GEARS AND GEAR-CUTTING
By the Same Author
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Workshop Calculations, Tables and Formulae
Practical Wireless Encyclopaedia
Practical Motorist's Encyclopaedia
Practical Wireless Service Manual
Radio Engineer's Vest-Pocket Book
The Superhet Manual
Practical Wireless Circuits
Wireless Transmission
Radio Training Manual
Everyman's Wireless Book
Wireless Coils , Chokes and Transformers
Watches: Adjustment and Repair
Newnes Short-Wave Manual
Motor Car Principles and Practice
ft
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Newnes Engineer's Vest-Pocket Book
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GEARS AND
GEAR-CUTTING
PRINCIPLES, PRACTICE AND
FORMULAE
EDITED BY
F. J. CAMM
Editor of “Practical Engineering * 0
WITH OVER 100 ILLUSTRATIONS
Second Edition
LONDON
GEORGE NEWNES LIMITED
TOWER HOUSE, SOUTHAMPTON STREET
STRAND, W.C.2
First Published .
Second Edition .
. October 1940
. April 1941
MAD* AND PRINTED IN GREAT BRITAIN BY
MORRISON AND GIBB LTD*, LONDON AND EDXNBOROXC
PREFACE
Gear design and methods of cutting gears have progressed
considerably during the last thirty years. Accurate gears
are so essential to a Vide variety of machine tools and
mechanisms that the time seemed opportune to produce
this practical work dealing with gear design, types, functions
and methods of cutting them, together with formulae
necessary for the designer and the operator.
Additional material deals with the making of end-mills,
hobs, and gear-generating cutters. The old method of
cutting spur gears by means of a rotary milling cutter, one
tooth at a time, is still used in some shops and, therefore,
is included in this book. Gear measurement is an important
aspect of inspection, which is also dealt with.
This book includes the articles on gear-cutting contri¬
buted to Practical Engineering by W. A. Tuplin, D.Sc.
F. J. CAMM.
CONTENTS
PAGE
I. Types, Cutting Methods and Terms . , 9
II. Bevel Gears ..... 29
HI. Worm Gears ..... 34
IV. Gear Generation . . . .41
V. Gear-Wheel Forms - . . . .66
VI. Epicyclic Gear Trains . . .71
VII. Methods of Mounting . . . .74
VIII. Measuring Gears . . . .78
IX. Hobs, End-Mills and Generating Cutters ioo
X. Load Capacity of Gears . . .113
XI. The Efficiency of Gears . . .117
XII. Useful Formulae for Gears . . .119
Index. 141
7
GEARS AND GEAR-CUTTING
CHAPTER I
Types, Cutting Methods and Terms
The toothed gear has long been a familiar element in
mechanical engineering and its importance seems likely
to increase as development proceeds. Early examples of
gears were, cumbrous, noisy and inefficient, owing to the
lack of suitable materials, design technique and manu¬
facturing facilities. Improvements in these respects have
more than kept pace with advancing demands and toothed
gearing is-now confidently employed at speeds and power
outputs that could not have been attempted a few decades
ago.
Gears are also used to transmit power between rotating
shafts, which are not in the same straight line and which
have, in general, different speeds. Other means are possible
in some circumstances, but where a compact drive is
Fig. t.— The shafts and pinions of an automobile gearbox.
9
10
G
[Fig. 2.—This type is known as the double-helical gear with
continuous teeth.
necessary, or where “ timing ” of the shafts is essential,
the toothed gear usually surpasses all competitors in
simplicity, reliability and efficiency.
Applications. —Examples of the use of gearing occur
TYPES, CUTTING METHODS AND TERMS II
in the conventional auto¬
mobile. For example, the
camshaft which operates
the valves has to run at
half the crankshaft speed,
and in many instances it
is driven from the crank¬
shaft through gearing.
In order to permit the
engine to run within a
reasonable economical
speed-range whilst driving
the vehicle at different
speeds, the gearbox con¬
tains several pairs of gears,
the drive being taken
through any particular
pair at will, to suit the
conditions under which
the vehicle is running.
From the propeller shaft
which runs parallel to the
length of the vehicle, the
drive is taken to the rear
axle through another pair of gears, in which there is a
reduction in speed besides a change in the direction of the
drive. The rear axle gear assembly also contains the
“ differential,” which is a gearing assembly designed to
divide the driving effort equally between the two rear wheels.
In contrast with the small gears used in automobiles,
gearing is used in diameters up to 20 ft., for example, in?
ship propulsion, where the economical speed of the steam
turbine may be about 3000 r.p.m., and that of the propeller
about 250 r.p.m.
In industrial service, gearing is now very extensively
used, largely because of the widespread application of
electric-motor drives. The economical running speeds of
electric motors of moderate output range from about
750 r.p.m. to 1500 r.p.m., whereas the greater proportion
of industrial machinery runs at considerably lower speeds,
usually from about 50 to 200 r.p.m. In such applications,
toothed gearing forms a compact and efficient transmission.
Fig. 3.—An example of a single-
helical gear.
12 GEARS AND GEAR-CUTTING
whose availability makes it possible to design driving and
driven machines each to run at its own most economical
speed.
Main Types of Gear. —Toothed gearing is first of all
divided on the basis of relative shaft position. The
sim plest examples are those used for parallel shafts as,
for instance, in the conventional clock or in the automobile
gearbox (see Fig. i). These are classed as spur gears if
their teeth are parallel to the shaft centre lines, otherwise
they are helical gears which may be either “ single-helical ** or
double-helical” (Figs. 3 and 2). Triple-helical teeth (Fig. 4)
are also occasionally used but are not to be recommended.
If the shaft centre lines intersect, the bevel gear is
employed. Here again there is a subdivision associated
with tooth formation, “ straight bevel ” gears having
teeth whose centre lines converge at the apex, i.e. the
Fig. 4.—Triple-helical teeth are occasionally used, but are not to be
recommended.
Fig. 5.—An example of gears in which the centre lines of the
teeth converge towards the apex (straight bevel).
intersection of the shaft centre lines (see Fig. 5). When,
as in Fig. 7, the tooth centre lines are set at an angle to
the straight lines which converge at the apex, the gear is
classed as a “ spiral bevel ” or as a “ single-helical bevel.”
Bevel gears are usually applied to shafts whose centre lines
are perpendicular to one another, but they can be manu¬
factured to suit almost any shaft angle.
If the shaft centre lines are not parallel and do not
intersect, the connection between the shafts may be
effected by spiral gears (Fig. 6), which have, however, a
comparatively small load capacity. Each gear in such a
combination is actually a helical gear, but the fact that the
shaft centre lines are not parallel is the distinction that
leads to the description " spiral gears.” It also causes the
contact between any pair of mating teeth to be confined
to a theoretical point, this being the^ reason for the small
load capacity.
Of a pair of gears, the one with the smaller number of
teeth is called the “ pinion ” and the other is the “ wheel.”
By forming one member of a pair of spiral gears so that
the teeth “ envelop ” the other gear, line contact between
pairs of mating teeth is secured and the worm gear (Fig. 8)
GEARS AND GEAR-CUTTING
*4
is produced. The gear of parallel formation is called the
“ worm.” Its teeth are helical and are called " threads,”
and it may have
(and often has) only
one thread. If the
worm has more
than one thread, the
number of threads
is often described as
the “ number of
starts.” This is to
avoid confusion
with screw-thread
terminology. Occa¬
sionally the worm
is made " hollow
faced ” so as to
envelop the wheel
which also envelops
the worm, but this
" hour-glass ” worm
gear has no advant¬
age that justifies the
extra cost and com¬
plication involved
in manufacturing it.
Essential Re¬
quirements. —The
first essential of any
pair of gears is that
of uniform angular
velocity transmis¬
sion. This means
that for each degree
of angular move¬
ment of the driving
gear the mating gear
Fig. 6. —Spiral gears. These have a com- rotates through a
paxatively small load capacity. certain multiple of
one degree.
The second is that despite the distortion of the
teeth under load, the gears must be capable of
Fig. 7.—A spiral bevel-gear unit.
transmitting power without appreciable shock or
noise.
The third requirement is that the gears must transmit
the full rated torque without fear of tooth breakage and
without showing signs of excessive wear on the working
surfaces of the teeth.
Temperature rise resulting from heat generation by
friction must be kept within due limits and, particularly
with high-speed gears, this may demand special cooling
arrangements.
Uniform Angular Velocity Transmission. —Smooth
cylinders mounted on shafts and pressed into contact with
each other will rotate at relative speeds depending on their
diameters, and will transmit power up to the relatively
low limit which causes slip to occur. Toothed gears are
required to operate in the same way as such “ rolling
cylinders ” except that the intermeshing of the teeth
prevents any possibility of slip. The rolling "cylinders
l6 GEARS AND GEAR-CUTTING
corresponding to any pair of toothed gears are called the
“ pitch cylinders.” Fig. g shows the pitch cylinders
corresponding to a pair of helical gears, the flat strip
representing the " imaginary common rack. The corre¬
sponding geometrical forms in the case of bevel^ gears are
cones (see Fig. io) with a disc representing the imaginary
common crown-wheel. A worm-wheel has a corresponding
pitch cylinder which rolls, however, with a plane and this
is the “ pitch plane ” associated with the worm. A worm
has no pitch cylinder, properly so called, although it does
contain an imaginary cylinder which is a useful basis on
Fig. 8.—A typical example of a worm gear.
which to work in calculating detail dimensions of the
worm; it may be regarded as a " nominal pitch cylinder,”
and is indicated in Fig. II.
The mating pitch elements (cylinder, cone, plane, etc.)
generally make contact along a line which is called the
" pitch line ” ; in the case of spiral gears the contact is
confined to a single " pitch point.”
It can be shown that if a pair of gears is to transmit
unif orm angular velocity the line which lies perpendicular
to the tooth surfaces at any point of contact must intersect
the pitch fine. Many geometrical forms will meet this
requirement, and choice amongst them is influenced by
ease of production with the necessary degree of accuracy.
TYPES, CUTTING METHODS AND TERMS 1 7
Fig. 10 {right ).—The cor¬
responding geometrical J Q
forms in the case of '
bevel gears are cones.
The tooth form used
almost exclusively in
modern practice is based
on the involute. .
Generation of Gear
Teeth. —A small part
of the pitch cylinder of
a gear of very large
size appears as a flat
surface. If the gear
were of infinite size, a
small part of it would
have a flat “ pitch
surface ”; such a gear
is known as a rack.
A rack may mesh
with a gear (usually
called a pinion in this
connection), and, rota-
Nominal Pilch. GjUndei
1 of Worn,
Pitch. Plane,
of Norm. '
Fig. 11 {above ).—The
"nominalpitch cylinder’’
of a worm.
Fig. 9 {left ).—A diagram¬
matic {sketch, showing
the pitch cylinders corre-
. sponding to a pair of
helical gears.
2
i8
GEARS AND GEAR-CUTTING
rack.
If we start with an actual rack, having straight-sided
teeth as shown in Fig. 12, we may use this to " generate ”
teeth in a cylindrical blank of plastic material. The rack
may be set with its teeth lying in a direction parallel to
the axis of the blank and situated in such a position that
when the rack is moved parallel to its length its teeth will
penetrate the blank.
Now if the rack be given a uniform motion of this sort,
and the blank be simultaneously rotated at a uniform
rate, the relative movement of rack teeth and blank will
cause the teeth to mould or generate teeth in the blank
in the manner shown in Fig. 12 and Fig. 13.
TYPES, CUTTING METHODS AND TERMS 19
It can be proved that, within certain limitations, the
teeth generated in this manner are of involute form.
The Involute. —Fig. 14 represents the end view of a
cylinder (radius r Q ) from which a string AB is being un¬
wrapped, the end B having been originally at C. The
curve CB traced by the end of the string is an involute to
the circle, which is called the “ base circle ” of the involute.
When an involute gear is generated from a rack in the
manner already described, the size of the base circle
associated with the involute tooth profiles is determined
by the relative motions of rack and blank and by the
spacing of the rack teeth.
The distance between corresponding flanks of adjacent
rack teeth measured in a direction parallel to the plane
containing the tips of the teeth is called the normal pitch.
20
GEARS AND GEAR-CUTTING
The distance between corresponding flanks of adjacent
rack teeth measured in the direction perpendicular to the
flanks is called the " base pitch/'
If the gear blank, during generation, makes one revolution
whilst the rack is moved through a distance equal to t normal
pitches, the gear produced will have t teeth. The base circle
of the involute profiles will be the one whose circumference
is equal to t times the base pitch of the rack teeth.
The " pitch circle of generation ” of the gear will be the
circle whose circumference is equal to t times the linear
pitch of the rack.
The angle between a line drawn perpendicular to the
flank of the rack tooth and a line joining the tips of the
rack teeth is called the “ pressure angle/’ and it can be
shown that diameter of the base circle of the gear is equal
to the diameter of the pitch circle multiplied by the cosine
of the pressure angle.
It will be noticed that, so far, no relation has been speci¬
fied between the diameter of the blank and the diameter
of the pitch circle.
The root circle of the gears is defined by the distance
by which the rack teeth are allowed to penetrate into the
blank. If the
root circle is
larger than the
base circle, the
tooth profiles are
purely involute.
If, however, the
root circle is
much smaller
than the base
circle, part of the
tooth profile is
not involute, be¬
cause the in¬
volute can have
no existence
inside its base
circle.
If the root
circle is appreci-
TYPES, CUTTING METHODS AND TERMS 21
ably smaller than the base circle, the part of the tooth
which lies inside the base circle is liable to be “ under¬
cut ” ; it is mechanically weak and liable to produce
unsatisfactory meshing of the gear with its mating
gear.
Tooth Proportions. On the British Standard System,
the total depth of a tooth from tip to root is made 0*716
times the normal pitch of the tooth. The depth to which
the tip of a gear tooth penetrates into the space between
adjacent teeth in the mating gear is 0-636 times the pitch.
The difference between these fractions, i.e. 0-080 times the
pitch, is called the “ clearance.”
The distance from the circumference of the pitch circle
of generation to the pitch circle is called the ,f addendum ”
of tooth. The distance from the pitch circle to the root
circle is called the “ dedendum.”
Methods of Defining Pitch. —The method of describ¬
ing the spacing of the teeth by specifying the distance
between them on the pitch circle is the most straightfor¬
ward one. This is sometimes called the “ circular pitch,”
or more often simply the ** pitch.” When dealing with
helical gears, 4 ‘ transverse pitch ” is a more suitable term.
Other methods of specifying pitch are by. module and
diametral pitch.
Module is equal to circular pitch divided by 71.
22 GEARS AND GEAR-CUTTING
The advantage of defining pitch in this way is that the
pitch diameter of a gear is simply the product of module
and number of teeth. This system of measurement has
been almost confined to continental practice, using the
millimetre as the unit of length, but it is equally applicable
on the basis of the inch.
Diametral Pitch. —This is equal to n divided by the
circular pitch. The pitch diameter of a gear is equal to
the number of teeth divided by the diametral pitch. For
most practical purposes it is possible to use a diametral
pitch which is a whole number, whereas with the inch as the
unit of length, pitch and module are very often fractional.
Table I. gives standard pitches expressed in all three ways.
Determination of Blank Diameter. —When the num¬
ber of teeth in the gear and the pitch of the generating
rack are fixed, the size of the pitch circle is fixed and the
diameter of the tip circle [i.e. the blank diameter) is
found by adding twice the addendum to the diameter of
the pitch circle.
In the past it was the practice to make the addendum
equal to the module, and although this practice had the
merit of minimising calculation, it often led to weak and
inefficient tooth shapes that could easily have been avoided.
There is no need to restrict the addendum in this way,
and more satisfactory tooth shapes are obtained if the
addendum is varied according to the numbers of teeth in
the mating gears. Too large an addendum produces
pointed teeth; too large a dedendum produces “ under¬
cut” teeth with weak roots. For a pressure angle of 20
degrees and a tooth depth equal to 0716 times the pitch,
addenda defined by the following formulae lead to satis¬
factory tooth shapes.
Addendum of pinion—(1-4—ovJi)-P
T 7C
Addendum of wheel—(o-6+0-4 i) -
T 7C
where t —number of teeth in pinion.
T=number of teeth in wheel,
p—pitch of cutter.
In each case
Dedendum—07i6p—Addendum.
TYPES, CUTTING METHODS AND TERMS 23
Fig. 15 represents the British Standard basic rack-form,
and Figs. 16 and 17 show typical involute teeth generated
from it.
The greater the number of teeth in a gear, the greater
is the size of the base circle for a given pitch, and as the
depth of tooth is the same in all cases, it becomes only a
small fraction of the involute when the number of teeth
is very large. Consequently, gears of large numbers of
teeth have profiles which are only slightly curved, and
the tooth of the rack (which is really an infinitely large
Fig. 16.—Typical involute teeth generated from the British Standard
rack-form.
gear) has no curvature at all, i.e., it is straight-sided as
already mentioned.
Generating Rack. —The fundamental principle in¬
volved in moulding a gear in the manner previously
described is used in nearly all modern gear-cutting
processes. The generating member is made in the form
of a cutter, and besides its moulding motion has another
which causes its teeth to cut.
In some cases the cutter itself is actually of rack-form.
In other cases the cutter corresponds to some particular
form of gear, and in the process of generating the teeth.
types, cutting methods and terms 25
the cutter and the gear are rotated just as the correspond¬
ing gears would actually do when in mesh.
In all cases, however, it is convenient to examine the
conditions of generation and meshing of gears as if a rack
were present. Actual tooth forms are most easily deter¬
mined from that of the imaginary basic rack.
Meshing of Double-Helical Gears.—If two helical
gears are generated by the same basic rack, and have
equal and opposite spiral angles, they may be meshed
Fig. 17.—Another example of involute teeth generated from the
British Standard rack-form.
together with their shafts parallel. On any plane perpen¬
dicular to the axes of the gears, the tooth action is exactly
the same as in a pair of spur gears. At any instant the
teeth in one transverse plane are in a different phase of
engagement from those in other transverse planes. If the
width of the gears is great enough, every phase of engage¬
ment is occurring in some transverse plane or other.
Consequently, helical gears operate much more smoothly
than do spur gears of comparable dimensions.
This can be appreciated also by considering how the
2 6 GEARS AND GEAR-CUTTING
load is applied to the tooth of a rotating gear. In the case
of spur gears the full width of the tooth enters the zone of
engagement at some particular instant, so that load is
applied to the tooth over its whole width at once. Similarly,
when the tooth passes out of the zone of engagement the
load is removed instantaneously.
In a pair of helical gears, however, one end of the tooth
enters the zone of engagement first, and as it passes
through, other sections of the tooth enter the zone, one
after another. If the width of the gears is great enough,
one part of the tooth may have passed completely through
the zone of contact before Other parts have entered it at
all. Thus the load is applied to each tooth first at one
end and then over a line of gradually increasing length
which passes uniformly along the length of the tooth
before diminishing gradually to a point and disappearing
at the other end of the tooth. This action is in contrast
to the sudden application and sudden removal of load in
the case of a spur-gear tooth.
Noise in gears is produced by the rapid succession of
slight impacts as load is imposed on the various teeth,
and the smooth mode of application in the case of helical
gears causes a very substantial reduction in the amount
of noise produced. Consequently, helical gears may be
run quietly at much higher speeds than would be per¬
missible in the case of spur gears.
The essential advantage of the helical gear may be
attributed to the fact that the longitudinal centre line of
tooth is not perpendicular to its direction of motion, and
this feature is advantageously copied in other types of
gear.
Overlap of Helical-Gear Teeth.—In order to make sure
that all phases of engagement do occur simultaneously, it
is necessary that one end of any tooth should overlap, or
should^ at least come in line with the opposite end of the
adjacent tooth. The condition for this is that the face-
width should be at least equal to the transverse pitch
multiplied by the cotangent of the spiral angle. If for
any reason the face width cannot be made so great as this,
the smoothest possible action will not be obtained, but
the gears will, nevertheless, be superior in action to spur
gears.
TYPES, CUTTING METHODS AND TERMS 27
End-Thrust.—The effective force exerted by one gear
tooth on the mating tooth is perpendicular to the centre
lines of the shafts. If the teeth are of helical form, however,
equal and opposite forces are exerted on the two gears
in the direction parallel to the shaft centre lines. These
forces would cause the gears to slide axially out of mesh if
nothing were done to prevent it, and consequently a thrust
bearing is necessary on each shaft.
As in some instances it may be difficult or impracticable
to provide suitable thrust bearings, the double-helical
form of tooth has been developed. A gear of this type has
helical teeth of a particular spiral angle in one half of its
face-width, whilst the other half has similar teeth of the
opposite hand. Thus the end-thrust produced by tooth
load in one half of the gear acts in the opposite direction
to that produced in the other half, and if either of the
mating gears is free to move axially, it will do so and will
take up the position which causes the two end-thrusts to
be equalised.
Thus, not only is it unnecessary to provide a thrust
bearing, but the tooth formation ensures that the total
28 GEARS AND GEAR-CUTTING
load is equally divided between the two halves of the face-
width. Owing to bending and twisting of slender pinions
under load, there is not the same certainty in this respect
with spur gears or single-helical gears.
Direction of Rotation.—In one direction of rotation,
the teeth of a pair of helical gears move with the apex
leading, whilst if the direction of rotation is reversed, the
apex follows the remainder of the tooth. It is sometimes
supposed that there is a particular advantage in one
direction of rotation over the other, but this is not the
case.
There is a certain disadvantage if the apices " trail,”
because there is then a tendency for the oil which lies on
the tooth surfaces to be trapped at the centre of the face-
width, and in some circumstances this can l$ad to appreci¬
able loss of power and to objectionable noise.
Triple-Helical Gears.—The assumption that there is
an advantage in having the apex leading has led to the
use of the triple-helical gear for frequently-reversing
drives, as, for example, to mine winders (see Fig. 4). The
idea is that the advantage is gained in each direction of
rotation when triple-helical gears are used., The truth is
that the disadvantage of oil trapping may occur in both
directions of rotation, and the triple-helical gear is in no
way superior to the double-helical gear.
On the contrary, it is impossible to manufacture the
triple-helical with the degree of accuracy easily attainable
in double-helical gears, and the only method of cutting
that can be adopted is relatively slow and expensive.
Triple-helical gears are now never adopted, except in a
few special fields of application where tradition alone
continues to demand them. <
CHAPTER II
Bevel Gears
Just as a pair of spur gears is the toothed equivalent of
a pair of rolling cylinders, a pair of bevel gears is the
counterpart of a pair of cones which roll together whilst
rotating on fixed axes. It may be noted here that since
the surface-speed of each cone is zero at its apex, the two
cones cannot roll together unless the apices coincide. In
other words, the apex of a bevel gear must be at the point
of intersection of the shaft centre lines.
The general principles involved in the generation of
spur gears apply with slight modifications to the generation
of bevel gears. Instead of a basic rack we have an imaginary
basic crown-wheel whose teeth are straight-sided, and this
corresponds to a flat disc capable of rolling simultaneously
with two cones (see Fig. io).
The pitch of the teeth of the crown-wheel must clearly
diminish as the centre is approached and, so that there
may be similar tooth action ■ at all points, the depth of
tooth is made proportional to the pitch. Thus the pitch
and depth of a crown-wheel tooth at any section are each
proportional to the distance of that section from the apex.
The pressure angle is the same everywhere.
In generating spur gears a rack-shaped cutter reciprocates
30 GEARS AND GEAR-CUTTING
in the direction of its teeth and produces teeth which are
of the same shape and size at all sections. It is not possible,
in generating bevel gears, to reciprocate the whole crown¬
wheel in a similar manner, because different teeth run in
different directions. It is not even possible to reciprocate
a single complete tooth because depth and thickness have
to diminish as the apex is approached. It is possible,
however, to sweep out the surface of one flank of a straight-
aded tooth of the crown-wheel by a straight-sided cutter
blade reciprocated so that its tip follows the root-surface
between two teeth of the crown-wheel. As the cutter
blade moves towards the apex, its depth of immersion in
the blank diminishes, but since the profile of the blade is
BEVEL GEARS
31
straight, it produces a complete straight-sided profile
everywhere. Here an important virtue of the straight¬
sided basic crown-wheel tooth is that while the profiles at
large and small ends of the tooth are similar to each other,
the smaller profile is simply a part of the larger one, so
that formation of the entire profile by a single reciprocating
cutter is possible. (See Fig. 19.)
The fillet at the root of the tooth must, of course, have
the same radius at each end, and in this respect the tooth
is not geometrically similar throughout its length, but this
is unimportant from the point of view of tooth action.
Generation of Bevel-Gear Teeth.—The process of
generating bevel-gear teeth from a crown-wheel is examined
in a plane perpendicular to that of the crown-wheel. In
the neighbourhood of the point of contact of crown-wheel
plane and pitch cone of bevel gear, the section of the
crown-wheel tooth on the plane mentioned may be regarded
as moving in a straight line. The associated part of the
bevel gear may be regarded as rotating about the point of
32 GEARS AND GEAR-CUTTING
intersection of the gear’s axis with the plane mentioned
(see Fig. 20). The shape of tooth generated in the pinion
is the same as that in a spur gear of pitch radius BA, and
likewise the wheel-tooth shape is that associated with a
spur gear of radius CA. These two radii are the “ virtual
radii ” of the gears. Similar generating action on smaller
scales takes place on parallel planes lying nearer to the apex.
Tooth Form of Straight Bevel Gears.—The simplest
type of bevel gear is that in which the centre lines of all
the teeth converge at the apex, i.e. the point of inter¬
section of-the centre lines of the mating gears when they
are correctly in mesh. To avoid excessive difference in
size between the inner and outer ends of the teeth, it is
usual to limit the face-width of bevel gears to about one-
third of the “ cone distance.”
The nominal dimensions (pitch, addendum, dedendum)
of bevel-gear teeth apply to the outer end ; the correspond¬
ing dimensions at other points in the face are proportional
to their distances from the apex, but the pressure angle is
the same at all points in the length of the tooth. The
tooth form of the imaginary crown-wheel is similar to
that of the basic rack for spur and helical gears.
Owing to the fact that the same cutting tool generates
teeth of different pitch at different points in the face-
width, it is not necessary to adhere to ' • standard ” pitches.
Although bevel gears are most frequently used to
transmit power between shafts whose centre lines are
perpendicular, they can be equally well designed for a
wide range of shaft angle. The limit to what can be done
in this respect is the pitch angle of the pinion, for in many
types of bevel-gear-cutting machine this cannot be less
than about 5 degrees. Subject to certain limitations in
reduction ratio, shaft angles varying from about 30 degrees
to about 150 degrees are permissible.
Like straight *spur gears, straight bevel gears are apt to
be unduly noisy at high speeds and for the same reason,
i.e. that the full length of each tooth enters the zone of
engagement at the same instant. To overcome the noise
difficulty recourse is had to single helical bevel gears or
spiral bevel gears.
Spiral Bevel Gears.—If the teeth of a bevel gear have,
their centre lines not perpendicular to their direction of
BEVEL GEARS
33
motion they are called " spiral bevels ” or “ single-helical
bevels.” In the latter case the centre line of each tooth is
straight but does not pass through the apex. In the spiral
bevel gear (Fig. 7) each tooth centre line is a curve, which
is usually a circular arc.
In neither type of gear is the spiral angle exactly the
same at all points in the face-width. This is not important,
provided, of course, that the mating gears have the same
spiral angles at corresponding points. The reason for
the adoption of the circular arc form for the spiral bevel
tooth is that it permits
the use of a cutter hav-
ing’a number of blades
set *in a circular body
and that this can oper¬
ate without reversal of
the motion of the
cutter. The teeth of
straight bevel gears
and single-helical bevel
gears are generated by
reciprocating cutters.
The nominal spiral
Fig. 21A.—Diagram showing the prin¬
ciple of the zerol bevel gear, the
spiral angle of which is zero at mid¬
point of face.
angle of spiral bevel
gears is the inclination
of the centre line*of
the tooth at the mid¬
point of the face-width to the line joining that point to
the apex. This is approximately the mean spiral angle
and is used in calculating the end-thrusts and journal
loads on the bearings.
The “ Zerol ” Gear.—Comparatively recently there
has been introduced the “ Zerol ” spiral bevel gear, so
called because although the teeth are of the curved spiral
type, the spiral angle at the middle of the face is zero.
This confines end-thrust to the amount produced in the
corresponding straight bevel gears, but nevertheless gives
the smoother running of the spiral bevel gear. The zerol
gear thus corresponds to the double-helical gear.
Double-helical bevel gears have been used but the only
practicable way of cutting the teeth (by means of end-
mills) is comparatively slow, expensive and inaccurate.
3
CHAPTER III
Worm Gears
The tooth action in worm gears (Fig. 8) involves com
siderably more sliding than occurs in the types of gear
already considered, and in the past this caused worm
gearing to be regarded with disfavour. Excessive friction
, led to serious loss of power and to heavy wear of the tooth
surfaces. Careful study of contact conditions and the use
of appropriate materials have combined to permit the
economical use of worm gears in a wide range of practical
applications.
If the worm be imagined to be cut by a plane passing
Fig. 22. —Section of worm and worm-wheel on central plane.
34
GEARS
35
through its axis and lying perpendicular to that of the
worm-wheel the threads appear as a series of rack teeth.
Rotation of the worm gives these teeth an apparent motion
parallel to the axis of the worm, and the form of the worm-
wheel teeth examined in the same plane must be such as
to mesh accurately with such rack teeth (see Fig. 22).
Fig. 23.—Section of worm and worm-wheel on offset plane.
36
GEARS AND GEAR-CUTTING
On planes parallel to this central one, the worm threads
show different shapes and the worm-wheel teeth must also
differ in form in such planes from that which they have
in the central plane (see Fig. 23).
The complex shape of worm-wheel teeth makes it
impracticable to produce them by any method other than
generation by a cutter of .the same form as the mating
worm. In this respect the generation of worm-wheel
teeth differs from the normal method of producing spur,
WORM GEARS
37
helical or bevel gear teeth, for mating gears of these types
are each generated by a cutter having a tooth form
depending only on the normal base pitch required. One
cutter can produce gears of any size provided only that
the base pitch is specified. A hob for generating worm-
wheel teeth can be used only if the mating worm is of the
same essential dimensions as the hob.
It is not difficult to choose a worm thread form which
will lead to satisfactory worm-wheel tooth shapes in the
central plane, but shapes in other planes may be unsuit¬
able unless special care is taken. The British Standard
worm thread is of involute form in a plane perpendicular
to the axis of the worm, or, in other words, the worm is
actually a single-helical involute pinion. The kinematic
PITCH - CYLINDER
n\
1
1
L ^
LEAD L
DEVELOPED
PITCH-CVL.INOC FT
Fig. 26. — Develop¬
ment of pitch
cylinder of worm.
38 GEARS AND GEAR-CUTTING
properties of the involute are not utilised in the worm
gear and this thread form is adopted .because it offers
certain advantages in manufacture and checking for
accuracy. The normal- pressure angle is 20° except in
cases where certain geometrical considerations make a
greater pressure angle necessary.
As indicated in Fig. 24, an accurate thread profile of
this type contains a straight line and consequently shape
errors may be detected.by comparison with a straight¬
edge. No such facility exists in the case of the worm-
wheel teeth, and the only practicable method of checking
them is by mounting the worm-wheel in correct relation to
an accurate worm and observing the nature of the contact
between them by the marking on the worm-wheel teeth.
Dimensions of Worm Gears.—The distance measured
parallel to the worm axis between adjacent threads of a
worm is called the " axial pitch.” The axial distance in
which each thread makes a complete revolution is called
the <f lead ” ; it is equal to the axial pitch multiplied by
the number of threads. The angle between the thread
at the surface of the nominal pitch cylinder and a plane
perpendicular to the axis of the worm is called the “ lead
angle.”
It will be seen that if the lead angle is small, the sliding
velocity of the worm threads on the wheel teeth is high
compared with the apparent axial velocity of the threads.
Thus frictional losses are comparatively large and the
efficiency of the gears is low.
A high lead angle is necessary for high efficiency.
The trace of a complete revolution of a worm thread on
the pitch cylinder is indicated in Fig. 26, from which it
may be seen that
tan
where A=lead angle.
L=lead.
d=diameter of nominal pitch cylinder of
worm.
The velocity ratio of the gears is equal to the number
of worm-wheel^ teeth divided by the number of worm
threads, and this is equal to the circumference of the pitch
WORM GEARS
39
circle of the worm-wheel divided by the lead of the worm.
Hence the lead is fixed if the wheel diameter is fixed, and
therefore the highest value of X is secured by using the
smallest worm diameter.
Contact Between Worm and Wheel.—At any instant
each worm thread touches one or more worm-wheel teeth
along a curved line. As the gears revolve, the various lines
of contact change their positions in space in such a way as
always to lie on a particular surface called the “ zone of
contact.” The boundaries of the zone of contact are
decided by the- thread form and by the diameters and
widths of the worm and wheel. In some particular angular
position of the gears the total length of‘the lines of contact
is a minimum which is used in determining the load
capacity of the gears. Consideration of the shape of the
zone of contact makes it possible to decide the maximum
useful face-widths of worm and wheel.
Fig. 27 shows the shape of the zone of contact for a
typical worm and worm-wheel.
40 GEARS AND GEAR-CUTTING
Each worm-wheel tooth enters the contact zone obliquely,
with the result that load is applied to it in a gradual
manner and smooth action is easily obtained. Given
suitable materials and lubrication, worm gearing is the
quietest type of toothed gear. The curvature of the
surfaces at any line of contact is less than in spur, helical
or bevel gears of the same general dimensions and the
better " fitting ” of the surfaces leads to substantial load-
capacity despite the fact that the worm-wheel is usually
made from phosphor bronze, which is a comparatively soft
material. ,
Owing to the fact that the worm-wheel teeth are pro¬
duced by a rotating cutter whose axis does not change its
position, the axial position of the worm-wheel must be
accurately adjusted in relation to the worm if correct
contact is to be obtained. Distortion of worm and wheel
and of the members which support them has a serious
influence here, and special attention has to be directed to
this point when worm gears are being mounted before
setting to work. As the cross-sectional dimensions of the
worm are the same throughout its length, accurate axial
location is unimportant.
CHAPTER IV
Gear Generation
Sunderland Process for Spur Gears.—The moulding
generation process depicted in Figs. 12 and 13 appears in
the Sunderland spur gear generator, with the single
modification that the rack is made with sharp cutting
edges and is reciprocated parallel to the axis of the blank so
that the teeth cut their way into the metal instead of being
pushed into the plastic blank in the manner already described.
In the machine shown in Fig. 28 the cutter-box moves
horizontally to and fro across the whole width of the
Fig. 28.—Piano-generating machine for spur gears.
41
42 GEARS AND GEAR-CUTTING
blank, and the slide which guides it is given a slow vertical'
movement in unison with rotation of the^ blank. Whilst
the cutter moves downwards through a distance equal to
the pitch of its teeth the blank turns through an angle
equal to one revolution divided by the number of teeth
required in the work, the “ index change gears ” and the
“ pitch change gears ” being selected to secure this result.
When the cutter has been fed so far downwards that its
uppermost tooth is beginning to cut, the generating-action
GEAR GENERATION
43
Fig. 30.—A Fellows-type machine cutting cluster gears.
is automatically stopped. The cutter is arrested at one
end of its cutting stroke, clear of the blank, and is moved
upwards through a distance equal to a whole number of
pitches without rotation of the blank. Its feed mechanism
is then reconnected to that which rotates the blank, a
short generating movement is given to cutter and blank
to take up backlash, and the cutting strokes and “ rolling ”
motions are resumed. This cycle is automatically repeated
until the entire circumference of the blank has been treated.
If the pitch of the teeth is small compared with the
capacity of the machine, the full depth of tooth may be
cut at once, and complete generation of all the teeth
effected in one revolution of the blank. In general, how¬
ever, two or more revolutions are necessary, the cutter being
reset after each revolution to sink more deeply into the blank.
44
GEARS AND GEAR-CUTTING
Fig. 31.—A Fellows-type machine cutting an internal spur gear.
In some circumstances this type of machine may use
a special cutter-box mounting, two cutters facing in
opposite directions so that cutting takes place in both
directions of motion of the cutter-box across the face of
the gear blank.
The cutter is sharpened by grinding its front face, the
form of the cutting edges being unaffected.
Fellows Process for Spur Gears.—The Fellows
process resembles the Sunderland process in that it uses a
cutter which reciprocates parallel to the axis of the gear,
but differs from it in the form of the cutter. The effective
form of the Fellows cutter is that of a spur gear, the teeth
being tapered to give cutting relief on the profile and the tip.
The necessary generation motion of cutter and blank
is secured by continuous rotation of each of them at rates
inversely proportional to their numbers of teeth, and t'here
GEAR GENERATION 45
is nothing to correspond to the periodic resetting of the
cutter that occurs in the Sunderland process. ’
To avoid rubbing of the edges of the cutter on its idle
return strokes the work is drawn a short distance away
from it at the end of each cutting stroke and replaced at
the beginning of the next one.
The Fellows process can be applied to a gear which is
situated close to a shoulder and it can, therefore, handle
cluster gears (Fig. 30). The Sunderland process suffers
some restriction in this direction (unless a specially shaped
cutter is used) because part of the cutter-box must be in
front of the cutter.
The Fellows process can be used to generate internal
gears (Fig. 31), and except for the Sykes process, which
differs from it in detail, is the only true generating process
applicable to that type of gear.
The Hobbing Process for Spur Gears.—The most
accurate spur gear generating process is that based on the
use of the cutting tool known as the "hob” (Fig. 32).
Fig. 33.—Hobbing a spur gear.
Fundamentally this is a spiral gear or worm provided with
cutting edges. Very, often—as in the case of the one
illustrated—the hob has only one thread (if it be regarded
as equivalent to a worm) or one tooth (if it be looked upon
as a spiral gear). In this connection it may be observed
that the provision of “ gashes ” (or “ flutes ”) in the hob
produces a number of cutting teeth, but this number has
no essential relation to the number of teeth in the gear to
which the hob corresponds.
In the hobbing machine the hob is mounted in the
same relation to the blank to be cut as would be the corre¬
sponding spiral gear to mesh.with the finished.spur gear.
For starting the cut, the hob is set clear of the end face
of the blank, and the two are then rotated, the hob at a
rate which gives its edges a suitable cutting speed and the
blank at the corresponding rate inversely proportional to
its number of teeth. The hob is slowly fed parallel to the
GEAR GENERATION
47
axis of the blank until it has traversed the whole face-
width. If the pitch of the gear is small enough, the hob
may be set to the full depth of tooth, and one traverse
suffices to complete the gear; usually, however, two or
more cuts are required.
The fundamental accuracy of the hobbing process is
associated with its continuous nature. For each cut the
whole width and circumference of the blank are covered
without change in speed in any part of the hobbing machine,
and there is thus no possibility of introduction of error by
variation of stress at any point.
The hobbing process applied to the generation of spur
gear teeth is shown in progress in Fig. 33. The cutting of
the teeth has commenced at the upper edge of the blank
and part of the face-width has been covered. The bottom
surfaces of the grooves cut by the hob teeth slope gradually
to the outer surface of the blank, this “ run-out ” resulting
from the circular path of the tips of the hob teeth. When
the hob has been moved vertically downwards until full-
depth teeth have been cut throughout the width of the
S^IND'HS
48 GEARS AND GEAR-CUTTING
blank, the run-out of the hob teeth demands clearance
which has to be taken into account when setting-up the
blank on the hobbing machine. Also, in the case of a
pinion made solid with its shaft, this consideration limits
the diameter of collars situated near to the ends of the
gear teeth.
Grinding of Spur Gear Teeth.—To secure the greatest
load capacity in spur gears, it is necessary to make them
of case-hardened steel, and if a high degree of accuracy is
demanded, the profiles must be finished by a grinding
process in order to remove the inaccuracies resulting from
distortion in hardening. Several types of such grinding
machines operate on the generation principle, and a typical
one is the Maag, illustrated in Fig. 34.
The grinding of the teeth is done by the outer faces of
the abrasive wheels, each of which is set to correspond to
one flank of a tooth of the imaginary basic rack. The
abrasive wheel is moulded to a saucer-shape, contact with
the gear tooth being limited to a narrow band near the
edge of the outer face. This band is trimmed by a
mechanically controlled diamond and, when viewed in
a direction parallel to the axis of the gear, appears as a
straight line, just as would the flank of the basic rack
tooth. The active band of the abrasive wheel thus lies in
the plane flank of the basic rack tooth and by feeding the
blank parallel to its axis the band has ultimately the
same effect as if it were a continuous abrasive face but
with the advantage that as contact between wheel and
gear tooth is confined to two very small areas at any one
instant, the load on the wheel is small. This is particularly
important because the wheel is necessarily thin, to pass
between adjacent gear teeth, and any considerable lateral
load would deflect it appreciably, thus introducing
inaccuracy.
Generating action, as already described in. connection
with the Sunderland gear generating machine, would
require the abrasive wheels to be moved perpendicularly
to the axis of the gear in unison with its rotation. It is
in general desirable, however, that a precision grinding-
wheel spindle should remain stationary in order that the
utmost degree of rigidity may be obtained. In the Maag
machine, this ideal is achieved and the necessary generating
GEAR GENERATION
49
action is secured by giving the blank both straight-line
motion perpendicular to its axis and rotation about that
axis at the correct relative rate.
Obtaining the Double Motion.—This double motion
is obtained, simply and accurately, by means of the
segment and tapes shown in Fig. 34. The radius of the
segment is that of the desired pitch circle of the gear teeth.
The segment is mounted on the end of the work spindle
into which the work arbor is fitted and four steel tapes
are attached to it, two passing horizontally to a fixed
bracket on one
side of the mach¬
ine, the other two
lying between
them and extend¬
ing to a tension¬
ing lever on the
other side of the
machine. (The
tapes are dupli¬
cated as a safe¬
guard against
damage should
one of them
break.) Thus
when the work
head moves per¬
pendicularly to
the axis of the Fig. 35.—Formed-wheel grinding of a spur gear,
work spindle, the
tapes cause the spindle to rotate as if the segment were rolling
without slip on a flat surface. Hence the gear teeth have the
same rolling action, and contact with the appropriately posi¬
tioned abrasive wheels thus produces ground involute profiles.
Generating action is produced by lateral reciprocation
of the work head, and to cover the whole face-width of the
teeth concerned, the gear is slowly fed parallel to its axis.
After it has moved clear of the abrasive wheels an auto¬
matic indexing motion occurs and on the next traverse
two other tooth profiles are treated. When dealing with
gears of large pitch the abrasive wheels may be situated
in the same tooth space, but usually they are in adjacent
4
50
GEARS AND GEAR-CUTTING
Fig. 36.—Spur and helical Fellows-type cutters.
spaces and when grinding gears of very small pitch they
may have to be even more widely separated.
The Maag machine incorporates special mechanism to
offset inaccuracy that would otherwise result from wear
of the abrasive wheels. A diamond-faced feeler periodically
follows a path in which it should make contact with the
active band of the abrasive wheel. If it fails to do so by
reason of wear of the wheel, the position of the wheel is
automatically adjusted until contact is established. By
this means the effective face of the abrasive wheel is kept
within one-quarter of a thousandth of an inch of its correct
position, despite wear.
The Formed-Wheel Process .—Certain types of spur-
gear tooth grinding machines do not use the generating
principle but depend upon direct reproduction of the shape
of the abrasive wheel in the gear teeth. Machines of this
nature are described as operating by the “ formed-wheel
process.” The wheel is trimmed by diamonds under the
control of mechanism whose movements depend on the
guidance of plates cut to the shape of the required tooth
GEAR GENERATION 51
profile enlarged about five times. It is usual to grind the
two profiles and the root surface of each tooth space at
once, and this makes the form-grinding process more rapid
than generation grinding, where the abrasive wheels are
less effectively used. (See Fig. 35.) The gear is moved
parallel to its axis so that the abrasive wheel may grind the
whole face-width ; after that, rotation of the gear through
an appropriate angle causes the wheel to grind a different
tooth space on the next traverse.
An advantage of the formed-wheel process is that it
facilitates slight modifications of tooth profile such as
are sometimes found necessary in heavily loaded gears.
Generation of Helical Gear Teeth .—The spur gear
Fig. 37.—Hobbing a helical gear.
52 GEARS AND GEAR-CUTTING
generating processes already described are all adaptable,
with some complication in mechanism, to the generation
of helical gear teeth. For example, in an elaborated form
of the machine shown in Fig. 28 the cutter-slide can be
set at any desired angle to the axis of the gear and then
helical teeth are generated with involute profiles on
transverse sections.
A similar adaptation is possible in the case of the
Fellows process (Fig. 29). A helical guide has to be pro¬
vided so that vertical movement of the cutter gives it
proportional angular movement, and the cutter itself
must have helical teeth. Fig. 36 shows a group of Fellows-
type cutters, some for producing straight tooth gears, the
others for helical gears. The helix angle of the cutter (and
therefore of the gear produced) depends on its diameter,
because the lead of the cutter must be that of the helical
guide.
Hobbed Helical Gears. —Adaptation of the hobbing
process to the generation of helical gear teeth is com¬
paratively easy. The angular position of the hob spindle
is adjusted so that the teeth, when cutting, are moving
in the direction of the tooth spirals of the gear to be
generated (see Fig. 37). Feed of the hob parallel to the
axis of the blank introduces a complication because it
must be accompanied by rotation of the blank through
an angle inversely proportional to the lead of its teeth.
Thus if
T—number of teeth in gear
L=lead of gear teeth (in.)
t—number of threads in hob
f=feed of hob (in.) per revolution of hob
N=rotational speed of gear
n=rotational speed of hob
then if there were no feed of hob
Since feed of the hob through a distance equal to the
lead of the gear would require one additional revolution of
the gear, the additional .speed of the gear corresponding to
feed of the hob at nf inches per minute is -j- r.p.m.
GEAR GENERATION
53
Hence the speed of the
gear is
N=f±£=n (*± *)
and this relation deter-
mines the ratio of the
change-gears required in
setting up the machine.
The double sign is used
because, according to the
relation of direction of Fig. 38. — Staggered arrangement of
feed of hob to hand of teeth in a hobbed double-helical
gear teeth and direction gear '
of rotation it may be necessary in one case to use -f- and
in another to use the — sign.
By adding a differential gear to the mechanism of the
gear-cutting machine it is possible to change the rate of
feed or even to discontinue it altogether without stopping
the rotation of hob and blank. This is often a great con¬
venience in practice, but the additional complication in
the machine tends to have an adverse effect on the accuracy
of the product.
The hobbing process is widely used for the production
of double-helical gears, and the Cf run-out ” of the hob
demands a gap between the right- and left-hand helices.
By adopting staggered teeth and permitting a chamfering
of the ends of the teeth the loss of effective face-width at
the centre is minimised (see Fig. 38).
Plano-Generating Processes for Double-Helical
Gear Teeth. —The Sunderland process, with rack-shaped
cutters, may be used for double-helical gears, but it
involves a special type of machine of which an example is
shown in Fig. 39. Here the cutter-boxes are guided to
move in directions oppositely inclined to the axis of the
blank, but the cutters themselves are set perpendicular
to it. One cutter moves towards the centre of the face-
width as the other recedes from it; cutting occurs on the
“ in ” strokes only. By setting the cutters so that they
very slightly overrun the central plane, each cuts a
clearance for the other, and it thus becomes possible to
54
GEARS AND GEAR-CUTTING
Fig. 39.—Piano-generating machine for double-helical gears.
produce continuous double-helical teeth by a generating
process. This type of tooth is somewhat stronger than
the double-helical tooth with a central gap, but the ad¬
vantage is slight in gears of normal proportions. Further¬
more, the load capacity of few double-helical gears is
limited by tooth strength.
The Sykes System.-— The Sykes process for producing
double helical gears is the counterpart of the Fellows
process; it uses simultaneously two helical pinion-type
cutters, one of each hand. The mechanical construction
of Sykes machines is elaborate, inasmuch as there is a
spiral guide for each cutter, a relieving motion for each
cutter to avoid rubbing on the idle stroke, and a hollow-
shaft mounting for one cutter encircling the shaft which
drives the other cutter. Nevertheless, machines of this
type can cut accurate gears at high rates of production,
and this is a tribute to care in detail design and to excellence
of manufacture.
GEAR GENERATION 55
Profile-Grinding of Helical Gears— The Maag, and
similar generation-grinding processes for spur gears, may¬
be adapted to profile-grinding of helical gears at the cost,
of course, of some complication in machinery. The
formed-wheel process may be similarly adapted, but a
difficulty is that the normal section of the abrasive wheel
is not exactly reproduced in the gear owing to interference
between the wheel and the helical teeth. Whilst it is
possible to determine by calculation the shape of an
abrasive wheel necessary to grind a helical tooth which is
of involute form on the transverse section, the operation
is more involved than in the case of spur gears, and the
formed-wheel process has had comparatively little appli¬
cation to the grinding of helical gear teeth.
Generation of Straight Bevel Gear Teeth. —The
fundamental principle underlying the operation of most
bevel gear generating machines is that of “ rolling” the
Fig. 40.—Straight bevel gear generating machine.
56 GEARS AND GEAR-CUTTING
blank with a crown-wheel which is imaginary save for a
cutter blade or blades continually sweeping two of its
togth flanks.
In Fig. 40 two straight-sided cutter blades can be
distinguished. These are reciprocated along lines con¬
verging at the apex of the machine and the cutter head
which carries their guides is given partial rotation about
the imaginary crown-wheel’s axis, which passes through
the apex. Simultaneously the blank (whose axis also passes
through the apex) is given proportional rotation so that
blank and imaginary crown-wheel are correctly “ rolled ”
together and the reciprocating cutter blades then form teeth
which will mesh correctly with teeth in other gears generated
in the same way.
After the rolling has proceeded so far that the tooth
flanks concerned are fully generated, the blank is with¬
drawn from the cutters by rotation of the work head about
a vertical axis passing through the apex, and the blank is
rotated through an angle corresponding to one tooth
pitch. At the same time cutter head and blank are
rotated back to the initial generating position. The
blank is then returned to the cutting position, and genera¬
tion of another pair of tooth profiles begins. These opera¬
tions are entirely automatic and the cycle is repeated
until all the gear teeth have been generated.
The generation of the teeth of the mating gear proceeds
in exactly the same way except, of course, that the machine
is “set up ” to suit the different pitch angle and the
different number of teeth. Thus the two gears are generated
so as to mesh with a common imaginary crown-wheel,
and as the teeth of such a wheel are symmetrical (neglect¬
ing clearance) about the pitch plane, the two gears will
mesh correctly with each other.
Generation of Single-Helical Bevel Gears. —The
process just described may be applied with slight modi¬
fication to the production of bevel gears with helical
teeth. The path of each cutter blade is still a straight line
which, however, does not intersect the axis of the imaginary
crown-wheel. The teeth of that wheel are thus straight
but not radial, and the teeth of bevel gears generated from
it are of tapered helical form. Such gears are described as
“ single-helical bevel ” gears.
GEAR GENERATION
57
Generation of Spiral Bevel Gears. —Consideration of
tooth action in helical bevel gears does not demand that
the teeth of the imaginary crown-wheel be straight. The #
advantages of helical tooth formation are secured if the*
crown-wheel teeth are of almost any form other than the
straight radial one that characterises the straight bevel
gear. A bevel gear whose corresponding crown-wheel
teeth are not straight is called a ft spiral bevel ” gear.
The Gleason spiral bevel gear is based on a crown-wheel
whose teeth are of circular arc form (Fig. 41). The
advantage of this is that the cutter blade may be carried
by a revolving disc and may thus return to take further
cuts by virtue of successive rotations of the disc. In
practice the disc is fitted with a number of equally spaced
cutter blades, so that cuts on any one tooth flank take
place in quick succession and cutting proceeds more,
smoothly than is the case in a machine having recipro¬
cating cutters. Each cutter blade acts on one tooth
58 GEARS AND GEAR-CUTTING
flank only, and successive blades deal with opposite
flanks. This feature can be distinguished in Fig. 42,
which shows a typical Gleason spiral bevel cutter having
“ inside” and “outside” blades placed alternately along
the edge of the circular body. Fig. 43 shows a Gleason
spiral bevel gear generating machine operating on a wheel,
and Fig. 44 shows a similar machine generating teeth in
a small pinion.
The rotation of the cutter about its axis causes its
blades to sweep out one tooth space of the crown-wheel.
Simultaneously the cutter is moved bodily in a manner
corresponding to a partial rotation of the imaginary
crown-wheel and the gear blank is given the appropriate
rotation in unison with it. These movements continue
far enough for complete generation of the flanks adjacent
to one tooth space. Then the cutter is withdrawn from*
the blank, the “ rolling ” motions are reversed and the
blank “ indexed ” so that on the next cycle another tooth
space is generated. The machine is fully automatic ;
once set to work it generates all the teeth in the gear and
then brings itself to rest.
The Gleason process is the most widely employed method
of producing spiral bevel gears, and it finds special favour
in dealing with large quantities of the small gears of this
Fig. 42.—A Gleason-type spiral bevel gear cutter.
gear generation
59
Fig. 43.—Gleason-type spiral bevel gear generating machine.
type used in automobile rear axles. By certain modi¬
fications in the machine it can be arranged to produce
“ offset spiral bevel ” gears to which the name “ hypoid ”
has been applied. An example of this type of gear is
shown in Fig. 45.
Another modification of the Gleason spiral bevel gear
is the “ formate ” gear. In this the teeth of the wheel are
not generated at all, the blank and cutter head being
held stationary whilst each tooth space is treated and the
teeth, therefore, have the same shape as the cutter blade.
The pinion teeth are generated substantially in the stan¬
dard Gleason manner. This process is a specialised
one and can be applied econoifiically only when several
6o
GEARS AND GEAR-CUTTING
Fig. 44.—Generating a Gleason-type spiral bevel pinion.
hundreds of pairs of gears are to be made in one
batch.
Worm-Wheels. —The only practicable way of produc¬
ing accurate worm-wheel teeth is by generation with a
hob whose essential dimensions are the same as those of
the worm which is to mate with the wheel. A typical
worm-wheel generator is shown in Fig. 46. The hob is
mounted horizontally and rotates at a rate which imparts
a suitable cutting speed to its teeth. .
The worm-wheel blank is arranged
with its axis vertical, and its speed
of rotation is that which the com- llg
pleted worm-wheel would have when j||g ( |B
meshed with a worm running at the l g|| \Jl
hob’s speed. The head carrying the wSl fL
hob and its driving mechanism is
able to slide horizontally, and during
the generating process it is caused to Fig. 45.—A hypoid gear.
GEAR GENERATION bl
do this so that the hob penetrates more and more deeply
into the blank until finally the distance between their
axes is equal to the centre distance of the finished gears.
A hob is an expensive tool to manufacture, and its cost
may be prohibitive if it is required to produce only one or
two worm-wheels. In such cases it is common to use a
“ fly-hob,” which corresponds to one. tooth of the com¬
plete hob (see Fig. 47). In order that this tool may effect
complete generation of the worm-wheel, the two must
have the appropriate relative speeds as for worm and
worm-wheel, and in addition the hob must be fed along
its own axis so that it gradually penetrates more deeply
into the blank, the centre distance between them remain¬
ing constant throughout the generating operation. This
62
GEARS AND GEAR-CUTTING
axial feed of the hob means that the worm-wheel blank
must be given the corresponding rotation superimposed
on the one which it already has in association with
rotation of the hob.
Generation by “ fly-hobbing ” is naturally much slower
than is possible with a complete hob, because the single
tooth has to do the amount of work that is spread over
perhaps fifteen teeth in the complete hob. To obtain
smoothly finished teeth the axial feed of the hob has to
be slow.
Axial feed is often adopted when using complete hobs
with the object of securing a higher degree of tooth finish
than is practicable with the direct “ in-feed.” For such
applications the hob is usually made of tapered form over
part of its length, the roughing-out of the worm-wheel teeth
being effected by the teeth at the small end of the hob.
Fig. 47.—The generation of a worm-wheel by means of a " fly-hob."
GEAR GENERATION
63
Fig. 48.—Cutting a worm in a thread-milling machine.
When quantities of worm-wheels are being generated it
is usual to finish the cutting of the teeth in a separate
operation from the initial rough-cutting. In such cases
the finishing of the teeth is often carried out by means
of a “ serrated hob ” which has a large number of finely-
pitched teeth.
Cutting of Worm Threads. —The worm threads are cut
in a thread-milling machine (Fig. 48) or (in certain circum¬
stances) by hobbing. The thread-milling _ cutter rotates
about an axis whose angular position is adjustable to suit
the lead-angle of the worm to be produced. The table
which carries the work spindle can slide parallel to the
spindle and in actual operation this sliding occurs in unison
with rotation of the blank, change-gears being provided in
order to afford means of accommodating any required lead
of worm. This combination of rotation and axial move¬
ment of the blank past the rotating cutter causes it to cut
64 GEARS AND GEAR-CUTTING
a helical groove in the blank. When the whole width of
the blank has been covered, the blank is quickly returned
to the starting-point, and after " indexing/' the process is
repeated to cut another thread. Further repetition takes
place until all the threads have been completed.
Grinding Worms. —Worm threads are nearly always
Fig. 49.—A worm-thread grinder, finishing the rough-cut worm.
subjected to a grinding operation, and this is carried out
on the type of machine shown in Fig. 49. The movements
of the machine are similar to those of the thread-miller,
but the milling cutter is replaced by an abrasive wheel
and the operation is entirely automatic. Each flank of
each thread has usually to be ground several times to
bring the thread thickness down to the required figure. *
In grinding worm threads of the involute helicoid type
gear generation 65
(the British Standard) the abrasive wheel may be flat-sided
and this gives the advantage that the wheel may be
trimmed by a diamond moving in a straight line. By
setting the axis of the abrasive wheel at the appropriate
angle it touches^ the worm thread along the straight line
which characterises the involute helicoid worm. On this
basis only one thread flank can be ground at once.
Alternatively, two flanks may be ground at the same time,
but in this case the abrasive wheel has to be shaped to
two curved contours by diamonds moving under the
control of specially shaped former plates. The shape of
the plates is determined by calculation according to the
particular thread form required.
Finishing Worms. —In order to obtain the necessary
mirror finish on worm threads the grinding operation
must be followed by treatment with fine abrasive. To
effect this the worm is rotated in mesh with a lap in the
form of a wooden worm-wheel charged with abrasive
paste. Power is applied to the worm which thus rotates
the worm-wheel against the resistance of an adjustable
brake and is consequently polished by the paste. To
cover the whole face-width of the worm it is reciprocated
in the direction parallel to its axis.
5
CHAPTER V
Gear-Wheel Forms
Small gears may be made in the form of solid discs, but
as this is not economical of material the armed wheel is
common in large sizes. An example is the cast-steel
wheel shown in Fig. 2, and it will be noticed that the
arms are of H-section to give adequate strength with a
comparatively small amount of material. The particular
gear shown has a
“ split boss,” this
form of construction
being adopted in
large gears in order
to reduce the strains
caused by contrac¬
tion of the metal as
it cools in the mould
when in the solid
state immediately
after casting. When
the boss is made in
one piece the arms in
the finished wheel
are in a state of
tension; splitting of
the boss avoids
this. After the boss
has been rough-
machined, the splits
are filled with molten
brass, and after this
has cooled the boss is consolidated with two steel rings,
one shrunk on to each projecting end.
An alternative way of diminishing contraction strains
is to make the wheel in halves, and although this is more
expensive by reason of extra material and machining, the
reduction in dimensions of the largest piece to be handled
often brings important advantages in transport and
Fig. 50.—Cast-steel double-helical gear¬
wheel in halves.
GEAR-WHEEL FORMS 67
Fig. 51.—A double-helical gear-wheel built up by welding.
erection. Fig. 50 shows a large gear-wheel made in halves
and fastened together by means of bolts.
Built-up Gears.—When loading conditions demand
the use of a material of such hardness that it cannot be
obtained as a casting, the wheel may consist of a forged
steel rim shrunk on to a " centre ” made from cast-iron
which is, of course, comparatively inexpensive.
There is nowadays a tendency to build up steel gear¬
wheels by welding the rim to arms cut out of plates, and
very often the design is such that the wheel is not easily
distinguishable from a cast wheel of conventional design.
Alternatively the arms may be made from standard
rolled-steel sections (see Fig. 51).
Because of the high cost of the phosphor bronze used
68
GEARS AND GEAR-CUTTING
for worm-wheel teeth, such wheels are usually made as
rims secured to cast-iron centres by shrinking and pegging,
by bolting, or by welding.
Pinions. —The pinion of a high-ratio pair has often to
be made solid with its shaft because a bore large enough
to accommodate a separate shaft of adequate strength
would leave insufficient metal below the roots of the teeth.
In some cases ( e.g „ Fig. 52) the effective face-width
necessary for load capacity is so great in relation to the
diameter of the pinion that extra support must be provided
by a third bearing separating the two halves of the
face-width.
Automobile Gears. —Weight considerations demand
Iftw;-:
mm
Fig. 52.—Turbine reduction gears, showing each pinion arranged
for a third bearing between the right-hand and left-hand helices.
GEAR-WHEEL FORMS
69
Fig. 53.—Spiral bevel gears for automobile rear-axle drives,
the minimum dimensions in automobile gears, and case-
hardened steel is almost invariably used in such cases
because of its high load capacity. Fig. 53 shows a typical
group of rear-axle spiral bevel gears, the pinion being
made solid with its shaft and the wheel bored to fit on
to an intermediate shaft. In heavy automobiles the rear-
axle drive is usually a pair of worm gears, such as is
shown in Fig. 54. This type of gear is specially suited to
the comparatively high reduction ratios (between 6 to 1
and 10 to 1) required in such drives, and its quiet running
qualities are specially valuable in trolley-buses because in
that class of vehicle all other noises have been reduced
to a negligible total.
Automobile Change Gears. —These are now so accur¬
ately cut that the noisiness associated with them is a thing
of the past. This is due not only to the use of helical
gear teeth but also to ground teeth, the use of shafts of
stouter diameter which do not whip, and to improved
methods of mounting.
The use of constant mesh gears instead of sliding dogs
has also contributed to gearbox efficiency.
In the synchromesh gearbox the constant mesh gears
70 GEARS AND GEAR-CUTTING
are fixed on the primary shaft and countershaft as also
are. all the countershaft gears, but the second and third
mainshaft gears are free to revolve on the mainshaft,
and are engaged by means of dogs formed on the gears.
It must be remembered that in replacing worn gears
it is advisable to renew them in pairs, as replacement of
one of the pair invariably tends to create noise and cause
accelerated wear of the new gear.
Mention must also be made of the overdrive. One
type takes the form of two bevel pinions in the rear axle.
Fig. 54.—Worn gears for automobile rear-axle drive.
meshing with crown-wheels of different diameter. Either
of the bevel pinions can be locked to the driving shaft by
means of a dog clutch, so that either a high or a low back-
axle ratio is available. It will be seen that with this
arrangement, not only are two top gears available but
also a low and high ratio for each of the other gears.
The same effect is obtained in the case of certain large
passenger and commercial vehicles by providing an
auxiliary two-speed box behind the gearbox. A further
alternative is to provide an additional ratio providing an
overgeared top in the gearbox itself.
CHAPTER VI
Epicyclic Gear Trains
A gear train in which any gear rotates about a centre line
which is not fixed is called an “ epicyclic ” train. In the
example in Fig. 55 the gear B runs on a pin carried by arm
D which rotates about the axis XX. Gear A is carried by
another shaft whose centre line is also XX. Gears A and B
mesh together, and B also meshes with the internal gear C
whose centre line is also XX. If C is prevented from
rotating, rotation of A will cause rotation of D with a
definite velocity ratio. Similarly, if any one of the members
A, B, C, and D is prevented from rotating, the motions of
the other three are governed by fixed relations. In the
following table, each of the letters A, B, C, and D refers to
the number of teeth in the corresponding gear:
Revolutions of
Gear A
Gear B
Gear C
Arm D
Case 1.
D fixed
1
A
B
A
C
0
Case 2.
C fixed
, A
I+ C
A A
C B
0
A
C
Case 3.
B fixed
0
A A
B — C
A
B
Case 4.
A fixed
0
A ,
B +I
A ,
C +I
1
Case 3 is included for the sake of completeness, although
it is unlikely to occur in practice. The condition is that B
is prevented from rotating although permitted to move in
a circle in accordance with the rotation of D.
71
72 GEARS AND GEAR-CUTTING
To illustrate the use of the table, consider the assembly
in which
A =20 B=i8 C=56
and C is fixed. Then
A makes i-f ~ = 1*357 revolutions while
5b
B makes ^ = -0-754 revolutions and
56 18
D makes
20
56
= 0*357 revolutions
If A is the driving gear, and D the driven member, then
* 1 * 3^7
Velocity ratio = b 7 ^^ 3 * 8 '
r _j Gear A is called the
" sun/' and B is a
“ planet/' In order to
obtain greater load-
capacity and to im¬
prove the balance of D,
more than one planet
may be used. Such an
arrangement cannot be
assembled, however,
unless (A+C) is exactly
divisible by the number
of planets.
Compound epicyclic
1 -1 gear trains are often
used for obtaining very
high velocity ratios. The revolutions of the gears in the
particular compound train, shown in Fig. 56, are given, for
several important cases, in the table overleaf.
Case 2 is the one most frequently used in practice, with
A the driving gear and E the driven gear. If B and C are
nearly equal, the velocity ratio between A and E becomes
very high.
The method of constructing tables, such as those given
above, is simple and is applicable to any epicyclic gear
assembly. In the first instance, the member carrying the
Fig. 55. — An
example of
an epicyclic
gear train.
EPICYCLIC GEAR TRAINS
73
A
B and C
D
E
F
Case 1.
F fixed
1
A
B
A
D
A
B
C
E
0
Case 2.
D fixed
1+ D
A A
D~B
0
A A
D~B
C
E
A
D
Case 3.
A fixed
;
0
A ,
B + 1
A
D + I
A C
B E
+ 1
X
Case 4.
E fixed
, A C
I+ B E
A /C \
B (e z )
A C
B E
A
D
0
A C
B E
Table showing the revolutions of the gears in the compound train
illustrated in Fig. 74.
planetary spindle is assumed to be fixed. Any other
member is imagined to have one revolution, and the corre¬
sponding revolutions
of the remaining
members are deter¬
mined exactly as
for an ordinary (not
epicyclic) train.
This is Case 1. Now
if any particular
member is to be con¬
sidered at rest, the
numbers of revolu¬
tions of all members
are each reduced by
the amount neces¬
sary to make that
particular member’s
revolutions equal to Fig. 56.—A compound epicyclic gear train,
zero. Thus, in the
first table, Case 2 is derived from Case 1 by subtracting
A A\
— ^ {i.e. adding to each quantity.
Although the compound epicyclic gear is a means of
obtaining a high velocity ratio in small space, its efficiency
is low if the ratio is high, and on that account it is suitable
only for small powers.
CHAPTER VII
Methods of Mounting
Gears have to be manufactured to a high degree of
accuracy for two main reasons :
(a) to permit of reasonably quiet running, and
(b) to permit of adequate distribution of load
across the face-width.
The first of these requires that errors of tooth shape be
kept within two or three ten thousandths of an inch, that
pitch errors between adjacent teeth be less than about
half a thousandth of an inch, and that the pitch error
between any two teeth in the gear be less than one.or two
thousandths of an inch. A reasonable approximation to
uniformity of distribution of load requires that the tooth
Fig. 57.—A self-contained single-reduction turbine gear unit (hobbed
double-helical gears).
74
METHODS OF MOUNTING 75
spiral shall be correct within about one thousandth of an
inch across the width of the gear.
These degrees of accuracy can be achieved by appropriate
care in manufacture, but they are quite unavailing unless
the gears are mounted with equal accuracy. In some
instances, particularly highly-stressed units such as auto¬
mobile rear-axle assemblies, distortion of the mounting
under the bearing-loads produced by the gears is sufficient
to affect the accuracy of meshing to a serious extent, and
the initial “ setting-up ” has to be adjusted so as to
counteract the full-load distortion. Such adjustment
Fig. 58 .—A self-contained single-reduction double-helical gear unit.
means that spread of contract over the whole face-width
can occur only at full load or some selected fraction of
full load and emphasises the desirability of the utmost
rigidity in the mounting.
Spur- and Helical-Gear Units.—This consideration in
conjunction with the necessity for protection of the gears
from dust and dirt and for their lubrication has led to the
use of self-contained gear units. These are standardised
in type and size so that manufacturing costs are minimised.
Bearing details and lubricating arrangements are settled
once and for all, and the only variations between indi¬
vidual units lie in the gear ratio.
76 GEARS AND GEAR-CUTTING
The parallel-shaft type of gear unit (Fig. 58) may contain
spur, single-helical or double-helical gears and the bearings
are usually of the ball or roller type. Unless the peripheral
speed is higher than about 2500 feet per minute, lubrication
of the gears is effected by allowing them to dip in a bath
of oil contained in the lower part of the case. Oil is
splashed on to the walls of the case, and troughs are pro¬
vided to guide into the bearings the small amount of
lubricant that they need.
Lubrication Problems. —Gear units of this type may
be used in conjunction with steam turbines, but the high
speeds concerned necessitate some modification in detail.
White-metal-lined journal bearings are the rule in appli¬
cations of this sort and oil is fed to them under pressure
to ensure the flow required to avoid excessive rise in
temperature. Oil is sprayed on to the gears near to the
point of mesh, oil bath lubrication being objectionable
owing to the considerable power loss which occurs at very
high peripheral speeds. In turbine reduction gears the
peripheral speed may be as high as 15,000 feet per minute.
Bevel-Gear Units. — For connecting shafts whose
centre-lines are perpendicular, either bevel-gear or worm-
METHODS OF MOUNTING 77
gear units may be used, the former having some prefer¬
ence if the gear ratio required is lower than about 4 to 1.
An example is shown in Fig. 7, the horizontal shaft
being driven by an internal-combustion engine and the
vertical shaft transmitting power to a centrifugal pump
situated some distance below the engine-room* Installa¬
tions of a similar character are associated with deep bore¬
hole pumps, whose speeds are often three or four times as
high as those of the engines which supply the power.
Worm-Gear Units. —Largely because of the wide
range of ratios that it can conveniently afford the worm
drive is extensively used in industrial service. It also
gives high torque capacity in small space and is con¬
sequently an inexpensive form of gear transmission. Its
mechanical efficiency is lower than that of a helical-gear
unit, but this disadvantage is insufficient to outweigh the
worm gear’s compactness and low initial cost. A rough
estimate of the overall efficiency of a worm-gear unit may be
obtained by subtracting half the reduction ratio from 100 ;
this rule apples for ratios between about 10 to 1 and 50 to 1.
It is very often found that the load capacity of a worm-
gear unit with natural cooling is limited by temperature-
rise rather than by endurance of the teeth. In order to
raise this limitation it is now usual to provide artificial
cooling by mounting a fan on the worm shaft and arranging
to direct its draught between fins cast on the gearcasing.
Fig. 59 illustrates a typical worm-gear unit of this type.
CHAPTER VIII
Measuring Gears
A Gear dimension that is invariably checked either
directly or otherwise is the tooth thickness. Given an
accurate machine and cutter, the desired tooth thickness
may be secured by sinking the cutter to the appropriate
depth in the blank, and whilst in practice this is often
done (with the aid of measuring devices built into the
machine and indicating the movement of cutter-mounting
relatively to the blank), the final check is usually made by
Fig. 6o.— A vernier caliper for measuring the thickness of gear teeth.
means of a gear-tooth caliper (Fig. 60). This has a vernier
adjustment, setting the edge of a blade at the required
height above two gauging points. The distance between
these points is measured by a second vernier. The caliper
thus measures the thickness of a tooth in a plane at any
required distance from the tip of the tooth.
The vernier graduations correspond to steps of *ooi in.,
but owing to the difficulty of reading such fine graduations
it is not usually possible to guarantee a setting to less than
78
MEASURING GEARS
79
*002 in. Even this order of accuracy more than meets
nornal requirements, any error occurring on this account
being easily absorbed in the backlash allowance which is
rarely less than about 0*005 in.
The gear-tooth caliper may be applied to any type of
gear tooth and usually, though not essentially, it is set to
measure the tooth thickness on the pitch cylinder of
generation.
Micrometer Measurement of Tooth Thickness.—
In certain circumstances the tooth thickness of involute
spur gears may be determined from micrometer measure¬
ment over the teeth. In Fig. 61 are shown a number of
tangents to the base circle of a spur gear. From the
construction of the involute it follows that each of the
tangents shown intersects the two " opposed ” involutes
at right angles. Also the distance between the points of
intersection of a tangent with the two involutes is equal
So GEARS AND GEAR-CUTTING
to the distance PQ (measured round the base circle)
between the origins of the two involutes. Hence the
distances AB, CD, EF are all equal and consequently a
micrometer measurement over any two " high points ”
such as A and B on the opposed involutes X and Y gives
the same dimension (see Fig. 62). This can be calculated
from the tooth thickness and the number of teeth between
X and Y. Obviously this method can be used only where
the dimensions of the gear are such that a tangent to the
base circle can be drawn to intersect two involute profiles
in points to which a micrometer can be applied. Alterna¬
tively use may be made of a vernier caliper, but this is
less accurate than a micrometer.
Fig. 63.—Checking tooth thickness by micrometer measurement
over rollers.
Tooth thickness may also be determined from micro¬
meter measurement over rollers placed in two tooth spaces
(see Fig. 63).
Checking by Comparison.—When gears are being
made in quantities, it is often more convenient to check
tooth thickness by comparison with a master gear than
by direct measurement. Fig. 64 shows this principle
applied to the checking of thickness of worm threads.
The worm is mounted between centres, and a ball held
between two threads presses on the flat gauging anvil of
a dial indicator as the worm is slowly rotated. The read¬
ing of the indicator is compared with that given in similar
circumstances by a master worm. Repetition of the
operation with the ball in each thread-space in turn affords
a check on concentricity.
MEASURING GEARS
8l
For checking the thickness of worm-wheel teeth a ball
micrometer (Fig. 64) is often used ; here again the reading
obtained is compared with that derived from a master gear.
Tooth Shape.—The simplest method of checking the
tooth shape of spur, helical or straight bevel gears is to
lay against a profile a correctly formed plate gauge. Such
a gauge is made by filing until its optical projection (at
25 to 50 magnifications) matches an accurate large-scale
drawing.
A comparable direct check on tooth form is afforded by
using a gear-tooth caliper to measure tooth thickness on
various planes be¬
tween tip and root
and comparing the
figures with those
obtained by calcu¬
lation.
These methods
are, however, in¬
sufficiently accur¬
ate for modern
precision gears, and
instruments have
therefore been de¬
veloped by which
the departure of
the profile from
the desired form is
directly indicated.
An example is the
instrument shown
in Fig. 66. Here
the gear under in¬
spection is mounted
on the same spindle
as a disc of dia¬
meter equal to that
of the base circle
of the gear. The
disc is pressed into
contact with a p IG . g 4 .—Checking the thickness of worm
straight-edge C by threads by means of a ball and indicator.
6
82
GEARS AND GEAR-CUTTING
means of screws acting through the rods A A and the
bar B. Endwise movement of the bar causes the gear to
roll along the straight-edge and (looking parallel to the
axis of the gear) the point of intersection of a true
involute profile with the straight-edge will not change its
position.
The checking of the profile is effected by a stylus placed
so as to touch it on the line of the straight-edge and
operating the plunger of a dial indicator reading in ten-
thousandths of an inch. Departure of the tooth profile
from the true involute form is shown directly by the
reading of the dial indicator.
A rapid check on accuracy of profile and concentricity
of teeth and bore is afforded by the type of gear-tester
shown in Fig. 67. Here the gear under test is slipped on
to a spindle supported in a movable carriage spring-loaded
in such a way as to press the gear into contact with a
master gear mounted on a parallel spindle. The fit of
the spindles in the gears is such that they can be
rotated by hand, when any error in profile or concen¬
tricity causes variation in centre distance between the
Fig. 65.—Checking the thickness of worm-wheel teeth by the use
of a ball micrometer.
MEASURING GEARS
83
AflKSON GEAR TESTER
Fig. 66 .—An involute-testing instrument.
spindles. This variation is shown on a dial indicator,
and a permament record may be taken by a pen which
traces a line on a disc rotating in unison with the master
gear.
A perfect combination of gears results in a circular
track; any variation in centre distance as the gears
revolve is reproduced with suitable magnification as a
radial departure from the ideal circular form.
The instrument shown in Fig. 67 is designed to accom-
84 GEARS AND GEAR-CUTTING
modate spur or helical gears. The same general principle
applies in other machines arranged to take bevel gears and
worm gears.
Profile Checking of Worm Gears.—Some worm-
thread profiles show straight sections in planes passing
through the centre-line of the worm. In such cases,
checking of shape may be effected by a gauge fixed to a
V-block placed in contact with the tips of the worm
Fig. 68 .—Checking a worm thread of the involute helicoid type
with a straight-edge.
threads. A more accurate method, however, is to mount
the worm between centres and to apply to it a stylus
pivoted on a block constrained by an adjustable guide to
move parallel to the correct thread profile. Any error in
profile is detected by the stylus and is made visible on a
dial indicator.
A worm thread of the involute helicoid type may be
checked for shape in the manner shown in Fig. 68. Here
a straight-edge is positioned so as to correspond to a
generator of the thread, and the worm is rotated so as
measuring gears 85
S a thr , e ^ “ t0 contact with the straight-edge.
Examination of the line of contact with a light placed on
5®. ^ ar slde °f the straight-edge reveals proffle errors
which, if large enough, may be estimated by use of feeler
gauges. J
The form of worm-wheel teeth is so complex that the
Fig. 69.—The principle of an instrument for testing
spur or helical gears.
only practicable way of checking it is to mount the
wheel in correct relation to a master worm whose
threads have been lightly smeared with “blue marking”
and after rotation of the gears to observe the distribution
of marking on the wheel-teeth.
86 GEARS AND GEAR-CUTTING
Pitch-Testing.—The principle involved in the pitch¬
testing instrument shown in Fig. 69 is to bring into contact
with two gear teeth a pair of styluses mounted in such a
way as to show on dial indicators the amount by which
the distance between the gauging points differs from the
original setting. These errors are recorded for consecutive
pairs of teeth until the whole circumference of the gear has
been covered. The algebraic
sum of the true errors (the
accumulated error) is, of
course, zero for the whole
circumference, and the actual
sum in any particular instance
is a measure of the unavoid¬
able errors in setting the in¬
strument. The necessary
corrections to the individual
readings are conveniently
made by recording them
graphically as shown in Fig.
70. The straight line AB
joining the first and last points
is the base line from which the
accumulated pitch errors are
measured. Thus the accumu¬
lated error between tooth 1
and tooth 12 is represented
by CD, i.e, 0-0003 in-
In the instrument shown in
Fig. 69 the gauging elements
are carried on a pillar which
is swung about its anchor-pin
to bring them into and out
of the measuring position.
This is done by means of the
handle shown, which is held
against a fixed pin by a
spring-loaded pawl to ensure
uniformity of location when
measuring. This move¬
ment of the gauging
elements is necessary
5 * * £
MEASURING GEARS 87
to permit of rotation of the gear for successive
readings.
In order to deal with large gears which could not con¬
veniently be rotated about a fixed axis for measuring
purposes, the pitch-testing instrument shown in Fig. 71
may be employed. The framework is held in contact with
the tips of the teeth of the gear to ensure its correct location
in relation to the centre-line of the gear. Circumferential
location relative to a tooth is by means of a ball attached
rigidly to the frame, whilst a dial indicator measures the
adjacent error between any two teeth that are not too
widely separated.
A check on accumulated error in gears which are small
enough to be rotated on centres is shown in principle in
88 GEARS AND GEAR-CUTTING
Fig. 72. The instruments are set to measure the accumu¬
lated error between two teeth nominally separated by
180 deg. By rotating the gear through an angle which
causes the gauged teeth to interchange their positions, the
instruments may be made to measure the errors in the
two different half circles. The difference between these
readings is equal to the difference between the true errors,
whose sum must be zero. Hence the difference between
the readings is equal to twice the numerical value of the
accumulated pitch error in either half circle.
Angular Division Tester.—Measurement of pitch
errors in gear teeth may be effected by an optical method.
A theodolite is mounted on the gear in such a position
that its centre-line intersects that of the gear at right-
angles. The mounting is such that relative angular move¬
ment of gear and theodolite can be accurately measured
by means of a finely graduated dividing plate and a
“ built-in ” microscope.
The gear is first located angularly, by allowing a tooth
to bear on a stylus, such as one of those shown in Fig. 69,
accurate positioning being ensured by use of a sensitive
dial indicator. The theodolite is sighted on to an illu¬
minated target situated at least ten feet away, and the
reading on the dividing plate is recorded. The gear is
then rotated until a different tooth is located by the
stylus, the theodolite is redirected on to the target, and
the difference between the new reading on the dividing plate
and the previous one is the angle between the selected teeth.
'An important point is that this procedure measures
• accumulated error directly and not merely as the sum of
measured adjacent errors. On this account the gain in
accuracy of measurement is considerable.
Spiral Angle of Helical Gears.—The instrument
shown in Fig. 73 is used to check spiral angle and con¬
centricity of helical gears. It employs a gauging member
in the form of a tooth of the basic rack corresponding to
the gear teeth. This member is mounted at one end of a
spindle which carries on its other end an arm having two
projecting pins. These permit a micrometer or slip-gauges
to be used to effect accurate angular setting of the gauging
member, and when this has been done a dial indicator is
set with its operating plunger in contact with one pin.
MEASURING GEARS
89
The spindle assembly is then moved bodily towards the,
gear until the gauging member has penetrated as far as
possible into a tooth space. Any error in spiral angle acts
on the gauging member to rotate the spindle, the amount
of such movement being shown on the dial indicator.
The distance between the spindle mounting and the
axis of the gear when the gauging member is fully engaged
is measured with a second dial indicator, and variation of
this reading from one tooth space to another is a measure
of eccentricity.
Checking by Running Together.—The most trouble¬
some effect of errors in high-speed gears is that of noisy
running. Consequently, if it is convenient to mount a
pair of gears in bearings and to run them at full speed,
this is the most satisfactory method of checking them.
If they pass such a running-test it may be concluded that
any errors they do possess are too small to be of any
practical importance.
90 GEARS AND GEAR-CUTTING
Gear-Tooth Comparators. — Until comparatively
recently the only means of measuring gear teeth was by
means of the gear tooth-caliper, an instrument necessitating
two settings for each pitch measured, and also a different
setting for any variation in the number of teeth for a given
pitch. That is to say, if a gear of 20 teeth, six d.p,, is being
measured, a different setting is required for a gear of 30
teeth of the same pitch. This complication arises out of the
fact that the gear-tooth caliper measures the thickness at
the pitch line, so that for gears of a given pitch, but of
varying numbers of teeth, a calculation has to be made of
the chordal thickness and also the chordal height.
Fig. 74.—A diagram explaining the principle of the Sykes gear-tooth
comparator or caliper.
All workmen, unfortunately, are not able to make this
calculation, and they are rather apt to set the instrument to
measurements given in tables, which, of course, require
correction according to the number of teeth; Another dis¬
advantage of this instrument is the fact that the measure¬
ment of the tooth thickness is taken on the comers of the
two jaws, so that the wear is concentrated at these points,
and leads to a rapid deterioration of the instrument.
Two instruments which have largely overcome these
difficulties are the Sykes' comparator and the Sykes' caliper
for measuring gear teeth. The comparator and its method
of application is shown in the accompanying illustrations.
It will be observed that, in fine, the instrument consists
of a substantial frame or beam carrying one fixed and one
MEASURING GEARS 91
adjustable jaw, in addition to a specially designed dial test
indicator. The movable jaw is provided with means for
fine adjustment, and between the jaws the plunger of the
dial indicator projects. Fig. 75 shows the instrument set to
a master gauge block. Fig. 78 shows the instrument
complete in case, with a set of master gauge blocks.
The Working Principle.—The principle on which the
instrument is based will be obvious to those possessing a
knowledge of the property of the involute. Each gauge
block represents an involute rack tooth of a particular
pitch and pressure angle, and the inclined faces of the jaws
are made to correspond. It is well known that any gear of
a given pitch, irrespective of the number of teeth, will
accurately gear with a rack of the same pitch and pressure
angle. Therefore, when the comparator is set for a rack
tooth (represented by the gauge block), the setting is
correct for all teeth of the same thickness, pressure angle
and addendum. It will readily be apparent that any tooth
9 2 GEARS AND GEAR-CUTTING
thicker or thinner than that setting will cause the dial
indicator to show plus or minus, owing to the teeth entering
to a lesser or greater distance between the jaws.
Using the Comparator.—For convenience the dial of
the indicator can be rotated so as to alter the position of the
zero mark. When it is only desired to test teeth for
uniformity, gauge blocks may be dispensed with, as a
master gear may be used to set the comparator. The jaws
of the instrument and also the gauge blocks are glass hard
and are ground and lapped to a minute degree of error.
The standard gauge blocks are supplied at 144 degrees
pressure angle, but of course, blocks of any pressure angle
can be supplied.
MEASURING GEARS 93
One division on the dial represents *0005 in. for teeth of
14J degree pressure angle. The instrument has a capacity
for 12 d.p. to 1 d.p., or from J in. to 3 in. circ ular pitch.
It must not be assumed that a gauge block is necessary
for every pitch to be measured. Seven blocks will measure
all pitches ; six blocks will measure all pitches between
12 d.p. and ij d.p. and four blocks will measure every
pitch between 12 d.p. and 3 d.p.
Fig. 76 shows the method of testing for uniformity of
tooth thi ckn ess; a series of three instruments may be
used for testing tooth curves. Due to the principle of the
ins tr um ent, no alteration or adjustment in any _way is
necessary in cases where the addendum of the pinion is
increased or decreased, provided that the addendum of the
94 GEARS AND GEAR-CUTTING
wheel into which the pinion gears is similarly decreased or
increased. In either case the jaws automatically bear on
the theoretical con¬
tact points of the
tooth curves.
Zeiss Optical
Gear - Tooth
Caliper.—This in¬
strument is similar
in principle to the
vernier gear-tooth
caliper but differs
from it in that
the “ height ” and
“ thickness ” scales
are on glass and can
be seen only through
a magnifying glass
built into the in¬
strument. They lie
at right angles to
each other and their
Fig. 78.—The comparator complete in case, point of intersection
with a set of master gauge blocks. is at the centre of
the field of view. The longitudinal line of each scale acts
as the fiducial line for the other.
The Sykes Gear-Tooth Caliper.—The diagram. Fig. 74,
also explains the principle of the caliper shown in Fig. 79,
designed by the originator of the comparator. The lines
aa , aa represent a rack-tooth space, and the dotted lines
the jaws of the Sykes caliper, which it will be seen directly
represent the rack. The line c is the pitch line of the rack,
and d the pitch line of a particular pinion or wheel, bb and
bb are the lines of pressure or lines of action, on which tooth
contact must always occur, whatever the size or number
of teeth of the gear. These lines are always the same for any
given pressure angle, and it is well known that in the in¬
volute system the rack tooth has straight sides, and the first
law of gear-tooth contact is the common normal to the tooth
curves must pass through the pitch point. In the diagram
the pitch point is fi } and the lines b pass through it and are
normal to the rack-tooth profile. It will, therefore, be
MEASURING GEARS
95
apparent that contact must always take place on the lines
b t and that the position of the point of contact e depends
only on the thickness of the tooth. In gears of any particular
pitch the profile of the tooth is different for each number of
teeth, but the point of contact, with the centre-line of the
tooth and tooth space coincident, never varies.
Base Circle.—In measuring helical or spur-gear teeth
an important dimension is the base diameter of the gear.
This is denoted by d 0 and its value is given by
d 0 =—* cos <J/ t or Sfr -——^-—7 — r - r
TZ 7TT v (cos 2 CT-ftan 2 ^ a )
where t —number of teeth in gear.
p t =transverse pitch of teeth (measured in plane
perpendicular to axis).
p n =normal pitch of teeth (measured in plane per¬
pendicular to tooth spiral).
t{; t =transverse pressure angle (measured in plane
perpendicular to axis).
^ n =normal pressure angle (measured in plane per¬
pendicular to tooth spiral).
o- =spiral angle at pitch cylinder.
o- 0 == spiral angle at base cylinder.
L=Lead of spiral.
GEARS AND GEAR-CUTTING
The base spiral angle cr is determined from the given
quantities by the relation
sin c^-sin <r cos
and
J tp t • tp n
tan or sm cr~-jr~
a ^ so tan ^ t =tan sec cr.
Another important mathematical quantity is the " in¬
volute function ” of an angle.
Thus the involute function of 0 is denoted^ by inv. 0 and
is equal to tan 0—0, where 0 is measured in radians, i.e.
it is equal to the number of degrees in the angle multiplied
by tc/i8o. Very often .this function is introduced in a
form such as inv. sec -1 5 which means " Determine the
d 0
angle whose secant is g- (or whose cosine is -g°) and then
find the involute function of that angle by subtracting its
value in radians from the value of its tangent.”
To determine the tooth thickness of involute spur or
helical gears from micrometer measurement over the
teeth (see Fig. 62):
Let t=number of teeth in gear.
T=number of teeth contained between micrometer
anvils.
M=micrometer measurement over teeth.
S 1 =tooth thickness measured perpendicular to tooth
spiral on cylinder of diameter d x .
spiral angle at diameter d x .
L=lead.
d # —base diameter.
<r 0 —spiral angle at base cylinder.
Then „ Ted.
tan
and o / ' M . d tc„ ,
Si= id^;- mv - sec d„- 5t<r—cos or 1
This gives the normal tooth thickness on any desired
cylinder.
For the determination of tooth thickness of involute
MEASURING GEARS
97
spur gears from micrometer measurement over rollers (see
Fig. 63):
Let d—diameter of circle on which tooth thickness is
measured.
d^diameter of base circle.
Z=measurement over two equal rollers in opposite
tooth spaces.
D =diameter of each roller.
W—circumferential thickness of tooth on circle of
diameter d.
t—number of teeth in gear (must be even).
Then
\xt a ( n 1 • -1 . d D\
W=d mv. sec 1 -mv. sec” 1 -^ J
Assuming that the profile of the teeth is accurately
involute, this formula applies with any roller diameter
that causes the roller to make contact with the teeth at
points on the involute part of the profile.
If the roller is required to touch the teeth on the pitch
circle where the pressure angle is i|» and where the width
of tooth is equal to width of space, then the roller diameter
may be determined from
D=d 0 | tan (^-j- 2 L)—tan 4 » j-
The expression for D is approximately equal to
d, tan 22. and this value may be used in all ordinary cases.
2 t
The measurement of tooth thickness of involute spur or
helical gears at the pitch cylinder by use of vernier calipers
is accomplished in the following manner :
Measurement of Tooth Thickness by Vernier Caliper
(Fig. 60 ).—If tooth and space are to be of equal width.at
the pitch cylinder, the vernier caliper will fit the teeth with
the settings specified below—
Height Setting=Addendum—0-196 £2.
Thickness Setting—^ sin ( 2 ?)
where p n =normal pitch
and t=number of teeth in gear.
7
98 GEARS AND GEAR-CUTTING
Alternative Formulae.—As an alternative to the
measurement formulas given, the following formulae are
offered as being simpler to apply. The fundamental
difference is that they apply to measurement of tooth
thickness at points which do not lie on the pitch cylinder.
The formulae do not involve any mathematical approxi¬
mation but are quite- accurate.
To determine the tooth thickness of involute spur
or helical gears from micrometer measurement over
teeth by the base tangent method (Fig. 62):
Let p n =normal pitch.
^—normal pressure angle.
^—transverse pressure angle,
c-—spiral angle,
t—number of teeth.
L=lead.
w—normal thickness of tooth at pitch cylinder.
n=number of teeth between micrometer anvils.
M—micrometer measurement over teeth.
Then
sin <r — t p n /L
tan (p t =tan sec a-
and M—cos |w-fp n [ (n—i)inv. <J> t Jj j-
Here inv. tan <J; t —the angle t|/ t being measured
in radians, i.e. (71/180) times the angle measured in degrees
and decimals of a degree.
To determine the tooth thickness of involute spur
or helical gears by micrometer measurement over
rollers (Fig. 63) ;
The symbols are as above, except that w represents the
required normal thickness of tooth at the pitch cylinder
and then
Roller diameter — (p n —w) cos
and the micrometer measurement over the rollers, when the
normal tooth thickness is actually w, is given by
N=^- n sec cr—{— (p n —w) cos
7 T
This applies only to gears with even numbers of teeth.
MEASURING GEARS 99
To determine the tooth thickness of involute gears
of odd numbers of teeth by micrometer measure¬
ment over rollers:
The symbols are as above, and micrometer measurement
n =~F ? (i —3 sin 2 ^-)+(p„—w) cos <J>„.
This applies to straight tooth gears (so that £*=circular
Fig. 79a.—D iagram showing the principle of the involute gear¬
testing instrument. See also Fig. 66.
pitch and \p n —transverse pressure angle) and to the con¬
dition in which one roller is in a tooth space as nearly as
possible opposite to that occupied by the other roller. The
number of teeth must be odd, and w represents the required
tooth thickness at the pitch cylinder.
Micrometer measurement of tooth thickness cannot be
applied to helical gears with odd numbers of teeth.
CHAPTER IX
Hobs, End-Mills and Generating Cutters
The hobbing process is the one that produces the most
accurate cut gears, and it does demand a high standard of
accuracy in the hob itself. To meet modern requirements
the hob teeth must be finished by grinding on a machine
in which the abrasive wheel head has a relieving motion
whilst the hob is rotated and moved endwise.
The material used for hobs of normal dimensions is high¬
speed tool steel, containing about 075 per cent, carbon,
18 per cent, tungsten, 4 per cent, chromium, and 1-3 per
cent, vanadium. This needs special care in forging to
avoid the formation of cracks and, after turning and boring,
careful inspection is required to detect any defect of this
nature.
A boss about a quarter-inch wide is turned at each end
of the hob in order to provide cylindrical surfaces for check¬
ing the true running of the hob in service. The keyway or
driving slot is cut at this stage.
Thread-Milling. —The thread (or threads) are then
produced in a thread-milling machine. If quantities are
involved, it may be worth while to make a milling cutter
specially formed to produce the required thread section.
Otherwise the most closely approximating available cutter
is used to rough out the thread, which is afterwards modified
to the required shape by use of a form-tool in the lathe. The
thread shape is checked by comparison with a gauge or,
if it is straight-sided, by the use of a sensitive tracer and dial
indicator traversed in a straight line along a slide at the
appropriate angle.
Milling Flutes. —The flutes are then produced on a
universal milling machine using a cutter of 30 deg. V-
section, one side of which is perpendicular to the axis of the
cutter. The vertical line through the plane containing
these perpendicular edges is set to intersect the centre-line
of the hob flank so that the cutting face of each flute
contains radial straight lines. The radius at the tip of.the
100
IOI
HOBS AND END-MILLS
cutter is equal to about one-eighth of the normal pitch of
the hob.
The angular setting of the table of the milling machine
is the spiral angle of the flute at the pitch cylinder of the
hob. The change-gears between dividing head and horizontal
feed screw are selected to produce the lead of the flutes.
The hob is indexed from one flute to the next by use of
the dividing head. The cutter is finally sunk into the blank
so that the root of the flute is at a distance from the outer
surface of the hob blank equal to
Depth of hob tooth-frise of relieving cam-f- tip
radius of cutter.
Relieving. —The relieving of the teeth is carried out on
a relieving lathe ; the tool is formed to match the thread
form of the hob, a witness about 0*020 in. wide being left
on each tooth.
The hob is then hardened in accordance with the steel¬
maker's instructions for the particular brand of steel used,
double-quenching and tempering frequently being specified.
Grinding. —The next operation is to grind the bore,
end faces and cylindrical registering surfaces. These
surfaces are required to run truly in relation to each other
to an accuracy of o*oooi in., and the diameter of the bore
must not exceed the nominal size by more than 0*0002 in.
The cutting faces of the flutes are next ground on a hob-
sharpening machine set up for the lead of flute specified for
the hob. Indexing of the flutes is required to be accurate
within 0*0005 in. measured at the pitch cylinder.
The relieved surfaces of the teeth are then ground on a
hob-profile grinding machine. After the tip (which is
usually straight) has been treated, the abrasive wheel is
trimmed by successive trial to give the required hob tooth
form. If this is of simple straight-sided type it may be
checked by use of a fixture arranged to run a tracer-point
along the required path. Otherwise an optical projector is
used to compare a magnified shadow of the tooth with an
accurate, drawing of the required normal section on the
same enlarged scale.
After one set of flanks has been satisfactorily^ shaped, the
others are similarly treated. Further grinding is then done
to bring the teeth to the required thickness.
The error of axial position of any tooth relative to its
102 GEARS AND GEAR- CUTTING
neighbours should not exceed 0*0002 in. and the error of
relative position between any two teeth should not exceed
o*ooo2 in. plus o-oooi in. for every inch of axial distance
between them.
Sharpening. —The hob is sharpened when necessary by
grinding the radial faces of the flutes on a hob-sharpening
machine, accurately set up to give correct lead and indexing.
This operation must be carried out with care if the hob is to
reproduce the accuracy with which it was manufactured.
Formulae used in Hob Design and Manufacture
Let p n =normal pitch of tooth to be produced,
t =number of threads in hob.
d =pitch diameter of hob.
j = outside diameter of hob.
A. —lead angle of hob thread,
or ^spiral angle of flutes.
F =*=number of flutes.
L —lead of threads.
I —lead of flutes.
For hob made with solid shank
Minimum desirable value of d—4p a .
For hob bored to mount on arbor
Minimum desirable value of d—4p a +diameter of arbor.
Outside diameter of hob j
=d-|-2 X dedendum of gear tooth.
Lead of hob thread L
~W VCMla 2 ) 2 ]
. , L
tan
Tip radius of flute-cutter—0-125 Pu
Lead of flutes
Spiral angle of flutes 0-=A..
Number of flutes F= ~^ ^
, , 1,3 P*
(taken to nearest whole number).
The number of flutes F and the number of threads t should
preferably have no common factor.
HOBS AND END-MILLS
103
Rise of relieving cam^ ^^J j (approx.).
Ratio of change-gears for relieving cam:
Product of numbers of teeth in driving gears
Product of numbers of teeth in driven gears
T _ / Number of flutes \ Z-fL , Tr . . .
= K x 'number of lobes in cam' T where K15 a constant
for the machine. The number of lobes in the cam is usually
unity.
The Manufacture of End-Mills for Gear-Cutting.—
The end-mill process is usually employed for producing
gear teeth only when more accurate processes are im¬
practicable, e.g. in the manufacture of triple-helical gears
with continuous teeth.
The end-mill cutter (see Fig. 80) of the shape necessary
for cutting average tooth forms is a comparatively fragile
and short-lived tool, and more than one may have to be
used in completing a large gear. In the smaller sizes the
cutter is made in one piece with the shank, which is tapered
to fit into the spindle of the machine. In large sizes it is
practicable to make cutter and shank as separate pieces.
104 GEARS AND GEAR-CUTTING
the one from high-speed tool steel and the other from plain
carbon steel.
Turning. —The tapered shank is turned in the lathe and
a tapped hole made for reception of a draw-bolt when the
cutter is in use. The body of the cutter is turned to a profile
which fits a “ half-shape ” gauge, the diameter at a specified
distance from the end being measured by a vernier gear-
tooth caliper, and made to leave an allowance of about
o-oio in. for grinding. The profiling is accomplished by
feeding the cutting tool inwards by hand, whilst the saddle
has a slow traverse away from the headstock ; the final
shape is achieved by applying a hand-scraper, ground to a
suitable radius, to the work whilst running in the lathe.
Milling. —Cutting edges are next produced by a milling
operation. As the edges must continue almost to the centre¬
line they are usually limited to four ; in large cutters four
extra intermediate edges are sometimes formed on the
curved part of the profile. A plain cylindrical milling cutter
is employed and is guided by hand control of the vertical
and horizontal movements of the milling-machine table.
The shape of the surfaces produced by the cutter is not
important provided that the radial cutting faces and the
" lands " behind the cutting edges are each about J in. wide.
The lands are then filed with a relief angle of about 10
deg., leaving a witness about 0-020 in. wide at the cutting
edges. This operation is not critical, as the edges are
subsequently finished by grinding.
Grinding. —After hardening, the tapered shank is
ground to fit a standard taper gauge.
The final operation of profile grinding the cutting edges
is effected by mounting the cutter in the taper bored
spindle of a fixture arranged to be moved by hand over a
horizontal table. The spindle carries a dividing plate by
which it may be indexed into four equidistant angular
positions.
The table carries an abrasive wheel head arranged with
the centre-line of the wheel slightly lower than that of the
centre-line of the cutter. Vertically beneath the periphery
of the abrasive wheel is a stop of the same shape as the
plan view of the edge of the wheel.
A former plate is attached to the cutter fixture in such a
position as to correspond with the desired profile and to
HOBS AND END-MILLS 10$
bear on the fixed stop. Thus when the fixture is guided by
hand so that the former plate always makes contact with
the stop, the abrasive wheel produces an edge of the
correct shape on a cutting face which lies in a hori¬
zontal plane. The difference in height of centre-lines of
cutter and abrasive wheel is such as to give a suitable
relief angle.
The grinding is continued equally on each cutting edge
until the diameter at a certain distance from the end
corresponds to the calculated figure. As a check on the
shape of the profile, the diameter should be measured in two
other planes as widely separated as possible and the figures
compared with those obtained by calculation.
Determination of the Shape. —The shape of the end-
mill required to produce any helical gear teeth of involute
form can be determined by calculation. Except in the case
of spur gears the end-mill shape is not involute, although
it approximates to it if the helix angle is small. The shape
may be taken as that corresponding to a spur gear having
pitch and pressure angle equal to the normal pitch and
normal pressure angle of the actual gear and a number of
teeth equal to the actual number multiplied by the cube
of the secant of the spiral angle. If this latter number
exceeds about 60 the approximation is adequate for good-
class commercial gears.
The cutter is re-sharpened when necessary by the method
used for grinding its form in the first instance.
Gear-Generating Cutter.— The four principal methods
practised for cutting gear teeth are : (i) by means of a
rotary disc cutter formed to the tooth profile ; (2) by means
of an end-mill; (3) by means of a planer; and (4) by
generation.
The rotary disc cutter has the disadvantage that practical
considerations compel the use of the same cutter through
a considerable range of sizes of gears, whereas to obtain a
true tooth shape a cutter would be required for every size ;
in one well-known system only eight cutters are supplied
to cut all gears of a given pitch from 12 teeth to a rack,
which latter may be considered as a gear of infinite radius.
For the second method, it may be said that it is chiefly
used for herring-bone teeth, and would not be used except
in special circumstances for straight-cut teeth.
106 GEARS AND GEAR-CUTTING
The third practice is chiefly of a makeshift nature. The
fourth, however, has now been accepted as the only correct
method of cutting gears, and its introduction has made
possible the cutting of gears of greater accuracy than by
any other process.
Before dealing with the actual manufacture of gear
cutters it is necessary to understand certain principles
and functions relating to generation.
There are two principal methods of machining a con¬
toured or other surface, namely, forming or generating.
To illustrate this
point, a surface plate
may be planed by
using a tool having
a perfectly straight
cutting edge, the full
width of the plate,
feeding the tool
gradually at every
stroke; this is an
example of forming a
surface.
On the other hand,
if a tool is fed
across the work, and
moved slightly in a
parallel plane at each
successive stroke, the
,„ j surface would be
Fig. 8i.—A typical gear cutter. generated. Similarly
a cylinder may be
formed on the milling machine, or generated in the lathe.
It is evident that the generating method of obtaining a
surface is the more accurate, whether applied to gears or
cylinders.
Gear Generation.—The generation of gears is made
possible by the use of a cutter which is in itself a gear
(see Fig. 81), and the action of the tool is clearly shown in
rig. 82. This illustrates a metal gear being rolled and in
mesh with a material of plastic nature, such as wax. When
rolled together as two gears in mesh, the generating gear
will mould teeth in the plastic material of correct shape;»
HOBS AND END-MILLS I©7
therefore, all gears so shaped will mesh with one another,
or with their generator.
Fig. 83 shows this principle as applied to the cutting of
metal blanks. The cutter is reciprocated vertically by the
ram of the gear shaper on which it is mounted. During this
motion the tool is first gradually fed inwards until the
correct depth has been reached, then the cutter and the
’generated gear are revolved together exactly as two gears
in mesh. The teeth of the revolving and reciprocating
cutter thus form teeth of the correct form in the blank.
Obviously, the teeth of the generating cutter must be
extremely accur¬
ate, with proper
clearance angles,
in order to en¬
sure free cutting.
The sequence of
the tool move¬
ments will be
gathered by re¬
ferring to Fig. 84.
Fig. 82 (above ).—-A
sketch showing the
action of the tool
illustrated in Fig.
81.
Fig. 83 (left ).—The
cutter at work on a
metal blank.
108 GEARS AND GEAR-CUTTING
Making the Cutter.— As is the practice with other gears,
the cutter is firstly roughly machined, bored and faced on
the turret lathe. The teeth are then profiled roughly either
on the millin g machine or on the gear shaper by means of
cutters, which leave a sufficient tolerance on the tooth
thickness for final grinding to shape.
After these preliminary roughing operations have been
effected the cutter is hardened, and it will thus be seen that
on account of the following finishing and final grinding
operations there is no risk of the tool, being inaccurate.
As the accuracy of subsequent operations is dependent
on the bore, this is
atank s' finished first, after
\ which, the front and
f^C\ \ back faces of the
K\\ \ teeth and blank and
/Hill 1 A \ an y succeeding
\ \ operation that may
, uTS "" I \ be necessary, are
\ completed.
\ The Involute
/ ( / Grinder. — Figs. 85
/ \ Cutter an( i 86 outline the
Shape of \ machine and prin-
chip ciples for grinding the
' profile of the teeth
^ , to a true involute
FiG.84.—The sequence of the tool move- This opera _
ments, showing the shape of the chip ,. . ~ ^ ,
removed. tion is effected by the
generating process,
and it is this function which defines a generated cutter.
Cutter
Shape of
chip
Fig. 84.—The sequence of the tool move¬
ments, showing the shape of the chip
removed.
Referring to the illustration, the cylinder A represents
the rolling circle of the cutter, and it is, therefore, of the
same diameter as the pitch circle of the generating gear.
This cylinder oscillates on a plane surface B, and is ensured
against slipping by the action of the steel tapes C, which
are fixed to the cylinder and positively control the rolling
motion. The cutter D (Fig. 85) is mounted on the end of the
spindle, which is rigidly secured to the cylinder. As the
cylinder oscillates up and down on the inclined plane the
cutter is caused, by means of the steel tapes, to roll in
exactly the same way as if it were a gear in mesh with
HOBS AND END-MILLS IO9
the imaginary rack, F, as denoted by dotted lines in
Fig. 88.
The straight face of the grinding wheel, E (Fig. 85),
realty represents one of the flat faces of the rack teeth,
and since the rolling motion of the blank is controlled by
the cylinder,, together with the angular plane surface and
steel tapes, it will cause the revolving grinding wheel to
profile a true involute curve on the face of the cutter tooth
or work. In order that one face only of each of the teeth
of the cutter may be ground in turn, a dividing head is
fitted so that the blank can be indexed in relation to the
spindle. When this operation is completed the cutter is
taken off and placed on a machine which is “ left hand ”
to the first, and the same procedure is repeated on the
opposite faces or sides of the teeth.
The face of the grinding wheel should be dressed to great
accuracy by means of a diamond, which should be truly
traversed across the face of the wheel.
A string being unwound from a cylinder describes the
involute curve, and in this grinding operation the steel
bands impart a similar movement; thus, since the cutter
motion is synchronous with that of the rack the correct
no
GEARS AND GEAR-CUTTING
shape must necessarily be transferred to the tooth being
ground.
Pressure Angle Corrections.—The head which carries
the grinding wheel has a swivelling movement, so that the
cutters for generating helical gears may be ground. It must,
of course, have a relative motion in a right-angled direction
which permits the relief angles to be ground on each tooth
flank.
With regard to relief angles, these are involved in the
calculations for determining the pressure angle. For
example, if the tooth has a 2 deg. relief angle, the pressure
angle of the grinding wheel
^£=====5^. is 14J- deg. However, the
// relief angle will increase
ff __ fK the pressure angle of the
II \\ C cutter by 12' approxi-
fHHHi Jr mately; thus, in order
\\ If cu ^ er luay be
\ produced with the required
pressure angle, this discrep-
ancy must be adjusted by
inclining the grinding wheel
head 14J 0 —I2'=I4° 18'.
j In setting the grinding
l _head to correspond to the
Secf/b /7 X—X s P iral for a helical
Fig. 86 .—A section on the line fatter, the pressure angle
X_X in Fig. 85 . has again to be corrected ;
it has to be transferred
from the linear pressure angle into the normal pressure
angle.
This is found by applying the following formula : tan of
normal pressure angle=tan of linear pressure angle X cos
spiral angle.
In checking the chordal thickness of a helical cutter, it is
easier and more convenient to measure it normally to the
spiral angle. The normal thickness (NT)—circular thickness
X cos spiral angle.
It frequently happens that a cylinder, to which the steel
tapes are fitted, required with a pitch diameter correspond¬
ing to that of the cutter desired to be produced, is not in
stock. The cylinder having a diameter nearest to that
HOBS AND END-MILLS
III
figure required should be used and the pressure angle
corrected to correspond. The following formula will give
the corrected pressure angle : cos press, angle required x
correct pitch radius divided by pitch radius of cylinder used.
The normal pitch—cos spiral angle x circular pitch.
The following pressure angle data will be of great assist¬
ance in the design of gear-generating cutters.
Tan normal pressure angle (NPA)=tan linear pressure
angle (LPA) x cos spiral angle (SA).
tan NPA
cos SA
=tan LPA
tan NPA
tan LPA
—cos SA
tan LPA x cos SA
tan NPA
tan NPA
cos SA x tan LPA'
Addendum of the Cutter.— In measuring the thickness
of these cutters it should be remembered that the addendum
112 GEARS AND GEAR-CUTTING
of the cutter is the dedendum of the gear tooth ; calipers
should therefore be set to a corrected addendum and not
to that obtained by the formula,
diametral pitch (DP)
which only gives the addendum of the teeth cut. The rule
for the addendum of gear-generating cutters is ~jyp
Chordal Thickness and Height. —Fig. 87 indicates
clearly what is meant by the chordal thickness—the usual
method of measuring the thickness on the pitch line, with
the gear-tooth calipers set to the addendum of the gear.
Standard formulas only give the length of the arc cut by
the tooth on the pitch Ime, and it mil at once be evident
that this is incorrect, as the true thickness must be the
chordal length of the arc. Again, the curvature of the pitch
line virtually increases the addendum at the point of
measurement, and this increase is equal to the height of the
arc. In making these corrections the following formulas
should be applied:
angle B “timber 0 f teeth
Chordal thickness=sin BX2 rad.
or—sin Bx pitch diam.
Chordal height —pitch rad. x (i-cos B)
CHAPTER X
Load Capacity of G ears
When power is being transmitted by a pair of gears a force
is exerted by each tooth on the one with which it makes
contact, and the load capacity of the pair is determined bv
the ability of the teeth to resist the effects of such forces.
For example the force P applied to the gear tooth
shown in Fig.. 89 causes local “surface stress” in the
material pear its point of application and also tensile and
compressive bending
stresses” at A and B
respectively. These
stresses are applied and
removed every time the
tooth passes through the
zone of contact, and the
intermittency of the
loading makes it more
destructive than a con¬
stant loading of the same
amount.
Repetition of excessive
surface stress usually
leads to destruction of
the tooth surface by
“pitting,” small pieces
of metal breaking away
to leave depressions of
approximately spherical
shape. If lubrication is
inadequate, high surface pressure on the parts of the
tooth profile which slide over the mating profile causes
wear to occur in such parts. In extreme cases, the
lubricant may be squeezed out, when sliding contact of
metal on metal under heavy pressure may lead to partial
welding together and “ dragging ” of the surfaces, leaving
them too rough and inaccurate for smooth meshing.
8 IJ 3
Fig. 89.—The force P causes inter¬
mittent tensile and compressive
bending stresses at A and B.
114 GEARS AND GEAR-CUTTING
In most cases the load capacity of a pair of gears is
limited by surface pressure rather than by bending stress.
In other words, the teeth are more likely to fail by
destruction of the working, surfaces than by breakage of
the teeth. If the material is case-hardened steel, however,
the reverse may be true, because the surface is much
harder than that of steel in its normal unhardened con¬
dition, and so the teeth can withstand relatively high
surface stress whilst their bending strength is not
proportionately increased. Also, teeth made from a
brittle material such as cast-iron may snap off under a
shock load which, if only occasionally repeated, might
not have had any serious effect on the working surfaces.
Shock Loading.—In designing gears it is necessary to
take into account intermittency of loading, the possibility
of shock loading such as occurs, for example, in rolling
mill drives and the occurrence of continual load pulsation
as, for example, in drives associated with engines or
reciprocating pumps.
The magnitude of the load that may safely be imposed
on (say) one-inch width of a gear tooth depends on the
following factors:
(a) Frequency of application of load, i.e. speed of
rotation of gear.
(b) Nature of material of gear.
(c) Shape and size of tooth.
(d) Rigidity of mounting of gear.
(e) Duration and frequency of loading periods.
No further comment is needed on (a) and (b). So far
as surface pressure is concerned the important feature of
(c) is the “ relative radius of curvature ” of the teeth at
the line of contact. This varies to some extent according
to the position of the line of contact on the profile, but its
average value is used as a means of determining the
allowable surface pressure. Broadly it may be said that
for gears of given normal pressure angle and spiral angle,
the permissible load per inch width is proportional to the
o*8th power of the pitch diameter of the pinion. The
limiting figure in any particular instance may be calculated
from, the dimensions of the gears with the aid of formulae
published in the appropriate British Standard specification.
Bending Stress.—So far as bending stress is con-
LOAD CAPACITY OF GEARS 115
cerned, the important features of (c) are the depth of the
tooth and its thickness at the root. These are controlled
largely by the pitch, but even for a given pitch the tooth
thickness varies somewhat according to the number of
teeth. Here again the British Standard specification
indicates how to determine load capacity on the basis of
" strength ” (as this is conveniently described in distinction
from “ wear ”) of any pair of gears.
It may be observed that if the materials, diameters and
width of a pair of gears are fixed, the load capacity on the
basis of " strength ” may be varied by varying the numbers
of teeth, or, in other words, varying the pitch. Such a
change, however, has no appreciable effect on the load
capacity on the basis of “ wear,” and as this is usually
the limiting factor it follows that nothing is to be gained
by adopting excessively large pitches.
Smoothness of running is improved by the use of a
fine pitch, and as a general rule the best design practice
is to use the finest pitch that will give adequate strength.
Except when case-hardened steel is used this means that
the most suitable number of teeth in the pinion is rarely
less than about 25.
Working Life of a Gear. —The rigidity of mounting
of the gears affects the load capacity, because it has an
influence on the distribution of loading across the face of
the gear. It is not easy, or usual, to attempt any numerical
assessment of this factor, and indeed its variation between
different examples is not great if care is taken in design to
provide a mounting of the utmost practicable rigidity.
The standard basis of gear design gives a normal life-
expectation of 25,000 hours, or, roughly, 2,000 working
days of 12 hours each. If the operating time in any
particular instance is known to be different from 12 hours
per day, it is usual to modify the working stresses so as to
keep the normal expectation of life at 2,000 working days.
The probable life is approximately proportional to the
reciprocal of the third power of the working stress; for
example, if the required length of life were only one-
eighth of 25,000 hours the working stresses might be
double those applying to the standard 25,000-hour rating.
In some instances, particularly in the case of worm
gears, continuous operation may lead to such a rise of
Il6 GEARS AND GEAR- CUTTING
temperature (due to generation of heat by friction and oil
turbulence) that the oil tends to lose its lubricating quali¬
ties, when rapid wear of the teeth commences. It can
therefore happen that the limit of load that can be safely
imposed on gears is set by temperature-rise rather than
by stresses in the teeth.
Fig. 89A. —Comparative Sizes of Involute Gear Teeth.
Worm gears are especially prone to this limitation
because of the relatively great amount of sliding associated
with their tooth action, whilst high-speed gears of any
type are liable to encounter it by reason of power loss in
oil-churning.
In order to raise this limitation, recourse may be had to
The Efficiency of Gears 1x7
artificial cooling devices in the form of fans or cooling
coils. Alternatively, the oil may be sprayed on to the
gears in the zone of contact (thus avoiding the considerable
power loss otherwise caused by turbulence in the oil-bath),
drained from the bottom of the gearcase, and passed
through an oil-cooler before being returned to the gears.
The accepted formulse relating to the strength and load
capacity of gears are published in British Standard Speci¬
fications Nos. 436, 545 and 721. These have been prepared
in collaboration with the leading gear manufacturers. They
are obtainable from the British Standards Institution, 28
Victoria Street, London, S.W.i.
CHAPTER XI
The Efficiency of Gears
The amount of power lost by friction between the teeth
of most gears is usually less than x per cent, of the
transmitted power, but in estimating the efficiency of
the whole gear drive it is necessary to take into account
the power lost in bearing friction, oil-drag, and windage.
Bearing friction may absorb only about the same amount
of power as tooth friction, but oil-drag may be more
serious, particularly if the peripheral speed is high. Windage
is unimportant except at very high speeds.
The overall efficiency of a single-stage gear assembly
of any of the types mentioned above may be expected to
lie between 97 and 98 per cent, at full load under favourable
conditions. At fractional loads the efficiency is lower,
largely because oil-drag (on the teeth and in the bearings)
does not diminish as the load drops and therefore becomes
a greater fraction of the transmitted power.
The power lost by friction between the teeth of worm
gears depends on the lead angle of the worm, and therefore
indirectly on the reduction ratio. In this type of gear
there is a greater amount of relative sliding than in other
Il8 GEARS AND GEAR-CUTTING
forms, and consequently tooth efficiency is lower. Oil-drag
is also a cause of appreciable loss of power if the worm-shaft
speed is high. To take all the factors into account is
difficult, but as an approximate guide the overall efficiency
of a worm-gear unit with a worm-shaft speed not higher than
about iooo r.p.m, may be taken as equal to 100 minus
half the reduction ratio, e.g. 30 to 1 would give 85 per cent.
This assumes the use of a case-hardened and ground steei
worm in conjunction with a phosphor-bronze worm-wheel.
At higher worm-shaft speeds the efficiency tends to be
slightly lower because of greater oil-drag. At worm-shaft
speeds below about 100 r.p.m the efficiency again tends to
be lower because of the rise in co-efficient of friction at
low rubbing speeds.
Lubrication of Gears.—Gears which run only very
intermittently may be lubricated by grease, but in general
oil lubrication is essential if the full load-capacity of the
gear-materials is to be realised. Oil of high viscosity is
necessary when the tooth-loading is heavy, but is unsuitable
if the speed is high because, on the one hand, it may lead to
senous power loss by churning, or may " channel ” to such
an extent under the action of the gears that they are not
satisfactorily lubricated. Fortunately, consideration of
fatigue makes it impracticable to subject high-speed gears
to very heavyloading, and it is therefore possible to lubricate
them with the low-viscosity oil which is necessary to keep
oil-drag loss within reasonable bounds.
Up to peripheral speeds of about 2500 f.p.m., lubrication
of gears is adequately effected by allowing one of a pair to
dip into an oil-bath. At higher speeds oil-drag loss tends
to become serious, and on this account it is usual to employ
anoil-spray, keeping the gears clear of the oil in the s um p
The value of a particular oil as a gear lubricant is not
shown by viscosity alone, and it is necessary to make sure
that the oil used for any important gears carries the
approval of the gear manufacturer. This is especially
important in the case of worm gears because they are
liable to complete failure in a very short time if lubrication
is defective, whereas the other conventional types of gear
are rather less sensitive in this respect.
CHAPTER XII
Useful Formulae for Gears
Spur Gearing.—The Standard gear terms and symbols,
forming the basis of calculations concerning their tooth
proportions, are shown in Figs. 90 and 91. It should be noted
that circular pitch refers to the distance, measured round
the pitch line, from the centre of one tooth to the centre
of the next tooth; whereas diametral pitch refers to the
number of teeth per inch of the pitch diameter. It is usual
to refer to gears according to their diametral pitch.
Module is found by dividing the pitch diameter by the
number of teeth. Thus a 40-tooth gear with 20-in. pitch
diameter would have 2 teeth per inch of its pitch diameter
and would thus be :
42=2DP.
20
The module of the same gear would be :
P/tcmC/pclE Dedemdl/m
Fig. go.—Diagram explaining terms used in gearing.
It is thus obvious that the module is the reciprocal of the
diametral pitch, and vice-versa. The following formulae
apply to spur gears, where the addendum is equal to 0-3183
x circular pitch :
N—number of teeth.
s=addendum or module.
119
120
GEARS AND GEAR-CUTTING
2=thickness of tooth on pitch line.
chordal thickness of tooth.
/—clearance at bottom of tooth.
D=outside diameter.
D'—pitch diameter.
D"=working depth of tooth.
D"+/=whole depth of tooth.
D'*—bottom diameter.
P==diametral pitch.
P'—circular pitch.
H—height of arc.
s /l '=distance from chord of pitch line to top of tooth,
0—J angle subtended by circular pitch.
The following formulae apply to the system in which the
addendum is equal to 0*3183 x circular pitch.
Fig. 91.—Diagram illustrating symbols used in spur gearing formulae.
USEFUL FORMULA FOR GEARS
121
/= *
^ 10
1-5708
p '
s +/=^r?»' or 0-3683 P\
B"= 2s; or|.
D"-f-/==?^ 3 Z; or 0-6866 P'.
D '=p , or 521 ; or 0-3183 P'N ; or .
F 7 T N + 2
D=D'+2s; or N+2
0 = 22 .°.
N
r=D' sin 0.
D"’=D-2(D"+/); or
N =D'P ; or DP-2.
H=P-' (?~ C ?- S - 9 j. ; s'=s+H.
B. & S. Involute Gear Cutters
No. 1 will cut wheels from 135 teeth to a rack inclusive.
2 „
55 „
„ 134 teeth
»>
3 »
35 »
>, 54 „
M
4 „
26 „
t* 34 >>
,,
5 „
21 „
„ 25 „
6 „
17 »
„ 20 „
»»
7 »
14 »
„ 16 „
8 „
12 „
13 ..
Bevel Gears
(Axes at Right Angles)
N.=Number of teeth on gear.
N* “Number of teeth on pinion.
P=diametral pitch.
P'—circular pitch.
122
GEARS AND GEAR-CUTTING
a,,—centre angle—angle of edge or pitch angle of gear,
a*—centre angle=angle of edge or pitch angle of pinion,
d—angle of top.
p'=angle of bottom.
angle of face of gear.
g h —angle of face of pinion.
^—cutting angle of gear.
Zk=cutting angle of pinion.
A—apex distance from pitch circle.
A'=apex distance from large bottom of tooth.
D=outside diameter,
D'—pitch diameter.
D"—working depth of tooth.
5—addendum or module.
2 —thickness of tooth at pitch line.
/^clearance at bottom of tooth.
D*+/=whole depth of tooth.
2#—diameter increment.
b= distance from top of tooth to plane of pitch circle
F—width of face.
Fig. 92 .—Diagram of symbols relating to right-angle bevel gears.
N
tan a.—tan a a =
N*
N„*
tan
2 sin a
~~W~ P
or
5
A’
USEFUL FORMULAE FOR GEARS I23
tan or£±- f
N A
g.= 90 °—(o.X/S); g s =90°—(a—/S).
h=a-£\
A=== ^n/N?W.
N
A=—- *
2 P sin a
Ai=—^ or
N
cos /?" 2P sin a cos /?’
A=
P=
4D
sin (a+P)
N__
2 A sin a' J
TC
COS p.
N-f-2 cos a tc
° r-p-,or p .
p# TC
n ,_N nr NP ' nr DN
JL) —-=i or f or ijj~r 1
P iv N+2 cos a
, or D=
2 cos a
P *
Cutters for Mitre and Bevel Gears
Diametral Pitch
Diameter of Cutter
Hole in Cutter
4
3i in.
i£ in.
5
3 i i n -
1} in.
6
3} in.
in.
7
3 i i 11 *
1} in.
8
3 i in-
ij in..
10
3 im.
I in.
12
3
I in.
14
3
Jin.
16
2J in.
Jin.
20
2J in.
Jin.
34
2| in.
Jin.
Table for Obtaining Set-Over for Cutting Bevel Gears
124
GEARS AND GEAR-CUTTING
00 1 M
t 1- ■*f* 04 m to 04 vO O
O CO CTi M w VO 4>.CO
04 CM M CO CO ro CO CO
06666666
!>>, M
Ot CO O OO CO VO 0 0
<0 OO O'. O M to t-*00
04 04 04 COCOCOCOCO
6 6 6 6 6 6 6 6
\Q , W
O O N >0 0 N OOO
VOOOOO 0 O too vO
04 04 04 CO CO CO CO CO
66666666
oife
Ov 0^3 N 4>*0O O w
IOJ>00 0 O Th «0 VO
04 04 04 CO CO CO CO CO
66666666
< 1 )
«+-«
tOi M
OO t>» COCO 04 CO CO 04
0 0.00 0.0
040I<M<NC0<OCOCO
66666666
F
■4*1«
00i00400 0 000 ^j*
to {''•OO Ov O -tt* CO '«t*
04 04 04 04 CO CO CO CO
6666 6 6 66
<w
1
"$1 w
NtCOvOOO C-. -rt-oo
to 0-.00 O' Q\ CO CO CO
04 04 04 04 04 CO CO CO
66666666
1
8
^IH
4>* '^“OO CO VO Tf* O'. M
tO O'- O- O'. O'. CO 04 CO
04 04 04 04 04 CO CO CO
66666666
S'
O
O
•rt
^f-J M
NC0<0 w toO Tf" Ot
O O. tN O O <0 C4 M
0404040404<OC0CO
66666666
i
pa
«H*. i_j
CO
VO 04 CO 0> COOO 0 Q\
to O' O'OO O'. 04 H 0
04 04 04 04 04COCOC0
66666666
to M M 10 O coco VO
»o t"> i>.oo 0 n 0 a
0404040404COC004
66666666
*S!I«
t}- 0 O OO 0 to OO OO VO
tovo vo 00 00 m avoo
04 04 04 04 04 CO 04 04
66666666
CO» H
Tt-VD VO to O H O' to
too o x>-oo H OO o
04 04 04 04 04 CO 04 04
66666666
jo^no jo -ojvt
M 04 CO Tf too 0"0O
1
S 3
O
CD
O
H
a
s
•tJ *
CD
_ »
53 8>
£ 0
|fc
o'a
$13
§43
s s
11
. .a .2
g HCV
*
03
Relevant formulae are
on next page.
126
GEARS AND GEAR-CUTTING
D=D'+20.
2(2=25 COS a.
Bevel Gears
(Axes at any Angle)
The formulae for tooth parts on pages 121 to 124 apply in
both cases. C=angle formed by axes of gears.
tan a t = * m . C ; cot cot C.
£-‘+cos C N « sm C
N.
tan <x» =g ^— - y cot — ^.^,-^ - 1 cot C.
^•+cos C
Where the indicating letters a and b are not used the
formulae apply to gear and pinion.
In the case of Fig. 93 the formulae given on pages 124 to
126 apply.
USEFUL FORMULA FOR GEARS 12?
In the case shown by Fig. 95 :
tan a , = - - sin - ( l8o - C ) ; tan *-**&£=£ L_
COS (180-C) Jj£—,cos (180-G)
In the case shown by Fig. 94:
a a = 90° ; a*=C—90°
In the case shown by Fig. 96 :
tan a a —
sin E
cosE—
N.
sin E
—cos E
Fig. 97 .—Symbols used in calculations relating to bevel gears
with axes at any angle.
Spiral or Screw Gearing
N*
N*
=Number of teeth in gears.
=centre distance.
P' = circular pitch.
P M = normal diametral pitch.
P'„= normal circular pitch.
y- wangle of axis.
L= lead of tooth spiral.
T = number of teeth marked on cutter.
128
GEARS AND GEAR-CUTTING
D'=pitch diameter.
D=outside diameter.
angle of teeth with axis.
t —thickness of tooth,
addendum or module.
D''+/==whole depth of tooth.
y=a fl +aa.
P.'»=P/".
P'N
ND
P' n N
: —--, Ui i r=--.
7 u cos a N+2 cos a
D=D , +2 s, or
P'=^E.or P '
N ' cos a
P *=P' cos a.
P n =pr-(pitch of cutter).
-U M
P' n . i D—D'
or pT' ——*
TC V H 2
, P'n
D*+/- 25 +-L.
J 10
T N
cos a 3 '
L—
NP'
Ntc
tan a* ° r P tan a
In many small shops the Brown & Sharpe system of
milling involute gear teeth is still used. In this system
a series of eight cutters are required to cut all gears having
12 or more teeth in any one pitch. The list of eight cutters
supplied for each pitch appears on page 121.
USEFUL FORMULAE FOR GEARS
129
Worm and Worm Wheels
L=lead of worm.
N =Number of teeth in gear.
M —turns per inch of worm.
diameter of worm.
&' =pitch diameter of worm.
d"=diameter of hob.
D—throat diameter.
D'=pitch diameter of worm-wheel.
B=blank diameter (to sharp corners).
C=distance between centres.
P=diametral pitch.
Fig. 98 .—Worm proportions
and symbols.
9
P'—circular pitch for worm
wheels or axial pitch
for worms.
r *j see Fig. 98.
5= addendum or module.
£=thickness of tooth at
pitch line.
4 =normal thickness of
tooth.
,/= clearance at bottom of
tooth.
D"—Working depth of tooth.
Fig. 99 .—Hob proportions.
130 GEARS AND GEAR-CUTTING
D" -)-/=■whole depth of tooth.
b =pitch circumference of worm,
a—width of worm thread tool at end.
w —width of worm thread of top and width of hob
tool at end.
S=angle of tooth of worm-wheel with its axis.
F —min. length of worm.
F 1 —length of hob.
If the lead is for single, double or multiple start threads,
L=P' 2 P', 3 P', etc.
Fig. ioo . — Diagram
illustrating hob and
worm relations.
a= 6o° to 90°.
P'
£'
_ 7 zP
N-F2
NP
s-5-
6 = 7 u (d— 25), or 7 i i'.
-tan S=t.
0
Jt
—
- T~ ‘
=1=11
Fig. ioi . — Diagram
illustrating angle of
tooth of worm-wheel
with its axis.
2
/W'—D"+/.
c=^- s , or£±l
2 2
B— D+2 (y’— r' cos
d”=d-\- 2 f.
^=0-3095 P'
0-3354 P'.
F=5s+2 v 's(D'- s ).
F'=F+i.
2
cos S.
USEFUL FORMULA FOR GEARS 131
Chordal Thickness of Teeth of Gears
(Basis of 1 Diametral Pitch)
N =Number of teeth in gear,
t"—Chordal thickness of tooth.
H 2 ==Height of arc.
D'— Pitch diameter.
R2—pitch radius.
^'=90° divided by the number of teeth.
t'^D 1 sin 8 .
H 2 =R (1—cos /i 1 ).
Chordal Thickness of Teeth of Gears
(involute)
Cutter
f
H
s"
No. 1—135 T—1 P
1-5707
0-0047
1-0047
No. 2— 55 T—1 P
i*57°6
0-0II2
X-0II2
No. 3— 35 T—-1 P
1*5702
0-0I76
I-OI76
No. 4— 26 T—1 P
1-5698
0-Q237
I-.0237
No. 5—- 21 T—1 P
1-5694
0-0294
1-0294
No. 6— 17 T—1 P
1-5686
0*0362
I-0362
No. 7— 14 T—1 P
1-5675
0-0440
1-0440
No. 8— 12 T—r P
1-5663
0-0514
I-05I4
11 T—1 P
1-5654
0*0559
1-0559
10 T—1 P
1-5643
0-0616
1*0616
9 T—1 P
1-5628
0-0684
1-0684
8 T—1 P
1-5607
0-0769
1-0769
Velocity Ratios of Gear Trains.—If all the gears in
the train rotate about fixed centre 'lines, then
Velocity ratio =
Pr oduct of numbers of teeth in all driv en gears
Product of numbers of teeth in all driving gears
and
Revolutions of first driving gear=
Velocity ratio x Revolutions of last driven gear.
132 GEARS AND GEAR-CUTTING
Sprocket Wheels.—Chain or sprocket wheels are gears
made to mesh with chains, by means of which motion
may be transmitted from one point to another. The teeth
are of different formation
from the ordinary involute
gear, as they have to accom¬
modate the circular rollers of
the chain. Ordinary gearing
calculations for pitch line
functions do not apply, for
whereas with two gears in
mesh with one another the
circular pitch is the distance
between two tooth centres
measured round the pitch
line, with a sprocket the
circular pitch refers really to
the chordal distance between
Fig. 102. — Proportions of a two tooth centres. As in-
sprocket wheel on block-centre formation relating to these
calculations is extremely
scanty, the following formulae for block-centre and roller-
chain may be of interest.
Sprocket Wheels for Block-Centre Chains.—Fig. 102
shows a wheel as used for block-centre chains, and the
following formulae give proportions and diameters relevant
, l8o° _ 0 sin a
to them, Ian p=g -
5 + cos a
Pitch diam. — . ^ —5-
sm p
Outside diam.—pitch diam.-ffc. Bottom diam.=pitch
diam. —b.
In calculating the diameter of sprocket wheels the bottom
diameter is most important. In the above calculations
N=number of teeth, b =diameter of round part of chain
block, B=centre to centre of holes in chain block, A—centre
to centre of holes in side-links.
USEFUL FORMULA FOR GEARS I33
British Standard Roller Chain Wheels
Measuring pin diameter
Maximum roller diameter
Bottom diameter=Pitch diameter minus roller diameter.
Chordal distance—[| (pitch diameter of 'wheel having
twice the number of teeth) minus roller diameter].
Measurement over Pins
The bottom diameter of wheels with even numbers of
teeth is checked by measuring over pins inserted in opposite
tooth spaces.
The bottom diameter of wheels with odd numbers of
teeth is checked from chordal distance by measuring over
pins inserted in the tooth spaces most nearly opposite.
For even numbers of teeth =pitch diameter
-t , plus roller diameter.
Measurement p Qr num fo ers 0 f teeth=[ J (pitch diameter
over pins. 0 f w h ee l having twice the number of
teeth) plus roller diameter].
Tolerance on Cutting Sizes —
Bottom Diameter and Measurement over Pins
Plus o-ooo inch.
Minus 0-004 inch per inch (mm. per mm.) of pitch dia-
134 GEARS AND GEAR-CUTTING
meter, with a maximum of 0-020 inch (0-51 mm.), subject
to the provision that a tolerance up to 0-005 inch (0-13 mm.)
shall be permitted on all chain wheels with, pitch diameter
less than 5 inches (127-00 mm.).
Formula for Centre Distances
(Two point drives)
N=number of teeth in wheel.
n=number of teeth in sprocket.
D== distance between centres.
L=number of links in chain.
P=pitch.
L=
2 xD N+n Px(N-n) 2 '
rT + ~ + —^oD~-.
(approx).
The result obtained from this expression will not
ordinarily be a whole number of links; in that event the
nearest even number should be taken and a recalculation
made to find the corresponding centre distance from the
following :
\^(A+p'9B) x (A—o* 9B) J
where A=2L—(N+n) and B—(N—n).
This formula gives results correct to within 0-05 per cent,
for wheels having a ratio of 2:1 at minimum centres. This
is sufficiently ■ accurate for drives where adjustment is
provided.
USEFUL FORMULAE FOR GEARS 135
Block-Centre Chains j
Circular
pitch
Thickness
of block
Diameter
of cutter
Centre to
centre of
block
i
Hole in j
cutter J
1
In.
15/16
1 1/2
In.
•4375
17/32
In.
3 1/2
3 3/4
In.
•53X3
•5625
In.
1 1/4
1 1/4
Seven cutters are made for each pitch, for Numbers of
teeth as follows : 8, 9, 10, 11, 12, and 13, 14 to 16, 17 to
20, 21 and over.
Roller Chains
Circular
pitch
Diameter
of rolls
Diameter
of cutter
Hole in
cutter
In.
In.
In.
In.
i
*306 or -308
' at
1
1
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1
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3 i
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4i
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2
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" Whitney Standard."
measuring along the Pitch Circle.
136
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GEARS AND GEAR-CUTTING
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USEFUL FORMULAE FOR GEARS
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Diametral Pitch
Diametral Pitch is the Number of Teeth to each Inch of the Pitch Diameter.
138
GEARS AND GEAR-CUTTING
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I40 GEARS AND GEAR-CUTTING
Pulley Calculations : General Rule.—A general rule
which is not affected by the system of gears and/or pulleys
employed—it applies whether the train is simple or
compound—is that the product of the diameters (or the
number of teeth in the case of gears) of the driving wheel
and the number of revolutions per minute of the first
driver is equal to the product of the diameters or the
number of teeth of the driven wheels and the number of
revolutions per minute of the last driven wheel.
In the workshop, calculations regarding speeds of
machine countershafts, etc., are frequent, and the following
formulas apply :
R.P.M. of Driven Pulley or Gear
__ Diam. of Driver x R.P.M. of Driver.
Diam. of Driven.
Diam. of Driven Pulley or Gear
__Diam. of Driver X R.P.M. of Driver.
R.P.M. of Driven.
R.P.M. of Driver Pulley or Gear
__ Diam. of Driven x R.P.M. of Driven .
Diam. of Driver.
Diam. of Driver
_ Diam. of Driven X R.P.M. of Driven.
RJ^feMfilJQnver.
INDEX
A
Addendum, 21.
Angle, base spiral, 96.
_involute, function of, 9*>-
British Standard rack-form, 21.
— Standard system, 21.
Brown and Sharpe involute
gear cutters, 121.
Built-up gears, 67.
— pressure, 20, 27.
.— spiral, 27. >
Angular division tester, 08.
— velocity, 15-
Automobile gearbox, 9.
— gears, 68.
Axial feed, 62.
— pitch of worm, 38.
B
Ball and indicator, checking
worm teeth by, 81.
— micrometer, checking worm
teeth by, 82.
Base circle, 20, 95.
— pitch, 20.
— spiral angle, 96.
Bending stress, 113.
Bevel-gear units, 76.
— gears, 29, 121.
-cutters for, 123.
-double-helical, 33.
-formulae for, 121.
— — generation of, 31.
-principles of, 29.
-set over for, 124.
-single-helical, generating,
56-
-spiral, 13, 15, 3 2 *
-generating, 57.
-straight, 12.
-tooth form of, 32.
Blank diameter, determining
size of, 22.
Block - centre chains, formula
for, 132.
C
Caliper, gear-tooth vernier, 78.
— Zeiss, 95.
Camshaft, gearing of, n.
Chain sprocket, formula for, 132.
— wheels, block-centre, formula
for, 135.
Chordal thickness, formula for,
98, 131-
Circular pitch, 21.
-tables, 24, 136.
Cluster gears, cutting, 43.
Concentricity of teeth and bore,
83-
Contractions, strains, 66.
Cutters for mitre gears, 123,
— generating, making, 108
B
Dedendum, 21.
Diameter of blank, 20.
— of pitch circle, 20.
Diametral pitch, 22, 138.
Double-helical bevels, 33.
-gears, 12.
-meshing of, 25.
— piano-generating, 53.
E
Efficiency of gears, 117.
Electric motors, speed of, 11.
End-mills, making, 103.
-thrust, 27.
Epicyclic gear trains, 71.
141
142
INDEX
F
Fellows gear generator, 42.
— process, 44.
Fly hob, 61.
Formate gears, 59-
Formed wheel, grinding, 50.
Formulae for spur gearing, 120.
Friction, 117.
G
Gear cutters, B. and S., in¬
volute, 123.
— generation, 17, 41.
— generator. Fellows, 42.
— grinder, Maag, 47.
— single-reduction, 75.
— teeth, checking by com¬
parison, 80.
-Checking shape of, 81.
-helical, overlap of, 26-
-measuring, 78, '
-- comparative sizes of, 116.
-tooth caliper, Sykes, 94.
-Zeiss, 94.
■-comparator, using, 90.
— vernier, 78.
— trains, epicyclic, 71.
-velocity ratio of, 140.
— types, 12-
— units, bevel, 76.
-worm, 77.
— wheel, definition of, 13.
-form, 66.
-types of, 66.
— units, spur, 75.
-worm, 76, 77.
— working life of, 118.
— Zerol, 33.
Gearbox, 9.
Gears, applications of, 10.
— automobile, 68.
— bevel, 29.
-cutters for, 123.
-generating, 55
-generation of, 31.
— built-up, 67.
— cause of noisy, 26.
— checking by running together,
89.
— chordal thickness of, 135.
— double-helical, meshing of, 25.
Gears, efficiency of, 1x7.
— formate, 59.
— helical, generating, 51.
-hobbing, 52.
-profile grinding, 55.
— hypoid, 59.
*— load capacity of, 113.
— loading of, 14.
— lubrication of, 76, rxS.
‘— measuring, 78.
— mitre, cutters for, 123.
— mounting, 74.
— requirements of, 14.
— root circle of, 20.
— screw, formula for, 126.
— shock loading of, 114.
— spiral, 13.
-formula for, 126.
-Gleason, 57.
— straight bevel, 12.
-tooth form of, 32.
— triple-helical, 12.
— turbine reduction, 68.
— types of, 9.
— velocity ratio of, 38.
— welded, 67.
Generating cutters, making, 108.
•— helical gears, 51.
— rack, 23.
— single-helical bevels, 56.
— system, Sykes, 54.
Generation of bevel gears, 31.
— of gear, 17, 41.
— pitch circle of, 20.
Gleason spiral gear teeth, 57.
Grinding, formed wheel, 50.
— spur gears, 48.
— worms, 64.
H
Helical bevels, single, 33.
— gear, single, 11.
-teeth, overlap of, 26.
-units, 75.
— gears, direction of rotation of,
28.
-double, 12.
-generating, 51.
-hobbing, 52.
-piano-generating, 53.
INDEX
143
Helical gears, profile grinding,
55 * .
-spiral angle of, measuring,
88 .
-Sykes generating system,
54*
-triple, 28.
Helicoidal worm thread, 36.
Hob, fly, 61.
Hobbing helical gears, 52.
— spur gears, 45.
Hobs, making, 100.
Hypoid gears, 60.
I
Internal spur gear, 44.
Involute curve, 19.
— function of angle, 96.
— gear cutters, B. and S., 121.
— tester, 83, 99.
L
Lead of worm, 38.
Load capacity of gears, 113.
Loading of gears, 14.
Lubrication of gears, 76, 118.
M
Maag spur gear grinder, 47.
Meshing of double-helical gears,
25-
Micrometer measurement of
tooth thickness, 79.
Milling worm threads, 63.
Mitre gears, cutters for, 123.
Module pitch, formula for, 21.
Mounting gears, 74.
P
Parkson gear tester, 83.
Pinion, definition of, 13.
Pinions, making, 68.
Pitch, 21.
— base, 20.
Pitch circle, diameter of, 20.
-of generation, 20.
— circular, 21, 136.
— cylinder of worm, 37.
— cylinders, 16.
— diametral, 22.
— line defined, 16.
— methods of defining, 21.
— module, formula for, 21.
— of worm, axial, 38.
— point defined, 16.
— testing, 86.
-- by optical method, 88.
— transverse, 21.
Pitches, tables of standard, 24.
Piano-generating machine, 41.
— processes, 53.
Pressure angle, 20, 27.
Profile grinding gears, 55.
— of teeth, checking, 83.
— of worm gears, checking, 84.
Proportions of teeth, 21.
Pulley calculations, 140.
R
Rack - form, British Standard
basic, 21.
— generating, 23.
— method of generation, 18.
Roller Chain Wheels, 132.
Rollers, measuring tooth thick¬
ness by, 80.
Root circle of gear, 20.
— of tooth, fillet at, 31.
Rotation, direction of, 28.
S
Screw gears, formula for, 126.
Shafts and pinions of car gear¬
boxes, 9.
Shock loading, 114.
Single-helical bevels, 33.
— gear, 11.
-reduction gear units, 75.
Spiral angle, 27.
-measuring, 88.
— bevel gears, 13, 15, 33.
— -generating, 57.
-- Zerol, 33.
— gear, Gleason, 57.
INDEX
144
Spiral gears, 14.
-formula for, 126.
Sprocket wheels, formula for,
132.
Spur gear generator. Fellows,
42.
-■ grinder, Maag, 47.
-internal, 44.
-units, 75.
— gears, definition of, 12.
-generating, 41.
-Fobbing, 45.
Standard pitches, table of, 24.
Stress, bending, 113.
Sunderland process, 41.
Sykes generating system, 54.
T
Teeth, measuring by rollers,
80, 97-
Thread-milling, 63.
Thrust, end-, 27.
Tooth and bore, checking con¬
centricity of, 83.
— depth of, 21.
— fillet at root of, 31.
— form of straight bevel gears,
32.
— proportions, 21.
— shape, checking, 81.
— thickness, finding from micro¬
meter measurements, 97.
-measurement of, 79.
Toothed gearing, 12.
Transverse pitch, 21.
..gears, 12, 28.
:ion gears, 68,
U
V
Velocity, angular, 15.
— ratio of gears, 38, 131.
Vernier, caliper, 78.
W
Welding gears, 67.
Worm and wheel, contact be¬
tween, 39.
— dimensions of, 36.
— gear, 14.
—> -— units, 76, 77.
— gears, 16, 34.
-checking profile of, 84,
-dimensions of, 38.
-tooth action of, 34.
— lead of, 38.
— pitch cylinder of, 37.
— teeth, checking by ball and
indicator, 81.
-micrometer, 82.
— thread, helicoidal, 36.
— threads, milling, 63.
— wheel generator, 61.
-pitch cylinder of, 16.
— wheels, cutting, 60.
-formula for, 129.
-producing, 60.
Worms, finishing, 65.
— formula for, 129.
— grinding, 64.
Z
Zeiss optical gear-tooth caliper,
94 -
Zerol, gear, 33.
— spiral bevel, 33.
Undercut, 21.
