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Argus Books Ltd.
14 St James Road, Watford
Hertfordshire, England
Second Edition 1947
Third Edition I951
Second impression 1960
Third impression I968
Fourth impression 1971
Fifth Impression 1973
Sixth impression 1975
Fourth Edition 1977
ISBN 0 85242 532 5
0 1977 Argus Books Ltd.
Ail rights reserved. No part of this
publication may be reproduced in
any form without the prior
permission of the publisher.
Printed ORset Lieho in Gym BrimIn
by Cox (L Wyman Ltd. Londbn, Fakenham and Rudin#
II Proportions and Form of Teeth of Gear Wheels
I First Principles Explained
III Lantern or Pin Wheels and Racks
IV Bevel Gears
:, ‘,
VI Helical Gears-Spiral Gears-Chain GearingEpicyclic Gears-Special Forms of Gear
Wheels . . . . . . . . . . . .
VII Definitions and Calculations for Gear Cutting
: VIII Gear Cutting in Milling Machines
‘, 1% Cutting Spur Gears in the Lathe . .
X Gear Generating Methods
Frontispiece.-Cutting the teeth of a small spnr gear in the lathe
First Principles Explained
ONE of the common methods of transmitting motion is by
means of wheels which make contact or gear, as it is called,
with one another. Movement being given to the first wheel
is communicated by it to the second wheel. Any number of
such wheels can be geared together-the movement of the
first wheel can be communicated to the second wheel, and by
the second to the third, and so on. Two such wheels are called
:, a pair ; if there are more than two, the arrangement is caBed a
train of wheels. If the edges or surfaces by which contact is
made between one wheel and another are smooth, the power
is transmitted by means of the
friction existing between the
surfaces. The wheel which is
transmitting the power is
called the driving wheel, and
the one receiving it is called
the driven or following wheel,
x-Diagram showkg pair
or just the follower. The Fig.
of gear wheels and their
relative motion
wheels may be of equal size
or one may be larger than the
other. In this latter instance the smaller wheel is called a pinion.
Fig. I shows a diagram of a pair of wheels in gear. If D is the
driver, F is the driven wheel or follower. Fig. 2 shows a diagram
of a train of ,wheels ; if A is the driver, its motion will be transmitted by B and C in turn to D. Any one of the wheels can be
made the driver ; for example, B, which will then communicate
its movement to A, and to D through C. Fig. 3 shows a wheel W
and pinion p. The driven wheel will resist the action of the
driving wheel. It will do this because some friction must exist
at its bearings, even if no
transmitted by the driver
will vary according to the
Fig. %-Train of gear wheels
and relativewrezmn of each
resistance to motion offered
by the driven wheel. If
this resistance is too great to be overcome by the frictional grip
existing between the contact surfaces of the wheels, the driven
wheel will lose movement and there will be slip between the
contact surfaces. To prevent slip the surfaces are cut into teeth,
and made to engage positively with each other. By this means
an accurate transmission of the motion is ensured. Such wheels
are called spur or gear wheels.
Imagine a pair of gear wheels, A and B (Fig. 4) ; B is the
driver giving motion to A. If we fix a tooth T upon B to prevent
slip, we must cut a groove G in A for it to engage with, or the
wheels cannot continue to rotate. A series of such teeth, spaced
at equal distances, may be fixed upon the circumference of B,
and a series of grooves to receive them cut in the circumference
of A. Slip cannot then take place. B is geared into A and
drives that wheel positively, or A may be the driver and give
motion to B. This positive engagement between the two
wheels is entirely due to the teeth T
projecting beyond the circumferential
surface of B. Matters will be equalised,
and the time during which any particular
tooth of one wheel is engaged with the
other wheel will be prolonged, if teeth
Fig. S.-Gear wheel
are placed upon the circumferences of
andpinion, and their
relative motion
both wheels. In this instance we
should place teeth V upon wheel A for this purpose. We must
then cut grooves W in B to receive these teeth. As A is already
cut with a series of grooves,
and B is provided with a series
of teeth, the new grooves and
teeth must be placed at the
unoccupied parts of the respective circumferences. The
teeth till therefore be placed Fig. q.-Showlng action of tooth
and space
the grooves cut in the parts G of B, as indicated by the dotted lines.
The Pitch Circle
The teeth of a gear wheel are, therefore, made up of two
parts, one of which
is inside and the other
outside the true circumference o f t h e
wheel, as indicated by
ing a pair or train
of toothed wheels,
we should therefore first imagine them to be without teeth
and merely tolling against one another with frictional contact
only. In fact, we should plan them as friction gearing and
merely add the teeth to the plain wheels thus designed. The
circumference of such a plain wheel is called the pitch surface,
usually referred to as the pitch circle because, when setting
out the gear upon
paper, circles are first
Fig. 5.--Explaining the parts of a
these pitch surfaces.
The pitch circles
shown in Fig. 6 represent the contact surfaces
between a pair of
plain wheels.
Fig. 6.-Principle of pitch circles
part of the contact surface of the tooth which is outside the
pitch circle is called the face, and that part inside the pitch
circle is called the flank. The entire portion of a tooth which
is outside the pitch circle is called the addendum, and that
inside it is the dedendum, When planning a pair or train of
wheels, the first consideration is the value or ratio of the gearing.
This means the relation between the number of complete
revolutions made by the first and last wheels respectively in
any given interval of time ; or time can be left out of consideration
and the value of the gearing be regarded as the number of
complete revolutions which the last wheel will make whilst the
first wheel makes one complete revolution. The first wheel is
considered to be the one which sets the whole train in motion.
If the last wheel makes one complete revolution whilst the first
wheel also makes one revolution the train is said to be of equal
gear ratio. But if we arrange the sizes of the wheels in suitable
proportion, the last wheel can be made to give more or less than
one revolution for each revolution of the first whee!. If the last
wheel has rotated more than once when the first wheel has made
one complete revolution, the train is said to be geared up ; if
less, it is said to be geared down.
Gear Ratios
The ratio of revolutions is determined by the diameters of
Fig. T.-Pitch circles to
increase revolutions
Fig. S.-Pitch circles to
decrease revolutions
the pitch circles. Thus, if the wheels A and B (Fig. 6) are
to make equal revolutions, B making a complete revolution for
each complete revolution of A, the pitch circles must be equal
in diameter. If B is to make two revolutions for each one
made by A, the pitch circle of B must be exactly one-half the
diameter of the pitch circle of A (Fig. 7). Suppose that A
(Fig. 8) is to make two revolutions for one revolution of B, the
pitch circle of B must be twice as large a s that of A. Thus
the required ratio of revolutions between the driver and driven
wheel is determined not by their diameters, as measured over
the points of the teeth, but by temporarily leaving the teeth
out of consideration and
calculating the sizes of
the pitch circles alone.
Having decided the
diameters of the pitch
circles, the diameters of
the wheels, measured
over the tops of the
Fig. g.-Determination of overall
teeth, are determined
diameter of wheels
by adding an allowance
equal to ihat part of the teeth which projects beyond the pitch
circles. This is shown by Fig. g, the pitch circles being the
dotted lines and the full circles the over-all diameters of the
wheels. The part of the teeth which projects beyond the pitch
circle is shaded. Patterns or blanks from which the wheels will
be made would, therefore, be turned to this over-all diameter
which thus provides the requisite allowance to complete the
teeth. When machining the wheels in the lathe, it is often the
practice to mark a line representing the pitch circle upon the
side of the wheel. This serves as a guide when cutting the teeth
and also for meshing the wheels correctly.
The Tooth Pitch
The ratio of revolutions between one wheel and another
also depends upon the relative number of teeth. If wheel A
has 10 teeth and wheel B 30 teeth, A will rotate one and a half
turns to, one complete revolution of B. Therefore we must
not only design the pitch circles so that their diameters are
of the correct ratio, but we must also make the numbers of the
teeth to correspond. To some extent this question decides
itself, because the teeth upon A must be spaced at a distance
apart to correspond with the spacing of the teeth upon B, or
the two sets will not fit properly together ; the numbers of
teeth should, however, always be calculated and made to correspond with th.e diameters of the pitch circles. The distance from
the centre of one tooth to the centre of the next is called the
pitch and is measured along the pitch circle. If the two wheels
are to gear properly together, the pitch of the teeth upon A
must be of the same pitch as those upon H. When determining
the number of teeth for, say, wheel B, it may be found that
any number which gives a reasonable pitch and is a convenient
fraction of an inch, such as * in. or $ is., %wil! not divide the
pitch circle of A into the correct number of teeth. If the wheel
centres are not fixed, the matter may perhaps be adjusted by a
slight alteration in the sixes of the pitch circles, still keeping
them to the desired proportion. If the whc:el centres cannot be
altered, the pitcl: of the teeth will have to be adjusted accordingly.
There is another method of reckoning the pitch. Instead of
measuring it along the circumference, it is measured as so many
teeth per inch diameter of the pitch circle. Thus, if a wheel
having a pitch circle diameter of 3 in. is to have 24 teeth, they
are said to be of 8 diametral pitch, because there are 8 teeth in
I in. of the pitch circle diameter. Awkward fractions of an inch
can thus be deait with in a simple way; No. 8 diametral pitch
would be .3g3 c,ircumferentiai pitch. If the circumference of
the pitch circle is made of such a size ,that fractions are avoided,
the diameter may be some awkward dimension. By working to
diametral pitch, the pitch circle diameter can be easily measured
and set at. Tool makers generally use this method and supply
a variety of cutters made to diametral pitch. Therefore, as a rule,
there is no difficulty in planning the gear teeth and obtaining the
corresponding gear cutters.
Rules tar Calculatitig Gear Wheels
The follow:ng formula? are useful for calculating gear wheels:T HE CIRCULAR PITCH MULTIPJ~I~D BY THE NUMBER OF TEETH
3.1416 BY T H E D I A M E T R A L P I T C H.
The quantity 3.1416 (“,” to four decimal places) is the ratio
between the circumference and diameter of any circle, named
and denoted by the Greek letter pi (7~).
These and other formulae are shown in greater detail, and
with examples, in Chapter VII.
Proportions and Form of Teeth
,q+,; , ,_
$&;;, , :
THE size of gea; teeth is determined according to the power
they have tc transmit. They tend to break at the lowest
portion-that is, at the root. If the wheels are well fitted, and
the teeth make contact throughout the entire width, thev will
resist the stress much better than if they are inaccurately fitted.
I,n the latter case, they may make contact. at some place near the
edge so that the stress is concentrated mainly upon a small
portion of the metal. The power which a gear wheel can transmit
increases with the speed of the rotation. If a wheel has a slow
speed of rotation, to transmit a given amount of power its teeth
must be larger than they need be if the speed were higher. Generally, there will be two or more pairs of teeth in contact simultaneously, so that the pressure is distributed over two teeth.
All the small gearing likely to be used will probably have sufficient
strength when the teeth are made of standard proportions, so
that no calculations need be made for this. If the teeth are to
be cut from the solid-and this is the best method for small
wheels-the blanks can be given to a gear-cutter who will select
a suitable tool, &nd it will only be necessary to state the sizes
of the pitch circles and the number of teeth to be cut in each
wheel. When deciding upon the number of teeth, arrange to
have as many teeth as possible consistent with strength and
wear. It is not advisable to have less than seven teeth in any
Proportions of Teeth
The teeth are usually proportioned according to the length
of the pitch, but different makers vary the dimensions to a small
extent. The well-known authority, Professor Unwin, in his
Elements of Machine Destgn, gave the proportions shown in Fig.
IO, the ~unit being the pitch. These dimensions show that the
thickness of a tooth measured on the pitch circle should be
slightly less than the width of the interval between the teeth
(called the space). Also that
there wili be a clearance
space between the point of
the tooth of one wheel and
the bottom of the space of the
wheel into which it is geared.
Fig. xo.-Tooth proportions
The width of the wheel is
usually about 2 to 2; times
the pitch. When a pair of wheels is in gear, the pitch circles
should touch. If such proportions for the teeth and spaces as
are given in Fig. IO have been adopted, there will then be a small
amount of play between the teeth, as the thickness of the teeth is
slightly less than the width of the spaces, and the height above the
$,.“‘, pitch circle is less than the depth below. This clearance allows
i{!:f!,,’ for very small irregularities, and enables the wheels to run
,; ; ;l:, , without jamming ; it should not be gained by extending the
distance between the centres of the wheels.
$ ‘I , Mortise Wheels
There is a kind of gearing used in mill work called mortise
, i ,‘, ,‘,
,‘, ‘, ; ‘, wheels. Here, one wheel of a pair is fitted with wooden teeth
‘dovetailed into slots in the rim. When these wheels are used the
: 2,
proportions of the teeth are altered, the wooden teeth being made
thicker than the metal teeth of the wheel with which it is geared.
The object of the gearing is to reduce noise, and it would hardly
be used for small gearing except as a model of a large gear.
Cycloidal Teeth
The teeth of gear wheels have to be made to an exact shape,
and it is not ,sufficient to make them of any pattern that will
allow of engagement and disengagement during rotation without
binding. There is a further consideration : the teeth must be of
such a shape that the relative velocities of the pitch circles will
not be disturbed as they roll one against the other. The pitch
circles should continue to roll as if there were no teeth and no
slip. Mathemati.cians have discovered that if the teeth are
shaped according to certain well-known geometrical curves,
this condition will be fulfilled. The three curves which are
used in practice are the epicycloid, the hypocycloid, and the
involute. When made on the cycloidal principle the contact
surfaces of each tooth are composed of two curves. That part
which is outside the pitch circle is an epicycloid curve, and that
part which is inside the pitch circle is a hypocycloid curve
(see Fig. I I) . A cycloid is the curve which is described
by a point fixed at the circumference of a circle when that
circle is rolled in contact with a straight line. An epicycloid
is the curve which would be described by the point if the circle
were rolled upon the circumference of another circle. A hypocycloid is the curve which would be described if the circle
were rolled in contact with, but inside, the circumference of another
circle. The rolling circle is termed the generating circle.
If the curves of the faces of the teeth on one wheel are formed
by the same rolling circle as that used to form the flanks of
the teeth on the wheel with which it is to gear, the relative
velocities of the pitch circles will not be disturbed by the engagement of the respective teeth. This is actually done in practice
and the curves are then produced by rolling a disc, representing
the curve-generating circle, upon another disc or template
representing the pitch circle, or these curves are drawn with
compasses to some geometrical construction which gives arcs of
circles very closely approximating to the real curve. Methods
of this kind are given in text-books on gearing and machine
construction and in engineering pocket-books. The same
generating circle can be used to describe the curves for the faces
and flanks of the teeth of each wheel ; this is convenient aiid
usual in practice, though two generating circles could be used-
one for the flanks of the driver teeth and faces of:he driven teeth,
and the other for the faces of the driver and flanks of the driven
teeth. If more than two wheels are in gear together, or if a
number of wheels are required to gear indiscriminately with
one other-as in the case of a set of change-wheels for a lathe
or other machine-it is necessary
to use one circle only to generate
the curves for the faces and
flanks of the teeth of all the
Teeth with Straight Flanks
Many wheels are made with
teeth which have straight radial
instead of curved flanks. This
Figs. 11 and x2.-Tooth forms
is quite correct, because a hypocycloid generated by a circle
whose diameter is equal to the
radius of the pitch circle inside
which it rolls is a straight line
The generating Circle
Fig 13-jveak tooth form
should not be made larger than
this, as the straight line then becomes a reverse curve, producing a weak form of tooth at the root, as indicated by Fig.
1 3 . For this reason the diameter of the generating circle to
form the teeth of a set of wheels of different sizes is usually
made equal to the radius of the pitch circle of the smallest
wheel. The flanks of the teeth of that wheel will then be
straight lines, and those of all the others will be curves. But
all will be hypocycloids, and the teeth will not be weak at the
root. According to Molesworth, the best diameter of the
generating circle is given by 2.22 times the pitch, provided the
number of teeth in any one of the wheels is not less than fourteen.
If the number be less, the diameter of the generating circle should
be equal to the number of teeth multiplied by the pitch and
divided by 6.3,
Involute Teeth
Teeth shaped upon the cycloidal principle preserve the
relative velocities of the pitch circles only if the wheel centres are
at the proper distance apart. If the centres are further apart
so that the pitch circles do not rotate in contact, the relative
velocity is not maintained. There wiii, however, still be the
ratio of revolutions between the wheels-obviousiy this cannot
alter; but the wheels themselves will not revolve with an even
motion, Each will be intermittently going faster and slower
as the teeth come into and pass out of engagement. This will
tend to set up vibration and noise and may be a disturbing
factor in delicate mechanisms. If the teeth are shaped on
the involute principle, the relative velocities of the wheels will
not be affected if the centres are spread apart or one brought
more closely together, provided the change of position is
small. Any normal wear of the bearings would thus not interfere with the proper action of the teeth. The involute is a
geometrical curve produced by the end of a stretched cord which
is being unwound off a cylinder or the circumference of a circle ;
it would also be produced by the end of a straight line which is
being rocked upon the circumference of a circle. In either case
the circle is called the base circle,
called the involute of that circle.
principle do not have two curves fo
face and flank at each side of a to
same involute curve (see Fig. 14).
end of the line B, which represents the cord or straight line
rocking upon the base circle. Such teeth are of strong shape,
and all wheels with involute teeth will work correctly together
if the teeth are of the same pitch and the same angular line of
contact. That is a line L C (Fig. 14) making contact with the
base circle and passing through the pitch point P. For any pair
of a set of wheels to work together, the radii of the base circles
must bear the same proportion as the radii of the pitch circles. A
curve consisting of an arc of a circle can be produced which is
very near to the,true involute curve ; geometrical methods for this
are given in text-books on gearing and machine construction
and in engineering pocket-books. According to some authorities, teeth shaped to the involute curve exert a thrust along the
line joining the centres of the wheels (called the line of centres),
thus exerting extra pressure upon the bearings ; therefore the
involute curve should not be adopted if the wheels have to transmit a considerable amount of force. Others do not agree with
this. Hawkins, in his book on the teeth of wheels (largely
translated from the work of the French mathematician Camus),
describes an experiment made by him with a pair of wheels
having involute teeth. The wheels were tried with the teeth
engaging at various depths, and did not show any tendency to
thrust the centres apart until they were placed with the teeth
only engaged to a depth of $ in. out of a total depth of r4 in. ;
even then the tendency to separate the centres was very slight.
The involute form of teeth is favoured in modern practice, and
is now very generally used. If the angle A (Fig. 14) be made as
large as practicable, involute teeth appear to give good results
in working. The effect of increasing the angle A is to bring the
circumference of the base circle close to that of the pitch circle:
and so shorten the teeth.
Internal Toothed Gears
Gear wheels can be made in
the form of a ring with teeth
inside the circumference instead
of outside. These wheels are
called internal or annular gear
Such a wheel can
Fig. x4.-Ezute tooth
obviously only gear with another
which has external teeth and
is smaller in diameter, because the second wheel is placed
within the first. These wheels are calculated and set out according to the diameters of their pitch circles, as in the case of external
toothed gears. The pitch circle of the pinion P is inside
that of the wheel W (Fig. IS) and touching at the pitch point.
If the diameter of the pitch circle of the
wheel W is four tiines that of *&e pinion
w P, the latter will make four revolutions for
one revolution of the wheel, and so on.
The numbering and proportions of the
teeth apply as in the case of external gears.
Cycloidal or involute curves can be
used for the shape of the teeth.
-~~~~C~~i~~&~i~~~~ cycloidal curves are used, they will
internal gear
be transposed in the case of the
teeth of the wheel.
The curvegenerating circle for the faces of the teeth will
roll inside the pitch circle, and therefore produce a
hypocycloid, and that producing the flanks will roll
outside the pitch circle and therefore produce an epicycloid
(see Fig. 16). An internal gear wheel can engage with several
pinions simultaneously placed at different positions along its
pitch circle ; the pinions may be of various sizes. In such an
instance the curvegenerating circle for
all the teeth should
have a diameter equal
to half that of the
Fig. x6.-Geometric
curves of internal
gear teeth
pitch circle of the
smallest pinion. There
is an important difference between the
working of an internal
and external pair of toothed wheels.
If the wheels are
external, they rotate in opposite directions, but if internal,
they rotate in the same direction, as indicated by the arrows
(Fig. 15).
The Contact of Teeth
If the teeth are made as
so far explained, with a
part projecting beyond the
pitch circle as well as a
part inside it, any pair will
come into contact as they
Fig. x7.--Principle of engagement and disengagement of teeth
approach the line of
centres, and this contact
will be maintained to some
distance after they have
passed the line of centres.
For example, a pair of
wheels A B, Fig. 17) are in
gear. The teeth of wheel
A come into contact with those ofB to the left of the line of centres
X, Y, as the wheels rotate in the direction indicated by the arrows.
They are then said to be engaging. After passing the lint of centres
they are said to be disengaging. The contact is maintained for
some distance, but finally ceases as each pair of teeth passes out
of gear. The distance through which they make contact when
engaging is called the arc or angle of approach, and that through
which they move when disengaging is called the arc or angle of
recess. The actual path followed by the contact point is a curve in
the case of cycloidal teeth, which consists of arcs of the toothgenerating circles C, C. (Fig. r7a, Chap. III), any particular tooth
commencing to make contact at a point in one circle, and leaving
contact at a point in the other circle, as indicated by the full
line which is termed the path of contact, also the arc of action.
In the case of involute teeth the path of tooth contact is along
a~straight line, such as L C (Fig. 14), commencing inside one
pitch circle and ceasing inside the other pitch circle.
Lantern Or Pin Wheels and Racks
THE teeth of gear wheels rub together whilst in motion, causing
friction and wear of the surfaces, but these are reduced by
designing the teeth so that the path of contact is as short as
possible. The friction arising during engagement in the arc of
approach-this would be to the left of X, Y (Fig. IL/a)-is
considered to be more detrimental than that which occurs
during engagement in the arc of recess-this would be to the
right of X, Y (Fig. r7a). The teeth rub to greater disadvantage
when coming into ’ contact than
when disengaging. On this account,
designers of wheel gearing, in
which it is of special importance
that friction and wear should be
eliminated as much as possible, such
as watch and clock gearing, prefer
teeth which only make contact
when they have reached the line
of centres. Engaging friction is
thus reduced to a minimum. This
action will be accomplished if the
Fig. ITa.-Path of contact driven-wheel teeth are made withof cycloidal teeth
out points-that is, they would not
project beyond the pitch circle. For example, wheels having
complete teeth, such as Fig. 17, come into engagement before
the line of centres X, Y. At this stage, the Banks of the teeth on
wheel B make contact with the faces of the teeth on wheel A,
assuming that B is the driver. After passing the line of centres
the faces of teeth on wheel B make contact with the flanks of the
teeth on wheel A. Therefore, if we desire that contact shall
only be made after the line of centres, the points of the teeth on A
should be removed. It f o l l o w 5
from this that the roots of the
teeth on B will not be required,
and as the engagement is only to
take place at, or after, the line of
centres, B must be the driver. If
A be the driver, the engagement
Fig. I&-Principle of
will take place only before the
wheel and lantern
line of centres, which is, here, the
condition to be avoided. The
teeth having points are therefore put on the driver, and those
having no points are on the follower. If cycloidal teeth are
used, as they should be, the curve of those upon the driver
will be an epicycloid, and that of the teeth upon the driven
wheel a hypocycloid. This leads to two distinct forms of teeth
for the driven wheel. If the curve-generating circle has a
diameter equal to the radius of the pitch circle of the driven
wheel, the hypocycloid becomes a straight line (as previously
explained), and the teeth have merely straight, radial lines for the
contour of their flanks. In the second case, the curve-generating
circle is made to a diameter equal to that of the pitch circle of
the driven wheel; the hypocycloid then becomes a point and
the teeth may be pins projecting at right angles to the side of the
This is the only instance in which the hypocycloid
formed by a generating circle of such a proportion to the pitch
circle is used for wheel teeth.
Pin Wheels or Lantern Pinions
When the curve-generating circle is made of a diameter equal
to that of the pitch circle in which it rolls, and the hypocycloid
becomes a point, some practic-al modification is necessary,
because the teeth are, theoretically, merely points, which,
according to Euclid’s definition, have position but no magnitude.
Obviously the teeth must have some thickness, and in practice
they become pins. Fig. 18 is a diagram showing the teeth of
a pinion (P) as points, which represent rhe hypocycloid we
have produced by using a curve-generating circle with a
diameter equal to that of P. The teeth on the wheel W are
entirely outside the pitch circle, and the faces are epicycloids
produced by rolling the circle which has generated the point
teeth of P upon the pitch circle of W. To make a practical
working construction, we fit cylindrical pins to P to form teeth.
‘This is quite justifiable, as a circle is the equivalent of a point,
and therefore in the particular instance is logically a hypocycloid.
The teeth on W as originally formed to engage with points are
represented by dotted lines (Fig. IS). If we enlarge the points
so that they become pins, it will be necessary to cut away some
portion of the teeth of W to provide room for the pins to engage
between the teeth; as already explained, we may not alter the
distance between the centres of the wheels. Space for the pins is
provided by cutting away a portion equal to half the diameter
of one of the pins from
each face of a tooth along a
line parallel to the original
curve of the face. This will
leave the teeth with the size
and shape indicated by the
fidl lines (Fig. Ig), the curves
still being epicycloids. To
complete the clearance space,
a semi-circular space is cut
Fig. Ig.-Practical gear
away below the pitch line of
wheel and lantern pinion
W between each pair of teeth.
This procedure may be understood by imagining the pin to be
a milling cutter moving with its centre coinciding with the
original line of the teeth, and thus cutting away the amount of
metal necessary to allow the pins to engage. In such a gear, if
W is the driver, the engagement of the teeth of the pair of wheels
will take place principally after the line of centres. As this is really
the object for which the particu‘*
lar arrangement is designed,
-x-‘-~. . \
whenever it is used, the wheel
having the projecting teeth is
always made to drive the one
.o Q 0
provided with pins. If the pins
are made to drive the teeth, the
engagement takes place before
Fig. zo.-Lantern pinion
or pin wheel
the line of centres, and the
object of the design is lost. The
arrangement is much used in clockwork, the wheels with the
pins appearing in the familiar form shown in Fig. 20 ; these are
called lantern pinions, from the resemblance to a lantern. In
the main train of wheels of a clock, the driving force passes
through the gearing from the great wheel, which is driven by the
spring or weight to the escapement. The wheels, therefore,
drive the pinions, and the latter can be provided with pins as
teeth, hence the extensive use of lantern pinions for clockwork.
Incidentally, these have the advantage of being very strong and
durable. The name pin wheel is also applied to this form and
the pins are termed rungs, staves, or rounds. Gears of this kind
may also be made in the form of a rack and pinion, as will be
described later.
Clock and Watch Gearing
It is also possible to design a pinion with teeth hatig radial
tlanks to engage after the line of centres. The number of teeth
must be at least ten, and it may be necessary to cut the spaces
between the teeth of the pinion with extra width. There would
thus be a certain amount of play between the wheels, but this is
permissible in clock-gearing where the teeth move slowly, and
are kept in contact by a steady constant pressure. Though the
teeth of such pinions need not project beyond the pitch circle,
they are usually made with a round end projecting beyond the
pitch circle by an amount equal to half the thickness of the
tooth, in order to ensure smooth engagement as the teeth come
into action. Clock and watch gearing reqGre large wheels
driving very small pinions, and work under special conditions.
Anybody contemplating the construction of a time-keeping or
similar mechanism should consult a treatise on clock and watch
gearing. The principles upon which the teeth of the wheels are
constructed are, however, precisely those which govern the design
of wheel-gearing for machinery in general. As smooth action is
very important, the teeth must be of correct design to preserve the
relative velocities of the pitch circles. Practical. modifications
would be introduced to meet the peculiar conditions of clock
gearings. For example, the spaces at the roots of the teeth of W
(Fig. 19) would probably be cut somewhat deeper thanindicated,
and made rectangular instead of being semicircular; the sides
are then radial lines. The teeth of W would be of less width
than the spaces between the pins on P, this amount of play
permitting engagement to take place at or very near to the line
of centres. The best length for the teeth of W would probably
be found by experiment.
Rack and Pinion Gears
A rack and pinion gear may be considered as a pair of toothed
wheels, one of which, the rack, has a pitch circle of infinite
radius represented by a straight line. The teeth can therefore
be shaped according to the principles already explained. Provided the rack is made of sufficient length, the pinion can be
made to give any desired number of revolutions for one stroke
of the rack, or the rack made to move for any desired length of
stroke for one revolution of the pinion. The gear is planned in
accordance with the principle used for designing a pair of toothed
wheels. The pinion I? is represented by its pitch circle, and the
rack R by a pitch line (Fig. 21), which is really the pitch circle
of a second wheel stretched out to form a straight line. The two
pitch lines touch one another at the pitch point X. The pinion
may drive the rack, or, conversely, the rack may drive the
pinion. As with a pair of wheels, the teeth should be of such a
shape that the relative velocities of the pitch circle and line are
maintained. The length of stroke which the rack will make for
one revolution will depend upon the diameter of the pitch circle
of the pinion. When planning a rack
and pitiOn, therefore, the positions of
the pitch circle of the pinion and pitch
P ‘\
line of the ,rack should be determined
first without regard to the teeth of
!J Pm either. The distance the rack will
move for one revolution of the pinion
Fig. ax,-Pitch circle
principle applied to will be equal to the circumference of
rack and pinion
the pitch circle of the pinion. Thus,
if the diameter of the pitch circle of
the pinion is 4 in., the rack will move nearly 124. in. for each
revolution of the pinion. Conversely, if the rack drives the pinion,
the latter will be rotated one complete revolution if the rack be
moved through a stroke of approximately 12+ in.
Driver or Follower
If the rack or pinion is to be indiscriminately either driver or
follower, the teeth should be partly formed outside the pitch
lines and partly inside, as
in the case of a pair of
wheels. The curves of the
teeth of the pinion will be
formed with the faces epicycloid and the flanks
hypocycloid. The curves p,
of the teeth of the rack will make the faces
cycloid, and the flanks
curves of rack
cycloid also, because in Fig. az.-Geometric
and pinion teeth
each instance the curvegenerating circle is rolled upon a straight line (Fig. 22). The
diameter of this generating circle may be anything not exceeding
the radius of the pitch circle of the pinion, and the same
generating circle can be used to form the whole of the curves. If a
set of wheels is required of different diameters and numbers of
teeth, any one to work with the rack, the generating circle should
be equal to the radius of the pitch circle of the smallest wheel. The
pitch of the teeth is measured on the circumference of the pitch
circle of the pinion and along the pitch line of the rack. The pitch
may be expressed as diametral pitch in terms of per jnch of the
pitch circle diameter of the pinion, as previously explained ; it
will thus also apply as pitch in the number of teeth per inch
length of the rack. When the curve-generating circle has a diameter
equal to the radius of the pinion pitch circle, the teeth of the
pinion will have straight radial
lines for the flanks, as previously explained, and the rack
teeth will have curved lines
for both faces and flanks, the
curves being a cycloid. The
teeth of the rack may, however, be made to have straight
Fig. q.-Teeth of rack and
radial flanks. As the radius
pinion with straight flanks
of a straight line is of infinite
length, the flanks of such teeth will be straight lines perpendicular to the pitch line. The faces will be a cycloid formed
by the generating circle, which produces the straight radial flanks
of the pinion. The faces of the pinion teeth, however, should
not be an epicycloid curve, because they should be produced by
the generating circle which has produced the so-called radial
flanks of the rack. But this circle is one of infinite radiusin fact, it is a straight line. Therefore, the faces of the pinion
teeth should be curves produced by rolling a straight line upon
the circumference of the pitch circle ; the curve should thus be
an involute of the pitch circle of the pinion (see Fig. 23).
Pin Teeth for Rack and Pinion Gear
The rack or the pinion may be fitted with teeth in the form
of pins on the principle explained with reference to Figs. 18 to
As in the case of a wheel and pinion, the pin teeth should
be on the follower and not on the driver. If they are placed
upon the pinion, the rack should therefore be the driver ; if they
are placed upon the rack, the
pinion should be the driver.
The pins represent a hypocycloid produced by a curvegenerating circle having a diameter equal to that of the pitch
circle of the wheel upon which
they are placed. Therefore, if
they are placed upon the
Fig. &+-Rack and lantern
pinion, the teeth of the rack will
pinion ; tooth shapes
be formed by a cycloid curve
produced by a point on the pitch circle of the pinion when
it is rolled upon the pitch line of the rack. The rack teeth will
be composed of the part which projects above the pitch line,
so that they will have faces only and no flanks (see Fig. 24).
If the pin teeth are placed upon the rack they represent, as
before, a hypocycloid
produced by a curvegenerating circle
will have a diameter
equal to the pitch
circle upon which
the teeth are placed.
In this case, the pitch
circle is a straight line,
therefore the curves of
Fig. q.-Pinion ‘- and rack with pinthe teeth of the ,pinio n
teeth ; tooth shapes
should be produced
by rolling a straight line upon its pitch circle. The curve thus
produced will be an i nvolute. of that pitch circle. The teeth of
the pinion are therefore shaped to an involute curve, and
consist of the part which projects beyond the pitch circle, no
flanks being required (see Fig. 2 5 ) . Gears on this principle
should work very smoothly if properly made, as the friction
between the teeth takes place principally after they have passed
the line of centres. Theoretically, the pin teeth are points, as
in Fig. 18 ; the spur teeth of the driver are cut away to allow for
the thickness of the pins, as reference to Fig. 19 will explain.
Involute Teeth for Rack and Pinion Gear
The teeth of both rack and pinion may be formed on the
involute principle, and
either may be the driver.
In this instance, the teeth of
the pinion have both faces and
flanks formed by one curve ;
namely, an involute formed
by rolling the pitch line of the
’ rack upon a suitable base
circle (see Fig. 26). The teeth
of the rack must also be an
pig. a6.-Involute form of rack involute in theory. But the
and pinion teeth
involute of a circle of infinite
radius is a straight line. The
teeth will therefore have a straight line for both faces and flanks,
and the line should be inclined so that it is a tangent to the
curve of the teeth of the pinion. As in the ,case of a pair of
wheels, the pitch circle of the pinion and the pitch line of the
rack must be in contact,or the relativevelocity will not be constant, unless the teeth are formed upon the involute principle.
This is the modern practice, and forms the basis of the generation of gears in gear-cutting machines, as will be seen in the
later pages of this book. The rack teeth are formed at two
standard included angles : 2g deg., which is the angle of the
Acme thread, and 45 deg. Half these angles, 144 deg. or 224 deg.,
are called the Pressure Angle. The effect of weakening
undercutting of the flanks with a 145 deg. pressure angle is very
obvious, but the teeth of the 224 deg. gear are much stronger.
Pinions having 1esCthan 12 teeth are liable to this defect.
Bevel Gears
two shafts are not parallel with one another, conical
toothed wheels, called bevel wheels, may be used to transmit
the drive. These wheels are difficult to construct so that they
will work properly together and maintain the relative velocities
of the shafts, If the shafts are at a right angle and the wheels
are of equal size, the gears are then called mitre wheels. When
planning a pair or train of bevel wheels, first imagine them as
cones with smooth surfaces rolling against each other and transmitting the motion by frictional contact (Fig. 27). The relative
velocity of cone W to cone P will depend upon the relative
diameter A A of cone W to the diameter B B of we P. If these
diameters are equal, cone P will make one revolution for each
revolution of cone W. If any other diameters, such as C C, D D,
are selected, they will be in the same proportion to one another
as the large diameter A A is to B B. We can imagine a series
of such pairs of diameters between the bases and points of
the cones, and each pair will bear the same proportion to one
another. The entire surfaces, therefore, of the two cones
roll together. with the proportional velocity of the large
circles A A, B B, and the entire surface of each cone forms a
pitch surface of that cone. We could thus select any pair of
diameters upon which to form the pitch circles of the cones. In
practice, the circles formed upon the largest diameters A A and
B B are selected as the pitch circles. To drive one shaft by the
other at any relative n$mber of revolutions, the sizes of the circles
forming the bases of the cones are made in proportion to the
gear ratio between the shafts. Thus, if shaft B is to make one
complete revolution -whilst shaft A makes one revolution, the
base A A of cone W will have a diameter equal to the base B B
of cone P ; if shaft B is to make one revolution whilst shaft A
makes two revolutions, the base of cone P should be designed
Fig. 27.~Bevel gears ;
principle of rolling
with a diameter twice as large as the diameter of the base of cone
W (Fig. 28). The bases of the cones are equivalent to the pitch
circles of flat gear wheels, and the shafts which they connect
will rotate with relative velocities proportional to the diameters
of the bases of the cones. In these explanations it is assumed
that the axes of the shafts intersect, as this is the condition
usually met with in practice.
The Shape of Bevel Gear Teeth
A pair of cones, made of wood, metal, or other material,
will transmit the motion of one shaft to the other by contact
friction between the surfaces. If these cones are large in proportion to the amount of power to be transmitted, and conditions
of working are favourable, the friction may be sufficient for
driving. To prevent slip,
teeth may be provided as
in the case of spur gear
wheels. This introduces a
difficulty, as the teeth and
spaces must be conical and
follow the shape of the pitch
surfaces of the cones. For
Fig. 28.-Bevel gears
example, if we construct
and shafts rotating
teeth of similar shape and
at different speeds
character to those used for
spur wheels, these teeth
must be made to taper from the base to the point of the cone, as
indicated by the shaded surfaces (Fig. 29). If the teeth are made
of uniform height and thickness, or of less angle of taper than
would terminate in the point of intersection of the pitch cones,
they could not work together, but would foul and break off if
sufficient power were
applied to drive the
shafts. Eveiy part of
the surface of each tooth
-the faces and flanks
as well as the tops-must
be conical, the taper
coming to a point at the
intersection point of the
pitch cones. The teeth,
if properly made, will
therefore become very
thin near to the points of
Fig. tg.-Form;zeti;of bevel-gear
the pitch cones, finally
vanishing away. Only a
portion of the tooth length is of practical use, and bevel wheels
are never made to the complete theoretical form of the pitch
cones ; the breadth is usually made equal to one-third the
distance D (Fig. 30). Each wheel thus becomes a truncated
cone, but is actually
part . . C a complete cone,
as indicated by the
dotted lines (Fig. 30).
Fig. so.--Practical
form of bevel gears
The Sizes of Bevel
This principle of rolling cones permits considerable latitude in
selecting the size of
the wheels, and.in this
respect the problem
differs from that of connecting two parallel shafts by spur
wheels. In the latter case, the size of the whee!s is limited by
the distance between the shafts, but when the shafts are at an
angle, the wheels may
be of any size within
the space available.
Fig. 3x.-Bevel
gears ; relative
For example, in Fig.
diameters for
31 the shafts A and B
,.&f in-,- :
equal speed
are to be connected by
in ...~..-~j
r-; ” $ ‘1
the wheels so that they
rotate with equal velo’ -It
~ :<;‘! ; c I
cities. Wheels of size
,J /
C C may be used, or
L/ -hY ,&I;]
those of size D D, or
//g@- ~---1~~~~
any intermediate size,
without affecting the
relative speed of the
shafts. Both pairs of
wheels could be used simultaneously, because all bevel wheels on
either shaft, having pitch surfaces meeting on the line of the two
cones indicated by the dotted
lines, are really a portion of one
large conical wheel, the teeth and
pitch surfaces of which extend
from the point to the base of the
largest wheel. The principle is
not affected if the shafts rotate
at different relative velocities.
For example, in Fig. 32 shaft A
makes two revolutions to one
revolution of B. The wheels
may be of size C C or D D, or
any other size, provided their
pitch surfaces form part of the
cones indicated by the datted
Fig. St.-Bevel gears ; relative diameters for differing
lines. The relative numbers of
teeth must remain the same or
be in the same ratio. If Cr has 30 teethand Cz 60 teeth, DI
must have 30 teeth, and D2 60 teeth, or numbers of teeth having
a ratio of I to 2 ;
.~. -~‘-‘-~~
,,q <
: 1
thus DI could have
15 and D2 30 teeth,
and so on. Obvi-
Fig. 33.--Principle of
rolling cones ; shafts
placed at less than
90 deg.
ously, a large pair of
I i 1
wheels can have teeth
of greater size and
strength than a
smaller pair. When the shafts are not at a right angle to one
another the principle of rolling cones is still applicable, if the
axes of the shafts intersect. Fig. 33 is a diagram showing two
shafts intersecting at an angle of less than 90 deg., and Fig. 34
shows the shafts intersecting at an angle greater than 90 deg. ;
in each instance, the cones have equal diameters, so that the
two shafts will rotate at equal speeds. The shafts may be made
to rotate at different speeds by designing the cones so that their
diameters are of corresponding proportions to the speeds, as in
the case of shafts at a right angle. Fig. 35 shows the principle
of rolling cones applied to an internal gear connecting two shafts
Fig. 3+-Rolling cones :
shafts placed at angle
greater than 90 deg.
S, S, which are at an angle. In this arrangement the wheel P
must be smaller than the other, as it is a pinion working inside
an annular wheel.
Crown Wheel and Pinion
The term crown wheel and pinion is sometimes given to
the gear shown in Fig. 76. Correctly speaking, crown wheel
is another name for bevel wheel, and the gear shown in Fig. 36
should be formed on the principle of rolling cones. If the
wheel W is made with straight teeth and a cylindrical pinion P
is used to gear with it, the arrangement will not work correctly.
The wheel may be represented by a flat disc W (Fig. 37), and
its pitch surface would be a part of the disc. The pinion would
be represented by a cylinder P, rotating in contact with the disc, ;
such a cylinder would form the pi,tch sudace of the pinion.
Obviously ail parts of the circumference of P will move with
the same velocity. But all parts of the surface of the disc
will not move with
the same velocity.
.;-- /
That part represented
-==-z--: - - -,- - - -IQ-=*-~.,
I w
by the dotted circle C
Fig. 36.-Crown wheel
an,d pinion
will have a much
greater surface speed
than the part represented by the dotted
circle D. As both parts
are in contact with t,he circumference of P, the circle D will be
trying to drive P at a slower speed than it is being driven b’y P.
Every part of the surface between C and D will therefore be
trying to rotate P at a different rate of speed. As P can only
rotate at one speed at any instant, a slipping and grinding action
must take place between the surfaces. If the surface of the disc
is cut away so that only a circular ridge is left in contact with P,
such as would be represented by the circle C, this action would
be reduced to a minimum. Therefore the
gear shown in Fig.
Fig. 3y.-Emphasising
narrowness of -‘own
wheel teeth
36 can be made to
work if the teeth on
W have very small
breadth, as indicated by the sketch, so that they make very
narrow contact with P. Such a wheel can then only transmit
or receive a very small amount of power or its teeth will soon
wear away. When the diameter of the wheel is large compared
to that of the pinion, the error in the shape of the teeth as regards
taper is small, as indicated by the dotted sketch Fig. 36, but it
still exists.
The Shape of Bevel Wheel Teeth
The teeth .’ bevel wheels are formed on the same principle
as the teeth
Lt wheels, but the shapes of the faces and flanks
are not dev, 1 id upon the actual pitch circles. A section
through a tooth which would show its actual shape would not
be in a pl~.~ parallel to the base of the pitch cone, as indicated
Fig. 38.-Design of
bevel wheel teeth
by the teeth shown by Fig. 29, but would be in a plane perpendicular to the conical pitch surface. The teeth are placed so
that they are perpendicular to this pitch surface ; therefore,
it would not be correct to develop their shape by curves generated
on the circumferences of the pitch circles. They are developed
upon circles C, C of larger diameter, as indicated in Fig. 38,
the centres and radii being found by drawing lines B at a right
angle to the pitch surfaces, and meeting the centres of the shafts
at D, D.
Worm Gears
of gearing, which is frequently adopted when the driven
wheel is required to give a much lower number of revolutions
in a given interval of time than the driver, is shown in Fig. 39.
The arrangement is called worm gearing. The driving wheel
is a screw S, and is called the worm. The driven wheel W
is provided with teeth, and is
P-E--tailed a worm wheel. Imagine
the wheel W to be fixed so
that it cannot rotate. If the
worm, S, is rotated, and can
move also in a direction along
the line of its axis, it will act
as if W was a nut through
which it was being screwed,
because the thread of the
worm i s engaging with the
teeth of the wheel. The worm
will therefore move in a forward or backward direction,
depending upon the direction
Fig. 39.-Worm gearing
in which it is being rotated.
If, on the contrary, the shaft of the worm is held between thrust
bearings so that it cannot move in an endwise direction, and the
wheel W is free to rotate, it will do so if the worm is rotated.. AS
the worm is unable to screw itself past the wheel, the latter will
rotate owing to the sliding action of the worm thread upon the
wheel teeth. The rotation of the shaft S wifi be thus transmitted
to the shaft upon which W is fixed.
Worm and Spur Gearing Compared
Worm gearing, though equivalent to a pair of spur wheels
in its action, differs to some extent. Either wheel of a pair
of spur wheels may be ;nzde to drive the other, but though
the worm can always be made to be the driver, the wheel will
not necessarily drive the worm. As in the case of spur wheels,
the gear is designed in accordance with pitch lines and surfaces.
There is this difference, however-the pitch surfaces of spur
wheels roll together and, as already explained, one would drive
the other by contact friction if the load upon the driven wheel
were not excessive. The pitch surfaces of the worm gear (Fig.
39) are represented in Fig. 40 ; obviously, if S is rotated its
effort will be expended entirely in a iine parallel to the shaft
of W, and will not produce any rotating effect on W. The
surfaces will merely grind together without producing any
turning effort upon W. Similarly, if W is rotated, the effort
will be expended entirely in a line with the shaft of S, and no
rotating effect will be produced. Any rotary effort can therefore
only be produced by providing S and W with teeth which are
placed at an angle to the axes of the shafts and can slide against
one another. This is effected in practice by means of a screw
thread upon S and teeth upon W, which are machined at an angle
to correspond with the inclination of the screw so that the two
will engage in gear. The amount of rotation which will be given
to W for each complete revolution of S will therefore depend
upon the pitch of the screw thread. The pitch, divided into the
circumference of the pitch circle of the worm wheel, gives the
number of revolutions which the worm will make to produce
one complete revolution of the wheel.
Multiple-thread Worms
The relative number of revolutions made by S and W is
therefore quite independent of the diameter of the pitch surface
of S, and is also, in a sense, independent of the number of teeth
upon W. For example, suppose the screw S to have a single
thread of r-in. pitch and the pitch circle of the wheel W to have a
circumference of 20 in. The
wheel should then have
twenty teeth of I.-in. circular
pitch. As the pitch of a
screw is the distance through
which the thread advances
whilst making one complete
turn round its axis, S must
make twenty complete revolutions to drive W through
one complete revolution. If
the screw thread is made to
have a pitch of 2 in., the
ao.-8Worm gearing: prinwheel W would then be Fig.ciple
of pitch surfaces
made to have ten teeth of
z-in. circular pitch. The screw would then make ten revolutions
to drive W through one complete revolution. So far, the number
of teeth on W has been made proportional to the ratio of the gear,
that is, we have halved the rmmber of teeth whilst obtaining half
the number of revolutions of S required to obtain one complete
revolution of W. But we need not have reduced the number of
teeth on W. We could have allowed the screw to gear into alternate
teeth, half of the number of teeth thus being unused. The arrangement would effect the desired result as, for each revolution,
the screw would move the circumference of W twice as far as
with a screw pitch of I in. The screw would thus give one
revolution to W when its own shaft had made ten instead of
twenty turns. But it would not be a good arrangement to
permit half of the number of teeth to be idle. The whole
twenty teeth can be utilised by providing a second thread
upon S, interspaced with the first thread, to gear with the
idle teeth. Each thread will then take a share in driving the
wheel, and the pressure and wear will be distributed over double
the amount of contact surface. It would be necessary for
the teeth to be re-shaped to fit the altered curve of the screw
due to the increased pitch. The wheel W thus rettins its
previous number of teeth, namely, twenty, and yet makes one
revolution for every ten instead of every twenty revolutions of
the screw. Similarly, the screw may have three or more threads.
The number of teeth upon the wheel therefore does not determine the ratio of the gear which, as already stated, is determined
by the pitch of the worm.
When considering worm gearing, it is therefore advisable
to call the pitch of the worm the lead, as it will differ from the
circular pitch of the teeth on the w h e e l if the. worm has more
than one thread. The term “ lead ” also expresses more correctly
the function of the worm thread in its relation to the wheel,
as represented in Fig. 39, if the worm has a single threed its
pitch will be the distance D, and this will be equal to the circular
pitch of the teeth on W. But if the worm has a double thread,
its pitch will be the distance E ; this will be twice the length
of the circular pitch of the teeth on W.
Involute Worm Wheel Teeth
When preparing the worm blank and wheel blank for machining, allowance must be made for the distance beyond the pitch
line by which both the thread of the worm and teeth of the wheel
will project. If a section be taken through the centre line of the
worm, the teeth of the wheel and the thread of the worm can be
regarded as a pinion and rack at that line. The screw thread then
represents the rack, and it is designed in accordance with the
method used for determining the shapes of the teeth of a pinion
and rack. These teeth may be curved upon the cycloidal or
involute principle. If the former method be used, both the teeth
and screw thread will have curved sides ; if the involute method
be adopted, the sides of the screw thread will be straight
lines, as already explained in an earlier chapter. The
involute princple is usually adopted, because it is easier to
cut the worm thread if it has straight sides. Fig. 4 1 shows
a section of a worm gear through the centre line L M.
Cutting Worm Wheels with a Hob
If the teeth of the wheel accurately fit the spaces between
the thread of the worm throughout the entire breadth of the
wheel and follow the true curve of the screw, their shape will alter
in section according as the distance from the centre line L M
is increased. A section of any tooth taken on any line except L M
will show a different shape to that taken on line L M. In
addition to this, the circumference of the wheel must be hollowed
Fig. 4x.-Worm gearing : shapes of wheel
teeth and worm thread
to fit the worm at the points of the teeth and bottoms of the
spaces between the teeth, the curves being arcs of circles of two
different radii, as indicated by P and R (Fig. 41). On this account,
shaping and cutting the teeth 0’1‘ a worm wheel correctly is a
difficult matter. The method adopted in practice, especially
for wheels of small or comparati,vely small sizes, is to shape them
by means of a cutter which is a facsimile of the screw intended
to gear with the wheel. This cutter is called a hob, and consists of a steel worm of exactly the shape of the worm which is
t o gear with the wheel ; it is provided with cutting edges and
hardened. The teeth of the wheel are first cut nearly to size
by means of an ordinary circular cutter ; the hob is then geared
with the wheel, and the two are run together until the hob has
cut the teeth to the true shape. Obviously, if the hob is a correct
representation of the worm it will remove all irregular places
from the teeth and leave them a perfect fit to the actual worm.
It is sufficient, therefore, to plan the thread and teeth upon a
single section L M taken through the centrc line of the worm
and wheel.
Other Methods of Cutting Worm Wheel Teeth
A correctly shaped worm wheel will have the appearance of
Fig. qz.-Shapes of worm wheel
A (Fig. 4 2 ) . On account of the expense of making a hob, the
form of worm gears is often a compromise. The circumference
of the wheel is not hollowed at all, but straight, as in the case of
an ordinary flat spur wheel. The teeth are cut on the slant, as
indicated by B (Fig. 42), and at an angle to correspond with
the inclination of the worm thread. Another method is to make
the circumference of the wheel straight and to cut the teeth
with a circular milling cutter, as indicated by C (Fig. 42). If
the axis of the worm is not at a right angle to that of the wheel,
an ordinary flat spur wheel can be *used by slanting the
worm until its thread meshes with the teeth of the wheel, as
indicated by Fig. 43. Any one of these methods (Figs. 42 and
43) may be used successfully, and for the transmission of a
very small amount of power the worm can be an ordinary
Whitworth or similar screw thread. In practice, the edges of
the wheel are usually bevelled off, as indicated at C (Fig. 44),
except in the case of wheels like C (Fig. 42). This diminishes the
inaccurate portion of
the teeth and removes the weak
A blank
worm and wheel,
prepared for cutting
Fig. 43.-Worm wheel with
straight teeth
a gear such as Fig.
39, would have the appearance indicated in Fig. 45 if the teeth
were to be of perfect form and shaped by means of a hob. The
shaped by means of a hob. The dotted
lines show the pitch lines and the allowance of metal required to form the part
of the teeth projecting beyond the pitch
line. The part P is called the “ throat,”
and should be curved to a circle equal
to the diameter of the worm at the bottom
of its thread. As a small amount of clearance
should exist between the worm and whee! at
the tops of the teeth and bottom of the
spaces, the hob shou!d be made slightly
larger in dia-meter than the worm, so
that it will produce this clearance. Fig. 46
shows a blank wheel similar to Fig. 45, but
Fig. jq.-Pracwhich is to be cut with straight-,through
tical form of
worm wheel
teeth, as B (Fig. 42). The throat P is now
made straight, and not curved, as in Fig. 45.
Gear Ratios and The Worm Lead
The diameter of the worm has no influence upon the ratio
of the gearing. was already explained, this is determined by the
lead of the worm thread and the circumference or the diameter
of the wheel. If the distance between the shaft centres of
worm and wheel is fixed, you must select a lead for the worm
thread that will give the ratio desired. The diameter of the
worm can be made greater or less to accommodate the wheel
selected. For example, suppose the distance between the
centres will admit a wheel having a circumference of IO in. ;
the pitch circle for this will have a diameter of 3 i: in.
the worm thread is made with a lead of one turn in 4 in. (+-in
pitch), the ratio of the gear will be I to 20, because each revoiution of the worm will rotate the wheel by j- in. As there
are twenty half-inches in the circumference of the wheel,
twenty revolutions of the worm are required to rotate the wheel
through a distance of IO in. If a ratio of I to IO be required, it can
be obtained by making the worm thread with a lead of one turn
in I in. Ten revolutions of the worm will then produce one
revolution of the wheel. If a greater ratio than I to 2 0 be
required, it can be obtained by decreasing the lead of the worm
thread. For example, if the lead is one turn in $ in., the ratio
of the above gear will be 1 to 40, as there are forty $- in. in a
circumference of I0 in. ; therefore, forty revolutions of the worm
will be required to produce one revolution of the wheel. If
the worm thread is made with a lead of ,‘0 in., the ratio of the
gear will be I to I O O. The lead may be one turn in 2 in. ; the
ratio will then be I to 5. As the lead of the worm thread is
increased, the thickness and height of the wheel teeth must
be increased also. It may therefore be necessary to cut several
threads upon the worm to enable the teeth to be made of
reasonable proportions, as previously explained. In the
instance given above, where the worm thread has a lead of
2 in., it may be advisable to cut four or five th.reads. The
wheel would then have twenty or twenty-five teeth respectively.
Should the circumference of the wheel pitch circle, a s first
determined, prove not to contain the exact number of teeth,
it may be increased or decreased to a limited extent and the
Fig. 4g.-Determination of blanks for worm
and wheel
diameter of the worm altered to make good the discrepancy.
When the centres are not faxed, the wheel size is only limited
by constructional requirements. It is, however, advisable to
Fig. 46.-Blank for worm wheel with
straight-through teeth
have at least thirty teeth in the wheel. When a smaller number
must be used, the tips of the worm thread will not properly
clear the wheel teeth. This interference can be avoided by a
slight rounding off towards the tops of the thread, or by increasing
the diameter of the wheel, so that the teeth project almost
entirely outside the pitch circle. Messrs. Brown and Sharpe,
in their treatise on gearing, give the following rule for this
increase of diameter. The pitch diameters to be multiplied
by .g37; add to the product four times the addendum, that
is, the part which in the ardinary way would be outside the
pitch circle. The sum gives the diameter of the blank at the
throat P (Fig. 46). The whole diameter of the wheel is obtained
by making a drawing to this rule and measuring off the dimension Messrs. Brown and Sharpe say, however, that it is not
practicable to finish wheels, sized by this rule, with a hob when
they have twelve to eighteen teeth ; unless the wheel is driven
by a separate mechanism the hob must not be relied upon to
drive the wheel.
Proportions and Efficiency of Worm Gearing
Professor Unwin gives the following proportions for worm
gearing, I? being the circumferential pitch of the wheel teeth.
Thickness of tooth on pitch line, .48 p ; height outside pitch
line, .3 p ; depth below pitch line, .4 p ; length of worm, 3 to
6 p (usually 4 p) ; width of wheel face, 1.5 to 2.5 p. The worm
is frequently made of some different metal to that used for the
wheel. For example, a steel worm and gun-metal wheel, a
hardened steel worm and a phosphor-bronze wheel give good
results ; a wrought-iron or steel worm and a cast-iron wheel are
also used ; a cast-iron worm can be used with a cast-iron wheel.
When the gear is used for continual running and transmitting
power for driving purposes, the shape and materials are of
much greater importance than when the gear serves only for
adjustment purposes and occasional use. A hardened steel worm
and a phosphor-bronze wheel are a very good combination for
transmitting power, but efficient lubrication of the contact surfaces is the most important factor. The gear should run in an oil
bath if possible. Worm gearing was at one time regarded as a very
inefficient means of transmitting power, but during recent years
it has come into extensive use for this purpose and, if well
designed and run in oil, it is found to have a high efficiency.
The maximum so far recorded, we believe, is still 97.3 per cent.,
with D.B.S. glass, when tested at the National Physical Laboratory. The loss of power due to friction between the worm thread
and wheel teeth decreases with increase of the inclination of the
thread ; that is, a coarse lead will give a higher efficiency than
a fine lead pitch. Worms having multiple threads thus give
a higher efficiency than single-threaded worms, and a small
diameter worm gives a higher efficiency than one of corresponding
lead, but of larger diameter. In addition to the friction at the
worm thread, there is friction set up by the end-thrust on the
worm shaft. This is also of importance, and some form of thrust
Fig. q;r.-Right-hand
Fig. q&-Left-hand
bearing is required if high efficiency is to be maintained; a ball
thrust bearing is, here, advisable. As previously explained, the
worm must be prevented from moving end-wise, if it is to exert
pressure upon the wheel teeth and rotate the wheel ; therefore,
the wheel teeth will press against the worm thread with a force
proportional to the driving load. The teeth, therefore, thrust
the worm shaft against the bearing in which it runs. The
direction in which the wheel rotates for a given direction of
rotation of the worm depends upon whether the worm thread
is right- or left-handed. It is possible on this account to combine two worm gears so that the end thrusts of the worms
oppose each other, and no thrust is imposed on the bearings.
Fig. 47 is a diagram of a worm and wheel, in which the thread
is right-handed, and Fig. 48 is a similar design, in which the
thread is left-handed. The arrows indicate the direction of
rotation and thrust. The worm shaft rotates in the same
direction in either instance, but the wheels rotate in opposite
directions, and the direction of thrust exerted upon the thread
of the worm, and therefore upon its bearings, is also opposite, as
indicated by the straight arrows. In Fig. 4g the gears ase
combined, and the two worms are cut on one shaft. Each
worm exerts a thrust in the opposite direction to that produced
by the other; the bearings are thus relieved of end pressure,
which is taken by the part of th.e shaft connecting the two worms.
Fig:. 49.-Neutralising end thrust of
wcorm gearing
The driving force of the wheels can be combined and transmitted
to a single shaft by means of spur wheels.
A distinctive feature of worm gearing is that it is not always
reciprocal ; that is, the worm will al,ways drive the wheel, but
the wheel may not drive the worm. If the lead of the worm
is small, and therefore its angle small, the friction between the
surfaces in contact will be so great that the worm cannot be
rotated by the wheel. The critical lead of thread to enable the
wheel can drive the worm will depend upon the friction between
the surfaces in any particular instance. Generally speaking,
single-thread worms cannot be driven by the wheel, but if the
lead required is sufficiently great to necessitate a multiple thread,
the wheel may drive the worm ; the greater the lead, the more
likely is the gear to be reversible.
Special Forms of Gear Wheels
elical Gear Wheels
I T should be noted that with straight-toothed gear wheels,
that is, those having teeth parallel to the axis of the wheel, the
teeth when in motion slide upon one another. The pitch circles
roll together, but the teeth rub and slide upon one another as they
enter into and leave engagement, resulting in wear and tending
to produce noise when running at high speed. To minimise
this, “ stepped gearing ” was devised, and is applicabic both
to wheels and racks and pinions. Imagine several thin gear
wheels placed side by side on a shaft, but the teeth, instead of
being set in a straight line, are p!aced SO that they form a series
of steps (Fig. 50). If the wheels are fastened together they
will form one gear wheel with
stepped teeth. If such a wheel is
geared with another similar wheel,
the load transmitted by the driving
wheel to the driven wheel will be
applied progressively instead of
suddenly as when straight teeth
Fig. So,-Stepped teeth
are used. Obviously, with a given
breadth of wheel, the greater the number of steps the move
gradual and smoother will be the application of the driving
force. If the number of steps is made infinitely great the steps
vanish and become merged into a curved line, and the application
of the force will therefore be infinitely gradual. Actually, the
curved line becomes a helix or
screw thread, hence the term
“ helical ” applied to this
method of constructing gearwheel teeth (Fig. 51). In
action, the sliding of the teeth
is abolished and replaced by
an effect of pressure and rolling only. A practical advantage is that, with a wheel-andpinion drive, a pinion of
smaller diameter can be used
with good results ; there is
also diminution of shock at
high speed, and with an unFig. q.-Single helical gear wheel even load, there is less vibraljy couriesy o f IhUid t.irorwl & SOliS
(tIz!ddcrs/ield) Lid.
tion and noise, and at. the
same time the efficiency of
the transmission is also increased. The curves for any
pair of wheels in gear must be
of opposite hands ; that is, if
the driver, for example, has
its teeth on the curve of a
right-hand helix, those of the
driven wheel mttst be on the
curve of a left-hand helix and,
further, the angle or slope
must be the same in each
wheel, see Fig. 51. There is
a disadvantage in the use of
helical teeth, for the conversion of the sliding action,
which obtains with straight
teeth, into a rolling pressure
Fig. p-Double helical gear
Hy Ccwrfesy of Dauid Brown & Sons
action results in an endwise
(Huddersficld) Lid.
thrust upon the shaft ; in effect the wheels try to push themselves
apart sideways. On this account “ single ” helical teeth, as they
are termed, are not suitable if a heavy driving force is to be transmitted. If, however, a second pair of wheels, with teeth sloped
so that the endwise thrust is in the opposite direction to that
exerted through the first pair, is placed upon the shafts, the two
thrust effects will be in opposition and there will be no endwise
thrust upon the shafts or bearings. The endwise thrust of one
pair of wheels will neutral&e that given by the other pair, whatever may be the amount of variation in force applied through the
gearing. Instead of having two pairs of wheels, the two drivers
and the two driven wheels might respectively be fastened together,
so as to form one driver and one driven wheel. Usually, in
__---_- ---- _ -.-r’;------. ~~--__
1 &&..
/;it;;% ;
I ,&P
,,’ ‘.
--._ . .
Ii-;-‘:--__-’ _..___ -._ l-l -------~---~=-Y~
Fig. S3.-Derivation of curves of helical teeth
practice, this is effected by making each wheel as one, the teeth
being then termed “ double helical ” (Fig.. 5 2 ) . This type of
gearing is much used in mills and factories where heavy duty is
required. To visualise the formation of the curved teeth in a
double-helical gear wheel imagine that the wheel is a slice cut
from the middle of a cylinder upon which are traced two helices
of screw threads (Fig. 53), one being left hand and the other right
hand. The portions of the curves near to the point of intersection
represent the curves of one tooth. Every tooth in the wheel is
composed of portions of two similar curves. Double or single
helical teeth are applied to bevel wheels and pinions, and also
to racks and pinions. The teeth of a double-helical wheel are not
always .placed so that the two portions meet; these are o f t e n
“ staggered,” that is, one row is halfway in advance of the other
row. Helical teeth are made by the processes of cutting or
casting, special appliances being used.
Screw or Spiral Gear Wheels
These are another special kind of gear wheel having screw
or spiral teeth, but the term is particularly applied when
the axes or shafts are not in the same plane ; for example, if
the shafts cross each other at a right angle and at some distance
apart. The shafts may, however, be at any angle to one another.
When the shafts cross at~a right angle the teeth of each wheel
must be the same hand, that is, they are curved to a screw or
helical line which must be right-handed or left-handed in both
wheels. Whether the curve is to be right- or left-handed depends
upon the direction of motion required between the respective
shafts. Fig. 54 shows an example of spiral gear wheels. Strictly,
the term screw gearing or helical gearing includes all the forms
in which the teeth are made with a helical or screw curve, but in
practice these are divided into worm gearing, helical gearing with
singie or double teeth, and spirai gearing. The subject is wide
and involved and various arrangements of teeth have been devised
and adopted for use under special names, but the principles as
explained in Chapters I to IV apply to them all.
Chain Gearing
Transmission of power and motion through gearing from
one shaft to another can be effected by a combination of toothed
wheels and chains ; this is termed chain gearing. Imagine a
pair of toothed gear wheels fixed upon parallel shafts and each
engaging with a rack which is free to move in the direction of its
length. If one of these wheels is made a driver and power is
applied to it, the rack will be moved along and will rotate the
other wheel. Obviously, if the transmission of power and motion
is to be continuous and in one direction, the rack must be of a
length too great to be practicable. If, however, the rack is made
flexible and is carried round the wheels and joined at its ends, the
motion may be continued indefinitely. Conceivably, such a
flexible rack might be made ; in practice, it is resolved into links
and becomes a chain ; therefore a gear chain may be conveniently
regarded as an endless flexible rack. The principles which
apply to wheel and rack gearing, as explained in Chapters I to
III, form the basis upon which chain gearing is designed. The
distance between the pins of the chain must be of some definite
pitch, similar to that of the teeth or pins of a rack ; such chains
are therefore termed “ pitch chains,” as distinct from the
ordinary hoisting chain working on a smooth drum. Sometimes
notches or recesses into which the links of this chain engage are
provided upon the pulley to prevent slip, but these are entirely
Fig. 5q.---Spiral gear wheels
for the purpose of obtaining a sure grip and not for the accurate
transmission of motion.
Forms of Pitch Chains
There are various forms of pitch chains. The familiar cycle
chain has links consisting of flat plates connected by studs or
-‘eeves are mounted (Fig. 5s). The
pins upon wb?ch rollers o,,. DI
teeth of the chain wheels fit between the sides of the links and
press against the pins. In action, there is no relative movement
between the teeth of the wheel and the pins or rollers of the
chain whilst thej are in contact, except at the start and end of
engagemen,t. The teeth and pins do not slide upon one another
as do those of gear wheels and a rack and pinion. The teeth
of the wheel therefore need only be made of a shape which will
permit them to enter and leave the links without interfering
with the engagement of the pins. Owing to the links being
rigid, however, the curye is an arc of a circle struck from the
centre of the pin of the link preceding the one just leaving
engagement, or, at the engaging side, from the centre of the
pin of the link in advance of that just coming into engagement.
The general shape of the teeth, therefore, is composed of these two
arcs joined at the root by another portion of a circle which corresponds to the periphery of
one of the pins or rollers.
The points of the teeth may
be sharp or blunt according
to the general design of the
wheel and chain ; the
essential consideration is
that they clear the pins as
the latter come into and
leave engagement. In practice, the teeth of chain
wheels are shaped accordFig. 55.-Chain and sprocket drive
ing to the experience of the
maker to suit practical conditions of service. Whilst the pins and teeth are in contact, there is
pressure bttween them tending to crush in the surface ; there may
also be grit due to conditions of working, and there is slight movement between them at moments of engagement and disengagement. The teeth, therefore, tend to wear away and become
undercut at the roots. The pitch of the chain tends to lengthen,
due partly to wear of the journals of the pins or bushes, and
possibly by actual stretch of the side plates. Makers of chain
wheels therefore endeavour to shape the teeth so that they will
accommodate to some extent the effect of wear and stretch in
Frequently, straight lines are substituted for
the chain.
arcs of circles at the faces of the teeth, but this is a matter
which depends upon the
opinion of the maker.
Another type of pitch
chain. is used for transmission of power instead
of belting ; this will work
at higher speeds than
would be sound, practice
with open link chains.
Several constructions have
Fig. j6.-Chain drive with
been invented and painverted teeth
tented, the terms silent
chain, high-speed pitch chain or certain proprietary names,
These three chains
being applied to distinguish them.
have, as shown in Fig. 56, inwardly projecting teeth, which
engage with the teeth of the wheel, both being shaped to
the ideas of the particular maker, so as to comply with the
condition that the teeth engage and disengage with freedom.
The chain is usually of considerable width as compared with an
open-link chain, and the links consist of a number of thin plates
connected by pin joints. The links are designed so that the
effects of wear are automatically neutralised. Chain gear wheels
are termed sprocket wheels.
lEpicyclic Gearing
When there are two or more wheels in gear to transmit motion,
the arrangement is termed a “ train.” If the wheels are all in
line with one another it is termed a simple train ; if the motion
is transmitted through wheels placed side by side upon one or
more of the shafts the gearing is termed a compound train.
Arrangements of gear wheels in trains form a section of “ Kinematics ” (the science of movement) ; this is too wide a subject
to be dealt with in this book except in a merely introductory
manner. Included in this subject is a section relating to Epicyclic
Trains. A brief mention and explanation is given here in order
that the reader may be familiar with the term “ epicyclic ” as
applied to gearing. An ordinary train of gear wheels in certain
arrangements may easily be mistaken for an epicyclic gear unless
the observer understands the basic underlying principle.
In an ordinary train of gear wheels the motion is derived
by one or other of the wheels being rotated about its axis. In
an epicyclic train the motion is derived by rotating a bar or arm,
carrying at least one 0f the wheels, around the axis of another
wheel of the train. The result of this is that the wheels receive
a compound rotation ; in addition to the amount due to the value
of the train, one revolution is given due to the rotation of the bar.
The epicyclic principle may be applied to external and to internal
gears, and there may be external rotary motion applied to one
or more of the wheels in addition to that derived from the rotation
of the arm.
The simplest instance of an epicyclic train is that of
tW0 wheels, one of which is carried around the other,
mounted upon an arm which can rotate concentrically with
the firsL wheel. This illustrates the basic principle of all
epicyclic trains (Fig. 57). Power is applied to the arm A, and
the wheel B is carried around C. Assume that the wheels
are of equal size. If B is kc2 to revolve upon its axis and
C is fixed so that it cannot revolve, B will make two revolutions upon its own axis for each complete revolution of the
arm around C. One of these revolutions will be due to the
gear value or ratio of the train, and the other will be due to
the movement of the arm. If the wheel B is one-half the
diameter of C, it will make three revolutions about its own axis
whilst the arm makes one revolution around C. Two of these
revolutions will be due to the gear value of the train and the other
will be due to the revolution made by the arm. The operation
may be mversed, C being ailowed to rotate about its own axis,
and B constrained by some means so that it cannot rotate
upon the stud or journal upon which it is mounted. That
is to say, a mark upon it will remain pointing in the same
direction throughout the revolution of the arm. If the wheels
are of equal diameters C will make two revolutions during one
revolution of the arm. If the wheel B is one-half the diameter
of C, the latter will make one and a half revolutions about its
own axis whilst the arm makes one revolution. The half
revolution will be due to the gear value o:f the train, t h e one
revolution will be due to the revolution of the arm.
Fig. ST.-Epicyclic train of two wheels
Any number of wheels may be arranged along the arm, but
unless some independent movement is given to one or other
of them, those which are between the first and last wheels of
the train act merely as idle or carrier wheels. For example,
if a wheel D (Fig. $3), free to rotate upon its stud or journal, is
interposed between B and C, the relative motions of B and C
due to the epicyclic principle will not be affected. The only
effect of intermediate wheels is upon the relative direction of
rotation between B and C. But if some independent movement
is given to B or C, or to one or more of the intermediate wheels,
the rotations of the last wheel of the train, that is B or C, will be
altered accordingly. In the general applications of epicyclic
trains of gearing, the motion is derived only from the movement
of the arm, but in all cases the fundamental principle is that
there will be one rotation of the last whee!, either plus or minus,
due to the rotation of the arm in addition to that given by the
gear ratio of the train.
With internal gears a disc is frequently used instead of an
arm ; nevertheless it is still the arm, but in tbe guise of a disc.
For example, A (Fig. 59) is a disc carrying wheels D in gear
with wheels B and C, the former being an internally toothed
wheel. Either B or C may be a fixed wheel ; DI, D2, D3, D4, so
far as the relative motion between B and C is concerned, are
merely idle or carrier wheels. The motion therefore will be one
revolution, due to the rotation of the disc, plus the motion due
to the gear value of the train. The four intermediate wheels, D,
are used for mechanical reasons in order to distribute the wear,
but this has no effect upon the epicyclic value of the train which
Fig. j8.-Epicyclic train of three wheels
really consists of any one of the intermediate wheels together
with the centre and outside wheels, and the arm in the form of a
disc. The epicyclic principle is applicable to trains of bevel
gearing ; for example, three bevel gears, one of which is carried
upon an arm attached to a central shaft and gearing with two
others placed side by side. The wheel on the arm can rotate
upon its own axis ; of the others, one is fixed and one is free to
rotate upon the shaft. When the shaft is rotated the wheel
which is free upon it receives a compound motion ; it will receive
one revolution due to the motion of the arm, and also rotation due
to the gear value of the train. As the wheels are of equal size,
it will receive two revolutions for each revolution of the shaft.
The arrangement of two wheels in an epicyclic train is frequently
termed a “ sun and planet ” gear.
Fig. 59.~-Internal epicyclic gearing
The “ Marlborough ” Gear
There is an arrangement of
gearing which should not be mistaken for an epicyclic train ; it is used
for gearing two parallel shafts when
their axes are very close together;
Fig. 60 shows an example. Shafts
A and B are each provided with a
Fig. 6o.-Marlborough
wheel gearing
gear wheel ; a pinion or wheel C upon
another shaft gears into both of these. The rotation of A is thus
transmitted to B through the medium of C, which is merely an
idle or carrier wheel. If, however, C were mounted upon an
arm and the motion derived from the rotation of the arm around
the other wheels, the ‘arrangement would then be an epicyclic
train, and the shafts A and B would have to be in the same axial
line. G,ear wheels are not always circular; there are elliptic,
square, heart-shaped, lobed, and other forms, the object of their
use being to obtain varying velocities between the driving and
driven shafts.
Knuckle Gearing
There is a type of gear wheel used where great strength is necessary for the teeth in order that they may stand heavy and rough
working. The teeth are of simple form, being merely shaped to
arcs of circles, and are very short
(Fig. 61). This type of gearing
is only suitable for slow motion,
and when noise and jar and an
uneven velocity ratio between
the shafts can be tolerated on
account of the gain in strength.
It is a makeshift, wrong in theory,
Fig. 61.-“ Knuckle ” teeth
but usable for practical reasons
and because there is very little oblique thrust. Strictly it is
equivalent, so far as the form and action of the teeth are concerned,
to lantern or pin-wheel gearing. But, as explained in Chapter III,
with gearing of this kind to be correct in principle, the pins
should be on one wheel only, and that should be the driven
wheel. With knuckle gearing there are really two pin wheels in
gear with each other; the principle is incorrect. In practice,
the teeth are sometimes modified in shape ; but the more this
departs from the simple circular form, the less will be the advantage of strength and simplicity for which knuckle teeth are adopted.
Knuckle gearing is an example of construction, wrong in principle,
yet of utility, applicable to and successful for purposes within its
efinitims am! Calc~ulsdons for Gutting Invohte Gem-s
describing gem cutting, it will be as well to give a
resume of definition;, adding to these a few simple practical
Diameter of a gear invariably refers to the pitch circle; the
Outside Diameter is the size of the blank required.
Diametral Pitch is the number of teeth in each inch of pitch
For example, a gear with 48 teeth has a pitch diameter of
6 in.
The diametral pitch is 4* = 8.
The gear is thus 8 pitch.
If the outside diameter is known, and also the number of
teeth ; to get the diametral pitch, add 2 to the number of teeth
and divide by the outside diameter. Thus : if the outside diameter-is 6$ in. and there are 48 teeth ; 48 + a = SQ, divide by
6&, result, 8 pitch.
Circular Pitch is the distance along the circumference of the
pitch circle between the centres of adjacent teeth.
If the diametral pitch is known, then circular pitch is found
dividing this into 3.1416. Thus 8 pitch measures 3:rz?6 =
in. between adjacent teeth.
Conversely, diametral pitch is found by dividing 3.r4r6 by
circular pitch.
M o d u l e is the pitch diameter in millimetres divided by the
number of teeth, and the pitch diameter in millimetres is equal
to the module x number of teeth in the gear. The module can
be converted into inch diametral pitch by dividing it into 25.4.
Thus Module I = 25.4 diametral pitch, 2 = 12.7, 3 = 8.46,
4 =: 6.35 and so on.
Pitch Diameter can be found by dividing the number of teeth
by the diametral pitch.
Thus : number of teeth 48, and diametral pitch 8 give ‘8” = 6 in.
pitch di.ameter.
Outside Diameter, equal to the diameter of the wheel blank, is
found from the number of teeth and diametral pitch by adding
2 to the number of teeth and dividing by the pitch.
Thus : number of teeth 48 i- 2 = 50 ; diametral pitch ;= 8,
outside diameter = 5’ = 69 in
Number of Teeth, given pitch cliameter and. diametral
pitch, is
found by multiplying these two numbers together.
If the diametral pitch and the outside diameter are known,
then. multiply the two figures together and subtract 2, and the
result is the number of teeth wanted.
Distances between Centres of Gears, knowing their diametral
pitch and numbers of teeth, can be found by adding together the
numbers of teeth, halving the sum and dividmg this by the diametral pitch. Thus : two gears of 8 pitch have 48 and 16
teeth ; adding these 48 -+
=z 64 : halve this,64 = 32, divide
by pitch, 7 == 4 in. between centres.
Dimensions of Teeth. Thickness at the pitch line is half the cir-
cular pitch, or 1.57 divided by the diametral pitch. Thus : for
diametral pitch 8, Ii7 =- 0.196 in.
Tlle whoie depth of the tooth is 0.686 x circuiar pitch, or
2.157 + diametral pitch.
Thus : for diametral pitch 8, 2*27 = 0.269 in. or circular
pitch 0.3927 x 0.686 = 0.269 in.
The convenience of the system of reckoning in terms of
diametral pitch can be seen at once, for diametral pitches are
standardised in whole numbers, and in fractions in the larger sizes;
circular pitches can only be reckoned to several places of
decimals. The reason
for this is that the
factor which governs
the relation between
the diameter and the
circumference of a
circle is not a whole
number, nor is it an
exact one, for though
it has been taken to a
hundred decimal
places and more, yet
Fig. 62.-Measur;z;;ts of gear wheel
it has never been precisely established.
Sufficient exactness, however, is obtained by using standardised diametral pitch. The makers of milling cutters provide a
range of cutters of from I to 48 pitch, and as it is use&l to know
the vzriety of circular pitches these provide, these pitches as well
as the tooth depth are shown in the table on the following page.
All gears of any one given diametral pitch will mate together
satisfactorily so long as they have been cut correctly. When
cutting gears by the milling process, if a variety of sizes have to
be produced, several cutters will be needed for each diametral
pitch. The reason for this is shown in the diagram, Fig. 63,
which illustrates a portion of a IO-tooth pinion and part of a
36-tooth wheel. Very little observation is required to note the
difference in profile of the tooth spaces which are formed by
the miliing cutter. For this reason, cutter makers standardise
eight shapes of cutter for each diametral pitch and, in addition,
a series of half sizes where greater accuracy is needed. The
wheels that the standard cutters are designed to cut are also shown
Depth of
o. 196
No. 8 cuts gears of 12 to 13 teeth
No. 7 cuts gears of 14 to 16 teeth
No. 6 cuts gears of 17 to 20 teeth
No. 5 cuts gears of 21 to 25 teeth
No. 4 cuts gears of 26 to 34 teeth
No. 3 cuts gears of 35 to 54 teeth
No. 2 cuts gears of 55 to 134 teeth
No. I cuts gears of 134 to a rack
Fig. 6j.-Varied shapes of cutters for differing diametrical
Simple Gearing Pro’blems
Velocity Ratio of Wheels in Gear
R.P.M. of driver of 27 teeth is 1,800, at what speed will it
turn the driven wheel that has 108 teeth ?
Theequationis: D xR = d x r
which becomes 27 x 1,800 = I 0 8 x r
27 x 1,800
sor= --. __-~ = 450 r.p.m.
In a compound train the driver wheels have 27, 36 and 48
teeth and the r.p.m. of the first is 1,800. The driven wheels
have 108,120 and 180 teeth ; what is the speed of the final shaft ?
As before : D x D x D x R L=: d x d x d x r
which comes to 27 x 36 x 48 x 1,800 =: 108 x 120 x 180 x r
27 x 36 x 48 x 1 , 8 0 0
so r = ~~.~.~~~ ~~_~_
= 36 r.p.m.
108 x 120 x 180
The layshaft in a gear box has to run at 1,150 r.p.m. (v) and
the driving shaft at 2,000 r.p.m. (V). If they are at I O inch
centres (D), what must be the diameters of the connecting gears ?
Let R and r be the radii of the gears : then
x D=
---+ IO =
3.65 in.
R = D - r = 6.35 in.
Diameters required are 7.3 and 12.7 in.
To achieve the object set out in the above problem, design a
suitable pair of gear wheels.
Trying first 8 diametral pitch :
Number of teeth in both gears = 2 x D x P = 20 x 8 =160.
(N + n)
160 x 1,150
Ntmrber of teeth in small gear (n) = T&y = ~--~~~
= 58
So number of teeth in large gear= 160 - 58 = 102.
Speed of layshaft is ,b”,
X 2,ooo
Trying IO pitch next :
200 x v
200 x 1,150
n = ~v-$ b
N = 200- 73 = 127.
= 73 teeth
Speed of layshaft is ::7 x 2,000 =
This is as near as can be expected, so the terms of the problem
are solved by the use of gears of IO pitch, having 73 and 127
teeth respectively. As will be shown later, these are rather
awkward numbers, but they serve as examples of how the calculation is made.
As sprocket wheels for chain driving are frequently employed
in machine work, a problem dealing with these may be added.
To calculate the diameter of a roller chain sprocket, Chordal
Pitch is used. Chordal pitch is measured in a straight line,
along the pitch circle, between the centres of adjacent teeth,
that is, the pitch of the chain.
If N = number of teeth in sprocket
P = pitch of chain
d = chain roller diameter
As shown in Fig. 63a,
Angle A = $$ and the pitch diameter = ~-Psin A
Outside Diameter will be Pitch Diameter + D.
Diameter at bottom will be Pitch Diameter - D.
Find the pitch diameter of a sprocket having ~8 teeth, chain
4 inch pitch, roller diameter 0.30 in.
A = -!% = 10’
2 x 18
0.500 = _ 0.500
and P.D. = ..--~~
sm 10’
= 2.88 in.
Outside diameter = 2.88 -t 0.30
= 3.18 in.
Bottom diameter == 2.88 - 0.30
L- 2.58 in.
Facts in Brief
The weakest point in a train of gearing is the smallest pinion
in that train.
The wider the rim of the wheel, the stronger are the teeth. A
gear with teeth of fine pitch and a broad rim is stronger than one
with teeth of coarse pitch and a narrow rim.
The velocity ratio between the first and last wheels in a simple
train is not affected by the insertion of intermediate wheels.
These wheels do however affect the direction of rotation of the
first and last wheels, but not the relative number of revolutions
made by the driving and driven shafts. An odd number of
intermediate wheels preserves the direction of rotation ; an
even number reverses it.
In a compound train of wheels, the first and last wheels
revolve in the same direction. To reverse the relative motion
of these wheels a single intermediate wheel may be inserted
anywhere in the train without affecting the velocity ratio of the
first and last wheels. This is useful when cutting left-hand
threads in a screw-cutting lathe.
Gear Cutting in Milling Machines
FROM the foregoing chapters in this book enough will have been
learned to show that a properly designed gear is a matter of
precision, and that exact methods of manufacture are essential.
There are, speaking generally, two main methods of making
gear wheels in use-Today. The enormous demands of the motor
industry for gears that will run silently, and will be strong
enough to withstand the stresses in the motor-car gearbox, have
produced a large variety of gear cutting machines. Such gears
as these are invariably cut from solid blanks, and this cutting
may be carried out in one of two different ways : direct cutting
by a milling cutter, or generating by hob or planer.
Die-casting is a process also used for producing small gears ;
here liquid metal is forced into a mould, under considerable
pressure, resulting in a clean profile that needs no machining.
Gears of this sot+. ..,e
e- relatively expensive, and only a very large
production figure would justify the expense of the dies. Another
method of producing small gear wheels or pinions for clocks,
watches, and scientific instruments, where a single wheel or a
very small number is re+iired, is by the use of a fly-cutter. This
is a single-pointed cutter, accurately shaped to the space between
the teeth, and driven at a high speed by suitably arranged belting.
Wheels can be cut in an ordinary lathe by this process, in conjunction with a dividing head.
Gear Cutting by Milling
The time-honoured method of gear cutting, still much used,
is by employing a milling machine, using specially formed cutters,
and a dividing head to move the blank round step by step at the
correct intervals. A sketch (Fig. 64) shows the general arrangement. The blank is mounted upon an arbor, adequately supported
against the pressure of the cutter and carried at one end by the
Fig. 6+-Gear cutting in a milling machine
‘headttock, with its dividing mechanism, and at the other end by
a tailstock centre. An adjustable strut can be brought into contact with the rim of the wheel to counteract the cutting pressure.
The feed is provided in the usual way by the table screw,
actuated by suitable change gears. In some quarters, where
cutting down of production time is important, the feed is
increased to the maximum, short of heating the cutter unduly,
but instead of proceeding immediately to the next tooth space,
the blank is rotated by several spaces. This gives the part of
the blank heated by the machining a chance to cool down.
Fixed directly to the spindle of the dividing head is a disc
having 24 notches around its rim, and by the use of this any
number of divisions can be made that are factors of 24, that is :
2,3,4,6, 8, 12. This is the plain dividing head.
Universal Head
The universal head is fitted with a number of steel discs, in
each of which a number of circles of accurately spaced holes
has been. drilled, usually six. A typical set will includes plates
having 15,16,17,18,1g and 20 holes ; 21,23,27,2g, 31 and 33
holes ; and 37, 39, 41, 43, 47 and 4g holes.
The spindle of the universal head is driven by a worm and
worm wheel, with a ratio of 40 to I, through a crank and handle.
Thusj one turn of the crank advances the spindle 1/40 of a turn,
so by this means a gear of any number of teeth that will divide
exactly into 4 0 can be accurately spaced out : that is to say,
5, 8, IO, 20 and 40 teeth, by making 8,5,4,2 or I t u r n s of the
crank respectively. The idea of the index plate can now be
seen, for it enables the operator to make exact part-turns of the
crank. A peg that locks the crank handle in any position can
be pushed into any of the holes in the plate (see Fig. 65).
When using this plate the rule is : divide 40 (the worm and
wheel ratio) by the number of spaces to be set out in order to
obtain the number of turns or parts of a turn to be given to the
crank handle. Consider a gear of 28 teeth. Dividing 40 by 2 8
we get I and r2/28 of a turn, but there is no index circle with 28
holes, so some other similar fraction must be worked out, thus :I! = 3 = T_’ , and as we have a 4g hole circle, each adjustment is
7 49
made with one whole turn and 21 holes in addition.
Another example may show the idea more fully. The number
of divisions wanted is, say, 60. The division shows that each
new cut must be made after 40 60 of a turn has been given to the
crank. But there is no disc with a 60 hole circle, so we must
find one with a circle divisible by 3, such as 39. The calculation
is as follows :
‘!o = 2 = 26, so each movement is made
60 3
by advancing the crank handle 26 holes.
To facilitate spacing out the teeth, a pair of arms is pivoted
upon the crank spindle, and these can be clamped at any angle
relative to each other, so that they span an exact number of
holes. Each forward step is made within the space included
between the two arms.
With 18 circles of holes a very wide range of divisions can be
made, but it is by no means complete. Every division up to 50
can be made, but thenceforward to IOO and beyond, there are
l ..
:.’ l
. .*
. ..*
l .;. 0 .l ..*
l . . .
.a* :.
Fig. 6j.-Diagram of universal head and
division plate
many gaps that can only be covered by the system known as
differential indexing. Here, the crank handle is geared to the
worm spindle, but the method is a little complicated, and its
description must be left to a more advanced book than this.
The example given at the end of the last chapter, in which two
gears were designed, resulted in wheels having 73 and 127 teeth.
B o t h of these would need differential indexing, as practically
all prime n.umbers-numbers that have no factors-are unobtainable by direct indexing.
In milling gear teeth it is possible, of course, to clamp several
blanks together, thus producing more than one wheel at a time.
This depends upon the capacity of the machine. But if the
need is for a considerable production of a variety of gears,
then the use of a gear-cutting machine is to be recommended.
Bevel Gear Cutting
The more difficult process of bevel gear cutting on a
plain milling machine is one that must ble mentioned, even
though that machine is not the ideal one fc+r this purpose. A
little consideration of the form of these gears will show why
this is so. The blank for a bevel gear is a part of a cone, consequently both the teeth and the spaces are tapering in shape.
As a result of this, no cutter of rotary type can be devised to mill
out this space in a single operation, and even when the necessary
cuts have been made, the teeth will have to be finished by hand to
get an exact profile.
The tooth space is narrower at one end than at the other. The
cutter cannot be wider than the narrower end, and if it is made
to give the correct profile there, then the profile will not be correct
at the larger end. Two cuts have to be taken, with the gear blank
set off from the true centre, first to one side and then to the
other ; the blank also has to be rotated in order to give the correct
width at the larger end (see Fig. 66 and Fig. 67). The cutter
will give the proper profile at this end, and so, if bevel gears have
to be produced by milling, some hand-finishing will be essential.
The amount of the set-over can be calculated with the aid of a
table of constants. These constants are based upon the width
of the tooth faces compared with the whole length of the cone
side of which the surface is a part.
The selection of cutters is not a straightforward matter, as a
different cutter will be used for each wheel, that is, of course, if
they are of different sizes. Cutter makers publish tables for this
purpose, but these are too extensive to insert here. Further
information on the generation of bevel gears is given in the next
Laying Out Blanks
It is as well that the reader
should know how to lay out
the shape of blanks for a bevel
Fig. 66.-Milling the
teeth of a bevel wheel
gear and pinion ; this is by
no means as simple a matter
as for the spur gear blank.
Assume that a pinion of
32 teeth is to drive a gear
of 72 teeth, 8 diametral
pitch. The process is shown
in Fig. 68.
First draw two centre lines
intersecting at right angles, PP for the pinion, GG for the
gear. On either side of these draw lines QQ and RR for the
pinion, spaced apart as many
eighths of an inch as there
Fig. 67.-The two cuts in
milling bevel-wheel teeth
are teeth in the pinion-32 in
this case-and HH and JJ
for the gear, 72 eighths apart
and parallel to the centre
lines. Join up to the centre C
the diagonals CD, these
represent the pitch lines.
Lines through these intersections, at right angles to the
diagonals, represent the backs, and i in. on each side of these
indicate the top and bottom of the teeth. The width of the tooth
faces is a matter that depends upon the load they are to carry
and the speed at which the wheels are to run. In the same
way the thickness of the webs and the, length of the shaft
bosses also depend upon the work the wheels have to do.
Fig. 68.-Laying out blanks for bevel wheel and pinion
Gear wheel bianks in cast iron OK gun-metal may be cut dry.
Steel blanks should be cut with a plentiful supply of a suitable
cutting lubricant.
It is important that the bore of a gear blank should be perfectly
concentric with the circumference of the blank, to ensure even
meshing of the teeth and smoothness of working. The bore
should also be of the correct size to fit accurately on to the
mandrel or other fixture carrying the blank in the machine.
Where silent working is especially desired, pinions with
machine-cut teeth of rawhide are sometimes used. They work
smoothly and quietly, require little or no lubrication, and are
approxim;*ely equal in strength to cast iron gears. These pinions
can be used to mesh with gears cut in any metal. The layers of
rawhide are clamped together by metal discs on either side of the
pinion to p:event spreading of the hide through the lateral strain
incurred in working.
Modern practice, however, tends to favour wheels made from
fabric impregnated with synthetic resins. This material provides
the necessary silence in operation; in addition it has good
mechanical strength and does not readily deform.
Cutting Spur Gears in the Lathe
IN Chapter
VIII it was pointed out that gear wheels can be cut in
the lathe. This may be done in two ways. In Method I the
gear blank is supported in a fixture attached to the lathe cross
slide and the cutter is rotated by the lathe mandrel. In Method 2
the exact opposite procedure is adopted, the work being held in
the chuck or mounted between centres on an arbor and the
cutter is revolved in an attachment bolted to the top slide. If
gears of heavy section are to be cut, the first method is preferable
as there is usually greater inherent rigidity with this arrangement, to say nothing of the increased driving power available.
The second method is, however, very satisfactory for cutting the
lighter type of gear and it has the merit that the operator’s view
of the work is generally less restricted.
Dividing Devices
In both the above methods some form of dividing attachment
is required and this may be either simple or elaborate as the work
demands. The simplest form of dividing device is a master gear
attached to the spindle carrying the gear blank. A detent to
engage the tooth spaces of the master gear is provided and the
gear is rotated progressively tooth by tooth until all teeth in
the blank have been cut. This will give an exact replica of the
master gear or will enable gears to be cut that are multiples of
the master gear. Fig. 6g shows a simple arrangement of this
type. The principle of using a master gear can be applied to
either of the aforementioned methods of gear cutting ; in the
Fig. 69.~Simple in$;3qehead on cross-slide
first method the device can be used as shown, and in the second
the master gear can be mounted on a suitable adapter, forming
an extension to the lathe mandrel. The gear blank is then held
in the chuck or mounted on an arbor. In this case, the detent
to engage the teeth of the master wheel is carried in an extension
of the change wheel quadrant, as shown in Fig. 70.
A more elaborate device for dividing is that shown in Fig. 71.
This is a combination fixture
and is intended to be bolted
to a vertical milling slide
, . attached to the lathe cross,
slide. It does, in fact, form a
complete milling attachment,
as it is provided with a worm
Fig. To.-Indexing by change
gear on lathe mandrel
Fig 7x.-Two views of the Myford dividing attachment to
fit cross-slide of lathe
Fig. 7z.-Worm-geared d.i~l.i~e,p, a p p l i a n c e o n l a t h e
dividing head, as described in Chapter VIII, and also a means
of holding the gear blanks.
A worm Dividing Head may also be applied to the lathe
mandrel and this is shown set up in Fig. 72 with its worm engaging
a standard change wheel mounted on the mandrel extension.
With such an arrangement it is possible to put any of the lathe
change wheels on to the mandrel extension and to engage them
73.-Details of worm dividing appliance
with the worm of the dividing head. This greatly facilitates
dividing and avoids the use of a multiplicity of the division
plates described in the previous chapter.
As will be seen in Figs. 73,74, and 75, such a dividing head
is of relatively simple construction and is well within the capabilities of the amateur machinist. This attachment is carried on
a mounting that can be bolted to the change-wheel bracket,
thus making its adjustment for mesh with the change wheel a
simple and rapid process.
With regard to the mandrel extension or change wheel
adapter, the form shown in Fig. 76 has proved very reliable and
is suitable for any lathe having a mandrel bore of -3 in. and
upwards. Below this size, it may not be possible to accommodate
this form of adapter, in which case it will be necessary to thread
the bore accurately and to screw in a stub extension to take the
change wheel. It must be emphasised that accuracy is essential
in this matter as eccentric running of the wheels cannot be
It will be appreciated that, with a little ingenuity it is possible
to drive the leadscrew of the lathe independently and so to
Fig. 74.-The dividing head mounted on the ML7 lathe
Fig. 75.-Division-plate and sectors
provide an automatic feed when cutting gears by the second
method ; that is when the cutter is rotated in an attachment
bolted to the saddle and independently driven. When cutting
by the first method, this form of drive can be used provided
the cross-slide has power feed.
In order to design such an arrangement, it is necessary to work
out the correct rate of feed for the cutter, having regard to its
diameter and the material being machined. It is assumed that
the peripheral speed of the cutter will have been correctly
adjusted to suit the material, in accordance with the methods
which will be described later.
As will be seen in Fig. 72, in addition to the change wheel
bracket, there is a further quadrant mounted on the end of the
lathe carrying a reduction gear box and a train of gears to drive
the leadscrew of the lathe. The pulley on the gearbox is driven
from the lathe countershaft from which the belt to the lathe
mandrel has been unshipped as it is not required.
Calculating Correct Feeds
The correct train of gears to drive the leadscrew is arrived at
in the following way : the amount of metal removed by a
cutter of I& in. diameter (this being a convenient size fo: gesar
cutting in the small lathe) in one revolution varies from about
0.001 in. for tool steel to some 0.030 in. for aluminium. Assume
the r.p.m. of the cutter at the lowest driven speed is 60 r.p.m.
and the highest spee:d 2 0 0 r.p.m. Therefore, when cutting at
0.001 in. per revolution, the cutter will remove 0.060 in. per
minute at the low speed and 0.240 in. per minute at the high ;
that is, the amount removed per minute is the amount removed
per revolution multiplied by the r.p.m. of the cutter.
Now suppose the feed screw has a Q in. lead, then one turn of
Fig. T&-Expanding plug mandrel to carry
change gears
the feed screw per minute will give & in. or 0.125 in. of feed per
minute. So to feed 0.060 in. per minute, the lead screw must
revolve - turns per minute or .48 of a turn, say half a turn.
Similarly if it is required to feed at 0.240 in. per minute the
lead screw must revolve at ‘$ turns per minute that is I.925
turns, say 2 turns.
The worm reduction gear, which is driven from the lathe
countershaft, has a reduction of eighty to one and the lead screw
is driven from this through a train of gears for final adjustment
of the rate of feed. Let us suppose that the lathe countershaft
rotates normally at 4oo r.p.m., then the worm wheel shaft
of the reduction gear will make 5 turns per minute. To make
the leadscrew revolve at two turns a minute, a train of gears,
after the worm reduction gear, having a ratio of 5:~ will be correct
and two gears of IOO T and 40 T can be used. Similarly, if
half-a-turn per minute is required, the gear train must be in
the ratio of 5 : 0.5. Unfortunately, no standard set of lathe
change wheels will satisfy this, so the speed of the worm wheel
shaft must be reduced to 200 r.p.m. by changing the ratio of the
cone pulley on the countershaft, from which the reduction box
is driven, to 2 : I from the original I : I ratio. This will leave a
2 . 5 : 0.5 reduction between the feed screw and the worm
reduction box and this can be met by using two wheels of IOO T
and 20 T.
To reduce the calculations to a formula, the number of turns
the leadscrew must make is equal to the feed per minute required,
divided by the pitch of the leadscrew, i.e.
Feed per minute required
Number of turns equals -Pitch of lead Screw
both feed and pitch being expressed in inches, and the ratio of
reduc:tion will be :
Turns per minute of feed screw
R.p.m. of countershaft
from which ratio the necessary train of gear; can be set up.
Calculating Correct Cutter Speed
To revert now to the calculation of correct peripheral cutter
speed, to which reference has already been made.
Correct cutting speeds are a Amatter of finding from the chart
on page 80 the speed in feet per minute at which the cutter
should run, having regard to the rigidity of the gear cutting
attachment and the nature of the workpiece. The speed of the
cutter is then adjusted to the figure found and this may be
reduced to the formula :Feet per minute
R.p.m. equals
The figure 0.295 is a constant, depending on the diameter of
the cutter used (in this case I& in.), and is given by the formula
equals the number of feet travelled in one turn of the
I2 cutter, where rr is 3.~42 and d is the diameter of the
cutter in inches.
Conversely, if the r.p.m. are known, the number of feet per
minute the cutter shall run can be found from :
- nxd
_. ,__ x r.p.m. equals feet per minute.
or 0.295 x r.p.m. equais feet per minute.
Peripheral speed
of curter
ft. per min.
Soft grey iron . .
Machine casting Cast-i&
Cast steel
Malleable iron
S o f t m a c h i n e s&l : :
T o o l steel:)an&led : :
1: 1: 1: 1:
: : !
: : i
1: 1: 1: 1:
Cutters are mounted on arbors, carried between centres, in
the first method and on an accurately made adapter fitting the
milling spindle in the second method, Details of this adapter
and the method of mounting the cutter are shown in the frontispiece of this book. It will be seen that, in order to impart greater
rigidity, a steady bearing, similar to the overarm bearing of the
milling machine proper, is fitted. The desirability of ensuring
that the cutter runs absolutely true cannot be too strongly
emphasised, for a wobbling cutter will at once destroy any
accuracy in the machined gear teeth.
Driving the Cutters
When the first method of gear cutting in the lathe is employed,
the cutter is, of course, driven by the lathe mandrel in the usual
way, but in the second method an independent drive is required.
This may take the form of an elaborate overhead gear, allowing
for automatic adjustment of the belt tension, like that shown in
Figs. 77 and 78.
Mounting and Centring the Work
Whichever method of cutting is used it is of the utmost
importance to see that the gear blanks are secure on their
mounting, whether it is a mandrel running between centres or
an arbor held in the chuck. If there is any slackness in the
mounting, the blank may move under the pressure of the cut
and the tooth spacing will then be upset. Similarly, the mandrel
Fig. 77.-Overhead for gear cutting
must run perfectly true or a gear form eccentric to the blank
diameter will be cut.
Of equal importance is the need to ensure that the centre of
the cutter falls on the centre line of the gear blank ; an exaggerated error in this respect is shown in Fig. 79. A fairly simple
method of ensuring this is shown in Fig. 80. This shows a
blank in position on the mandrel ready for cutting and the gear
Fig. @.-Set-up for milling gears in lathe
cutter on its arbor ready for centring, the set-up being as for
method 2. It is necessary to measure accurately the lettered
dimensions shown in Fig. 80 so that an estimation of the measurement “ e ” can be obtained. When this has been arrived at, it
is quite easy to set the cutter’to this measurement from the lathe
bed by means of inside callipers to which a micrometer measurement has been transferred.
The measurement “ e ” equals C - ;
where “ c ” equals A + 2
aud a, b, c, d are the measurements indicated in Fig. 80.
An alternative method of centring is to use a simple height
gauge, having a blade provided with a V-notch which may be set
to centre height from the lathe tailstock centre. This method is
open to objection as, with fine
-I- _
diametral pitches, the accuracy of
the setting may be in doubt owing
Fig. 79.
to difficulty in determining exactly if;zztttfe -,~-,-JL- ,_
when the cutter is touching both
of cutter
sides of the V-notch.
The first method of centring is
-/also applicable when the cutter is
rotated by the lathe mandrel, all measurements shown as being
taken from the lathe bed are then taken from the face of the
chuck or from the driver plate, according to the way in which
the cutter is mounted.
Cutting The Teeth
In Chapter VII we have seen that in a spur gear the whole
depth of tooth is given by the formula :2.157
-- equals whole depth of tooth.
This figure is required for setting the depth of cut and,
whether the operation is carried out by Method I or Method 2 ,
the cut must be set to this depth so that the gear tooth can be
fully formed at a single pass. This holds good for cutting small
gears in the lathe, though for larger gears the practice is different.
In any case the slide controlling the depth of cut must be set
to zero, This setting is carried out preferably by inserting a
Fig. 6o.-Method of centring cutter with gear blank
feeler gauge between the work and the top of one of the cutter
teeth, and the slide is then advanced by an amount equal to
the thickness of the feeler gauge. At this stage, the slide index
is at zero and the correct depth of cut is then put on. After
making sure that the dividing device is also set at zero, the
actual cutting may be started. When one tooth space is completed the cutter should be returned ready for the next cut, the
dividing device is moved round for the next tooth, the cut is
then taken and the whole procedure repeated until all the teeth
are machined.
Generally, cutting should be carried out at as high a speed as
the inherent rigidity of the machine and its attachments will
As the illustration, Fig. 8oA (on the back cover) will show, it is
not impossible to cut an internal ring gear in the lathe. To do so
the work itself is mounted in the d-jaw chuck so that the flank
from which the gear is to be cut can be set to turn truely. The
operation is one of shaping each tooth separately by means of a
form tool set in the lathe toolpost and traversed by its lathe saddle
which is moved by hand against a simple stop. The work is, of
course, indexed by th,e methods described earlier, while the whole
depth of cut, obtained from the formula on the previous page, is
set progressively by means of the cross slide index, cuts of
ooo~ in. to 0.003 in. at a time being taken. Again a simple stop
is set with reference to readings taken from the cross-slide index
showing when the full depth of cut has been attained.
Gear Generating Methods
IT is generally admitted that, if accurate gear teeth are wanted,
the best system of producing them is by one of the generating
methods. There are various processes in which the form of the
tooth is generated by a rack tooth cutter, the wheel blank being
rotated at the same time as the cutting is done. As every
involute gear is able to mesh truly with an involute rack, and the
rack tooth, as we have seen, has easily-formed straight sides, the
method is excellent in principle.
Webbing, Planing and Shaping
There are three methods of gear generating in use-hobbing,
planing, shaping, and to a certain extent, grinding. The
hob is actually a milling cutter in the form of a worm with
a single thread, having straight sides making an included angle
of 29 deg. Nicks are cut across to form cutting faces and the
Fig, 8x.--End. and sectio&Z views of bob
worm is relieved a little behind these faces to give cutting clearance. These hobs can be used in a milling machine or in specially
designed hobbing machines. Fig. 81 shows such a hob.
As the hob has a worm thread, the blank will have to be set
over to ensure that the cut teeth will be truly parallel to the axis
when ordinary spur gears have to be cut. Once cutting has
Fig. &-Generation of teeth by hob
commenced, the hob rotates the blank just as a worm rotates its
worm wheel, and the teeth are thus truly formed in the manner
shown in the drawing (Fig. 82).
This method can produce as nearly perfect a tooth form as the
condition of the cutting tool and the machineability of the metal
Fig. 83.~Principle of Sunderland gear planer
will permit. It suggests the mouhhng of teeth in a plastic
material blank, rolled in contact with a set of teeth of perfect form.
Fortunately, this latter condition is easy to attain because it
exists in the straight-sided rack tooth. The plastic wheel blank
is perhaps not yet available, but it is possible to use a planing
cutter formed with rack teeth and to plane out the spaces in a
rotating blank with it.
Fig. 84.~-Principle of Fellows gear shaper
This is the principle of the Sunderland gear planer, shown
diagrammatically in Fig. 83. On the left is seen, in plan view.
the cutter after having made a full depth cut with the blank
held stationary. This stage completed, the blank is rotated and
the result is to produce a truly formed tooth in much the same
-way as does the hob. On the right are shown two stages of the
blank, removed from contact with the cutter. The cutter is
moved to and fro to make successive strokes, as well as up or
down in gear with the rotating blank.
A similar idea is applied in the Fellows gear shaper, but here
the cutter is in the form of a complete gear that rotates with the
wheel blank after a preliminary cut has been made. This also
is shown (see Fig. 84) with four sections of the blank, A, B, C, II,
indicating progressive stages in machining. The left-hand flanks
of the teeth are correctly formed first, and the right side is not
completed until the teeth come into position after making the
first revolution.
Bevel wheels are best produced upon a planer-a machine
with a somewhat complex action, in which two cutters each
machine one side of a tooth. Their motion is controlled by
moulded templates to give the correct contour, and a device
brings them gradually closer together, so thae they converge upon
the apex of oae wheel cone, as they are given a forward motion
on the cutting stroke. There are several such machines on the
market, but as they are highly specialised in their action they do
not come within the scope of this book.
elical Gears
Another important cutting process is the production of the
helical, or so-called “ spiral ” gears. These are not difficult to
make, and the work can be done either in a lathe or in a milling
machine, the latter for preference. When the nature of the
teeth of a helical gear is understood, as it should be after reading
the section devoted to it, it becomes clear that the wheels are
simply sections cut off a bar on which a multi-start thread has
been turned. If the operator can turn such a thread as this
he can equally well produce helical gears. The number of
starts is equal to the number of teeth in the wheel, and this means
that a dividing head will have to be used, though the competent
lathe hand knows how to mark the teeth on the change wheels to
produce similar results.
In milling helical gears, the blank has to be rotated at the same
time as it is carried forward under the cutter, and the degree of
rotation must be strictly governed. To produce this result,
change wheels have to be used just as in a lathe. The calculation
is simple. The lead of the helix (or spiral) and the lead of the
table screw are the two factors. The formula is as follows :Lead of required helix
Driven gears
Lead of machine table = Dnvmg
--. ’ gears
Lead screws of milling machines and lathes usually have four
threads to the inch, and the index head a 40 to I worm drive.
If the two are geared together by equal wheels, then ono turn of
the lead screw will produce r/40 turn of the index head spindle
and, by the time 5 complete turn of the latter has been made, the
table will have been moved IO in. The lead of the table is IO in.
Four change wheels have to be considered, one on the lead
Fig. Sj.-Use.of idler wheel for left-hand helix
screw, one on the worm, and the two intermediate wbeels on
the studs. An idler wheel will be needed if the helix is lefthanded and this is in centre as the diagram Fig. 85 shows. The
*“‘S fOi milling machines is two
usual equipment of change v:h--i
of 24 and one each of 28, 32, 40,44. 48, 56, 64, 72, 86 and IOO
teeth respectively.
As an example, consider a gear with a helix of 32 in. lead.
From the formula just given :Lead
..-- of
..- .-,,.,~.. ~.. ~ 32
- which can be expressed as
Lead of table
3-is, or as 4 ' '2 x 8 - X - 8
2 x 12
::= 48 i< 5?
5 x 8’ 24 40
Out of these, 24 and 40 will be the drivers, and 48 and 64 the
followers arranged as in the diagram Fig. 86 : 24t. on the lead
screw, 64t. on the worm spindle, 4ot. the first and 48t. the second.
As another example, consider a helix of 6 in, to be cut on the
same machine. Proceeding as before
Lead of helix
6 -2x3_- 40- x48
Lead of table
5 x 2
So the drivers are xoo and 32 and the followers 40 and 48.
Fig. 86.-Arrangement for milling helical gears
In addition to giving this motion to the blank, the table will have
to be set over at an angle equal to the helix angle, a term that needs
explanation. In the diagram (Fig. 87) a blank wheel is shown,
with one tooth line drawn across it and proiected on either side
to show the true course of the helix. The pitch or lead of the
curve is equal to the distance marked AB, and the helix angle is
that at which the straight line AC crosses the wheel. This line
indicates the angle made by the helix with a line at right angles to
the wheel axis at any point. The distance BC is equal to the
pitch circumference of the wheel. If these details were plotted
for a lead of 6 in., it would be possible to measure the angle
with a protractor, but such a method is not sufficiently accurate.
Consider a wheel in
which the pitch diameter is to be 3 in. and
the lead 6 in. This gives
us enough information
to calculate the tangent
Fig. @.-The helix
of the angle, by dividing the pitch circumference-3 x 3.1416by the lead-6 in.
Tangent of angle BAC
zz CB
_ zzz 3 I~~,~x~ ~...._,_.~.
= 1.5708.
A table of tangents,
usually given with workshop handbooks shows that this represents an angle of 57” 3x’, which is the angle at which the table
will have to be set over to the helix.
Conversely, if the angle and the lead are given, then the
pitch circumference is found by multiplying the tangent of the
given angle by the lead, and the diameter by dividing this
product by 3.1416.
In selecting a cutter for
machining this type of wheel,
another point presents itselfFig, 88.-Difference between
circular and normal pitches
the difference between the circular pitch and the normal
pitch. This is illustrated in
Fig. 88. Seen endwise, the
wheel teeth have an apparent
pitch that is unreal, though it is
clearly the circular pitch. The
other view shows the same wheel in plan, from which can be seen
the normal pitch, or width between successive teeth; this is
clearly the measurement needed to select the cutter required.
Grinding Gear Teeth
The use of the grinding process for accurate gear-cutting
is increasing, though generally speaking this may be said to be
limited to precision work. The process can be used in two ways :
either for finishing gears, cut in the milling machine, to a high
degree of accuracy, or for cutting them entirely by grinding. In
either case, the profile of the grinding wheel has to be very closely
controlled by a diamond truing device, governed in its turn by
accurate templates. Here, again, there are two systems at least :
the first is the simplest possible, where the wheel face is given a
shape that is true to the finished involute, and the second provides
a true rack shape. In this case, the teeth are generated by a
rolling motion of the head carrying the wheel blank, while the
wheel is moved to and fro by a ram at the prescribed rate to give
the correct cutting speed..
Gears that have to be specially hardened before use are ground
to a high degree of finish after the hardening process, thus removing any traces of distortion. Though grinding produces a
higher degree of finish than almost any other workshop process,
it does not require a glass of high magnifying power to reveal
the real nature of this finish. The use of burnishing to smooth
out even the finest scoring produced by grinding is worth while.
Here the cut gear is rotated between three hardened steel gears
while pressure is applied. After only a few seconds, a very
finely burnished surface is given to the soft gear, consisting
of a skin of compressed metal that makes for an increased life.
However, the finished tooth profile will have been checked and,
if necessary, corrected. Burnishing is usually carried out on a
milling machine. The two lower gears are supported in a
fixture secured to the table ; the upper driving gear is fixed to
the spindle, and the gear under treatment is held under pressure
between the three.
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