderate pressure should be applied, so as
to permit of erasures at any time, whether for the
purpose of making alterations, or to free the draw-
ing of pencil marks after inking.
The drawing pens should be kept sharp, though
not so sharp as to cut the paper, and their ends
should present a neat oval shape. The needle
points of the compasses should also be kept sharp
to avoid the tendency to slip when doing work
where it is undesirable to prick through the pa-
per. A small Arkansas stone will be found useful
for this purpose. Where much use is made of a
given center, it may be desirable to employ a horn
or metal center, such as are kept in stock by deal-
ers in artists' supplies, to avoid the troublesome
enlargement of the center in the paper which the
points of the compasses would otherwise make.
In making a drawing, care should be taken to
have the preliminary pencil work done correctly.
It is a mistake which beginners are likely to make,
to think that errors in the pencil work may be
readily corrected in the inking. This, however,
is usually another case where "haste makes
waste." It is much better to spend a little extra
time on the pencil work, than to have to throw
away a nearly finished ink drawing and do the
work all over again. In locating the various
6
SELF-TAUGHT MECHANICAL DRAWING
views of a drawing upon the paper, it will fre-
quently be found to be well to make rough sketches
of it on scrap paper. These sketches can then be
moved around on the drawing paper until the best
arrangement is secured.
In making a drawing, it will be found most con-
venient, ordinarily, to limit the use of the T-square
to horizontal lines, the head of the square being
kept pressed firmly against the left-hand end of
the drawing board. Vertical lines are then made
FIG. 5. —Appearance of Carelessly made Drawing.
with the aid of the triangles resting against the
blade of the T-square. Vertical lines which are
too long to be made in this way, are, of course,
made with the T-square itself. In inking in a
drawing, it is best to draw all curved or circular
lines first, as it is easier to join straight lines onto
curved lines than to join curved lines onto straight
lines. Care should also be taken to have meeting
lines just meet, whether they meet end to end or
at an angle. Carelessness in this respect gives a
drawing a very bad appearance, as shown by Fig.
5, A and B.
INSTRUMENTS AND MATERIALS 7
In using the pens, whether the ruling or the com-
pass pens, care should be taken to see that both nibs
rest upon the paper, otherwise lines such as shown
in Fig. 6 may result. If the pen does rest squarely
upon the paper, and such lines continue to appear,
it is fair to infer that the paper has become some-
what greasy, perhaps from too much handling.
This trouble may be avoided, and the work kept
cleaner, by having a piece of thin paper inter-
posed between the hands and the drawing paper.
The cross hatching work, such as is shown at A
in Fig. 5, is frequently done by simply using one
of the triangles resting against the blade of the
FlG. 6.— Line Resulting from not Having both Pen Points or
Nibs Resting on the Paper when Inking.
T-square, the same as is done for vertical lines, the
spacing being done entirely by the eye; but unless
one is doing a good deal of this work, so as to
keep in practice, he will find it very difficult to
make the spacing regular. There are various sec-
tion-lining devices on the market for doing this
work, some of them quite expensive. Fig. 7 shows
a simple device for cross-sectioning, which serves
the purpose as well as any of the more elaborate
ones, and possesses the additional advantage that
anyone may readily make it for himself. This
instrument was shown by Mr. E. W. Beardsley in
Machinery, September, 1905. An old instrument
screw, B, is screwed into a slightly smaller hole
in a piece of wood, A, shaped as shown, and of a
8
SELF-TAUGHT MECHANICAL DRAWING
thickness a little in excess of the diameter of .the
screw-head. This combination is then used in the
central hole in a triangle, as shown. Then, with
one finger on the triangle itself, and with another
one on A, the two may be moved along, first one
and then the other, for section lining, the desired
width of space being secured by the adjustment
given to B.
For making erasures of ink lines on paper, a
steel scraping eraser or a sharp knife blade is usu-
FIG. 7. — Simple Cross-section Liner.
ally the best, the roughened surface being after-
wards rubbed down smooth with some hard sub-
stance. When making erasures of either pencil
or ink with a rubber eraser, an erasing shield,
such as is shown in Fig. 8, is useful for prevent-
ing rubbing out more than is intended. These
shields are made both of thin sheet metal and of
celluloid; the metal ones, being the thinner, are
the more convenient to use.
Tho paper used, if good work is desired, should
INSTRUMENTS AND MATERIALS 9
be regular drawing paper, whether it be white or
brown. This has an unglazed surface, and will be
found much more satisfactory in every way than
common paper. The glazed surface of the cheaper
paper does not take pencil marks well, and is torn
up badly in making erasures. Such paper, if used
at all, should be used only on the most temporary
FIG. 8.— Erasing Shield made from Sheet Metal
or Celluloid.
work. Of white drawing papers, the smooth sur-
faced kinds should be selected. For making ink
drawings, it will be found most satisfactory to use
the prepared drawing inks, rather than to go to
the trouble of preparing it oneself from the stick
India ink.
For fastening the paper on to the board, common
one-half-ounce copper tacks are as good, if not
preferable, to other fastening means.
CHAPTER II
DEFINITIONS OF TERMS USED IN GEOMETRICAL
AND MECHANICAL DRAWING
1. A Point has position, but not magnitude.
2. A Line has length, but neither breadth nor
thickness.
3. A Surface has length and breadth, but not
thickness.
4. A Solid has length, breadth and thickness.
5. A Plane is a surface which is straight in
every direction; that is, one which is perfectly
flat.
6. Parallel lines are such as are everywhere
equally distant from each other. Circular lines
which answer to this condition are also said to be
concentric.
7. An Angle is the difference in the direction
of two lines. If the lines meet, the point of meet-
ing is called the vertex of the angle, and the lines
ab and ac, Fig. 9, are its sides.
8. If a straight line meets another so that the
adjacent angles are equal, each of these angles is
a right angle, and the two lines are perpendicular
to each other. Thus the angles acd and deb, Fig.
10, are right angles, and the lines ab and dc are
perpendicular to each other. A distinction is to
be made here between the words perpendicular
10
DEFINITIONS OF TERMS
11
and vertical. A vertical line is one which is per-
pendicular to the plane of the earth's horizon ; that
is, to the surface of still water.
9. An Obtuse Angle is one which is greater
than a right angle, as ace, Fig. 10.
10. An Acute Angle is one which is less than a
right angle, as ecb, Fig. 10.
11. It is obvious that the sum of all the angles
which may be formed about the point c, Fig. 10,
above the line ab will be equal to the two right
angles acd and deb.
FIG. 9.— Angle.
FIG. 10. —Illustration for Making
Clear the Terms Right, Acute
and Obtuse Angles.
12. The Complement of an angle is a right angle,
less the given angle. Thus bcet Fig. 10, is the
complement of dee.
13. The Supplement of an angle is two right
angles less the given angle. Thus bee, Fig. 10, is
the supplement of ace.
14. A Circle is a continuous curved line, Fig. 11,
or the space enclosed by such line, every point of
which is equally distant from a point within called
the center.
15. The distance across a circle, measured
through the center, is the diameter. The distance
around the circle is the circumference. The dis-
12 SELF-TAUGHT MECHANICAL DRAWING
tance from the center to the circumference is the
radius.
16. The ratio between the circumference and
the diameter, that is, the circumference divided
by the diameter, is 3.1416. While this is not exact
(Bradbury's Geometry states that it has been car-
ried out to two hundred and fifty places of deci-
mals), it is near enough for practical purposes.
This ratio is frequently represented by the Greek
letter TT (pi).
17. A circle is considered as being equally divided
FIG. 11. -Illustration for FIG. 12. -Similar Triangles.
Making Clear the Terms
Relating to the Circle.
into three hundred and sixty degrees (360°), each
degree into sixty minutes (60'), and each minute
into sixty seconds (60").
18. If two diameters cross each other at right
angles, the circle is divided into four equal parts;
hence a right angle contains ninety degrees.
19. An Arc of a circle is any part of its circum-
ference, as abc, Fig. 11.
20. A Chord is a straight line joining the ends
of an arc, as ac, Fig. 11.
21. Two triangles, as abc and dec, Fig. 12, hav-
ing like angles are similar triangles. The corre-
DEFINITIONS OF TERMS
13
spending sides of similar triangles have the same
ratio. Thus if ac were twice as long as dc, ab
would be twice as long as de, and be would be
twice as long as ec.
22. The sum of the angles of a triangle is equal
to two right angles. Let abc, Fig. 13, represent
any triangle. Extend one side, ac, as shown, and
make cd parallel with ab. Then the angle dee is
equal to the angle bac, for their sides have the
same direction, and the angle bed is equal to the
FIG. 13.— Illustration for
Showing that the Sum of
the Angles in a Triangle
equals Two Right Angles.
FIG. 14.— Tangent and Nor-
mal to a Curve.
angle abc, for their sides have opposite directions;
hence the sum of the three angles formed about
the point c is equal to the sum of the three angles
of the triangle abc, and these are equal to two
right angles (11).
23. A Tangent is a line which touches another,
but does not, though extended, cross it. Thus, a,
b and c, Fig. 14, are tangent lines. A line, d,
perpendicular to the straight line 6, at the point
of tangency, is called a normal. If one of the
14
SELF-TAUGHT MECHANICAL DRAWING
lines, as a, is circular, the normal will pass through
its center.
24. A Parallelogram is a figure whose opposite
sides are parallel, as ab and cd, or eb and fd in
Fig. 15. The sides may all be of equal length,
FIG. 15. —Parallelograms.
FIG. 16. -Square.
(See
when the parallelogram is called a square.
Fig. 16.)
25. Figures having five, six or eight sides are
called respectively Pentagon, Hexagon and Octagon.
These, and all figures having more than four sides,
are called Polygons. If the sides in a polygon are
FIG. 17.— Regular Polygon.
FIG. 18.— Ellipse.
all of equal length, and all the angles equal, the
polygon is called a regular polygon. (See Fig. 17.)
26. An Ellipse, Fig. 18, is a continuous curved
line, or the space enclosed by such line, of such
shape that the sum of the distances from two
DEFINITIONS OF TERMS
15
points within, as a and 6, called the foci (singu-
lar: focus), to any point upon its circumference
is constant. Thus al plus bl equals a2 plus b2 or
a3 plus b3.
27. An Involute is a line of such shape (as a in
FIG. 19.— Involute.
FIG. 20.— Cycloid.
Fig. 19) as might be made by a pencil at the end
of a string which is unwound from a circle.
28. A Cycloid is a line of such shape (as a in
Fig. 20) as might be made
by a pencil fastened to the
circumference of a circle
which is being rolled upon
a straight line. If the circle
was being rolled upon the
convex side of a circular
line the line traced by the
pencil would be an epicy-
cloid. If it was being rolled
upon the concave side of a
circular line, the line traced
by the pencil would be a
hypocycloid. The involute
and cycloidal curves are used in gear outlines.
29. A Parabola is a curve which may be ob-
FIG. 21. Method of Sec-
tioning a Cone to Ob-
tain a Parabola.
16 SELF-TAUGHT MECHANICAL DRAWING
tained by cutting a cone so that the exposed
sectional surface will be parallel with one of the
sides of the cone, as shown in Fig. 21. This
curve, as shown in Fig. 22, is of such shape that
lines drawn to it from a certain point within,
called the focus, shown at / in the illustration,
niake the same angle with it as lines drawn from
/7
/\7
FIG. 22. Parabola.
the intersection points parallel with the axis ax.
Thus the line fm makes the same angle with the
parabola, at the point of intersection, as the line
ml. Because of this property of the parabola,
mirrors of this shape are used in headlights of
locomotives, in search lights," and in many light-
houses ; because, if a light be placed at the focus,
its rays, when reflected from the mirror, will be
thrown out in parallel lines.
CHAPTER III
GEOMETRICAL PROBLEMS
Prob. 1, Fig. 23. To bisect a line, either curved
as abc, or straight as ac. — With centers at a and c
and with a radius somewhat greater than half the
length of the line, describe the arcs d and e. A
line passing through the intersections of these arcs
bisects either line. It will also pass through the
center of the circle of which the arc abc is a part.
Prob. 2, Fig. 24. To bisect an angle. — With
FIG. 23. — Bisecting a Line. FIG. 24.— Bisecting an Angle.
center at a, and with any convenient radius, de-
scribe the arc be. With centers at b and c, and
with a radius greater than half the arc, describe
the arcs d and e. A line from a through the inter-
section of these arcs bisects the angle.
Prob. 3, Fig. 25. To make an angle equal to a
given angle. — Let a be the given angle, and let it
be desired to make an angle equal to it on the line
dg. With center at a make the arc be, and then
with center at d make the arc eh with the same
17
18
SELF-TAUGHT MECHANICAL DRAWING
radius. Then with a radius equal to be, and with
center at h, make the arc /. A line from d through
the intersection of the arcs gives the required
angle.
Prob. 4, Fig. 26. To erect a perpendicular at the
end of a line, ab. — With any convenient center, c,
FIG. 25. — Making an Angle Equal to a Given Angle.
and with radius cb, draw a semicircle intersecting
ab at d. Draw a line from d through c intersect-
ing the semicircle at e. A line from 6 passing
through e is the required perpendicular.
Prob. 5, Fig. 27. To drop a perpendicular from
a point a, to a given line be. — With a as a center,
FIG. 26.— Erecting a Perpen-
dicular Line.
r
FIG. 27. — Drawing a Perpen-
dicular Line.
draw an arc intersecting be at d and e. With d
and e as centers draw the intersecting arcs / and
g. A line from a through the intersection of
these arcs is the required perpendicular. If a
were over one end of the line be the process shown
GEOMETRICAL PROBLEMS 19
in the preceding problem might be reversed by
drawing a line from a corresponding to de, Fig.
26, and upon this line drawing a semicircle, when
its intersection with the base line would give the
point to which the perpendicular from a should be
drawn.
Prob. 6, Fig. 28. To draw a tangent to a circle
at a given point. — Draw a radius of the circle to
the required point, and erect a perpendicular to it,
which will be the required tangent. To find the
point of tangency of a line to a circle, drop a per-
FIG. 28.— Drawing a Tangent FIG. 29.— Finding the Center
to a Circle. of a Circle.
pendicular to the tangent from the center of the
circle.
Prob. 7, Fig. 29. To find the center of a circle.—
Mark off two arcs as ab and ac upon the circumfer-
ence, and bisect these arcs as in Prob. 1. Where
these bisecting lines cross each other will be the
required center.
Prob. 8, Fig. 30. To draw a regular hexagon
upon a given base, ab. — With a radius equal to the
length of ab draw the arcs c and d. The intersec-
tion of these arcs will be the center of a circum-
scribing circle upon which the other sides may be
marked off.
20
SELF-TAUGHT MECHANICAL DRAWING
Prob. 9, Fig. SI. To draw a regular octagon in
a square. — Draw the diagonals of the square, ad
and be, and with a radius equal to half of a diago-
nal, and with centers at a, b, c and d, draw the
arcs e, f, g and h. The intersections of these arcs
FIG. 30.— Drawing a Regular
Hexagon.
FIG. 31. — Drawing a Regular
Octagon.
with the sides of the square give the corners of
the required octagon.
Prob. 10, Fig. 32. To draw a circle about a tri-
angle, as abc. — Bisect any two of the sides as in
Prob. 1. Where the bisecting lines cross each
32. — Drawing a Circle
about a Triangle.
FIG. 33.— Inscribing a Circle
in a Triangle.
other will be the center of the required circle. In
a similar manner a center may be found from
which to draw a circle through any three given
points, the given points in this case being the cor-
ners of the triangle.
GEOMETRICAL PROBLEMS
21
Prob. 11, Fig. 33. To draw a circle within a
given triangle, as abc. — Bisect any two of the angles
as in Prob. 2. Where the bisecting lines cross, will
be the center of the required circle. In a similar
manner a center may be found from which to draw
a circle tangent to any three given straight lines.
Prob. 12, Fig. 34. To find the foci of an ellipse. —
Draw the long and the short diameters of the
ellipse, ab and cd, and with a radius equal to half
of the long diameter, and with a center at c or d
FIG. 34.— Finding the Foci of
an Ellipse.
FIG. 35.— Simplified Method
of Drawing an Ellipse.
draw the arcs e and /. Where these arcs intersect
the long diameter will be the required foci.
Prob. 13, Fig. 35. To draw an ellipse with a
pencil and thread. — Having found the foci of the
ellipse, stick a pin firmly into each focus, and loop-
ing a thread around them, allow it to be slack
enough so that the pencil will draw it out to the
end of the short diameter. The thread will then
guide the pencil so that it will draw an ellipse. A
groove should be cut around the pencil lead to pre-
vent the thread from slipping off.
Prob. 14, Fig. 36. To draw an ellipse with a
trammel. — Lay out the long and the short diame-
ters of the ellipse, ab and cd, and on a strip of
paper, A, mark off 1-3 equal to half of the long diam-
22 SELF-TAUGHT MECHANICAL DRAWING
eter, and 2-3 equal to half of the short diameter.
Then, keeping point 1 on the short diameter, and
point 2 on the long diameter, mark off any desired
number of points at 3. A curved line passing
through these points will be the required ellipse.
The ellipsograph, an instrument for drawing el-
lipses, is made on this principle, points at 1 and
2 traveling in grooves which coincide with ab
and cd.
Prob. 15, Fig. 37. To draw an ellipse by tangent
lines. — Make ab equal to one-half of the long di-
d
FIG. 36.— Another Method of FIG. 37.— Drawing an Ellipse
Drawing an Ellipse. by Tangents.
ameter of the required ellipse, and be equal to one-
half its short diameter. Divide ab and be into
the same number of equal parts, and, numbering
them as indicated, connect 1 and 1' ', 2 and 2' and
so forth. A curved line starting at a, tangent to
these lines, and ending at c, is one-quarter of the
required ellipse.
Prob. 16, Fig. 38. To draw an approximate el-
lipse with compasses, using four centers. — Lay out
the long diameter ab, and the short diameter cd,
crossing each other centrally at o. From 6 meas-
ure off be equal to co, one-half of the short diam-
eter. The length ae will then be the radius gh
for forming the part hk of the ellipse. From e
GEOMETRICAL PROBLEMS
23
mark off the point/, making ef equal to one half
of oe. The point / will be the center, and fb the
radius for forming the end of the ellipse. Lines
drawn from the centers g through the points / de-
termine the points at which the different curves
meet. This method is not considered applicable
when the short diameter is less than two-thirds of
the long diameter.
FIG. 38. — Drawing an Approximate Ellipse by Four
Circular Arcs.
Prob. 17, Figs. 39 and 39a. To draw an approx-
imate ellipse with compasses, using eight centers.—
Lay out the long diameter ab, and the short diam-
eter cd crossing each other centrally at /. Con-
struct the parallelogram aecf, and draw the diago-
nal ac. From e draw a line at right angles to ac,
crossing the long diameter at h, and meeting the
short diameter, extended, at g. Point g is the center
from which to strike the sides of the ellipse, and
24 SELF-TAUGHT MECHANICAL DRAWING
h will be the center, subject to certain modifica-
tions for narrow ellipses, from which to strike the
ends of the ellipse. To get the radius of the third
curve for connecting the side and end curves, lay
off a base line ab, Fig. 39 A, of any convenient
length, and divide it into five equal parts by the
points 1, 2, 3 and 4. At one end of the line erect
the perpendicular ac, equal to the end radius ah,
and at the other end erect the perpendicular bd
equal to the side radius eg. Connect the ends of
these perpendiculars by the line cd, and at point
2 erect a perpendicular, meeting cd at e. The
length e2 will be the desired third radius. With
the compasses set to this radius, find a center i
from which a curve can be struck which will be
just tangent to the side and end curves. From
other centers similarly located the remainder of
the ellipse is drawn. Lines drawn from i through
h, and from g through i determine the meeting
points of the different curves.
For narrow ellipses the length of the end radius,
ah, should be increased as follows : For an ellipse
having its breadth equal to one-half of its length,
make ah one-eighth longer. For an ellipse having
its breadth one-third of its length, make ah one-
fourth longer. For an ellipse having its breadth
equal one-quarter of its length, make ah one-half
longer. For intermediate breadths lengthen ah
proportionately. With this modification of the
length of the end radius, this method gives curves
which blend well together so as to satisfy the eye,
and gives a figure which conforms quite closely to
the actual outlines of an ellipse.
GEOMETRICAL PROBLEMS
25
FIG. 39a.
FIGS. 39 and 39a. — Drawing an Approximate Ellipse by
Eight Circular Arcs.
26 SELF-TAUGHT MECHANICAL DRAWING
Prob. 18, Fig. 40. To draw a regular polygon of
any number of sides on a given base, ab.— Extend ab
as shown, and on it with one end as a center and
a radius equal to the length of the given side, draw
a semicircle. Divide this semicircle into as many
equal spaces as there are to be sides to the polygon.
A line from b to the second space, reckoning from
where the semicircle meets the extension of ab,
will be a second side of the required polygon.
Lines are then drawn from b through the remain-
ing divisions of the semicircle, and the remaining
FIG. 40. — Drawing a Regular FIG. 41. — Drawing a Spiral
Pentagon. about a Square.
sides of the polygon are marked off upon them as
indicated. If the polygon is to have many sides,
as an additional precaution against error, bisect ab
and b2, thus getting the center of a circumscribing
circle upon which the remaining sides may be
marked off.
Prob. 19, Fig. 41. To draw a spiral about a
square. — Lay out a square, 1-2-3-4, having the
length of each side equal to one-quarter of the de-
sired distance between the successive convolutions
of the spiral, and extend each side in one direction
as shown. With a center at 2, and with a radius
1-2 draw a quarter of a circle. With a center at 3
GEOMETRICAL PROBLEMS 27
draw another quarter of a circle, continuing the
first one, and so continue with successive corners
of the square for centers.
Fig. 42 shows how, by similarly extending one
end of each side, a spiral may be drawn about a
regular polygon of any number of sides. A curve
so formed determines the shape of the teeth of
sprocket wheels.
Prob. 20, Fig. 43. To draw an involute. — Upon
the circumference of the given circle mark off any
FIG. 42. — Drawing a Spiral FIG. 43.— Drawing an Invo-
about a Regular Polygon. lute.
number of equally distant points, as 0-1-2-3, etc.,
and draw lines tangent to the circle at these points,
beginning at point 1. Then with the compasses
set the same as for marking off the spaces on the
circle, mark off one space on line 1, two spaces on
line 2, three spaces on line 3t and so forth. A
curved line starting at 0 and passing through these
points will be the required involute. This curve
is used for the shape of the teeth of involute gears.
Prob. 21, Fig. 44. To draw a cycloid. — Upon the
base line ab mark off any number of equally dis-
tant points, as 0-1-2-3, etc., the distance between
28
SELF-TAUGHT MECHANICAL DRAWING
them being made, for convenience sake, about one-
sixth of half the circumference of the generating
circle. Beginning at 1 erect perpendiculars from
these points, and with centers on these lines draw
arcs of circles tangent to the base line to represent
FIG. 44.— Drawing a Cy-
cloid.
FIG. 45.— Drawing an Epicy-
cloid.
successive positions of the generating circle as it
is rolled along. With the compasses set as for
spacing off the base line, mark off one space on the
arc which starts from point 1, two spaces on arc
2, three spaces on arc 3, and so forth. A curved
line starting at 0 and pass-
ing through' the points
thus obtained will be the
required cycloid.
An epicycloid, Fig. 45,
or a hypocycloid, Fig. 46,
is formed in precisely the
same way, excepting that
as the base line, ab, is an arc of a circle, the center
lines from points 1-2-3, etc., are made radial.
These three cycloidal curves are used for the
shape of the teeth of epicycloidal gears, sometimes
called simply cycloidal gears.
FIG. 46.— Drawing a Hypo-
cycloid.
GEOMETRICAL PROBLEMS
29
Prob. 22, Fig. 47. To draw a parabola by means
of intersecting lines. — Draw the axis ax, and on it
mark the focus / and the vertex v, and at right
angles to it draw the line be at a distance from v
equal to the distance of v from/. Across the axis,
and at right angles to it, draw a number of lines,
1, 2, 3, 4, 5, 6. Then with radius al, and with
center at the focus /, draw arcs intersecting line
1; with radius a2, and with center again on /draw
arcs intersecting line 2, and so on. A curved line
FIG. 47.— Drawing a Parabola.
passing through these intersections will be a para-
bola. It will be seen from this method of drawing
a parabola that any point on it is equally distant
from the focus, and from the line be, called the
directrix.
Prob. 23, Fig. 48. To draw a parabola with a
pencil and string. — Lay out the axis, the focus, the
vertex and the directrix as before. Attach one
end of a thread to the focus, / by means of a pin,
and attach the other end of the thread to the
square shown at d, having the thread of such
30
SELF-TAUGHT MECHANICAL DRAWING
length that when the inner edge of the square is
on the axis, ax, the thread if drawn down with
a pencil will just reach to the vertex, v. Now
slide the square along be in the direction of the
FIG. 48.— Simplified Method of Drawing a Parabola.
arrow, keeping the pencil against the square; the
thread will cause the pencil to move along so as to
describe a parabola as shown.
Prob. 24, Fig. 49. To draw a parabola of a given
\
FIG. 49.— Another Method of Drawing a Parabola.
breadth of opening, ab, and of a given depth, cd.—
Draw ef parallel with ab, and draw ae and bf paral-
lel with cd, having ac and be equal. Space off dc
GEOMETRICAL PROBLEMS
31
and df into any number of equal parts, and also
space off ea and/6 into the same number of equal
parts, as shown. From d draw lines to the di-
visions on ea and/6, and from 1, 2, 3 and 4 on de
and df draw perpendicular lines to intersect the
lines drawn from d to 1, 2, 3 and 4 on lines ca and
fb. A curved line passing through these inter-
sections will be the required parabola.
Prob. 25, Fig. 50. To find the focus of a para-
bola.— Let abed be the given parabola, eft being its
FIG. 50.— Finding the Focus of a Parabola.
axis. Across the parabola at its vertex, v, draw
the line ij at right angles to the axis. From any
point, g, on the parabola, draw the line gh parallel
to the axis. With center at g.fmd a radius, by
trial, which will cut the axis as much inside the
vertex, v, as it cuts the line gh beyond the line ij.
The intersection at x will be the required focus.
CHAPTER IV
PROJECTION
Mode of Representing Objects. — In mechanical
drawing, machines, or parts of machines, are rep-
resented by views, generally three, in which per-
spective is ignored, and which show the object in
different positions at right angles to each other.
The mode of representing these views, and their
positions with regard to one another, which expe-
rience has shown to be most convenient is perhaps
best shown by means of the familiar cardboard
illustration. Let abcdefgh, Fig. 51, represent a
piece of cardboard, which we will suppose to be
transparent, creased on the dotted lines to permit
of the outer portions being turned back. Let us
now suppose that we have a prism shaped as shown
at C, and of the length shown at A. If the prism
is stood upright with its broad side facing the ob-
server, and the cardboard, being blank, is held up
in front of it, the prism will appear, if all its lines
are brought perpendicularly forward to the card-
board, as it is shown at A, lines on the prism
which would be hidden by its body, as the further
corner, being dotted. If section Cof the cardboard
is now turned backward through an angle of 90
degrees over the top of the prism we would get the
view shown in that part, all lines being brought
32
PROJECTION
33
perpendicularly forward from the prism to the
cardboard as before. Likewise if part D of the
cardboard were turned backward through an angle
of 90 degrees, and the lines of the prism were
brought perpendicularly forward onto it, we would
get the view shown in that part. The view shown
at A is called the elevation, that shown at C is
called the plan, and that shown at D is called the
side view. Occasionally a piece is so shaped, or
FIG. 51. — Principle of Projection.
has so much of detail to it as to make another side
view desirable; such a view would be placed at B.
In many other cases, as in the case of the prism
here shown, the plan and elevation views alone
will fully show the object.
The production of these views from one another
is called projection ; and by the use of connecting
lines, and also at times of temporary construction
views, objects may be shown at any desired angle,
irregular or curved lines may be traced, and sur-
faces may be developed.
34
SELF-TAUGHT MECHANICAL DRAWING
An Upright Prism. — Fig. 52 shows a prism in its
simplest position. A moment's examination will
show that the elevation cannot be drawn directly,
as the distance apart of the vertical lines which
represent the corners of the prism, cannot be deter-
mined without other aid; hence it is necessary to
draw the plan view first. Horizontal lines having
been made to give the height of the prism in the
elevation, the vertical lines may then be drawn in
FIG. 52.— Projections of
Prism.
FIG. 53.— Projections of
Tilted Prism.
from the plan, as indicated by the vertical dotted
line.
The Prism Inclined at One Angle.— Fig. 53 shows
the prism inclined to the right. A brief exami-
nation of these views will show that none of them
can be drawn directly, as the distance apart of
the vertical lines in the elevation and side views
is not known, and the lines of the plan view are
foreshortened; but the views can be developed
from Fig. 52. It is evident that as the prism is
tipped, the elevation view will remain unchanged,
PROJECTION 35
hence the first step will be to reproduce that view
inclined at the desired angle. As the prism is
tipped it is also evident that all points in the plan
view of Fig. 52 will move in horizontal lines to the
right, hence horizontal lines are drawn from these
points through the position which the plan will
occupy in Fig. 53. The intersection of these lines
with vertical lines from the corresponding points
in the elevation will determine the position of
each point in the plan. The points so determined
one by one being then connected by straight lines,
gives the plan view as shown. To make the side
view, horizontal lines are first drawn from the
various points of the prism as seen in the eleva-
tion through the position which the side view will
occupy. Then, bearing in mind that each point
of the prism in the side view will be as much to
the left of the vertical line ab as the same point
in the plan is below the line ccZ, the position of
each point on the horizontal lines is marked off
from ab.
The Prism Inclined at Two Angles. — Fig. 54 shows
the prism tipped forward after having been tipped
to the right as shown in Fig. 53. An examination
of these views will show that not only can they
not be drawn directly, but they cannot be devel-
oped from Fig. 52. They may, however, be de-
veloped from Fig. 53. It is evident that as the
prism is tipped forward, the side view of Fig. 53
will remain unchanged ; hence the first step will
be to reproduce that view inclined at the desired
angle. Next, horizontal lines are drawn from the
corners of the prism as seen in this view through
36
SELF-TAUGHT MECHANICAL DRAWING
the place which the elevation is to occupy, and the
perpendicular line gh is drawn. It is evident that
as the prism is tipped forward, the different points
of it as seen in the elevation of Fig. 53 do not
move any to the right or left, but forward only.
Hence, the distance of the corners of the prism
from the line ef may be taken by the compasses
and marked off from the line gh upon the proper
horizontal line. The new position of all of the
corners having thus been
determined, the con-
necting straight lines
are drawn, giving the
elevation as shown in
Fig-, 54. Vertical lines
are then drawn from the
different points of the
prism, as seen in this
view, through the posi-
tion which the plan is
to occupy, and the exact
position of each point
upon these lines is
marked off from mn at the same distance which
it is from the line jk in the side view.
An Upright Rectangular Prism. — The upright
rectangular prism shown in Fig. 55 is, of course,
drawn in the same way as was the prism shown
in Fig. 52.
The Prism of Fig. 55 Tipped Forward on One Edge.
— It is evident that if the prism were to be tipped
on its edge in the direction of the arrow No. 1, the
result would be the same as though it had been
FIG. 54.— Projections of Prism
Tilted in Two Directions.
PROJECTION
tipped first to the right, and then directly forward,
as was done to produce Fig. 54; but as those
angles are not given, the method employed in that
case is not readily available.
Fig. 56 shows the prism tipped to its new po-
sition, and shows, also, the method employed to
produce the views. Draw the line cd at the same
FIG. 55.— Upright Rectan-
gular Prism.
FIG. 56.— Rectangular Prism
Tipped Forward.
angle to the horizontal as the edge ab of the prism
in Fig. 55, and make e/at right angles to it. Upon
these lines draw the temporary side view of the
prism, A, tipped at the desired angle. With the
aid of this view the plan view is readily drawn.
Vertical lines are then drawn from the various
points of the plan view through the place which
the elevation is to occupy, and the exact location
of each point is marked off on these lines at the
38
SELF-TAUGHT MECHANICAL DRAWING
same height above the base line gh that it is above
the line ef in the temporary side view, A. The
permanent side view is then developed from
the plan and elevation in the same way as was the
side view of Fig. 53.
Let it now be required to tip the prism of Fig. 55
forward on one corner in the direction of arrow
No. 2.
It will be seen that tipping it in this direction
FIG. 57. — Rectangular Prism Tipped in Two Directions.
will cause a foreshortening of all of the lines in the
plan, hence the use of a single temporary view
such as was used in Fig. 56 will not solve the
problem; but it may be solved by the use of two
temporary views as shown in Fig. 57. Draw the
line ab in the direction in which the prism is to be
tipped, and the line cd at right angles to it. At A
reproduce the plan view of Fig. 55, and at B draw
PROJECTION
39
a side view of the prism as it would appear if A
were viewed in the direction of the arrow, but in-
clined to cd at the required angle. The intersec-
tion of lines drawn from the corners of A, parallel
with ab, with lines drawn from the same corners
of B, parallel with cd, will give their location
FIG. 58.— Projections of a Cube.
in the permanent plan view. This view being
finished, the elevation and the permanent side
views are drawn in the same way as were those
of Fig. 56.
Let a cube be set on one corner so that a diagonal
of it shall be horizontal; required to show the angle
which the edges that meet at that forward corner
make with a plane perpendicular to the diagonal,
the angle which the sides that have corners coming
together at the same point make with the plane, and
40
SELF-TAUGHT MECHANICAL DRAWING
also the am ^ ant of foreshortening of the lines which
will be caused.
In Fig. 58, A shows a face view of the cube set
on edge, B shows a side view of the same, and C
shows B inclined until the diagonal ke becomes
horizontal. The length of ke being laid out on the
center line, the position of the other corners is ob-
tained as indicated by the arcs a, b, c and d. The
angle geh is the required angle which the edges
• which meet at e make
with a plane perpendic-
ular to ek, of which fg is
an edge view ; the angle
fej is the angle which
the sides having corners
meeting at e make with
the plane. D is a face
view of C, and any of its
lines, when compared
with any of the lines of
A, will show the fore-
shortening caused by the
cube being put into this
position.
The Surface Develop-
ment of a Cone. — Let A
and B, Fig. 59, be the
plan and elevation views
of a cone. With a radius
equal to ab, and with a
center at c, draw the arc def, making it equal in
length to the circumference of the base of the cone,
as shown at A, This may be most conveniently
FIG. 59.— Development of
a Cone.
PROJECTION
41
done by spacing it off. Draw the lines cd and cf,
and the figure C thus formed will be the required
surface development.
The Surface Development of a Pyramid Having
Its Top Cut Off Obliquely.— In Fig. 60, A, B and C
show, respectively, the plan, elevation, and side
FIG. 60.— -Development of a Frustum of a Pyramid.
views of the pyramid, the top of which is cut off by
the plane ab. These views may be made by the
principles already explained, as may also the view
at D, which shows the pyramid as though B were
viewed in the direction of the connecting dotted
line, which is at right angles to ab, thus showing
the shape of the section exposed by cutting off the
top.
42 SELF-TAUGHT MECHANICAL DRAWING
To get the surface development, take a radius
equal to the length of one edge of the pyramid as
shown at cd in the elevation, this being the only
one which shows at full length, the others being
more or less foreshortened, and with a center at e
in view E, draw an arc of a circle upon which the
sides of the base are to be marked off. These
points are connected with one another and with e;
this gives the shape of the surface of the whole
pyramid. Upon the lines connecting the points
with e, as el, e2 and e3, the .lengths of the different
edges of the cut off pyramid are marked off. As
the edge which is seen at the left in the elevation
shows full length, its length, dl, may be taken di-
rectly and marked off on the line el. As the other
edges are seen foreshortened, their lengths cannot
be taken directly, but by horizontally transferring
the upper end of each edge to the line cd, their
actual lengths d2 and d3 may be obtained and then
marked off on the lines e2 and e3. The points so
obtained being connected, and the outer half sec-
tions being finished, gives the required surface
development.
If the cone shown in Fig. 59 were to have its top
cut off obliquely, the views of it corresponding to
A, B, Cand D, Fig. 60, and its surface develop-
ment, would be obtained by dividing off its base,
as seen in the plan, into any number of sides, and
then proceeding as though it were a pyramid of
that number of sides, until the points correspond-
ing to those of Fig. 60 had been located, but then
connecting them with curved lines instead of
straight lines.
PROJECTION
43
Intersecting Cylinders, Fig. 61.— Required the
line of the intersection, the surface development of the
branch, and the shape which the end of the branch
would appear to have as seen in the view at the
right.
First draw the elevation, A, in outline, and as
much of the end view, B, as can be directly drawn.
FIG. 61. — Intersecting Cylinders.
Opposite the end of the branch in each of these
views, and in line with it, draw circles of the same
diameter as the branch, and space off the semi-
circumference nearest to it into a number of
equal parts, the same number in both cases. From
the points so obtained draw lines parallel with the
center line of the branch, as shown. From the
points where these lines in the view B meet the
44 SELF-TAUGHT MECHANICAL DRAWING
circle representing the end of the large cylinder,
draw horizontal lines intersecting the lines drawn
from C. These intersections will be points through
which the line of the intersection of the cylinders
is to be drawn. From the points where the lines
drawn from C cross the end of the branch, draw
horizontal lines intersecting those drawn from D.
These intersections will be points through which
the line representing the end of the branch is to
be drawn.
To get the surface development of the branch,
first draw the line ab, in E, having it in line with
the end of the branch. Make this line equal in
length to the circumference of the branch, spacing
it off equally each way from the center line OX into
the same number of spaces as the semi -circumfer-
ence of C was divided into. From these points
draw lines parallel with OX, and from the points
in the intersection of the two cylinders, previously
obtained, draw lines parallel with ab, intersecting
these lines. These intersections will be points
through which a curved line is to be drawn, thus
giving the completed surface development of the
branch.
In drawing these curved lines through the points
of intersection, the irregular curves mentioned in
the early part of the chapter on instruments and
materials are used.
Intersecting Cylinder and Frustum of Cone, Fig.
62. — Required line of intersection and surface de-
velopment of branch, as before.
Draw the elevation, A, in outline, continuing the
sides of the conical branch either way until they
PROJECTION
45
meet at their vertex, a, on the one hand, and to
any convenient points, c and d, on the other. In a
similar manner draw as much of the end view, B,
FIG. 62.— Intersecting Cylinder and Cone.
as can be made directly. With centers at a and at
6, and with any convenient radius, draw the arcs
c'd and ef, intersecting the extended sides of the
conical branch. Then, with centers at the inter-
46 SELF-TAUGHT MECHANICAL DRAWING
section of these arcs with the center line of the
branch, draw the half ci cries shown, tangent to
the extended sides of the branch, and space them
off into a number of. equal parts, the same number
in each case. From these points draw lines to the
vertices a and b. From the points where these
lines in the end view, B, intersect the circle repre-
senting the end of cylinder, draw horizontal lines
to the elevation, A, intersecting the lines drawn
from the vertex a to the half -circle cd. The inter-
sections will be points through which the line rep-
resenting the intersection of- the cylinder and its
conical branch is to be drawn. The shape of the
end of the branch as seen in the end view, B, is
now obtained in the same manner as in the case of
the intersecting cylinders. From the points where
the lines drawn from the vertex, a, of the side ele-
vation A, to the half-circle at cd, cross the end of
the branch, draw horizontal lines intersecting the
lines drawn from the vertex b. These intersec-
tions will give points through which the line rep-
resenting the end of the branch in view B is to
be drawn.
To get the development of the branch as shown
at F take a radius equal to the distance from the
apex a to the end of the branch as seen in the side
elevation, A, and with a center at g draw an arc
hi, making the length of the arc equal to the cir-
cumference of the end of the branch as shown at
E, spacing equally each way from the center line
gj, the length and number of the spaces each way
being the same as those obtained in spacing off the
semicircle at E. Through these points draw lines
PROJECTION
47
radiating from g, as shown. On these lines dis-
tances are marked off from the arc hi through
which the irregular curved line is drawn which
gives the development of the branch. The lengths
at the middle and at the extremities may, of course,
be taken directly from the elevation A, the length
kl being the length on the center line, and the
length mn being the length at the extremities.
The other lengths, being foreshortened, as seen in
the elevation A, cannot be taken directly, but are
obtained by transferring the points to either kl or
mn as shown by the dotted lines, as was done in
the case of the pyramid, Fig. 60.
To Draw a Helix. — A helix is a line of such shape
as would be made by winding a thread around a
FIG. 63.— Drawing a Helix.
FIG. 64.— The Helix as
it Appears in a Screw
Thread.
cylinder, and having it advance lengthwise on the
cylinder at a uniform rate as it is wound around
it. In Fig. 63 we have the side and end views of
a cylinder upon which it is desired to draw a helix,
which shall advance from a to b in making a half
turn around it. Divide the space from a to b into
any number of equal parts, and at the points so
obtained erect perpendicular lines. Divide the
48 SELF-TAUGHT MECHANICAL DRAWING
semi-circumference of the end view of the cylinder,
toward the side view, into the same number of
equal parts, and from these points draw horizontal
lines to meet the perpendiculars previously erected.
Where these lines meet will be points through
which the helix is to be drawn.
The outlines of a screw thread are helices. Fig.
64 shows a double threaded Acme standard, or 29
degree threaded screw, the outline of which, on
its outside diameter, is the helix of Fig. 63.
Isometric Projection. — If a cube is tipped over on
one corner, so that the diagonal of it is horizontal
as shown at D, Fig. 58, and also in Fig. 65, the
FIG. 65.— Principle of FIG. 66.— An Example of
Isometric Projection. Isometric Projection.
lines of it will all appear of equal length. Draw-
ings made on this principle, as Fig. 66, are called
isometric drawings. Vertical lines remain ver-
tical. Horizontal lines become inclined to the
horizontal of the paper at an angle of 30 degrees.
Circles appear as ellipses, which may be drawn as
shown in the upper square of Fig. 65. From the
ends of the "short" diagonals, lines are drawn to
the middle of the opposite sides. Where these
lines cross the "long" diagonals are located the
centers from which the ends of the ellipse may
PROJECTION 49
be drawn. The ends of the short diagonals will
be centers from which to draw the sides of the
ellipse.
Irregular curves may be drawn as indicated in
Figs. 67 and 68. The figure 2 there shown is first
drawn in the desired position in a naturally shaped
square, which is then divided off by equally spaced
lines into smaller squares. The isometric square
is then similarly divided off, and the figure is
FIGS. 67 and 68.— Method of Transferring Irregular Lines in
Isometric Projection.
made to pass through the corresponding inter-
sections.
Isometric drawings differ from perspective draw-
ings in that receding lines remain parallel, instead
of converging to a vanishing point. They may be
measured the same as ordinary drawings in any
one of the three directions indicated by the lines
of the cube. The foreshortening of the lines caused
by tipping the cube into this position is generally
ignored. If an isometric drawing is to be shown
in connection with ordinary views, however, it
should be made on a scale of about 8-10 of an inch
to the inch, otherwise it would appear too large.
CHAPTER V
WORKING DRAWINGS
As the object of working drawings is to convey
to the workman a clear idea of the appearance and
construction of the piece to be made, and as the
whole "science" of mechanical drawing has been
developed primarily for the purpose of conveying
the ideas and thoughts of the designer and drafts-
man to the men who carry out these ideas in wood
and metal, the subject of working drawings is of
supreme importance to all mechanics. A working
drawing should be as complete as possible, so com-
plete, in fact, that when it has once passed out of
the draftsman's- hand into the shop, no further
questions will be necessary. In or.der to accom-
plish this, all necessary information, of whatever
kind, should be .included, and, if required, short
notes and directions may be written on the draw-
ing to prevent eventual misunderstandings.
The number of views necessary to properly rep-
resent an object must be left for the draftsman's
judgment to determine. Usually two views are
sufficient, when the object is simple, but when at
all complicated, three or more views will be found
necessary. Cylindrical pieces can often be ade-
quately represented by a single view, on which the
various diametral and length dimensions are given.
50
WORKING DRAWINGS 51
While it is customary to put the plan view of an
object above the elevation, it frequently becomes
necessary, in order to present the objects shown in
as clear a manner as possible, to deviate from this
rule. A case of this kind is shown in Fig. 69,
where the shaft hanger illustrated has been se-
lected as an example of the methods employed in
working drawings.
An examination of the hanger will show that if
the plan were placed above the elevation, and if it
were represented according to the methods already
explained, the box and the yoke with its adjusting
screws and check-nuts would have to be shown
mostly by dotted lines. Such a multiplicity of
dotted lines would tend to confusion; hence the
object in view, that of presenting the hanger in as
clear a manner as possible, is best accomplished in
a case like this by having the plan underneath the
elevation, and letting it be a bottom view instead
of a top view.
In designing a machine detail of this kind, the
starting point would of necessity be the shaft
itself, and the first step would be to design the
box; next would come the yoke, and lastly, the
frame. Much of the preliminary work may fre-
quently be done on scrap paper; having determined
the size and proper proportions of the various
parts, the position which the different views will
occupy in the finished drawing is easily ascer-
tained. The center lines are then laid out as
shown, and the drawing built up about these lines
as a base.
When a drawing is for temporary use only, and
52
SELF-TAUGHT MECHANICAL DRAWING
the mechanism represented on it of a simple nature,
the assembly drawing, corresponding to the three
views in Fig. 69, will answer all purposes, the di-
mensions being given directly on this drawing. In
FIG. 69.— Shaft Hanger.
some cases only the most important dimensions
would be given, those of secondary consequence
being left for the workman to be obtained by
" scaling' ' the drawing. This procedure, however,
is possible only when the drawing is made care-
fully to scale, and is not one that should be en-
WORKING DRAWINGS
53
couraged. In general, a drawing should be so di-
mensioned that it can be worked to without the
workman obtaining any measurements by "scal-
ing" the drawing.
In most cases it is not possible to show the de-
tails of a mechanism clearly enough in an assembly
drawing; for if the device shown is more or less
complicated, a hopeless confusion results from the
attempt to put in all the lines necessary to fully
show all the details ; neither would it be possible,
for the same reason, to give more than the princi-
CAST IRON, BABBITTED
FIG. 70.— Example of Working Drawing.
pal dimensions. In such cases it is, therefore, cus-
tomary, after the assembly drawing has been com-
pleted, and the proper sizes and proportions of the
various parts of the mechanism thus ascertained,
to make a separate drawing of each detail, either
on the same sheet of paper, or on separate sheets.
This permits the parts of the mechanism to be
clearly and completely shown and fully dimen-
sioned. Figs. 70 and 71 show two pieces of the
hanger in Fig. 69 detailed in this manner. These
detail drawings give all the required informa-
tion for the making of the pieces, and the assembly
54
SELF-TAUGHT MECHANICAL DRAWING
drawing merely shows, in a general way, how the
parts are to be assembled when completed.
In the case of jig and fixture drawings, it is the
practice in a great many large drafting-rooms to
show assembled views only, and to put all dimen-
CAST IRON,
Tap %"-
10 thd.
FIG. 71.— Example of Working Drawing.
sions directly on the assembly drawing; the argu-
ment advanced in favor of this practice is that ex-
perienced pattern-makers and tool-makers, who are,
as a rule, the only mechanics who will work on
the making of these tools, will find no difficulty in
reading the assembly drawing; besides, it is said,
WORKING DRAWINGS 55
as a drawing of this kind is, in most cases, used
but once, it would be waste of time to have the
draftsman detail the different parts of the tool.
While these arguments are undoubtedly true in
the case of very simple jigs and fixtures, there can
be little doubt that in the case of more complicated
ones, the comparatively short time required by the
draftsman to make detail drawings will be saved
many times over in the shop; for the pattern-
maker and tool-maker will not have to spend, in
the total, a number of hours puzzling over the draw-
ing, and even then being liable to make a mistake.
In making drawings, it is always a rule to work
from the center lines, when the outline of the
piece is such that it has a definite center line.
Dimensions in either direction from the center
line can be best marked off with the compasses.
This insures a symmetrical appearance to the fin-
ished drawing, such as might not be secured if the
dimensions are set off on either side of the center
line from the rule, it always being easy to then
introduce small errors which show plainly in the
finished work. If the piece is of such shape as to
have no center line, some one principal line may
be selected, one in each direction in each view,
and the remaining points and lines may be located
from these lines.
The various styles of lines ordinarily used in
working drawings are shown in Fig. 72. The
regular "full" line A A is used for the outlines of
objects, and when drawn rather "fine," for cross-
hatching or cross-sectioning. The heavy shade
line BB is used to represent lines assumed to sepa-
56 SELF-TAUGHT MECHANICAL DRAWING
rate the light surfaces of an object from the dark,
as will be explained in the following. The dotted
line CC, as has already been explained in the pre-
vious chapter, is used to represent lines obscured
or hidden from view. The line DD, called a
"dash" line, is used by a great many draftsmen
for dimension lines. Finally, the line EE, the
"dash and dot," or, simply, the "dash-dotted"
FIG. 72.— Styles of Lines Used on Working Drawings.
line, is used in common practice for center lines,
to indicate sections, etc. This line is also com-
monly used for construction lines, in laying out
mechanical movements.
The dimension lines may be made either fine
full lines or "dash" lines, the dashes being about
| inch long. A space is left open for the figures
giving the dimension. The witness points or ar-
row heads, showing the termination of the dimen-
sion, are made free hand. Many draftsmen draw
the extension and dimension lines in red ink, the
arrow heads, however, still being made black. It
is well to avoid, as far as possible, having the
WORKING DRAWINGS 57
dimension lines cross each other, as such crossing
tends to confusion; the difficulty can usually be
avoided by having at least one set of dimensions
placed outside or between the views, the larger di-
mensions being placed farther from the outline of
the object than the shorter ones, to avoid having
the extension lines of the latter cross the dimen-
sion lines of the former. Dimensions under 24
inches are most conveniently given in inches;
larger dimensions are given in feet and inches.
The usual practice is to indicate feet and inches
on drawings by short marks, " prime' ' marks ('),
placed at the right, and a little above the figure,
one mark (0 indicating feet, and two marks,
" double prime' ' marks ("), indicating inches, so
that 5' 7" would read 5 feet 1 inches. Some drafts-
men do not consider this method of marking safe
enough to eliminate mistakes, and prefer to write
dimensions of this kind in the form 5 ft. 7". A
method equally satisfactory in preventing possible
mistakes is to place a short dash between the
figure giving the number of feet and that giving
the number of inches, at the same time retaining
the "prime" marks; thus, 5'— 7". When feet only
are given, it is well, for the sake of uniformity
and to prevent any misunderstanding, to give the
dimension in the form 5' — 0".
A few examples showing the principles of the
usual methods of dimensioning drawings may be
of value. In Fig. 73 is shown a simple bushing.
The diameter of the hole or bore is given as 2
inches by a dimension line passing through the
center of the circles in the end view. It is con-
58
SELF-TAUGHT MECHANICAL DRAWING
fusing, however, to have more than one dimension
line passing through the same center, and, there-
fore, the outside diameters of the bushing have
been given on the side view. The lengths of the
various steps or shoulders of the bushing are given
below the side view, as is also the total length. It
will be noticed that the dimensions of the three
steps are slightly offset — that is, the dimension
FIG. 73. — Simple Example of Dimensioning a Drawing.
lines do not extend in one straight li'ne ; this makes
a very clear arrangement.
The method of dimensioning holes drilled in a
circle is shown in Fig. 74. Outside of the dimen-
sion for the holes themselves only the diameter of
the circle passing through the centers of the holes
is given, together with the number of holes. As
the holes, of course, are to be equally spaced, that
is* all that is required. When a great many bolt
holes or bolts occur around a flange, it is not nec-
essary to draw them all in on the working draw-
ing; a common method is to show a few, and to
WORKING DRAWINGS
59
draw the circle passing through their centers, the
pitch circle. The total number of bolts around the
flange is, of course, also given. A case of this
kind is illustrated in Fig. 75. When a great many
holes are drilled in a row, a similar expedient may
FIG. 74.— Dimensioning Holes Drilled in a Circle.
be adopted to avoid showing and dimensioning
all the holes; an illustration of this is shown in
Fig. 76.
In Fig. 77 are shown the common methods of
dimensioning screws and bolts. At A is shown a
hexagon head bolt, so drawn that three sides of
60
SELF-TAUGHT MECHANICAL DRAWING
the head are visible. Hexagon bolt-heads are
usually drawn in this manner in all views, irre-
spective of the fact that the rules of projection
would call for only two sides to be visible in one
view. The reason for this is partly that the bolt-
FiG. 75. — Simplified Method of Dimensioning Holes
Drilled in a Circle.
head looks better when three sides are visible, and
partly that when so drawn there can be no confu-
sion whether a hexagon or a square head is meant.
If only two sides were shown, as at B, the head,
especially if carelessly drawn, might be mistaken
for a square bolt-head. As a rule, the dimensions
WORKING DRAWINGS 61
of bolt-heads are standard for given diameters of
bolts, and no dimensions are required for the
head. In some cases, however, the head may be
required to fit a given size of wrench, or for some
other reason be required to be made different from
the standard size ; in such cases dimensions may
be given as shown at C, Fig. 77, the dimension
"V hex." indicating that the head is one inch
K 10-HOLE8-2-CENTER-DISTANCE »j
!
*
~t-
--$--&- -6
j< : 10-HOLES-2-CENTER-DISTANOE >j
T
FIG. 76. — Dimensioning Holes Drilled in a Row.
"across flats." In the same way, "f" sq." would
indicate that the head should be square, and three-
quarters inch "across flats."
The length of the bolt should be given as shown
in the lower view in Fig. 77. The dimensions
should be given "under the head," both the total
dimension, and the distance to the beginning of
the thread.
In general, full circles should be dimensioned by
their diameters ; an arc of a circle, again, should
be dimensioned by its radius. The center from
62
SELF-TAUGHT MECHANICAL DRAWING
which the arc is struck should preferably be indi-
cated by a small circle drawn around it. In small
dimensions, the arrow points are frequently placed
outside of the lines between which the dimension
is given, as shown in Fig. 71 in dimensioning
the narrow ribs; sometimes, the figures giving the
"A"7
^
JL
-a
8T D
HEAD
< -- --- 2M—
FIG. 77. — Dimensioning Screws and Bolts.
dimension are themselves placed outside of the
space between the arrow heads, because the space
is too small to permit the dimension to be clearly
written within it.
The principal dimensions should be so given
that the workman will not have to add a number
of other dimensions to get them. When the
dimensioning of a piece naturally divides itself
into several measurements, an over-all dimension
should always be given for verification. If, how-
WORKING DRAWINGS 63
ever, the piece terminates with a round end, as
the yoke in Fig. 71, the over-all dimension may
properly terminate at the center of curvature of
the end, the distance beyond being of entirely
secondary importance, and being taken care of by
its radius. If a dimension has been given in one
view, there is usually no reason for repeating it
in the other views; sometimes such repetitions
would cause too many dimensions to be given in
each view, so that confusion would arise, and in-
stead of making the drawing plainer, the repeti-
tion of dimensions might cause mistakes which
otherwise would have been avoided.
Drawings should always be dimensioned the full
size of the finished article, regardless of the scale
to which the drawing is made. If a drawing is
made to any other scale than full size, it is cus-
tomary to state on the drawing the scale to which
it is made, as " Scale, i inch=l ft."
A drawing should be so marked as to tell the
workman what surfaces are to be finished ; a fin-
ished surface is usually indicated by the letter
"f" placed either upon the line representing the
surface, or in close proximity to it. While the
amount and kind of finish is usually left to the
workman to determine, the best modern methods
require that the draftsman should indicate on the
drawing how closely the various parts are to be
machined. A very commendable method is to
give dimensions in thousandths of an inch, where
accuracy is required, and in common fractions in
cases where there is no need of working to thou-
sandths. In very highly systematized establish-
64 SELF-TAUGHT MECHANICAL DRAWING
ments, the limits of variation between which any
measurement is allowed to vary, are given with
each dimension, or, at least, with dimensions for
diameters which are to fit the holes or bores of
other pieces. The determination of the limits of
accuracy required calls for good judgment on the
part of the draftsman. Limits may be expressed
in two ways. For instance, a running fit on a
shaft to go into a li inch standard size hole may
be marked
-0.0005 max.
l5-0.0015min.
or it may be expressed
1.4995 max.
1.4985 min.
which means that the shaft must not be larger
than 1.4995 inch, and not smaller than 1.4985 inch.
On drawings, the tap drill size and the depth of
tapped holes should always be shown. Surfaces
to be ground to size should be marked " grind/'
If the surface is to be filed, the words "file finish' '
are substituted for the letter "f." Finishing
marks, as a rule, are used on castings and forg-
ings only. On work made from bar stock, every
surface is nearly always finished, so that here the
finishing marks are omitted. When a casting or
forging is finished on every surface, it is not nec-
essary to show finish marks, but the words "finish
all over" may be written in a conspicuous place,
so as to readily catch the eye of the workman. If,
on work made from bar stock, it is desired that
the piece be left rough at any point, the words
WORKING DRAWINGS 65
"stock size" may be applied to the figures giving
that particular dimension. For instance, on a
li-inch cold rolled shaft, turned for journals for a
short distance at each end, the central part would
be dimensioned "li-inch stock size/'
While the practice of indicating finished surfaces
by the letter "f" is by far the most frequently
met with, it is by no means universal. In some
shops the words " polish, " "ream," "finish," etc.,
are written near the lines representing the sur-
faces to be thus treated. Still another method
much in use is to draw a red line outside of the
line representing each surface to be finished. If
a blue-print is made from a tracing thus pre-
pared, the red lines will print fainter than the
black ones, and the finish lines on the blue-prints
are traced over with a red pencil or red ink before
being sent out in the shop. This method, how-
ever, is more expensive than that of indicating
the finished surfaces by the letter "f," and on
complicated drawings, the many additional red
lines tend to cause confusion. By whatever
method the finish is indicated, the finishing
marks should always be shown fully in every view
of the object.
It frequently happens that the representation of
an object is made clearer by the use of sectional
views, representing the object as having been cut
in two, either wholly or in part. Examples of this
are shown in Figs. 69, 70 and 71. From these
illustrations it is apparent that the construction of
the various pieces is much more clearly exhibited
when a section is shown. The surface "cut" or
66 SELF-TAUGHT MECHANICAL DRAWING
shown in section is cross-hatched or cross-sectioned
with fine lines at a distance apart varying from a
thirty-second to an eighth of an inch, according
to the size of the drawing and the piece. The
cross-sectioning brings the parts in section into
bold contrast with the remainder of the drawing,
and prevent all confusion as to what parts are in
section and what parts shown in full. All lines
beyond the sectional surface which are exposed to
view, should be shown in the drawing as usual.
Should it be deemed necessary, which it seldom is,
to show any parts that have been cut away for the
purpose of showing a section, such parts may be
drawn in by dash-dotted lines, this indicating that
the parts thus shown are in front of the section
and actually cut away.
When a mechanism is shown in section, the dif-
ferent parts of the same pieces should always be
cross-sectioned by lines inclined in the same direc-
tion, while separate pieces adjoining each other
should always, when possible, be cross-sectioned
by lines running in different directions. When a
solid round piece is exposed to view by a section,
it is customary to show, it solid, and not to section
it; the screw stud in Fig. 69 is an example of this
practice.
Sectional views may also be used for many pur-
poses where a slight deviation from the theory of
projection will tend to simplify the representation
of certain machine details. The shape of the arm
of a pulley or gear, or of any other part of a cast-
ing, may be conveniently represented in this way.
The cutting plane may be assumed to lie at any
WORKING DRAWINGS
67
angle necessary to bring out the details most clear-
ly. A sectional view, for instance, may represent
a casting as though it were cut through partly on
one plane and partly on another. In all such cases,
however, it should be indicated in another view of
the object just where the sectional views are sup-
SECTION AT
G-H
FIG. 78.— Methods of Showing Sections.
posed to be taken, so that no confusion may arise
on this account. The examples in the following
will serve to make clear the principles laid down.
In Fig. 78 are shown sections of two hand-
wheels. When an object is symmetrical it is
unnecessary to show more than one half in sec-
tion, although it is quite common to section gears,
pulleys, etc., completely on working drawings.
The hand-wheel at A in Fig. 78 is represented as
68
SELF-TAUGHT MECHANICAL DRAWING
though cut in two along its diameter BC. When
the section is taken along the center line, it is not
absolutely necessary to explain where the section
is taken ; but it can do no harm to make a practice
of in all cases to state where the section is made,
except when perfectly obvious. In this case it
would be clear that the section is taken through
SECTION AT A-B
FIG. 79.— A Gear-wheel in Section.
the center, and the legend "Section at BC'9 is
given only to show the principle. The hand-wheel
at D is provided with four arms, and the method
of representing the shape of the arms, hub and
rim are clearly indicated.
In Fig. 79 are shown two views of a gear-wheel,
indicating the conventional method of represent-
ing gears on drawings. The view on the left side
is the side view, and as all the teeth are, of course,
WORKING DRAWINGS 69
alike, it is unnecessary to draw more than a few
of them. The pitch line of the teeth is represented
by a dash-dotted line. In the part of the gear-
wheel rim where the teeth are not shown, the face
of the gear is indicated by a solid line, and the
bottom of the teeth by a dotted line. In the case
of machine-cut gearing, where the teeth are cut
by standard formed cutters, it is unnecessary to
show any teeth at all on the rim of the gear, it
being sufficient to state the pitch and the number
of teeth, as will be more fully explained later in
the chapter on gearing. To show the shape to
which the arms are formed, a sectional view of
one of the arms is drawn in the side view; the
ends of the shaft are supposed to be broken off,
and are, therefore, sectioned as shown. The right-
hand view of the gear is a section taken along the
line AB. It will be noted that the shaft and key
are not sectioned, usual practice being followed in
this respect. The gear shown has five arms, and
the line AB cuts through one of them only. This
arm, however, is not sectioned in the right-hand
view, and two opposite arms are drawn as though
both of them lay in the plane of the paper. While
this is not theoretically correct, it is the method
usually followed because of simplicity in drawing
and clearness of representation. The method of
representing the gear teeth in the sectional view
is the one commonly employed.
Sectional and top views of a cylinder end with
flange and cover are shown in Fig. 80. This
cylinder cover has only five bolts, and the plane
through which the section is taken cuts through
70
SELF-TAUGHT MECHANICAL DRAWING
only one of the bolts. It is common practice, how-
ever, to draw the section as shown at the left.
The bolts are shown as if two of them were in the
plane of the section. The bolts are not sectioned,
FIG. 80. — Section of Cylinder End with Flange and Cover.
but are drawn in full, as explained previously.
Dotted lines of the remaining bolts, or full lines
of their nuts, should not be shown, because this
detracts from the clearness of the drawing; the
top view shows clearly the number of the bolts
and their arrangement, and that is all that is nec-
essary. Some draftsmen prefer to draw sections
WORKING DRAWINGS
71
of this kind as indicated at the right in Fig. 80.
This method, however, is not as commonly used.
In a case where the object is rather unsymmet-
rical, as, for instance, in Fig. 81, the draftsman's
judgment must often be relied upon to decide how
|< A -»< A
FlG. 81. — Another Method of Showing Sections.
it shall best be shown in section. Usually the
sectional view is made symmetrical as shown, the
distances A in the lower view being made equal to
the radius A in the top view,
The materials for the various details making up
a complete mechanism are usually cross-sectioned
CF "HE
UNIVERSITY
OF
72
SELF-TAUGHT MECHANICAL DRAWING
in such a way as to indicate the material from
which each piece is made. There is, however, no
universally adopted or recognized standard for
cross-sectioning for the purpose of indicating dif-
ferent materials. In Fig. 82 is shown a chart,
FIG. 82. — Cross-sectioning used for Indicating Different
Materials.
published by Mr. I. G. Bayley in Machinery, Oc-
tober, 1906, which represents average practice,
although it must be distinctly understood that
there is no agreement in all respects between the
numerable charts in use in various drafting-rooms.
For this reason, cross-sectioning alone should
never be depended upon for indicating to the work-
WORKING DRAWINGS
73
man the kind of material to be used. Written
directions should also be given, the kind of mate-
rial for each part being plainly marked. Tool steel
may be abbreviated "T. S.", machine steel, "M.
ROUND BAR, SOLID
ROUND BAR, HOLLOW
SQUARE OR RECTANGULAR BAR
WOODEN BEAM
FIG. 83.— "Broken'
I-BEAM
Drawings of Long Objects.
S."; wrought iron, "W. L"; cast iron, "C. L",
etc. The less common materials in machine con-
struction, such as bronze, brass, copper, etc. , should
preferably be written out in full, in order to avoid
any chances for confusion. It is better to be too
74
SELF-TAUGHT MECHANICAL DRAWING
explicit as regards the information on the draw-
ing, than to risk misunderstandings and conse-
quent errors.
Long bars, shafting, structural beams, etc. , can-
not conveniently be shown for their full length on
the drawing. In such cases the pieces are drawn
as long as the drawing and the adopted scale per-
mit, and are broken as shown in Fig. 83, a part
between the two end portions shown being imag-
ined as broken out. The di-
mensions, of course, are given
for the full length of the piece,
as if not broken.
There are several conven-
tional methods for showing
screw threads; these methods
are adopted largely for saving
of time, as it would be out of
the question to spend the time
required for drawing a true
helical screw thread on a work-
ing drawing. A method for
very nearly approximating the
appearance of a theoretically correct screw drawing
is shown in Fig. 84, where the projection of the
screw helix is drawn by straight lines. The V-
shaped outline is first laid out, and the connecting
lines are then drawn. It will be noticed that the
lines representing the roots of the threads are not
parallel with those representing the tops or points.
This aids in making the drawing resemble that of
a true helix.
Usually, however, much simpler methods are
FIG. 84.— Method of
Drawing a Screw,
Giving Correct He-
lix Effect.
WORKING DRAWINGS
75
employed for indicating screw threads. In Fig.
85, A, B and C, some of these methods are shown.
When a long piece is threaded the entire length,
this fact can be indicated as at D, which saves
drawing the conventional thread for the full length
of the piece. The lines indicating the thread are
L.H
E F
FIG. 85. -Simplified Methods for Showing Screw Threads.
inclined, the same as would be the lines represent-
ing the true helix. At E in Fig. 85 is shown a
right-hand thread and at F a left-hand thread, the
different direction of inclination of the thread in-
dicating this fact. However, if a thread is to be
left-hand, it should always be so marked on the
drawing. It is usual to abbreviate left-hand, writ-
ing "L. H."
76
SELF-TAUGHT MECHANICAL DRAWING
Three methods of indicating tapped holes are
shown in Fig. 86, these being used when the holes
are obscured from view, and shown by dotted lines.
When a tapped hole is shown in section, and looked
upon from the top, it is shown as indicated at D,
while if seen from the side, in section, it is repre-
FIG. 86.— Simplified Methods for Indicating Tapped Holes.
sented as at E. A surface having tapped holes in
it, seen from above, is shown at F. At G and H
are shown the methods of representing bolts or
screws inserted in place in tapped holes. It will
be noted that when the threads of a tapped hole
are exposed to view by section, the lines repre-
senting the screw helix will be seen to slope in the
opposite direction to those of the screw, it being
WORKING DRAWINGS 77
the back side that is exposed to view. An example
of this is shown in Fig. 71 as well as in Fig. 86.
In drawings made for use in the shop it is cus-
tomary to make the lines of uniform thickness.
For shop use such drawings are as good as any.
When, however, the purpose of a drawing is chiefly
to show up the object which it represents, its ef-
fectiveness may be considerably enhanced by the
use of shade lines as shown in Fig. 87. In shade
line work, the light is usually assumed to come
from the upper left hand corner,
and to shine diagonally across
the paper at an angle of forty-
five degrees. Lines on the side
of the object away from the
light, or lines separating light
from dark surfaces, are made
extra heavy. This gives to the FlG 87>_Use Of
drawing a suggestion of relief. shade Lines.
An examination of the lines of
Fig. 87 taken in connection with the direction from
which the light is supposed to come will show,
without the aid of any other view, that the hex-
agonal part is raised above the surface of the
square, and that the circle in the center represents
a depression.
When a drawing is intended for permanent use
it is customary to make only a pencil layout on
paper, usually on brown paper, and from this to
make a tracing from which any number of blue
print copies may be made. The tracing is usually
made on the regular tracing cloth. This has one
glazed and one unglazed surface. Either surface
78 SELF-TAUGHT MECHANICAL DRAWING
may be used. The tracing cloth is drawn tightly
over the pencil drawing, and its surface is cleaned
of any greasiness with dry powdered chalk. This
insures a good flow to the ink. In doing the ink
work curved lines should be made first, straight
lines afterwards, as mentioned in Chapter I.
The blue prints are made in the same manner as
photographs are printed, the tracing taking the
place of the photographic negative. An exposure
of from three to ten minutes may be required, de-
pending on the freshness of the blue print paper
and the brightness of the sun. After the proper
exposure has been given, which may require some
experimenting at first, until one gets accustomed
to the change in the paper which the light makes,
the print is thoroughly rinsed out in clear water
and dried, by being hung up by one edge.
White writing may be made on a blue print with
saleratus water, the water being given all the sale-
ratus it will dissolve.
CHAPTER VI
ALGEBRAIC FORMULAS
IN order to be able to carry out the calculations
required in simple machine design, it is necessary
that a general understanding of the use of for-
mulas, such as are used in mechanical hand-books
and in articles in the technical press, is acquired.
Knowledge of algebra or so-called " higher mathe-
matics" is by no means necessary, although, of
course, such knowledge is very valuable ; but simple
formulas can be used, and the results of scientific
results employed in practical work to a very great
extent, by any man who understands how to use
the formulas given by the various authorities ; and
the knowledge required for an intelligent use of
algebraic formulas can be very easily acquired.
All the mathematical knowledge necessary as a
foundation is a clear understanding of the funda-
mental rules and processes of arithmetic.
A formula is simply a rule expressed in the sim-
plest and most compact manner possible. By using
letters and signs in the formula instead of the
words in the rule, it is possible to condense, in a
very small space, the essentials of long and cum-
bersome rules. The letters used in formulas sim-
ply stand in place of the figures which would be
used for solving any specific problem ; the signs
used are the ordinary arithmetical signs used in
79
80 SELF-TAUGHT MECHANICAL DRAWING
all kinds of calculations. As each letter stands for
a certain number or quantity, whenever a specific
problem is solved the figures for that case are put
into the formula in place of the letters, and the
calculation is carried out as in ordinary arithmetic.
This may, perhaps, be made clearer by means of a
few examples.
The circumference of a circle equals the diameter
times 3.1416. This rule may be written as a
formula as follows :
C= DX 3.1416.
In this formula C = circumference, and D =
diameter. No matter what the diameter is, this
formula says, the circumference is always equal to
the diameter (D) times 3.1416. Assume that the
diameter is 5 inches. Then, to find the circumfer-
ence, place 5 in the formula in place of D.
C = 5 X 3.1416 = 15.708 inches.
If the diameter of a circle is 12 feet, then
C = 12 X 3.1416=37.6992 feet.
This, of course, is the very simplest kind of a
formula, but it illustrates the principle involved,
and indicates how easily formulas may be em-
ployed.
One of the most well-known formulas in steam
engineering is that giving the horse-power of an
engine, when the average or mean effective pres-
sure of the steam on the piston, the length of the
stroke of the piston in feet, the area of the piston
in square inches, and the number of strokes per
minute, are known. Let
ALGEBRAIC FORMULAS 81
H.P. = horse-power,
P = mean effective pressure in pounds per
square inch,
L = length of stroke in feet,
A = area of piston in square inches, and
N = number of strokes per minute.
Then
PX LX A X N
H.P.
33,000
The rule conveying this information expressed
in words would require considerable space, and be
difficult to grasp immediately ; but the meaning of
the formula is quickly understood. If the pressure
(P) equals 75 pounds, the stroke (L) 2 feet, the
area of the piston (A) 125 square inches, and the
number of strokes per minute (N) 60, then
TT D 75 X 2 X 125 X 60 OA
H'P' = ~~
It will be seen that the values for the different
quantities are merely inserted in the formula in
place of the corresponding letters, and then the
calculation is carried out as usual. It will be
remembered that the line between numerator and
denominator in a fraction also means a division;
that is
i i OK f)(\f)
^ = 1,125,000 -*• 33,000 - 34.1.
It is very common in formulas to leave out, en-
tirely, the sign of multiplication ( X ) between the
letters expressing the values of the various quanti-
ties that are to be multiplied. Thus, for example,
82 SELF-TAUGHT MECHANICAL DRAWING
PL means simply P X L, and if P = 21 and L = 3,
then PL = P X L = 21 X3 = 63. If the multipli-
cation signs are left out in the formula for the
horse-power of engines just referred to, the for-
mula
PXLXA XN ,, , ' ... PLAN
- could be written
As a further example of the leaving out of the
multiplication sign in a formula, assume that D
= 12, R = 3, and r = 2, then
DRr DXRXr = 12 X 3 X 2 72 _
9 9 9 : 9 :
It must be remembered that no other signs, ex-
cept the multiplication sign, may thus be left out
between the letters in a formula.
From the examples given, the use of simple
formulas is clear; each letter stands for a cer-
tain number or quantity which must be known in
order to solve the problem ; when the formula is
used for the solution of a problem, the letters are
simply replaced by the corresponding number,
and the result is found by regular arithmetical
operations.
The expressions "square" and "square root"
and "cube" and "cube root" are frequently used
in engineering hand-books and technical journals.
It would seem, to one unfamiliar with these names
and their mathematical meaning, as well as the
signs by which they are indicated, that difficult
mathematical operations are involved; but this is
not necessarily always the case. The square of a
number is simply the product of that number mul-
ALGEBRAIC FORMULAS 83
tiplied by itself. Thus the square of 3 is 3 X 3 = 9,
and the square of 5 is 5 X 5 = 25. In the same
way, the square of 81 is 81 X 81 = 6561. Instead
of writing 81 X 81, it is common practice in
mathematics to write 812, which is read "81
square/' and indicates that 81 is to be multiplied
by itself. Similarly, we may write 72 = 7 X 7 =
49, and 12 2 = 12 X 12 = 144. The little "2" in the
upper right-hand corner of these expressions is
called "exponent." Nearly all mechanical and
engineering hand-books are provided with tables
which give the squares (and also the square root,
cube and cube root) of all numbers up to 1000, so
that it is usually unnecessary to calculate these
values by actual multiplication.
As the squares of numbers are frequently used
in formulas for solving problems occurring in
machine design and machine-shop calculations, a
few examples will be given below of formulas con-
taining squares.
The area of a circle equals the square of the
radius multiplied by 3.1416. Expressed as a for-
mula, if A = area of circle, R = radius, and the
Greek letter n (Pi) = 3.1416, we have:
A = E2 K.
If we want to know the area of a circle having a
5-foot radius, we have :
A = 527r=5X5X 3.1416 - 78.54 square feet.
As a further example, assume a formula to be
given as follows :
A _ D2N + R2n
A~ DR
84 SELF-TAUGHT MECHANICAL DRAWING
Assume that D = 3, N = 5, R = 4, and n (as
usual) =3.1416. What is the value of A? Insert-
ing the values of the various letters in the formula,
we have :
32X 5 + 42X TT 3X3X5 + 4X4X7T
A =
3X4 3X4
9 X 5 + 16 X n _ 45 + 50.2656 _ 95J2656 _ _ QQQQ
12 12 12
It will be seen in the example above that all the
multiplications are carried out before any addition
is made. This is in accordance with the rules of
mathematics. When several numbers or expres-
sions are connected with signs indicating that
additions, subtractions, multiplications or divisions
are to be made, the multiplications should be
carried out before any of the other operations,
because the numbers that are connected by the
multiplication sign are actually only factors of
the product thus indicated, and consequently this
product must be considered as one number by
itself. The other operations are carried out in the
order written, except that divisions when written
in line with additions and subtractions, precede
these operations. A number of examples of these
rules are given below:
12 X 3 + 7 X 2i - li= 36 + 17J - 1J = 52.
5 + 13X7-2=5 + 91-2 = 94. *
9-3 + 9X3=3 + 27 = 30.
9 + 9-3-2=9 + 3-2 = 10.
Sometimes, however, in formulas, it is desired
that certain operations in addition and subtraction
ALGEBRAIC FORMULAS 85
precede the multiplications. In such cases use are
made of the parenthesis ( ) and bracket [ ]. These
mathematical auxiliaries indicate that the expres-
sion inside of the parenthesis or bracket should be
considered as one single expression or value, and
that, therefore, the calculation inside the parenthe-
sis or bracket should be carried out by itself com-
plete before the remaining calculations are com-
menced. If one bracket is placed inside of another,
the one inside is first calculated, and when com-
pleted the other one is carried out. Some examples
will illustrate these rules and principles:
(6 - 2) X 3 + 4 =4 X 3 + 4 = 12+ 4 = 16.
3 X (12 + 7) - 28i = 3 X 19 - 28i = 57-28J =2.
3 + [5 X 3 (5 + 2) - 3] X 6 = 3 + [5 X 3 X 7 -3]
X 6 = 3 + [105 - 3] X 6 = 3 + 102 X 6 = 3 + 612
= 615.
Without the parentheses and brackets, the calcu-
lations above would have been as follows :
6-2X3 + 4 = 6-6 + 4 = 4.
3 X 12 + 7 H- 28i = 36 + 0.2456 = 36.2456.
3 + 5X3X5 + 2-3X6 = 3 + 75 + 2- 18 = 62.
These examples should be carefully studied until
thoroughly understood.
We are now ready to return to the question of
square roots. The square root of a number is that
number which, if multipled by itself, would give
the given number. Thus, the square root of 9 is 3,
because 3 multiplied by itself equals 9. The square
root of 16 equals 4, of 36 equals 6, and so forth. It
will be seen at once that the square root may be
86 SELF-TAUGHT MECHANICAL DRAWING
considered, or, rather, actually is the reverse of
the square, so that if the square of 20 is 400, then
the square root of 400 is 20. In the same way, as
the square of 100 is 10,000, so the square root of
10,000 is .100. The sign used_m mathematical
formulas for the square root is V . Thus V 9 = 3,
V 49 = 7, and . so forth. The process of actually
calculating the square root is rather cumbersome,
and it is very seldom required, because, as already
mentioned, the engineering hand-books usually
give tables of square roots for all numbers up to
1000, and for larger numbers the tables can also be
used for obtaining the square root approximately
correct, or at least near enough so for almost all
practical calculations.
The cube of a number is the product resulting
from repeating the given number as a factor three
times. Thus, the cube of 3 is 3 X 3 X 3 - 27, and
the cube of 17 is 17 X 17 X 17 = 4913. In the same
way as we write 22 = 2X2 = 4, for the square of
2, so we can write 23 = 2 X2 X 2 = 8, for the cube
of 2. The exponent (3) indicates how many times
the given number is to be repeated as a factor.
The cube of 4, for example, may be written 43 = 4
X 4 X 4 = 64. Similarly 17 3 - 4913. The expres-
sion 17 3 may be read " the cube of 17," "17 cube/'
or "the third power of 17." In the same way as
the square root means the reverse of square, so the
cube root (or "third root") means the reverse of
cube or "third power" ; that is, the cube root of a
number is the number which, if repeated as factor
three times, would give the given number. For
example, the cube root of 64 is 4, because 4 X 4 X
ALGEBRAIC FORMULAS 87
4 = 64. It is evident that if the cube of a number,
say 6, is 216 (6 X 6 X 6 = 216), then the cube root
of 216 is 6. The sign used injformulas for the cube
root is f^. For_example, f7 8 = 2 (because 2X2
X 2= 8), and 1^125 = 5 (because 5X5X5= 125).
Similarly, 1^3,723,875 = 155.
The use of the square and square root, and cube
and cube root in formulas may be shown by a few
examples :
V B X V C
Assume that B = 27, C = 25, and D = 2. Insert
these values in the formula. Then
X 125 3X5 15
252 + 22 " 125 +4 "" 129 "
As another example:
z?
A
B3 X V C
Assume B = 2, C = 9, and D = 4. Then
_ 22 + 92+42 _ 4 + 81 + 16 _. 101.
23XT/y 8X3 24
In the same way as 22 = 2 X 2 = 4, so 24 = 2 X 2
X 2 X 2 = 16, and 25 = 2X2X2X2X2 = 32.
The expression 24 is read the "fourth power of
2," and 2 5 the "fifth power of 2." The exponents
(4) and (5) indicate how many times the given
number is to be repeated as factor.
If, again, it is required to find the number which,
if repeated as factor four times, gives the given
number, we must obtain the "fourth root" or V~
88 SELF-TAUGHT MECHANICAL DRAWING
Thus, Vl6 = 2, ^because 2 X 2 X 2 X 2 = 16. In
the same way V256 = 4. The fifth root is writ-
ten V ; and \/243 = 3, because 3X3X 3X3X3
= 243.
These explanations, when fully understood, will
eliminate all difficulties with formulas of a simple
nature, and with such expressions as cube root,
exponents, etc.
An important method facilitating the use of
formulas, is commonly known as the transposition
of formulas. A formula for finding the horse-
power which can safely be transmitted by a gear
of a given size, running at a given speed, is :
D X NX PX FX20Q
H.P. =
126,050
In this formula H.P. = horse-power,
D = pitch diameter,
N = revolutions per minute,
P = circular pitch of gear,
F = width of face of gear.
Assume, for example, that the pitch diameter of
a gear is 31.5 inches, the number of revolutions
per minute 200, the circular pitch 1J inch, and the
width of the face 3 inches. Then, if these values
arc inserted in the formula, we have :
31.5 X 200 X lj X 3 X 200 ,c
H'R - ~ = 45
power, very nearly.
Assume, however, that the horse-power required
to be transmitted is known, and that the pitch of
the gear is required to be found. Assume that
ALGEBRAIC FORMULAS 89
tf.P = 30; Z> = 31.5; N = 200; F=3; and that P
is the unknown quantity; then, inserting the
known values in the formula, gives us:
31.5 X 200 XPX 3 X 200
126,050
In order to be able to find P, we want it given
on one side of the equals sign, with all the known
quantities on the other side. If we multiply the
expressions on both sides of the equals sign by
the same number we do not change the conditions ;
thus
on v i oa AKA - 31.5 X 200 X P X 3 X 200 X 126,050
1Zb'Ut 126,050
By canceling the number 126,050 on the right-
hand side we have :
30 X 126,050 = 31.5 X 200 X P X 3 X 200.
If we now divide on both sides of the equals
sign with 31.5 X 200 X 3 X 200, we have:
30 X 126,050 = 31.5X200 XPX 3X200
31.5 X 200 X 3 X 200 31.5 X 200 X 3 X 200
We can now cancel all numerical values in the
fraction on the right-hand side; then:
30 X 126,050 p
31.5X200X3X200
This is then the transposed formula giving P,
and from this we find that P = 1 inch.
In general, any formula of the form
B
A---C.
can be transposed as below :
A XC = B C = -
90 SELF-TAUGHT MECHANICAL DRAWING
It will be seen that the quantities which are in
the denominator on one side of the equals sign, are
transposed into the numerator on the other side,
and vice versa.
Examples:
BX C
A D .
Then:
n _ Bx c D AX D n AX D
A ' = C ' = ~~B '
A _ EXFX G
KXL
Then:
E_A XKXL F_A XKXL . r _A X KxL
FX G ' EX G ' EXF '
rf = EX FX G T EXFX G
AXL AXK
The principles of transposition of formulas can
best be grasped by a careful study of the examples
given. Note that the method is only directly ap-
plicable when all the quantities in the numerator
and denominator are factors of a product. If con-
nected by + or - signs, the transposition cannot
be made by the simple methods shown unless the
whole sum or difference is transposed. Example :
A =- ; then D =~and£ + C= A X D.
The most usual caclulations, perhaps, in some
classes of machine design, are those involving the
finding of the strength of certain machine mem-
bers ; and, in order to find the strength qf these
ALGEBRAIC FORMULAS 91
members, it is necessary to first find the cross-
sectional area of the part subjected to stress. For
this reason, the remainder of this chapter will be
largely taken up with rules and formulas for find-
ing the areas and other properties of various geo-
metrical figures. Rules and formulas for volumes
of solids will also be given. Examples have been
given in some cases merely to show the applica-
tions of the formulas.
The area of a triangle equals one-half the prod-
uct of its base and its altitude. The base may be
any side of the triangle, and the altitude is the
length of the line drawn from the angle opposite
the base, perpendicular to it.
Assume that A = area of triangle,
. B = base,
- H = altitude.
Then the rule above may be expressed as a
formula as follows :
A ^BXH
'
Let the base (B) of a triangle be 5 feet, and the
altitude (H) 8 feet. Then the area
5X8 40 ' .
A = — n — = -JT = 20 square feet.
z z
The area of a square equals the square of its
side. If A = the area, and S the side of the square,
then
If the side is 9.7 inches long, then
A = 9.7 2= 9.7 X 9.7 =94.09 square inches.
92 SELF-TAUGHT MECHANICAL DRAWING
The area of a rectangle equals the product of its
long and short sides. If A = area, L = length of
the longer side, and H= length of the shorter side,
then
A = L X H.
The area of a parallelogram equals the product
of the base and the altitude.
The area of a trapezoid equals one-half the sum
of the parallel sides multiplied by the altitude.
If A = area, B = length of one of the parallel sides,
C = length of the other parallel side, and H =
altitude, then
• B
Assume that the lengths of the two parallel
sides are 12 and 9 feet, respectively, and that the
altitude is 16 feet. Then
A = ^-^ X 16 = 10.5 X 16 = 168 square feet.
To find the area of an irregular figure bounded
by straight lines, divide the figure into triangles,
and find the area of each triangle separately. The
sum of the areas of all the triangles equals the
area of the figure.
The circumference of a circle equals its diameter
multiplied by 3.1416.
The diameter of a circle equals the circumfer-
ence divided by 3.1416.
The area of a circle equals the square of the
diameter multiplied by 0.7854.
The diameter of a circle equals the area divided
ALGEBRAIC FORMULAS 93
by 0.7854, and the square root extracted of the
quotient.
If D = diameter, C = circumference, and A =
area, these last rules may be expressed in formulas
as follows :
C-DX 3.1416. £=3ib-
A = D*X 0.7854. D =
The length of a circular arc equals the circum-
ference of the circle, multiplied by the number of
degrees in the arc, divided by 360. If L = length
of arc, C = circumference of circle, and N = num-
ber of degrees in the arc, then
r CXN
360
The area of a circular sector equals the area of
the whole circle multiplied by the quotient of the
number of degrees in the arc of the sector divided
by 360. If a = area of sector, A = area of circle,
and N = number of degrees in sector, then
a = A X N
360*
The area of a circular segment equals the area of
the circular sector formed by drawing radii from
the center of the circle to the extremities of the
arc of the segment, minus the area of the triangle
formed by these radii and the chord of the arc of
the segment.
The area of a pentagon (regular polygon having
94 SELF-TAUGHT MECHANICAL DRAWING
five sides) equals the square of the side times
1.720.
The area of a hexagon (regular polygon having
six sides) equals the square of the side times
2.598.
The area of a heptagon (regular polygon having
seven sides) equals the square of the side times
3.634.
The area of an octagon (regular polygon having
eight sides) equals the square of the side times
4.828.
The volume of a cube equals the cube of the
length of its side.
The volume of a prism equals the area of the
base multiplied by the altitude.
The volume of a cylinder equals the area of its
base circle multiplied by the altitude.
The volume of a pyramid or cone equals the area
of the base times one-third the altitude.
The area of the surface of a sphere equals the
square of the diameter multiplied by 3.1416.
The volume of a sphere equals the cube of the
diameter times 0.5236.
The volume of a spherical sector equals two-
thirds of the square of the radius of the sphere
multiplied by the height of the contained spherical
segment, multiplied by 3.1416. If V = volume of
sector, R = radius of sphere, and H= height of the
contained spherical segment, then
V = f-tf2 X HX 3.1416.
o
Assume that the length of the radius of a spheri-
ALGEBRAIC FORMULAS 95
cal sector is 6 inches, and the height of the con-
tained segment 2 inches. Then
V = -f-X 62 X 2 X 3.1416 = 150. 7968 cubic inches.
o
The volume of a spherical segment equals the
radius of the sphere less one-third the height of
the segment, multiplied by the square of the
height of the segment, multiplied by 3.1416. If R
= radius, H = height, and V = volume of segment,
then
(R- y)
XH2 X 3.1416.
Assume that the length of the radius is 4 inches,
and the height of the segment 3 inches. Then
V = (4 - y) X 32 X 3.1416 = 84. 8232 cubic inches.
The area of an ellipse equals the long axis multi-
plied by the short axis, multiplied by 0.7854. If
the area =A, the long axis =B, and the short axis
= C, then
A = BXCX 0.7854.
If the long axis is 12 inches and the short axis
8J inches, then
A = 12 X 8i X 0.7854 - 78.54.
Formulas and application of formulas have not
been given for such rules which are so simple and
easy to understand that the reader without diffi-
culty can formulate his own formula.
CHAPTER VII
ELEMENTS OF TRIGONOMETRY
TRIGONOMETRY is a very important part of the
science of mathematics, and deals with the deter-
mination of angles and the solution of triangles.
In order to fully understand the subjects treated
of in the following, it is necessary that the reader
is fully familiar with the usual methods of desig-
nating the measurements or sizes of angles. While
mathematicians employ also another method, in
mechanics angles are measured in degrees and
subdivisions of a degree, called minutes. The
minute is again subdivided into seconds, but these
latter subdivisions are so small as to permit of
being disregarded in general practical machine
design.
A degree is 1-360 part of a circle, or, in other
words, if the circumference of a circle is divided
into 360 parts, then each part is called one degree.
If two lines are drawn from the center of the
circle to the ends of the small circular arc which
is 1-360 part of the circumference, then the angle
between these two lines is a 1-degree angle. A
quarter of a circle or a 90-degree angle is called a
right angle. The meaning of obtuse and acute
angles has already been explained in Chapter II.
Any angle which is not a right angle is called an
oblique angle.
96
ELEMENTS OF TRIGONOMETRY 97
A minute is 1-60 part of a degree, and a second
1-60 part of a minute. In other words, one circle
= 360 degrees, one degree = 60 minutes, and one
minute = 60 seconds. The sign (°) is used for in-
dicating degrees; the sign (') indicates minutes,
and the sign ( " ) seconds. A common abbreviation
for degree is ' l deg. ' ' ; for minute, ' ' min. ' ' ; and for
second, "sec."
Two angles are equal when the number of de-
grees they contain is the same. If two angles are
both 30 degrees, they are equal, no matter how
long the sides of the one may be in relation to the
other.
Of all triangles, the right-angled triangle occurs
most frequently in machine design. A right-ang-
led triangle is one having the angle between two
sides a right angle; the angles between the other
sides may be of any size. In the calculations in-
volved in solving right-angled triangles, a useful
application of the squares and square roots of
numbers is also presented. Assume that the lengths
of the sides of a right-angled triangle, as shown
in Fig. 88, are 5 inches, 4 inches, and 3 inches,
respectively. Then
52 = 42 + 32, or 25 = 16 4- 9.
This relationship between the three sides in a
right-angled triangle holds good for all right-ang-
led triangles. The square of the side opposite the
right angle equals the sum of the squares of the
sides including the right angle. Assume, for ex-
ample, that the lengths of the two sides including
the right angle in a right-angled triangle are 12
98
SELF-TAUGHT MECHANICAL DRAWING
and 9 inches long, respectively, as shown in Fig.
89, and that the side opposite the right angle, the
hypotenuse, is to be found. We then first square
the two given sides, and from our rule, just given,
we have that the sum of the squares equals the
square of the side to be found. The square root
-*• H h •'
of the sum must then equal the side, itself. Carry-
ing out this calculation we have:
12 M-_92 = 144 + 81 = 225
V 225 = 15 inches = length of hypotenuse.
Similar methods may be employed for finding
any of the sides in a right-angled triangle if two
sides are given. If the hypotenuse were known
to be 15 inches, and one of the sides including the
right angle 9 inches, as shown at D in Fig. 90,
then the other side including the right angle can
be found. In this case, however, we must subtract
the square of the known side including the right
ELEMENTS OF TRIGONOMETRY
99
angle from the square of the hypotenuse to obtain
the square of the remaining including side. We,
therefore, have:
152-92 = 225-81 =
\/144 = 12 inches = length of unknown side.
In the same way, if the lengths 15 and 12 were
k; SIDE TO BE FOUND >
FIG. 90.
known, we could find the side AC, as shown at E,
Fig. 90:
15^- 122 = 225- 144 = 81
Vgf = 9 inches = length of AC.
From these examples we may formulate rules
and general formulas for the solution of right-
angled triangles when two sides are known. In
Fig. 91, at F, the square of AB plus the square of
AC equals the square of EC; the square of EC
minus the square of AC equals the square of AB;
and the square of EC minus the square of AB
100
SELF-TAUGHT MECHANICAL DRAWING
equals the square of AC. These rules written as
general formulas would take the form :
BC2-AB2 =
From these formulas we have, by extracting the
square root on each side of the equal sign :
EC = V~AB2 + AC2
AB = VBC2 - AC2
AC = V BC2 - AB2
These formulas make it possible to find the third
side when two sides are given, no matter what the
numerical values of the length of the sides may
be. Assume AB = 12, and BC = 20; find AC. Ac-
cording to the formula :
AC = \/202- 122 = V400- 144 = V^56 =16.
Assume that AB = 15 and AC = 20. Find BC.
BC =Vl52 + 202 = V 225 + 400 = \/625 = 25.
The rules and formulas given make it possible to
find the length of the sides in a right-angled tri-
angle. To -find the angles, however, use must be
ELEMENTS OF TRIGONOMETRY 101
made of the trigonometric functions, the meanings
of which will be presently explained. The trigo-
nometric functions are the sine, cosine, tangent, co-
tangent, secant and cosecant of angles. While these
functions are used in the solution of all kinds of
triangles, they refer directly to right-angled tri-
angles, and the meaning or value of each function
can be explained by reference to a right-angled
triangle as shown in Fig. 91, at G, where the side
BC is the hypotenuse, AC the side adjacent to
angle D, and AB the side opposite angle D. Of
course, if reference is made to angle E, then AB
is the side adjacent and AC the side opposite.
The sine of an angle is the length of the opposite
side, if the hypotenuse is assumed to equal 1. The
sine of angle D, then, is the length of AB if BC
equals 1. To find the sine of D when BC is any
other length, divide AB by the length of BC. To
find the sine of D, if BC equals 5, for example, it
is necessary to divide the length of AB by 5.
Find the sine of D, when AB = 15 and BC = 20.
The sine of D = 15 * 20 = 0.75.
The cosine of an angle is the length of the adja-
cent side, if the hypotenuse is assumed to equal 1.
The cosine of angle D, then, is the length of AC
if BC equals 1. To find the cosine of D when BC
is any other length, divide -AC by the length of
BC. To find the cosine of D, if BC equals 8, for
example, it is necessary to divide the length of
AC by 8.
Find the cosine of D, when AC = 12 and BC =
30. The cosine of D = 12 -*- 30 = 0.4.
The tangent of an angle is the length of th<
102 SELF-TAUGHT MECHANICAL DRAWING
posite side, if the adjacent side is assumed to
equal 1. The tangent of angle D is the length of
AB if AC equals 1. To find the tangent of D when
AC equals any other length, divide AB by the
length of AC. To find the tangent of D when AC
equals 3, for example, it is necessary to divide the
length of AB by 3.
Find the tangent of D, when AB = 16 and AC
= 12. The tangent of D = 16 + 12 = 1.333.
The cotangent of an angle is the length of the
adjacent side, if the opposite side is assumed to
equal 1. The cotangent of angle D is the length
of AC if AB equals 1. To find the cotangent of D
when AB equals any other length, divide AC by
the length of AB. To find the cotangent of D
when AB equals 12, for example, divide AC by 12.
Find the cotangent of D when AB = 3 and AC
= 36. The cotangent of D = 36 + 3 = 12.
The secant of an angle is the length of the hypo-
tenuse, if the adjacent side is assumed to equal 1.
The secant of angle D is the length of BC when
AC equals 1. To find the secant of -D when AC is
any other length, divide BC by the length of AC.
Find the secant of D when BC = 24 and AC = 9.
The secant of D = 24 -*- 9 = 2.666. . .
The cosecant of an angle is the length of the
hypotenuse if the opposite side is assumed to equal
1. The cosecant of angle D is the length of BC
when AB equals 1. To find the cosecant of D when
AB is any other length, divide BC by the length
of AB.
Find the cosecant of D when BC = 30 and AB
= 3.75. The cosecant of D = 30 ^ 3.75 = 8.
ELEMENTS OF TRIGONOMETRY 103
The expressions sine, cosine, tangent, cotangent,
secant and cosecant are abbreviated as follows:
sin, cos, tan, cot, sec, and cosec. Instead of writ-
ing tangent of D, for example, it is usual to write
tan D. By means of these functions, tables of
which are given in the following, the values of
angles can be introduced in the calculations of tri-
angles. The tables here used give the values of
the functions of angles for every degree and for
every ten minutes. Only three decimal places are
given, as that is enough for the great majority of
shop calculations. When very accurate calculations
are required, tables can be procured giving the
functions for every minute, and with five decimal
places. From the tables given, when the angle is
known, the corresponding angular function can be
found, and when the function is known, the cor-
responding angle can be determined by merely
reading off the values in the table. The tables in-
clude sines, cosines, tangents and cotangents only,
as these are most commonly used, and all problems
can be solved by the use of them. When the se-
cant is required, it can be found by dividing 1 by
the cosine. The cosecant is found by dividing 1
by the sine.
The tables of sines, cosines, etc., are read the
same as any other table. It will be seen that the
four tables given are headed Sines, Cosines, Tan-
gents, and Cotangents, respectively. At the bottom
of the table headed "Sines" is read the word
"Cosines," and at the bottom of the table headed
"Cosines" is read the word "Sines." In the same
way, at the bottom of the table headed "Tan-
104
SINES
MINUTES.
DEG.
DEG.
0'
10'
20'
30'
40'
50'
60'
0
0.000
0.003
0.006
0.009
0.012
0.015
0.017
89
1
0.017
0.020
0.023
0.026
0.029
0.032
0.035
88
2
0.035
0.038
0.041
0.044
0.047
0.049
0.052
87
3
0.052
0.055
0.058
0.061
0.064
0.067
0.070
86
4
0.070
0.073
0.076
0.078
0.081
0.084
0.087
85
5
0.087
0.090
0.093
0.096
0.099
0.102
0.105
84
6
0.105
0.107
0.110
0.113
0.116
0.119
0.122
83
7
0.122
0.125
0.128
0.131
0.133
0.136
0.139
82
8
0.139
0.142
0.145
0.148
0.151
0.154
0.156
81
9
0.156
0.159
0.162
0.165
0.168
0.171
0.174
80
10
0.174
0.177
0.179
0.182
0.185
0.188
0.191
79
11
0.191
0.194
0.197
0.199
0.202
0.205
0.208
78
12
0.208
0.211
0.214
0.216
0.219
0.222
0.225
"77
13
0.225
0.228
0.231
0.233
0.236
0.239
0.242
76
14
0.242
0.245
0.248
0.250
0.253
0.256
0.259
75
15
0.259
0.262
0.264
0.267
0.270
0.273
0.276
74
16
0.276
0.278
0.281
0.284
0.287
0.290
0.292
73
17
0.292
0.295
0.298
0.301
0.303
0.306
0.309
72
18
0.309
0.312
0.315
0.317
0.320
0.323
0.326
71
19
0.326
0.328
0.331
0.334
0.337
0.339
0.342
70
20
0.342
0.345
0.347
0.350
0.353
0.35)
0.358
69
21
0.358
0.361
0.364
0.367
0.369
0.372
0.375
68
22
0.375
0.377
0.380
0.383
0.385
0.388
0.391
67
23
0.391
0.393
0.396
0.399
0.401
0.404
0.407
66
24
0.407
0.409
0.412
0.415
0.417
0.420
0.423
65
25
0.423"
0.425
0.428
0.431
0.433
0.436
0.438
64
26
0.438
0.441
0.444
0.446
0.449
0.451
0.454
63
27
0.454
0.457
0.459
0.482
0.464
0.467
0.469
62
28
0.469
0.472
0.475
0.477
0.480
0.482
0.485
61
29
0.485
0.487
0.490
0.492
0.495
0.497
0.500
60
30
0.500
0.503
0.505
0.508
0.510
0,513
0.515
59
31
0.515
0.518
0.520
0.522
0.525
0.527
0.530
58
32
0.530
0.532
0.535
0.537
0.540
0.542
0.545
57
33
0.545
0.547
0.553
0.552
0.554
0.557
0.559
56
34
0.559
0.562
0.564
0.566
0.569
0.571
0.574
55
35
0.574
0.576
0.578
0.581
0.583
0.585
0.588
54
36
0.588
0.590
0.592
0.595
0.597
0.599
0.602
53
37
0.602
0.604
0.606
0.609
0.611
0.613
0.616
52
38
0.616
0.618
0.620
0.623
0.625
0.627
0.629
51
39
0.629
0.632
0.634
0.636
0.638
0.641
0.643
50
40
0.643
0.645
0.647
0.649
0.652
0.654
0.656
49
41
0.656
0.658
0.660
0.663
0.665
0.667
0.669
48
42
0.669
0.671
0.673
0.676
0.678
0.680
0.682
47
43
0.682
0.684
0.686
0.688
0.690
0.693
0.695
46
44
0.695
0.697
0.699
0.701
0.703
0.705
0.707
45
60'
50'
40'
30'
20'
10'
0'
MINUTES.
COSINES
COSINES
105
MINUTES.
DEC.
DEG.
0'
10'
20'
30'
40'
50'
60'
0
1.000
1.000
1.000
1.000
1.000
1.000
1.000
~~89~~
1
1.000
1.000
1.000
1.000
1.000
0.999
0.999
88
2
0.999
0.999
0.999
0.999
0.999
0.999
0.999
87
3
0.999
0.998
0.998
0.998
0.998
0.998
0.998
86
4
0.998
0.997
0.997
0.997
0.997
0.996
0.996
85
5
0.996
0.996
0.993
0.995
0.995
0.995
0.995
84
6
0.995
0.994
0.994
0.994
0.993
0.993
0.993
83
7
0.993
0.992
0.992
0.991
0.991
0.991
0.990
82
8
0.990
0.990
0.989
0.989
0.98)
0.988
0.988
81
9
0.988
0.987
0.987
0.986
0.986
0.985
0.985
80
10
0.985
0.984
0.984
0.983
0.983
0.982
0.982
79
11
0.982
0.981
0.981
0.980
0.979
0.979
0.978
78
12
0.978
0.978
0.977
0.976
0.976
0.975
0.974
77
13
0.974
0.974
0.973
0.972
0.972
"0.971
0.970
76
14
0.970
0.970
0.969
0.968
0.967
0.967
0.966
75
15
0.966
0.965
0.964
0.964
0.963
0.962
0.961
. 74
16
0.961
0.960
0.960
0.959
0.958
0.957
0.956
73
17
0.956
0.955
0.955
0.954
0.953
0.952
0.951
72
18
0.951
0.950
0.949
0.948
0.947
0.946
0.946
71
19
0.946
0.945
0.944
0.943
0.942
0.941
0.940
70
20
0.940
0.939
0.938
0.937
0.936
0.935
0.934
69
21
0.934
0.933
0.931
0.930
0.929
0.928
0.927
68
22
0.927
0.926
0.925
0.924
0.923
0.922
0.921
67
23
0.921
0.919
0.918
0.917
0.916
0.915
0.914
66
24
0.914
0.912
0.911
0.910
0.909
0.908
0.906
65
25
0.906
0.905
0.904
0.903
0.901
0.900
0.899
64
26
0.899
0.898
0.896
0.895
0.894
0.892
0.891
63
27
0.891
0.890
0.888
0.887
0.886
0.884
0.883
62
28
0.883
0.882
0.880
0.879
0.877
0.876
0.875
61
29
0.875
0.873
0.872
0.870
0.869
0.867
0.866
60
30
0.866
0.865
0.863
0.862
0.860
0.859
0.857
59
31
0.857
0.856
0.854
0.853
0.851
0.850
0.848
58
32
0.848
0.847
0.845
0.843
0.842
0.840
0.839
57
33
0.839
0.837
0.835
0.834
0.832
0.831
0.829
56
34
0.829
0.827
0.826
0.824
0.822
0.821
0.819
55
35
0.819
0.817
0.816
0.814
0.812
0.811
0.809
54
36
0.809
0.807
0.806
0.804
0.802
0.800
0.799
53
37
0.799
0.797
0.795
0.793
0.792
0.790
0.788
52
38
0.788
0.786
0.784
0.783
0.781
0.779
0.777
51
39
0.777
0.775
0.773
0.772
0.770
0.768
0.766
50
40
0.766
0.764
0.762
0.760
0.759
0.757
0.755
49
41
0.755
0.753
0.751
0.749
0.747
0.745
0.743
48
42
0.743
0.741
0.739
0.737
0.735
0.733
0.731
47
43
0.731
0.729
0.727
0.725
0.723
0.721
0.719
46
44
0.719
0.717
0.715
0.713
0.711
0.709
0.707
45
T^FT1
60'
50'
40'
30'
20'
10'
0'
J-^-h-O .
MINUTES.
DEG.
SINES
106
TANGENTS
DEG.
MINUTES.
DEG.
0'
10'
20'
30'
40'
50'
60'
0
0.000
0.003
0.006
0.009
0.012
0.015
0.017
89
1
0.017
0.020
0.023
0.026
0.029
0.032
0.035
88
2
0.035
0.038
0.041
0.044
0.047
0.049
0.0£2
87
3
0.052
0.055
0.058
0.061
0.064
0.067
0.070
86
4
0.070
0.073
0.076
0.079
0.082
0.085
O.C87
85
5
0.087
0.090
0.093
0.096
0.099
0.102
0.105
84
6
0.105
0.108
0.111
0.114
0.117 0.120
0.123
83
7
0.123
0.126
0.129
0.132
0.135 0.138
0.141
82
8
0.141
0.144
0.146
0.149
0.152 0.155 o.l£8
•81
9
0.158
0.161
0.164
0.167
0.170 0.173 0.176
80
10
0.176
0.179
0.182
0.185
0.188 0.191
0.194
79
11
0.194
0.197
0.200
0.203
0.206 0.210
0.213
78
12
0.213
0.216
0.219
0.222
0.225 0.228 0.231
77
13
0.231
0.234
0.237
0.240
0.243 0.246 0.249
76
14
0.249
0.252
0.256
0.259
0.262 0.265
0.268
75
15
0.268
0.271
0.274
0.277
0.280
0.284
0.287
74
16
0.287
0.290
0.293
0.296
0.299
0.303
0.306
73
17
0.306
0.309
0.312
0.315
0.318
0.322
0.325
72
18
0.325
0.328
0.331
0.335
0.338
0.341
0.344
71
19
0.344
0.348
0.351
0.354
0.357
0.361
0.364
70
20
0.364
0.367
0.371
0.374
0.377
0.381
0.384
69
21
0.384
0.387
0.391
0.394
0.397
0.401
0.404
68
22
0.404
0.407
0.411
0.414
0.418
0.421
0.424
67
23
0.424
0.428
0.431
0.435
0.438
0.442
0.445
66
24
0.445
0.449
0.452
0.456
0.459
0.463
0.466
65
25
0.466
0.470
0.473
0.477
0.481
0.484
0.488
64
26
0.488
0.491
0.495
0.499
0.502
0.506
0.510
63
27
0.510
0.513
0.517
0.521
0.524
0.528
0.532
62
28
0.532
0.535
0.539
0.543
0.547
0.551
0.554
61
29
0.554
0.558
0.562
0.566
0.570
0.573
0.577
60
30
0.577
0.581
0.585
0.589
0.593
0.597
0.601
59
31
0.601
0.605
0.609
0.613
0.617
0.621
0.625
58
32
0.625
0.629
0.633
0.637
0.641
0.645
0.649
57
33
0.649
0.654
0.658
0.662
0.666
0.670
0.675
56
34
0.675
0.679
0.683
0.687
0.692
0.696
0.700
55
35
0.700
0.705
0.709
0.713
0.718
0.722
0.727
54
36
0.727
0.731
0.735
0.740
0.744
0.749
0.754
53
37
0.754
0.758
0.763
0.767
0.772
0.777
0.781
52
38
0.781
0.786
0.791
0.795
0.800
0.805
0.810
51
39
0.810
0.815
0. 819 ! 0.824
0.829
0.834
0.839
50
40
0.839
0.844
0.849
0.854
0.859 0.864
0.869
49
41
0.869
0.874
0.880
0.885
0.890 0.895
0.900
48
42
0.900
0.906
0.911
0.916
0.922 0.927
0.933
47
43
0.933
0.938
0.943
0.949
0.955 0.960
0.966
46
44
0.966
0.971
0.977
0.983
0.988
0.994
1.000
45
60'
50'
40'
30'
20'
10'
0'
DEG.
MINUTES.
DEG.
COTANGENTS
COTANGENTS
107
MINUTES.
DEC.
DEG.
0'
10'
20'
30'
40'
50'
60'
0
oo
343.8 171.9
114.6
85.94
68.75
57.29
89
1
57. 29
49.10
42.96
38.19
34.37
31.24
28.64
88
2
28.64
26.43
24 . 54
22.90
21.47
20.21
19.08
87
3
19.08
18.07
17.17
16.35
15.60
14.92
14.30
86
4
14.30
13.73
13.20
12.71
12.25
11.83
11.43
85
5
11.43
11.06
10.71
10.39
10.08
9.788
9.514
84
6
9.514
9.225
9.010
8.777
8.556
8.345
8.144
83
7
8.144
7.953
7.770
7.596
7.429
7.269
7.115
82
8
7.115
6.968
6.827
6.691
6.561
6.435
6.314
81
9
6.314
6.197
6.084
5.976
5.871
5.769
5.671
80
10
5.671
5.576
5.485
5.396
5.309
5.226
5.145
79
11
5.145
5.066
4.989
4.915
4.843
4.773
4.705
78
12
4.705
4.638
4.574
4.511
4.449
4.390
4.331
77
13
4.331
4.275
4,219
4.165
4.113
4.061
4.011
76
14
4.011
3.962
3.914
3.867
3.821
3.776
3.732
75
15
3.732
3.689
3.647
3.606
3.566
3.526
3.487
74
16
3.487
3.450
3.412
3.376
3.340
3.305
3.271
73
17
3.271
3.237
3.204
3.172
3.140
3.108
3.078
72
18
3.078
3.047
3.018
2.989
2.960
2.932
2.904
71
19
2.904
2.877
2.850
2.824
2.798
2.773
2.747
70
20
2.747
2.723
2.699
2.675
2.651
2.628
2.605
69
21
2.605
2.583
2.560
2.539
2.517
2.496
2.475
68
22
2.475
2.455
2.434
2.414
2.394
2.375
2.356
67
23
2.356
2.337
2.318
2.300
2.282
2.264
2.246
66
24
2.246
2.229
2.211
2.194
2.177
2.161
2.145
65
25
2.145
2.128
2.112
2.097
2.081
2.066
2.050
64
26
2.050
2.035
2.020
2.006
1.991
1.977
1.963
63
27
.963
1.949
1.935
1.921
1.907
1.894
1.881
62
28
.881
1.868
1.855
1.842
1.829
1.816
1.804
61
29
.804
1.792
1.780
1.767
1.756
1.744
1.732
60
30
.732
1.720
1.709
1.698
1.686
1.675
1.664
59
31
.664
1.653
1.643
.632
1.621
1.611
1.600
58
32
.600
1.590
1.580
.570
.560
1.550
1.540
57
33
1.540
1.530
1.520
.511
.501
1.492
1.483
56
34
1.483
1.473
1.464
.455
.446
1.437
1.428
55
35
1.428
1.419
.411
.402
.393
1.385
1.376
54
36
1.376
1.368
.360
.351
.343
1.335
1.327
53
37
1.327
1.319
.311
.303
.295
1.288
1.280
52
38
1.280
1.272
.265
.257
.250
1.242
1.235
51
39
.235
1.228
1.220
.213
.206
1.199
1.192
50
40
.192
1.185
.178
.171
.164
1.157
1.150
49
41
.150
1.144
1.137
.130
.124
1.117
1.111
48
42
.111
1.104
.098
.091
.085
1.079
1.072
47
43
.072
1.066
.060
.054
.048
1.042
1.036
46
44
.036
1.030
.024
.018
.012
1.006
1.000
45
60'
50'
40'
30'
20'
10'
O'
DTTP
T^T1/~l
-L/rj(j .
MINUTES.
J_JEG.
TANGENTS
108 SELF-TAUGHT MECHANICAL DRAWING
gents, " we read "Cotangents/' and at the bottom
of the table headed " Cotangents, " we read " Tan-
gents/' The object of this will be presently ex-
plained. The extreme left-hand column, we find, is
headed "Deg.," and the following seven columns
are headed 0', 10', 20', 30', 40', 50' and 60', re-
spectively, these columns indicating the minutes.
At the bottom of the pages the same numbers are
found but reading from the right to the left. The
values of the functions marked at the top are read
in the table opposite the degrees in the left-hand
column and under the minutes at top. The values
of the functions marked at the bottom are read
opposite the degrees in the right-hand column and
over the minutes at the bottom. For example, the
sine of 39 ° 40 ' or sin 39 ° 40 ', as it is written in
formulas, is thus found to be 0.638, and the sine
of 64° 10' is 0.900, this latter value being read off
in the second table, reading it from the bottom up,
and locating the number of degrees in the right-
hand column.
As further examples, we find
tan 37° 40 ' = 0.772
cot 37° 40 ' = 1.295
tan80° 0' = 5.671
cos 75° 30 ' = 0.250
We are now ready to proceed to solve right-ang-
led triangles with regard both to the sides and the
angles. In any right-angled triangle, if either two
sides, or one side and one of the acute angles are
known, the remaining quantities can be found. As
a general rule, in any triangle, all the quantities
ELEMENTS OF TRIGONOMETRY
109
can be found when three quantities, at least one
of which is a side, are given. In a right-angled
triangle the right angle is always known, of
course, so that here, therefore, only two additional
quantities are necessary. If all the three angles
are known, the length of the sides cannot be de-
termined; one side, at least, must also always be
known in order to make possible the solution of
the triangle.
The following rules should be used for solving
right-angled triangles.
Case 1. Two sides known. — Use the rules al-
ready given in this chapter for finding the third
T
i
r ' 1
GIVEN
ANGL
< ADJACENT SIDE >j
FIG. 92.
side when two sides in a right-angled triangle are
given. To find the angles use the rules already
given for finding sines, cosines, etc., and the
tables.
Case 2. Hypotenuse and one angle given. — Call
the side adjacent to the given angle the adjacent
side, and the side opposite the given angle the
opposite side (see Fig. 92.) Then the adjacent
side equals the hypotenuse multiplied by the cosine
110 SELF-TAUGHT MECHANICAL DRAWING
of the given angle; the opposite side equals the
hypotenuse multiplied by the sine of the given
angle ; and the unknown angle equals 90 degrees
minus the given angle.
Case 3. One angle and its adjacent side given.
—The hypotenuse equals the adjacent side divided
by the cosine of the given angle ; the opposite side
equals the adjacent side multiplied by the tangent
of the given angle; and the unknown angle is
found as in Case 2.
Case 4. One angle and its opposite side known.
—The hypotenuse equals the opposite side divided
by the sine of the given angle; the adjacent side
equals the opposite side multiplied by the cotangent
of the given angle; and the unknown angle is
found as in Case 2.
These rules may be written as formulas as fol-
lows (see Fig. 93) :
Case 1. For formulas for the sides see the first
part of this Chapter. For the angles we have:
sin B = - sin C = — .
a • a
Case 2. Here, when a and B are given, we have :
c = a cos B; b = a sin B; C = 90° - B.
When a and C are given, we have :
b = a cos C; c = a sin C; £ = 90° - C.
Case 3. Here, when B and c are given, we have:
a = — C-^; b = c tan B; C = 90° - B.
cos B
When C and 6 are given, we have:
a = — ~; c = 6 tan C; B - 90° - C.
cos C'
ELEMENTS OF TRIGONOMETRY
111
Case 4. Here, when B and b are known, we
have:
,; c = b cot B; C = 90° - B.
sin
When C and c are known, we have :
a
- ~\ 6 - c cot C;
sin o
90° - C.
These rules and formulas, while not including all
possible combinations for the solution of right-
angled triangles, give all the information neces-
sary for the solution of any kind of a right-angled
A
C Bf
r-
FIG. 94.
FIG. 95.
triangle. A few examples of the use of these rules
and formulas will now be given, so as to clearly
indicate the mode of procedure in practical work.
Example 1. — In the triangle in Fig. 94, side A C
is 12 inches long and angle D is 40 degrees. Find
angle E and the two unknown sides.
This is an example of Case 3, one angle and its
adjacent side being given. Angle E equals 90 de-
grees minus the given angle, or
#=90° -40° -50°
112 SELF-TAUGHT MECHANICAL DRAWING
The hypotenuse BC equals the adjacent side
divided by the cosine of D, or
BC - - = 15-666 inches-
Side AB equals the adjacent side multiplied by
the tangent of D, or
AB = 12 X tan 40° = 12 X 0.839 = 10.068 inches.
The cosine and tangent of 40 degrees are found
in the tables of trigonometric functions as already
explained.
Example 2. — In the triangle in Fig. 95, the
hypotenuse BC = 17i inches. One angle is 44 de-
grees. Find angle E and the sides AB and AC.
This is an example of Case 2, the hypotenuse
and one 'angle being given. Using the rules or
formulas given for Case 2, we have :
AC = 174 X cos 44° = 17.5 X 0.719 = 12.5825
inches.
AB = 174 X sin 44° = 17.5 X 0.695 = 12.1625
inches.
E =90° -44° =46°.
Example 3. — In the triangle in Fig. 96, side AC
= 208 feet, and the angle opposite this side = 38
degrees. Find angle E, and the two remaining
sides.
This is an example of Case 4, one side and the
angle opposite it being known. From the rules or
formulas given for Case 4, we have:
BC = 208 - sin 38° = 208 - 0.616 = 337.66 feet.
AB = 208 X cot 38° = 208 X 1.280 = 266.24 feet.
#=90° -38° = 52°.
ELEMENTS OF TRIGONOMETRY
113
Example 4. — In the triangle in Fig. 97, side AC
= 3 inches, and the hypotenuse #C=5 inches. Find
side AB and angles D and E.
This is an example of Case 1. According to a
formula previously given in this chapter
AB = VBC'2- AC2 =
Vl6 = 4.
AB
sin E =
BC
\/52- 32 = \/25 - 9 =
=0.800.
From the tables we find that the angle corre-
FIG. 96.
FIG. 97.
spending to a sine which equals 0.800 is 53° 10'.
Consequently :
# = 53° 10', and D = 90° -53° 10' = 36° 50'.
Example 5. — In the triangle in Fig. 98, side BC,
the hypotenuse, is 1| inch long. One angle is 65
degrees. Find angle E and the remaining sides.
114 SELF-TAUGHT MECHANICAL DRAWING
This is an example of Case 2. We have:
E= 90° -65° =25°.
AB= 1| X cos 65° = 1.375 X 0.423 - 0.5816 inch.
AC = 1| X sin 65° = 1.375 X 0.906 =1.2457 inch.
Example 6.— In the triangle in Fig. 99, side AB
= 0.706 inch, and the angle adjacent to this side is
60 degrees. Find angle E and the sides AC and EC.
A B
T
U ia£
FIG. 99.
This is an example of Case 3. We have :
#=90° -60° = 30°.
EC = 0.706 •*- cos 60° - 0.706 - 0.500 = 1.412
inch.
AC= 0.706 X tan 60° = 0.706 X 1.732= 1.2228
inch.
The previous examples, carefully studied, will
give a comprehensive idea of the methods used for
solving right-angled triangles, no matter which
parts are given or unknown.
A triangle which does not contain a right angle
is called an oblique triangle. Any such triangle
can be solved by the aid of the formulas given for
the right triangle, by dividing it into two right-
angled triangles by means of a line drawn from
the vertex of one angle perpendicular towards the
opposite side. Formulas can be deduced which do
ELEMENTS OF TRIGONOMETRY
115
not require that the triangle be so divided, but for
elementary purposes, the method indicated is the
most easily understood.
In Fig. 100, for example, a triangle is given as
shown. One angle is 50 degrees, and the sides in-
cluding this angle are 4 and 5 inches long, respec-
tively. Draw a line from A perpendicular to the
side EC. We have here two right-angled tri-
angles, and can now proceed by using the formulas
previously given. In triangle ADB, the hypoten-
use AB and one angle are given. We then find
side AD by means of the formulas for Case 2, and
also angle BAD and side BD. Next we find CD
= 5- BD. We then, in the triangle A CD know
two sides AD and CD, and can thus find side AC
as in Case 1, as well as angles A CD and CAD.
The angle BAG finally is found by adding angles
116 SELF-TAUGHT MECHANICAL DRAWING
BAD and CAD and, then, all the angles and sides
in the triangle are found.
The successive calculations would be carried out
as follows:
AD = 4 X sin 50° = 4 X 0.766 - 3.064.
£D = 4 X cos 50°= 4 X 0.643 = 2.572.
Angle BAD = 90° - 50° = 40°.
DC = 5 - BD = 5 - 2.572 = 2.428.
AC = AD* + DC* = aoe 2.428* =- 3.91.
Sine of angle ACD = ~ = ~ - 0.784.
Angle A CD = 51° 40'.
Angle CAD = 90°- 51° 40' = 38° 20'.
Angle BAG = 40° + 38° 20'= 78° 20'.
In order to check the results obtained, add angles
ABC, BAG and ACD. The sum of these angles
must equal 180 degrees if the results are correct:
50° + 78° 20' + 51° 40' = 180°.
This method, with such modifications as are
necessary to meet the different requirements in
each problem, may be used for solving all oblique-
angled triangles, except in the case where no angle
is known, but only the lengths of all the three
sides. In this case the use of a direct formula
will prove the best and most convenient. Let the
three known sides be a, b and c, and the angles
opposite each of them A, B and C, respectively, as
in Fig. 101 ; then we have :
b2 + c2 - a2 b sin A
180°- (A + B).
ELEMENTS OF TRIGONOMETRY
117
As an example, assume that the three sides in a
triangle are a = 4, 6 = 5, and c = 6 inches long.
Find the angles.
52 + 62- 42 = 45_
60
Cos 4 =
2X5X6
,4 = 41° 25'.
0.750.
Sin B =
_
4 4
B = 55° 50'.
C = 180° - (41° 25' + 55° 50') = 82° 45'.
As only the first principles of trigonometry have
here been treated, some of the more advanced
^
problems have, by necessity, been omitted. For
ordinary shop calculations the present treatment
will, however, be found more satisfactory, as some
of the matter which would unnecessarily burden
the mind has been left out. If the student only
first acquires a thorough understanding of the first
118 SELF-TAUGHT MECHANICAL DRAWING
principles of mathematics and their application to
machine design, it is comparatively easy to broaden
the field of one's knowledge; it is, therefore, of
extreme importance that these first principles
be thoroughly understood and digested. The ap-
plication will then be found comparatively easy.
The trigonometric functions afford a convenient
means for laying out angles ; and when the sides
Ar —
r 60 >j
^FiG. 102.— Method of Laying Out Angles by Means of
Natural Functions.
of the angle laid out are much extended, it can
be laid out more accurately in this manner than
by the use of an ordinary protractor. Let it be
required, for instance, to lay out an angle of 37
degrees, one side of the angle being 60 inches long.
Lay out the side AB, Fig. 102, 60 inches long.
Then with a radius equal to the sine of 37 degrees
multiplied by 60, and with a center at B, draw an
ELEMENTS OF TRIGONOMETRY
119
arc C. Then draw a line from A, tangent to arc
C. This line forms an angle of 37 degrees with
line AB. If the required angle is over 45 degrees,
then it is preferable to lay out the complement
angle from a line perpendicular to the original
-B
FIG- 103.— Laying Out an Angle Greater than 45 Degrees.
line, as shown in Fig. 103, where an angle of 70
degrees is to be laid out, but the 20-degree comple-
ment angle is actually constructed. Many other
methods for use in laying out angles, arcs, etc.,
will readily suggest themselves to the student who
thoroughly understands the relation of the trigo-
nometric functions in a right-angled triangle.
CHAPTER VIII
ELEMENTS OF MECHANICS
MECHANICS is defined as that science, or branch
of applied mathematics, which treats of the action
of forces on bodies. That part of mechanics which
considers the action of forces in producing rest or
equilibrium is called statics; that which relates to
such action in producing motion is called dynamics;
the term mechanics includes the action of forces
on all bodies whether solid, liquid or gaseous. It
is sometimes, however, and formerly was often,
used distinctively of solid bodies only. The me-
chanics of liquid bodies is called also hydrostatics
or hydrodynamics, according as the laws of rest or
motion are considered. The mechanics of gaseous
bodies is called also pneumatics. The mechanics
of fluids in motion, with special reference to the
methods of obtaining from them useful results,
constitutes hydraulics.
The Resultant of Two or More Forces.— When a
body is acted upon by several forces of different
magnitudes in different directions, a single force
may be found, which in direction and magnitude
will be a resultant of the action of the several
forces. The magnitude and direction of this single
force may be obtained by what is known as the
parallelogram of forces. Let A and B, Fig. 104,
120
ELEMENTS OF MECHANICS 121
represent the direction of two forces acting simul-
taneously upon P, and let their lengths represent
the relative magnitude of the forces ; then, to find
a force which in direction and magnitude shall
be a resultant of these two forces, draw the line C
parallel with B, and draw the line D parallel with
A. A diagonal of the parallelogram thus formed,
drawn from Pto E, will give the direction, and its
F
FIG. 104.— Parallelogram of Forces.
length as compared with A and B, the relative
magnitude, of the required force.
That this is so may be seen by considering the
two forces as acting separately upon P. Let A be
considered as acting upon P to move it through
a distance equal to its length. Then P would be
moved to F. If the force B is now caused to act
upon P to move it through a distance equal to its
length, P will arrive at G. As FP has the same
length and direction as A, and as GFhas the same
length and direction as B, the distance from G to
P would be the same as the distance from P to E;
therefore, PE, the diagonal of the parallelogram
formed by the lines A, B, C, and D, represents the
required new force or resultant.
If there are more than two forces acting upon
the point P, first find a resultant of any two of the
forces; then consider this resultant as replacing
122
SELF-TAUGHT MECHANICAL DRAWING
FIG. 105. -Resultant of Three
Forces.
the first two, and find the resultant of it and an-
other of the original forces ; continue this process
until a force is obtained which will be the resultant
of all of the original forces. Thus, in Fig. 105, if
A, B and C be considered as representing in di-
rection and magnitude
three forces which are
acting simultaneously
uponP; then, if we draw
a parallelogram upon A
and B, we have its diag-
onal PD as the resultant
of A and B. A parallel-
ogram is now drawn
upon PD and C, giving PE, its diagonal, as the
resultant of these two, and, consequently, of the
three original forces.
This principle holds true whether the original
forces are acting in the same plane or not. Thus,
in Fig. 106, let A, B
and C be three forces
acting simultaneously
upon P. Then the re-
sultant of A and B
would be the diagonal
PD. Considering this
as replacing A and J5,
a resultant of it and C
would be a diagonal drawn from P to the further
corner E; PE would then be the resultant of A, B
andC.
This operation may, of course, be reversed to
allow of finding two or more forces in different
FIG. 106. -Resultant of Three
Forces in Different Planes.
ELEMENTS OF MECHANICS
123
directions which in magnitude shall be equivalent
to a single known force. Thus in Fig. 107, if PA
represents the direction and magnitude of a given
force which it is desired to replace by two others
acting in the direction of PB and PC, respectively,
then draw a line from A to PB parallel with PC,
and draw another from A to PC parallel with PB.
The lengths Pa and Pb
thus determined will
represent the relative
magnitudes, as com-
pared with PA, of the
required new forces.
Parallel Forces.— Let
A and B, Fig. 108,
represent the direction
and magnitude of two
parallel forces acting
together upon the bar DE. These two forces may
be replaced or counterbalanced by a single force,
equal in magnitude to A and B combined. To de-
termine the point of application of this new force
produce A to a, making Da equal in length to B.
Also make bE equal in length to A. The inter-
section of the line connecting a and b with DE, at
F, will be the required point of application. The
lengths DF and FE will be inversely proportional
to the forces A and B. That is, the length FE will
be to the force A as the length DF is to the force
B. The product of DF multiplied by A will be
equal to the product of FE multiplied by B.
Fig. 109 shows how several parallel forces, act-
ing in the same direction, may be replaced or
FIG. 107.— Resolution of Forces.
124 SELF-TAUGHT MECHANICAL DRAWING
counterbalanced by a single force. Let A, B and
C represent the relative magnitudes of the forces.
A resultant of B and C would be D, equal in
/
xi.
FIG. 108.— Parallel Forces.
FIG. 109.— Resultant of
Several Parallel Forces.
magnitude to B and C combined, and its point of
application, determined in the manner previously
described, would be at a. Regarding D as a single
force replacing B and C,
would give E, equal in
magnitude to A and D
combined, as the result-
ant of these two, and its
point of application, de-
termined as before, would
be at b.
Oblique Forces. — Let A
and B, Fig. 110, repre-
sent the directions and
relative magnitudes of
two forces acting simultaneously upon the barDE.
These two forces may be either replaced or counter-
FIG. 110.— Oblique- Forces
Acting at Different Points
on a Bar.
ELEMENTS OF MECHANICS
125
balanced by a single force, which in direction and
magnitude shall be a resultant of them. Produce
A and B until they meet at a. Draw the parallel-
ogram abed, making da equal to A, and ba equal
to B. The diagonal of this parallelogram will give
the direction and relative magnitude of the new
force, and if extended its intersection with DE
will give the point of application.
Opposing Forces. — Let A and J9, Fig. Ill, repre-
sent the directions and relative magnitudes of two
forces acting upon oppo-
site sides of the bar DE.
These two forces may be
replaced by a single force,
which in direction and
magnitude will be a re-
sultant of them. Produce
A and B until they meet
at a. Lay off ac equal to
the length of B, and make
be equal to and parallel
with A. A line drawn
from a to 6 will give the
direction of the new force, and the length of ab,
as compared with A and B will give its relative
magnitude. Its application on bar DE may be de-
termined by extending ab until it intersects DE.
Levers. — When a workman wishes to raise a
heavy object, he may insert one end of a bar un-
der it, and lift on the other end; or, pushing a
block of wood or iron in under the bar as close to
the object to be raised as he can, he presses down
upon the free end of the bar. A bar so used con-
FIG. 111. — Opposing Oblique
Forces.
126 SELF-TAUGHT MECHANICAL DRAWING
stitutes a lever, and the point where the bar rests
when the lever is doing its work, the end of the
bar in under the heavy object in the first case, or
the block on which the bar rests in the second
case, is the fulcrum of the lever.
Levers are of three kinds, as shown in Fig. 112:
First, where the fulcrum is between the power
J1ST
I 3RD
Ow
FIG. 112.— Classes of Levers.
and the weight; second, where the weight is
between the fulcrum and the power; and, third,
where the power is between the fulcrum and the
weight. A man's forearm furnishes a good illus-
tration of a lever of the third class, the fulcrum
being at the elbow, the weight at the hand, and
the muscle, being attached to the bone of the arm,
at a short distance from the elbow, furnishing the
power.
ELEMENTS OF MECHANICS 127
In all of these cases the gain in power is exactly
proportional to the loss in speed, or the gain in
speed is exactly proportional to the loss in power.
Also, in every case the product of the weight mul-
tiplied by its distance from the fulcrum, will equal
the product of the power multiplied by its distance
from the fulcrum, or, the weight and power will
balance each other when the weight multiplied by
the distance through which it moves, equals the
power multiplied by the distance through which it
moves.
If in Fig. 108 the bar DE is a lever, the fulcrum
will be at F, and the methods used in that figure
and in Figs. 109, 110 and 111 give solutions of dif-
ferent lever problems.
The length of the lever arm is independent of
the form of the lever. In Fig. 113 is shown a lever
FIG. 113. -Lever of Curved Shape.
of curved shape ; but the lever arms on which the
calculation as to the work that the lever is doing,
will be based, will be straight lines connecting the
point where the power is applied, or the point
which supports the weight, with the fulcrum.
The length of the lever arm is always at right
angles to the direction in which the power is being
128
SELF-TAUGHT MECHANICAL DRAWING
applied, or to the direction of the resistance of the
weight or load.
In Fig. 114 two cases are shown where the power
is applied obliquely on the lever; but the lever arm
on which the calculation is based will be the dis-
f\
FIG. 114.— Power Applied Obliquely on Lever.
tance Fa measured from the fulcrum, at right
angles to the direction of the power.
Compound Levers. — In Fig. 115 is shown a case
where the power gained with one lever is further
increased by' the use of a second lever, acting on
the first one. The weight and power will balance
ELEMENTS OF MECHANICS 129
each other when the product of the weight and the
lever arms ab and ef, multiplied together, equals
the product of the power and the lever arms gfand
be multiplied together. Thus, to find the weight
1 1
(_>
^F
9
V
I I
1 e >
FIG. 115.— Compound Levers.
which a given power will lift, divide the product
of the power and its lever arms &/*and be, multi-
plied together, by the product of the lever arms of
the weight, ab and ef, multiplied together. To find
6W
A B
FIG. 116.— Diagram for Lever Problem.
the power necessary to lift a given weight, divide
the product of the weight and its lever arms, ab
and ef, multiplied together, by the product of the
lever arms of the power, gf and be, multiplied
together.
130 SELF-TAUGHT MECHANICAL DRAWING
A few examples will illustrate these principles.
Assume that in Fig. 116 a weight at A must bal-
ance the 18-pound weight at B. The lever arms
are given as 12 and 5 inches, respectively. How
much must the weight W be, in order to balance
the weight at B ?
The weight at B (18 pounds) times its lever arm
(5 inches) must equal the weight W times its lever
arm (12 inches). In other words:
18 X 5 = W X 12.
90 = 12 W.
W = 12 = 7i P°unds-
In Fig. 117, two weights, 4 and 2 pounds, respec-
tively, are balanced by a weight W. Find what
r
./
c
FIG. 117. — Diagram for Lever Problem.
the weight of TFmust be with the lever arms
given in the engraving.
In this case the weight at A times its lever arm
plus the weight at B times its lever arm, will
equal weight W times its lever arm. The sum of
the products of the weights and leverages of the
weight at A and B is taken, because both these
weights are on the same side of the fulcrum F.
ELEMENTS OF MECHANICS 131
Carrying out the calculation outlined above, we
have:
4 X 16 + 2X8 = 6 W.
64 + 16 = 80 = 6 W.
W = 8^- = 134 pounds.
b
The product of a weight or force and its lever
arm is commonly called the moment of the force.
The moment of the force at A, for example, is 4
pounds X 16 inches = 64 inch-pounds. If the lever
arm were 16 feet instead of 16 inches, the result
would be 64 foot-pounds.
An interesting application of the lever, and the
moments of forces, is presented in calculations of
FlG. 118. — Diagram for Lever Problem.
weights for safety valves. A diagrammatical
sketch of a safety valve lever is shown in Fig. 118.
Assume that the total steam pressure, acting on
the whole area of the safety valve, is 300 pounds
when it is required that the steam should "blow
off." Find the weight W required near the end
of the lever to keep the valve down until the total
pressure is 300 pounds on the valve. Assume the
weight of the lever itself to be 6 pounds, con-
centrated at its center of gravity, 10 inches from
the fulcrum F.
132 SELF-TAUGHT MECHANICAL DRAWING
In this case we have that the moment of the
steam pressure, which acts upward, should equal
the sum of the moments of the weight of the lever
and the weight W. Therefore :
300 X 3 = 6 X 10 + 20 W.
900 = 60 + 20 W.
900 - 60 = 20 W.
840 = 20 W.
TJ7 840 ,0 ,
W = -gQ- = 42 pounds.
The calculation above has been carried out step
by step, so that students unfamiliar with the alge-
braic solution of equations may be able to under-
stand the principles involved in simple examples
of this kind. In the following, the calculations
have been carried out more directly, but the stu-
dent should use the "step by step" method until
thoroughly familiar with the subject.
Fixed and Movable Pulleys. — A fixed pulley is
frequently used to change the direction of the
power, as shown in Fig. 119, but there is no gain
in power with such a pulley, as there is no com-
pensating loss of speed ; the weight will move up-
ward at the same rate of speed as the power moves
downward.
If now a movable pulley be used in connection
with the fixed pulley as shown in Fig. 120, then as
the end of the rope to which the power is applied
is drawn downward, each of the two strands of
rope between the pulleys will take half of the
stress of the suspended weight, and the weight
will be raised only one-half the distance that the
ELEMENTS OF MECHANICS
133
power descends. The power will therefore need to
be only one-half of the weight. In Fig. 121, there
are three strands of rope between the pulleys, each
of which will be equally shortened when the free
end of the rope is pulled ; the power, therefore, is
only one-third of the weight. In Fig. 122, with
o
FIG. 119.— Fixed Pulley.
FIG. 120.— Fixed and Movable
Pulleys.
four strands of rope between the pulleys, each fur-
nishing an equal amount to the free end as it is
drawn out, the power need be only one-fourth of
the weight.
The law of the pulley, then, where a single rope
is employed, is that the power will be increased as
many times as there are lines of rope between the
pulleys to participate in the shortening. In a sys-
tem using more than one rope, as shown in Fig.
134 SELF-TAUGHT MECHANICAL DRAWING
123, each additional movable pulley doubles the
power, as it will move at only half the rate of the
preceding pulley.
Differential Pulleys. — Another form of pulley,
known as the differential pulley, much used in ma-
chine shops, is shown in Fig. 124. In this form of
w
FIG. 121.— Tackle where Load
is Taken on Three Strands
of Rope.
FIG. 122.— Tackle where Load
is Taken on Four Strands
of Rope.
pulley an endless chain replaces the rope, the pul-
leys themselves being grooved and toothed like
sprocket wheels. The two pulleys at the top are
of slightly different diameters, but rotate together
as one piece. In operation, as the chain is drawn
up by the large wheel it passes around in a loop to
the small wheel from which it is unwound, causing
the loop in which the movable pulley rests to be
ELEMENTS OF MECHANICS
135
shortened by an amount equal to the difference in
the pitch circumferences of the two upper wheels,
when they have made one revolution. This would
cause the weight to be raised one-half of that
amount. If in a given case the two upper pulleys
had respectively 20 and 19 teeth, then as the ap-
FlG. 123.— A Special Arrange-
ment of Movable Pulleys.
FIG. 124.— Differential
Pulley.
plied power was being moved through a distance
of 20 inches the small pulley would unwind 19
inches of the chain, causing a shortening of the
loop in which the movable pulley rests of one inch,
which would raise the weight one-half of an inch,
giving a ratio of load to power of 40 to 1.
In all of these cases the results actually attained
in practice will be somewhat modified from the
136 SELF-TAUGHT MECHANICAL DRAWING
theoretical results given by calculations, by the
losses occasioned by friction.
Inclined Planes. — In raising heavy weights
through short distances, as for instance in loading
barrels onto wagons, a plank may be used to facili-
tate the work by placing one end of it on the
ground and the other end on the wagon, and roll-
ing the barrel up the plank onto the wagon. Such
an arrangement is called an inclined plane. When
the force which is being applied to the rolling
FIG. 125.— Inclined Plane. FIG. 126.— Power Applied
Parallel to Base.
object is exerted in a direction parallel to the in-
clined surface, as in Fig. 125, it is evident that the
power must move through a distance equal to the
length of the incline in order to raise the weight
the desired height. The gain in power will then
be equal to the length of the incline divided by the
height.
If the power is applied in a direction parallel
with the base, as in Fig. 126, the power will have
to advance through a distance equal to the length
of the base to raise the object the desired height.
The gain in power will then be equal to the base
divided by the height. By considering Fig. 126
ELEMENTS OF MECHANICS
137
FIG. 127.— Power Applied Obliquely
to Surface of Incline.
further, it will be seen that in rolling the object
up the incline the power will have to advance from
the beginning of the
incline to a point
from which a line
may be drawn per- '
pendicular to its di-
rection to the top of
the incline. In any
case where the
power is applied in
any direction other
than parallel with
the incline, in roll-
ing the object to the top, the power will have to
advance to a point from which a line may be drawn
perpendicularly to its direction to the top of the
incline. In Figs. 127
and 128 are shown two
other cases where the
power is applied in
a direction obliquely
to the surface of the
incline. In either of
these cases, as in the
other two cases, the
gain in power will
be found by dividing
the distance through
which the force
distance through which the
FIG. 128.— Another Case where
Power is Applied Obliquely to
Surface of Incline.
moves, ab, by the
object is raised, cd.
It will be further seen that the gain in power is
138 SELF-TAUGHT MECHANICAL DRAWING
greatest when the direction in which the force is
being applied is parallel with the incline. When
the direction of the force is upward from the in-
cline, as in Fig. 127, part of the force is expended
in lifting the weight off from the incline, until,
when its direction is made vertical, it is all
expended in this way. When the direction of the
force is downward from the incline, as in Figs. 126
and 128, part of it is lost in pressing the object
against the incline.
The Screw. — The screw is a modified form of
inclined plane, the lead of the screw, the distance
FIG. 129.— Differential Screw.
that the thread advances in going around the
screw once, being the height of the; incline, and
the distance around the screw, measured on the
thread, being the length of the incline.
The Differential Screw.— The differential screw
is a compound screw having a coarse thread part of
its length, and a somewhat finer thread the rest of
its length, the object being to get a slow motion
combined with the strength of a coarse thread.
Fig. 129 shows such a screw. The piece A is a
fixed part of some machine. The piston B slides
within A, being prevented from turning by the pin
C which enters a groove in B. If that part of the
ELEMENTS OF MECHANICS 139
screw which engages in A has eight threads to
the inch, and that part of it which engages in B
has ten threads, then when the screw makes one
revolution, it will advance into A one-eighth of an
inch, and into B one- tenth of an inch ; the piston
B will therefore advance through a distance equal
to the difference between one-eighth of an inch
and one-tenth of an inch, or twenty-five one-thou-
sandths of an inch, requiring forty turns of the
screw to make the piston advance one inch.
Newton's Laws of Motion. — The relation which
exists between force and motion is stated by the
three fundamental laws of motion formulated by
Newton.
Newton's first law says that if a body is at
rest it will remain at rest, or, if it is in motion, it
will continue to move at a uniform velocity in a
straight line, until acted upon by some force and
compelled to change its state of rest or of straight-
line uniform motion. In a general way, this law is
self-evident, and based on daily experience. How-
ever, the part of the law stating that a body in
motion will continue indefinitely to move if not
acted upon by resisting forces, may not be so self-
evident; yet whenever a body is brought to a stand-
still after it has been in motion, such forces as
frictional resistance, gravity, etc., always have in
some way influenced the motion of the body.
Newton's second law of motion says that a
change in the motion of a body is proportional to
the force causing the change, and takes place in
the direction in which the force acts. If several
forces act on a body, the change is proportional to
140 SELF-TAUGHT MECHANICAL DRAWING
the resultant of the several forces, and takes place
in the direction of the resultant. This has been
clearly explained in the previous pages, in connec-
tion with the resolution and composition of forces.
The most important point to note in regard to
the second law of motion is that when two or
more forces act on a body at the same time, each
causes a motion exactly the same as if it acted
alone ; each force produces its effect independently,
but the total effect on the motion of the body, of
course, is a combination of all these independent
motions.
Newton's third law says that for every action
there is an equal reaction. This means that if a
force or weight presses downward on a support
with a certain pressure, the reaction, or resistance
in the support, must equal the same pressure. If
a bullet is shot from a rifle with a certain force,
there is a reaction, or "recoil," in the rifle, equal
to the force required to give the velocity to the
bullet. This law is very important, and many
failures in machine design have been due to
ignorance of the real meaning of the law of action
and reaction.
Newton's third law may be illustrated by a loco-
motive drawing a train of cars. The driving
wheels give as much of a backward push on the
rails as there is of forward pull exerted on the
train; and it is only because the rails are held in
place by their fastenings, and by the weight rest-
ing on them, that the locomotive is able to pull the
train forward. This principle of action and reac-
tion being equal and opposite is also an effectual
ELEMENTS OF MECHANICS 141
bar to any perpetual-motion machine, as such a
machine in order to work would have to produce
a greater action in one direction than the reaction
in the other direction.
The Pendulum. — A body or weight suspended
from a fixed point by a string or rod, and free to
oscillate back and forth is called a pendulum. The
center of oscillation is the point which, if all of the
material composing the pendulum, including the
sustaining string or rod, were concentrated at it
(the material so concentrated being considered as
being suspended by a line of no weight) would
vibrate in the same time as the actual pendulum.
The length of the pendulum is the length from the
point of suspension to the center of oscillation.
When the length of the pendulum is unchanged,
its time of vibration will be the same, if its angle
of vibration does not exceed three or four degrees,
and its time of vibration will be but slightly in-
creased for larger angles.
The time of vibration of a pendulum is not
affected by the material of which it is made,
whether light or heavy, except as the light mate-
rial will offer greater resistance to the air, by
presenting a greater surface in proportion to its
weight, than a heavy material.
The time of vibration of a pendulum of a given
length is inversely as the square root of the inten-
sity of gravity. As the intensity of gravity de-
creases with the distance from the center of the
earth it follows that a pendulum will vibrate faster
at the poles or at sea level than it will at the equa-
tor or at an elevation.
142 SELF-TAUGHT MECHANICAL DRAWING
The time of vibration of a pendulum varies di-
rectly as the square root of its length. That is, a
pendulum to vibrate in one-half or one-third the
time of a given pendulum will need to be only one-
quarter or one-ninth of its length.
Example l.—A pendulum in the latitude of New
York will require to be 39.1017 inches long to beat
seconds. Required the length of a pendulum to
make 100 beats per minute.
A pendulum to make 100 beats per minute will
have to make its vibrations in 60-100 of the time
of one which is making 60 beats per minute, and
its length will be equal to the length of one which
beats seconds, multiplied by the square of 60-100,
or:
39.1017 X 60 2 39.1017 X 3600 ,
~W~ 10,000 : 14076 mches-
Example #.— Required the time of vibration of
a pendulum 120 inches long. Letting x repre-
sent the required time, we have the proportion
V 120 :V 39. 1017 = x : 1, or 10.954 : 6.253 = x : 1.
10.954 ,
x = /? oco = 1-75 second.
b.Zoo
A short pendulum may be made to vibrate as
slowly as desired by having a second "bob" placed
above the point of suspension, which will partially
counteract the weight of the lower bob.
Falling Bodies. — A falling body will have ac-
quired a velocity at the end of the first second of
32.16 feet per second, under ordinary conditions.
If the body is of such shape or material as to pre-
sent a large surface to the air in proportion to its
ELEMENTS OF MECHANICS 143
weight, its velocity will, of course, be lessened, and
as its velocity depends upon the force of gravity,
its velocity will be affected somewhat by the lati-
tude of the place, and its distance above sea level.
During the next second it will acquire 32.16 feet
additional velocity, giving it a velocity of 64.32
feet at the end of the second second. Each suc-
ceeding second will add 32.16 feet to the velocity
the body had at the end of the preceding second.
To find the velocity of a falling body at the end
of any number of seconds, therefore, multiply the
number of seconds during which the body has
fallen by 32.16. This rule, expressed as a formula,
would be :
v = 32.16 X *
in which v = velocity in feet per second, t = time
in seconds.
The acceleration due to gravity, 32.16 feet, is
often, in formulas, designated by the letter g. As
an example, find the velocity of a falling body at
the end of the twelfth second:
v = 32. 16X12 = 385. 92 feet.
As the body falling starts from a state of rest,
its average velocity will be one-half of its final ve-
locity ; the distance through which it falls equals
the average velocity multiplied by the number of
seconds during which it has been falling. This
rule, expressed as a formula, is :
A-f X* .^
in which h = distance or height through which
144 SELF-TAUGHT MECHANICAL DRAWING
body falls, and v and t have the significance given
above. But v = 36.16 X t; if this value of v is in-
serted in the formula just given, we have:
This last formula, expressed in words, gives us
the rule that the distance through which a body
falls in a given time equals the square of the num-
ber of seconds during which the body has fallen,
multiplied by 16.08.
How long a distance will a body fall in 10 sec-
onds? Inserting t = 10 in the formula, we have:
h = 16.08 f = 16.08 X 102 = 16.08 X 100 = 1608
feet.
The time, in seconds, required for a body to fall
a given distance equals the square root of the
distance, expressed in feet, divided by 4.01. Ex-
pressed as a formula, this rule would be :
t = V^
4.01'
As an example, assume that a stone falls through
a distance of 3600 feet. How long time is required
for this?
Inserting h = 3600 in the formula, we have :
V3600 60
t = . = = 15 seconds, very nearly.
The velocity of a falling body after it has fallen
through a given distance equals the square root of
the distance through which it has fallen multi-
plied by 8.02.
ELEMENTS OF MECHANICS 145
This rule, expressed as a formula, is:
What is the velocity of a falling body after it
has fallen through a distance of 3600 feet?
Inserting h = 3600 in the formula, we have:
v = 8.02 X V3600 = 8.02 X 60 = 481.2 feet.
The height from which a body must fall to acquire
a given velocity equals the square of the velocity
divided by 64.32. As a formula, this rule is:
~ 64.32
From what height must a body fall to acquire a
velocity of 500 feet per second? Inserting v = 500
in the formula given, we have:
500 2 500 X 500
= 64.32 = .64.32
If a body is thrown upward with a given ve-
locity, its velocity will diminish during each second
at the same rate as it increases when the body
falls. A body thrown up into the air in a vertical
direction will return to the ground with exactly
the same velocity as that with which it was thrown
into the air. At any point, the velocity on the up-
ward journey will be equal to the velocity on the
downward journey, except that the direction is
reversed.
The acceleration of a falling body, 32.16 feet per
second, is the value at the latitude of New York,
at sea level.
The force required to give to a falling body its
146 SELF-TAUGHT MECHANICAL DRAWING
acceleration of 32.16 feet per second is the weight
of the body itself. The force required to give any
acceleration to a body, then, is to the weight of the
body as that acceleration is to the acceleration
produced by gravity. Therefore, to find the force
required to produce a given rate of acceleration to
a body, divide the weight of the body by 32.16,
and multiply the quotient by the required rate of
acceleration.
Example. — A body weighing 125 pounds is to be
lifted with an acceleration of 10 feet per second.
Required the strain on the sustaining rope.
125
00 .,„ X 10 = 38.8, the tension necessary to produce
oZ.lb
the acceleration.
To this must be added the pull necessary to lift
the weight without acceleration, or the weight of
the body itself. Thus 38.8 + 125 = 163.8 is the re-
quired tension on the rope.
The rate of acceleration which a continuously
acting force will produce is equal to the force
divided by the weight of the body,, multiplied by
32.16.
Energy and Work. — The unit of work, the stand-
ard by which work is measured, is the foot-pound,
or the amount of work done in lifting a weight or
overcoming a resistance of one pound through one
foot of space.
"Energy is the product of a force factor and a
space factor. Energy per unit of time, or rate of
doing work, is the product of a force factor and a
velocity factor, since velocity is space per unit of
time. Either factor may be changed at the ex-
ELEMENTS OF MECHANICS 147
pense of the other; i.e., velocity may be changed,
if accompanied by such a change of force that the
energy per unit of time remains constant. Corre-
spondingly force may be changed at the expense of
velocity, energy per unit of time being constant.
Example. — A belt transmits 6000 foot-pounds per
minute to a machine. The belt velocity is 120 feet
per minute, and the force exerted is 50 pounds.
Frictional resistance is neglected. A cutting tool
in the machine does useful work ; its velocity is 20
feet per minute, and the resistance to cutting is
300 pounds. Then the energy received per minute
- 120 X 50 = 6000 foot-pounds; and energy deliv-
ered per minute = 20 X 300 = 6000 foot-pounds.
The energy received therefore equals the energy
delivered. But the velocity and force factors are
quite different in the two cases." (Prof. A. W.
Smith.)
Force of the Blow of a Steam Hammer or Other
Falling Weight— The question, "With what force
does a falling hammer strike ?" is often asked.
This question can, however, not be answered
directly. The energy of a falling body cannot be.
expressed in pounds, simply, but must be expressed
in foot-pounds. The energy equals the weight of
the falling body multiplied by the distance through
which it falls, or, expressed as a formula:
E = WXh,
in which E = energy in foot-pounds,
W = weight of falling body in pounds,
h = height from which body falls in feet.
The energy can also be found by dividing the
148 SELF-TAUGHT MECHANICAL DRAWING
weight of the falling body by 64.32 and then mul-
tiplying the quotient by the square of the velocity
at the end of the distance through which it falls.
This rule, expressed as a formula, is:
in which E and W denote the same quantities as
before, and v = the velocity of the body at the end
of its fall.
Both of these formulas give, of course, the same
results. That the second method gives the same
result as multiplying the weight by the height
through which it falls, is evident from the fact,
stated under the head of "Falling Bodies/' that the
square of. the velocity of a falling body, divided
by 64.32, gives the height through which it has
fallen.
This second method allows of determining the
energy of any weight or force moving at a given
velocity, whether its velocity has been acquired by
falling, or is due to other causes.
Now assume that we wish to find the force of
the blow of a 300-pound drop hammer, falling 2
feet before striking the forging, and compressing
it 2 inches.
The energy of the falling hammer when reach-
ing the forging is:
E = W X h = 300 X 2 = 600 foot-pounds.
This energy is used during the act of compress-
ing the forging 2 inches or 0.166 of a foot. Con-
sequently, the average force of the hammer with
ELEMENTS OF MECHANICS 149
which it compresses the forging is 600 -*- 0.166 +
the weight of the hammer, or
Average force of blow = A «„,* + 300 =
U.lbb
3600 + 300 = 3900 pounds.
The general formula for the force of a blow is:
in which F = average force of blow in pounds,
W = weight of hammer in pounds,
h = height of drop of hammer in feet,
d = penetration of blow in feet.
A horse-power, in mechanics, is the power ex-
erted, or work done, in lifting a weight of 33,000
pounds one foot per minute, or 550 pounds one foot
per second. The power exerted by a piston driven
by steam or other medium during one stroke, in
foot-pounds, is equal to the area of the piston,
multiplied by the pressure per square inch, multi-
plied by the stroke in feet, the product of the area
by the pressure giving the force, and the stroke
giving the distance through which the force is
exerted. In the case of steam engines, where the
steam is cut off at one-quarter, one- third or one-
half of the stroke, the piston being driven the rest
of the way by the expansion of the steam, the
average pressure for the entire stroke, the ''mean
effective pressure" (M.E. P.), as it is called, is the
basis of calculations. As each revolution of the
engine equals two strokes of the piston, the number
of foot-pounds per minute an engine is developing
will be the product of the area of the piston in
150 SELF-TAUGHT MECHANICAL DRAWING
square inches, multiplied by the mean effective
pressure, multiplied by the stroke in feet, multi-
plied by the number of revolutions per minute
times 2. This product, divided by 33,000, gives
the indicated horse-power (I.H.P.) of the engine;
this name being derived from the fact that the
mean effective pressure is determined by the use
of the steam engine indicator. Therefore:
r TT D Area XM. E. P. X stroke X rev, permin. X2
LH'P' = 33,000
This formula may be transposed in various ways
to give other information. For instance, if the
piston area for a given horse-power is desired,
then
Area LH.P. X 33,000
M.E.P. X stroke X rev. per min. X 2.
If the volume of the cylinder is desired, then
A LH.P. X 33000
Area X stroke = , , ^ n — — rr^r
M.E.P. X rev. permin. X 2.
If the pressure to produce a given horse-power
is desired, then
MEp = LH.P. X 33000
Area X stroke X rev. per min. X 2.
The mean effective pressure in the cylinder of
the engine is, of course, considerably less than the
boiler pressure as shown by the steam gauge. The
indicated horse-power of an engine does not take
into account the losses caused by the friction of
the working parts. The power which the engine
actually delivers as shown by a brake dynamo-
meter or other contrivance at the flywheel is called
the brake horse-power.
CHAPTER IX
FIRST PRINCIPLES OF STRENGTH OF MATERIALS
Factor of Safety. — It is obvious that it would be
unsafe in designing a piece of construction work
to allow a strain of anywhere near the breaking
limit of the material it is to be made from. It is,
therefore, customary in making any calculations
for the size of the parts to use what is called a
factor of safety, by making the part from three or
four to ten or even more times the strength neces-
sary to just resist breaking with a steady load.
The factor of safety used will depend upon several
considerations. It will depend, first, upon the na-
ture of the material used. A wrought or drawn
metal, for instance, will be likely to be more uni-
form in its nature than a cast metal which may
contain air holes, or which may be more or less
spongy, or which may be under unequal strains in
cooling. The matter of strains in a casting due to
unequal cooling is to a considerable extent a mat-
ter of proper or improper design ; still it is not
possible to entirely avoid them.
Again the factor of safety to be used will depend
upon the nature of the work which will be re-
quired of the part. If the part has to simply sus-
tain a steady load it will not need to be as strong
as though the load was applied and reversed, or
151
152 SELF-TAUGHT MECHANICAL DRAWING
even as strong as though the load was applied and
released. To illustrate, it is a familiar fact that a
piece of wire which may be bent a given amount
without apparent injury, may be broken by repeat-
edly bending it back and forth the same amount
at one point. And, similarly, in machine parts,
rupture may be caused not only by a steady load
which exceeds the carrying strength, but by re-
peated applications of stresses none of which are
equal to the carrying strength. Rupture may also
be caused by a succession of shocks or impacts,
none of which alone would be sufficient to cause it.
Iron axles, the piston rods of steam hammers and
other pieces of metal subjected to repeated shocks,
invariably break after a certain length of service.
The factor of safety used will therefore vary
widely with the nature of the work required of the
part. For a steady or "dead" load, Prof. A. W.
Smith says: "In exceptional cases where the
stresses permit of accurate calculation, and the
material is of proven high grade and positively
known strength, the factor of safety has been
given as low a value as 1J • but values of 2 and 3 are
ordinarily used for iron or steel free from welds ;
while 4 to 5 are as small as should be used for cast
iron on account of the uncertainty of its composi-
tion, the danger of sponginess of structure, and
indeterminate shrinkage stresses." Others would
make 3 the lowest factor of safety that should be
used for wrought iron and steel.
Where the load is variable, but well within the
elastic limit of the material, that is where the load
is not so great but so that the part will immedi-
STRENGTH OF MATERIALS 153
ately resume its original shape when the load is
removed, a factor of safety of 5 or 6 might be
used. The part will need to be made stronger if
the load or force acts first in one direction and
then in the opposite direction, that is, if it acts
back and forth, than it will need to be if the same
force is simply applied and then released. Where
the part is subjected to shock, the factor of safety
is generally made not less than 10. A factor of
safety as high as 40 has been used for shafts in
mill-work which transmit very variable powers.
In cases where the forces are of such a nature
that they cannot be determined, then Prof. Smith
says: "Appeal must be made to the precedent of
successful practice, or to the judgment of some ex-
perienced man until one's own judgment becomes
trustworthy by experience. * * * In proportioning
machine parts, the designer must always be sure
that the stress which is the basis of calculation
or the estimate, is the maximum possible stress;
otherwise the part will be incorrectly propor-
tioned." And he cites the case of a pulley where
if the arms were to be designed only to resist the
belt tension they would be absurdly small, because
the stresses resulting from the shrinkage of the
casting in cooling are often far greater than those
due to the belt pull.
In many cases the practical question of feasi-
bility of casting will determine the thickness of
parts, independent of the question of strength.
For instance, on small brass work, such as plumb-
ers' supply, and small valve work, a thickness of
about 3-32 of an inch is as little as can be relied
154 SELF-TAUGHT MECHANICAL DRAWING
on to make a good casting on cored out work ; or
in the case of partitions in such work where the
metal has to flow in between cores, a thickness of
about J of an inch is as small as should be used;
yet such thicknesses may be much greater than
are required to give the necessary strength. On
larger cast iron work, the thickness to be allowed
to insure a good casting will, of course, depend
upon the size of the piece. The judgment of the
pattern-maker or foundry-man will naturally de-
termine the thickness in such cases.
Shape of Machine Parts. — While the size of ma-
chine parts will vary greatly with the nature of the
work required of them, their shape will depend
very much on the manner or direction in which
the load or strain is brought to bear upon them.
If the part is subjected to simple tension, that is,
merely resists a force tending to pull it apart, then
the shape of the member which serves this purpose
is not very material, though a round rod, being most
compact and cheapest, is best. Almost any shape
will answer, however, though it is well to avoid
using thin and broad parts, as a strain, though not
greater than that which the part as a whole might
bear safely, might be brought upon one edge, pro-
ducing^a tearing effect beyond the safe limit. For
resisting simple tension the part should be made of
uniform size its entire length, of a size to be deter-
mined by the tensile strength of the material and
the factor of safety used.
If the part is to resist compression, then when
the proportion of its length to its diameter or
thickness is such that it will " buckle' ' or bend,
STRENGTH OF MATERIALS 155
instead of crushing, that is when its length ex-
ceeds five or six times its diameter, it becomes
desirable to use a hollow or cross-ribbed form of
construction, so as to get the metal as far from the
axis of the piece as possible. The hollow cylind-
rical form, by getting all of the metal equally dis-
tant from the axis is, of course, most effective,
but considerations of appearance may make a hol-
low square form more desirable, while considera-
tions of cost may make a cross-ribbed form to be
preferred, as such a form can be cast without the
use of cores. In cases where a wrought metal must
be used a solid form is often the only practicable
one. When it becomes important to keep the
weight down to the lowest point, it is common to
have the piece slightly enlarged in the middle
of its length, as in the case of connecting rods of
steam engines. In the case of steam engine con-
necting-rods, the tendency to buckle is least side-
ways, as the cross-head and crank-pins tend to
hold it in line this way, while the rotary motion of
the crank-pin tends to produce buckling the other
way. Connecting rods are therefore frequently
made somewhat flat, of a breadth about twice
their thickness.
When a piece is designed to resist bending, it
becomes desirable to get a good depth of material
in the direction in which the force is applied, as
the capacity of a piece to resist bending increases
as the square of its thickness or depth in the di-
rection of the force, but only directly as its breadth
or width, so that to increase the thickness of a
piece two or three times in the direction of the
156 SELF-TAUGHT MECHANICAL DRAWING
force would increase its capacity to resist bending
four or nine times; while to increase its breadth
two or three times would only increase its strength
two or three times. The proportion of depth to
breadth which can be used will, of course, depend
upon the length of the piece, as if the piece is long
and its depth is made large in proportion to its
thickness the tendency will be for the piece to
buckle, or yield sideways. To resist this tendency
it is customary to put ribs on the edges of such a
FIG. 130. FIG. 131.
FIGS. 130 and 131.— Beam Cross-sections of
Different Types.
piece, giving it the form shown in Fig. 130. The
hollow box-form shown in Fig. 131 is, of course,
equally effective to resist combined bending and
buckling stresses, and in some cases may be pref-
erable as a matter of appearance on account of
the impression of solidity which it gives.
A projecting beam, like that shown in Fig. 132,
designed to resist a force or sustain a load at
its end, would need to have its lower edge made
of the form of a parabola, if made of uniform
thickness. If the edges were ribbed to prevent
buckling, then material might be taken out of
the middle portion, as shown in Fig. 133, without
weakening it.
STRENGTH OF MATERIALS
157
Strength of Materials as Given by Kirkaldy's
Tests. — A very large number of tests of cast iron
made by Kirkaldy gave results as follows : Tensile
strength per square inch, necessary to just tear
asunder, from about 10,000 or 12,000 pounds to
about 28,000 or 32,000 pounds, or an average
strength of about 20,000 pounds. Tests on the
ability of cast iron to resist crushing gave results
vary ing from about 50, 000 to about 150, 000 pounds,
FIG. 132. — Cantilever of
Uniform Strength, when
Loaded at End.
FIG. 133. — Common Design
of Cantilever of Uniform
Strength.
or an average strength of about 100,000 pounds
per square inch. These tests indicate that cast
iron has about five times the capacity to resist
crushing that it has to resist tension. They also
indicate that cast iron is a somewhat uncertain
material.
Tests of wrought iron indicated a tensile strength
of between 40,000 and 50,000 pounds per square
inch, the elastic limit being reached at about one-
half the tensile strength. Tests on steel castings
gave results for tensile strength ranging from
55,000 to about 64,000 pounds per square inch,
158 SELF-TAUGHT MECHANICAL DRAWING
the elastic limit being reached at about 30,000
pounds.
Tests of wire gave results as follows : Brass,
from 81,000 to 98,000 pounds per square inch of
area. Iron, from 59,000 to 97,000 pounds. Steel,
from 103,000 to 318,000 pounds.
The tensile strength of regular machine steel
(low carbon steel) is generally given at about
60,000 pounds per square inch.
Size of Parts to Resist Stresses. — To resist ten-
sion it is, of course, only necessary to have the
piece of such a size that each square inch shall not
have a stress greater than the average strength
of the material (as 20,000 pounds for cast iron)
divided by whatever factor of safety may be
selected.
To Resist Crushing.— Prof. Hodgkinson's rule
for the strength of hollow cast iron pillars is as
follows : To ascertain the crushing weight in tons
multiply the outside diameter by 3.55; from this
subtract the product of the inside diameter multi-
plied by 3.55, and divide by the length multiplied
by 1.7. Multiply this quotient by "46.65. Ex-
pressed as a formula this rule would be:
(D X 3.55) - (d X 3. 55)
L X 1.7
Sc = 46.65 X
in which
Sc = ultimate compressive (crushing) strength
of hollow column, in tons,
D = outside diameter in inches,
d = inside diameter in inches,
L = length of column in feet.
STRENGTH OF MATERIALS
159
Any desired factor of safety may be introduced
in the above formula by dividing the factor 46.65
by the factor of safety. In this case the formula
would be:
Q 46J>5_Xl CD ._X_SL55) - (d X 3.55)]
F X L X 1.7
in which
S« = safe compressive strength in tons,
F = factor of safety, and
D, d and L have the same meaning as above.
This rule and formula assumes that the ends of
the column are perfectly flat and square, and that
the load bears evenly on the whole surface.
If the ends are rounded, the column yields at
about one-half the -stress of one with fixed square
ends.
To Resist Bending. — In the following commonly
given rules for the strength of beams or bars to
— -i
K-H
I
FIG. 134.— Rectangular Cantilever.
resist breaking by transverse stresses, the tensile
strength of cast iron is assumed at 20,000 pounds
per square inch. Divide 20,000 in the formulas
>
160 SELF-TAUGHT MECHANICAL DRAWING
by the desired factor of safety. The breadth and
depth of rectangular bars, the diameter, if the bar
is round, and the length, are all in inches.
For rectangular bars fixed at one end with the
force applied at the other, Fig. 134, the breaking
load equals
bX d2X 20,000
I
:
For round bars under the same conditions, Fig.
135, the breaking load equals
JL / 0.59 X d*X 20,000
6 I
If the rectangular bar is hollow, as shown in
FIG. 135. — Circular Section Cantilever.
Fig. 131, subtract the internal b X d2 from the
external b X d2.
If the round bar is hollow subtract the internal
d 3 from the external d 3.
The case of a bar of the I-section shown in Fig.
130 is similar to that of the hollow rectangular bar
of Fig. 131, the depressions in its sides correspond-
ing to the hollow part of Fig. 131, the sum of their
STRENGTH OF MATERIALS 161
depths corresponding with the internal width b of
the hollow rectangular bar.
If a beam is fixed at one end and the load is
evenly distributed throughout its entire length,
it will bear double the weight it will if the load
is supported at the outer end.
If the beam is supported at the ends and loaded
in the middle it will bear four times the weight of
the beam of Fig. 134, or, if the load is evenly dis-
tributed throughout the length of the beam, eight
times.
If the beam, instead of being simply supported
at- the ends, has the ends fixed and is loaded at the
center, its ability to resist breaking will be doubled
as compared with that when loaded at the center
and with the ends only supported.
Regarding the safe load that beams or bars of
different material may bear Griffin says that "with
but a general knowledge of the elastic limit, ordi-
nary steel is good for from between 12,000 to 15,000
pounds per square inch non-reversing stress, and
from 8000 to 10,000 pounds reversing stress. Cast
iron is such an uncertain metal, on account of its
variable structure, that stresses are always kept
low, say from 3000 to 4000 for non-reversing stress,
and 1500 to 2500 for reversing stress. "
Again, though the tests of wrought iron show it
to have a much higher tensile strength than cast
iron, Nystrom, in formulas for lateral strength,
gives wrought iron but little more than three-quar-
ters the value of cast iron, probably because it
bends so readily.
162
SELF-TAUGHT MECHANICAL DRAWING
A table is appended giving the average breaking
strength, in pounds per square inch, of some com-
monly used materials in engineering practice.
Tension.
Compression.
Aluminum
15,000
12,000
24,000
30,000
Copper cast
24 000
40,000
Iron cast
15,000
80,000
Iron wrought ....
48,000
46,000
Steel castings
70,000
70,000
Structural steel ....
60,000
60,000
Stresses in Castings. — Reference has been pre-
viously made to stresses in castings, due to shrink-
age in cooling. If all parts of a casting could be
made to cool equally fast there would not be much
trouble in this respect, but as different parts of a
casting vary in thickness, the time they require
to cool will vary, and the thick parts remaining
fluid the longest, will, on cooling, cause a strain on
the already cool thin parts. In the case of a pulley,
where the rim and arms are much lighter than the
hub, the hub on cooling will tend to draw the arms
to itself and away from the rim, and if the differ-
ence in thickness is great, they may be even found
to be pulled away so as to show a crack where they
join the rim. The"remedy in such a case would, of
course, be first, to take out as much of the metal
from the center of the hub as possible by means
of a core, and second, to keep the outside of the
hub as small as would be consistent with strength,
getting necessary thickness for set screws by hav-
ing a raised place or boss at that point.
STRENGTH OF MATERIALS 163
As these strains are primarily due to unequal
cooling, it is evident that in order to reduce them
to the lowest point the first thing to do is to make
the different parts of the casting of as nearly uni-
form thickness as possible. Where different parts
of the casting vary in thickness, the change from
one thickness to the other should be made as grad-
ual as possible. Sharp internal corners should also
be avoided, as such places are very liable to be
spongy; the sand from the sharp corner in the
mould is also very liable to wash away when the
metal is poured in, and lodge in some other place,
causing a defective casting. A good ' ' fillet, " as an
internal round corner is called, which the pattern-
maker may put into the pattern with wax, putty or
leather, will not be very expensive, and will save
much trouble in the casting.
Besides possessing a knowledge of factors of
safety, proportioning parts to resist various
stresses and the like, a general knowledge of the
principles of foundry and machine shop practice is
essential to properly design machine work. If one
does not understand foundry work, he will be con-
stantly designing castings which it will be im-
practicable to mould ; if not actually impossible of
moulding, they will be needlessly expensive. And
in like manner, unless he understands the general
principles of machine shop practice, his work will
be giving trouble at that end of the line.
CHAPTER X
CAMS
General Principles. — In designing machinery it
is frequently desirable to give to some part of the
mechanism an irregular motion. This is often
done by the use of cams, which are made of such
form that when they receive motion, either rotary
or reciprocating, they impart to a follower the
desired irregular motion.
The follower is sometimes flat, and sometimes
round. When the follower is round it is usually
made in the form of a wheel or roller, so as to les-
sen the wear and the friction. The follower may
work upon the edge of the cam, or if round, it
may work in a groove formed either on the face
or on the side of the cam.
The working surfaces of cams with round fol-
lowers are laid out from a pitch line, so called,
which passes through the center of the follower.
The shape of this pitch line determines the work
which the cam will do. The working surface of
the cam is at a distance from the follower equal to
one-half the diameter of the follower. This prin-
ciple of a pitch line holds good whether the cam
works only upon its edge like the one shown in
Fig. 139, or whether it has an outer portion to
insure the positive return of the follower. This
164
CAMS
165
outer portion is frequently made in the form of a
rim of uniform thickness around the groove.
Design a Cam Having a Straight Follower Which
Moves Toward or From the Axis of the Cam, as
Shown in Fig. 136. — Let it be required that the
follower shall advance at a uniform rate from a to
FIG. 136.— Cam with Straight Follower having Uniform
Motion.
6 as the cam makes a half revolution, this advance
being preceded and followed by a period of rest of
a twelfth of a revolution of the cam.
Divide that half of the cam during the revolu-
tion of which the follower is to be raised from a to
&, in this case the half at the right of the vertical
center line, into a number of equal angles, and
166 SELF-TAUGHT MECHANICAL DRAWING
divide the distance from a to b into the same num-
ber of equal spaces. Mark off the points so ob-
tained onto the successive radial lines as indicated
by the dotted lines, and at the points where these
dotted lines intersect the radial lines draw lines at
right angles to the radial lines to represent the
position of the follower when these radial lines
become vertical as the cam revolves.
A period of rest in a cam is represented by a cir-
cular portion, having the axis of the cam as its
center. In order, therefore, to obtain the required
periods of rest, the distances of a and b from the
center are marked off upon the radial lines c and
d, these lines being made a twelfth of a revolution
from the vertical center line, and lines represent-
ing the follower are drawn at these points as be-
fore. To get the return of the follower the space
from c to d is divided into a number of equal
angles, and the distance from e to /is divided off
to represent the desired rate of return of the
follower. In this case the rate of return is made
uniform, so the distance ef is spaced off equally.
The distance of these points from the axis is marked
off upon the radial lines between c and d, and lines
representing the follower are drawn.
A curved line, which may be made with the
aid of the irregular curves, which is tangent to all
of the lines representing the follower, gives the
shape of the cam.
Fig. 137 shows a cam having the conditions as to
the rise, rest and return of the follower the same
as the one shown in Fig. 136, the follower, how-
ever, being pivoted at one end.
CAMS
167
Draw the arc ab representing the path of a point
in the follower at the vertical center line, and
divide that part of the arc through which the fol-
lower rises into the same number of equal spaces
as the half circle at the right of the vertical cen-
ter line is divided into angles. Through these
FlG. 137.— Cam with Pivoted Follower.
points draw lines, as shown, representing consecu-
tive positions of the working face of the follower.
The various distances of the follower from the axis
of the cam are now marked off upon the corre-
sponding radial lines as before. Lines to represent
the follower are now drawn across each of these
radial lines, at the same angle to them that the
follower makes with the vertical center line when
168 SELF-TAUGHT MECHANICAL DRAWING
at that part of its stroke corresponding to the par-
ticular radial line across which the line represent-
ing the follower is being drawn. A curved line
passing along tangent to all of these lines gives
the shape of the cam as before.
Design a Cam with a Round Follower Rising Ver-
tically.— In Fig. 138 the follower has the same uni-
form rise, and the same periods of rest as before.
FIG. 138.— Cam with Roller Follower.
A cam with a round follower is less limited in its
capabilities than one with a straight follower ; in
the one here shown the follower on its return
drops below the position in which it is shown.
That part of the cam during which the conditions
are the same as in the others is divided off and
CAMS 169
the position of the center of the follower upon the
radial lines is obtained in the same manner as
before. That part of the cam representing the
return of the follower is divided into such angles
as desired, and the distance through which the fol-
lower is to drop as the cam revolves through each
of these angles is marked off upon the proper
radial line. A curved line which is now made to
pass through all of the points so obtained gives
the pitch line of the cam.
In drawing such a cam it is not always neces-
sary to fully draw the working faces. The pitch
line and the method of obtaining it being shown,
a number of circles representing consecutive posi-
tions of the follower may be drawn. This will
usually be sufficient. The side view of the cam,
which in a case like this would naturally be made
in section, will give opportunity to show any fur-
ther detail that may be desired.
Design a Cam with a Round Follower Mounted on
a Swinging Arm. — Fig. 139 shows such a cam, all
of the conditions as to rise, rest and return of the
follower being the same as in the cam shown in
Fig. 138. The cam is divided into the same angles
as before, and the position of the follower is laid
out on these radial lines as though it moved ver-
tically. These positions are then modified in the
following manner : Draw the arc ab representing
the path of the center of the follower as it rises,
and extend the dotted circular lines, which repre-
sent successive heights of the follower, from the
vertical center line to this arc. The distance of
each of the intersections of the dotted circular
170 £ELF-TAUGHT MECHANICAL DRAWING
lines with the arc a&, from the vertical center line
is then taken with the compasses and is marked
off upon the same dotted line from the radial line
at which it terminates, or, where the follower has
a period of rest, from both of the radial lines
FIG. 139.— Cam with Roller Follower Mounted on
Swinging Arm.
where the period of rest takes place. Thus the dis-
tance of the point 1 from the vertical center line is
marked back upon the dotted circular line from the
radial lines ra and n. Point 2 is marked back from
the radial line o. Point 3 is marked back from the
line p. By this means the position which the fol-
lower will occupy, when each of the radial lines
has become vertical, as the cam revolves, is deter-
CAMS 171
mined. A curved line which is made to pass
through all of these points will be the required
pitch line of the cam. The method of getting the
working face of the cam is indicated by the small
dotted circular arcs, which are drawn with a radius
equal to that of the follower. It will be noticed
that, as the follower, on its return, drops below
the position in which it is shown, it passes to the
other side of the vertical center line, so that in
marking off its position from the radial lines x and
y this must be borne in mind. The question as to
FIG. 140.— Reciprocating Motion Cam.
on which side of a radial line the new position of
the follower will be, may be readily determined by
imagining the cam to revolve so as to bring that
particular line vertical.
Reciprocating Cams. — Fig. 140 shows a straight
cam, which by a reciprocating motion imparts a
sideways motion to its follower. The pitch line
of such a cam may be determined by intersecting
lines at right angles to each other. As here shown
the distance through which the follower is to be
raised is divided into a number of equal spaces by
horizontal lines, and the distance through which it
is desired to have the cam move in order to raise
the follower from one horizontal line to the next
one is indicated by vertical lines. A curved line
172 SELF-TAUGHT MECHANICAL DRAWING
which is made to pass through the intersections of
these lines will be the required pitch line of the
cam.
If the follower, instead of rising vertically, rose
at ah angle, or if it were mounted on a swinging
arm, the pitch line would be modified in the same
manner as that of the cam shown in Fig. 139.
Cams With a Grooved Edge. — It is sometimes de-
sired to have a revolving cam impart a sideways
FIG. 141.— Cam with Grooved Edge.
motion to a follower. This is done by having a
groove in the edge of the cam, as shown in Fig.
141. Such a cam may be considered as a modified
form of a reciprocating cam, and its pitch line may
be determined in the same way.
By laying out a development of the pitch lino, or
of that part of it which is to operate the follower,
as shown in Fig. 142, horizontal lines, that is, lines
parallel with the pitch line, may be drawn to indi-
cate successive stages in the movement of the fol-
lower, and lines at right angles to these to indicate
CAMS
173
the desired movement of the cam. The pitch line
is then drawn through the intersections of these
lines as before.
A Double Cam Providing Positive Return. — In a
cam like that shown in Fig. 138, where the return
FIG. 142. — Development of Cam Action of Grooved-Edge
Cam in Fig. 141.
of the follower is insured by a groove in the face
of the cam, the groove must be slightly broader
than the diameter of the cam roller to insure free-
dom of action, as, when the cam is forcing the rol-
FIG. 143.— Double Cam Providing Positive Return.
ler away from the center, the roller will revolve in
the opposite direction to that in which it revolves
when the other face of the cam groove acts on it
to draw it toward the center, so that unless clear-
174 SELF-TAUGHT MECHANICAL DRAWING
ance is provided, there will be a grinding action
between the roller and the faces of the cam groove.
This clearance, however, causes the cam to give a
knock or blow on the roller each time its action is
reversed, and the reversal of the direction of the
revolution of the roller itself causes a temporary
grinding action. These actions may become ob-
FlG. 144.— Positive Return Cam with Rollers Mounted on
Swinging Arms.
jectionable, especially at high speeds. A method
which overcomes these objections, and which is
preferred by some for such work, is shown in Fig.
143, where the return is secured by a secondary
cam mounted on the same shaft as the primary
cam, but acting on a roller of its own. In this case
there is no reversal of the direction of the revolu-
tion of the rollers, so that the necessity of provid-
CAMS 175
ing clearance does not exist. Where the forward
and backward motion of the rollers is in a straight
line passing through the center of the cam shaft,
as in this case, it is only necessary in designing
the secondary cam to preserve the distance be-
tween its pitch line and the pitch line of the prim-
ary cam constant, measuring through the center of
the cam shaft, as shown at x and y.
If, however, the rollers are mounted on swing-
ing arms, as shown in Fig. 144, so that their for-
ward and backward motion is not in such a straight
line, then the shape of the secondary cam will be
subject to modification on principles previously
explained. It is obviously necessary where this
method of operation is used, that provision be made
to absolutely prevent any change in the relative
position of the two cams, as by bolting them to-
gether, or, better still, by having them cast
together in one piece.
Cams for High Velocities. — In machinery work-
ing at a high rate of speed, it becomes very im-
portant that cams are so constructed that sudden
shocks are avoided when the direction of motion
of the follower is reversed. While at first thought
it would seem as if the uniform motion cam would
be the one best suited to conditions of this kind, a
little consideration will show that a cam best suited
for high speeds is one where the speed at first is
slow, then accelerated at a uniform rate until the
maximum speed is reached, and then again uni-
formly retarded until the rate of "motion of the
follower is zero or nearly zero, when the reversal
takes place. A cam constructed along these lines
176 SELF-TAUGHT MECHANICAL DRAWING
FIG. 145.— Uniformly Accelerated Motion Cam.
CAMS 177
is called a uniformly accelerated motion cam. The
distances which the follower passes through during
equal periods of time increase uniformly, so that,
if, for instance, the follower moves a distance equal
to 1 length unit during the first second, and 3
during the second, it will move 5 length units
during the third second, 7 during the fourth, and
so forth. When the motion is retarded, it will
move 7, 5, 3 and 1 length units during successive
seconds, until its motion becomes zero at the re-
versal of the direction of motion of the follower.
In Fig. 145 is shown a uniformly accelerated
motion plate cam. Only one-half of the cam has
been shown complete, the other half being an exact
duplicate of the half shown, and constructed in the
same manner. The motion of the follower is back
and forth from A to G, the rise of the cam being
180 degrees, or one-half of a complete revolution.
To construct this cam, divide the half-circle, AKL,
in six equal angles, and draw radii HB± , HC^ ,
etc. Then divide AG first in two equal parts AD
and DG, and then each of these parts in three
divisions, the length of which are to each other as
1:3:5, as shown. Then with H as a center draw
circular arcs from J5, C, Z>, etc., to B± , d , A , etc.
The points of intersection between the circles and
the radii are points on the cam surface.
If the half-circle AKL had been divided into 8
equal parts, instead of 6, then the line AG would
have been divided into 8 parts, in the proportions
1:3:5:7:7:5:3:1, each division being the same
amount in excess of the previous division while
the motion is accelerated, and the same amount
178 SELF-TAUGHT MECHANICAL DRAWING
less than the previous division while the motion is
being retarded. With a cam constructed on this
principle the follower starts at A from a velocity
of zero ; it reaches its maximum velocity at D ; and
at G the velocity is again zero, just at the moment
when the motion is reversed.
A graphical illustration of the shape of the uni-
formly accelerated motion curve is given in Fig.
i
BC DE F GHL
\
/
FIG. 146. — Development and Projection of Uniformly
Accelerated Motion Cam Curve.
146. To the right is shown the development of
the curve as scribed on the surface of a cylindrical
cam. This development is necessary for finding
the projection on the cylindrical surface, as shown
at the left. To construct the curve, divide first
the base circle of the cylinder in a number of equal
CAMS 179
parts, say 12; set off these parts along line AL, as
shown ; only one division more than one-half of the
development has been shown, as the other half is the
same as the first half, except that the curve to be
constructed here is falling instead of rising. Now
divide line AK in the same number of divisions as
the half-circle, the'divisions being in the proportion
1:3:5:5:3:1. Draw horizontal lines from the
divisions on AK and vertical lines from B, C, D,
etc. The intersections between the two sets of
lines are points on the developed cam curve. These
points are transferred to the cylindrical surface at
the left simply by being projected in the usual
manner.
In order to show the difference between the uni-
form motion cam curve, and that illustrating the
uniformly accelerated motion, a uniform motion
cylinder cam has been laid out in Fig. 147. The
base circle is here divided in the same number of
equal parts as the base circle in Fig. 146. The
divisions are set off on line AL in the same way.
The line AK, however, is divided into a number of
equal parts, the number of its divisions being the
same as the number of divisions in the half -circle.
By drawing horizontal lines through the division
points on AK, and vertical lines through points B,
C, D, etc., points on the uniform motion cam curve
are found. It will be seen that this curve is merely
a straight line AM. The curve is transferred to its
projection on the cylinder surface at the left, as
shown.
It is evident from the developments of the two
curves in Figs. 146 and 147, that the uniform motion
180
SELF-TAUGHT MECHANICAL DRAWING
curve, Fig. 147, causes the follower to start very
abruptly, and to reverse from full speed in one
direction to full speed in the opposite direction.
The uniformly accelerated motion curve, Fig. 146,
permits the follower to start and reverse very
smoothly, as is clearly shown by the graphical
A B C DEFGHIL
FIG. 147. — Development and Projection of Uniform Motion
Cam Curve.
illustration of the curve. The abrupt starting and
reversal of the follower in the uniform motion
curve is the cause why this form of cam, while
the simplest of all cams to lay out and cut, cannot
be used where the speed is considerable, without
a perceptible shock at both the beginning and the
end of the stroke.
CAMS
181
Besides the uniformly accelerated motion cam
curve, quite commonly called the gravity curve,
on account of it being based on the same law of
acceleration as that due to gravity, there is another
curve, the harmonic or crank curve, which is quite
often used in cam construction. The harmonic
motion curve provides for a gradual increase of
speed at the beginning, and decrease of speed at the
end, of the stroke, and in this respect resembles
xV
\
EI. FI GI H!
FIG. 148. — Lay-out of Harmonic Motion Cam Curve.
the uniformly accelerated motion curve; but the
acceleration, not being uniform, does not produce so
easy working a cam as the gravity curve provides
for. The harmonic motion curve is, however, very
simple to lay out, and for ordinary purposes, where
excessively high speeds are not required of the
mechanism, cams laid out according to this curve
are very satisfactory.
The harmonic curve is laid out as shown in Fig.
148. Draw first a half-circle AEL Divide the
182 SELF-TAUGHT MECHANICAL DRAWING
circle in a certain number of equal parts. Draw a
line AI /! , and divide this line in a number of equal
parts, the number of divisions of Av Jt being the
same as that of the half -circle. Now draw hori-
zontal lines from the divisions A, B, C, etc., on the
half-circle, and vertical lines from the divisions on
line AI /! . The points where the lines from corre-
sponding division points intersect, are points on
the required harmonic cam curve.
An approximation of the uniformly accelerated
motion or gravity curve can be drawn as shown in
\
\n
\
\
/
^
yr
^^
^^
Ai Bj c, D, ei F., G! Hj i,
FIG. 149.— Approximation of Uniformly Accelerated Motion
Curve.
Fig. 149. By using this approximate method, any
degree of accuracy can be attained without the
necessity of dividing the vertical line AK, Fig.
146, in an excessively great number of parts. The
approximate curve in Fig. 149 is constructed as
follows: Draw a half-ellipse AEI, in which the
minor axis is to the major axis as 8 to 11. Divide
this half-ellipse in any number of equal parts, and
divide the line Ailt in the same number of equal
parts. Now draw horizontal lines from the division
CAMS 183
points on the ellipse, and vertical lines
BI, Ci, etc. The points of intersection between
corresponding horizontal and vertical lines, are
points on the cam curve. This cam curve, as well
as the one in Fig. 148, can be transferred to the
cylindrical surface of a cylinder cam by ordinary
projection methods, as shown in Figs. 146 and 147.
In Figs. 150 and 151 are shown two plate cams
for comparison. The one in Fig. 150 is a uniform
V
FIG. 150.— Plate Cam Laid Fia. 151.— Plate Cam Laid
out for Uniform Motion. out for Uniformly Accel-
erated Motion.
motion cam. The dwell is 180 degrees, the rise, 90
degrees, and the fall, 90 degrees. As shown by
the sudden change of direction of the cam curve
at A and B, there is considerable shock when the
follower passes from its "dwell" to the " rise, " as
well as at the end of the ' ' fall. ' ' A sudden reversal
takes place at C, which also causes a shock in
the mechanism connected with the follower. In the
uniformly accelerated motion cam, Fig. 151, the
184 SELF-TAUGHT MECHANICAL DRAWING
passing from " dwell" to "rise, " the reversal of the
direction of motion, and the return to the "dwell"
position, is accomplished by means of smoothly
acting curves, and, even at high speeds, no per-
ceptible shock will be noticed.
The examples given will show the necessity of
careful analysis of conditions, before a certain type
of cam curve is selected. In machinery which
works at a low rate of speed, it is not important
whether the follower moves with a uniform, har-
monic, or uniformly accelerated motion ; but when
the cam has a high rotative speed, and the follower
a reciprocating motion, it often becomes practically
impossible to make use of the uniform motion
curve in the cam. In such cases, as already men-
tioned, the harmonic, or, preferably, the uniformly
accelerated motion curve should be used in laying
out the cam.
CHAPTER XI
SPROCKET WHEELS
WHEN it is desired to transmit power from one
shaft to another one quite near to it, especially if
the power to be transmitted is considerable, so as
to preclude the use of belting, sprocket wheels
with chain are frequently used, if the speed is not
high. Bicycles afford a familiar illustration of
this sort of power transmission.
Fig. 152 shows a sprocket wheel of a type similar
to those used on bicycles and shows the method of
getting the shape of the teeth. The chain is shown
with the links (on the side toward the observer)
removed so as to allow of showing the teeth with-
out dotted lines. The size of a sprocket wheel to
fit a given chain may be determined graphically as
follows : A circle, not shown in the illustration, is
first drawn of a diameter about equal to that of
the desired wheel, and this circle is spaced off into
as many divisions as the wheel is to have teeth.
Lines corresponding to the dotted radial lines in
the upper half of the wheel shown, are drawn from
these division points to the center of the circle. A
templet, similar in shape to that shown in Fig. 154,
is next cut out of paper, the lines ab and cd being
at right angles to each other, and the length of a
link of the chain, measured from center to center
185
186 SELF-TAUGHT MECHANICAL DRAWING
of the pins as shown at a, Fig. 152, is marked off
upon the line ab, measuring equally each way from
the center line cd. In getting the length of the
link in the chain it will be best, for the sake of ac-
curacy, to measure off the length of a considerable
portion of the chain, and with the spacing com-
passes divide this length into twice as many spaces
as there are links in the measured portion of the
FIG. 152.— Sprocket Wheel and Chain.
chain. The compasses, being then set to exactly
half the length of a link, may be used to mark off
the length of the link, 1 — 2, upon the templet.
Now letting the angle abc, Fig. 155, represent one
of the angles into which the circle has been di-
vided, bisect it to get a center line bd, and placing
the templet so that its line cd shall coincide with
this center line move it along until the points 1 — 2
shall coincide with the lines ab and cb of the angle.
These points being now marked off upon the lines,
give the location of the centers of the pins in the
chain, and a line connecting them will be one side
SPROCKET WHEELS
187
of the polygon which forms the pitch line of the
wheel. A spiral may now be formed upon this
polygon (see geometrical problem 19, Figs. 41 and
42) , and will give the path of the pin as the chain
FIG. 153.— Sprocket Wheel Designed for Common
Link Chain.
unwinds from the wheel when the latter revolves,
as shown in Fig. 152. The working face of that
part of the tooth in the wheel lying outside of the
pitch polygon is now struck from such a center as
will cause it to fall slightly within the path of the
chain, as just obtained, so that the link may fall
FIG. 154. FIG. 155.
FIGS. 154 and 155.— Graphical Method of Laying Out
Sprocket Wheel.
freely into place as it enters upon the tooth. Of
course allowance must be made all around for the
natural roughness of the casting if the wheel is to
be left unfinished. The length of the tooth is
usually made about equal to the width of the chain.
188 SELF-TAUGHT MECHANICAL DRAWING
If a wheel is to have many teeth, it will gener-
ally be accurate enough to consider the pitch line
as a circle of a circumference equal to the number
of the teeth multiplied by the length of the link.
Its diameter will then, of course, be found by
dividing the circumference by 3.1416.
In the case of the wheel shown in Fig. 152,
should the pitch line be regarded as such a circle
it would have a diameter a little over a thirty-
second of an inch too small, if the length of the
link is taken at three-quarters of an inch. If the
wheel were to be made twice as large, the error
would be a little less than a sixty-fourth of an inch,
as it would decrease at a slightly faster rate than
that at which the number of the teeth increased. An
error of a sixty-fourth of an inch in the diameter
of such a sprocket would be of but very little
moment. Where a sprocket has but few teeth,
however, it will be on the side of safety to always
give to the pitch line its true polygonal form, and
the only way by which its diameter could be ascer-
tained with any greater accuracy than by the
method here given would be to calculate it, as may
be done by trigonometry. When the pitch line of
a sprocket is regarded as a circle, the path of the
chain as it unwinds will be regarded as an involute
(see geometrical problem 20).
The shape of the rim of a sprocket wheel will be
governed by the style of the chain for which it is
designed. Fig. 153 shows a portion of the rim of
a wheel which is designed for a common link
chain ; but whatever the general shape of the rim
may be, the working faces of the teeth, or of the
SPROCKET WHEELS 189
projections. which correspond to teeth, will always
be made on the principles here explained.
The speed ratio of the two wheels of a pair of
sprockets will be inversely as the number of teeth
in each. For instance, if the large and the small
wheels have respectively 13 and 7 teeth, then the
speed of the large wheel will be to the speed of
the small wheel as 7 to 13.
CHAPTER XII
GENERAL PRINCIPLES OF GEARING
Friction and Knuckle Gearing. — In machinery
it is frequently necessary to transmit power from
one shaft to another near to it. For this purpose
gears are generally employed. Let a and 6, Fig.
156, be two such shafts. If now disks c and d are
mounted upon these shafts, of such diameters as
FIG. 156. -Friction Wheels.
FIG. 157. -Knuckle Gears.
to give the required speed ratio, we will have
gearing in its simplest form. Such disks, having
their edges covered with leather or other equiva-
lent material, are called friction gears and are
sometimes employed on light work. At best, how-
ever, they will transmit but little power.
If now we make semi -circular projections at
equal distances apart upon the outside of the cir-
cles c and d, and cut out corresponding depressions
inside of the circles, as shown in Fig. 157, we will
have a simple form of toothed gearing and the cir-
190
GENERAL PRINCIPLES OF GEARING 191
cles c and d will be the pitch circles. Such gears,
called knuckle gears, are sometimes employed on
slow-moving work where no special accuracy is
required. They will not transmit speed uniformly.
If the driver of such a pair of gears rotated at a
uniform rate, the driven gear would have a more
or less jerky movement as the successive teeth
came into contact, and if run at high speed they
would be noisy. Various curves may be employed
to give to gear teeth such an outline that the
driver of a pair of gears will impart a uniform
speed to the driven one, but in common practice
only two kinds are used, the cycloidal, or, as it is
sometimes called, epicycloidal, and the involute.
Epicycloidal Gearing. — Let the circles a, b and
c, Fig. 158, having their centers on the same
straight line, be made to rotate so that their cir-
cumferences roll upon each other without slipping.
If the circle c has tracing points 1, 2, 3 upon its
circumference, and when we start to rotate the
circles point 1 is half way around from the posi-
tion in which it is shown, then in rotating the cir-
cles sufficiently to bring the tracing points to the
position in which they are shown, point 1 will
trace the line 1 ' inwardly from the circle a, and the
line 1 " outwardly from the circle b. Point 2 will
trace the two lines which are shown meeting at
that point, one inwardly from the circle a, and one
outwardly from the circle 6. Point 3 will similarly
trace the two lines which met at that point. Inas-
much as these lines were traced simultaneously by
points at a fixed distance apart, it is evident that
if the circle c were to be removed, and the circles
192 SELF-TAUGHT MECHANICAL DRAWING
a and b were rolled back upon each other, these
lines would work smoothly together, being in con-
tact and tangent to each other at all times upon
the line of the circle c. If the circle c is now placed
beneath the circle b in the position shown, and the
three circles are rolled together as before, the tra-
cing points would trace lines inwardly from 6, and
FIG. 158.— Principle of Epi-
cycloidal Gearing.
FIG. 159.— Principle of Invo-
lute Gearing.
outwardly from a, which would also work together
smoothly if the circle c were removed and the cir-
cles a and 6 were rolled back upon each other. It
is evident that as the three circles are rolled
together the lines formed by the tracing points are
the same as though either a or 6 were taken by
itself, and the circle c were rolled either within
or upon it, hence the lines formed by the tracing
points are either epicycloids or hypocycloids as
the case may be, and so could be formed by the
GENERAL PRINCIPLES OF GEARING 193
plotting method described in the geometrical
problems.
If these two sets of lines are now joined together
so that the lines which extend inwardly from a or
b form a continuation of those which extend out-
wardly and reverse curves are made at a distance
from the first set equal to the thickness of a gear
tooth, and they are the-n cut off at such a distance
both outside and inside of the circles a and 6 as to
give to the teeth the proper
length, it is evident that
we will have a pair of per-
fectly working gears. The
circles a and b would roll
upon each other without
slipping and hence would FlG ^.-Definitions' of
be true pitch circles. The Gear Tooth Terms.
teeth would work smoothly
together in constant contact, the point of contact
being always on the line of the generating circle.
The length of the point of the gear tooth, that
is the portion lying outside of the pitch line, is
usually made one-third of the circular pitch^the
latter being the distance between the teeth meas-
ured from center to center on the pitch line. The
distance below the pitch line is made somewhat
greater for the sake of clearance. For the names
of the various parts of a gear tooth see Fig. 160.
Cast gears have some backlash between the teeth
to allow for the roughness of the castings, as
shown in Figs. 161 and 163.
It is evident that if another circle, either larger
or smaller, were substituted for b in Fig. 158, the
194 SELF-TAUGHT MECHANICAL DRAWING
lines formed by the generating circle c either
within or upon the circle a would remain unchanged.
Or if a different circle were substituted for a, the
curves formed within or upon 6 would remain un-
changed. Hence it follows that all gears in the
epicycloidal system, having their teeth formed by
the same generating circle and made of the same
FIG. 162.-Rack with
Epicycloidal Teeth.
FIG. 161.— Gears with Epicycloidal
Teeth.
size, will work together correctly, or% as it is com-
monly expressed, are interchangeable.
In standard interchangeable gears the generat-
ing circle is made one-half the diameter of the
smallest gear of the set, which has twelve teeth.
This smallest gear will have radial flanks, as that
part of the working surface lying within the pitch
line is called, because the hypocycloid of a circle
formed by a generating circle of half its size will
be a straight line passing through its center.
Fig. 161 shows a portion of a pair of such gears,
Fig. 162 showing the rack.
GENERAL PRINCIPLES OF GEARING
195
Gears with Strengthened Flanks. — A further ex-
amination of Fig. 158 will show that the curves
formed by the generating circle when it is in the
upper of the two positions in which it appears,
work together by themselves, and those formed
when it is in the lower position work similarly, so
that it is not necessary that the same sized gener-
r\
\J
FIG. 164.— Rack with
Involute Teeth.
FIG. 163.— Gears with Involute Teeth.
ating circle should be used in both positions, unless
the gears are to be members of an interchangeable
set of gears. Advantage may be taken of this fact
to strengthen the roots of the teeth in a pinion.
If, for instance, in Fig. 161, a smaller generating
circle were used in the upper position, the effect
would be to broaden out the roots of the teeth in
the pinion, and to correspondingly round off the
points of the teeth of the other gear.
Gears with Radial Flanks. —Another modification
which may be made is to have the teeth of both
gears with radial flanks. If, for instance, in Fig.
161 a generating circle were to be used in the
196 SELF-TAUGHT MECHANICAL DRAWING
lower portion, of half the pitch diameter of the
large gear, the effect would be to give to that gear
radial flanks, and to make the points of the teeth
of the small gear broader in order to work properly
with them. Then both gears would have radial
flanks. Such gears have been considerably used.
They are not as strong as gears of the standard
shape, and the only advantage is that it is easier
to make the pattern, the teeth being all worked out
with a flat-faced plane; but as the teeth of in-
volute gears, described in the next section, can be
worked out in the same way, and as such gears are
interchangeable, the advantage is obviously in
favor of the involute system for such work.
Involute Gears. — In involute gears the working
surfaces of the teeth are involutes, formed not
upon the pitch circles, but upon base circles lying
within the pitch circles and tangent to a line,
called the line of action, which passes obliquely
through the point where the pitch circles cross the
line connecting their centers. Let a and b, Fig.-
159, be pitch circles, and let the line cd be the line
of action. Then e and /, being made tangent to
the line cd, will be the base circles upon which
the involutes are to be formed. If now this line
of action be considered as part of a thread which
unwinds from one base circle and winds up on the
other, as the pitch circles are revolved back and
forth upon each other, then if tracing points were
attached to the thread at points 1, 2, 3, 4, 5 and 6,
these points would describe involutes outwardly
from the base circles, which, being formed simul-
taneously in pairs and each pair being formed by
GENERAL PRINCIPLES OF GEARING 197
a common point, would work together smoothly
like those formed by the generating circles of the
epicycloidal system. That the base circles are of
such size as to just pass the thread as the pitch
circles roll upon each other is proven by the fact
that their radii, gd and gi, and he and hi, the radii
gd and he being made at right angles to the line of
action, are corresponding sides of similar triangles,
the segments into which the line of action is di-
vided by the line of centers being the other sides,
and hence have the same ratio. It would only then
FIG. 165. -Modified Form of Involute Rack Teeth.
be necessary to reverse the direction of the thread
to get curves for the other side of the teeth, and
to give to the teeth their proper length inside and
outside of the pitch line to obtain a pair of cor-
rectly working involute gears. That part of the
tooth of an involute gear which may lie within the
base line is made radial.
In the standard interchangeable involute gears
the line of action is given an obliquity of 15
degrees (cut gears, 14J degrees) . This angle may
be readily obtained by the combination of the
triangles resting against the blade of the T-square
shown in Fig. 166. The point of contact of the
198 SELF-TAUGHT MECHANICAL DRAWING
teeth is always upon the line of action and the
push of one tooth against another is in its direc-
tion, hence its name.
The teeth of the 15-degree involute rack have
straight sides, inclined to the pitch line at an angle
of 75 degrees as shown in Fig. 164. This shape,
however, is subject to a slight modification to avoid
interference of the points of the teeth with the
radial flanks of small gears.
Interference in Involute Gears. — The points c
and d, Fig. 159, where the line of action is tangent
to the base circles, are called the limiting points.
If the involutes which spring from either base cir-
cle are so long as to reach
beyond these limits on the
other base circle, they will
interfere with the radial
flanks of the mating teeth.
At A; is shown an elongated
involute interfering with
the radial flank of the
mating tooth. This is, of
course, a highly exagger-
ated case. The interfer-
FIG. 166.— Obtaining a 15-
or 75-degree Angle by
30- and 45-degree Tri-
angles.
ence will occur sooner as the line of action is made
to cross the line of centers at a less oblique angle,
as in standard gears, and still earlier as the pitch
circle b is made larger. In gearing of standard pro-
portions, a gear of 30 teeth is the smallest that will
work correctly with a straight toothed rack. In
the gears shown in Fig. 163, the teeth of the large
gear pass beyond the limiting point of the small
gear, and hence, if made of , true involute shape,
GENERAL PRINCIPLES OF GEARING 199
their extremities will not work properly with the
flanks of the small gear.
There are three methods available to overcome
this interference. First, to hollow out the flanks
of the teeth of the small gear. Second, to round
off the points of the teeth of the large gear. This
is the method usually adopted, in interchangeable
gears, the point being rounded off enough to clear
the flanks of the smallest gear of the set. Fig. 165
shows the teeth of the rack so corrected in larger
scale. Third, to cut off that part of the tooth in
the large gear which extends beyond the limiting
point of the small gear. This is done in special
cases.
The Two Systems Compared. — The great point
in favor of epicycloidal gearing would appear to be
in its freedom from interference. It is necessary,
however, in order to have epicycloidal gears run
well, to have the pitch circles of the two gears of
a pair just coincide, as shown in Fig. 161; but
with involute gears the distance between centers
may be varied somewhat without affecting their
smoothness of operation, though where the points
of the teeth are rounded off to avoid interference,
as previously explained, the amount of variation
which can be allowed is not great. As no value
has been given to the angle at which the line of
action crosses the line of centers in Fig. 159, it is
evident that whether the base circles are brought
nearer together or are carried further apart, circles
which might then be drawn through the point
where the line of action crosses the line of centers,
would roll upon each other while the base circles
200 SELF-TAUGHT MECHANICAL DRAWING
passed the thread as before, and hence would be
true pitch circles for the time being. The amount
of backlash, that is, the space between the faces
of the teeth, would vary, but the smoothness of
operation would not be affected. This property of
involute gears is very valuable in cases where the
distance between centers is variable, as in rolling
mill gearing. In such cases, however, interfer-
ence must be avoided by the first of the three
methods explained, that of hollowing out the flanks
of the teeth of the mating gear.
The epicycloidal system is the older of the two,
and cast gears are still quite largely made to this
system, there being so many patterns of that sys-
tem on hand. But though the epicycloidal system
once had the field to itself, the fact that the invol-
ute system has so largely replaced it, having al-
most wholly superseded it for cut gearing, shows
the trend of modern practice. It is sometimes
urged against the involute system that the thrust
on the shaft bearings is greater than with the epi-
cycloidal system, on account of the obliquity of its
line of action. But though the line of action is at
an angle to the direction of the motion of the teeth
when they are on the line connecting their centers,
it is a constant angle; while it is never less, it is
never more. With the epicycloidal system, on the
other hand, though the teeth of the driver give a
square push to the teeth of the driven gear when
they are in contact on the line of centers, yet
the direction of this pushing action being on the
line of the generating circle, is variable, so that
when the teeth are first coming into contact with
GENERAL PRINCIPLES OF GEARING 201
one another they have an obliquity of action fully
as great, if not greater, than standard involute
gears. For this reason such authorities as the
Brown & Sharp Co., Grant and Unwin, do not con-
sider this objection as being of great weight.
Twenty-Degree Involute Gears. — It has been al-
ready shown how the teeth of epicycloidal gears
may be considerably strengthened where it is not
necessary to have them interchangeable. In invol-
ute gearing, when a stronger gear is desired than
the standard 15-degree tooth provides for, recourse
may be had to increasing the obliquity of the line
of action. This makes the tooth considerably
broader at the base, and correspondingly narrower
at the point. The angle usually adopted in such
cases is 20 degrees, and some makers report an
increasing demand for such gears.
Shrouded Gears. -When it is desired to strengthen
the teeth of cast gears without increasing their
size, or without using any other than a standard
shape or tooth, the practice of shrouding them is
sometimes resorted to. This consists in casting
a flange on one or both sides of the gear. Full
shrouding consists in having the flanges extend to
the points of the teeth as shown in Fig. 167 ; half
shrouding is where the flanges extend only to the
pitch line as shown in Fig. 168. When the two
gears of a pair are of nearly equal size so that
their teeth would be of about the same strength
it would be natural to use half shrouding on both
gears as shown.
When, however, there is much difference in the
size of the gears, as shown in Fig. 167, it would be
202 SELF-TAUGHT MECHANICAL DRAWING
natural to use full shrouding on the small gear, as
otherwise its teeth would be weaker than those
of the large gear. Shrouding is estimated to
strengthen the teeth from 25 to 50 per cent.
FIG. 167.
FIGS. 167 and 168. -Shrouded Gears.
Bevel Gears. — In cylindrical or spur gears the
pitch surfaces are cylinders of a diameter equal to
the pitch circle; in bevel gears the pitch surfaces
are cones, having their apices coinciding.
In designing a pair of bevel gears as shown in
Fig. 169, the center lines ab and cd are first drawn,
and the pitch diameters then laid out from these
GENERAL PRINCIPLES OF GEARING
203
lines as indicated. From the point where the lines
of the pitch diameters meet at e, a line is drawn to
the point where the center lines intersect at k.
This gives one side of the pitch cone of each gear
and from this the other sides of the cones are
FIG. 169.— Bevel Gears.
readily drawn. All lines of the working surfaces
of the gears meet at the point h.
To lay out the teeth, the line/gr is first drawn
through the point e and at right angles to eh. This
gives the outside face of the teeth, and the points
/and g become the apices of cones upon the devel-
opment of which the teeth are laid out. With cen-
ters at /and g the pitch line developments ei and
ej are drawn, and upon these lines the teeth are
laid out the same as for ordinary gears. When
the two gears of a pair are of the same size
they are called miter gears.
204
SELF-TAUGHT MECHANICAL DRAWING
Worm Gearing. — In worm gearing, as shown in
Fig. 170, a screw having its threads shaped like
the teeth of a rack engages with the teeth of a
gear having a concave face and teeth of such shape
as to fit the threads of the screw. If the screw is
single threaded, one rotation of it will cause the
gear to revolve the distance of one tooth ; if double
threaded, the gear will turn two teeth, and so on.
In worm gearing, the worm wears much faster
than the gear; it is, therefore, frequently made of
FIG. 170.— Worm and Worm-Gear.
steel while the worm-wheel is made of bronze, to
give the combination increased durability.
In involute worm gearing interference is com-
monly avoided by the last of the three methods
already mentioned. The points of the thread of the
screw in Fig. 170 project but little beyond the
pitch line, the root spaces of the gear being made
correspondingly shallow. At the same time, the
points of the teeth in the gear are made long
enough to preserve their total length the same as
usual, and the depth of the screw thread inside the
pitch Iin2 is made sufficient for clearance. But un-
GENERAL PRINCIPLES OF GEARING 205
less the worm-gear has less than 30 teeth, the
standard shape of tooth will be satisfactory.
Circular Pitch. — In designing gearing, the old
method (the one which is given in the older trea-
tises on the subject) is to use the circular pitch;
that is, the distance between the teeth, measured
from center to center on the pitch circle. This
method has many disadvantages. For instance, if
it is required to make a pattern of a gear to mesh
with one already on hand, the natural thing to do
in measuring up the old gear is to first guess at
where the pitch line is, and then measure straight
across from one tooth to the next. This leads to
two errors in the result; first, the probably incor-
rect location of the pitch line, and, second, the dis-
tance measured is the chordal pitch instead of the
circular pitch. A noisy pair of gears would quite
likely be the result.
Again, as the ratio between the circumference
and the diameter of a circle is not an even num-
ber, but a troublesome fraction, the use of the cir-
cular pitch method will give the pitch diameter of
the gear in inconvenient fractions of an inch, un-
less an equally inconvenient circular pitch is used.
This method has so many disadvantages that it
has been largely replaced by the more convenient
"diametral pitch'* method. For cut gears the dia-
metral pitch method is used almost exclusively;
but for cast gears there are so many patterns on
hand, made by the circular pitch method, that that
method is still used considerably on such work,
especially on the larger sizes of gears.
Where one is designing new work, however,
206 SELF-TAUGHT MECHANICAL DRAWING
where no old gear patterns made by the circular
pitch method are used, the diametral pitch method
will be by far the most convenient to use, which-
ever style of tooth, whether involute or epicy-
cloidal, may be adopted.
PITCH DIAMETERS OF GEARS FROM 10 TO 100
TEETH, OF 1-INCH CIRCULAR PITCH.
No.
of
Teeth
Diam. in
Inches
No.
of
Teeth
Diam. in
Inches
No.
of
Teeth
Diam. in
Inches
No.
of
Teeth
Diam. in
Inches
10
3.183
33
10.504
56
17.825
79
25.146
11
3.501
34
10.823
57
18.144
80
25.465
12
3.820
35
11.141
58
18.462
81
25.783
13
4.138
36
11.459
59
18.781
82
26.101
14
4.456
37
11.777
60
19.099
83
26.419
15
4.775
38
12.096
61
19.417
84
26.738
16
5.093
39
12.414
62
19.735
85
27.056
17
5.411
40
12.732
63
20.054
86
27.375
18
5.730
41
13.051
64
20.372
87
27.693
19
6.048
42
13.369
65
20.690
88
28.011
20
6.366
43
13.687
66
21.008
89
28.329
21
6.685
44
14.006
67
21.327
90
28.648
22
7.003
45
14.324
68
21.645
91
28.966
23
7.321
46
14.642
69
21.963
92
29.285
24
7.639
47
14.961
70
22.282
93
29.603
25
7.958
48
15.279
71
22.600
94
29.921
26
8.276
49
15.597 .
72
22.918 '
95
30.239
27
8.594
50
15.915
73
23.236
96
30.558
28
8.913
51
16.234
74 .
23.555
97
30.876
29
9.231
52
16.552
75
23.873
98
31.194
30
9.549
53
16.870
76
24.192
99
31.512
31
9.868
54
17.189
77
24.510
100
31.831
32
10.186
55
17.507
78
24.828
When the pitch of a gear is given in inches or
fractions of an inch, the circular pitch is always
meant; as, for instance, where a gear is said to be
of 1-inch pitch, or IJ-inch pitch. To get the pitch
diameter in such a case, it is necessary to multiply
GENERAL PRINCIPLES OF GEARING 207
this pitch by the number of teeth in the gear, and
then divide this product by 3. 1416, the ratio be-
tween the circumference and the diameter. For
ascertaining the pitch diameter of gears when
using the circular pitch, the accompanying table
will save much time. If the gear is of any other
than 1-inch circular pitch, multiply the diameter
here given for the required number of teeth, by
the circular pitch to be used.
Proportions of Teeth. — The proportions of the
teeth of gears where the circular pitch method is
used, are given slightly different by various writ-
ers. The length of the teeth is entirely arbitrary
and therefore this discrepancy is quite natural.
It is also unimportant, excepting as uniformity
is desirable. The proportions as given by Grant
are as follows : The addendum and dedendum are
each made one-third of the circular pitch; the
clearance, the distance of the root line below the
dedendum line, is made one-eighth of the adden-
dum; the backlash, the space which is allowed be-
tween the sides of the teeth in cast gears, is made
about the same as the clearance. This presents the
proportions in fractions which are convenient to
use, and at the same time makes the proportions
practically the same as those of the diametral pitch
method. Cut gears are made without backlash.
Diametral Pitch. — In the diametral pitch method
the gear is considered as having a given number
of teeth for each inch of pitch diameter. Gears
having three, four, or five teeth to each inch of
their pitch diameters are said to be of three, four,
or five pitch. With this method the addendum
208 SELF-TAUGHT MECHANICAL DRAWING
(the distance which the teeth project beyond the
pitch line) is made equal to one divided by the
pitch, so that the addendum on gears of three, four
or five pitch would be, respectively, one-third, one-
fourth or one-fifth of an inch. The advantages of
this method are numerous.
To get the diametral pitch of a gear it is only
necessary to divide the number of teeth by the
pitch diameter, or to divide the number of teeth
plus two, by the outside diameter. A complete
set of rules, as well as formulas and examples for
calculating spur gear dimensions, will be given in
the next chapter.
It is quite a common practice in figuring gears
made by diametral pitch to give only the pitch and
the number of teeth, as 4 pitch, 18 teeth, or 4 D.
P., 18 T. The letters D. P. stand for diametral
pitch, the letters P. D. standing for pitch diameter.
The pitch diameter is then found by dividing the
number of teeth by the diametral pitch. When
this method is used, the circular pitch becomes
of secondary importance, but may be found by di-
viding 3.1416 by the diametral pitch. When the
circular pitch is given and the diametral pitch is
desired, divide 3.1416 by the circular pitch. The
diameter of a gear, unless otherwise specified, is
always understood to be the pitch diameter. With
the diametral pitch method, the pitch diameter,
unless in even inches, will be in fractions of an
inch corresponding to the pitch, so that the frac-
tional parts of the diameter of gears of three, four
or five pitch, for instance, would be thirds, fourths
or fifths of an inch.
GENERAL PRINCIPLES OF GEARING 209
The Hunting Tooth. — It is a common practice in
making gear patterns to have the teeth of the two
gears of a pair of such numbers that they do not
have a common divisor. For instance, instead of
having 25 and 35 teeth in the gears of a pair, one
may give to one of them one more or one less
tooth, so as to insure all of the teeth of one gear
coming into contact with all of the teeth of the
other as they run together.
This practice is condemned by some, however,
on the ground that if any of the teeth are of bad
shape it would be better to confine their injurious
action within as narrow limits as possible, rather
than to have them ruin all of the teeth of the other
gear ; but the shape of badly formed teeth should
be corrected as soon as the error is discovered.
Approximate Shapes for Cycloidal Gear Teeth.—
That part of the~cycloidal curve which is used in
the formation of gear tooth outlines is so short
that it may be replaced with a circular arc which
will very closely approximate it, and such arcs are
generally used in the practical construction of gear
patterns. In the following is given a table of such
arcs with the location of the centers from which
they are struck. The center from which that part
of the tooth lying outside of the pitch line is
drawn, the face of the tooth, will be inside of the
pitch line, while the center from which that part
of the tooth lying inside of the pitch line is drawn,
the flank of the tooth, will be outside of the pitch
line. These radii and center locations were ob-
tained directly from a set of tooth outlines of
3-inch circular pitch, formed by rolling a genera-
210
SELF-TAUGHT MECHANICAL DRAWING
ting circle, drawn upon tracing paper, upon a set
of pitch circles, correct rotation being assured by
the use of needle points pricked through the gen-
erating circle into the pitch circle, the needle
points serving as pivots upon which the genera-
ting circle was swung through short successive
stages, the forward movements of the tracing
point in forming the cycloidal curves being also
pricked through. Needle points were also used in
the instruments which were used for tracing this
curve when the radius and center location were
determined.
CYCLOIDAL TOOTH OUTLINES
Radii and center locations for one-inch circular pitch. For
any other pitch multiply the given figure by the required
pitch.
Number of
Teeth.
Face
Radius.
Inside of
Pitch Line.
Flank
Radius.
Outside of
Pitch Line.
12
0.625 ins.
0.016 ins.
Radial
14
0.666
0.021
4.00 ins.
2.35 ins.
16
0.697
0.026
2.80
1.33
18
0.724
0.031
2.37
_
0.96
20
0.750
0.036
2.14
0.73
25
0.802
0.042
1.91
0.58
30
0.844
0.052
1.79
0.48
40
0.906
0.062
1.64
0.375
60
0.958
0.083
1.50
0.29
100
1.010
0.095
1.33
0.21
200
1.040
0.120
1.23
0.177
Rack
1.080
0.127
1.08
0.127
If the diametral pitch method is being used, the corresponding circular
pitch may be found by dividing 3.1416 by the diametral pitch, as already
mentioned.
Involute Teeth.— The construction of a correct
involute tooth outline is so simple a matter as to
make the use of tables of approximate circular
GENERAL PRINCIPLES OF GEARING 211
arcs .unnecessary. An involute may be formed by
the plotting method given in the geometrical prob-
lems, but in most cases it may be more readily
formed by the use of a sharply pointed pencil
guided by a strong thread as shown in Fig. 171,
where ab represents the pitch line of a gear, and
cd represents the base circle, having a number of
pins stuck into it at short distances apart. The
thread being doubled, forms a loop to hold the pen-
cil point. The thread being drawn tightly around
the pins, the pencil is swung outward from the
FIG. 171. — Laying out an Involute Gear Tooth.
base circle, forming the required involute. When
gears of over thirty teeth are to mesh into others
of less than that number, it will be necessary to
slightly round over the points of the teeth to avoid
interference with the radial flanks of the mating
gear. For this purpose use a radius of 2.10 inches
divided by the diametral pitch, with a center on
the pitch line as shown in Fig. 172. This radius,
2.10 inches divided by the diametral pitch, is the
same as that given by Grant for rounding off the
points of the teeth of racks ; but actual trial on
teeth of large size shows it to be correct for gear
wheels also, giving a curve which coincides very
closely with the epicycloidal shape which the point
212 SELF-TAUGHT MECHANICAL DRAWING
should have to work correctly with the radial flank
of the mating gear.
That part of an involute tooth lying within the
base circle is made radial, as previously stated, and
a good fillet should be drawn in at the root. For
this purpose use a radius of one-twelfth of the
circular pitch. A templet which is fitted to this
FlG. 172.— Modified Tooth Form to Avoid Interference.
outline is used to finish the drawing, and to mark
out the teeth on the pattern.
On large work the size of the base circle may be
obtained by calculation more readily than by the
use of the triangle, as shown in Fig. 166. When
the line of action has an obliquity of 15 degrees,
the diameter of the base circle will be equal to
0.966 of the pitch diameter. For 20-degree invo-
lute gears the diameter of the base circle will be
0.94 of the pitch diameter.
With the 20-degree involute system the teeth of
the rack have an inclination of 70 degrees to the
pitch line. With this system there will be no ne-
cessity for rounding off the points of the teeth of
the rack or of a large gear unless it meshes with
GENERAL PRINCIPLES OF GEARING 213
a gear of less than 18 teeth. When, to avoid inter-
ference, it does become necessary to round off the
points of the teeth of the rack or of large gears,
the same radius, 2.10 inches divided by the diam-
etral pitch, is to be used, as in the 15-degree
system, the center being on the pitch line as
before.
Proportions of Gears. — A somewhat common rule
is to make the rim and the arms of about the same
thickness as the teeth at the root, though some
make the thickness of the rim equal to the height
of the tooth ; and to make the diameter and length
of the hub about equal to about twice the diameter
of the shaft. On spoked gears, the rim is also stiff-
ened by ribbing it between the arms. On a light
gear mounted on a relatively large shaft it would
be natural to lighten the hub somewhat. The
width of tthe face of cast gears is usually made
from two to three times the circular pitch. The
face of bevel gears should not exceed one-fifth of
the diameter of the large gear, and the face of
worm gears should not exceed one-half of the
diameter of the worm.
Strength of Gear Teeth. — When a gear is to be
designed for a given work, the first question is
how large to make the teeth to give the required
strength. On their size will also depend the gen-
eral proportions of the gear.
It is comparatively easy to determine the work
which the teeth are doing, that is, the strain or
load which they are bearing, when the power
which the gear transmits is known. A horse-power
being the power required to lift 33,000 pounds one
214 SELF-TAUGHT MECHANICAL DRAWING
foot in one minute, the load on the teeth will be
33,000 multiplied by the horse-power which is
being transmitted, and divided by the velocity of
the pitch line of the gear in feet per minute ; or,
what is the same thing, 126,050 multiplied by the
horse-power, and divided by the product of the
pitch diameter in inches multiplied by the number
of revolutions per minute. This latter figure,
126,050, takes into account the fact that in the first
case the velocity is expressed in feet, while in this
case the diameter is in inches, and also the fact
that the velocity is a factor of the circumference
instead of the diameter.
While the load on the teeth may be readily
determined, the question of how large they should
be made to bear it is one where authorities have
differed very much on account of the number of
factors involved. First of all is the question of
the material, usually cast iron, which is a variable
quantity, both on account of the nature of the
material itself, different grades varying greatly as
to strength, and the liability of defects in the cast-
ing. Then there is the question of whether the
load should be considered as divided between two
or more teeth or carried by one tooth, or the cor-
ner of a tooth.
Then there is the nature of the work: whether,
the load will be uniform or whether the teeth will
be subject to severe strain or shock. There are
questions of the shape of the tooth, and the velocity
at which the gear is running, the teeth having
greater strength at slow speeds than at high speeds
due to the shocks accompanying high velocities.
GENERAL PRINCIPLES OF GEARING 215
To show the different results given by different
writers we may take the case of a gear 24 inches
diameter, 2 inches circular pitch, 4 inches face,
running at 100 revolutions per minute. A rule
given by Box in his treatise on mill gearing, and
quoted by Grant and Kent, would make the gear
safe for 9.4 horse-power. The rule in Nystrom's
Mechanics gives 12.2 horse-power. Rules by other
writers, quoted by Kent, give results as follows:
Halsey, 22.6; Jones & Laughlin, 35; Harkness, 38;
Lewis, 65.2. The rule by Prof . Harkness is the
result of investigations conducted by him in 1886.
He examined a great many rules, largely, how-
ever, for common cast gears. Mr. Lewis's method,
the result of his investigations of modern machine
molded and cut gears, though giving much higher
results than the others, is said to have proved sat-
isfactory in an extensive practice, and so may be
considered reliable for gears which are so well
made that the pressure bears along the face of the
teeth instead of upon the corners.
It is customary in calculating gears to proceed
on the assumption that the load is borne by one
tooth, and in ordinary work, the size of the tooth
may be determined by the load it may safely bear
per inch of face and per inch of circular pitch.
In 1879, J. H. Cooper selected an old English rule
giving the breaking load of the tooth as 2000 X
pitch X face, which, allowing a factor of safety of
10, would give us a safe load of 200 X pitch X face.
Kent says of this rule that for rough ordinary work
it "is probably as good as any, except that the fig-
ure 200 may be too high for weak forms of tooth,
216 SELF-TAUGHT MECHANICAL DRAWING
and for high speeds. " Lewis also considers this
rule as a passably correct expression of good gen-
eral averages.
The value given by Nystrom and those given by
Box for teeth of small pitch, are so much smaller
than those of other authorities that Kent says they
may be rejected as giving unnecessary strength.
Accepting the factor 200 as a good average would
leave one room for the exercise of individual judg-
ment for the particular case in hand. If the speed
were slow and the teeth were of strong shape, as
where both the gears of a pair, or all of the gears
of a train, have a reasonably large number of
teeth, a higher figure, perhaps 225 or more, might
be taken; while if the speed were higher and one
of the gears had but few teeth, giving them a
weak form, or if they were to be subject to much
vibration or shock, a lower figure, perhaps as low
as 125, might be taken.
To ascertain the horse-power safely transmitted
by an existing gear, we would then multiply to-
gether its diameter, pitch (circular) and face,
taken in inches, and the number of revolutions
per minute, and multiply their product by 200, or
whatever figure is selected, and divide the total
product by 126,050. This may, perhaps, be ex-
pressed clearer, as follows:
diam. X rev. X circ. pitch X face X 200
Horse-power= "
The figure 200 would give to the 24- inch gear
previously considered 30.5 horse-power. The fig-
ure 125 would give 19.0 horse-power.
GENERAL PRINCIPLES OF GEARING 217
To ascertain the size of the teeth to transmit a
given horse-power we may transpose the above rule
and say that the product of the pitch multiplied
by the face would be equal to 126,050 multiplied
by the horse-power, and divided by the product of
the diameter in inches, the number of revolutions
per minute, and 200, or the figure selected; that is:
~. , , , , 126^050 X horse-power
Circ. pitch X face :=--
Assuming some pitch and dividing this result
by it would give the breadth of face. A few
trials will give the desired ratio between pitch
and breadth of face. If one has a table of square
roots at hand, the work may be simplified by
assuming some desired ratio, when the pitch will
be the square root of the quotient of this figure,
pitch multiplied by the face, divided by the ratio.
If, for instance, the pitch multiplied by the face
were found to be 12, and we desired them to be
in the ratio of 2J to 1, the pitch would be equal to
the square root of the quotient of 12 divided by
2i, or 2. 191, which would be about the same as li
diametral pitch.
Example. — Required the size of the teeth of a
gear 18 inches in diameter, to run 120 revolutions
per minute, which shall transmit five horse-power,
allowing 200 pounds load per inch of face, and
inch of pitch. Then :
126,050 X 5 630,250
Pitch X face =- = "432^000 = L46
nearly. A circular pitch of 0.785 inch, correspond-
218 SELF-TAUGHT MECHANICAL DRAWING
ing to 4 diametral pitch, would give a breadth of
face of about 15 inches. For bevel gears take the
diameter and pitch at the middle of the face.
Mr. Lewis's method differs from the preceding
in that instead of using a single constant, as 200
pounds per inch of pitch and inch of face, two
constants are used, one, Y, a factor of strength
depending on the number of teeth in the gear, and
another, S, a safe working stress for different
speeds of the pitch line, in feet per minute. The
values of these constants are given in the accom-
panying tables.
The rule to get the horse-power of a given gear
is:
TJ p = circ. pitch X face X velocity X S X Y
33,000
the velocity being that at the pitch line in feet per
minute, and the values of S and Y being taken
from the tables. The velocity is, of course, the
diameter in feet X 3.1416 X number of revolu-
tions. If the diameter were taken in inches then
the total product would be divided 'by 12. The
product of the pitch multiplied by the face, to
determine the size of teeth to transmit a given
power, would then be
33,000 X H. P.
Circ. pitch X face = — v .T— ~ "0 ~- ^
velocity X S X Y.
The calculation should be made for the gear of
the pair or train having the fewest teeth, as it
would be the weakest, unless it were made of some
stronger material as steel, or unless it were
GENERAL PRINCIPLES OF GEARING
219
WORKING STRESS, S, FOR DIFFERENT SPEEDS
AT PITCH LINE IN FEET PER MINUTE,
FOR CAST IRON.
Speed.
s.
Speed.
s.
100 or less
200
300
600
8000
6000
4800
4000
900
1200
1800
2400
3000
2400
2000
1700
shrouded. If made of steel S might be taken 2i
times the tabulated values.
As a gear with cut teeth has from two to three
times the strength of one with cast teeth, because
of the more perfect contact, Mr. Lewis's method
might be adapted to common cast gears by taking
the value of S at from one-half to one-third of the
tabulated value. By so doing one could bring into
the calculation the question of shape of teeth and
FACTOR FOR STRENGTH, Y, TO BE USED IN
LEWIS'S FORMULAS.
0.078
0.083
0.088
0.092
0.094
0.096
0.098
0.100
13 3
0.3 -S-i
0.067
0.070
0.072
0.075
0.077
0.080
0.083
0.087
0.102
0.104
0.106
0.108
0.111
0.114
0.118
0.122
0.090
0.092
0.094
0.097
0.100
0.102
0.104
0.107
43
50
60
75
100
150
300
Rack
0.126
0130
0.134
0.138
0.142
0.146
0.150
0.154
0.110
0.112
0.114
0.116
0.118
0.120
0.122
0.124
220 SELF-TAUGHT MECHANICAL DRAWING
speed, which would be especially desirable if the
speed were high or the teeth of weak form. Tak-
ing S at one-half the tabulated value would give
to the 24-inch gear previously considered about
the same power as allowing 200 pounds per inch
of pitch and face, which Mr. Lewis considers a
fair value. With cast gears where interchange-
ability is not a necessary feature, the teeth of a
small gear could of course be considerably strength-
ened in the manner previously indicated for epicy-
cloidal gears; or the 20-degree system might be
used if the teeth have the involute form.
Thurston's Rule for Shafts.— The size of shaft
which the gear will require may be found by the
rule given by Thurston. Multiply the horse-power
to be transmitted by 125 for iron, or by 75 for cold
rolled iron, and divide the product by the number
of revolutions per minute. The cube root of the
quotient will be the size of the shaft.
. The size of gear to give a required speed may
be readily determined from the fact that the prod-
uct of the speed of the driving shaft multiplied by
the size of the driving gear or gears, should be
equal to the product of the speed of the driven
shaft, multiplied by the size of the driven gear or
gears. This, perhaps, may be made clearer by
placing the driving members on one side of a line,
and the driven members on the other side, as in
the following example.
A shaft making 75 turns per minute has on it a
gear of 200 teeth. Required the size of gear to
mesh with it which shall drive its shaft 120
GENERAL PRINCIPLES OF GEARING 221
revolutions per minute. Letting x represent the
size of the required gear we have
Rev. driving shaft = 75
Size driving gear = 200
x = size driven gear.
120 = rev. driven shaft.
Then as the product of the numbers on one side
of the line equals the product of those on the other
side, 75 X 200 -5- 120 will give the value of x, the
number of teeth in the driven gear. This method
applies to a train of gears as well as a pair.
CHAPTER XIII
CALCULATING THE DIMENSIONS OF GEARS
IN the previous chapter, the general principles
of gearing have been explained. The three kinds
of gearing most commonly in use, spur gearing,
bevel gearing and worm gearing, have been
touched upon, and the fundamental rules for the
dimensions of gear teeth have been given. In
this chapter it is proposed to give in detail the
rules and formulas for these three classes of gears,
so as to enable the student to calculate for himself
any general problem in gearing with which he
may meet.
Spur Gearing. — In the following, machine cut
gearing is, in particular, referred to; but the gen-
eral formulas are, of course, of equal value for use
when calculating cast gears. The expressions pitch
diameter, diametral pitch and circular pitch have
already been explained, and rules have been given
for transferring circular pitch into diametral
pitch, and vice versa. These rules, expressed as
formulas, would be:
in which P = diametral pitch, and
P'= circular pitch.
Assume as an example that the diametral pitch
222
CALCULATING THE DIMENSIONS OF GEARS 223
of a gear is 4. What would be the circular pitch
of this gear?
Using the formula given, we have:
pf = == 0.7854 inch.
When the diametral pitch and the pitch diameter
are known, the .number of teeth may be found by
multiplying the pitch diameter by the diametral
pitch, as already mentioned in the previous chap-
ter. This rule, expressed as a formula, would be :
N=PXD
in which N = number of teeth,
D = pitch diameter, and
P = diametral pitch.
Assume that the diametral pitch of a gear is 4
and the pitch diameter 6i inches. What would be
the number of teeth in this gear?
By inserting the given values in the formula
above, we would have :
N = 4 X 6i = 25 teeth.
If the number of teeth and pitch diameter of the
gear are known, and the diametral pitch is to be
found, a rule and formula for this may be arrived
at by merely transposing the rule and formula just
given. The diametral pitch equals the number of
teeth divided by the pitch diameter, or, expressed
as a formula:
in which P, N and D signify the same quantities
as in the previous formula.
224 SELF-TAUGHT MECHANICAL DRAWING
Assume, for an example, that the number of
teeth in a gear equals 35 and that the pitch diam-
eter is 3J inches. What is the diametral pitch?
If we insert the known values in the given for-
mula, we have :
35
P = ^f = 10 diametral pitch.
05
Finally, if the diametral pitch and the number
of teeth are known, the pitch diameter is found
by dividing the number of teeth by the diametral
pitch, which rule expressed as a formula, would be:
As an example, assume that the number of teeth
in a gear is 58 and the diametral pitch 6. What is
the pitch diameter of this gear?
By inserting the known values in the formula,
we find :
D = 5|- =9.667 inches>
If it now be required to find the outside diam-
eter of the gear, that is, the diameter of the gear
blank, we make use of the following rule : The
outside diameter equals the number of teeth plus
2, divided by the diametral pitch. Expressed as
a formula, this rule is:
TV N+2
P
in which D ' = outside diameter of gear, and N
and P have the same significance as before.
As an example, assume that the number of teeth
CALCULATING THE DIMENSIONS OF GEARS 225
is 58 and the diametral pitch 6. By inserting these
values in the formula, we find the outside diameter:
rv 58+2 60
D' = — - — = -^-= 10 inches.
b b
When the pitch diameter and the diametral pitch
are known, the outside diameter is found as
follows: Add the quotient of 2 divided by the
diametral pitch to the pitch diameter; the sum is
the outside diameter. This rule, expressed as a
formula, is:
in which the letters have the same significance as
before.
Assume that the pitch diameter of a gear is 9.667
inches, and the diametral pitch 6. Find the out-
side diameter.
By inserting the given values in the formula, we
have:
Df= 9.667 + ~ = 9.667 + 0.333 = 10 inches.
o
By a transposition of the rule and formula just
given, we find that the pitch diameter equals the
outside diameter minus the quotient of 2 divided
by the diametral pitch. This rule, written as a
formula, is-
D = u- Jr
Assume that the diametral pitch of a gear is 8,
and the outside diameter 12 inches. What is the
pitch diameter?
D = 12-~f-=12-i = 111 inches.
o
226 SELF-TAUGHT MECHANICAL DRAWING
When the number of teeth and outside diameter
are known, the diametral pitch may be found by
adding 2 to the number of teeth and dividing the
sum by the outside diameter; or, expressed as a
formula:
N + 2
P =
D'.
If the number of teeth in a gear is 96 and the
outside diameter is 14 inches, what is the diame-
tral pitch?
If the known values are inserted in the given
formula, we have :
-D 96 + 2 98 . ,
P ~TZ~ = IT = diametral pitch.
When the outside diameter and the number of
teeth are known, the pitch diameter may be found
by multiplying the outside diameter by the number
of teeth, and dividing the product by the sum of
2 added to the number of teeth; or, as a formula:
ZXX N
"WTz
Find the pitch diameter for the gear having 96
teeth and an outside diameter of 14 inches.
14X96 1344 ,
^==96T2~ ~98~ = 13.714 inches.
When it is required to find the center distance C
between two gears in mesh with each other, we
must first know the pitch diameters of, or the
number of teeth in, the two gears. The center
CALCULATING THE DIMENSIONS OF GEARS 227
distance equals one-half of the sum of the pitch
diameters of the two gears :
„ D + d
2
in which D and d denote the pitch diameters in
the large and small meshing gears, respectively.
The pitch diameters of two gears equal 9. 5 and 7
inches, respectively. Find the center distance
between them when in mesh.
•„ 9.5 + 7 16.5 .
O = ~^ ~~^~ = o.Zo inches.
The center distance is also equal to the sum of
the numbers of teeth in the two gears divided by
two times the diametral pitch; or, as a formula:
2P
in which N and n denote the numbers of teeth in
the meshing gears.
As an example, assume that the number of teeth
in each two gears equals 95 and 75. The diametral
pitch is 10. What is the center distance?
n 95 + 75 170
: 2~~X~10 = "20"
We will now find the dimensions of the tooth
parts. The addendum (see Fig. 160) equals 1
divided by the diametral pitch. Expressed as a
formula:
in which A = addendum.
228 SELF-TAUGHT MECHANICAL DRAWING
What is the addendum or height above the pitch
line of a 5 diametral pitch gear tooth?
A =4-= 0.2 inch.
o
The dedendum (see Fig. 160) equals the ad-
dendum.
The clearance, c, equals 0.157 divided by the
diametral pitch, or:
c_0157
P.
What is the clearance at the bottom of the gear
tooth (see Fig. 160) of a 4 diametral pitch gear?
c = ^p - 0.039 inch.
The full depth of the tooth equals the sum of the
addendum, dedendum, and clearance, or
JL JL , 0.157 2J.57
P ' P P P
in which d ' = full depth of gear tooth.
What is the full depth of a 4 diametral pitch
tooth?
d,= =.a539inche
The thickness of a cut gear tooth at the pitch
line equals 1.5708 divided by the diametral pitch;
or, as a formula :
T 1.5708
~1T
in which T = thickness of tooth at pitch line.
CALCULATING THE DIMENSIONS OF GEARS 229
What is the thickness at the pitch line of a 4
diametral pitch gear tooth?
T == ~ == 0.3927 inch.
As a general example, let it be required to de-
termine the various dimensions for a pair of gears,
the one having 36 and the other 27 teeth. The
gears are of 8 diametral pitch.
By using the formulas given, we have :
For the larger gear :
Pitch diameter = -5 = -5- = 4.5 inches.
r o
Outside diameter = ~~w^ - 86 J" 2= 4.75 inches.
Jr o
For the smaller gear:
n 27
Pitch diameter = — = -— = 3.375 inches.
Jr o
Outside diameter = — 5~~ = — o — = 3.625 inches.
Jr o
For both gears :
Addendum = ~ = - - = 0.125 inch.
f O
Dedendum = ^ = -5- = 0.125 inch.
Jr o
~ 0.157 0.157 ftmofi- v.
Clearance = — =r~ = — ^~~ = 0.0196 inch.
Full depth of tooth = — = -~ - 0. 2696 inch.
X O
36 + 27 63 015 .
Center distance = —- - 2x g == 16 = 3|| inch.
230 SELF-TAUGHT MECHANICAL DRAWING
This concludes the required calculations neces-
sary for a pair of spur gears.
Bevel Gears. — Bevel gears are used for trans-
mitting motion between shafts whose shafts are
not parallel, but whose center lines form an angle
with each other. In most cases this angle is a
--o—
FIG. 173. — Diagram for Calculation of Bevel Gearing.
right, or 90-degree, angle. The formulas for the
dimensions of bevel gears are not as simple as
those for spur gears, and an understanding of the
trigonometrical functions, explained in Chapter
VII, is necessary, as well as the use of trigonomet-
rical tables. As bevel gears with a 90-degree angle
between their center lines are the most common,
CALCULATING THE DIMENSIONS OF GEARS 231
formulas will be given for this case only, in the
following.
In Fig. 173 a pair of bevel gears are shown, the
dimensions of which are to be determined. The
letters in the formulas below denote the following
quantities :
P = diametral pitch,
Di = pitch diameter of large gear,
D2 = pitch diameter of small gear,
01 = outside diameter of large gear,
02 = outside diameter of small gear,
NI = number of teeth in large gear,
AT2 = number of teeth in small gear,
NI = number of teeth for which to select cut-
ter for large gear,
N2' = number of teeth for which to select cut-
ter for small gear,
flu &i, Ci, 02, &2, c2, d and e = angles as shown in
Fig. 173.
A = addendum,
A + C = dedendum = addendum plus clearance.
If the pitch diameter and diametral pitch are
known, the number of teeth equals the pitch
diameter multiplied by the diametral pitch, or:
N, = D, X P
N2 = D2 X P
If the number of teeth and the diametral pitch
are known, the pitch diameter equals the number
of teeth divided by the diametral pitch, or:
232 SELF-TAUGHT MECHANICAL DRAWING
Angles at and a2 can be determined if either the
numbers of teeth or the pitch diameters of both
gears are known. The tangent for these angles,
the pitch cone angles, equals the number of teeth
in one gear divided by the number of teeth in the
other, or the pitch diameter in one gear divided by
the pitch diameter in the other, according to the
following formulas :
2 ,
tan a2= ^ = ^
Angle a2 also equals 90° - alt
The outside diameter equals the pitch diameter
plus the quotient of 2 times the cosine of at or a2,
respectively, divided by the diametral pitch, or:
0,=
Angles d and e are determined by -the formulas:
, 2 sin a,! 2 sin a2
tancZ= ^~ -~
2.314 sin a^ 2.314 sin a2
tan e • -^r- ^
Angles 6t, Ci, 62 and c2 are determined by the
formulas :
&! = O-! + d
Ct = di — e
&2 = a2 + d
c = a - e
CALCULATING THE DIMENSIONS OF GEARS 233
The number of teeth for which the cutter for
cutting the teeth should be selected is found as
follows :
COS
cosa2
Finally the addendum, dedendum and clearance
are found as in spur gears.
As a practical example, assume now that two
bevel gears are required, 8 diametral pitch, with
24 and 36 teeth, respectively. Find the various
dimensions.
n Ni 36
D1 = -p -- ~^ =4.5 inches.
N2 24
D2= -p- = -g- = 3 inches.
tan ttl == ^ == H = 1.5; a, = 56° 20'.
tan a2= ^ = ^ = 0.667; a2= 33° 40'.
JMi ob
Qi=Di+ 2^0, =45+ 2X|554 = 46W incheg_
02= Z)2+ ^f^ = 3 + ^^2 _ 3 m incheg_
-L O
tan d =
234
SELF-TAUGHT MECHANICAL DRAWING
b, = a, + d = 56° 20' 4- 2° 40' = 59° 0'.
Cl = a,- e = 56020'-3°0' = 53°20'.
62 = a2 + d = 33° 40' + 2° 40' = 36° 20'.
c2 = a2 - e = 33 ° 40 ' - 3 ° 0 ' = 30 ° 40 '.
ZV'= N* = -36
1 COS ttj
AT2' =
0.554
N2 24
cos a2 0.832
= 65 approximately.
= 29 approximately.
A - -4- - 4- - 0.125 inch.
Jr o
= 00196inch
o
Whole depth of tooth = y 4- y +
0.2696 inch.
Worm Gearing.— Worms and worm gears are
used for transmitting power in cases where great
FIG. 174. -Worm.
reduction in velocity and smoothness of action are
desired. They are also used when a self -locking
CALCULATING THE DIMENSIONS OF GEARS 235
power transmission is desirable, that is, when
it is required that the mechanism itself, due to
the friction between the worm and worm-wheel,
should support the load without slipping if the
driving power be rendered inoperative.
In Figs. 174 is shown a worm and in Fig. 175 a
worm-wheel; the dimensions to be found are, in
most cases, given in these illustrations. The fol-
lowing notation has been used in the formulas
given below for worm and worm-wheels :
P = circular pitch of worm-wheel = pitch of the
worm thread,
N = number of teeth in worm-wheel,
Z>! = pitch diameter of worm-wheel,
DT = throat diameter of worm-wheel,
01 = outside diameter of worm-wheel (to sharp
corners) ,
R = radius of worm-wheel throat,
C = center distance between worm and worm-
wheel axes,
D2 = pitch diameter of worm,
02 = outside diameter of worm,
DR = root diameter of worm,
A = addendum, or height of worm tooth above
pitch line,
d = depth of worm tooth,
a = face angle of worm-wheel.
If the pitch of the worm and the number of
teeth in the worm-wheel are known, the pitch
diameter of the worm-wheel may be found by
multiplying the pitch of the worm by the number
236 SELF-TAUGHT MECHANICAL DRAWING
of teeth, and dividing the result by 3.1416, or, as
a formula :
D-
3.1416
The outside diameter of the worm, 02, is usually
assumed. To find the pitch diameter of the worm,
the addendum must first
be found. The addendum
equals the pitch of the
worm thread multiplied
by 0.3183, or:
A = P X 0.3183.
Now the pitch diameter
of the worm equals the
outside diameter minus 2
times the addendum, or:
£>2 = 02 - 2A.
The root diameter of
the worm can be found
first after the full depth
of the worm-wheel thread
has been found. The full
depth of the worm-wheel
thread equals the pitch
multiplied by 0.6866, or:
d = PX 0.6866.
Now the root diameter
of the worm thread equals
the outside diameter of
the worm minus 2 times the depth of the thread, or:
DR = 02 - 2d.
FIG. 175.— Worm-wheel.
CALCULATING THE DIMENSIONS OF GEARS 237
The throat diameter of the worm-wheel is found
by adding 2 times the addendum of the worm
thread to the pitch diameter of the worm-wheel, or:
DT = A + 2A.
The radius of the worm-wheel throat is found
by subtracting 2 times the addendum from the
outside diameter of the worm divided by 2, or:
R = Y2- 2A.
The outside diameter of the worm-wheel (to
sharp corners) is found by the formula below :
G! = DT + 2 (R - tfcos-
The angle a is usually 75 degrees.
Finally, the center distance between the center
of the worm and the center of the worm-wheel
equals the sum of the pitch diameter of the worm
plus the pitch diameter of the worm gear, and this
sum divided by 2, or:
Find, for an example, the required dimensions
for a worm and worm-wheel, in which the worm-
wheel has 36 teeth, the pitch of the worm thread
is J inch, and the outside diameter of the worm is
3 inches. We have given P = J; N = 36; 02 = 3.
X 36
3.1416 3.1416 =
A = P X 0.3183 = i X 0.3183 = 0.15915 inch.
D2 = 02 - 2A = 3 - 0.3183 = 2.6817 inches.
238 SELF-TAUGHT MECHANICAL DRAWING
d = P X 0.6866 = i X 0.6866 = 0.3433 inch.
DR = 02 ~ 2d = 3 - 0.6866 = 2.3134 inches.
DT = D, + 2A = 5.730 + 0.3183 = 6.0483 inches.
R=^-2A=~- 0.3183 = 1.1817 inch.
0,= DT + 2 (R - R cos y) == 6.0483 + 2 X
(1.1817 - 1.1817 X cos 37° 30X) = 6.5375 inches.
r A+A 5.730 + 2.6817
C = ~— — — - = 4.2058 inches.
CHAPTER XIV
CONE PULLEYS
WHEN it is desired to have a variable speed ratio
between two shafts which are belted together, the
method of having reversed conical cylinders or
drums mounted on the shafts, as shown in Fig. 176
and 177, is sometimes used. These permit any
FIG. 176. -Simplest
Form of ' ' Cone-
Pulley."
FIG. 177.— An Im-
proved Form of
"Cone-Pulley."
FIG. 178.— The Mod-
ern Type of Stepped
Cone Pulley.
desired change of speed, but they have disadvan-
tages which on most work offset this advantage.
It would be necessary, in the first place, to use a
narrow belt to avoid undue stretching at the edges.
Then, as the tendency of a belt is to mount to the
largest part of a pulley, this tendency, acting in
239
240 SELF-TAUGHT MECHANICAL DRAWING
the same way on the cones, would produce undue
tension on the belt. If a crossed belt is used on
such cones their faces would be made straight, as
the belt would be equally tight in any position.
This may be seen by an inspection of Fig. 179,
where circles A and B represent sections of such
cones on one line, and circles C and D represent
sections on another line. If the cones have the
FIG. 179. — Diagram Showing relative Influence of Open and
Crossed Belt on Pulley Sizes.
same taper it is evident that the circle D will be
as much larger than B as C is smaller than A, the
gain in one diameter being offset by the loss in the
other. Then, as the circumferences of circles vary
directly as their diameters (the circumference of
a circle having twice the diameter of another, for
instance, will be twice as long as the circumfer-
ence of the other) , whatever is gained on one cir-
cumference_will be lost on the other. For a crossed
belt then, it is only necessary that the cones have
the same taper.
When, however, an open belt is used, it becomes
necessary to have the cones slightly bulging in the
CONE PULLEYS 241
middle as shown in Fig. 177. By again inspecting
Fig. 179 it will be seen that it is only when the
belt is crossed that one cone gains as fast in size
as the other loses, because it is only when the belt
is crossed that the arc of contact of the belt on the
pulleys is the same on all steps of the cone.
In practice these cones are usually replaced by
stepped or cone pulleys as shown in Fig. 178, so as
to avoid the troubles with the belt previously
mentioned.
Applying the principles mentioned to cone pul-
leys, we see that when a crossed belt is used, all
that is necessary is that the sum of the diameters
of any pair of steps shall be equal to the sum of
the diameters of any other pair of steps. For
instance, the sum of the diameters of steps 1 and
r must be equal to the sum of the diameters of
steps 2 and 2'. When, however, an open belt is
used, as is usually the case, the sum of the diam-
eters of the steps at or near the middle of the cone
will have to be somewhat greater than the sum of
the diameters of those at or near the ends.
What is generally considered to be the best
method of determining the size of the various
steps of cone pulleys is that given by Mr. C. A.
Smith in the ' 'Transactions of the American Society
of Mechanical Engineers, ' ' Vol. X, page 269. Make
the distance C, Fig. 180, equal to the distance
between the centers of the shafts, and draw the
circles A and B equal to the diameters of a known
pair of steps on the cones. At a point midway
between the shaft centers erect the perpendicular
ab. Then, with a center on ab at a distance from
242 SELF-TAUGHT MECHANICAL DRAWING
a equal to the length of C multiplied by 0.314, draw
the arc c tangent to the belt line of the given pair
of steps. The belt line of any other pair of steps
will then be tangent to this arc.
If the angle which the belt makes with the line
of centers, de, exceeds 18 degrees, however, a
slight modification of the above is made as follows :
Draw a line tangent to the arc at c at an angle
of 18 degrees with de', and with a center on a&, at
FIG. 180.— Method of Laying out Cone Pulleys.
a distance from a equal to the length C multiplied
by 0.298 draw an arc tangent to this 18-degree
line.
All belt lines which make an angle with de
greater than 18 degrees are made tangent to this
new arc.
The sizes of the steps so obtained maybe verified
by measuring the belt lengths of each pair. For
this purpose a fine wire may be used, the wire
being held in place by pins placed at close intervals
on the outer half circumference of each pulley of
the pair.
CHAPTER XV
BOLTS, STUDS AND SCREWS
SCREWS for clamping work together are of three
classes: through bolts, Fig. 181; studs, Fig. 182;
cap screws, Fig. 183. In Fig. 181 the bolt is put
entirely through both of the two pieces to be
FIG. 181.— Through Bolt for
Holding two Pieces to-
gether.
FIG. 182.— Stud used for
Clamping one Piece to
another.
clamped together, and a nut is put onto the
threaded end. This is considered to be the best
method on cast iron work, both as regards efficiency
and cheapness, as there is no tapping of any holes
243
244 SELF-TAUGHT MECHANICAL DRAWING
in the cast iron. A tapped hole in cast iron is to
be avoided, if possible, as, on account of the brittle
nature of the material, the threads are liable to
crumble or wear away easily.
In many cases, however, it is not practicable to
avoid tapping holes in cast iron, or questions of
appearance may make the broad flange which is
necessary when through bolts are used, undesirable.
In such cases studs should be used. A stud consists
of a piece of round stock threaded on both ends,
and having a plain portion
in the middle. The studs
are screwed firmly into the
tapped holes, which should
be deep enough to prevent
the studs from bottoming
in them, the studs instead
binding or coming to a
bearing at the end of the
threaded portion. The
loose piece is then put on
over the studs, and is held
in place by the nuts. By
using studs, any further wear of the tapped hole
is avoided, as, when removing the loose part, the
nuts only are taken off, the studs being left in
the body piece.
When the material of the parts which are being
clamped together is of such a nature that threads
formed in it are not liable to crumble or to rapid
wear, then cap screws, Fig. 183, may be used to
advantage. They give a neat appearance to a
piece of work, and the nut is entirely eliminated.
FIG. 183.— Cap Screw used
for Clamping Purposes.
BOLTS, STUDS AND SCREWS 245
United States Standard Screw Thread.— The
most commonly used of all screw threads is the
United States standard thread. A section, indicat-
ing the form of this thread, is shown in Fig. 184.
The thread is not sharp neither at the top nor at
the bottom, but is provided with a flat at both of
these points, the width of the flat being one-eighth
of the pitch of the thread. The sides of the thread
x^
FIG. 184.— Form of the United States Standard Thread.
form an angle of 60 degrees with each other. The
"pitch" and the "number of threads per inch"
should not be confused. The pitch is the distance
from the top of one thread to the top of the next.
If the number of threads is 8 per inch, then the
pitch would be 4 inch ; and the flat on the top of
a United States standard thread, which, as men-
tioned, is one-eighth of the pitch, would be 1-64
inch. If the number of threads per inch is known,
the pitch may be found by dividing 1 by the num-
ber of threads per inch, or
No. of threads per inch.
If, again, the pitch is known and the number of
threads per inch required, then
No. of threads per inch = p. ,
246 SELF-TAUGHT MECHANICAL DRAWING
U. S. STANDARD SCREW THREADS.
BOLTS AND THREADS
HEX. NUTS AND
HEADS.
SQUARK
NUT AND
HEAD.
•8 -
1
ll
*o .
'o
i-s
2 M
o 3
||
£
0)
c •
1-
8-s
1
3
11
J3
I
'^
•si
P
<3
&*l
^S
T3 «
Across Coi
Rougl
[o J
|l
v
j
20
0.185
0.0062
0.049
0.027
ft
1?
J
ft
If
JL
18
0.240
0.0074
0.377
0.045
3"f
H
11
Tti
T5<>
i
||
|
16
0.294
0.0078
0.110
0.068
:.|
fa
fl
1
ft
ft
14
0.344
0.0089
0.150
0.093
If
If
ft
1
l/¥
£
13
0.400
0.0096
0.196
0.126
1
If
i
^>
Iff
ft
12
0.454
0.0104
0.249
0.162
II
H
T^T
§
f
11
0.507]0.0113
0.307
0.202
1ft
Is2
1/2
|
T96
1^
10
0.620
0.0125
0.442
0.302
lj3G
|
H
HI
£
9
0.731
0.0138
0.601
0.420
1ft
If9
If 2
|
11
2A
1
8
0.837
0.0156
0.785
0.550
If
li
1
ii
2|f
H
7
0.940
0.0178
0.994
0.694
144
1|
2^37
H
1ft
2T9*
H
7
1.065
0.0178
1.227
0.893
2
IT 6
2ft
H
lT3g
m
6
1.160
0.020S
1.485
1.057
2ft
2*
if
It5?
3ft
if
6
1. 28410.0208
1.767
1.295
2f
2|
H
1ft
3f|
if
5£
1.3890.0227
2.074
1.515
2ft
2i6
2H
if
iT9;
3f
5"
1.4910.0250
2.405
1.746
2|
2fi
3ft
if
IB
3 |j
1J
5
1.6160.0250
2.761
2.051
2H
2|
3JH
11
IB
43-52"
2
4*
1.7120.0277
3.142
2.302
3ft
3f
2
Ifo
4 1|
21
4*
1.9620.0277
3.976
3.023
3}
3ft
4ft
21
2r3o
4fi
2i
4
2.176|0.0312
4.909
3.719
3f
3rt
4i
2ft
5fl
2!
3
4
3*
2.4260.0312
2.6290.0357
5.940
7.089
4.620
5.428
4f
4^
4.11
5|
2|
2 i-i
6
6H
31
3*
2.879
0.0357
8.296
6.510
5
4rl
5H
31
3j3G
7ft
3|
31
3
3.100
3.317
0.0384
0.0413
9.621
11.045
7.548
8.641
5|
51i
6 A
6H
3|
3H
P
4
3
3.567
0.041312.566
9.963
6*
6ft
7¥
4
3}|
8*1
41
2J
3.798
0.043514.18611.32)
6ft
41
4T3g
Q_3_
1 6
4*
4|
21
2f
4.028
4.256
0.0454
0.0476
15.90412.753
17.72114.226
71
s}|
4V
4f
til
9|
5
2*
4.480
0.0500
19.635,15.763
7f
7ft
8fl
5
4fti
lot!
2-i
4.730
0.0500 21. 648' 17. 572
8
7yf
9s9,
51
5y3g
nil
1
6
2f
2|
21
4.953
5.203
5.423
0.052623.758
0.052625.967
0.055528.274
19.267
21.262
23.098
Ico oooo
ot*-iHcooo|w
41
9f!
10ft
5i
5f
II
n|
BOLTS, STUDS AND SCREWS 247
For example, assume that the pitch is 0.0625
inch. Then
No. of threads per inch = . = 16.
The accompanying table of United States stand-
ard screw threads gives the standard number of
threads per inch, corresponding to given diame-
ters, the diameter at the root of the thread, the
width of the flat at the top and bottom of the
thread, the area of the full bolt body, and the
area at the bottom of the thread. These dimen-
sions are, of course, always the same with all
manufacturers. As regards the sizes for hexagon
nuts and heads, and square nuts and heads also
given in the table, it may be said that all makers
do not conform strictly to the sizes as given. The
catalog of one large bolt manufacturing concern,
which is at hand, gives the width across flats of
finished bolt heads and nuts the same as the rough
sizes given in the table, which, it will be seen, are
founded on the rule that the width across the flats
of the heads and nuts should equal one and one-
half times the diameter of the body of the bolt,
plus one-eighth of an inch. It will also be noticed
that the thickness of the head or nut is the same
as the diameter of the body of the bolt.
With cap screws, although the length of the head
is made the same as for bolts, or equal to the di-
ameter of the bolt body, the diameter of the head,
and the distance across flats, is made different as
shown in table on the following page :
248
SELF-TAUGHT MECHANICAL DRAWING
CAP SCREW SIZES.
(From catalog of Boston Bolt Co. )
Size of Screw
I
T7*
I
T56
i
TV
1
A
I
A
A
&
\-
I
f
A
it
H
f
1
t
1
1
i
Width Across Flats
Hex. Head
i
1
i*
i*
it
H
Width Across Flats
Square Head
Check or Lock Nuts. — When a bolt is subjected
to constant vibrations there is a tendency for the
nut to work loose. To overcome this tendency it
is customary to employ a second nut, called a check
or lock nut, which is screwed down upon the first
one as shown in Fig. 185. When the first nut is
screwed down to a bear-
ing, the upper surfaces of
its thread are in contact
with the under surfaces
of the bolt thread. When
the check nut is screwed
down, however, it forces
the first nut -down so that
the under surfaces of its
thread come into contact
with the upper thread sur-
faces of the bolt. This
means that the check nut has to bear the entire load.
When, therefore, the two nuts are of unequal
thickness, as is frequently the case, the thick
nut should be on the outside.
Bolts to Withstand Shock.— When a bolt which
is subjected to shocks fails, it breaks, of course,
FIG. 185.— Correct Arrange-
ment when Using Check
or Lock Nut.
BOLTS, STUDS AND SCREWS 249
at the part having the least cross sectional area,
that is, at the bottom of the thread. If now the
body of the bolt be reduced so that its cross section
is of the same area as the area at the bottom of
the thread, a slight element of elasticity is intro-
duced, and the bolt is likely to yield somewhat
instead of breaking. This is considered very im-
portant in some classes of work. The reduction of
area may be accomplished by turning down the
body of the bolt, or, according to some authorities,
the same object is attained by removing stock from
the inside by drilling into the bolt from the head
end.
Either method, it is stated, gives the same degree
of elasticity to the bolt, but as the drilling method
takes the stock from the center, the bolt is left
stiffer to resist bending or twisting than when the
stock is taken off the outside by turning.
Wrench Action. — When bolts or any form of
screws are used to hold machine parts together,
they must be strong enough not only to withstand
the strain which is put upon them by the operation
of the machine, but also to withstand the strain
which is put upon them by the wrench in setting
or screwing them up. In the case of a cylinder
head, for instance, the strain upon the bolts due to
the working of the engine will be the exposed area
of the head, multiplied by the pressure per square
inch. This divided by the number of bolts used
will give the proportional part of this strain which
each bolt must sustain. But in order to insure a
tight joint, it is necessary that the bolts be not
merely brought up to a bearing, but that they be
250 SELF-TAUGHT MECHANICAL DRAWING
set up hard enough so as to press the cylinder and
cylinder head surfaces firmly together. The force
which the wrench exerts in doing this work will
be equal to the circumference of the circle through
which the hand moves in turning the wrench
through one revolution, multiplied by the force in
pounds exerted at the handle, and this product
divided by the distance through which the nut
advances in one revolution, that is, by the lead of
the screw. This theoretical result is, of course,
modified by the friction between the nut and the
bolt, and between the nut and washer. In addition
to this direct strain, there is also a twisting strain
in the bolt, caused by the friction between the bolt
and nut.
To insure the bolts being sufficiently strong to
resist these various forces, it is customary to make
them somewhat more than double the strength
that would be necessary to enable them to safely
resist the pressure of the steam or other fluid in
the cylinder; that is, they are made about double
strength to enable them to resist the direct strain
of the wrench action, and then this amount is in-
creased about 15 or 20 per cent, to allow for the
twisting action of the wrench. Allowing that a
factor of safety of 4 would be sufficient to allow
for the steam pressure only, a factor of safety of
not less than about 9 or 10 would therefore be used
to provide for the added strain on the bolt due to
the wrench action. In the case of small bolts,
where the workman might set them up much harder
than is really necessary, a factor of safety of about
15 may be used.
BOLTS, STUDS AND SCREWS
251
The distance apart which bolts can be spaced
without danger of leakage is given by Prof. A. W.
Smith as between 4 or 5 times the thickness of the
cylinder flange for pressures between 100 and 150
pounds per square inch.
In the case of bolts which are not under strain
as a result of the wrench action, as in the case of
FIG. 186.— Example of Thread
not under Stress due to
Wrench Action.
FIG. 187.— Square Threaded
Screw, such as is Generally
used for Power Transmis-
sion.
the hook bolt shown in Fig. 186, a factor of safety
as low as 4 might be properly used, if the load is
steady.
Assuming that the material of which the bolts
are made has an ultimate strength of 40,000 to
60,000 pounds per square inch, the factors of
safety previously indicated would give allowable
working stresses of from 4000 to 15,000 pounds per
square inch.
252 SELF-TAUGHT MECHANICAL DRAWING
Screws for Power Transmission. — In Fig. 187 is
shown a square threaded screw such as is generally
used for power transmission. In such a screw the
depth of the thread is made one-half of the pitch.
The size of the body of the screw, assuming that
the work which the screw is doing brings a ten-
sional stress on the screw, will be determined by
the tensile strength of the material of which it is
made and the factor of safety which is used. As
a screw which is used for power transmission is
subjected to constant wear when in use, the ques-
tion of the proper amount of bearing surface in the
threads of the nut is of first importance, in order
that it may not wear out too rapidly. The area of
the thread surface in the nut on which the pressure
bears will be equal to the difference in area of a
circle of a diameter equal to the outside diameter
of the screw, and one of a diameter equal to the
diameter at the root of the thread of the screw,
multiplied by the number of threads ; or, letting D
represent the outside diameter of the screw, and d
represent the diameter of the body, the area will be:
(D2 - d2) X 0.7854 X No. of threads in the nut.
The allowable pressure per square inch of working
surface will vary with the nature of the service
required, whether fast or slow, and also with the
lubrication, and with the material used. Where
the speed is slow, say not over 50 feet per minute,
and the service is infrequent, as in lifting screws,
a pressure of 2500 pounds for iron or 3000 pounds
for steel is allowable, while for more constant
service some authorities limit the pressure to
about 1000 pounds per square inch even when the
BOLTS, STUDS AND SCREWS 253
lubrication is good. For high speeds a pressure of
about 200 or 250 pounds is considered to be as
much as should be allowed.
For a screw which, fitting loosely in a well lubri-
cated nut, is to sustain a load without danger of
running down of itself, the pitch of the screw
should not, according to Professor Smith, be greater
than about one-tenth of its circumference.
Efficiency of Screws. — A square-threaded screw
has a greater efficiency than a V- threaded one, as
the sloping sides of the V-thread cause an increase
of friction. Square threads are therefore preferable
for power transmission. Experiments show that
in the case of bolts used for fastenings, the friction
of the nut on the bolt and washer may absorb 90
per cent, of the power applied to the wrench,
leaving only 10 per cent, for producing direct com-
pression. For square-threaded screws an efficiency
of about 50 per cent, is considered fair, if the
screws are well lubricated.
Acme Standard Thread. — While the square thread
gives the greatest efficiency in a screw it is not as
strong as one having sloping sides. Fig. 188 shows
a section of a screw thread called the Acme or 29-
degree thread, which is often used for replacing
the square thread for many purposes, such as in
screws for screw presses, valve stems, and the
like. The use of such a screw permits the employ-
ment of a split nut, when such construction is
desirable, which would not be practicable with a
perfectly square thread, and for this reason, as
well as for the reason that it can be cut with
greater ease than the square thread, it has of late
254 SELF-TAUGHT MECHANICAL DRAWING
become widely used. In the Acme standard thread
system the threads on the screw and in the nut are
not exactly alike. A clearance of 0.010 inch is
provided at the top and at the bottom of the thread,
so that if the screw is 1 inch in diameter, for
example, then the largest diameter of the thread
in the nut would be 1.020 inch. If the root diam-
eter of the same screw were 0.900 inch, then
the smallest diameter of the thread in the nut
would be 0.920 inch. The sides of the threads,
however, fit perfectly.
The depth of an Acme thread equals one-half the
pitch of the thread plus 0.010 inch. The width
FIG. 188. -Shape of Acme Screw Thread.
of the flat at the top of the screw thread equals
0.3707 times the pitch; and the width of the flat
at the bottom of the thread equals 0.3707 times the
pitch minus 0.0052 inch.
Miscellaneous Screw Thread Systems. — Besides
the screw thread systems already mentioned, a
great many other systems are in more or less
common use. Leading among these is the sharp
V-thread, which, previous to the introduction of the
United States standard thread, was the most com-
monly used thread in this country. This thread
is, theoretically at least, sharp at both the top and
the bottom of the thread, the angle between the
BOLTS, STUDS AND SCREWS 255
sides of the thread being the same as in the United
States standard system, or 60 degrees. In ordinary
practice, however, a small flat is provided on the
top of the thread, because it would be almost impos-
sible to commercially produce the thread otherwise ;
and even if the thread could be produced, the sharp
edge at the top would rapidly wear away. The
sharp V-thread is being more and more forced
out of use by the United States standard thread,
although it must be admitted that it will probably
long hold its own in steam fitting work, because of
being especially adapted for making steam-tight
joints. It answers this purpose probably better
than any of the other common forms of threads.
The Whitworth standard thread is not used to a
very great extent in the United States, but it is the
recognized standard thread in Great Britain. In
this form of thread the sides of the thread form
an angle of 55 degrees with each other, and the
tops and bottoms of the threads are rounded to a
radius equal to 0.137 times the pitch. This round-
ing of the thread at the top provides for a thread
which does not wear rapidly, and screws and nuts
made according to this thread system will work
well together in continuous heavy service for a
longer period than would screws and nuts with any
of the other standard thread forms. The fact that
the threads are rounded in the bottom is advan-
tageous on account of the elimination of sharp
corners from which fractures may start. The main
disadvantage of the thread, and the reason why
the United States standard thread was adopted in
this country in preference to the Whitworth stand-
256 SELF-TAUGHT MECHANICAL DRAWING
ard, which is the older of the two, is to be found
in the fact that it is more difficult to produce than
a 60-degree thread with flat top and bottom. The
Whitworth form of thread is used in this country
mostly on special work and on stay-bolts for loco-
motive boilers. •
A thread perhaps more commonly used than any
of the others, with the exception of the United
States standard thread, is the Briggs standard
pipe thread, which is used, as the name indicates,
for pipe fittings. This thread is similar to the
sharp V-thread, having an angle of 60 degrees
between the sides, and nearly sharp top and bottom ;
instead of being exactly sharp at the top and bot-
tom, however, it is slightly rounded off at these
points. The difficulty of producing these slightly
rounded surfaces has brought about a modification,
at least in the United States, so that a small flat is
made at the top, and the thread made to a sharp
point at the bottom. It appears that a thread cut
with these modifications serves its purpose equally
as well as a thread cut according to the original
thread form.
Besides these systems, there are the metric screw
thread systems. These use the same form of thread
as the United States standard system, but the
thread diameters and the corresponding pitches
are, of course, made according to the metric system
of measurement.
Other Commercial Forms of Screws. — Set-screws,
shown in Fig. 189, are usually made with square
heads, and have either round or cup-shaped points,
and are generally case hardened. They are used
BOLTS, STUDS AND SCREWS
257
for such work as fastening pulleys onto shafts,
etc. Some set-screws are made headless, and are
slotted for use with a screw-driver in places where
it is undesirable that the _
screw projects beyond the
work.
The term machine screws
covers a number of styles of
small screws made for use
with a screw-driver. Fig.
190 shows the principal styles.
Machine screw sizes are usu-
ally designated by numbers,
the size and the number of
threads per inch being usually given together, with
a "dash" between ; thus a 10 — 24 screw would be a
number 10 screw with 24 threads per inch. There
are two standard systems for machine screw
FlG. 189.— Forms of
Set-screws.
FILLISTER
HEAD
FIG. 190.— Forms of Machine Screws.
threads, the old, which until recently was the only
system, and the new, which was approved in 1908
by the American Society of Mechanical Engineers.
The standard thread form of the old sy"stem was
the sharp V-thread, with a liberal but arbitrarily
258
SELF-TAUGHT MECHANICAL DRAWING
selected flat on the top. The basic thread form
of the new system is that of the United States
standard thread.
The accompanying tables give the numbers and
corresponding diameters and number of threads
per inch of the old as well as the new system for
machine screw threads.
MACHINE SCREW THREADS, OLD SYSTEM.
Number.
Diameter.
Threads
per inch.
Number.
Diameter.
Threads
per inch.
1
0.071
64
12
0.221
24
1*
0.081
56
13
0.234
22
2
0.089
56
14
0.246
20
3
0.101
48
15
0.261
20
4
0.113
36
16
0.272
18
5
0.125
36
18
0.298
18
6
0.141
32
20
0.325
16
7
0.154
32
22
0.350
16
8
0.166
32
24
0.378
16
9
0.180
30
26
0.404
16
10
0.194
24
28
0.430
14
11
0.206
24
30
0.456
14
MACHINE SCREW THREADS, NEW SYSTEM.
Number.
Diameter.
Threads
per inch.
I Number.
Diameter.
Threads
per inch.
0
0.060
80
12
0.216
28
1
0.073
72
14
0.242
24
2
0.086
64
16
0.268
22
3
0.099
56
18
0.294
20
4
0.112
48
20
0.320
20
5
0.125
44
22
0.346
18
6
0.138
40
24
0.372
16
7
0.151
36
26
0.398
16
8
0.164
36
28
0.424
14
9
0.177 32
30
0.450
14
10
0.190
30
!
CHAPTER XVI
COUPLINGS AND CLUTCHES
A COUPLING is a device for connecting together
the ends of two shafts or axles for the purpose
of making a longer shaft, the term being usually
limited to those devices which are intended for
permanent fastening. The term clutch is used to
designate a disengaging coupling.
The simplest form of coupling consists simply of
a sleeve or muff, made of a length about three
times the diameter of the shaft, bored out to fit
the shaft, and provided with a keyway its entire
length, made to receive a tapering key. The ends
of the shafting are, of course, also provided with
keyways, and are inserted into the sleeve; then
the key is driven in. In some couplings the sleeve
is made tapering on the outside at both ends, and,
being split, is clamped upon the shafts by means
of rings or hollow conical sleeves which are driven
onto the tapered ends, or drawn together by means
of bolts.
One of the most common forms of coupling is the
flange coupling shown in Fig. 191. In this case a
flanged hub is keyed to each of the shaft ends, and
the flanges are then held together and prevented
from turning relative to each other by bolts, as
shown. In some cases the bolt heads and nuts are
259
260 SELF-TAUGHT MECHANICAL DRAWING
provided with a guard by having the rim on the
outer edge of the flange made deep as shown by
the dotted lines on one side. This construction
also allows the coupling to be used as a pulley, if
necessary. In a coupling of this kind, the chief
problem is to get the bolts of such size that their
combined strength to resist the shearing action
to which they are subjected equals the twisting
FIG. 191.— Flange Coupling.
strength of the shaft. Letting d represent the
diameter of the shaft in inches, its internal resist-
ance to twisting is given by the formula
~ 5.1
in which T equals the internal resistance to twist-
ing, or the twisting moment, and S the shearing
strength per square inch of area in pounds.
Regarding the shearing strength of materials
Kent says : ' ' The ultimate torsional shearing re-
sistance is about the same as the direct shearing
resistance, and may be taken at 20,000 to 25,000
pounds per square inch for cast iron, 45,000 pounds
COUPLINGS AND CLUTCHES 261
for wrought iron, and 50,000 to 150,000 pounds for
steel according to its carbon and temper."
The torsional and direct shearing resistance being
the same, this quantity may be neglected if the
shaft and coupling bolts are of the same material,
and
ds
5.1
the internal resistance factor or torsion modulus
of the shaft, should be equal to the product of the
radius of the bolt circle of the coupling, the number
of bolts used, and the area of each bolt. Or, letting
a represent the area of each bolt, R the radius of
the bolt circle of the coupling, and n the number
of bolts used, we would have :
a = -~- + (R X ri).
£). -L
Example. — Required the size of the bolts for a
flange coupling for a 2-inch shaft. The radius of
the bolt circle is 3 inches, four bolts being used.
Using the notation in the formula given, our
known values are:
d = 2 inches,
R = 3 inches,
n = 4 bolts.
If we insert these values in the formula we have:
03 o
a = -g-y -^ (3X4)= g-^^ 12 =0.13 square inches.
This area corresponds to a diameter of about A
of an inch. To allow for the strain on the bolt
caused by the action 01 the wrench, the next size
262 SELF-TAUGHT MECHANICAL DRAWING
larger bolt, at least, or a i inch bolt, will be se-
lected. The capacity of the bolt to resist shear-
ing will be considerably increased by having the
corners of the holes at those faces of the flanges
which come together, somewhat rounded. If this
is not done, the action of the flanges on the bolts
will be like that of a pair of sharp shears. Experi-
ments have shown that with the corners rounded,
the capacity of the bolt to resist shearing may be
increased 12 per cent.
If the shaft and bolts are of different materials
then the modulus
^
5.1
should be multiplied by the shearing strength of
the shaft in pounds per square inch and the product
t=
3 d »
FIG. 192.— Clamp Coupling.
RXn should be multiplied by the shearing strength
of the bolts per square inch, before dividing in
the formula to get the bolt area.
In Fig. 192 is shown another form of coupling
much used. It consists of two parts bolted together
over the joint in the shafting, a key and keyway
being provided to prevent the slipping of the shafts.
COUPLINGS AND CLUTCHES
263
By having a thickness of heavy paper interposed
between the two parts of the coupling when it is
bored out, it may be made to clamp very tightly
onto the shafts.
With either form of coupling, the length is made
such that each shaft end is held by the coupling
by a length of about one and one-half times its
diameter, as indicated in the engravings.
Oldham's Coupling. — Fig. 193 shows a form of
coupling which may be used for shafts which are
FIG. 193.— Oldham's Coupling.
parallel, but slightly out of line. In this coupling
each shaft end has a flanged hub attached to it.
Across the face of each flange is planed a single
groove passing through its center. Interposed
between the two flanges is a disk, shown at the
right, having tongues on both faces at right angles
to each other, to engage in the grooves in the
flanges.
Hooke's Coupling or Universal Joint is used for
connecting two shafts whose axes are not in line
with each other, but merely intersect. The shafts
A and B, and B and C, in Fig. 194, are thus con-
nected by universal joints. If the shaft B is made
telescoping, as is very often the case, a solid part
264 SELF-TAUGHT MECHANICAL DRAWING
entering into and being keyed in a sleeve so as
to prevent independent rotation, but yet permit
a sliding action, then the two shafts A and C may
move independently of each other within certain
limits, the distance between their ends being
capable of variation. The arrangement shown in
Fig. 194 is used on various machine tools, notably
on milling machines, flange drilling machines, etc.
Many designs of flexible shafts are really only a
combination of a great number of universal joints.
FIG. 194. — Application of Universal Joints and Telescoping
When this coupling is employed for driving only
one shaft at an angle with another, as if shaft
A simply drove shaft B which, of course, is the
fundamental type of universal coupling, then, if
the driving shaft has a uniform motion, the driven
shaft will have a variable motion, and so cannot be
used in such cases where uniformity of motion of
the driven shaft is necessary ; but where there are
three shafts, as shown in the illustration, A will
impart a uniform motion to C provided the axes of
A and C are parallel with each other, as shown ; for
if A, having a regular motion, imparts an irregular
COUPLINGS AND CLUTCHES
265
motion to B, then if B, with its irregular motion,
is made the driver, it will impart a regular motion
to A, and as C is parallel with A it will also impart
a regular motion to C.
This form of coupling does not work very well
if the angle a is more than 45 degrees.
Clutches are of two general classes, toothed
clutches and friction clutches. An example of a
toothed clutch is shown in Fig. 195. In this clutch
the part at the left is fastened to its shaft; the
part at the right is free to slide back and forth upon
FIG. 195.— A simple Form of Toothed Clutch.
its shaft, but is prevented from turning on the
shaft by a key. The sliding motion for engaging
or disengaging this part of the clutch is accom-
plished by means of the forked lever and jointed
ring, shown at the right, which latter engages in
the groove A. Such a clutch, while giving a pos-
itive drive, cannot, of course, be thrown in or out
while the driving shaft is running at a high rate
of speed. By having the back faces of the teeth
beveled off as shown by the dotted lines, this diffi-
culty is partly overcome, although the shock caused
by the sudden engaging of the teeth still renders
266
SELF-TAUGHT MECHANICAL DRAWING
the clutch unsuitable for operating at very high
speed. To facilitate uncoupling, the driving faces
may also be given an angle of about 10 or 12
degrees.
Friction Clutches are generally made in one of
the two styles shown in Figs. 196 and 197. The
power which a clutch of the type shown in Fig. 196
will transmit, depends upon the power which is ap-
plied to force the sliding part against the fixed part,
FIG. 196.— Friction -Disk Clutc.h.
and the efficiency of the frictional force between
the rubbing surfaces. As to the efficiency of the
clutch, therefore, much depends upon the nature of
the engaging surfaces, whether metal comes in
contact with metal, or whether one of the surfaces
has a facing of leather or wood. The efficiency is,
of course, much increased by either a leather or
wood facing. Professor Smith gives the efficiency
of these different surfaces as follows: Cast iron on
cast iron, 10 to 15 per cent. ; cast iron on leather,
COUPLINGS AND CLUTCHES 267
20 to 30 per cent. ; cast iron on wood, 20 to 50 per
cent.
The horse-power which such a clutch will trans-
mit will be found by multiplying the velocity of the
parts in contact, in feet per minute, taken at their
mean diameter as indicated at D, by the force
which is being applied at this diameter in the
direction of revolution, and dividing this product
by 33, 000. The force which is acting at the diameter
D to produce revolving motion is equal to the pres-
sure which is being applied to force the two parts
of the clutch together, multiplied by the coefficient
of friction (as the frictional efficiency between the
surfaces in contact, as given above, is called) of the
materials which form the driving surfaces.
Example. — What power will a clutch of the type
shown in Fig. 196 transmit if running at a speed
of 250 revolutions per minute? The diameter D is
18 inches, and a pressure of 50 pounds is exerted
to force the two clutch faces together. One of
the clutch parts has a leather facing, and the
coefficient of friction is 0.25.
The general formula for finding the horse-power
of a clutch of this type is:
IT p £>X3.1416XnXPX/
33,000
in which H.P. = horse-power transmitted,
D = mean diameter of friction sur-
faces in feet,
n = revolutions per minute,
P = pressu/e between clutch surfaces
in pounds,
/ = coefficient of friction.
268 SELF-TAUGHT MECHANICAL DRAWING
The values to be inserted in the formula, which
are given in this problem, are as follows :
D = ]n = 1.5 foot,
n = 250 revolutions,
P = 50 pounds,
/ = 0.25.
Inserting these values in the formula we have :
1.5 X 3.1416 X 250 X 50 X 0.25 n ,_
"33,000
The formula given may be transposed in various
ways according to the requirements of the problem ;
if, for instance, it is desired to know what pressure
must be applied to transmit a given horse-power,
then:
H.P. X 33,000
D (in feet)" X 3.1416 X n X /.
If the pressure is known, and it is required to
find what diameter the clutch must be made to
transmit a given power, then :
D (in feet) - -
~ 3.1416 X n X P X /.
If the pressure and diameter are both known,
then the number of revolutions which the clutch
must make per minute to transmit a given horse-
power will be :
H.P. X 33,000
n =
D (in feet) X 3.1416 X P X /.
It may be said that the capacity of the clutch
to transmit power is independent of the area of the
COUPLINGS AND CLUTCHES
269
friction surfaces; for, if the friction surface is
increased the pressure which is applied to force
the two parts of the clutch together is simply dis-
tributed over a much greater area, giving a smaller
pressure per square inch. The durability would be
increased, but the horse-power capacity would re-
main unchanged.
The conical clutch shown in Fig. 197 may be
made to run metal to metal, or the hollow part may
FIG. 197.— Friction Cone Clutch.
be made larger to allow of the insertion of wooden
blocks. This would increase the efficiency, but at
the expense of the durability. The principle of
this form of clutch may be explained by referring
to the diagrammatical sketch at the right of Fig.
197, where the angle ACB represents the angle
which the opposite sides of the clutch make with
each other, the line DC representing the axis of
the shaft. If now line bd of the small triangle
abd be considered as representing the magnitude
270 SELF-TAUGHT MECHANICAL DRAWING
of the force acting in the direction of the axis
of the shaft to force the two parts of the clutch
together, then if ab is at right angles to AC, ab
will represent the resultant magnitude of the force
acting on the face of the clutch at right angles to
its surface, according to the principles explained
in the chapter on the elements of mechanics. The
efficiency of the clutch will therefore be as much
greater than that of a flat-faced clutch as ab is
greater than bd. The horse-power of such a
clutch, using the same notation as before, would,
therefore, be:
_ D (in feet) X 3.1416 X n X P Xf v ab^
33,000 K bd.
But from the chapter on the solution of triangles
we know that
— T = sine of angle bad.
Hence
ab 1
bd sin bad.
But angle bad equals angle x, the angle which
the conical surface of the clutch makes with the
axis of the shaft.
Therefore
ab = 1
bd sin x
and our original formula takes the form:
TJ P D On feet) X 3.1416 X n X P Xf
' 33,000 X sin x.
COUPLINGS AND CLUTCHES 271
Transposing this formula as before for the flat-
faced clutch, gives us:
H.P. X 33,000 X sin a?
~ D (in feet) X 3.1416 XnXf.
nr - ,, H.P. X 33,000 X sin x
D (in feet) - ~ "
n = -— = ,____^ srn^
D (in feet) X 3.1416 X Pxf.
The sine of x may be taken from the tables of
trigonometric functions previously given in the
chapter on the solution of triangles, or it may be
found by dividing the length bd (Fig. 197) by the
length ab.
The power necessary to force the two parts of
the clutch together may be neglected, as the slip-
ping which occurs as they are engaging allows
them to come together with but little pressure
beyond what is required for power transmission
purposes. The angle which the face of the clutch
makes with the shaft (the angle x in the diagram
at the right in Fig. 197) should be such that the
clutch does not grip too quickly when thrown into
gear, nor require too much pull to release. Making
this angle between 7 and 12 degrees conforms to
the average given by different authorities.
CHAPTER XVII
SHAFTS, BELTS AND PULLEYS
Shafts. — The twisting strength of a shaft, as
stated in the preceding chapter, is given by the
formula
T =
5.1
in which T = twisting moment, or force which
acting at a distance of one inch
from the center of the shaft would
produce in it a torsional shearing
stress of S pounds per square inch,
d = diameter of shaft in inches,
S = torsional shearing stress in pounds
per square inch.
Expressing this formula in words we may say
that the cube of the diameter in inches multiplied
by the torsional shearing stress, and this product
divided by 5.1, gives the force which acting at a
distance of one inch from the center of the shaft
would produce in it the given torsional shearing
stress.
The twisting moment T equals, therefore, the
force Fl9 acting at a distance of one inch from
the center of the shaft, times 1 ; it also equals any
other force F exerting a twisting action on the
272
SHAFTS, BELTS AND PULLEYS 273
shaft multiplied by its distance from the center of
the shaft. The formula given can hence be written
5.1
vn which F = any force acting at a distance r from
the center of the shaft.
Transposing this formula to obtain the distance
from the center (r) at which a given force would
have to act to set up a torsional shearing stress S
in the shaft, we would have:
r "= KlXF.
The force which would be necessary to set up
a stress S in the shaft when acting at a given
distance would be :
= 5.1 Xr.
The diameter of shaft to resist a given force
acting at a given distance would be:
d -- ^F
r X5.1
The torsional shearing strength of ordinary
shafting is about 45,000 pounds to the square inch,
and of steel shafting from about 50,000 to 150,000
pounds, according to its quality; these figures
should be divided by five or six to give a safe
working stress.
The above formulas, however, are based on the
assumption that the force acting is of a purely
274 SELF-TAUGHT MECHANICAL DRAWING
twisting nature, as if a hand-wheel were put onto
the end of the shaft, and the tendency to bend the
shaft, caused by the pull of one hand, were counter-
acted by the push of the other hand. In the case
of a shaft actuated by a rocker arm, as sometimes
occurs in machines, the tendency to bend the shaft
caused by the push on the arm could be provided
for by using a somewhat higher factor of safety.
If the arm were placed at some distance from the
bearing, however, the tendency to bend the shaft
might be greater than the twisting effect.
The methods of calculating the size of shafts for
transmitting a given power, so as to take into
account both the twisting and bending effects pro-
duced by the pull of the belt are quite complicated,
and the beginner will ordinarily find it best to use
some of the empirical formulas for that purpose
which are intended to take into account both of
these effects.
The following rules by Thurston are considered
to afford ample margin for strength for shafts
which are well supported against springing:
To find the diameter of a cold rolled iron shaft to
transmit a given horse-power, multiply the horse-
power to be transmitted by 75, and divide the product
by the number of revolutions per minute that the
shaft is to make. The cube root of this quotient will
be the diameter of the shaft.
If the shaft is to be of turned iron, proceed as
above, except that the horse-power to be trans-
mitted is to be multiplied by 125 instead of 75.
This rule is "for head shafts, supported by bear-
SHAFTS, BELTS AND PULLEYS 275
ings close to each side of the main pulley or gear, so
as to wholly guard against transverse strain. " If
the main pulley is at a distance from the bearing,
the size of the shaft will need to be increased,
while for ordinary line shafting, with hangers 8
feet apart, the size may be reduced, figures of 90
for turned iron, and 55 for cold rolled iron shafting
being substituted for those given in the rule ; or, in
the case of shafting for transmission only, without
pulleys, figures of 62.5 for turned iron, and 35 for
cold rolled iron are substituted.
To find the horse-power which a given shaft will
transmit, multiply the cube of its diameter by the
number of revolutions per minute, and divide the
product by 125 for turned iron, or by 75 for cold
rolled iron.
For line shafting substitute the figures given by
90 and 55, respectively.
The horse-power which is being transmitted is
determined by multiplying the pull in pounds
which the belt exerts (or the push which the teeth
of the driving gear exert, if gears are used) by
the diameter of the pulley in inches (or the pitch
diameter of the gear in inches) and multiplying
this product, again, by the number of revolutions
per minute of the shaft; then divide this product
by 126,050, and the quotient gives the horse-power
transmitted.
Expressed as a formula this rule would be:
n P X D X N
n'^' 126,050
276 SELF-TAUGHT MECHANICAL DRAWING
in which P = pull on belt or push on gear teeth in
pounds,
D = diameter of pulley or pitch diameter
of gear in inches,
N = number of revolutions per minute of
pulley or gear.
Belts. — The theoretical horse-power which a belt
will transmit is equal to the pull which the belt
exerts in pounds, multiplied by its velocity in feet
per minute, and this product divided by 33,000.
The question then arises as to what is the allowable
stress to be put upon a belt.
A common rule of practice for ordinary belting
is that for single thickness belts the horse-power
transmitted equals the breadth of the belt in inches,
multiplied by its velocity in feet per minute, this
product being divided by 1,000. This rule assumes
a belt pull of 33 pounds per inch of width. Many
authorities, however, would allow a much higher
tension. The higher the tension, however, the
narrower the belt for a given horse-power, and
the greater the stretch, the more frequent the
necessity for relacing, and the shorter the life of
the belt.
Allowing 33 pounds tension per inch in width for
the thinnest commercial single belt, and allowing
the tensions for increased thicknesses given by a
large belt manufacturing concern, would give the
following formulas for the transmission capacities
of given belts:
SHAFTS, BELTS AND PULLEYS 277
Single belt, A inch thick, H.P. = Breadth^velocity.
Single belt, J inch thick, H.P. = Breadth X velocity.
800
Light double, # inch thick, H.P. = Breadth X velocity,
733
Heavy double, & inch thick, #. P. = Breadth X velocity.
687
Heavy double, A inch thick, H.P. = Breadth X velocity.
660
Heavy double, | inch thick, H.P. =Breadth X velocity.
550
Heavy double, if inch thick, H.P. = Breadth X velocity.
500
In these formulas the breadth of the belt is
understood to be in inches, and its velocity in feet
per minute, the letters H.P. meaning horse-power.
Transposing the above formulas to ascertain the
breadth of belt required to transmit a given power,
we would have :
Single belt, T\ inch thick, Breadth = H' R x 100°
Velocity
Single belt, i inch thick, Breadth =.H-P- x 80°
Velocity
Light double, U inch thick, Breadth = H" P' x 73B
Velocity
Heavy double, & inch thick, Breadth - H' P' x 687
Velocity
Heavy double, T\ inch thick, Breadth = H' P' x 66°
Velocity
Heavy double, | inch thick, Breadth =-H'P- x 55°
Velocity
Heavy double, ifinch thick, Breadth ^ H- P' x 50Q
Velocity
278 SELF-TAUGHT MECHANICAL DRAWING
These formulas are all for laced belts. A belt
made endless by being lapped and cemented or
riveted is considered to be nearly 50 per cent,
stronger than a laced belt, and is thus capable of
transmitting nearly 50 per cent, more power; or
the breadth of an endless belt to transmit a given
power would not need to be more than between
two- thirds to three-quarters of the breadth of a
laced belt. Metal fastenings are not considered to
make as strong a belt as lacings.
If the foregoing formulas had been made on the
basis of an allowable stress of 45 pounds for each
inch in width of a single belt, a figure which many
consider perfectly safe for a belt in good condition,
they would have shown the belts as being capable
of transmitting one-third more power than at 33
pounds stress per inch; to transmit a given power
a belt would then need to be not more than three-
quarters of the width.
It will be seen from these formulas that the
power transmitting capacity of a belt depends upon
its breadth (a wide belt allowing an increased
tension) or on its velocity. Increasing the width
of the belt without increasing the tension to corre-
spond would not give any increase of power trans-
mitting capacity, as the given tension would simply
be distributed over so much more pulley surface ;
but a tight belt means more side strain on shaft
and journal. Therefore, according to Griffin, from
the standpoint of efficiency, use a narrow belt under
low tension at as high a speed as possible. The de-
sired high speed is, of course, secured by simply
putting on large pulleys.
SHAFTS, BELTS AND PULLEYS 279
Speed of Belting. — The most economical speed
is somewhere between 4000 and 5000 feet per
minute. Above these values the life of the belt is
shortened ; ' ' flapping, " ' ' chasing, ' ' and centrifugal
force also cause considerable loss of power at higher
speeds. The limit of speed with cast iron pulleys
is fixed at the safe limit for the bursting of the
rim, which may be taken at one mile surface speed
per minute.
The formulas given for the horse-power trans-
mitted assume that the belt is in contact with just
one-half of the pulley ; or, in other words, that the
arc of contact is 180 degrees. If the arc of contact
is increased, as it might be in the case of a crossed
belt, until it becomes 240 degrees, or two- thirds of
the circumference of the pulley, it is stated that
the adhesion of the belt to the pulley, and conse-
quently the efficiency of the belt, will be increased
50 per cent. If, on the other hand, the arc of con-
tact should be reduced to 120 degrees, or one- third
of the circumference of the pulley, as might be the
case with open belts where the shafts were near
together, and the pulleys were very unequal in
size, the efficiency is stated to be only 60 per cent,
of what the formulas would show ; if the arc of con-
tact should be reduced to 90 degrees, the efficiency
is stated to be only 30 per cent.
From these percentages one can form a fairly
good idea of what percentage to allow for varying
arcs of contact. In most cases, however, it will
probably be correct enough to assume the arc of
contact to be 180 degrees.
In all cases of open horizontal belting the lower
280 SELF-TAUGHT MECHANICAL DRAWING
run of the belt should be made the working part,
so that the sag of the upper run will increase the
arc of contact.
In the location of shafts that are to be connected
with each other by belts, care should be taken to
secure them at a proper distance from one another.
It is not easy to give a definite rule what this dis-
tance should be. Some authorities give this rule:
Let the distance between the shafts be ten times the
diameter of the smaller pulley ; but while this is
correct for some cases, there are many other cases
in which it is not correct. Circumstances generally
have much to do with the arrangement; and the
engineer or machinist must use his judgment, mak-
ing all things conform, as far as may be, to general
principles. The distance should be such as to allow
a gentle sag to the belt when in motion. The Page
Belting Co. states that if too great a distance is
attempted, the weight of the belt will produce a
very heavy sag, drawing so hard upon the shafts
as to produce considerable friction in the bearings,
while at the same time the belt will have an un-
steady, flapping motion, which will destroy both
the belt and the machinery.
As belts increase in width they should be made
thicker. It is advisable to use double belts on
pulleys 12 inches in diameter and larger. If thin
belts are used at very high speed, or if wide belts
are thin, they almost invariably run in waves on
the slack side, or "travel" from side to side of the
pulleys, especially if the load changes suddenly.
This waving and snapping that occurs as the belts
straighten out, wears the belts very fast, and
SHAFTS, BELTS AND PULLEYS 281
frequently causes the splices to part in a very short
time, all of which is avoided by the employment of
suitable thickness in the belts. The Page Belting
Co. states that driving pulleys on which are to be
run shifting belts should have a perfectly flat sur-
face. All other pulleys should have a convexity in
the proportion of about -fj of an inch to one foot
in width. The pulleys should be a little wider than
the belt required for the work.
Pulley Sizes. — The sizes of pulleys to give a re-
quired speed, or the speed which will be obtained
with given pulleys may be readily found from the
fact that the product of the speed of the driving
shaft, in revolutions per minute, and the diameters
of all driving pulleys, on the main and on counter-
shafts, multiplied together, will be equal to the
product of the diameters of all driven pulleys and
the speed of the last driven shaft, in revolutions
per minute, multiplied together; so that if the size
of one driven pulley, for instance, is required, its
size may be found by dividing the product of the
speed of the driving shaft and all driving pulleys
multiplied together, by the product of speed of the
final driven shaft and the diameters of such driven
pulleys as are given, multiplied together. The re-
sult will be the required pulley size.
Example. — A shaft making 200 revolutions per
minute has mounted on it a pulley 18 inches in
diameter which belts onto a 6-inch pulley on a
countershaft. The countershaft has mounted on it
a 20-inch pulley which belts to a pulley on the
spindle of a machine which is to make 3000 revolu-
tions per minute. What size pulley will be required
on the spindle.
CAH?
rrv
282. SELF-TAUGHT MECHANICAL DRAWING
Placing the speed of the driving shaft, and the
sizes of all driving pulleys on one side of a vertical
line, for convenience sake, and the sizes of all
driven pulleys and the speed of the last driven
shaft (or spindle) on the other side, and letting x
represent the required size we would have:
Speed of shaft = 200
Pulley on shaft = 18
Driving pulley on
countershaft = 20
6 = Driven pulley on coun-
tershaft.
x = Required size of pulley
on spindle.
3000 = Speed of spindle.
i
Then 200 X 18 X 20 = 6 X x X 3000
= 200J<_18_X_20 _ _721000
6 X~3000 18,000
The diameter of the pulley on the spindle would
therefore have to be 4 inches. If this size had
been given, and the speed of the spindle had been
required, x might have been taken to represent the
required speed, when the same process would have
given the desired information.
Twisted and Unusual Cases of Belting. — It
frequently happens that, in transmitting power,
conditions present themselves in which ordinary
straight belting, either open or crossed, will not
serve the purpose, and recourse must be had to
some form of twisted belting, either quarter turn
belting or belting guided by idler pulleys. In the
following are given some of the principal con-
ditions.
Fig. 198 shows a quarter turn belt, by which
power can be transmitted from one shaft to another
at right angles to it. The condition necessary for
SHAFTS, BELTS AND PULLEYS
283
the successful working of this arrangement is that
the middle of the face of the pulley toward which
FIG. 198.— Arrangement of
Pulleys for Quarter-Turn
Belt.
FIG. 199. — Another Arrange-
ment for Transmitting
Power between Shafts at
Right Angles.
the belt is advancing shall be in line with the edge
of the pulley that the belt is leaving. An exami-
284 SELF-TAUGHT MECHANICAL DRAWING
nation of both the plan and elevation views will
make this clear.
While this is the simplest arrangement for this
purpose, it has several drawbacks. The edgewise
stress on the belt as it is leaving either pulley is
very severe on the belt. It also causes a consider-
able loss of contact with the pulley face, with
corresponding loss of power transmission capacity.
The edgewise stress also makes it necessary, if
durability is to be considered, to have the belt
relatively narrow. Incidentally, also, any reversal
of the motion will cause the belt to immediately
run off the pulleys.
Fig. 199 shows another arrangement for trans-
mitting power from one shaft to another at right
angles to it, which overcomes all of the objections
mentioned to the arrangement shown in Fig. 198,
but at the expense of a double length belt and an
extra pair of pulleys.
As shown in the illustration, A and B are tight
pulleys, and C and D are loose pulleys. The belt,
as it leaves the tight pulley A, passes down under
the loose pulley D, up over the loose pulley C, down
under the tight pulley B, and then up over the
tight pulley A, making a complete circuit. The
loose pulleys, it will be seen, revolve in an opposite
direction to the shafts on which they are mounted.
Fig. 200 shows an arrangement by which, by
employing loose guide pulleys, power may be trans-
mitted from one shaft to another so close to it as
to prohibit direct belting. If the main pulleys are
of the same size, and their shafts are in the same
plane, the guide pulleys may be mounted on a
SHAFTS, BELTS AND PULLEYS 285
single straight shaft at right angles to a plane
passing through the axes of the shafts on which
the main pulleys are mounted. If, however, the
main pulleys are of unequal size, as shown in the
illustration, the guide pulleys will have to be in-
clined to such an angle that the center of the face
FIG. 200. —Arrangement of Belt Transmission Using
Loose Guide Pulleys.
of the pulley toward which the belt is advancing
shall be in line with the edge of the pulley that
the belt is leaving, the same as in the case of the
quarter turn belt shown in Fig. 198.
It is not necessary that the shafts on which the
main pulleys are mounted be in the same plane ;
their direction may be such that their relation to
286 SELF-TAUGHT MECHANICAL DRAWING
each other is similar to that of those shown in
Fig. 198, or at any intermediate angle.
Again, if they are in the same plane, it is not
necessary that they should be parallel with each
other; they may be at any angle with each other.
Fig. 201 shows a case which is a modification of
Fig. 200. The main shafts are at right angles to
SHAFTS, BELTS AND PULLEYS
287
each other. The main pulleys, being of the same
size, permit the guide pulleys to be mounted on
a single shaft. This arrangement is a common
method of transmitting power around a corner.
Fig. 202 shows a case where the direction of the
shafts with regard to each other is the same as in
FIG. 202.— A Case where the Guide Pulleys would be Mounted
in an Adjustable Frame.
Fig. 198, but where shop conditions are such that
it is not practicable to bring the lower shaft under
the upper one to permit of belting by either of the
methods shown in Figs. 198 or 199. The guide
pulleys are, therefore, mounted on a frame which
can be raised or lowered in guides by means of an
adjusting screw, permitting of an easy adjustment
of the belt tension.
288 SELF-TAUGHT MECHANICAL DRAWING
Fig. 203 shows a case which is similar to Fig.
200 in that it permits the belting together of shafts
which are at angle to each
other, but accomplishes this
result by the use of only one
guide pulley. The shafts,
though at an angle to each
other, are in the same plane.
This, however, is not neces-
sarily so. The shafts may
be twisted around until they
are at right angles to each
other, as in Fig. 198. As
shown in Fig. 200, the belt
may be run in either direc-
tion as long as the shafts are
in the same plane; but as
shown in Fig. 203, it is nec-
essary that the belt should
be run in the direction in-
dicated by the arrows.
An examination of the en-
gravings will show that the
condition necessary for the
proper working of guide
pulleys is that the shaft on
which the guide pulley is mounted shall be at right
angles to a line drawn from the edge of the pulley
that the belt is leaving in its advance toward the
guide pulley, to the middle of the guide pulley
face.
FIG. 203.— An Arrange-
ment in Which but One
Guide Pulley is Used.
CHAPTER XVIII
FLY-WHEELS FOR PRESSES, PUNCHES, ETC.
IN a great many different classes of machinery,
the work that the machine performs is of a variable
or intermittent nature, being done, in the case, for
example, of punches and presses, during a small
part of the time required for the driving shaft or
spindle of the machine to make a complete revolu-
tion. If this work could be distributed over the
entire period of the revolution, a comparatively nar-
row belt would be sufficient to drive the machine;
but a very broad and heavy belt would otherwise be
necessary to overcome the resistance, if the belt
only be depended on to do the work. It is, of
course, in a sense, impossible to distribute the work
of the machine over the entire period of revolution
of the driving shaft of the machine, but by placing
a large, heavy-rimmed wheel, a fly-wheel, on the
shaft, the belt is given an opportunity to perform
an almost uniform amount of work during the
whole revolution. During the greater part of the
revolution of the driving shaft the power of the
belt is devoted to accelerating the speed of the fly-
wheel. During that brief period of the revolution
of the shaft when the work of the machine is being
done, the energy thus stored up in the fly-wheel is
given out at the expense of its velocity. The
289
290 SELF-TAUGHT MECHANICAL DRAWING
energy a fly-wheel would give out if brought to a
standstill would be (neglecting the weight of the
arms and hub, as the efficiency of the wheel depends
chiefly on the weight of the rim), expressed in
foot-pounds, equal to the weight of the rim in
pounds multiplied by the square of its velocity at
its mean diameter in feet per second, and this
product divided by 64.32, the same as in the case
of a falling body moving at the same velocity, as
explained in the section on mechanics.
Expressed as a formula this rule is :
= Wv* Wv2
2g 64.32
in which E = total energy of fly-wheel,
W = weight of fly-wheel rim in pounds,
v = velocity at mean radius of fly-wheel
in feet per second,
g = acceleration due to gravity = 32. 16.
If the speed of the fly-wheel is only reduced, the
energy which it would give out would be equal to
the difference between the energy which it would
give out if brought to a full stop, and that which
it would still have stored up in it at its reduced
velocity. Therefore, to find the energy in foot-
pounds which a fly-wheel will give out with an
allowable loss of speed, subtract the square of the
velocity of the rim in feet per second at its reduced
speed from the square of its velocity in feet per
second at full speed, multiply this difference by
the weight in pounds, and divide the product by
64.32. The result will give the loss of energy in
foot-pounds.
FLY-WHEELS 291
This long and cumbersome rule is expressed
much more simply by the formula:
ElS= ""64732"
in which El = energy, in foot-pounds, fly-wheel
gives out while speed is reduced
from Vi to v2 ,
Vi = speed before any energy has been
given out, in feet per second,
v2 = speed at end of period during which
energy has been given out, in feet
per second,
W = weight of fly-wheel rim in pounds.
This rule and formula may be transposed as fol-
lows : To find the weight of a fly-wheel to give out
a required amount of energy with an allowable loss
of speed, multiply the required amount of energy
in foot-pounds by 64.32, and divide the product by
the difference between the square of the velocity
of the rim, at its mean diameter, in feet per second
at full speed, and the square of its velocity in feet
per second at its reduced speed ; or, expressed as a
formula, using the same notation as above :
v i2 — v ?
When the mean diameter of the fly-wheel is
known, the velocity of the rim at its mean diameter
in feet per second will be
Diameter in feet X 3.1416 X rev, per minute
~60
It is evident that in designing a fly-wheel for a
292 SELF-TAUGHT MECHANICAL DRAWING
machine, there is an opportunity for a wide range
in the weight, from a wheel heavy enough, when
once it has been brought to its full speed, to do, by
means of the energy stored in it, the work without
assistance from the belt, the belt being only just
wide enough to restore the speed of the wheel in
time for the next operation, to a wheel where the
belt is wide enough to do the most of the work
directly, the stored energy in the fly-wheel merely
assisting it somewhat. Perhaps the best way would
be to have the wheel heavy enough so that its
stored energy could do the bulk of the work, the
belt assisting it, and at the same time have the
latter wide enough to quickly restore the speed of
the wheel, so that, in case its velocity should be
reduced beyond that calculated, there would be a
margin of available power in the belt.
Example. — Let it be required to design a fly-
wheel for a press to cut off one-inch round bar
steel, the press making 30 strokes per minute.
Soft steel having a shearing resistance of about
50,000 pounds per square inch, and a one-inch bar
having an area of cross-section of 0.7854 square
inch, the shearing resistance of the bar will be
50,000 X 0.7854 = 39,270 pounds, or practically
40,000 pounds. This resistance varies,, however,
during the process of shearing, being greatest near
the beginning of the cut, and decreasing as the
cutting progresses. In the case of a round bar it
could not decrease uniformly, because of the shape
of the cross-section. For the sake of getting the
decrease in resistance as nearly uniform as possible,
we will assume that the work of cutting off a one-
FLY-WHEELS 293
inch round bar is the same as the work of cutting
off a square bar of the same area ; though this may
not be quite exact, it would probably not be far
out of the way. The length of the sides of a square
of the same area as a given circle, is equal to the
diameter of the circle multiplied by 0.886. There-
fore, our equivalent square bar will be 0.886 of
an inch square. The mean resistance to cutting,
assuming that the resistance decreases uniformly
as the cutting progresses, would be 40,000 ^ 2 =
20,000 pounds. As the cutting operation continues
through a space of 0.886 of an inch, the power
required would be 20,000 X 0.886 = 17,720 inch-
pounds, or 1476.6 foot-pounds. Let us plan to have
the belt do one-fifth of the work of cutting direct-
ly, leaving four-fifths to be done by the stored up
energy of the fly-wheel. One-fifth of 1476.6 equals
295.3. Subtracting this from 1476.6 leaves 1181.3
foot-pounds to be supplied by the energy of the
fly-wheel. As a preliminary calculation let us find
what would have to be the weight of the wheel if
it were to be placed upon the crank-shaft, the shaft
which operates the plunger of the press. Assuming
the mean diameter of the fly-wheel rim to be 4
feet, the circumference would be 4 X 3.14 = 12.56
feet, and, as the shaft makes 30 revolutions per
minute, the velocity of the rim in feet per second
would be :
12.56 X 30 c 00 ,
— gg = 6.28 feet.
If we expect the fly-wheel to suffer a loss of,
say, 10 per cent, while doing its work, then its
velocity at its reduced speed will be 6.28 - 0.628 =
294 SELF-TAUGHT MECHANICAL DRAWING
5.65 feet. The weight of the fly-wheel to give out
1181. 3 foot-pounds under these conditions will then
be, according to the rule and formula already
given :
1181.3 X 64.32_ 75,981.2 75.981.2 _ 1 n
6.282-5.652 39.44-31.92 = 7.52
nearly.
A wheel weighing 10, 100 pounds would, of course,
be out of the question ; but as the energy increases
as the square of the velocity, the weight may be
very rapidly reduced by mounting the wheel upon
a higher-speeded secondary shaft, connected with
the crank-shaft by reducing gears. If the speed
of the secondary shaft is to the speed of the crank-
shaft as 6 to 1, the weight of the wheel, if the
mean diameter be kept the same, will need to be
only about one thirty-sixth of what it would need
to be if mounted on the crank-shaft. At thisjhigher
speed, however, it might be desirable to somewhat
reduce the diameter of the wheel. Let us assume
that the mean diameter be made 3 feet. If the
ratio of speeds is 6 to 1, the wheel will make 180
revolutions per minute, and the velocity of the rim
in feet per second will be :
3 X 3.14 X 180 ' 0 ,
To" =28-3 feet.
If the wheel suffers a loss of 10 per cent., its
velocity at its reduced speed will be 28.3 - 2.83 =
25.5 nearly. The weight of the wheel will then
be:
1181.3 X 64.32 75,981.2 Kn A ,
28.3 2- 25.5 2 = 15064^ = 5°4 P°Unds'
FLY-WHEELS 295
As a cubic inch of cast iron weighs 0.26 pound,
the wheel will contain . 504 -5- 0.26 = 1938 cubic
inches. The mean circumference of the rim in
inches will be 3 X 12 X 3.14 = 113 inches. The
cross-section of the rim will then be :
1938 - 113 = 17.1 square inches.
This would mean a rim about 4 by 4J inches.
The outside diameter of the wheel would then be
40 inches.
We planned to have the belt do one-fifth of the
work, and this we found to be 295.3 foot-pounds.
If the crank has a radius of li inch, the cutter will
have a stroke of 2J inches, and if the cutters over-
lap each other one-quarter of an inch at the end of
the stroke, the crank will have to swing through
an angle of about 54 degrees in order to make the
cutters advance the one inch necessary to cut off
the one-inch bar, as a simple lay-out will show.
The belt must then develop 295.3 foot-pounds while
the crank swings through 54 degrees. It will then
develop 295.3 + 54 = 5.5 foot-pounds, nearly, in one
degree, and in a complete revolution it will develop
5.5 X 360 = 1980 foot-pounds. As the press makes
30 strokes per minute, the belt will develop 30 X
1980 = 59,400 foot-pounds per minute. If a driving
pulley 18 inches in diameter is used, the belt speed
in feet per minute will be :
18 X 3.14 X 180 a ,
— ^— = 848 feet.
If a single thickness belt, one-inch wide, at
1000 feet per minute, transmits 33,000 foot-pounds
296 SELF-TAUGHT MECHANICAL DRAWING
per minute, the same belt at 848 feet per minute
will transmit TWo as much, or 33,000X0.848 =
27,984 foot-pounds. The width of belt necessary
to transmit 59,400 foot-pounds per minute at this
speed will then be 59,400 -*- 27,984 = 2.1 inches.
No account has so far been taken of the power
necessary to drive the machine itself. To allow
for this the belt should evidently be not less than
2i inches wide. A 3-inch belt would allow consid-
erable of a margin of safety, and further calculation
will show that such a belt would develop, during
about one- third of a revolution of the crank, the
amount of energy which the fly-wheel had lost, so
that, as the cutting operation takes about one-sixth
of a revolution, the fly-wheel would be running at
full speed for about one-half of a revolution of the
crank, previous to the beginning of the cut, pro-
vided that it had not suffered any greater reduction
of velocity than the 10 per cent, planned for.
If the press was employed doing punching the
same method of procedure would be employed in
the calculations, the area in shear in such a case
being equal to the circumference of the hole mul-
tiplied by the thickness of the plate. The end of a
punch . is usually made slightly conical or slightly
beveling, the effect in either case being to increase
the shearing action, and make the work of punch-
ing easier.
CHAPTER XIX
TRAINS OF MECHANISM
FOR obtaining high speeds without the use of
unduly large driving pulleys or gears, for securing
gain in power by sacrificing speed, for securing
reversal of direction, or for obtaining some par-
ticular velocity ratio between the driver and some
part of the mechanism, pulleys, gears, worm-gears,
or the like, may be substituted for direct acting
driving-mechanisms.
To Secure Increase of Speed. — Let a shaft making
100 revolutions per minute be required to drive the
spindle of a machine at 2000 revolutions per minute,
the pulley on the spindle being 3 inches in diam-
eter. If a direct drive were to be used, the pulley
on the shaft would have to be as many times greater
than the pulley on the spindle as 2000 is greater
than 100, or 20 times.
This would mean a pulley on the shaft 60 inches
in diameter. Practical considerations, such as the
weight of the pulley, size of hangers and the like,
would make such a pulley out of the question.
By interposing an intermediate countershaft be-
tween the first shaft and the spindle of the machine,
however, having pulleys of such size that the
product of the ratio of the pulley on the first shaft
and the one to which it is belted on the counter-
shaft, multiplied by the ratio of the second pulley
297
298 SELF-TAUGHT MECHANICAL DRAWING
on the countershaft and the pulley on the spindle
to which it is belted is equal to the ratio which
it is desired to have between the first shaft and
the spindle, the same speed may be secured by the
use of pulleys of convenient size. Thus, if the
ratio between the pulley on the first shaft and the
one on the countershaft is as 1 to 4, and the ratio
between the driving pulley on the countershaft
and the one on the spindle of the machine is as
1 to 5, the product of these two ratios, 1 to 4 and 1
to 5, is 1 to 20, and the arrangement will give the
FIG. 204.— Reversal of Direction Obtained by Crossed Belt.
required speed. The pulley on the spindle being 3
inches in diameter, the driving pulley on the coun-
tershaft will be 15 inches in diameter, and if the
driven pulley on the countershaft is 4 inches in
diameter the pulley on the first shaft to which it is
belted will be 16 inches in diameter, instead of 60
inches, as would be required with direct belting.
If the spindle of the machine, instead of being
driven were made the driver, as it would be if it
were the armature shaft of a motor, then this ar-
rangement would give gain in power with con-
sequent loss of speed.
To Secure Reversal of Direction.— In cases where
shafts are belted together, reversal of direction of
TRAINS OF MECHANISM
299
rotation is secured by simply using a crossed belt
instead of an open one, as shown in Fig. 204.
When gears are used, reversal of direction of rota-
tion follows as a natural condition of their meshing
together, as shown in Fig. 205. In order that the
two gears A and B shall rotate in the same direc-
tion, it is necessary to separate them slightly, and
interpose an intermediate gear, or idler, between
FIG. 205.— Relative Direc-
tion of Rotation in a
Pair of Gears.
FlG. 206.— Influence of Idler
on Direction of Rotation.
them as shown in Fig. 206. The rates of rotation
of A and B with regard to each other is not affected
by the idler gear, whether the idler be large or
small. That this is so may be seen by direct exam-
ination. If A is the driver, its circumference will
impart to the circumference of C its own rate of
motion, and C will in turn impart to B the same
rate of motion, which is the same as it would have
if in direct connection with A.
If, now, another idler be interposed between A
and B, making four gears in the train, A and B
will again rotate in opposite directions. From this
it will be seen that when a train is composed of an
300 SELF-TAUGHT MECHANICAL DRAWING
even number of gears, the first and last members
rotate in opposite directions ; but when the train is
composed of an odd number of gears, the first and
last members rotate in the same direction.
In Fig. 207 is shown the mechanism used in
engine lathes to secure either direct or reversed
motion, by having the working train consist of
either an even or an odd number of gears. In this
FIG. 207.— Principle of Turn- FIG. 208.— Principle of Com-
bler Gear. pound Idler.
arrangement A is a gear on the head-stock spindle,
and B is a gear on a stud below. Pivoted on the
axis of B is a triangular piece of metal, or bracket,
shown in dotted lines, which can be swung back
and forth by the handle E. Mounted on this
bracket are the idler gears C and D, C being con-
stantly in mesh with B, and D being in mesh with
C. When it is required that B shall rotate in the
same direction as A, the handle E is lowered until
C meshes with A. The working train then consists
TRAINS OF MECHANISM 301
of three gears, A, C and B, D being out of mesh with
A, revolving by itself, but not forming a part of
the working train. When it is desired that B shall
rotate in the opposite direction to A, the handle E
is raised until D meshes with A, C being thrown
out of mesh with it. The working train then con-
sists of four gears, A, D, C and B, and the desired
reversal is secured.
The Compound Idler. — It has been shown that
when a train consists of simple gears the relative
rates of rotation of the first and last members re-
main unchanged, regardless of the number or size
of the idlers that may be interposed. When it is
desired to secure a different rate of rotation be-
tween two members of a train than that which
they would have if meshing directly together, a
compound idler is used, as shown in Fig. 208. Such
a gear is used on many screw cutting lathes. For
cutting threads up to a certain number per inch
the screw cutting train consists of simple gears.
A compound idler may then be introduced into
the train, when without other change additional
threads may be cut. If with screw cutting trains
of simple gears a lathe will cut all whole numbers
of threads up to 13 threads per inch, then, by adding
a compound idler to the train, having its two steps
in the ratio of 2 to 1, threads from 14 to 26 per inch
(except odd numbers) may be cut with the same
gears as previously used for cutting up to 13 threads
per inch. If the compound idler forms an additional
member of the train, the reversal of direction of
rotation which would take place in the motion
of the lead-screw of the lathe may be taken care of
302 SELF-TAUGHT MECHANICAL DRAWING
by the reversing gears between the spindle of the
head-stock and the stud, previously described, and
shown in Fig. 207.
The Screw Cutting Train.— In Fig. 209 is shown
the screw cutting mechanism found on engine
lathes. The reversing mechanism shown in Fig.
FIG. 209. FIG. 210.
FIGS. 209 and 210.— Arrangement of Lathe Change Gearing.
207 is reproduced entire, and these gears — the gear
A on the lathe spindle, the gear B on the stud,
which is connected with A by the idlers C and D—
are all permanent gears. These gears are usually
on the inside of the head-stock as shown in Fig.
210. The stud reaches through the head-stock, and
on its outer end is the change gear F, connecting
with the change gear G on the lead-screw of the
lathe by means of the intermediate idler H. The
idler H is mounted on a slotted swinging arm as
shown, so as to allow of gears F and G being
TRAINS OP MECHANISM 303
replaced by others of such size as may be required
to cut the particular screw desired. The carriage of
the lathe, carrying the screw cutting tool, is driven
directly by the lead-screw. On large lathes this
screw is quite coarse, four threads per inch being
common, while on smaller lathes a finer thread is
used. The gear A on the spindle and the fixed gear
B on the stud are sometimes of the same size, and
sometimes of different sizes.
The problem met with in screw cutting is to find
what sizes change gears, F and G, must be used
so that the lead-screw shall drive the carriage along
one inch while the spindle of the lathe is making
a number of revolutions equal to the number of
threads to be cut per inch. Let us take as an
example the assumed case of a lathe in which the
lead-screw has 9 threads per inch, and in which
the number of teeth in the gear on the spindle is
to the number of teeth in the fixed gear on the stud
as 3 to 4; required the size of change gears to cut
23 threads per inch. Then, as the lead-screw has
9 threads per inch, the spindle of the lathe must
make 23 revolutions while the lead-screw is making
9 revolutions. The method used in a previous
chapter for obtaining the size of pulleys to give
required speeds will give us the solution of this
problem; if the speed of the first driving member
of the train, together with the number of teeth or
relative sizes of all other driving members be placed
on one side of a vertical line, and the speed of the
last driven member, together with the number of
teeth or relative sizes of all other driven members
be placed on the other side of the line, the product
304 SELF-TAUGHT MECHANICAL DRAWING
of the numbers on one side of the line multiplied
together will equal the product of the numbers on
the other side of the line multiplied together. The
spindle of the lathe is, of course, the first driving
member of the train, and the lead-screw is the
last driven member. As the spindle is to make 23
revolutions while the lead-screw makes 9 revolu-
tions, 23 will be the first number on the side of the
line on which the driving members are placed, and
9 will be the last number on the side of the line on
which the driven members are placed. Next, as
the ratio between the sizes of the driving gear on
the lathe spindle and the fixed gear on the stud
below which it drives is as 3 to 4, these numbers
will be placed against each other on opposite sides
of the line.
The ratio between the numbers of teeth or sizes
of the two change gears, F and G, whose sizes it
is required to find, being unknown, may be said to
be as 1 to the unknown number x. These numbers,
1 and 05, are now placed on their proper sides of
the line, and the problem appears as shown below.
The size of the idler gear H does not enter into the
question, because, as has been previously shown, a
simple idler gear does not affect the relative rates
of rotation of the gears between which it transmits
motion.
Speed of spindle 23
Ratio of size of spindle gear 3
4 to size of fixed stud gear.
Ratio of number of teeth
in change gear F 1 ! x to number of teeth in
change gear G
9 speed of lead-screw.
69 = ZGx
TRAINS OF MECHANISM 305
Multiplying together the numbers on both sides
of the line gives the equation 69 = 36x. It is evi-
dent that if 69 equals 36x, x must be equal to 69
divided by 36, or f|. The ratio between sizes of
the gear F and the gear G is then as 1 to ||.
Eliminating the fraction by multiplying both terms
of the ratio by 36 gives the ratio as 36 to 69. If,
then, F has 36 teeth, and G has 69 teeth, the lathe
will cut the required number of 23 threads per
inch.
In Fig. 211 is shown how a compound idler gear
is sometimes used in a screw cutting train. The
FIG. 211.— Compound Gearing.
change gear G and the idler H have long hubs on
one side. When it is desired to cut finer threads
than what the gears E and G with the idler H will
give, H and G, are put on with the long hubs
toward the lathe, throwing them out of line with
E. The gear E then meshes into the large step of
7, the small step of / meshes into H, and H meshes
306 SELF-TAUGHT MECHANICAL DRAWING
into G. The ratio between the large and the small
steps of / must then be taken into account in the
calculation. For cutting the coarser threads H and
G are put on with the short hubs toward the lathe,
bringing them into line with E. The idler / is also
turned over, so that its large step is on the outside
and out of line with E and H. It is then swung
back out of the way.
When the gearing is fully compounded the two
gears at / are separate from each other but keyed
together on the same stud and mounted in the
same manner as shown in Fig. 211. By varying
the sizes of these gears, almost any screw thread
may be cut within reasonable limits. In this case,
of course, there are four gears to be determined in
our calculations. Simplified rules are given in the
following for this case, as well as for the regular
simple trains.
Large lathes are provided with change gears for
cutting threads from about 2 to about 20 threads
per inch, smaller lathes being provided with gears
for cutting from about 3 or 4 to 40 or 50 threads per
inch, in either case including a pair of gears for
cutting 11J threads per inch, this being the stand-
ard thread for iron pipes from one to two-inch sizes
inclusive. The smaller lathes would also naturally
be provided with gears for cutting 27 threads per
inch, this being the number of threads on i-inch
iron pipes.
Simplified Rules for Calculating Lathe Change
Gears. — The following rules for calculating change
gears for the lathe have been published by Ma-
chinery (Reference Series Book No. 35, Tables
TRAINS OF MECHANISM 307
and Formulas for Shop and Draftingroom) , and
are here given because of their concise form and
simplicity.
Rule 1. — To find the "screw-cutting constant*' of
a lathe, place equal gears on spindle stud and lead-
screw; then cut a thread on a piece of work in the
lathe. The number of threads cut with equal
gears is called the " screw-cutting constant " of
that particular lathe.
Rule 2. — To find the change gears used in simple
gearing, when the screw-cutting constant as found
by Rule 1, and the number of threads per inch to
be cut are given, place the screw-cutting constant
of the lathe as numerator and the number of threads
per inch to be cut as denominator in a fraction, and
multiply numerator and denominator by the same
number until a new fraction is obtained represent-
ing suitable numbers of teeth for the change gears.
In the new fraction, the numerator represents the
number of teeth in the gear on the spindle stud,
and the denominator, the number of teeth in the
gear on the lead-screw.
Rule 3. — To find the change gears used in com-
pound gearing, place the screw-cutting constant as
found from Rule 1 as numerator, and the number
of threads per inch to be cut as denominator in a
fraction ; divide up both numerator and denomi-
nator in two factors each, and multiply each pair
of factors (one factor in the numerator and one in
the denominator making a pair) by the same num-
ber, until new fractions are obtained, representing
suitable numbers of teeth for the change gears.
The gears represented by the numbers in the new
308 SELF-TAUGHT MECHANICAL DRAWING
numerators are driving gears, and those in the
denominators driven gears.
Two examples, showing the application of these
rules, will be given in the following.
Example 1. — Assume that 20 threads per inch are
to be cut in a lathe having a "screw-cutting con-
stant," as found by the method explained in Rule
1, equal to 8. The numbers of teeth in the avail-
able change gears for this lathe are 28, 32, 36, 40,
44, etc., increasing by 4 up to 96.
By applying Rule 2, we have then :
S_ _8_X_4 = 32
20 == 20 X 4 80
By multiplying both numerator and denominator
by 4 we obtain two available gears having 32 and
80 teeth. The 32-tooth gear goes on the spindle
stud and the 80-tooth gear on the lead-screw. It
will be seen that if we had multiplied by 3 or by 5
instead of by 4, we would not have obtained avail-
able gears in both numerator and denominator, as
8X3 would have given 24 and 20 X 5 would have
given 100, both of which gears are not in our given
set of gears. The proper number by which to
multiply can be found by trial only.
Example 2. — Assume that 27 threads per inch are
to be cut on the same lathe as assumed in Example 1.
In this case the calculation must be made for
compound gearing, as so fine a pitch could not be
cut by simple gearing in this lathe. By applying
Rule 3 we have :
_8_ _2_>^4 (2 X 20) X (4 X 8) = 40 X 32
27 : = 3 X 9 "(3~X 20) X (9 X 8) = 60 X 72
TRAINS OF MECHANISM
309
The four numbers in the last fraction give the
numbers of teeth in the required gears. The gears
in the numerator (40 and 32) are the driving gears,
and those in the denominator (60 and 72) are the
driven gears.
It makes no difference which one of the driving
gears is placed on the spindle stud or which one of
the driven gears is placed on the lead-screw.
Back-Gears.— Nearly all engine lathes and many
other machine tools are provided with a set of re-
c =
FIG. 212.— Principle of Back -Gearing.
ducing gears, called back-gears, by means of which
double the range of speeds that can be obtained by
direct driving may be given to the spindle of the
machine. Fig. 212 illustrates such a set of gears,
and the method of applying them to the machine.
The large gear A is fastened to the spindle of the
machine, but the cone pulley, with the gear B
attached to it, is loose on the spindle. The back-
310 SELF-TAUGHT MECHANICAL DRAWING
gear shaft with gears C and D is mounted in
brackets on the back side of the head-stock, and
is provided with eccentric bearings, by means of
which the gears on it can be thrown into or out of
mesh with the gears on the head-stock spindle.
When direct driving is desired, the back-gears are
thrown back, out of the way, and the cone pulley
and the large gear are clamped together by means
of a screw pin or stud passing through the gear
into the cone. They then revolve together as one
piece.
Let us assume the case of a lathe having a cone
with four steps, the largest step being 6 inches in
diameter, and the smallest 4 inches in diameter,
with the intermediate steps in proper proportion.
If the cone pulley on the countershaft is of the
same size as the one on the spindle, then, if the
countershaft runs 300 revolutions per minute, direct
driving will give about the following speeds to the
spindle: 450, 345, 260 and 200. Let it now be
required to find the sizes of gears to be used so
that with the back-gear driving, a proportionately
slower rate of speeds may be obtained. We may
solve the problem by giving to the gears some
arbitrary sizes, and finding what speeds such sizes
will give, and then modify these sizes until the
required speeds are obtained. For trial purposes
let us make the pitch diameter of the gear A the
same as the diameter of the large step of the cone
pulley, or 6 inches, and the pitch diameter of the
gear B the same as the diameter of the small step
of the cone pulley, or 4 inches. Arranging driving
and driven members on opposite sides of a vertical
TRAINS OF MECHANISM 311
line, the speed of the first driving member of the
train, the countershaft, being 300, the required
speed of the last member, the lathe spindle, being
represented by x, and having the belt on the largest
step of the countershaft cone so as to obtain the
highest speed with back-gears, gives an arrange-
ment of the case as below. The sizes of the back-
gears are the same as those on the lathe spindle,
the gear C being 6 inches in pitch diameter, and
the gear D 4 inches in pitch diameter.
Speed of countershaft 300
Pulley on countershaft 6
Gear B on lathe 4
Back-gear D 4
4 Pulley on lathe
6 Back -gear C
6 Gear A on lathe
x Speed of spindle
28,800= 144s
From this it is seen that with the sizes of the
gears as above, the highest speed with back-gears
would be the same as the lowest speed without
the back-gears. This, of course, would be useless
duplication of speeds.
For another trial we. will make the sizes of the
gears B and D each 3J inches in pitch diameter.
The calculation then becomes:
Speed of countershaft 300
Pulley on countershaft 6
Gear B on lathe 3.5
Back-gear D 3,5
4 Pulley on lathe
6 Back-gear C
6 Gear A on lathe
x Speed of spindle
) nearly.
312 SELF-TAUGHT MECHANICAL DRAWING
A speed of 153 revolutions per minute for the
fastest back-gear speed follows quite regularly the
series of speeds which the direct drive gives.
Instead of using the pitch diameters of the gears
in making the calculations the number of teeth
which the gears would have, the pitch being first
decided on, might be used. In this manner it is
possible to make slight changes in the diameters of
the gears without bringing troublesome fractions
into the calculations.
Many lathes and other machine tools have trains
of mechanism much more complicated than any
here shown, but the method of procedure here
outlined can be applied to all of them.
CHAPTER XX
QUICK RETURN MOTIONS
IN a large class of machinery the work is done
during the forward motion of a reciprocating part;
the return of the part to its starting point is then
a question of time. The quicker the part can be
returned to its starting point, the more efficient
becomes the machine. When the stroke is long, as
in the case of the bed of an iron planer for large
work, this rapid return motion is usually obtained
by means of shifting the driving belt onto a return
pulley so arranged that a higher ratio of speed is
procured; but in other cases, where the recipro-
cating motion is shorter, and the stroke is actuated
by means of a crank, the actuating mechanism is
made such that the crank gives a slow forward
and a quick return motion to the reciprocating
part. Iron planers for small work, shapers, and
the like, and some classes of engines and pumps,
use such quick return motions. Below are described
the principal devices used for such purposes.
Fig. 213 shows a method of securing a quick
return by having the axis of the crank outside of
the path of the reciprocating end of the connecting-
rod. Let A be a crank, the crank-pin of which, a,
acting upon the connecting-rod B represented by
the heavy line, causes the block b to move back and
313
314 SELF-TAUGHT MECHANICAL DRAWING
forth in the path CD. When the crank is in the
position shown the block is at the extreme left of
its stroke, the connecting-rod and crank being in
the same straight line, the center line of the con-
necting-rod coinciding with the axis of the crank.
As the crank swings downward, the block b is
driven to the right; but an examination . of the
illustration will show that the crank must make
FIG. 213.— Simple Quick Return Motion.
more than a half revolution before it again forms
a straight line with the connecting-rod, which it
will do when the block has reached its extreme
position to the right. As, therefore, the block
makes its movement to the right while the crank
is swinging through the lower angle included be-
tween these two positions, and as it makes its
return stroke while the crank is swinging through
the upper angle included between these same two
positions, the time of the forward stroke of the
block -will be to the time of its return stroke as
the lower angle is to the upper angle.
QUICK RETURN MOTIONS 315
The upper angle being the smaller of the two,
the block has a quick return motion. To secure
ease of motion to the block as it starts on its stroke
to the right, the angle abC, the angle which the
connecting-rod makes with the path of the block,
should not be more than about 45 degrees.
To design a quick return motion of this type, lay
out a horizontal line ab, Fig. 214, and on it mark
off cb equal to the required length of stroke. From
c draw the line cd of indefinite length at such an
fl b
FIG. 214.— Lay-out of Quick Return Motion in Fig. 213.
obliquity that the angle acd shall not be more than
45 degrees. From b draw the line be at the angle
required to give the desired quick return. The
intersection of these two lines at /will be the axis
of the crank. The length bf will be seen by re-
ferring back to Fig. 213 to be equal to the length
of the crank plus the length of the connecting-rod.
The length of cf will be seen to be equal to the
length of the connecting-rod minus the length of
the crank. If in a given case the length cb is
made 12 inches, and cf is found to be 10 and bf 21
inches, which they would be if the angles were as
316 " SELF-TAUGHT MECHANICAL DRAWING
shown in Fig. 214, then, letting x represent the
length of the connecting-rod and y the length of
the crank, we would have x + y = 21 inches, and
x - y = 10 inches. Adding the left-hand and the
right-hand members, respectively, of these two
equations, we would have x + y + x-y = 21 + 10
= 31 inches. As + y - y= 0 we may eliminate
these expressions, and the equation will read 2x =
31 inches, and x, the length of the connecting-rod,
will thus be 15J inches. The length of the crank
will then be 21 inches (the length of bf) minus 15J
inches, or 5J inches.
It will be seen that if the length of the stroke is
made variable by having the crank-pin, a, adjust-
able to different positions on the crank A, Fig. 213,
the difference between the time of the forward
and of the return stroke of the sliding block b will
be lessened, because the two positions which it
will occupy at. the extremes of its stroke will be
nearer together, and the lower and upper angles
which the crank passes through in giving to the
block its forward and return movements will be
more nearly equal.
Fig. 215 shows a quick return motion device
especially adapted to cases where the horizontal
space is limited, and which is much used on shapers.
The illustration shows a shaper in outline. The
ram of the shaper is given its forward and return
motion by means of the rocking arm A, which
swings on a fulcrum at B. The rocking arm is
given its motion by means of a crank-pin on the
disk C, the pin engaging in a sliding block which
travels in a slot in the arm A.
QUICK RETURN MOTIONS
317
Let BC and BD, Fig. 216, represent the extreme
positions of the rocker arm A. Draw the lines OF
and OG from the center of the crank disk at O at
right angles to BC and BD. It is evident that in
order that the crank, on its upper sweep, shall
FIG. 215.— Diagram of Quick Return Arrangement
in a Shaper.
move the rocker arm from C to Z), it must move
through the arc FAG, while to return the arm
from D to C, on its lower sweep, it must move only
through the lower arc FG. The time of the return
motion will therefore be to the time of the forward
motion as the lower arc or angle FG is to the arc
318 SELF-TAUGHT MECHANICAL DRAWING
or angle FAG. If the crank is shortened so as to
give a shorter stroke to- the ram of the shaper,
then the rocker arm will swing through a smaller
angle, as from H to /, and lines drawn from 0 at
FIG. 216.— Diagram of Speed Ratios in Shaper Motion.
right angles to HB and IB will be more nearly in a
straight line than OF and OG. There will, there-
fore, be less difference between the time of forward
and return motions on short strokes than on long
ones.
QUICK RETURN MOTIONS
319
The Whitworth Quick Return Device.— Let A,
Fig. 217, be a slotted arm revolving on its axis at
B. Above A is the driving crank C, having a pin
engaging in the slot at the left in the arm A. The
slot at the right in the arm A is provided for an
adjustable stud which drives the reciprocating
parts, through the medium of the connecting-rod
_D
FIG. 217. -Whitworth Quick Return Motion.
D. It will be seen that, as shown, the connecting-
rod is at the extreme right of its motion, forming
as it does a straight line with the revolving arm
A, which latter is at the same time at right angles
with the center line cd. It will be seen that in
order that the arm A may move through half a
revolution so as to bring the connecting-rod to the
extreme left of its motion, it will be necessary for
the actuating crank C to revolve either through the
320 SELF-TAUGHT MECHANICAL DRAWING
upper angle x or through the lower angle y, so as
to form again the same angle with the center line
cd, but at the right of it, as it is now shown form-
ing with it at the left. The forward and return
motions will, therefore, be to each other as the
angle x is to the angle y. To design a quick return
motion of this type it is, therefore, necessary to
first lay out the angles x and y of such relative
sizes that x is as many times greater than y as the
time of the forward motion is to be greater than
the time of the return motion, having them, of
course, central on the line cd. The distance apart
of the f ulcrums of the crank C and of the revolving
arm A will be partly determined by the sizes of
their shafts. The location of the crank-pin, de-
termining the length of the crank, will then be at
the intersection of the horizontal center line of the
revolving arm A with the dividing line ef between
the angles x and y. The length of the crank must,
of course, be sufficient so that the crank pin will
swing under the hub of the arm A, and the length
of the crank-pin slot in A must be. sufficient for
the motion of the pin relative to the arm.
It will be noticed that, unlike the two preceding
quick return devices, varying the stroke of the
reciprocating parts does not alter the relative time
of the forward and return motions ; for such change
does not affect the angles x and y upon which the
time of the forward and return motions depends.
If, however, the length of the crank C is varied,
then the angles x and y are altered, and the time
of the forward and return motions will be affected.
It will be seen upon examination that with the
QUICK RETURN MOTIONS
321
construction shown the revolving arm A must be
made in two parts, one at each end of its shaft, in
order to avoid interference of the parts of the
mechanism with one another as they revolve. This
trouble is overcome by replacing the crank C with
a crank disk which fits over and revolves upon a
fixed stud or hub large enough to receive the stud
at B upon which the arm A revolves.
The Elliptic Gear Quick Return. — If two ellipses
of equal size, Fig. 218, having foci at w and x and
FIG. 218.— Quick Return Motion by Means of Elliptic Gears.
at y and z, be placed in contact with each other
with their long diameters forming a continuous
straight line as shown; then if the ellipses are
caused to, revolve freely upon their correspond-
ing foci, w and y, they will roll upon each other
perfectly, without slipping. From the nature
of an ellipse as shown by its construction with a
thread and pencil (see Chapter III, Problem 13) it
will be seen that if the ellipse at the left were
being formed in this manner and the pencil were
at D, the intersection of the circumference of the
322 SELF-TAUGHT MECHANICAL DRAWING
ellipse with the long diameter, the length of the
thread would be equal to the sum of the distances
wD and Dx. But the distance Dx is the same as
the distance Dy\ therefore, the length of the thread
would be equal to the distance wy, the distance
between the foci upon which the ellipses are re-
volving. If, now, the ellipses are revolved until
the points A and B, vertically over the foci x and
y, are in contact with each other, the sum of the
distances wA and By will be equal to the distance
between the foci w and y, for their sum is equal to
the length of the thread, and the length of the
thread is equal to wA plus Ax, and Ax is equal to
By, as points A and B are both vertically over the
foci of the ellipses. In a similar manner any pair
of points may be selected on the two ellipses equally
distant from the point D. The distance from the
point on the ellipse at the left, to the focus w, will
be equal to the length of the thread at the left of
the pencil, and the distance from the point on the
ellipse at the right, to the focus y, will be equal to
the length of the thread at the right of the pencil,
and their sum will be equal to the distance between
the foci w and y. This distance between the foci
w and y will be seen on further examination to be
equal to the long axis of the ellipse. This property
of the ellipse has been taken advantage of to secure
a quick return motion to a reciprocating part of a
machine. If in Fig. 218 the two ellipses represent
the pitch lines of elliptic gears; with the gear at
the left as the driver with a uniform motion, the
one at the right will have an ununiform motion.
If, now, a crank is mounted on the same shaft as
QUICK RETURN MOTIONS 323
the driven elliptic gear, the crank having its center
line at right angles to the long axis of the ellipse,
and this crank actuates a sliding block back and
forth in the direction of the center line of the two
gears, then this block will have a slow motion in
one direction, and a quick motion in the other
direction. If, now, the gears are revolved from the
position in which they are shown until A and B
are in contact, the gear at the right will have made
a quarter of a revolution and the sliding block will
be at the extreme right of its stroke; but while
this gear has made a quarter of a revolution, the
driving gear has revolved through the angle AwD
only. If, now, the gear at the right is revolved
another quarter of a turn, the points E and F will
be in contact, and the crank will be directed ver-
tically upward. The driving gear will, however,
have revolved through the angle AwF. The forward
and return motions of the sliding block will, there-
fore, be to each other as the angle AwF is to the
angle AwD. In designing a pair of elliptic gears,
therefore, the first thing to do is to determine the
size of the angle Awx. To find the distance be-
tween the foci w and x first lay out on a large scale
a triangle similar to the triangle Awx. Then the
sum of its hypothenuse and the perpendicular will
be to the length of its base as the sum of wA and
Ax (the long axis of the ellipse) is to wx, the dis-
tance between the foci of the ellipse. The length
of the short axis may then be found by reversing
Problem 13, Chapter III. The problem may be
solved even more accurately by the rules given for
the solution of right-angled triangles. The length
324 SELF-TAUGHT MECHANICAL DRAWING
of wA will be to Ax as 1 is to the sine of the angle
Awx. Dividing the long axis of the ellipse into
two parts in this proportion gives the length of wA
and Ax. The length of wx will then be equal to
the length of Aw multiplied by the cosine of the
angle Awx. Then to find the short axis of the
ellipse, divide the distance wx into two equal parts
and construct the triangle wgh. The length wh
will be half of the distance between the foci, and
the length of wg will be half of the long axis. The
length gh, half of the short axis, may then be found.
Calculations made in this manner give the follow-
ing proportions to ellipses for quick return ratios
as indicated in the first column :
Ratio of Forward
to Return Motion.
Long Axis.
Short Axis.
Distance Between
Foci.
2 to 1
1.000
0.963
0.268
2* to 1
1.000
0.936
0.351
3 to 1
1.000
0.910
0.414
4 to 1
1.000
0.860
0.509
5 to 1
1.000
0.817
0.577
There appear to be two difficulties with elliptic
gearing. The first is that if a high quick return
ratio is attempted, so as to make considerable dif-
ference between the long and the short axes, the
obliquity of the action of the teeth upon each
other, and the consequent great amount of friction
between the teeth as they come together, becomes
so great as to be troublesome. This may, to a con-
siderable extent at least, be overcome by using a
train of gears, each gear but slightly elliptic, in
place of one pair of decidedly elliptic form. Thus
QUICK RETURN MOTIONS 325
a train of three gears having their long and short
axes in the proportion required to give a quick
return of 3 to 1, with one pair of gears, will give
a quick return of 9 to 1. If three gears of the 4 to
1 proportion are usad, a quick return of 16 to 1
will result.
The second difficulty is that of correctly cutting
the teeth. To work properly, the teeth should be
cut on a machine having a special elliptic gear
cutting attachment, otherwise the gears are likely
to be expensive and unsatisfactory. Such an ellip-
tical gear cutting arrangement is described, and
the subject of elliptic gearing is quite fully dis-
cussed, in Grant's treatise on gearing. Not being
within the territory of this elementary treatise on
machine design, the subject cannot here be dealt
with in detail.
INDEX
Accelerated motion cams, 176
Acceleration of falling bodies,
143
Acme standard screw thread,
253
Addendum of gear teeth, 193
Aluminum, strength of, 162
Angle, definition of, 10
Angle of cone clutches, 271
Angle, to bisect an, 17
Angles, laying out, 118
Areas of plane figures, 92
A. S. M. E. standard machine
screws, 258
Assembly drawings, 52
B
Back gears, 309
Beams, cross-sections of, 156
Beams, strength of, 159
Belt for reversal of motion,
crossed, 298
Belting, horse-power of, 277
Belting, speed of, 279
Belting, twisted and unusual
cases of, 282
Belts, 276
Belts, endless, 278
Belts, laced, 278
Belts, width and thickness of,
277
Bending, shape of parts to
resist, 155
Bending strength of beams,
159
Bevel gearing, calculating,
230
Bevel gears, 202
Blue printing, 78
Bolt heads, table of United
States standard, 246
Bolts, studs and screws, 243
Bolts to withstand shock,
248
Brass, strength of cast, 162
Brass wire, strength of, 158
Broken drawings of long ob-
jects, 73
Cam curve for harmonic mo-
tion, 181
Cams, comparison between
uniform motion and accele-
rated motion, 183
Cams for high velocities, 175
Cams, general principles, 164
Cams with grooved edge, 172
Cams with pivoted follower,
167
Cams with positive return,
double, 173
Cams with reciprocating mo-
tion, 171
Cams with roller follower,
168
Cams with straight follower,
165
Cams with uniform motion,
165
Cams with uniformly accele-
rated motion, 176
Cap screw sizes, 248
327
328
INDEX
Case for drawing instru-
ments, 4
Cast iron, strength of, 157
Castings, stresses in, 162
Change gears, for screw cut-
ting, 302
Check or lock nuts, 248
Chord of circle, definition of,
12
Circle, area and circumfer-
ence of, 92
Circle, area of, 83
Circle, circumference of, 80
Circle, definition of, 11
Circle, to find center of a,
19
Circles, circumscribed and in-
scribed, 20
Circles, concentric, 10
Circles in isometric projec-
tion, 48
Circular pitch, 205
Circular sector, area of, 93
Circular segment, area of, 93
Clamp coupling, 262
Clutches, friction cone, 269
Clutches, friction disk, 266
Clutches, toothed, 265
Compasses, 3
Complement angle, definition
of, 11
Composition of forces, 120
Compound idler gear, 301
Compound gearing for screw
cutting, 305
Compression of machine
parts, 154
Compressive strength of ma-
terials, 158
Concentric circles, 10
Cone and cylinder intersect-
ing, 44
Cone clutches, angle of, 271
Cone clutches, friction, 269
Cone pulleys, 239
Cone pulleys, method of lay-
ing out, 242
Cone, surface development
of a, 40
Copper, strength of cast, 162
Cosecant of an angle, 102
Cosine of an angle, 101
Cosines, table of, 105
Cotangent of an angle, 102
Cotangents, table of, 107
Coupling, Hooke's, 263
Couplings, 259
Couplings, clamp, 262
Couplings, flange, 260
Crank motion, quick return,
313
Cross-sectioning device, 7
Cross-sections of beams, 156
Cube, projections of a, 39
Cube root, 82
Cube, volume of, 94
Cutting screw threads, gear-
ing for, 302
Cylinder and cone, intersect-
ing, 44
Cylinder, volume of, 94
Cylinders, intersecting, 43
Cycloid, definition of, 15
Cycloid, to draw a, 27
Cycloidal gear teeth, approx-
imate shape of, 209
D
Dedendum of gear teeth, 193
Definitions of terms, 10
Degree, definition of, 96
Detail drawings, 53
Diametral pitch, 207
Differential pulleys, 134
Disk clutches, friction, 266
Dimensions on drawings, 56
Double cams with positive
return, 173
Drawings, assembly, 52
Drawing board, 1
Drawings, classes of lines on,
55
Drawings, detail, 53
Drawings, dimensions on, 56
Drawing instruments, 1
Drawing paper, 8
INDEX
329
Drawing pens, the use of, 7
Drawings, sectional views on,
66
Drawings, working, 50
E
Efficiency of screws, 253
Elevation, definition of, 33
Ellipse, area of, 95
Ellipse, definition of, 14
Ellipse, to draw an, 21
Elliptic gear quick return
motion, 321
Elliptic gear return motion,
table for lay-out of, 324
Energy and work, 146
Energy of fly-wheel, 290
Engines, horse-power of
steam, 81
Epicycloid, definition of, 15
Epicycloidal gearing, 191
Epicycloidal and involute
systems of gears, compari-
son between, 199
Erasing shield, 9
Factor of safety, 151
Falling bodies, 142
Finishing marks on drawings,
63
Flange couplings, 260
Foot-pound, definition of, 146
Force of a blow, 147
Forces, oblique, 124
Forces, opposing, 125
Forces, parallel, 123
Forces, resultant of, 120
Forces, resolution of, 123
Formulas, algebraic, 79
Formulas, transposition of, 88
Friction cone clutch, horse-
power of, 270
Friction cone clutches, 269
Friction disk clutch, horse-
power of, 267
Friction disk clutches, 266
Fulcrum, definition of, 126
Fly-wheel, energy of, 290
Fly-wheels for presses,
punches, etc., 289
Fly-wheel, weight of, 291
Gear, compound idler, 301
Gear, influence of the idler,
299
Gear quick return motion,
elliptic, 321
Gear teeth, approximate
shape of, 209
Gear teeth, laying out invo-
lute, 210
Gear teeth, Lewis' formula
for strength of, 218
Gear teeth, pitch of, 205
Gear teeth, proportions of,
207
Gear teeth, strength of, 213
Gear teeth systems, compari-
son between, 199
Gear tooth, hunting, 209
Gear tooth terms, definitions
of, 193
Gear, tumbler, 300
Gearing, back, 309
Gearing, calculating bevel,
230
Gearing, calculating dimen-
sions of, 222
Gearing, calculating spur,
222
Gearing, calculating worm,
234
Gearing, epicycloidal, 191
Gearing for reversal of direc-
tion of motion, 299
Gearing for screw cutting,
302
Gearing, general principles
of, 190
Gearing, worm, 204
Gears, bevel, 202
330
INDEX
Gears, interference in in-
volute, 198
Gears, involute, 196
Gears, knuckle, 190
Gears, method of drawing, 68
Gears, proportions of, 213
Gears, shrouded, 201
Gears, speed ratio of, 220
Gears, twenty degree invo-
lute, 201
Gears with radial flanks, 195
Gears with strengthened
flanks, 195
Geometrical problems, 17
Grooved edge cams, 172
Guide pulleys for belts, 285
Instrument case, 4
Involute and epicycloidal sys-
tems of gears, comparison
between, 199
Involute, definition of, 15
Involute gears, 196
Involute gears, interference
. in, 198
Involute gear teeth, laying
out, 210
Involute gears, twenty de-
gree, 201
Involute rack teeth, modified
form of, 197
Involute, to draw an, 27
Iron wire, strength of, 158
Isometric projection, 48
Harmonic motion cam curve,
181
Helix, to draw a, 47
Heptagon, area of, 94
Hexagon, area of, 94
Hexagon, definition of, 14
Hexagon, to draw a, 19
Hoisting pulleys, 132
Hooke's coupling or universal
joint, 263
Horse-power, 149
Horse-power of belting, 277
Horse-power of friction cone
clutch, 270
Horse-power of friction disk
clutch, 267
Horse-power of shafting, 274
Horse-power of steam en-
gines, 81
Hunting tooth, 209
Hypocycloid, definition of, 15
Hypotenuse, definition of, 98
I
Idler gear, compound, 300
Idler gear, influence of the,
299
Inclined .plane, 136
K
Kirkaldy's tests on strength
of materials, 157
Knuckle gears, 190
Lathe back gearing, 309
Lathe change gears, 302
Lathe change gears, simpli-
fied rules for calculating,
306
Levers, 125
Levers, compound, 128
Lewis' formula for strength
of gear teeth, 218
Line, definition of, 10
Line, to bisect a, 17
Lines on drawings, classes of,
55
Lock or check nuts, 248
M
Machine parts, shape of, 154
Machine screws, 257
Machine steel, strength of,
158
INDEX
331
Mechanics, elements of, 120
Materials, indicating, 72
Mechanism, trains of, 297
Metric screw thread, form
of, 256
Minute, definition of, 97
Moment, twisting or torsion-
al, 272
Motion, Newton's laws of,
139
N
Newton's laws of motion, 139
Nuts, check or lock, 248
Nuts, table of United States
standard, 246
0
Oblique-angled triangles, 114
Octagon, area of, 94
Octagon, definition of, 14
Octagon, to draw an, 20
Oldham's coupling, 263
Oscillation, center of, 141
Paper, drawing, 8
Parallel forces, 123
Parabola, definition of, 15
Parabola, to draw a, 28
Parallelogram, area of, 92
Parallelogram, definition of,
14
Parallelogram of forces, 121
Parallel lines, 10
Parenthesis in formulas, 85
Pencils, 4
Pendulum, 141
Pens, the use of drawing, 7
Pentagon, area of, 93
Pentagon, definition of, 14
Pentagon, to draw a, 26
Perpendicular lines, 10
Perpendicular lines, to draw,
18
Pitch, circular, 205
Pitch diameters, table of, 206
Pitch, diametral, 207
Plane, definition of, 10
Plane, inclined, 136
Point, definition of, 10
Polygons, definition of, 14
Positive return cams, 173
Power transmission, screws
for, 252
Presses, fly-wheels for, 289
Prism, projections of a, 34
Prism, volume of, 94
Projection, 32
Projection, isometric, 48
Pulley diameters, 281
Pulley diameters, to calcu-
late, 297
Pulleys, cone, 239
Pulleys, differential, 134
Pulleys, guide, 285
Pulleys, hoisting, 132
Punches, fly-wheels for, 289
Pyramid, surface develop-
ment of a, 41
Pyramid, volume of, 94
Q
Quarter-turn belting, 283
Quick return device, Whit-
worth, 319
Quick return motions, 313
R
Rack teeth, modified form of
involute, 197
Rack with epicycloidal teeth,
194
Reciprocating motion cams,
171
Resolution of forces, 123
Resultant of forces, 120
Return device, Whitworth
quick, 319
Return motion, elliptic gear
quick, 321
332
INDEX
Return motions, quick, 313
Reversal of direction of mo-
tion, to secure, 298
Right-angled triangles, 97
Safety, factor of, 151
Scales, 2
Screw cutting, gearing for,
302
Screw, differential, 138
Screw, in mechanics, 138
Screw thread, Acme stand-
ard, 253
Screw thread, form of met-
ric, 256
Screw thread, sharp V, 254
Screw thread, Whitworth,
255
Screw threads, drawing, 74
Screw threads, table of
United States standard, 246
Screw threads, United States
standard, 245
Screw threads, wrench action
on, 249
Screws, bolts and studs, 243
Screws, dimensioning, 62
Screws, efficiency of, 253
Screws for power transmis-
sion, 252
Screws, machine, 257
Screws, set, 256
Screws, square threaded, 251
Secant of an angle, 102
Second, definition of, 97
Sections on drawings, 66
Set-screws, 256
Shade lines, 77
Shafting, horse-power of, 274
Shafts, 272
Shafts at right angles, belt-
ing between, 283
Shafts, Thurston's rule for
strength of, 220
Shapers, quick return mo-
tion for, 316
Sharp V-thread, 254
Shearing strength of mate-
rials, 240
Shearing strength of shaft-
ing, torsional, 273
Shears, fly-wheels for power,
289
Shrouded gears, 201
Sine of an angle, 101
Sines, table of, 104
Solid, definition of, 10
Speed of belting, 279
Speed ratio of gears, 220
Speed ratio of sprocket
wheels, 189
Speed, to secure increase of,
297
Sphere, area and volume of,
94
Spherical sector, volume of,
94
Spherical segment, volume
of, 95
Spiral, to draw a, 26
Sprocket wheels, 185
Sprocket wheels, graphical
method of laying out, 187
Sprocket wheels, speed ratio
of, 189
Spur gearing, calculating,
222
Spur gears, method of draw-
mg, 68
Square root, 82
Square threaded screws, 251
Steel castings, strength of,
157
Steel, strength of machine,
158
Steel, strength of structural,
162
Steel wire, strength of, 158
Stepped cone pulleys, 239
Strength of gear teeth, 213
Strength of gear teeth,
Lewis' formula for, 218
Strength of materials, 151
Strength of: materials, Kirk-
aldy's tests on, 157
INDEX
333
Strength of materials, shear-
ing, 260
Strength of shafting, tor-
sional shearing, 273
Strength of shafts, twisting,
272
Stresses in castings, 162
Studs, screws and bolts, 243
Supplement angle, definition
of, 11
Surface, definition of, 10
Tangent, definition of, 13
Tangent of an angle, 101
Tangent to a circle, to draw
a, 19
Tangents, table of, 106
Tensile strength of materials,
158
Tension in belts, 276
Tension, machine parts sub-
jected to, 154
Thickness of belts, 277
Thread, Acme standard
screw, 253
Thread cutting, gearing for,
302
Thread, form of metric screw,
256
Thread, sharp V, 254
Thread, Whitworth screw,
255
Thread, drawing screw, 74
Threads, screws with square,
251
Threads, United States
Standard screw, 245
Thurston's rule for strength
of shafts, 220
Toothed clutches, 265
Torsional strength of shafts,
272
Trains of mechanism, 297
Transposition of formulas, 88
Triangle, area of, 91
Triangles, solution of, 96
Trigonometry, elements of,
96
Tumbler gear, 300
Twisting strength of shafts,
272
U
Uniform motion cams, 165
Uniformly accelerated mo-
tion cams, 176
United States standard screw
thread, 245
Universal joint, 263
V-Thread, sharp, 254
Vertex of angle, definition
of, 10
Views on working drawings,
number of, 50
Volume of solids, 94
w
Weight of fly-wheel, 291
Whitworth quick return de-
vice, 319
Whitworth screw thread, 255
Width of belts, 277
Wire, strength of, 158
Work and energy, 146
Working drawings, 50
Worm gearing, 204
Worm gearing, calculating,
234
Wrench action on screw
threads, 249
Wrought iron, strength of,
157
CATALOGUE OF
STANDARD
PRACTICAL AND
SCIENTIFIC
BOOKS
PUBLISHED AND FOR SALE BY
The Norman W, Henley Publishing Go,
132 Nassau St., New York, U. S. A.
INDEX OF SUBJECTS
Brazing and Soldering 3
Cams ii
Charts 3
Chemistry 4
Civil Engineering 4
C oke 4
Compressed Air 4
Concrete 5
Dictionaries 5
Dies— Metal Work 6
Drawing— Sketching Paper 6
Electricity 7
Enameling 9
Factory Management, etc 9
Fuel 10
Gas Engines and Gas 10
Gearing and Cams u
Hydraulics n
Ice and Refrigeration .'....- 1 1
Inventions— Patents 12
Lathe Practice 12
Liquid Air 12
Locomotive Engineering 12
Machine Shop Practice 14
Manual Training 17
Marine Engineering 17
Metal Work-Dies 6
Mining 17
Miscellaneous 18
Patents and Inventions 12
Pattern Making 18
Perfumery , 18
Plumbing , 19
Receipt Book 24
Refrigeration and Ice n
Rubber 19
Saws , '. 20
Screw Cutting 20
Sheet Metal Work 20
Soldering 3
Steam Engineering 20
Steam Heating and Ventilation 22
Steam Pipes ' 22
Steel 22
Watch Making 23
Wireless Telephones 23
Any of these books will be sent prepaid to any part of
the world, on receipt of price.
REMIT by Draft, Postal Money Order, Express Money Order
or by Registered Mail.
GOOD, USEFUL BOOKS
BRAZING AND SOLDERING
BRAZING AND SOLDERING. By JAMES F. HOB ART.
The only book that shows you just how to handle any job of
brazing or soldering that comes along; tells you what mixture
to use, how to make a furnace if you need one. Full of kiaks.
4th edition. 25 cents
CHARTS
BATTLESHIP CHART. An engraving which shows the
details of a battleship as if the sides were of glass and you could
see all the interior. The finest piece of work that has ever been
done. So accurate that it is used at Annapolis for instruction
purposes. Shows all details and gives correct name of every
part. 28 x 42 inches — plate paper. 50 cents
BOX CAR CHART. A chart showing the anatomy of a box
car, having every part of the car numbered and its proper name
given in a reference list. 20 cents
GONDOLA CAR CHART. A chart showing the anatomy
of a gondola car, having every part of the car numbered and its
proper reference name given in a reference list. 20 cents
P ASSEN GER CAR CHART. A chart showing the anatomy
of a passenger car, having every part of the car numbered and its
proper name given in a reference list. 20 cents
TRACTIVE POWER CHART. A chart v/hereby you can
find the tractive power or drawbar pull of any locomotive,
without making a figure. Shows what cylinders are equal, how
driving wheels and steam pressure affect the power. What sized
engine you need to exert a given drawbar pull or anything you
desire in this line. 50 cents
WESTINGHOUSE AIR-BRAKE CHARTS. Chart I.—
Shows (in colors) the most modern Westinghouse High Speed
and Signal Equipment used on Passenger Engines, Passenger
Engine Tenders, and Passenger Cars. Chart II. — Shows (in
colors) the Standard Westinghouse Equipment for Freight
and Switch Engines, Freight and Switch Engine Tenders, and
Freight Cars. Price for the set, 50 cents
CHEMISTRY
HENLEY'S TWENTIETH CENTURY BOOK OF
RECEIPTS, FORMULAS AND PROCESSES. Edited by
GARDNER D. Hiscox. The most valuable Techno-chemical
Receipt Book published, including over 10,000 selected scientific
chemical, technological, and practical receipts and processes.
See page 24 for full description of this book. $3.00
CIVIL ENGINEERING
HENLEY'S ENCYCLOPEDIA OF PRACTICAL EN-
GINEERING AND ALLIED TRADES. Edited by JOSEPH
G. HORNER, A.M.I., M.E. This set of five volumes contains
about 2,500 pages with thousands of illustrations, including dia-
grammatic and sectional drawings with full explanatory details.
It covers the entire practice of Civil and Mechanical Engineering.
It tells you all you want to know about engineering and tells it
so simply, so clearly, so concisely that one cannot help but
understand. $6.00 per volume or $25.00 for complete set of five
volumes.
COKE
COKE— MODERN COKING PRACTICE; INCLUDING
THE ANALYSIS OF MATERIALS AND PRODUCTS.
By T. H. BYROM, Fellow of the Institute of Chemistry, Fellow
of The Chemical Society, etc., and J. E. CHRISTOPHER, Member
of the Society of Chemical Industry, etc. A handbook for
those engaged in Coke manufacture and the recovery of By-
products. _ Fully illustrated with folding plates.
The subject of Coke Manufacture is of rapidly increasing in-
terest and significance, embracing as it does the recovery of
valuable by-products in which scientific control is of the first
importance. It has been the aim of the authors, in preparing
this book, to produce one which shall be of use and benefit to
those who are associated with, or interested in, the modern de-
velopments of the industry.
Contents: Chap. I. Introductory. Chap. II. General Classi-
fication of Fuels. Chap. III. Coal Washing. Chap. IV. The
Sampling and Valuation of Coal, Coke, etc. Chap. V. The
Calorific Power of Coal and Coke. Chap. VI. Coke Ovens.
Chap. VII. Coke Ovens, continued. Chap. VIII. Coke Ovens,
continued. Chap. IX. Charging and Discharging of Coke Ovens.
Chap. X. Cooling and Condensing Plant. Ch'ap. XI. Gas Ex-
hausters. Chap. XII. Composition and Analysis of Ammoniacal
Liquor. Chap. XIII. Working up of Ammoniacal Liquor.
Chap. XIV. Treatment of Waste Gases from Sulphate Plants.
Chap. XV. Valuation of Ammonium Sulphate. Chap. XVI.
Direct Recovery of Ammonia from Coke Oven Gases. Chap.
XVII. Surplus Gas from Coke Oven. Useful Tables. Very
fully illustrated. 83.50 net
COMPRESSED AIR
COMPRESSED AIR IN ALL ITS APPLICATIONS By
GARDNER D. Hiscox. This is the most complete book on the
subject of Air that has ever been issued, and its thirty-five
chapters include about every phase of the subject one can think
of. It may be called an encyclopedia of compressed air. It is
written by an expert, who, in its 665 pages, has dealt with the
subject in a comprehensive manner, no phase of it being omitted.
Over 500 illustrations, sth Edition, revised and enlarged.
Cloth bound; $5.00, Half morocco, *6.5O
CONCRETE
ORNAMENTAL CONCRETE WITHOUT MOLDS, By A. A.
HOUGHTON. The process for making ornamental concrete with-
out molds, has long been held as a secret and now, for the first
time, this process is given to the public. The book reveals the
secret and is the only book published which explains a simple,
practical method whereby the concrete worker is enabled, by
employing wood and metal templates of different designs, to
mold or model in concrete any Cornice, Archivolt, Column,
Pedestal, Base Cap, Urn or Pier in a monolithic form — right
upon the job. These may be molded in units or blocks, and
then built up to suit the specifications demanded. This work
is fully illustrated, with detailed engravings. 83.00
POPULAR HAND BOOK FOR CEMENT AND CON-
CRETE USERS, By MYRON H. LEWIS, C.E. This is a con-
cise treatise of the principles and methods employed in the
manufacture and use of cement in all classes of modern works.
The author has brought together in this work, all the salient
matter of interest to the user of concrete and its many diversified
products. The matter is presented in logical and systematic
order, clearly written, fully illustrated and free from involved
mathematics. Everything of value to the concrete user is given.
Among the chapters contained in the book are: I. Historical
Development of the Uses of Cement and Concrete. II. Glossary
of Terms employed in Cement and Concrete work. III. Kinds
of Cement employed in Construction. IV. Limes, Ordinary and
Hydraulic. V. Lime Plasters. VI. Natural Cements. VII.
Portland Cements. VIII. Inspection and Testing. IX. Adul-
teration; or Foreign Substances in Cement. X. Sand, Gravel
and Broken Stone. XI. Mortar. XII. Grout. XIII. Con-
crete (Plain). XIV. Concrete (Reinforced). XV. Methods
and Kinds of Reinforcements. XVI. Forms for Plain and Re-
inforced Concrete. XVII. Concrete Blocks. XVIII. Arti-
ficial Stone. XIX. Concrete Tiles. XX. Concrete Pipes and
Conduits. XXI. Concrete Piles. XXII. Concrete Buildings.
XXIII. Concrete in Water Works. XXIV. Concrete in Sewer
Works. XXV. Concrete in Highway Construction. XXVI.
Concrete Retaining Walls. XXVII. Concrete Arches and
Abutments. XXVIII. Concrete in Subway and Tunnels.
XXIX. Concrete in Bridge Work. XXX. Concrete in Docks
and Wharves. XXXI. Concrete Construction under Water.
XXXII. Concrete on the Farm. XXXIII. Concrete Chimneys.
XXXIV. Concrete for Ornamentation. XXXV. Concrete
Mausoleums and Miscellaneous Uses. XXXVI. Inspection for
Concrete Work. XXX VII. Waterproofing Concrete Work.
XXXVIII. Coloring and Painting Concrete Work. XXXIX.
Method of Finishing Concrete Surfaces. XL. Specifications and
Estimates for Concrete Work. $3.50
DICTIONARIES
STANDARD ELECTRICAL DICTIONARY. By T.
O'CoNOR SLOANE. An indispensable work to all interested in
electrical science. Suitable alike for the student and profession-
al. A practical hand-book of reference containing definitions
of about 5,000 distinct words, terms and phrases. The defini-
tions are terse and concise and include every term used in electri-
cal science. Recently issued. An entirely new edition. Should
be in the possession of all who desire to keep abreast with the
progress of this branch of science. Complete, concise and con-
venient. 682 pages — 393 illustrations. 83.00
DIES— METAL WORK
DIES, THEIR CONSTRUCTION AND USE FOR THE
MODERN WORKING OF SHEET METALS. By J. V.
WOODWORTH. A new book by a practical man, for those who
wish to know the latest practice in the working of sheet metals.
It shows how dies are designed, made and used, and those who
are engaged in this line of work can secure many valuable sug-
gestions. $3.00
PUNCHES, DIES AND TOOLS FOR MANUFACTUR-
ING IN PRESSES. By J. V. WOODWORTH. An encyclo-
pedia of die-making, punch-making, die-sinking, sheet-metal
working, and making of special tools, subpresses, devices and
mechanical combinations for punching, cutting, bending, form-
ing, piercing, drawing, compressing, and assembling sheet-
metal parts and also articles of other materials in machine
tools. This is a distinct work from the author's book entitled
"Dies; Their Construction and Use." 500 pages, 700 engrav-
ings. $4.00
DRAWING— SKETCHING PAPER
LINEAR PERSPECTIVE SELF-TAUGHT. By HERMAN
T. C. KRAUS. This work gives the theory and practice of linear
perspective, as used in architectural, engineering, and mechanical
drawings. Persons taking up the study of the subject by them-
selves, without the aid of a teacher, will be able by the use of the
instruction given to readily grasp the subject, and by reason-
able practice become good perspective draftsmen. The arrange-
ment of the book is good; the plate is on the left-hand, while the
descriptive text follows on the opposite page, so as to be readily
referred to. The drawings are on sufficiently large scale to show
the work clearly and are plainly figured. The whole work makes
a very complete course on perspective drawing, and will be
found of great value to architects, civil and mechanical engineers,
patent attorneys, art designers, engravers, and draftsmen. $2.50
PRACTICAL PERSPECTIVE. By RICHARDS and COLVIN.
Shows just how to make all kinds of mechanical drawings in the
only practical perspective isometric. Makes everything plain
so that any mechanic can understand a sketch or drawing in
this way. Saves time in the drawing room and mistakes in the
shops. Contains practical examples of various classes of work.
50 cents
SELF-TAUGHT MECHANICAL DRAWING AND ELE-
MENTARY MACHINE DESIGN. By F. L. SYLVESTER, M.E.,
Draftsman, with additions by Erik Oberg, associate editor of
"Machinery." A practical elementary treatise on Mechanical
Drawing and Machine Design, comprising the first principles of
geometric and mechanical drawing, workshop _ mathematics,
mechanics, strength of materials and the calculation and design
of machine details, compiled for the use of practical mechanics
and young draftsmen. $2.00
A NEW SKETCHING PAPER. A new specially ruled paper
to enable you to make sketches or drawings in isometric per-
spective without any figuring or fussing. It is being used for
shop details as well as for assembly drawings, as it makes one
sketch do the work of three, and no workman can help seeing
just what is wanted. Pads of 40 sheets, 6x9 inches, 25 cents.
Pads of 40 sheets, 9x12 inches, 50 cents
ELECTRICITY
ARITHMETIC OF ELECTRICITY. By Prof. T. O'CoNOR
SLOANE. A practical treatise on electrical calculations of all
kinds reduced to a series of rules, all of the simplest forms, and
involving only ordinary arithmetic; each rule illustrated by
one or more practical problems, with detailed solution of each
one. This book is classed among the most useful works pub-
lished on the science of electricity Covering as it does the mathe-
matics of electricity in a manner that will attract the attention
of those who are not familiar with algebraical formulas. 160
pages. $1.00
COMMUTATOR CONSTRUCTION. By WM. BAXTER,
JR. The business end of any dynamo or motor of the direct
current type is the commutator. This book goes into the de-
signing, building, and maintenance of commutators, shows
how to locate troubles and how to remedy them; everyone who
fusses with dynamos needs this. 25 cents
DYNAMO BUILDING FOR AMATEURS, OR HOW TO
CONSTRUCT A FIFTY WATT DYNAMO. By ARTHUR
J. WEED, Member of N. Y. Electrical Society. This book is a
practical treatise showing in detail the construction of a small
dynamo or motor, the entire machine work of which can be done
on a small foot lathe.
Dimensioned working drawings are given for each piece of
machine work and each operation is clearly described.
This machine when used as a dynamo has an output of fifty
watts; when used as a motor it will drive a small drill press or
lathe. It can be used to drive a sewing machine on any and all
ordinary work.
The book is illustrated with more than sixty original engrav-
ings showing the actual construction of the different parts. Paper.
Paper 50 cents Cloth 81.00
ELECTRIC FURNACES AND THEIR INDUSTRIAL
APPLICATIONS. By J.WRIGHT. This is a book which will
prove of interest to many classes of people; the manufacturer
who desires to know what product .can be manufactured success-
fully in the electric furnace, the chemist who wishes to post
himself on the electro-chemistry, and the student of science
who merely looks into the subject from curiosity. 288 pages.
$3.00
ELECTRIC LIGHTING AND HEATING POCKET
BOOK. By SYDNEY F. WALKER. This book puts in conven-
ient form useful information regarding the apparatus which is
likely to be attached to the mains of an electrical company.
Tables of units and equivalents are included and useful electrical
laws and formulas are stated. 43 8 pages, 3 oo engravings. $3.00
ELECTRIC TOY MAKING, DYNAMO BUILDING, AND
ELECTRIC MOTOR CONSTRUCTION. This work treats
of the making at home of electrical toys, electrical apparatus,
motors, dynamos, and instruments in general, and is designed to
bring within the reach of young and old the manufacture of gen-
uine and useful electrical appliances. 185 pages. Fully illus-
trated. $1.00
ELECTRIC WIRING, DIAGRAMS AND SWITCH-
BOARDS. By NEWTON HARRISON. This is the only complete
work issued snowing and telling you what you should know
about direct and alternating current wiring. It is a ready
reference. The work is free from advanced technicalities and
mathematics. Arithmetic being used throughout. It is in every
respect a handy, well-written, instructive, comprehensive
volume on wiring for the wireman, foreman, contractor or elec-
trician. 272 pages, 105 illustrations. 81.50
'ELECTRICIAN'S HANDY BOOK. By PROF. T. O'CpNOR
SLOANE. This work is intended for the practical electrician,
who has to make things go. The entire field of Electricity is
covered within its pages. It contains no useless theory; every-
thing is to the point. It teaches you just what you should
know about electricity. It is the standard work published on
the subject. Forty-one chapters, 610 engravings, handsomely
bound in red leather with titles and edges in gold. $3.50
ELECTRICITY IN FACTORIES AND WORKSHOPS,
ITS COST AND CONVENIENCE. By ARTHUR P. HASLAM.
A practical book for power producers and power users showing
what a convenience the electric motor, in its various forms, has
become to the modern manufacturer. It also deals with the
conditions which determine the cost of electric driving, and
compares this with other methods of producing and utilizing
power. 312 pages. Very fully illustrated. $2.50
ELECTRICITY SIMPLIFIED. By PROF. T. O'CoNOR
SLOANE. The object of "Electricity Simplified" is to make the
subject as plain as possible and to show what the modern con-
ception of electricity is; to show how two plates of different
metals immersed in acid can send a message around the globe;
to explain how a bundle of copper wire rotated by a steam engine
can be the agent in lighting our streets, to tell what the volt, ohm
and ampere are, and what high and low tension mean; and to
answer the questions that perpetually arise in the mind in this
age of electricity. 172 pages. Illustrated. $1.00
HOW TO BECOME A SUCCESSFUL ELECTRICIAN.
By PROF. T. O'CoNOR SLOANE. An interesting book from cover
to cover. Telling in simplest language the surest and easiest way
to become a successful electrician. The studies to be followed,
methods of work, field of operation and the requirements of the
successful electrician are pointed out and fully explained.
202 pages. Illustrated. $1.00
MANAGEMENT OF DYNAMOS. By LUMMIS-PATER-
SON. A handbook of theory and practice. This work is arranged
in three parts. The first part covers the elementary theory of
the dynamo. The second part, the construction and action of
the different classes of dynamos in common use are described;
while the third part relates to such matters as affect the prac-
tical management and working of dynamos and motors. 292
pages, 117 illustrations. $1.50
STANDARD ELECTRICAL DICTIONARY. By Prof. T.
O'CoNOR SLOANE. A practical handbook of reference contain-
ing definitions of about 5,000 distinct words, terms and phrases.
The definitions are terse and concise and include every term
used in electrical science. 682 pages, 393 illustrations. $3.00
8
SWITCHBOARDS. By WILLIAM BAXTER, JR. This book
appeals to every engineer and electrician who wants to know
the practical side of things. All sorts and conditions of dynamos,
connections and- circuits are shown by diagram and illustrate
just how the switchboard should be connected. Includes direct
and alternating current boards, also those for arc lighting, in-
candescent, and power circuits. Special treatment on high
voltage boards for power transmission. 190 pages. Illustrated.
81.50
TELEPHONE CONSTRUCTION, INSTALLATION,
WIRING, OPERATION AND MAINTENANCE. By W. H.
RADCLIFFE and H. C. CUSHING. This book gives the principles
of construction and operation of both the Bell and Independent
instruments; approved methods of installing and wiring them;
the means of protecting them from lightning and abnormal cur-
rents; their connection together for operation as series or bridg-
ing stations; and rules for their inspection and maintenance.
Line wiring and the wiring and operation of special telephone
systems are also treated. 180 pages, 125 illustrations. 81.00
WIRING A HOUSE. By HERBERT PRATT. Shows a house
already built; tells just how to start about wiring it. Where to
begin; what wire to use; how to run it according to insurance
rules, in fact just the information you need. Directions apply
equally to a shop. Fourth edition. 35 cents
WIRELESS TELEPHONES AND HOW THEY WORK.
By JAMES ERSKINE-MURRAY. This work is free from elaborate
details and aims at giving a clear survey of the way in which
Wireless Telephones work. It is intended for amateur workers
and for those whose knowledge of Electricity is slight. Chap-
ters contained: How We Hear — Historical — The Conversion of
Sound into Electric Waves — Wireless Transmission — The Pro-
duction of Alternating Currents of High Frequency — How the
Electric Waves are Radiated and Received — The Receiving
Instruments — Detectors — Achievements and Expectations —
Glossary of Technical Work. Cloth. 81.00
ENAMELING
HENLEY'S TWENTIETH CENTURY RECEIPT BOOK.
Edited by GARDNER D. Hiscox. A work of 10,000 practical
receipts, including enameling receipts for hollow ware, for
metals, for signs, for china and porcelain, for wood, etc. Thor-
ough and practical. See page 24 for full description of this book.
•3.00
FACTORY MANAGEMENT, ETC.
MODERN MACHINE SHOP CONSTRUCTION, EQUIP-
MENT AND MANAGEMENT. By O. E. PERRIGO, M.E. A
work designed for the practical and every-day use of the Archi-
tect who designs, the Manufacturers who build, the Engineers
who plan and. equip, the Superintendents who organize and
direct, and for the information of every stockholder, director,
officer, accountant, clerk, superintendent, foreman, and work-
man of the modern machine shop and manufacturing plant of
Industrial America. 85.00
FUEL
COMBUSTION OF COAL AND THE PREVENTION
OF SMOKE. By WM. M. BARR. To be a success a fireman
must be "Light on Coal." He must keep his fire in good con-
dition, and prevent, as far as possible, the smoke nuisance.
To do this, he should know how coal burns, how smoke is formed
and the proper burning of fuel to obtain the best results. He
can learn this, and more too, from Barr's "Combustion of Coal."
It is an absolute authority on all questions relating to the Firing
of a Locomotive. Nearly 350 pages, fully illustrated. 81.00
SMOKE PREVENTION AND FUEL ECONOMY. By
BOOTH and KERSHAW. As the title indicates, this book of 197
pages and 75 illustrations deals with the problem of complete
combustion; which it treats from the chemical and mechanical
standpoints, besides pointing out the economical and humani-
tarian aspects of the question. S2.5O
GAS ENGINES AND GAS
CHEMISTRY OF GAS MANUFACTURE. By H. M.
ROYLES. A practical treatise for the use of gas engineers, gas
managers and students. Including among its contents — Prepa-
rations of Standard Solutions, Coal, Furnaces, Testing and
Regulation. Products of Carbonization. Analysis of Crude Coal
Gas. Analysis of Lime. Ammonia. Analysis of Oxide of Iron.
Naphthalene. Analysis of Fire-Bricks and Fire-Clay . Weldom
and Spent Oxide. Photometry and Gas Testing. Carbur-
etted Water Gas. Metropolis Gas. Miscellaneous Extracts.
Useful Tables. $4.50
GAS ENGINE CONSTRUCTION, Or How to Build a Half-
Horse-power Gas Engine. By PARSELL and WEED. A prac-
tical treatise describing the theory and principles of the action of
gas engines of various types, and the design and construction of a
half-horse-power gas engine, with illustrations of the work in
actual progress, together with dimensioned working drawings giv-
ing clearly the sizes of the various details. 300 pages. $2.50
GAS, GASOLINE, AND OIL, ENGINES. By GARDNER D.
Hiscox. Just issued, i8th revised and enlarged edition. Every
user of a gas engine needs this book. Simple, instructive, and
right up-to-date. The only complete work on the subject. Tells
all about the running and management of gas, gasoline and oil
engines as designed and manufactured in the United States.
Explosive motors for stationary, marine and vehicle power are
fully treated, together with illustrations of their parts and tabu-
lated sizes, also their care and running are included. Electric
Ignition by Induction Coil and Jump Sparks are fully explained
and illustrated, including valuable information on the testing for
economy and power and the erection of power plants.
The special information on PRODUCER and SUCTION GASES in-
cluded cannot fail to prove of value to all interested in the gen-
eration of producer gas and its utilization in gas engines.
The rules and regulations of the Board of Fire Underwriters
in regard to the installation and management of Gasoline Motors
is given in full, suggesting the safe installation of explosive motor
power. A list of United States Patents issued on Gas, ^Gasoline
and Oil Engines and their adjuncts from 1875 to date is included.
484 pages. 410 engravings. S3. 50 net
MODERN GAS ENGINES AND PRODUCER GAS
PLANTS. By R. E. MATHOT, M.E. A practical treatise of
320 pages, fully illustrated by 175 detailed illustrations, setting
forth the principles of gas engines and producer design, the selec-
tion and installation of an engine, conditions of perfect opera-
tion, producer-gas engines and their possibilities, the care of gas
engines and producer-gas plants, with a chapter on volatile
hydrocarbon and oil engines. This book has been endorsed by
Dugal Clerk as a most useful work for all interested in Gas Engine
installation and Prodxicer Gas. 82.50
GEARING AND CAMS
BEVEL GEAR TABLES. By D. AG. ENGSTROM. No one
who has to do with bevel gears in any way should be without
this book. The designer and draftsman will find it a great con-
venience, while to the machinist who turns up the blanks or cuts
the teeth, it is invaluable, as all needed dimensions are given
and no fancy figuring need be done. 81. OO
CHANGE GEAR DEVICES. By OSCAR E. PERRIGO. A
book for every designer, draftsman and mechanic who is inter-
ested in feed changes for any kind of machines. This shows what
has been done and how. Gives plans, patents and all information
that you need. Saves hunting through patent records and rein-
venting old ideas. A standard work of reference. 81.00
DRAFTING OF CAMS. By Louis ROUILLION. The
laying out of cams is a serious problem unless you know how to
go at it right. This puts you on the right road for practically
any kind of cam you are likely to run up against. 25 cents
HYDRAULICS
HYDRAULIC ENGINEERING. By GARDNER D. Hiscox.
A treatise on the properties, power, and resources of water for all
purposes. Including the measurement of streams; the flow of
water in pipes or conduits; the horse-power of falling water;
turbine and impact water-wheels; wave-motors, centrifugal,
reciprocating, and air-lift pumps. With 300 figures and dia-
grams and 36 practical tables. 320 pages. 84.00
ICE AND REFRIGERATION
POCKET BOOK OF REFRIGERATION AND ICE MAK-
ING, By A. J. WALLIS-TAYLOR. This is one of the latest and
most comprehensive reference books published on the subject
of refrigeration and cold storage. It explains the properties and
refrigerating effect of the different fluids in use, the manage-
ment of refrigerating machinery and the construction and insula-
tion of cold rooms with their required pipe surface for different
degrees of cold; freezing mixtures and non-freezing brines,
temperatures of cold rooms for all kinds of provisions, cold
storage charges for all classes of goods, ice making and storage of
ice, data and memoranda for constant reference by refrigerating
engineers, with nearly one hundred tables containing valuable
references to every fact and condition required in the installment
and operation of a refrigerating plant. 81.50
II
INVENTIONS— PATENTS
INVENTOR'S MANUAL, HOW TO MAKE A PATENT
PAY. This is a book designed as a guide to inventors in per-
fecting their inventions, taking out their patents, and disposing
of them. It is not in any sense a Patent Solicitor's Circular,
nor a Patent Broker's Advertisement. No advertisements of any
description appear in the work. It is a book containing a quarter
of a century's experience of a successful inventor, together with
notes based upon the experience of many other inventors. $J .00
LATHE PRACTICE
MODERN AMERICAN LATHE PRACTICE. By OSCAR
E. PERRIGO. An up-to-date book on American Lathe Work,
describing and illustrating the very latest practice in lathe and
boring-mill operations, as well as the construction of and latest
devel9pments in the manufacture of these important classes of
machine tools. 300 pages, fully illustrated. 83.50
PRACTICAL METAL TURNING. By JOSEPH G. HORNER.
A work of 404 pages, fully illustrated, covering in a comprehen-
sive manner the modern practice of machining metal parts in
the lathe, including the regular engine lathe, its essential design,
its uses, its tools, its attachments, and the manner of holding the
work and performing the operations. The modernized engine
lathe, its methods, tools, and great range of accurate work. The
Turret Lathe, its tools, accessories and methods of performing
its functions. Chapters on special work, grinding, tool holders,
speeds, feeds, modern tool steels, etc., etc. $3.50
TURNING AND BORING TAPERS. By FRED H. COL-
VIN. There are two ways to turn tapers; the right way and
one other. This treatise has to do with the right way; it tells
you how to start the work properly, how to set the lathe, what
tools to use and how to use them, and forty^and one other little
things that you should know. Fourth edition. 25 cents
LIQUID AIR
LIQUID AIR AND THE LIQUEFACTION OF GASES.
By T. O'CoNOR SLOANE. Theory, history, biography, practical
applications, manufacture. 365 pages. Illustrated. $2.00
LOCOMOTIVE ENGINEERING
AIR-BRAKE CATECHISM. By ROBERT H. BLACKALL.
This book is a standard text book. It covers the Westinghouse
Air-Brake Equipment, including the No. 5 and the No. 6 E T
Locomotive Brake Equipment; the K (Quick-Service) Triple
Valve for Freight Service; and the Cross-Compound Pump.
The operation of all parts of the apparatus is explained in detail,
and a practical way of finding their peculiarities and defects,
with a proper remedy, is given. It contains 2,000 questions with
their answers, which will enable any railroad man to pass any
examination on the subject of Air Brakes. Endorsed and used
by air-brake instructors and examiners on nearly every rail-
road in the United States. 23d Edition. 380 pages, fully
illustrated with folding plates and diagrams. $2.00
AMERICAN COMPOUND LOCOMOTIVES. By FRED
H. COLVIN. The most complete book on compounds published.
Shows all types, including the balanced compound. Makes
everything clear by many illustrations, and shows valve setting,
breakdowns and repairs. 142 pages. $1.00
APPLICATION OF HIGHLY SUPERHEATED STEAM
TO LOCOMOTIVES. By ROBERT GARBE. A practical book.
Contains special chapters on Generation of Highly Superheated
Steam; Superheated Steam and the Two-Cylinder Simple
Engine; Compounding and Superheating; Designs of Locomotive
Superheaters; Constructive Details of Locomotives using Highly
Superheated Steam; Experimental and Working Results. Illus-
trated with folding plates and tables. 82.50
COMBUSTION OF COAL AND THE PREVENTION
OF SMOKE. By WM. M. BARR. To be a success a fireman
must be "Light on Coal." He must keep his fire in good con-
dition, and prevent as far as possible, the smoke nuisance.
To do this, he should know how coal burns, how smoke is formed
and the proper burning of fuel to obtain the best results. He
can learn this, and more too, from Barr's "Combination of Coal."
It is an absolute authority on all questions relating to the Firing
of a Locomotive. Nearly 350 pages, fully illustrated. $1.00
LINK MOTIONS, VALVES AND VALVE SETTING. By
FRED H. COLVIN, Associate Editor of "American Machinist.
A handy book that clears up the mysteries of valve setting.
Shows the different valve gears in use, how they work, and why.
Piston and slide valves of different types are illustrated and
explained. A book that every railroad man in the motive-
power department ought to have. Fully illustrated. 60 cents.
LOCOMOTIVE BOILER CONSTRUCTION. By FRANK
A. KLEINHANS. The only book showing how locomotive
boilers are built in modern shops. Shows all types of boilers
used; gives details of construction; practical facts, such as
life of riveting punches and dies, work done per day, allowance
for bending and flanging sheets and other data that means dol-
lars to any railroad man. 421 pages, 334 illustrations. Six
folding plates. $3.00
LOCOMOTIVE BREAKDOWNS AND THEIR REM-
EDIES. By GEO. L. FOWLER. Revised by Wm. W. Wood,
Air-Brake Instructor. Just issued 1910 Revised pocket edition.
It is put of the question to try and tell you about every subject
that is covered in this pocket edition of Locomotive Breakdowns.
Just imagine all the common troubles that an engineer may ex-
pect to happen some time, and then add all of the unexpected
ones, troubles that could occur, but that you had never thought
about, and you will find that they are all treated with the very
best methods of repair. Walschaert Locomotive Valve Gear
Troubles, Electric Headlight Troubles, as well as Questions and
Answers on the Air Brake are all included. 294 pages. Fully
illustrated. $1.00
LOCOMOTIVE CATECHISM. By ROBERT GRIMSHAW.
27th revised and enlarged edition. This may well be called an
encyclopedia of the locomotive. Contains over 4,000 examina-
tion questions with their answers, including among them those
asked at the First, Second and Third year's Examinations.
825 pages, 437 illustrations and 3 folding plates. $2.50
13
NEW YORK AIR-BRAKE CATECHISM. By ROBERT
H. BLACKALL. This is a complete treatise on the New York
Air-Brake and Air-Signalling Apparatus, giving a detailed de-
scription of all the parts, their operation, troubles, and the
methods of locating and remedying the same. 200 pages, fully
illustrated. 81.00
POCKET-RAILROAD DICTIONARY AND VADE ME-
CU!\I. ^ By FRED H. COLVIN, Associate Editor "American
Machinist." Different from any book you ever saw. Gives clear
and concise information on just the points you are interested in.
It's really a pocket dictionary, fully illustrated, and so arranged
that you can find just what you want in a second without an
index. Whether you are interested in Axles or Acetylene; Com-
pounds or Counter Balancing; Rails or Reducing Valves; Tires
or Turntables, you'll find them in this little book. It's very
complete. Flexible cloth cover, 200 pages. 81.00
TRAIN RULES AND DESPATCHING. By H. A. DALBY.
Contains the standard code for both single and double track and
explains how trains are handled under all conditions. Gives all
signals in colors, is illustrated wherever necessary, and the
most complete book in print on this important subject. Bound
in fine seal flexible leather. 221 pages. 81.50
WALSCHAERT LOCOMOTIVE VALVE GEAR. By
WM. W. WOOD. If you would thoroughly understand the
Walschaert Valve Gear, you should possess a copy of this book.
The author divides the subject into four divisions, as follows:
I. Analysis of the gear. II. Designing and erecting of the gear
III. Advantages of the gear. IV. Questions and answers re
lating to the Walschaert Valve Gear. This book is specially valu-
able to those preparing for promotion. Nearly 200 pages. $1.50
WESTINGHOUSE E T AIR-BRAKE INSTRUCTION
POCKET BOOK CATECHISM. By WM. W. WOOD, Air-Brake
Instructor. A practical work containing examination questions
and answers on the E T Equipment. Covering what the E T
Brake is. How it should be operated. What to do when de-
fective. Not a question can be asked of the engineman up for
promotion on either the No. 5 or the No. 6 E T equipment that
is not asked and answered in the book. If you want to thor-
oughly understand the E T equipment get a copy of this book.
It covers every detail. Makes Air-Brake troubles. and examina-
tions easy. Fully illustrated with colored plates, showing
various pressures. 82.00
MACHINE SHOP PRACTICE
AMERICAN TOOL MAKING AND INTERCHANGE-
ABLE MANUFACTURING. ^ By J. V. WOODWORTH. A
practical treatise on the designing, constructing, use, and in-
stallation of tools, jigs, fixtures, devices, special appliances,
sheet-metal working processes, automatic mechanisms, and
labor-saving contrivances; together with their use in the lathe
milling machine, turret lathe, screw machine, boring mill, power
press, drill, sub press, drop hammer, etc., for the working of
metals, the production of interchangeable machine parts, and
the manufacture of repetition articles of metal. 560 pages,
600 illustrations. *4.0O
HENLEY'S ENCYCLOPEDIA OF PRACTICAL EN-
GINEERING AND ALLIED TRADES. Edited by JOSEPH
G. HORNER. A.M.I.Mech.I. This work covers the entire prac-
tice of Civil and Mechanical Engineering. The best known ex-
perts in all branches of engineering have contributed to these
volumes. The Cyclopedia is admirably well adapted to the needs
of the beginner and the self-taught practical man, as well as the
mechanical engineer, designer, draftsman, shop superintendent,
foreman and machinist.
•It is a modern treatise in five volumes. Handsomely bound
in Half Morocco, each volume containing nearly 500 pages, with
thousands of illustrations, including diagrammatic and sectional
drawings with full explanatory details. $35.00 for the com-
plete set of five volumes. $6.00 per volume, when ordered singly.
MACHINE SHOP ARITHMETIC. By COLVIN-CHENEY.
Most popular book for shop men. Shows how all shop problems
are worked out and "why." Includes change gears for cutting
any threads; drills, taps, shink and force fits; metric system
of measurements and threads. Used by all classes of mechanics
and for instruction of Y. M. C. A. and other schools. Fifth
edition. 131 pages. 50 cents
MECHANICAL MOVEMENTS, POWERS, AND DE-
VICES. By GARDNER D. Hiscox. This is a collection of 1890
engravings of different mechanical motions and appliances, ac-
companied by appropriate text, making it a book of great value
to the inventor, the draftsman, and to all readers with mechanical
tastes. The book is divided into eighteen sections or chapters
in which the subject matter is classified under the following
heads: Mechanical Powers, Transmission of Power, Measurement
of Power, Steam Power, Air Power Appliances, Electric Power
and Construction, Navigation and Roads, Gearing, Motion and
Devices, Controlling Motion, Horological, Mining, Mill and
Factory Appliances, Construction and Devices, Drafting Devices,
Miscellaneous Devices, etc. nth edition. 400 octavo pages.
$3.50
MECHANICAL APPLIANCES, MECHANICAL MOVE-
MENTS AND NOVELTIES OF CONSTRUCTION. By
GARDNER D. Hiscox. This is a supplementary volume to the
one upon mechanical movements. Unlike the first volume,
which is more elementary in character, this volume contains
illustrations and descriptions of many combinations of motions
and of mechanical devices and appliances found in different lines
of Machinery. Each device being shown by a line drawing with
a description showing its working parts and the method of opera-
tion. From the multitude of devices described, and illustrated,
might be mentioned, in passing, such items as conveyors and
elevators, Prony brakes, thermometers, various types of boilers,
solar engines, oil-fuel burners, condensers, evaporators, Corliss
and other valve gears, governors, gas engines, water motors of
various descriptions, air ships, motors and dynamos, automobile
and motor bicycles, railway block signals, car couples, link and
gear motions, ball bearings, breech block mechanism for heavy
guns, and a large accumulation of others of equal importance.
1,000 specially made engravings. 396 octavo pages. $2.50
These two volumes sell for $2.50 each,
but when the twQ volumes are ordered
at one time from us, we send them prepaid to any address in the
world, on receipt of $4.00. You save $i by ordering the two
volumes of Mechanical Movements at one time.
15
MODERN MACHINE SHOP CONSTRUCTION, EQUIP-
MENT AND MANAGEMENT. By OSCAR E. PERRIGO.
The only work published that describes the Modern Machine
Shop or Manufacturing Plant from the time the grass is growing
on the site intended for it until the finished product is shipped.
Just the book needed by those contemplating the erection of
modern shop buildings, the rebuilding and reorganization of old
ones, or the introduction of Modern Shop Methods, Time and
Cost Systems. It is a book written and illustrated by a prac-
tical shop man for practical shop men who are too busy to read
theories and want facts. It is the most complete all-around book
of its kind ever published. 400 large quarto pages, 225 original
and specially-made illustrations. $5.00
MODERN MACHINE SHOP TOOLS; THEIR CON-
STRUCTION, OPERATION, AND MANIPULATION. By
W. H. VANDERVOORT. A work of 555 pages and 673 illustra-
tions, describing in every detail the construction, operation, and
manipulation of both Hand and Machine Tools. Includes
chapters on filing, fitting, and scraping surfaces; on drills, ream-
ers, taps, and dies; the lathe and its tools; planers, shapers,
and their tools; milling machines and cutters; gear cutters and
gear cutting; drilling machines and drill work; grinding ma-
chines and their work; hardening and tempering; gearing,
belting and transmission machinery; useful data and tables.
$4.00
THE MODERN MACHINIST. By JOHN T. USHER. This
book might be called a compendium of shop methods, showing a
variety of special tools and appliances which will give new ideas
to many mechanics from the superintendent down to the man
at the bench. It will be found a valuable addition to any machin-
ist's library and should be consulted whenever a new or difficult
job is to be done, whether it is boring, milling, turning, or plan-
ing, as they are all treated in a practical manner. Fifth edition.
320 pages, 250 illustrations. $2.50
MODERN MECHANISM. Edited by PARK BENJAMIN. A
practical treatise on machines, motors and the transmission of
power, being a complete work and a supplementary volume to
Appleton's Cyclopedia of Applied Mechanics. Deals solely with
the principal and most useful advances of the past few years.
959 pages containing over 1,000 illustrations; bound in half
morocco. $4.00
MODERN MILLING MACHINES : THEIR DESIGN,
CONSTRUCTION AND OPERATION. By JOSEPH G.
HORNER. This book describes and illustrates the Milling Ma-
chine and its work in such a plain, clear, and forceful manner,
and illustrates the subject so clearly and completely, that the
up-to-date machinist, student, or mechanical engineer can not
afford to do without the valuable information which it contains.
It describes not only the early machines of this class, but notes
their gradual development into the splendid machines of the
present day, giving the design and construction of the various
types, forms, and special features produced by prominent
manufacturers, American and foreign. 304 pages, 300 illustra-
tions. $4.00
" SHOP KINKS." By ROBERT GRIMSHAW. This shows
special methods of doing work of various kinds, and reducing
cost of production. Has hints and kinks from some of the largest
shops in th'is country and Europe. You are almost sure to find
some that apply to your work, and in such a way as to save time
and trouble. 400 pages. Fourth edition. $2.50
16
TOOLS FOR MACHINISTS AND WOOD WORKERS,
INCLUDING INSTRUMENTS OF MEASUREMENT. By
JOSEPH G. HORNER. A practical treatise of 340 pages, fully
illustrated and comprising a general description and classifica-
tion of cutting tools and tool angles, allied cutting tools for
machinists and woodworkers; shearing tools; scraping tools;
saws; milling cutters; drilling and boring tools; taps and dies;
punches and hammers; and the hardening, tempering and
grinding of these tools. Tools for measuring and testing work,
including standards of measurement; surface plates; levels;
surface gauges; dividers; calipers; verniers; micrometers;
snap, cylindrical and limit gauges; screw thread, wire and
reference gauges, indicators, templets, etc. 83. 50
MANUAL TRAINING
ECONOMICS OF MANUAL, TRAINING. By Louis
ROUILLION. The only book that gives just the information
needed by all interested in manual training, regarding buildings,
equipment and supplies. Shows exactly what is needed for all
grades of the work from the Kindergarten to the High and Nor-
mal School. Gives itemized lists of everything needed and tells
just what it ought to cost. Also shows where to buy supplies.
$1.50
MARINE ENGINEERING
MARINE ENGINES AND BOILERS, THEIR DESIGN
AND CONSTRUCTION. By DR. G. BAUER, LESLIE S.
ROBERTSON, and S. BRYAN DONKIN. This work is clearly
written, thoroughly systematic, theoretically sound; while the
character of its plans, drawings, tables, and statistics is without
reproach. The illustrations are careful reproductions from
actual working drawings, with some well-executed photographic
views of completed engines and boilers. $9.00 net
MINING
*ORE DEPOSITS OF SOUTH AFRICA WITH A
CHAPTER ON HINTS TO PROSPECTORS. By J. P. JOHN-
SON. This book gives a condensed account of the ore-deposits
at present known in South Africa. It is also intended as a guide
to the prospector. Only an elementary knowledge of geology
and some mining experience are necessary in order to under-
stand this work. With these qualifications, it will materially
assist one in his search for metalliferous mineral occurrences
and, so far as simple ores are concerned, should enable one to
form some idea of the possibilities of any they may find.
Among the chapters given are: Titaniferous and Chromif-
erous Iron Oxides — Nickel — Copper — Cobalt — Tin — Molyb-
denum— Tungsten — Lead — Mercury — Antimony — I r o n — Hints
to Prospectors. Illustrated. $2.00
PRACTICAL COAL MINING. By T. H. COCKIN. An im-
portant work, containing 428 pages and 213 illustrations, com-
plete with practical details, which will intuitively impart to the
reader, not only a general knowledge of the principles of coal
mining, but also considerable insight into allied subjects. The
treatise is positively up to date in every instance, and should
be in the hands of every colliery engineer, geologist, mine
operator, superintendent, foreman, and all others who are in-
terested in or connected with the industry. $2.50
17
PHYSICS AND CHEMISTRY OF MINING. By T. H.
BYROM. A practical work for the use of all preparing for ex-
aminations in mining or qualifying for colliery managers' cer-
tificates. The aim of the author in this excellent book is to place
clearly before the reader useful and authoritative data which
will render him valuable assistance in his studies. The only work
of its kind published. The information incorporated in it will
prove of the greatest practical utility to students, mining en-
gineers, colliery managers, and all others who are specially in-
terested in the present-day treatment of mining problems. 160
pages. Illustrated. $3.00
MISCELLANEOUS
BRONZES. Henley's Twentieth Century Receipt Book con-
tarns many practical formulas on bronze casting, imitation
bronze, bronze polishes, renovation of bronze. See page 24 for
full description of this book. 83.00
EMINENT ENGINEERS. By DWIGHT GODDARD. Every-
one who appreciates the effect of such great inventions as the
Steam Engine, Steamboat, Locomotive, Sewing Machine, Steel
Working, and other fundamental discoveries, is interested in
knowing a little about the men who made them and their achieve-
ments.
Mr. Goddard has selected thirty-two of the world's engineers
who have contributed most largely to the advancement of our
civilization by mechanical means, giving only such facts as are of
general interest and in a way which appeals to all, whether
mechanics or not. 280 pages, 35 illustrations. $1.50
LAWS OF BUSINESS, By THEOPHILUS PARSONS, LL.D.
The Best Book for Business Men ever Published. Treats clearly
of Contracts, Sales, Notes, Bills of -Exchange, Agency, Agree-
ment, Stoppage in Transitu, Consideration, Limitations, Leases,
Partnership, Executors, Interest, Hotel Keepers, Fire and Life
Insurance, Collections, Bonds, Frauds, Receipts, Patents, Deeds,
Mortgages, Liens, Assignments, Minors, Married Women, Arbi-
tration, Guardians, Wills, etc. Three Hundred Approved Forms
are given. Every Business Man should have a copy of this book
for ready reference. . The book is bound in full sheep, and Con-
tains 864 Octavo Pages. Our special price. $3.50
PATTERN MAKING
PRACTICAL PATTERN MAKING. By F. W. BARROWS.
This is a very complete and entirely practical treatise on the
subject of pattern making, illustrating pattern work in wood and
metal. From its pages you are taught just what you should
know about pattern making. It contains a detailed description
of the materials used by pattern makers, also the tools, both
those for hand use, and the more interesting machine tools; hav-
ing complete chapters on The Band Saw, The Buzz Saw, and The
Lathe. Individual patterns of many different kinds are fully
illustrated and described, and the mounting of metal patterns on
plates for molding machines is included. $3.00
PERFUMERY
HENLEY'S TWENTIETH CENTURY BOOK OF RE-
CEIPTS, FORMULAS AND PROCESSES. Edited by G. D.
Hiscox. The most valuable Techno-Chemical Receipt Book
published. Contains over 10,000 practical Receipts many of
which will prove of special value to the perfumer, a mine of in-
formation, up to date in every respect. Cloth, $3.OO; half
morocco. See page 24 for full description of this book. $4.00
18
PERFUMES AND THEIR PREPARATION. By G. W.
ASKINSON, Perfumer. A comprehensive treatise, in which
there has been nothing omitted that could be of value to the
Perfumer. Complete directions for making handkerchief per-
fumes, smelling-salts, sachets, fumigating pastilles; preparations
for the care of the skin, the mouth, the hair, cosmetics, hair dyes
and other toilet articles are given, also a detailed description
of aromatic substances; their nature, tests of purity, and
wholesale manufacture. A book of general, as well as profes-
sional interest, meeting the wants not only of the druggist and
perfume manufacturer, but also of the general public. Third
edition. 312 pages. Illustrated. $3.00
PLUMBING
MODERN PLUMBING ILLUSTRATED. By R M.
STARBUCK. The author of this book, Mr. R. M. Starbuck, is one
of the leading authorities on plumbing in the United States. The
book represents the highest standard of plumbing work. It has
been adopted and used as a reference book by the United States
Government, in its sanitary work in Cuba, Porto Rico and the
Philippines, and by the principal Boards of Health of the United
States and Canada.
It gives Connections, Sizes and Working Data for All Fixtures
and Groups of Fixtures. It is helpful to the Master Plumber in
Demonstrating to his customers and in figuring work. It gives
the Mechanic and Student, quick and easy Access to the best
Modern Plumbing Practice. Suggestions for Estimating Plumb-
ing Construction are contained in its pages. This book repre-
sents, in a word, the latest and best up-to-date practice, and
should be in the hands of every architect, sanitary engineer
and plumber who wishes to keep himself up to the minute on this
important feature of construction. 400 octavo pages, fully
illustrated by 55 full- page engravings. 84.00
RUBBER
HENLEY'S TWENTIETH CENTURY BOOK OF RE-
CEIPTS, FORMULAS AND PROCESSES. Edited by GARD-
NER D. Hiscox. Contains upward of 10,000 practical receipts,
...... ., , ..«'...,.. „
S3.00
including among them formulas on artificial rubber. See page
24 for full description of this book.
RUBBER HAND STAMPS AND THE MANIPULATION
OF INDIA RUBBER. By T. O'CoNOR SLOANE. This book
gives full details on all points, treating in a concise and simple
manner the elements of nearly everything it is necessary to under-
stand for a commencement in any branch of the India Rubber
Manufacture. The making of all kinds of Rubber Hand Stamps,
Small Articles of India Rubber, U. S. Government Composi-
tion, Dating Hand Stamps, the Manipulation of Sheet Rubber,
Toy Balloons, India Rubber Solutions, Cements, Blackings,
Renovating Varnish, and Treatment for India Rubber Shoes,
etc.; the Hektograph Stamp Inks, and Miscellaneous Notes,
with a Short Account of the Discovery, Collection, and Manufac-
ture of India Rubber are set forth in a manner designed to be
readily understood, the explanations being plain and simple.
Second edition. 144 pages. Illustrated. $1.00
19
SAWS
SAW FILING AND MANAGEMENT OF SAWS. By
ROBERT GRIMSHAW. A practical hand book on filing, gumming,
swaging, hammering, and the brazing of band saws, the speed,
work, and power to run circular saws, etc. A handy book for
those who have charge of saws, or for those mechanics who do
their own filing, as it deals with the proper shape and pitches of
saw teeth of all kinds and gives many useful hints and rules for
gumming, setting, and filing, and is a practical aid to those who
use saws for any purpose. New edition, revised and enlarged.
Illustrated. 81.00
SCREW CUTTING
THREADS AND THREAD CUTTING. By COLVIN and
STABEL. This clears up many of the mysteries of thread-
cutting, such as double and triple threads, internal threads, catch-
ing threads, use of hobs, etc. Contains a lot of useful hints and
several tables. 25 cents
SHEET METAL WORK
DIES, THEIR CONSTRUCTION AND USE FOR THE
MODERN WORKING OF SHEET METALS. By J. V.
WOODWORTH. A new book by a practical man, for those who
wish to know the latest practice in the working of sheet metals.
It shows how dies are designed, made and used, and those who
are engaged in this line of work can secure many valuable
suggestions. $3.00
PUNCHES, DIES AND TOOLS FOR MANUFACTUR-
ING IN PRESSES. By J. V. WOODWORTH. A work of 5.00
pages and illustrated by nearly 700 engravings, being an en-
cyclopedia of die-making, punch-making, die sinking, sheet-
metal working, and making of special tools, subpresses, devices
and mechanical combinations for punching, cutting, bending,
forming, piercing, drawing, compressing, and assembling sheet-
metal parts and also articles of other materials in machine tools.
$4.00
STEAM ENGINEERING
AMERICAN STATIONARY ENGINEERING. By W.
E. CRANE. A new book by a well-known author. Begins at
the boiler room and takes in the whole power plant. - Contains
the result of years of practical experience in all sorts of engine
rooms and gives exact information that cannot be found else-
where. It's plain enough for practical men and yet of value to
those high in the profession. Has a complete examination for a
license. 82.00
" BOILER ROOM CHART. By GEO. L. FOWLER. A Chart
— size 14x28 inches — showing in isometric perspective the
mechanisms belonging in a modern boiler room. Water tube
boilers, ordinary grates and mechanical stokers, feed water
heaters and pumps comprise the equipment. The various parts
are shown broken or removed, so that the internal construction
is fully illustrated. Each part is given a reference number, and
these, with the corresponding name, are given in a glossary
printed at the sides, 'ihis chart is really a dictionary of the
boiler room — the names of more than 200 parts being given.
It is educational — worth many times its cost. 25 cents
ENGINE RUNNER'S CATECHISM. By RpBERT GRIM-
SHAW. Tells how to erect, adjust, and run the principal steam
engines in use in the United States. The work is of a handy
size for the pocket. To young engineers this catechism will be
of great value, especially to those who may be preparing to go
forward to be examined for certificates of competency; and
to engineers generally it will be of no little service as they will
find in this volume more really practical and useful information
than is to be found anywhere else within a like compass. 387
pages. Sixth edition. 83.00
ENGINE TESTS AND BOILER EFFICIENCIES. By
J. BUCHETTI. This work fully describes and illustrates the
method of testing the power of steam engines, turbine and
explosive motors. The properties of steam and the evapora-
tive power of fuels. Combustion of fuel and chimney draft;
with formulas explained or practically computed. 255 pages,
179 illustrations. $3.00
HORSE POWER CHART. Shows the horse power of any
stationary engine without calculation. No matter what the
cylinder diameter or stroke; the steam pressure or cut-off; the
revolutions, or whether condensing or non-condensing, it's all
there. Easy to use, accurate, and saves time and calculations.
Especially useful to engineers and designers. 50 cents
MODERN STEAM ENGINEERING IN THEORY AND
PRACTICE. By GARDNER D. Hiscox. This is a complete and
practical work issued for Stationary Engineers and Firemen
dealing with the care and management of Boilers, Engines,
Pumps, Superheated Steam, Refrigerating Machinery, Dyna-
mos, Motors, Elevators, Air Compressors, and all other branches
with which the modern Engineer must be familiar. Nearly
200 Questions with their Answers on Steam and Electrical
Engineering, likely to be asked by the Examining Board, are
included. 487 pages, 405 engravings. S3.00
STEAM ENGINE CATECHISM. By ROBERT GRIMSHAW.
This volume of 413 pages is not only a catechism on the question
and answer principle; but it contains formulas and worked-out
answers for all the Steam problems that appertain to the opera-
tion and management of the Steam Engine. Illustrations of
various valves and valve gear with their principles of operation
are given. 3 4 tables that are indispensable to every engineer and
fireman that wishes to be progressive and is ambitious to become
master of his calling are within its pages. It is a most valuable
instructor in the service of Steam Engineering. Leading en-
gineers have recommended it as a valuable educator for the be-
ginner as well as a reference book for the engineer. Sixteenth
edition. S2.00
STEAM ENGINEER'S ARITHMETIC. By COLVIN-
CHENEY. A practical pocket book for the Steam Engineer.
Shows how to work the problems of the engine room and shows
"why." Tells how to figure horse-power of engines and boilers;
area of boilers; has tables of areas and circumferences; steam
tables; has a dictionary of engineering terms. Puts you onto
all of the little kinks in figuring whatever there is to figure
around a power plant. Tells you about the heat unit; absolute
zero; adiabatic expansion; duty of engines; factor of safety;
and 1,001 other things; and everything is plain and simple —
not the hardest way to figure, but the easiest. 50 cents
21
STEAM HEATING AND VENTILATION
PRACTICAL STEAM, HOT-WATER HEATING AND
VENTILATION. By A. G. KING. This book is the standard
and latest work published on the subject and has been prepared
for the use of all engaged in the business of steam, hot-water
heating and ventilation. It is an original and exhaustive work.
Tells how to get heating contracts, how to install heating and
ventilating apparatus, the best business methods to be used, with
"Tricks of the Trade" for shop use. Rules and data for esti-
mating radiation and cost and such tables and information as
make it an indispensable work for everyone interested in steam,
hot -water heating and ventilation. It describes all the principal
systems of steam, hot-water, vacuum, vapor and vacuum-
vapor heating, together with the new accelerated systems of
hot-water circulation, including chapters on up-to-date methods
of ventilation and the fan or blower system of heating and venti-
lation.
You should secure a copy of this book, as each chapter con-
tains a mine of practical information. 367 pages, 300 detailed
engravings. 83.00
STEAM PIPES
STEAM PIPES: THEIR DESIGN AND CONSTRUC-
TION. By WM. H. BOOTH. The work is well illustrated in regard
to pipe joints, expansion off sets, flexible joints, and self-contained
sliding joints for taking up the expansion of long pipes. In fact,
the chapters on the flow of Steam and expansion of pipes are most
valuable to all steam fitters and users. The pressure strength of
pipes and method of hanging them is well treated and illustrated.
Valves and by-passes are fully illustrated and described, as are
also flange joints and their proper proportions. Exhaust heads
and separators. One of the most valuable chapters is that on
superheated steam and the saving of steam by insulation with
the various kinds of felting and other materials, with comparison
tables of the loss of heat in thermal units from naked and felted
steam pipes. Contains 187 pages. $2.00
STEEL
AMERICAN STEEL WORKER. By E. R. MARKHAM.
The standard work on hardening, tempering and annealing steel
of all kinds. A practical book for the machinist, tool maker or
superintendent. Shows just how to secure best results in any
case that comes along. How to make and use furnaces and case
harden; how to handle high-speed steel and how to temper for all
classes of work. $2.50
HARDENING, TEMPERING, ANNEALING, AND
FORGING OF STEEL. By J. V. WOODWORTH. A new book
containing special directions for the successful hardening and
tempering of all steel tools. Milling cutters, taps, thread dies,
reamers, both solid and shell, hollow mills, punches and dies,
and all kinds of sheet-metal working tools, shear blades, saws,
fine cutlery and metal-cutting tools of all descriptions, as well
as for all implements of steel both large and small, the simplest,
and most satisfactory hardening and tempering processes are
presented. The uses to which the leading brands of steel may be
adapted are concisely presented, and their treatment for work-
ing under different conditions explained, as are also the special
methods for the hardening and tempering of special brands.
320 pages, 250 illustrations. 83.50
HENLEY'S TWENTIETH CENTURY BOOK OF RE-
CEIPTS, FORMULAS AND PROCESSES. Edited by GARD-
NER D. Hiscox. The most valuable techno-chemical Receipt
book published, giving, among other practical receipts, methods
of annealing, coloring, tempering, welding, plating, polishing
and cleaning steel. See page 24 for full description of this book.
$3.00
WATCH MAKING
HENLEY'S TWENTIETH CENTURY BOOK OF RE-
CEIPTS, FORMULAS AND PROCESSES. Edited by
GARDNER D. Hiscox. Contains upwards of 10,000 practical
formulas including many watchmakers' formulas. $3.0O
WATCHMAKER'S HANDBOOK. By CLAUDIUS SAUNIER.
No work issued can compare with this book for clearness and
completeness. It contains 498 pages and is intended as a work-
shop companion for those engaged in Watchmaking and allied
Mechanical Arts. Nearly 250 engravings and fourteen plates
are included. $3.00
WIRELESS TELEPHONES
WIRELESS TELEPHONES AND HOW THEY WORK.
By JAMES ERSKINE-MURRAY. This work is free from elaborate
details and aims at giving a clear survey of the way in which
Wireless Telephones work. It is intended for amateur workers
and for those whose knowledge of Electricity is slight. Chap-
ters contained: How We Hear — Historical — The Conversion of
Sound into Electric Waves — Wireless Transmission — The Pro-
duction of Alternating Currents of High Frequency — How the
Electric Waves are Radiated and Received — The Receiving
Instruments — Detectors — Achievements and Expectations — •
Glossary of Technical Words. Cloth. $1.0O
Henley's Twentieth Century
Book of
Recipes, Formulas
and Processes
Edited by GARDNER D. HISCOX, M.E.
Price $3. 00 Cloth Binding $4. 00 Half Morocco Binding
Contains over 10,000 Selected Scientific, Chemical,
Technological and Practical Recipes and
Processes, including Hundreds of
So-Called Trade Secrets
for Every Business
THIS book of 800 pages is the most complete Book of
Recipes ever published, giving thousands of recipes
for the manufacture of valuable articles forevery-day
use. Hints, Helps, Practical Ideas and Secret Processes
are revealed within its pages. It covers every branch of
the useful arts and tells thousands of ways of making
money and is just the book everyone should have at his
command.
The pages are filled with matters of intense interest and
immeasurable practical value to the Photographer, the
Perfumer, the Painter, the Manufacturer of Glues, Pastes,
Cements and Mucilages, the Physician, the Druggist, the
Electrician, the Brewer, the Engineer, the Foundryman,
the Machinist, the Potter, the Tanner, the Confectioner,
the Chiropodist, the Manufacturer of Chemical Novelties
and Toilet Preparations, the Dyer, the Electroplater,
the Enameler, the Engraver, the Provisioner, the Glass
Worker, the Goldbeater, the Watchmaker and Jeweler,
the Ink Manufacturer, the Optician, the Farmer, the Dairy-
man, the Paper Maker, the Metal Worker, the Soap Maker,
the Veterinary Surgeon, and the Technologist in general.
A book to which you may turn with confidence that you
will find what you are looking for. A mine of informa-
tion up-to-date in every respect. Contains an immense
number of formulas that every one ought to have that are
not found in any other work.
10781
212441
"
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11