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THE CALCULATION OF
CHANGE -WHEELS
FOR
SCREW-CUTTING ON LATHES
Digitized by tine Internet Arcinive
in 2007 witii funding from
IVIicrosoft Corporation
littp://www.arcliive.org/details/calculationofcliaOOdevruoft
9^
THE CALCULATION OF
CHANGE-WHEELS
FOR
SCREW-CUTTING ON LATHES
A PRACTICAL MANUAL
FOR THE USE OK
MANUFACTURERS, STUDENTS AND LATHEMEN
BY
D. DE VRIES
WITH 46 ILLUSTRATIONS
lonbon \
E. & F. N. SPON, Limited, 57 HAYMARKET
SPON & CHAMBERLAIN, 123 LIBERTY STREET
1908
PREFACE
It is a curious circumstance that the calculation of change-
wheels for the cutting of different pitches of thread on
a lathe, however simple such a calculation may be, is
comparatively but little known, being, for the majority of
those most closely interested in the subject, shrouded in
mystery.
Many whose theoretical knowledge is quite sufficient
to enable them to face the problem, have had so little
practical experience in screw-cutting that they are unable
to go deeply into the matter, and present, in a clear and
simple manner, the different variations which may possibly
occur.
The greater number of mechanics, even the younger ones,
possess too slight a theoretical knowledge to permit of their
building up a system by themselves.
There are, of course, mechanics who are quite capable
of working out the necessary calculation, but so many of
them — I speak from personal experience — regard their
knowledge as more or less of a secret, and say, at any rate
to themselves, "Why should I impart to others what has
taken me so much trouble and cost me so much money
to learn ? "
The purpose of the present treatise is to enable any
one, who is prepared to take the trouble to study it carefully
to learn how to calculate change-wheels properly.
vi The Calculation of Change- Wheels.
I have deemed it expedient, for the sake of those of
my readers who have but a superficial knowledge of the
lathe, to give a short description of this tool, in so far as it
is connected with screw-cutting, to which I have added a
description of the various types of thread to be met with,
with the necessary tables appended, as also a number of
practical hints, with reference to screw-cutting, together with
the operations connected therewith.
I have purposely refrained from including a number of
tables giving the change-wheels required for the various
pitches of threads on different lathes, in place of which a
large number of practical examples are given which cover
every possible variation likely to be met with in practical
work. Experience has taught me that the inclusion of such
tables only leads to purely mechanical work demanding
no effort of the mind, whereas, in each particular case, due
consideration should be given to the special work in hand,
so that in cases of exceptional difficulty, where one is obliged
to set to work without the assistance of such tables, the
manner of calculation may not be unfamiliar.
It is my earnest wish that the present work may proye
useful not only to students, but also to those engaged in
practical work.
D. DE VRIES.
CONTENTS
CHAPTER I.
PAGK
The Lathe . . . . i
CHAPTER II.
The Calculation of Change-Wheels —
(a) Systems . . . . . , , . . , . . i©
(d) What Change-Wheels are to be found on a Lathe . . 15
(c) The Cutting of Metric Threads on a Lathe with Metric
Leadscrew . .. .. .. .. .. 16
(d) The Cutting of English Threads- on a Lathe with
English Leadscrew . , . . . . . . 20
(e) To Cut English Threads on a Lathe with Metric
Leadscrew . .
23
(/) The Cutting of Metric Threads on a Lathe with
English Leadscrew .. . 27
(^) The Wheel with 127 Teeth 30
(A) Method for Calculating Approximate Fractions . . 32
(/) The Proof of the Sum 51
(k) Fixing up the Wheels . . . . . . . . 53
(/) Thread-Cutting with Double Compound Train 54
(m) The Cutting of Left-hand Threads . . 55
viii The Calculation of Change- Wheels.
CHAPTER III.
Threads and Their Construction — page
{a) Forms of Threads . . . . . . 56
{b) Types of Threads 56
{c) Screw-Cutting Tools . . . . . . 65
{d) Cutting the Thread 69
{e) The Cutting of Double and Multiple Threaded Screws 73
(J) The Cutting of very Coarse Thread 75
{g) The Hendey-Norton System . . 77
THE CALCULATION OF
CHANGE-WHEELS
FOR SCREW-CUTTING ON LATHES
CHAPTER L
THE LATHE.
Threads, both internal and external, can be obtained in two
different ways, the simplest of which is to cut the thread by
means of taps, dies and chasers. In the smaller sizes, the
majority of internal threads are tapped, whilst external threads
are cut with dies, but in the larger sizes too much material has
to be removed. Tapping, however, is far more general than the
use of dies, as in most cases, external threads can be obtained
in another way, viz. : on the lathe, whilst internal threads
can only be obtained on the lathe at considerable expense.
Moreover, internal threads are to be found in a number of
different places on the larger machine parts, and so it would
be well-nigh impossible to put these pieces on the lathe for
the purpose of cutting the threads. On the other hand, a
bolt or screw-spindle, as a rule, can be set on the lathe, and
threads may be cut by means of a common tool. It is just
for this reason that, whilst a large number of i in. external
threads are cut on the lathe, i in. threads in holes arc, with
but few exceptions, cut exclusively by tapping. The practice,
however, of cutting internal threads of more than 2 in.
diameter on the lathe, whenever the work-piece allows it, is
becoming more and more general.
The object of the present work is to give a detailed
description of the way in which it is possible to cut the various
B
2 The Calculation of Change-Wheels
threads on the lathe, and thus to answer, as fully as possible,
the question : " How are the change-wheels to be calculated
for screw-cutting on the lathe ? "
In order that this work may also be of service to those who
are not fully conversant with the lathe, the following points
will be treated successively, viz. : the general construction of
the lathe, more especially of those parts of the lathe used in
screw-cutting ; the theory of the calculation of change-wheels
and screw-cutting in practice.
Fig. I.
The lathe, as originally constructed, was not intended for
screw-cutting. Fig. i shows a lathe as it was first constructed.
On this lathe a rotary movement was imparted by means of
a driving belt to the headstock and workpiece only, all other
movements being executed by the operator himself.
Within a comparatively short time, however, more was
demanded of this machine, larger pieces were required to be
machined than was possible with direct belt drive, and the
for Screw-cutting on Lathes. 3
double back gear was introduced ; it was desired to move the
tool on the material automatically, and to obtain this, the
rest was mounted on a carriage and moved by means of a
leadscrew which motion was imparted by means of either
a belt or a train of gears from the headstock. The intro-
duction of a train of gears on the apron made it possible
not only to move the carriage over the whole length of the
bed for sliding, but also to move the rest automatically
in a transverse direction over the carriage itself for surfacing.
Fig. 2.
Finally, the leadscrew spindle, called for short the " leadscrew,"
was so arranged that by a set of gears of various diameters,
a variable, but at the same time for each train of gears
fixed ratio between the number of revolutions of the head-
stock, i.e. the workpiece, and the leadscrew was obtainable,
thus making it possible to cut different pitches of threads on
the lathe. Fig. 2 gives the general arrangement of such a
lathe.
B 2
4 The Calculatio7i of Change-Wheels
The leadscrew revolves in the leadscrew-nut, which is
fixed to the apron, and, as this nut cannot revolve, it travels
along the leadscrew, the carriage at the same time making a
corresponding movement.
The movement of the carriage already causes a considerable
pressure on the thread of the leadscrew and the nut, which
is still increased by the cutting of the tool on the material,
and, as a natural result, both the leadscrew and the nut are
exposed to a certain amount of wear. This wear is further
increased by swarf and chips falling on the leadscrew, and their
getting between the nut and thread.
It is evident, as far as the leadscrew is concerned, that
this wear will only affect that portion over which the nut
travels on the thread. As the work on the lathe varies in
length (but is as a rule considerably shorter than the maximum
distance between the centres), the wear of the thread is
greatest on those parts of the leadscrew where the nut moves,
and after being in use for a certain time, it is impossible to
prevent the leadscrew being scarcely worn at all at the end
but considerably worn in the centre, and worn most of all
close to the headstock. The wear of the nut, however, is
fairly even.
The nut was formerly made solid, consequently it was
impossible to repair the wear. It was soon seen, however,
that it was preferable to have half nuts, so that not only can
it now be repaired, but, by means of the lever a (Fig. 2), it
can also be opened and closed.
This has led to the attainment of a number of advantages.
First and foremost, the possibility of repairing the nut just
referred to. A downward pressure of the lever a keeps both
halves of the nut closed so as to grip the lead-screw. The
two halves of the nut ^ ^ move in a vertical direction at the
back of the piece c, and are provided with pins which fit in
eccentric slots in the circular plate which revolves on point e.
Fig. 3 shows these eccentric grooves in the plate. If the pins
of the half nuts are shifted by moving the lever a, the half
nuts travel the double distance A B (Fig. 3), viz. : the upper
nut up and the lower one down, the half nuts being thus
for Screw-cutting on Lathes.
I
entirely disengaged from the thread, causing the motion
imparted to the carriage by the leadscrew to cease im-
mediately.
In the earlier types of construction, with the solid nut, the
carriage had to be moved by hand by means of a handle
placed on a spindle in the apron, with a bevel gear on the
other side of the spindle to which this handle was attached ;
this in its turn meshed with another bevel gear fixed on the
hub of the nut. In this way the
nut was made to revolve over the
leadscrew and the carriage was
moved over the bed. But it took
far too long to move the carriage
any distance at all over the bed,
besides being very fatiguing work. ~
The nut, being in halves, can no
longer revolve, but it can be
opened. A rack is to be found on
the side of the bed in which a
pinion meshes to which motion is
imparted by the hand wheel h
(Fig. 2), by means of which the
carriage can be quickly disengaged
from the leadscrew, and a quick
and easy hand movement is secured.
Other advantages besides those
enumerated here have been derived
from the split nut. One great
difficulty, however, still remains,
viz., the different wear on a certain length of the leadscrew. If
this happens to be more worn in the middle than at the ends,
it is impossible to cut a true thread.
Now, in comparison to the work ordinarily performed on
a lathe, but little screw-cutting is done. The greater part of
the time the leadscrew is thus engaged for the feed motion of
the carriage and for surfacing. For this reason, the movement
imparted to the carriage for screw-cutting, has been separated
from that for feed motion. A separate shaft, provided with a
The Calculation of Change-Wheels
»> .,.r'."'';T«(.lil'Nil' 1''iii';,ii,r,Jll.:.l* .
for Screw-cutting on Lathes. 7
keyway, imparts motion to the pinion which meshes with the
rack (Fig. 4), by means of bevel and spur-gears. The sliding
movement of the carriage being accomplished in this manner,
the leadscrew is only used for screw-cutting. In still later, and
principally American constructions, the two shafts have finally
been united in one, the leadscrew being now provided with a
keyway ; for sliding and surfacing the leadscrew simply acts
as driving shaft, the thread of the leadscrew being only used
for screw-cutting, and so the same object is attained with one
shaft as is obtained in Fig. 4 with two, viz., the thread of the
leadscrew is used for screw-cutting only.
Fig. 5.
In Fig. 2 the gearing for the motion of the leadscrew from
the head spindle is clearly visible. Wheel i is keyed to the
head spindle ; rear wheels 2 and 3 run loose on studs fastened
to the lever 4, By means of knob 8, this lever can be raised
to hole 5 or lowered to hole 6. If the lever is placed in
position 5, wheels 3 and i become engaged, and wheel lo on
spindle 7 revolves by means of wheel 9. Wheel 2 now runs
to no purpose. If the lever is placed in position 6, wheels
2 and I become engaged, and wheel 3 is brought into play by
means of wheel 2, thus causing wheel 3, as well as wheel 9 and
8 TJie Calculation of Change-Wheels
spindle 7 to rotate in an opposite direction. In the illustra-
tion the lever stands midway, so that wheel i engages neither
of the wheels 2 or 3, consequently, although the lathe spindle
rotates, the leadscrew is not rotating. Wheels i, 2, 3 and 9
have the same number of teeth, so that the wheels on spindle 7
make precisely the same number of revolutions as the lathe
spindle. Wheels 10, 1 1, 12 and 13 are the actual change-wheels,
and can be easily mounted, dismounted, or changed. Wheels
II and 12 rotate on a sleeve on spindle 14, and consequently
make the same number of revolutions, so that wheel 12
transmits very slowly to wheel 13 the motion imparted to
wheel II. In the illustration the gearing between wheel 9 to
the leadscrew is accomplished by 4 wheels — wheels 10 and 12
being the driving wheels, il and 13 those driven. It is evi-
dent that the motion of wheel 9 on spindle 7 is imparted
but very slowly to the leadscrew, in the same ratio as the
'X=r^ZZZZ2ZZZZSZ^L
\M
itiin>lil»ll»l>/.
Fig. 6.
product of the number of teeth on wheels 10 and 12 to the
number of teeth on 11 and 13. Precisely the same is to be
seen in Fig. 4. Wheel 13 can, however, be driven by means
of a wheel engaging both wheels 10 and 13, without the
intermediate wheels 11 and 12, thus serving as an idle wheel,
in which case wheel 10 is the driving wheel and 13 the
one driven. The ratio between the number of revolutions of
the lathe-spindle and leadscrew is identical with the ratio
between the number of teeth on wheels 10 and 13.
Wheels 11 and 12 are mounted on a sleeve running on
stud 14. (See Fig. 6.)
This stud must be movable in accordance with the
dimensions of the wheels, and is consequently placed in a
casting called the shear or swingplate at the end of the lathe.
This shear (Fig. 7), has two long slots, so that the stud can
either be brought close to the leadscrew B, for small wheels*
for Screw-cutting on Lathes. 9
or more to the rear for larger wheels, at will. In order to per-
mit of working with five or six wheels, a second slot is to be
found in the shear. This shear turns on the leadscrew B,
and is held in position by means of the two bolts to be seen
in the circular slots. When the intermediate wheels have been
accurately set in the wheel on the leadscrew, the shear, which
was first lowered to its full extent, is raised till the inter-
mediate wheel engages the upper wheel properly, after which
the shear is fastened.
Fig. 5 shows an American type of lathe, on which it is
not necessary to change the wheels for different pitches of
Fig. 7.
threads. By means ot a cone-gear to be found under the
headstock and at the left-hand side of same, the ratio of speed
between the lathe-spindle and the leadscrew can be varied
by the simple movement of a lever. The necessity of calcu-
lating the change-wheels is done away with, all that is required
being the placing of two levers in a certain position indicated
in the table. The manner in which this result is attained will
be further described in Chapter III.
lO The Calculation of Change-Wheels
CHAPTER II.
THE CALCULATION OF CHANGE-WHEELS.
{a) Systems.
In the calculation of change-wheels for screw-cutting on the
lathe there is one difficulty, and that is, the difference between
the English and metric system of measurements. It is not
insurmountable, but it does not render the task any easier,
and has been the cause of a considerable amount of trouble.
In the calculation of change-wheels it is a matter of in-
difference whether a right- or left-handed screw is to be cut,
what form the thread has to take, whether the thread is
internal or external, or, finally, the exact internal or external
diameter of the thread. The one essential question to be
answered is : How many threads are required for a certain
unit of length ?
For this purpose two units exist ; 1st, the inch ; 2nd, the
centimetre.
For both these units of length the number of revolutions
of the thread are termed " number of threads."
The length of a single thread is spoken of as " pitch."
The member of threads is thus determined by the number of
revolutions per unit of length.
If the pitch is indicated with the inch as the unit of length,
we speak of " English thread," If the pitch is indicated with
the centimetre as unit of length, it is called a " metric thread."
Both, however, have a system, which is further treated of in
Chapter III., but which, as such, has nothing at all to do with
the calculation of the change-wheels.
If but one of these two units, either the inch or the centi-
metre, were exclusively adopted as the standard unit, then the
difficulty referred to at the beginning of this chapter would
for Screw-cuiiing mi Lathes. 1 1
entirely disappear. But the inch and the centimetre are em-
ployed together ; and not only that, but there is also a lack of
uniformity with regard to the leadscrew ; one maker cutting
the leadscrew according to the English, and another accord-
ing to the metric system. English and American lathes
usually have a leadscrew cut according to the English system ;
French and Swiss makers cut it almost exclusively according
to the metric system, whilst German manufacturers employ
both systems, though the preference is given to the English.
Four variations are thus possible : —
1. A metric thread to be cut on a lathe with metric
leadscrew.
2. An English thread to be cut on a lathe with
English leadscrew.
3. An English thread to be cut on a lathe with
metric leadscrew.
4. A metric thread to be cut on a lathe with English
leadscrew.
Briefly summarized : —
To cut : I. Metric on metric.
2. English on English.
3. English on metric.
4. Metric on English.
If one desires, once and for all, to be able to calculate
the change- wheels for every variety of pitch, it is imperative
to know these four varieties thoroughly, as they can occur
intermingled.
I St Axiom. — The number of threads is to be determined by
the pitch of the leadscrew and the ratio of the number of revo-
lutions of the lathe spindle to that of the leadscrew.
This axiom holds good for all four cases.
The ratio of the number of revolutions of the lathe-spindle
to that of the lead-screw is obtained by means of wheels
(change-wheels).
When the spindle of the lathe has completed one revolu-
tion, then the work on the lathe will have also completed
one revolution.
1 2 The Calctilation of Change- Wheels
If the number of revolutions of the lathe-spindle and lead-
screw are the same, so that the leadscrew has also completed
one revolution, then the carriage has moved a distance during
this one revolution equivalent to one thread of the leadscrew.
If a tool has been placed in the toolholder, so that it can cut
the work-piece, then precisely the same pitch will have been
cut on the work-piece as that on the leadscrew. With an
equal number of revolutions of the lathe-spindle and the lead-
screw, the thread cut on the work-piece will have the same
pitch as the leadscrew.
If the lathe-spindle has completed one full revolution,
but the leadscrew on the other hand only half a revolution,
then the carriage, and with it the tool, will have moved in a
straight line over a length equal to half a pitch of the
leadscrew. It is thus only when the lathe-spindle has
made two revolutions that the leadscrew will have completed
one full revolution ; two threads are now to be found
on the work-piece over a length equal to one pitch of the
leadscrew. The ratio of the number of revolutions of the
spindle to that of the leadscrew was 2:1; the ratio of the
number of threads per unit of length of the work-piece to
that of the leadscrew was also 2:1. Hence it follows : —
2nd Axiom. — The ratio of the number of revolutions of the
lathe-spindle to that of the leadscrew is the same as t/ie pro-
portion of the pitch of the thread to be cut to tliat of the lead-
screw.
Axiom 2 is also applicable to all four cases.
For example, the leadscrew of a lathe has a pitch of one
thread to the inch. It is required to cut two threads to the
inch. The proportion of the pitch to be cut to that of the
leadscrew is thus 2:1. According to axiom 2 the ratio
of the number of revolutions of the lathe-spindle to that of
the leadscrew must also be 2 : i.
The leadscrew has thus to complete one revolution to
two of the lathe-spindle. The leadscrew receives its motion
from the lathe-spindle, so that the rotation of the leadscrew
must be retarded accordingly. The rotation of the lathe-
spindle is transmitted to the leadscrew by wheels. The pro-
for Screw-cutting on Lathes. 13
portion of the number of teeth on wheel 10 (see Fig. 2), to
those on wheel 13 on the leadscrew must thus be in inverse
proportion to the ratio between the number of revolutions of
the lathe-spindle and the leadscrew, which, in the example
given, must be 2 : i ; the ratio of the wheels 10 and 13 thus
becomes i : 2. If then a wheel with 50 teeth be on the
sleeve of spindle 7, and one with lOO teeth on the leadscrew,
with any desired idle wheel, a screw of 2 threads to the inch
or j^inch pitch will be obtained on the work-piece with a
leadscrew having one-inch pitch. From this we arrive at
what is again applicable to all four cases : —
^rd Axiom. — The proportion of the fiumber of the threads to
be cut to those in the leadscrew is in inverse ratio to the pro-
portion of the number of teeth on the wheel on the lathe-spindle
to tlte number of t^th on the wheel of the lead-screw, or in
fractional form —
Number of threads to be cut
Number of threads in the leadscrew
No. of teeth on the leadscrew wheel
No. of teeth on the lathe-spindle wheel
In this manner the calculation of the change-wheels for
screw-cutting is reduced to the working out of a simple
fraction — the number of threads to be cut being the
numerator, those in the leadscrew being the denominator, or,
if it is desired to express the fraction in the same manner as
the wheels, i.e. the number of teeth on the lathe-spindle wheel
on top as numerator, that of the leadscrew underneath as
denominator, it is just the reverse. The number of threads
in the leadscrew will then represent the value of the
numerator, those of the thread to be cut representing the
denominator. As the pitch of the leadscrew on a certain
lathe is always the same, it follows that the value of the
numerator is always constant.
We must here call especial attention to a misunderstanding
which so often occurs in connection with the question as to
whether the number of threads in the leadscrew must form
1 4 The Calculation of Change- Wheels
the numerator or the denominator. A practical man can
generally tell fairly well which wheels have to be placed on
top and which underneath, but still, when the pitch of the
thread to be cut closely approximates that of the leadscrew,
mistakes can sometimes be made.
The screw may be denoted by the number of threads per
unit of length, in which case the number of threads in the
leadscrew is the numerator of the fraction.
The screw may also be denoted by the length of one pitch
of the screw ; in this case the length of pitch of the screw to
be cut will be the numerator, the length of pitch of the lead-
screw being the denominator of the fraction, the numerator of
which will indicate the number of teeth on the lathe-spindle
wheel, the denominator indicating the number of teeth of the
wheel on the leadscrew.
Should the number of threads of the screw to be cut be a
multiple of those in the leadscrew, one is naturally inclined to
express it in number of threads per unit ; for example, 4
threads per inch to be cut on a lathe with a leadscrew of
I thread per inch ; should it not be a multiple, as for
example, each thread having a length of 7 mm., one is then
inclined to denote it by the pitch. If, in both instances, the
number of threads in the leadscrew be i per inch, the fraction
in the first instance will be —
Number of threads in the leadscrew _ 1 _ driving wheel
Number of threads to be cut wheel to be driven
In the second instance, in which the pitch of the screw to
be cut must be 7 mm., the number of the threads to be cut
per unit is itself a fraction, viz. : ~ , the fraction thus being
I 7
= , 7 being the length in mm. of the pitch of the
25*4 25-4
7
screw to be cut, 25 4 the length in mm. of the pitch of the
lead-screw, so that, in this case, the length of pitch of the
screw to be cut can at once be placed in the numerator for
the driving wheel, the length of pitch of the leadscrew being
for Screw-cutting on Lathes. 15
placed in the denominator for the wheel to be driven. In
actual calculation the foregoing examples must be carefully
distinguished one from the other.
ijj) What Change-wheels are to be found on a Lathe.
This question presents itself each time change-wheels have
to be calculated, because the fraction which is formed by the
thread to be cut and the leadscrew, must be changed into one
formed from the wheels to be found on the lathe. These
wheels should have such a number of teeth as will, within
certain limits, include the indivisible factors, viz. : 2, 3, 5, 7,
II, 13, 17, 19, 23, etc. Some makers supply these wheels in
a progression of 5, others with another progression. The
following set of change-wheels is, or should be provided with
every lathe, viz. : —
15 = 3
X 5
60 = 2x2x3x5
20 = 2
X 2 X
5
65 = 5 X 13
25 = 5
X 5
75 = 3 X 5 X 5
30 = 2
X 3 X
5
85 = 5 X 17
35 = 5
X 7
95 = 5 X 19
40 = 2
X 2 X
2 X
5 100 = 2x2x5x5
45 = 3
X 3 X
5
105 = 3 X 5 X 7
50 = 2
X 5 X
5
115 = 5 X 23
55 = 5
X II
125 = 5 X 5 X 5
or
16 = 2
X 2 X
2 X
2 42 = 2x3x7
18 = 2
X 3 X
3
44 = 4 X II
20 = 2
X 2 X
5
56 = 2x2x2x7
21 = 3
X 7
60 =2x2x3x5
22 = 2
X II
66 = 2 X 3 X II
26 = 2
X 13
78 = 2 X 3 X 13
28 = 2
X 2 X
7
88 = 2X2X2X II
34= 2
X 17
96 = 2x2x2x2x2x3
38 = 2
X 19
108 = 2x2x3x3x3
One of the two foregoing sets is generally provided with
1 6 The Calculation of Change- Wheels
the lathe. English lathes usually have a set of 22 wheels
some of which have the same number of teeth.
It will be clear from what has been said, thus far, that
the easiest thread to be cut on a lathe, i.e. the thread causing
the least trouble in the calculation of the change-wheels, is
that having the same system as the leadscrew. This will be
the case with the ist and 2nd cases referred to on page 11.
if) The Cutting of Metric Threads on a Lathe with Metric
Leadscrew.
Take the case of a lathe with a leadscrew having I cm.
(10 mm.) pitch.
It is required to cut 4 threads per cm.
No. of teeth on driving wheel _ No. of threads in the leadscrew
No. of teeth on wheel to be driven ~ No. of threads to be cut.
= i = ^ = gear-wheel lo. 1 g^ ^. ^^
4 loo = gear-wheel on lead-screw. J
It is required to cut 7 threads per cm.
No. of threads in the leadscrew _ i _ IS = driving wheel.
No. of threads to be cut 7 105 = wheel to be driven.
To cut \\ thread per cm.
No. of threads in the leadscrew _ ji = 5° o ^ ~ driving wheel.
^No. of threads to be cut 1*5 75 90 = wheel to be driven.
To cut 3 threads per cm.
No. of threads in the leadscrew _ * _ 3^ = driving wheel.
No. of threads to be cut 3 90 = wheel to be driven.
To cut 5 threads per cm.
No. of threads in the J,eadscrew _ I _ 25^ = driving wheel.
No. of threads lo be cut 5 125 = wheel to be driven.
In the last example it is also possible to say, a pitch of
2 mm., in which case the fraction will be : —
Pitch in mm. to be cut _ 2 _ ^5 = driving wheel.
Pitch in mm. of leadscrew 10 125 = wheel to be driven.
In both cases the result will naturally be the same.
for Screw-cuttiiig on Lathes.
17
To cut a pitch of 7 mm.
Pitch in mm. to be cut _ 7 _ 70 = driving wheel.
Pitch in mm. of leads«;rew 10 100 = wheel to l)e driven.
To cut a pitch of 5^ mm.
Pitch in mm. to be cut _ 5-5
Pitch in mm. of leadscrew lo
To cut 7 threads per 22 mm.
55 = driving wheel.
100 = wheel to be driven.
22
Denoted in pitch = a pitch of - mm.
Pitch in mm. to be cut
Pitch in mm. of leadscrew
-y. _ 22 = driving wheel.
10 70 = wheel to be driven.
/a.
s5t:
/^
^^st.
A
xC.
CcoA
*>/yt<AA?^
r'>00t.
Fig, 8.
Fig, 9.
So far it ha.s always been possible to work with a single
train of wheels with any desired idle wheel. Fig. 8 shows a
single train.
C
1 8 The Calculation of Change-Wheels
In the set of wheels to be found on the lathe, wheels with
cither 22 or 70 teeth, as presumed were employed for the
preceding examples, were not included. A compound train
is now used.
22 _2Xil _ 20x55 = driving wheels.
70 7X 10 35X ic» = wheels to be driven.
Fig. 9 shows this compound train.
a and b are the drivers^ c and d those driven. The fixing
up of the wheels will thus be
55 X 20
ICXD X 35'
The wheels in the numerator, as well as those in the
denominator, can be interchanged ; a may thus be put in
place of ^, or c in place of d, or both may be changed ; but
interchanging of a driver with one to be driven may never
take place, as this would alter the value of the fraction and
an entirely different thread would be obtained.
It is always advisable to try to get the smallest of the
drivers on the lathe-spindle, and the largest to be driven on
the leadscrew, in order to obtain as rational a gearing as
possible.
To cut 1 1 threads per 14 mm. The pitch is thus 14/ 1 1 mm.
Pitch on leadscrew 10 mm.
Solution: M/il = "* =1217 = J^XH.
lo II X lo II X lo 55 X loo
To cut 3i thread per 40 mm. The pitch is thus 40/3 * 5
mm.
Solution : 42/3J = _A^. = 4X lO ^ 20 X 100^
10 35x10 5x7 50x35
To cut 4 threads on 15 mm. The pitch is thus 15/4 mm.
Solution : il/l = _i5 _ 3 X 5 ^ 30x50 .
10 4X10 4X10 4OXICX)
Should the lathe have another pitch than i cm., this will
only necessitate a change in the constant of the leadscrew in
the fraction.
for Screw-cutting on Lathes, 19
The following are a few examples with solutions, dealing
with different leadscrews : —
To cut 9 threads per 16 jum. ; leadscrew 2 threads per
I cm. The pitch of the thread to be cut in 16/9 mm.
The pitch of the leadscrew is 5 mm.
o 1 i.- 16/9 2x8 20x40 . ., , ,
Solution : — '— = = ^ = m case these wheels
5 9x5 45x50
,, 20x80
are too small
45 X IOC
To cut a pitch of 3 mm. Pitch of leadscrew being 7*5 mm.
Solution : — — = - .
7'S 75
To cut 8 threads per 13 mm. Pitch of leadscrew, 7*5 mm.
13/8 13 2x6*5 20 X 65
Solution :
7-5 8x7-5 8x7*5 80x75
In both the foregoing examples, a wheel with 75 teeth
appears among the wheels driven, but is not included in the
specification given on page 15. With a leadscrew having a
pitch of 7*5 mm. a wheel with 75 teeth will repeatedly
occur ; in such a case the manufacturer will be certain to
supply a wheel with 75 teeth.
To cut a pitch of 20 mm. Leadscrew pitch 25 mm.
c w 20 100
Solution: sT^T^'
To cut 3 threads per 20 mm. Leadscrew pitch 25 mm.
Solution: 52/i = .^^ = ?-2<_L° = ^°J<_ 5° .
25 3 X 25 3 X 25 60 X 125
To cut a pitch of 37*5 mm. Leadscrew pitch 25 mm.
Solution: 37 ' 5 ^ ■ 5 AIS ^ 3OXJ0O
25 10 X 25 40 X 50
To cut a pitch of 76 mm. Leadscrew pitch 25 mm.
Solution : 7^ = 4J<_19 ^ 40 X95 ^ 80 x 95 ,
25 2*5x10 25x50 25x100
C 2
20 The Calculation of Change-Wheels
{d) The Cutting of English Threads on a Lathe ivith
English Leadscrew.
In principle, this second case resembles the first. The
system of the leadscrew and the thread to be cut is the
same.
Most lathes have a leadscrew with \ in. pitch, thus 2
threads per inch. Heavy lathes have a leadscrew with i in.
pitch, the smaller sizes \ in., or 4 threads per inch, whilst in
exceptional cases 2\ threads per inch are to be found. Given
a certain pitch, the fraction can then be determined without
any difficulty.
Should the screw be denoted in a certain number of
threads per inch, the number of threads per inch of the lead-
screw is placed in the numerator, the number of threads per
inch to be cut in the denominator. Should the screw be
denoted in the length of the pitch, then the length in inches
of the pitch to be cut is placed in the numerator, the length in
inches of the pitch of the leadscrew being placed in the
denominator.
In practice the majority of threads are cut according to the
Whitworth system (see page 57), for which reason we shall
first of all give a number of problems with solutions for this
thread.
To cut I in. Whitworth thread. Leadscrew 2 threads per
inch. I in. Whitworth thread =16 threads per inch.
Solution
No. of threads in leadscrew per inch
No. of threads to be cut per inch
_ ^5^X SO
" 80 X 125*
To cut \\ in. Whitworth thread. Leadscrew 2 threads
per inch, i^ in. Whitworth thread = 7 threads per inch.
c. , ,. 2 2x1 20 X 40
Solution : - = = ^- .
'- ■- - 35 X 80
for Screw-cutting 07l Lathes. 21
To cut 2 in. Whitworth thread. Leadscrew 2 threads per
inch. 2 in. Whitworth thread = ^\ thread per inch.
2
=
2
X
10
=
40
X
50
4-5
5
X
9
45
X
100
Solution ;
To cut 3 in. Whitworth thread. Leadscrew 2 threads per
inch. 3 in. Whitworth thread = 3^ thread per inch.
Solution : — - = — .
3-5 70
To cut i^ in, Whitworth thread. Leadscrew 4 threads
per inch. \\ in. Whitworth thread = 7 threads per inch.
- 1 • 4 40
Solution : =^~ .
7 70
To cut i^ in. gas thread. Leadscrew 2^ thread per inch.
I i in. gas thread = 1 1 threads per inch.
o 1 .• 2i 25 20 X 50
Solution: -^ = — ^ = „ .
II 1 10 55 X 80
To cut 2^ thread per inch. Leadscrew 2 threads per
inch.
Solution :
2
2
X
9
4
=
2
3"
X
X
4
3
=
20
X
60
2i-
30
X
45
1 X
8
80
~35 "
40
X
100
7
25
X
70
To cut I thread per inch {not a J inch pitch). Leadscrew
2 threads per inch.
Solution : «
To cut 2| thread per inch. Leadscrew 2^ thread per inch.
Solution: "4-= - = — .
2| II 55
In the following examples, the length of pitch is given.
The pitch of the leadscrew will consequently appear in the
denominator.
22 The Calculation of Change-Wheels
To cut a I in. pitch. Leadscrew 2 threads per inch =
\^ in. pitch.
Solution : I = 3 _ 7_5 ^
i 2 50
To cut a ^ in. pitch. Leadscrew 2^ threads per inch
= —r in. pitch.
Solution : ii = liiili = i52iii ^ 75 X 100^
I I 16 X I 40 X 80
2i
To cut 19 threads on 11-5 in. Leadscrew 2^ threads per
inch. The pitch to be cut = — in. The leadscrew
19
I
pitch —r in.
^ 2i
Solution
"•5
19 _= II-5 X 2-5 ^ 115 XI125
I 19 ~ 95 X 100
2
To cut a pitch of 4| in. Leadscrew i pitch per inch.
Solution: if ^ 39 ^ 3ili3 ^ 65 X 75.
I 8 2X4 25 X 40
To cut a ^1 in. pitch. Leadscrew 2 threads per inch.
Solution: M='3X2^2X 13 65.
\ 32 2 X 16 80
To cut 9 threads per 5|| in. Leadscrew 2\ thread per
inch. The pitch to be cut = ^ in. The pitch of the
leadscrew = -r in.
2i
Solution : -2. ^ Sl^l X 2^ ^ 95x2-5 ^ l^i x^ ^
j^ 9 9 X 16 80 X 90
2i
In the foregoing examples practically every case which is
likely to occur, has been treated.
for Screw-cutting on Latfies. 23
{e) To cut English Threads on a Lathe with
Metric Leadscrew.
In the first and second cases considered, the system of
the thread to be cut and that of the leadscrew were identical,
viz., in the first case according to metric measurement, in the
second, according to the English measurement.
In the third case, however, the system of the thread to be
cut and that of the leadscrew are dissimilar. The leadscrew
is divided per cm. = 10 mm., or some part or multiple
thereof, the screw to be cut being divided per inch =25*4
mm., or some part or multiple thereof.
In the third case to be considered, this number 25*4 will
consequently appear regularly either in the numerator or
the denominator, and will invariably produce a fraction
which, with one exception, cannot be resolved into whole
numbers.
An equivalent must therefore be found, by means of which
it will be possible to form a divisible number from the nume-
rator and denominator of the fraction. I*^^'- W
This equivalent is to be found as follows : 6^ in. =
l6' 509675 cm.; taking for granted that 6^ in. = 16-5 cm.,
there is then a discrepancy of 0*09675 mm. per 165 mm.
of length, or rather less than o*o6 per cent., a difference of
practically no importance whatever.
If the number of threads to be cut be expressed in a
certain number per 6-5 in., and the number of threads of the
leadscrew be also expressed in a certain number per 6" 5 in.
or 16* 5 cm., the result will be an equivalent which can be
made use of.
As reference is here made to a certain number of threads
per unit of length, in this case, 6*5 in. or 16-5 cm., the numbers
of threads of the leadscrew will appear in the numerator, the
number of threads to be cut in the denominator.
The following comparison can thus be formulated —
No of threads in leadscrew per 16*5 cm. __ drivers
No. of threads to be cut per 6*5 in. wheels to be driven
24 Tlie Calculation of Change- Wheels
As the number of threads in the leadscrew remains in-
variable for the same lathe, the numerator is consequently a
constant factor for a certain lathe.
Should the leadscrew have a i cm. pitch, the leadscrew
will then have i6'5 threads per i6'5 cm., and the constant
factor of the numerator will be i6' 5, whilst, at the same time,
6*5 is to be found as a constant factor in the denominator,
and must constantly be multiplied by the number which
expresses the number of threads to be cut per inch. If both
these constant factors be multiplied by 10, the number 165
will always appear in the numerator and the number 65 in
the denominator, in this way —
constant factor of numerator . 165
„ „ denominator No. of threads to be cut per in. X 65
or L^-^iS
threads per in. X 65
The equivalent is now complete ; by replacing threads per
inch in the denominator by the actual number, a fraction is
obtained which will permit of the calculation of the wheels.
In the examples which follow, every possible variation
has been carefully worked out, from the simplest to the most
intricate.
To cut 6 threads per inch. Leadscrew 10 mm. pitch.
Numerator = 11x15 _IIXI5
Denominator = No. of threads per inch x 65 6x65
^ 55x75 =50x55
150x65 65x100*
To cut 4 threads per inch. Leadscrew 10 mm. pitch.
Solution: ^^^ = ^^ X 75 ^ 55 X75 .
4x65 20x65 65x100
To cut 2\ threads per inch. Leadscrew 10 mm. pitch.
Solution • ^^^'5 = 4x11x15 ^ 2JK2 X 3 X 5 X J^
:65 9x65
= 12x55 _ 55x60
45x13 45x65*
for Screw-cutting mi Lathes. 25
To cut si threads per inch. Leadscrew 10 mm. pitch.
Solution :
11x15 _iiX30_30
5-5x65 11x65 65
To cut I in. Whitvvorth-thread = 8 threads per inch. Lead-
screw 5 mm. pitch.
In this case the leadscrew has 2 threads per cm. Conse-
quently, for this particular lathe, the numerator is 2x165
= 330 or 1 1 X 30.
c 1 -• II X30 60x55
Solution: _^ = ^ -'-'.
8 X 65 65 X 80
To cut \ in. gas thread =14 threads per inch. Leadscrew
5 mm. pitch.
Solution: ' ii^f = 30^K 5S^
14x65 65x70
To cut \ in. Whitworth -thread = 20 threads per inch.
Leadscrew 5 mm. pitch.
Solution: 11x30^ 30x55,
20x15 65x100
To cut I in. gas thread = 1 1 threads per inch. Leadscrew
6 mm. pitch. No. of threads in leadscrew per cm., y^.
Solution: ^0" X 1 1 X 15 ^ \P X 15 ^jO^S^^S
11x65 65 6x65 65
To cut 36 threads per inch. Leadscrew 4 mm. pitch.
No. of threads in the leadscrew per cm., ^ or 2*5.
Solution: 2-5 X u^^5 ^ m^x 12-5 ^ 25 x 55 .
36x65 12x65 65 X 120
To cut I thread per inch. Leadscrew 10 mm. pitch.
Solution • iiiLLi= « X II X 15 ^ 1 1 X 120 ^ 55 X 120
■ ^x65 7x65 7x65 35x65 '
26 The Calculation of Change-Wheels
To cut a \ in. pitch. Leadscrew lo mm. pitch. No. of
I 8
threads per inch •= = - .
I 7
Solution :
11x15 _ 7x11x15 _ 10 5 X n o
8^6- "' «X65 ' 65x80"'
7
To cut 3 threads per 2 in. Leadscrew 6 mm. pitch.
No. of threads per inch |, No. of threads in the leadscrew
per cm. ^.
c 1 4.- \P X II X 15 10 X II X 5 55 X 100
Solution : -2 — i = ^ — ^ = ^ _- .
I X 65 3 X 65 30 X 65
To cut 36 threads per 7 in. Leadscrew, 7 mm. pitch.
No. of threads per inch '^^. No. of threads in the leadscrew
per cm. \^.
c 1 i.- V X II X 15 10 X II X 15
Solution : -t— r^ ^ = v^ -^—-^
^5x11 ^ 50 X 55
6 X 13 60 X 65*
To cut 9*5 thread per 8 inch. Leadscrew, 10 mm. pitch.
No. of threads per inch, ^ ^ .
Solution :
Ti X 15 _ 8 X II X 15 _ no X 120
9'Lx 65 9-5 -^^ '6rx"95" '
8
To cut 25 threads per 3f in. Leadscrew, 5 mm. pitch. No. of
25 _ 100
3T~ 15
threads per inch, -| = . No. of threads in the leadscrew
per cm.
Solution:24<lL4i5 = ?X.iX.5X.5
Vs^ X 65 100 X 65
^ 55 X 90
65 X 100*
for Screw-cutting on Lathes. 27
To cut a 2\i in. pitch. Leadscrew, 10 mm. pitch. No. of
threads t)er inch, -r = - •
^ 2i 5
Solution
II X 15 _ 5 X 1 1 X 1 5 _ 1 10 X 75
I X 65 ~ 2 X 65 ~ 2Cor65^
To cut 2 threads per 6^ in. Leadscrew, 25 mm. pitch.
No. of threads per inch, JL = ^. No. of threads in the lead-
r 2
screw per cm., XT" -•
„ , ^. f X II X 15 2 X 13 X II X 15 55 X 60
Solution : -^^7 ^ — - = ~ -> - = ' .
y*3 X 65 4 X 5 X 65 40 X 25
{/) The Cutting of Metric Threads on a Lathe with
English Leadscrew.
To some extent the fourth case resembles the third. The
proportion 10 : 25-4 also holds good, though with an opposite
meaning.
Use is also made in this instance of the fact that 6*5 in. is
equivalent to 16*5 cm.
Suppose, for example, that the leadscrew has a i inch
pitch and 10 threads per cm. have to be cut, i.e. a i mm.
pitch, then, when the leadscrew has completed 6*5 revolu-
tions, the lathe spindle should have made 165 revolutions,
which can be formulated
No. of threads in the leadscrew per 6" 5 in. _ 6*5
No. of threads to be cut per 165 mm. 165
The numerator of the fraction will thus, for a given lathe,
always be equivalent to the number of threads per inch in
the leadscrew x the factor 6*5; the denominator being
equivalent to a fraction, the numerator of which is the factor
165, and the length in mm. of the thread to be cut, the
denominator.
28 The Calculation of C/iange- Wheels
For example, a 2 mm. pitch is to be cut on a lathe
having a leadscrew of 2 threads per inch, then
the numerator will be 2 x 6*5 = 13
and the denominator will be
2
For this particular lathe the numerator will always be 1 3.
The first resolvent of the fraction is a whole number
obtained from the denominator by placing the denominator
of the fraction, which is the denominator of the compound
fraction in the numerator, thus ^ ^ ■
165
No useful purpose, however, is effected by this alteration
every time. The pitch of the thread to be cut is accordingly
placed directly in the numerator, the fraction then being
definitely formulated as follows : —
Numerator = Pitch in mm. of thread to be cut x No. of
threads in the leadscrew per inch x 6*5
Denominator = 165
Attention must here be directed to the fact that whenever
the length of the thread to be cut is a fraction, it must never
be resolved into a decimal, but must always be placed in the
numerator as a vulgar fraction, so that compound fractions
may be resolvable from numerator and denominator by multi-
plication of both.
The following examples, from the simplest to the most
complicated, will make clear what has been stated above : —
To cut a screw of 5 threads per cm. Leadscrew 2 threads
per inch.
To be cut a 2 mm. pitch.
Solution: g_X2X6j_ 2x13 ^,20x65
165 11x15 75x110
To cut 33-5 mm. pitch. Leadscrew 2 threads per inch.
Solution: 3-5X13^ 35x65.
II X 15 75 X no
for Screw-cutting on Lathes. 29
To cut a screw of 3 threads per cm. Leadscrew 2 threads
per inch.
To be cut a ^ mm. pitch.
CI.- V'xiS 10x13 10x13 20x65
Solution:-^ i= ^ = — "^ = ^ .
11x15 3x11x15 11x45 45x110
To cut a screw of 8 threads per 1 1 mm. Leadscrew 2
threads per inch.
To be cut a y mm. pitch.
c 1 .• y X 13 13 20x65
Solution : -^ i =? „ "^ = ^ .
11x15 8x15 lOOX 120
To cut a screw of 5 threads per 18 mm. Leadscrew 2
threads per inch.
c 1 .- ¥xi3 13x18 6x13 30x65
Solution: ^ •'-= -^ = •<-= "^ ^ .
11x15 5x11x15 11x25 55x125
To cut a screw of 4 threads per 7 mm. Leadscrew 2^
threads per inch.
Solution: |x4x6i^ 7x13 35x65 _
IIXI5 4X2x6x11 II0XI20
To cut a 7^ mm. pitch. Leadscrew 2^ threads per inch.
Solution : Zi2i£4i^ = 5 X 1 3 ^ 50 X 65
11x15 11x8 55 x8o
To cut a loi mm. pitch. Leadscrew i thread per inch.
Solution: i^iii^i^ 21x13 _=_^i3= 35x65
11x15 4x11x15 11x20 55x100
To cut a 42 mm. pitch. Leadscrew 1 inch pitch.
Solution: 42X6- 5= 42x13 ^7^S_^70X6S
11x15 2x11x15 5x11 50x55
To cut a screw of 1 3 threads per 5 mm. Leadscrew 4
threads per inch.
c 1 ♦• Ax4x6i 2x5 20x25
Solution : ^-^ — =• = -^ = ■* .
11x15 7-5x22 75 X 110
30 The Calculation of Change-Wheels
{g) The Wheel with i2y Teeth.
In addition to the equivalent 6-5 in. = 16*5 cm., which
has been employed in the third and fourth cases, there
is still another way of cutting English thread on a lathe
with metric leadscrew, or vice versd, which is, by making
use of a wheel with 127 teeth.
The proportion between the cm. and the inch of 10:25*4
can be resolved into one of 50 : 127.
127 is not divisible further, and so, if a wheel with
127 teeth be employed, this factor can be placed either in
the numerator or the denominator.
The third and fourth cases will then resemble the first,
seeing that it is now possible to express the English thread
in mm., whether it be the threads in the leadscrew or the
threads in the screw to be cut. The fraction will thus be —
Numerator = Pitch to be cut in mm.
or
Denominator = Pitch of leadscrew in mm.
Numerator = No. of threads in leadscrew per inch
Denominator = No. of threads to be cut per cm. X 2* 54
The following examples will clearly indicate what is
meant : —
To cut a 2 mm. pitch. Leadscrew 2 threads per inch.
Leadscrew pitch 12*7 mm. : —
Numerator = 2 _ 20
Denominator = I2"7 127
The foregoing example, when worked out as per the last
comparison, will yield the same result, seeing that : —
2 mm. = 5 threads per cm.
Numerator = 2 _ 2 _ 20
Denominator = 5 X2*54 ~ 12-7 ~ 127
for Screw-cutting on Lathes, 31
To cut 3 threads per cm. Leadscrew 2 threads per inch.
Solution: ^ - ^ - -^XSO _40X5o_
3X2*54 6X1*27 6X127 60X127
or, according to first comparison,
I pitch = y> mm.
Numerator = If _ 10 __ 40x50
Denominator = 12*7 3x12-7 60x127*
To cut 7 threads per 44 mm. Leadscrew 2 threads per
inch.
Solution: -^= 44 ^40 ^55^
12-7 7x12-7 35x127
To cut a 9 mm. pitch. Leadscrew 2\ threads per inch.
Solution: ^= 9X25_ 45 X 125.
25-4 254 50x127
2-5
To cut 28 threads per 45 mm. Leadscrew 4 threads per
inch.
Solution: ^t-= 45X4,^45X50
^5-4 28X25-4 70X127
~4~
To cut I in. Whitworth-thread = 8 threads per inch. Lead-
screw 10 mm. Pitch to be cut = -^— ^mm.
8
25 4
Solution: ^_= ^SH^. = 20X .27
10 8X10 10x100
When cutting metric thread on a lathe with English lead-
screw, the wheel with 1 27 teeth is always to be found amongst
the wheels driven, whilst, when cutting English thread on a
lalhc with metric leadscrew, it is found among the drivers.
To cut 3 in. Whitworth-thread = 3^ threads per inch.
Leadscrew 10 mm. pitch.
Solution: - J5:4_^ 20X .27
3-5X10 35X100
32 The Calculation of Change- W/teels
To cut 4 in. gas thread = 1 1 threads per inch. Leadscrew
lo mm. pitch.
Solution: _^i 'i. = _22 XJ27 ,
II X 10 loox no
■J
To cut 3 threads per 8^ in. = -~t- inch pitch. Leadscrew
10 mm. pitch.
25-4
Solution
_3_
^'5 _ 25-4xS'5 _ 85X 127
10 3x10 25x60
To cut 9 threads per 11 in. Leadscrew 25 mm. pitch.
25-4
II
Solution: __9_ ^ 9X 25 -4 ^ 45 X 127
25 11X25 55x125
To cut 7 threads per 3 in. Leadscrew 7 mm. pitch.
o 1 .• 3x25-4 30X 127
bolution : — "? ^— ^ = -^ ^ .
y'x? 35x70
To cut 24 threads per 9 in. Leadscrew 5 mm. pitch.
Solution: 9X 25-4 ^_45 X-27_
24X5 50X60
{h) Method for Calcjilating Approximate Fractions.
Before commencing with the actual calculation, the question
was propounded under heading {b) on page 15 : " What change-
wheels are to be found on a lathe ? " This was indeed im-
perative, as the change- wheels actually present on the lathe
have invariably to be taken into account, first of all because
the fraction must be resolved into numbers corresponding to
the change-wheels, and then, because the same factors which
go to make up the fraction must also be found in the change-
wheels. Should the fraction contain a factor not to be met
with in the change-wheels, then, according to the methods
now in vogue, a suitable set of wheels could not be found,
for Screw-cut tiftg on Lathes. 33
consequently, the thread in question could not be cut without
obtaining one or more wheels making up the requisite factors,
which, of course, would not be possible, as a certain thread is
generally required to be cut without notice, and there is,
therefore, no chance of either making or obtaining suitable
wheels.
Will such cases often occur ? Not as a rule. The
examples already given clearly show that even in the case
of threads which vary very considerably, the wheels necessary
for cutting a true thread can be found.
In the set of change-wheels, given on page 1 5, the following
factors were found : 2, 3, 5, 7, il, 13, 17, 19, 23; the factor
23 was not met with in the second set, whilst on many lathes
the factors 17, 19, and 23 are absent.
If factors appear in the fraction composed of the thread to
be cut and the leadscrew, which cannot be found in the cliange-
wlieels^ tJien such a thread cannot be cut accurately.
If it is absolutely necessary to cut such a thread, a fraction
must be sought for which approaches the correct fraction as
nearly as possible.
Lack of knowledge of the correct method of finding out a
fraction approximating the true fraction as closely as possible,
too often results in the calculation being skipped over, and a
fraction being chosen which actually gives a thread differing
considerably from the one required.
In addition, the fact is too often lost sight of that an
approximate fraction will still result in an unserviceable thread.
Suppose, for example, a fraction is found which yields a
thread differing only 0*05 mm. from the thread of the nut to
fit which the thread has to be cut. At first sight the differ-
ence appears trifling, but the error which has been made is
really very great, so great, indeed, that the thread obtained is
wholly useless. It must of course not be forgotten that each
thread increa.ses the error, which at the end of 20 threads will
result in a difference of 20 X 0*05 mm. = i mm. Suppose,
further, that a thread has to be cut of 23 threads per inch,
2 % 20
the pitch being ^ = I— mm. With a difference of
25-4 254
34 The Calculalio7i of Change-Wheels
0*05 mm. per thread, the diflference at the end of 10 threads
will be equivalent to one-half of the thread, whilst at the end
of 23 threads, the difference will amount to the entire thread.
The foregoing example clearly demonstrates that only
fractions differing by some thousandths of a millimetre, or
some ten thousandths of an inch, can be employed.
How can such an approximate fraction be arrived at ?
Regular practice often enables one to find a fraction which
approaches very closely, without the assistance of any method.
In one of his note- books the writer found a fraction which
had been discovered, apart from any method, for the cutting
of a 3 "7 mm. thread on a lathe with a leadscrew having a
pitch of 10 mm.
For this thread there were no change-wheels, for a wheel
in which the factor 37 appears, which is indivisible, is not to
be found among an ordinary set of change-wheels.
For this reason, according to the notes in question, the
fraction -^ was chosen, for which change-wheels could be
208 ' ^
r J • 77 7x11 35 X 55
found, smce ^ = —z. = "/ « •
208 13 X 16 65 X 80
^'7 77
Seeing that the difference between ^^-^ and ^ is simply
the difference between 3*7 and 3' 701 = O'OOI mm., so that
after 10 threads the difTerence is still only O'OI mm., which
may be considered near enough for all practical purposes.
Such groping about in the dark, however, is not at all
methodical, can take a very long time, and, finally, may not
lead to any actual result.
The compound fraction, however, supplies us with a ready
means of discovering a fraction which approximates suffi-
ciently to permit the obtaining of what is practically an
accurate thread.
Suppose the fraction to consist of two numbers, the
numerator and denominator of which are both positive.
for Screw-cutting on Lathes. 35
Let these numbers be represented by A and B, and A > B.
This can then be represented
A /'
p = rti + g ^1 < B or B > ;'i.
Taking the reverse of the last-named fraction, the reduction
can then be further continued,
B ra
= ^2 + r.i < r, or /-, > ^2-
''1 ''1
Continuing further
''^ = ^3 + ''? ra < r-i or i\ > r^
r-i r^
which can be continued ad infinitum, and can thus be
expressed
'l" -2 = a,. + ^,
^n- I f'n- \
in which
r„ </-„-i or r„_, > i\.
The quotients ^i, a^.a^. . . . a„, arc termed indicators.
By substitution can be obtained
A _L I
a^ +
'1
A , I
= «i +
B ' , 1
or,
r2
A , i.
g = ^. + ^
^3 H — ?, etc.. etc.
a-x
as
«4
«B
««
«»
, I
D 2
36 The Calculation of Change-Wheels
If — — = o, then the number of terms is finite, in which
case the fraction is determinable, in that it can finally be
divided without leaving a remainder.
If the proportion — be indeterminable, and cannot con-
B
sequently be expressed by a fraction with exactness, then
there will be no end to the divisions, in which case the
number of terms of the compound fraction will be infinite.
Every indeterminable number may be regarded as the limit
of an indefinite, non-recurring fraction. The limit of a
repeating decimal fraction is a determinable proportion, e.g.
the limit of 0*3 is ^.
To apply the foregoing to a definite fraction.
(i) Given A > B, for example.
To express the fraction ^L as a compound fraction.
9976^ J . 3015
6961 6961
+ I _
2 + 93L
3015
+ I
3 + 2_^
931
4-
44- 43_
222
+ I
7
5 +
43
6-f ^
7
The indicators are thus i, 2, 3, 4, 5, 6, 7.
for Screw-cutlifig on Lathes. 37
Consequently ^^ as a compound fraction =:
I +
2 +
3 +
4 +
5 + -'-
7
(2) Given A < B, for example.
To express — ~ as a compound fraction.
355
"3 _
I _
355
113
I
355
3 +113
+ I
I
7+ .^
If „ < I, the first indicator can then be expressed by o,
in which case the indicators will be
O, ^, ] and tV;
thus
113 r
rT7 = o + as compound fraction.
3 +
I
7 +
16
(3) Express the compound fraction 4 as an ordinary
fraction. 3
2
I
4
A I
B = 4 + \ = 4i^
■+4
The CalcuUition oj Change- PVheels
a:
I
J
i.
38
(4) Express the compound fraction \ as an ordinary
fraction.
\.
i
A
B
2 +
3 +
4 +
= 3 + -2% =
= 4 + t = ^^
' '' " w
^ + 5 =
A . , 93
g IS thus = -^.
The general formula can now be expressed by putting
letters in place of the figures given in the foregoing examples.
Given the compound fraction a, determine the ordinary
fraction. ^
c
d
A
B
b +
cd+i abcd + ab + ad-\-cd+i
bcd + b + d^ bcd-vb-Vd
d bed + b + d
cd + 1 ~ cd + 1
cd + I
= b +
^ + ^
~ d
given that
a= I
abed = 24
b = 2
c = 3
- then
ab = 2
ad = 4
^=4 J
cd = 12
42 + I
= 43
= the numerator.
bed = 24.
b= 2
d= 4
30 = the denominator.
A A',
so that in this case the value of the fraction ,, =
B 30
for Screw-aitting on LatJies. 39
For any given value of a, b, c, and </, the fraction can be
immediately determined from the fraction
abcd+ab-\-ad+cd+ i
~~^bcd -^ b-^-d
To take the reverse. Given the ordinary fraction
{ab ■\- \)c-\- a
determine the compound fraction.
{ab+i)c-^a_abc + c-\-a_ c
be + I ~ be -\- I ~ be + \
= a -f = indicators.
e
The indicators are thus a, b^ and e.
Given that in the foregoing fraction the indicators have
the following value : a — 2, b = i^ c =:^ y.
Then reversing the order of things in the foregoing
example
(^A+^1^^ ± rt _ (2 X 3 4- I) 4 + 2 ^ 28 + 2 ^ 30
be+ I 3x4+1 12 + I 13*
The indicators for the fraction are thus 2, 3, and 4.
13 » 0. ^
The foregoing consequently proves : —
( 1 ) That every determitiabk fraction may be expressed as
2t. finite compound fraction.
(2) That every y?«//^ compound fraction maybe expressed
as a determinable fraction.
Compound fractions may be divided into : —
(a) Symmetric
wholly.
(*) ^^^^ { ;Sy.
40 Tlie Calculation of C/iange- IV heels
If terms and compound fraction be expressed as
— = {(h^J^^z^J^ . ... an)
then
indicators
jj = \a\, ^2, ^3. • • • • ^3. ^2. ^l)
is termed, a symmetric compound fraction because the in-
dicators end in the same order of sequence as they began ; and
^ = {ai, a-i, as, a^, a.2. a^, ^4. a^, a^, at . . . .)
is termed a periodic compound fraction, because the indicators
a.2, a-j, a^ occur periodically. In both cases the number of
terms is infinite.
T/ie Finding-out of Approximating Fractions.
Whenever the factors of a fraction, according to which a
thread is required to be cut, are not represented by the
change-wheels belonging to the lathe, it is impossible, as has
already been demonstrated above, to cut a theoretically
accurate thread, but an attempt can be made to discover a
fraction, the value of which approaches that of the real
fraction so closely that the two may be regarded as practically
identical,
Such an approximating fraction can be found by resolving
the fraction into a compound fraction, and terminating this at
the second, third, fourth, fifth, etc., indicator.
For example —
B = '^^ + :
a-i ^
«3 +
for Screw-cutting on Lathes. 41
For the first quotient substitute
a - P' - "'
then the second quotient will be
Pa _ ^ j_ 1 — ^i ^2 -)- I
the third quotient being
P3 = a, 4- -^ = M^i^a+lHi^i^ etc. etc.
Qs ^^ ,1 «3 «3 + I
^3
P P P
7T' o'' r»^ ^^^ *-^^ reduced approximated fractions, the
\l\ Wa ys
values of which are alternately greater and smaller than the
A A
value of w, and they approach more and more closely to ^p,
which may consequently be regarded as their limit.
The greater the number of indicators, the smaller the
difference between the approximating fraction and the exact
value of ^.
13
The following connection can be established between the
approxifnating fractions and the indicators : —
P3 = rf3(«irt.^ + l) +^1
Q3 = ^3 a-i + I
Pi =rti) P2 = rtirt., -|- I
Ql = I 1 Q2 = '^3
consequently
P3 _ ^3 { ax a% + i) -f gj _ az P2 + Pi
Q3 ^3 <?a -I- I «3 Qa + Qi'
consequently
P3 = «3 Pa + Pi and Q3 = ^3 Qa + Qi.
It follows, therefore, as a general rule that
P„ = rt„ P„_, + P„_3 and Q„ = rt, Q„_, + Qn-,,
and this can be applied in the following manner: —
42
The Calculation of Change-WJieels
51
(i) Given the fraction ,. Determine the compound
fraction, i.e. the indicators, and find an approximative fraction.
B 16 "^
5 +
^1 = 3
«2 = 5
^3 = 3
— = -^^ = — = \ limit approached
B Qi I ""
A
B
~ = \ = - , limit approached still closer
Q2 5 5' ^^
A ^ P3 ^ 3ji6) + 3 ^ 51 the exact value.
^ Qz 15 + 1 16'
(2) Given the fraction ^^^^— .
399
A _ 3370 _ 1
B- 399-' + 2+ — '
4 +
7 +
/P, =ai = 8 Q,= i
P,
Qi
P. 17
P2 = a,a2+i = i7 Q, = a, = 2 (^=2
P.S 76
P8 = a3(aiaa+ 1) +rt, - 76 Q3 = «, + 1=9 Q3 = ^
P4 549
P4 = «4K(«i«a+ i) + «i'j;2+ = 549 Q4 = «4(«!t'^2+ 1) +«2 = 65 Q^ = 65"
^2 = 2
«8 =4 ^
«4 = 7
^5 = 6 V Pc = ^0 («4 («a («1 «2 + + '^l «2 + 0) + ^a ('^1«2 + ') + ''1 ^ 3370\
6 X 549 + 76
Qo = «6 («4 (<*8«2 + ') + f ii) + ^sO^+l
6 X 65 + 9 =399
, . 6 ^ 3370
399
The approximating fractions are thus
8 j7 76 549 3370
I ' 2 ' 9 ' 65 ' 399 '
for Screw-cuiling on LaiJies.
43
(3) Determine the compound fraction and the approxi-
mating fractions of the number 2-718281828459.
A _ 2718281828459
B "
10'
= 24-
I +
2 4-
I +
I +
ai = 2
rta = I
a^ = I
rtft = I
«6 = 4
P, ^2
P2=2Xl-fl = 3 Qa-I
Pi = 2 Qi = I (3
P2_3
Pg = 2x3 + 2 = 8 O3 =2X1-1-1=3
P^ = I X 8 + 3 = 1 1 O4 = I X 3 -|- I = 4
Pft =1X11+8= 19 Qs =1x4 + 3 = 7
Pe H7
P6 = 4X 19+ II =87 Q6 = 4X7 + 4 = 32 Qg = 32
(4) Determine the approximating fractions for the number
7r= 3-14159265359....
P3 _ 8
Qa"3
P^^M
Q4 4
_^ P6^ 19
Qs 7
A ^ 314159265359 __
B
10^
= 3 +
7 +
15 +
I +
292 +
I +
I +
I
«i = 3
«3 = 7
«3= 15
^4 = I
^6 = 292
<?6 = I
(1-, = I
rtw = 6
Pi = 3 Qi = I
Pa = 22 Q, = 7
P3 = 333 Q3=io6
P4 = 355 Q4=ii3
P6= 103993 Q6 = 33102
etc. etc.
6
etc.
44 Tfie Calculation of Change- Wheels
The approximating fractions are, consequently,
3, 2?, 333, 355, £03993 ,j,. .j,.
I 7 io6 113 33102
From which the following can be determined : —
Axiom I. — The difference between two successive approxi-
mating fractions is, the signs not being taken into considera-
tion, equal to the unit divided by the product of its numerators ;
or, in general,
V = — — "•*• ' = ( ~ 0"
Q« Q« + 1 Q« Q» + 1
Should there also be three successive approximating
fractions,
P P P
■•■ » — I ^ n ■*■ « + I
the first will then be greater than the second, the second
being smaller than the third, etc.
Example (see page 42) :
A ^ 3370.
B 399 '
the approximating fractions are
8 17 76 549
i' 2' 9' 65
V --^ +^ -'
"~ 2 ' 18' 585
Axiom 2. — The difference between the exact value of the
fraction „ and one of the approximating fractions will in-
variably be less than the unit divided by the product of the
denominators of this approximating fraction and those follow-
ing, and also less than the unit divided by the square of the
for Screw-cutting on Lathes. 45
denominator of the fraction under consideration ; or, in
general :
A
<
(-0"
A
B "
P«
<
(-0-
(-!)"_ (-I)»
A _ P I
plica
Ltior
1 :
A _ 3370
H 399
Pi
Qi"
_ 8
I
P2
Q2
17 P3 76
2 Qa 9
P4
549
65
3VO
399 '
- I 178
- 8 < = < -
^2 399^
3370 _ 17 ^ I
399 2 18
3370 _ 17 ^ I
399 2 4
I I
18 "^4
3370 _ 17 I
399 2 < 16 ^*^'
I
2
etc.
From which it follows that in order to obtain an approxi-
mating fraction, differing only a millionth part from the exact
value, the denominator must consist of at least 4 figures.
The differences between two successive approximating
fractions become continually smaller, and are alternately
positive and negative. The difference approaches «/7, and
consequently the limit of the approximating fraction to the
exact value of „•
46 The Calculation of Change-Wheels
By interpolation another fraction can still be found between
two approximating fractions.
General term : —
■T i» (If, Lf, _ J "T r„ _ a
By taking in place of a„ the values i, 2, 3
p
(an- ,), other fractions can be interpolated between ^ — " and
p
~^, both of which form an increasing or diminishing chain,
as they both have the same sign.
P 17
(1) Required, the interpolated fractions between ^ = —
and ,* = V^^ of the fraction ,j = ^^^ (page 42).
Q4 65 B 399 *^ ^ ^ ^
^4 = 7 a„-, = 6, 5, 4, 3, 2 and i.
Q» «« Q« - X + Q« - 2
P« = 549 Q« = 65
P«-x = 76 Q«-.=9
P«-3=I7 Q«-3 = 2
P„ ^ 6 X 76 + 17 _ 473 P« ^ 3 X 7<^> + ^7 ^ 245
Q„ ~ 6 X 9 + 2 56 Qn 3x9 + 2 29
P„ ^ 5 X 76 4- 17 ^ 397 P« ^ 2 X 76 4- 17 ^ 169
Q« ~ 5 X 9 + 2 47 Q„ 2x9 + 2 20
P„ ^ 4 X 76 + 17 _ 321 P„ ^ I X 76 + 17 ^ 93
Q„ "4x9 + 2 38 Q„ 1x9 + 2 II
The fractions 93, ^69 245 321 397 473 „, ^j,^, ^e-
u' 20' 29 38' 47 56
tween the fractions - and ^j^", which are approximating
fractions of ^^'-.
399
for Screw-cntting o?i Lathes. 47
P 8
(2) Required, the interpolated fractions between ^ =
P 76
and ;^ = of the same fraction.
Q3 9
rt, = 4 rt„ _ , = 3, 2, and I
P„ ^ rt« J\ - . + P^-a P„ ^ 3 XJ7 -f _^ ^ 59
Q« rt„ Q, - . 4- Q» - 2 Q„ 3x2 + 1 7
p„ = 76 Q„ = 9
P« -. = 17 Q«-. = 2
P„ _ 2 = 8 Q„ _ 5, = I
consequently, the approximating fractions •* , -, -^, lie
p«
2x17 + 8
2x2+1
_42
5
P«
I X 17 + 8
I X 2 + I
_25
3
between and — •
I 9
P 76
(3) Required, the interpolated fractions ^ = -
Wa 9
Ps _ 3370
Qs 399 '
and
«6 = 6 tf„ _
I = 5. 4. 3. 2 and i
P» = 3370
P«-. = 549
P«-. = 76
Q„ = 399
Q»-. = ^>5
Qn-, = 9
P» _ 5 ^ 549 4- 76 _ 2821
Q„ ~ 5 X ^^5 4- 9 ~ 334
P« _ 2 X 549 4- 76 _
Qn 2 X 65 + 9
1 174
139
P« _ 4 X 549 4- 76 _ 2272
Qn 4 X 65 + 9 269
P^ _ 1 X 549 4- 76 _
Q„ 1 X 65 + 9
625
74
P» ^ 3 X 549 + 76 ^ 1^23
Qn 3 X f^5 + 9 204
• u • ..• c 4.- 625 1 174 1723 2272 , 2821
the approximating fractions ■ , — ^ . ——'', —,— and
74 139 204' 269 334
76 3^70
thus lie between — and .
9 339
48 The Calculation of Change- Wheels
Application. — Determine the compound fraction and the
approximating fractions of the number 2*539954, so as to
obtain another proportion as -^-^ or -~ for expressing the
inch in cm.
A _ 2539954 ^ I
B ~ io«
1 +
1 +
5 +
1 +
3 +
2 +
3 +
2 +
1 +
1 +
2 + A
II
The indicators are consequently : —
2, I, I, 5. I. 3. 8. 2, 3, 2, 1, I, etc.
i 2 3 5 28 33 127 1049 2225
I I I 2 II 13 50 413 876
The following and the approximating fractions can be
28 i'^7
obtained by interpolation between fractions — and " : —
Pe=i27 Q6 = 50
«6=3
1^ = 33 Q6=i3
^6 _ 1 = 2 and I
P4 = 28 Q4 = II
P« a«P„_. + P«_,
2 X 33 + 28 _ 94
Q„ «„ Q„ _ , + Q„ - 2
2 X 13 + II 37
I X 33 + 28 61
I X 13 + II 24
p •? o Pi oao
By interpolation between ^ = ^ and ^ = ^^, the
y* ^3 \ii 413
following can be obtained : —
for Screw-cutting on Lathes. 49
P« a^ P,-
.+ P«.
., _ I X 127 + 33 _ 160
Q„-a„Q„_
. + Q«-
I X 50+ 13 63
It
_ 2 X 127 + 33 _ 287
2 X 50+ 13 113
»»
_3 X 127 + 33 _4i4
3 X 50+ 13 163
^4 X 127 + 33 ^ 541
4 X 50+ 13 213
^ 5 X 127 4- 33 ^ 6^8
5 X 50 4- 1 3 263
^ 6 X 127 + 33 ^ 795
6 X 50+ 13 313
^ 7 X 127 + 33^922
7 X 50 + 13 363
so that the following approximating fractions can be found
between ^3 'and ^^ ^•^^^. 160 287 414 54i 668 795
13 413 63' 113' 163* 213' 263' 313
and 922.
363
A few Examples in Coucltision.
(l) It is required to cut 34 threads per 2}^^ in. Lead-
screw J^ inch pitch.
Pitch to be cut = ^^. Leadscrevv \ inch pitch.
Solution : —
2^1
_3L ^ 2[i X 2 ^ 43 X 2 X 2 ^ 43 ^ 43
i 3i 7 X 16 7x4 28'
No wheel with 43 teeth is to be found, and the number
43 is indivisible. It will thus be necessary to find an approxi-
mating fraction.
43 . . I
Compound fraction = ^ = i 4-
28 - ' "^ I
7 + i
50
The Calculation of Change-Wheels
a, = I
^2 = I
Indicators : — i, i, i, 7, 2
Pi = I Qi = I
P2 = I X I -I- 1 = 2 Q2 = i
«3=I>P3 =1X2+1=3 Q3 =1X1 + 1=2
«4 = 7
^5 = 2
P4 = 7X3+2 = 23 Q4 =7x2+1 = 15
Ps = 2x23 + 3 = 49 Qe = 2x15 + 2 = 32
Q3~2
P6_49
Q5~32
Interpolating between - and .
^ ^ 2 32
P» _ «« P«-i + P«-2 _ I X 23 + 3 _ 26 .
i|= 1-5357.
^ = I -5312 which is 0*0045 l^ss than the actual fraction.
32
26
Yy = ^'5294
0*0063
This difference occurs in every 2 threads, so that the
actual difference per pitch is only o 00225.
49
32
approaches most closely to these two, so that the
wheels will consequently be
49 _ 7^X_7 __ 70 X 7
32 4x8 40 X 80*
(2) Required to cut a pitch of 3*7 mm. Leadscrew
10 mm.
Solution : — .
100
There being no wheel with 37 teeth, and the number 37
being indivisible, an approximating fraction will have to be
found.
for Screw-cutting on LatJus.
37
51
Compound fraction =
100
o 4-
2 +
I +
2 +
2 +
3
Indicators are thus o, 2, i, 2, 2, i, 3.
^2 = 2
Pi = o Qi = I
P2 = O X 2 + I = I Q2 = 2
^3=1 P3 =1x1+0=1 iQa =1X2+1=3
a^ = 2
P4 = 2x1 + 1=3 :Q4 =2x3 + 2 =
«6 = 2 I \\ = 2x3+1=7 Qb =2x8 + 3= 19
«6 = 1
«» = 3
Pe = I X 7 + 3 = 10 , Qe = I X 19 + 8 = 27
P7 = 3 X 10 +7 = 37 Q, = 3 X 27 + 19 = 100
Qi I
Q2 2
Ps^I
Q3 3
Pi = 3
Q4 8
Q5 19
P«^ 10
Qe ^7
P7 ^ 37
Q7 100
10
The approximating fraction — = 3 • 704, which only differs
27
from the actual fraction by 0*004 "^"f^' P^'* thread, may thus be
accepted for all practical purposes.
JO _ 2 X 5 _ 20 X S O
27 ~ 3 X 9 ~ 45 X 60*
{j) The Proof of the Sum.
The comparison that 6*5 in. = 165 mm., or an adopted
fraction, is not perfectly accurate. Should it be desired to
find out to what extent the fraction which has been arrived
at, and, consequently, the thread to be cut, deviate, this can
E 2
52 The Calculation of Change-WJteels
be done, when a metric thread has to be cut on a lathe having
an English leadscrew, by multiplying the numerator of the
fraction by the pitch of the leadscrew in mm. The pro-
duct thus obtained should coincide with the product of the
denominator of the fraction and the pitch to be cut, i.e.
numerator X pitch of leadscrew in mm. = denominator X pitch
of thread to be cut.
Numerator, denominator and leadscrew pitch being known,
the pitch of the thread to be cut can consequently be deter-
mined.
On page 28 the fraction ^^ has been determined for
a pitch to be cut of 2 mm., and a leadscrew of 2 threads
per inch.
The product of numerator and leadscrew pitch in mm. is
thus 26 X 12-69975 or 26 X 12*7 = 330-2. This product when
divided by the denominator of the fraction will give the pitch
in mm. to be cut with the wheels determined on, thus,
330-2: 165 = 2-OOI mm. The pitch is consequently exact
to within o*ooi mm.
7 x13 ^ 91
11x20 220
is given on page 29 for a pitch of 10^ mm., with a leadscrew
of I in. pitch.
91 X25-4 2311 -4 , ^^
^ -' ^ = -^ ^ =r 10-5063 mm.
220 220
The pitch is therefore exact to within 00063 mm. Both
these differences may practically be regarded as of no conse-
quence.
In the case of a lathe having a metric leadscrew on which
English thread is to be cut, the denominator should be
multiplied by 2*54. The numerator when divided by the
product thus obtained, gives the pitch to be cut in inches.
On page 24, the fraction for cutting 6 threads per inch
with a leadscrew of 10 mm. pitch is given as^-^ .
If the denominator be multiplied by 2-54, the result will
be '-^ = '^y.
6x65x2-54 990-6
for Screw-cutting 07i Latlies.
53
Each pitch cut is thus o* 1665656 in.
The exact pitch = \ in. = 016 in., so that the thread
cut differs only by o* 0001 010 in.
Note, that when cutting metric thread with an EngHsh
leadscrew, the thread cut is a fraction too coarse, whilst, on
the contrary, when cutting English thread with a metric lead-
screw, the thread obtained is a fraction too fine.
{k) Fixing up the Wheels.
It is not always possible to fix up the 4 wheels in the order
of sequence given in the examples.
Fig. 10.
Fig. II.
The following fraction may, for example, occur
50x30
125x55'
in which case the wheels must be placed as per Fig. 10,
although the wheels 30 and 55 cannot mesh.
The fraction can, however, be arranged in another order of
54 Tlie Calculation of Change-Wheels
sequence, viz. \ ^ -^ , which makes fixing up possible (see
Fig. ii), but care must be taken that the wheels of the
numerator are never placed in the denominator, or vice versd.
Should simple changing about of the factors in numerator
and denominator, or one of them, be impossible, the fraction
is then resolved into the lowest possible factors, and another
combination of wheels sought for, which will give the same
proportion between numerator and denominator, as, for
example :
30x50 ^2x2x3x5x5x5^ 30 X 40^j. 30 X 40
55x125 5x5x5x5x11 55x100 50x110"
(/) Thread-cutting with Double Compound Train.
Should it be necessary to cut a thread considerably coarser
or finer than that of the leadscrew, it can easily happen that
the necessary wheels are lacking.
For example, to cut 56 threads per inch, leadscrew
2 threads per inch.
2 10 X 1 1
The fraction is — r = ^. A wheel with 10 teeth is
56 70x120
lacking. If the numerator and denominator of the fraction
are once again multiplied by 2, a wheel with 140 teeth is
obtained in the denominator, which is also not at hand.
In such a case, the numerator and denominator of the
fraction are resolved into 3 factors, as, for instance :
2
20 _
2x2x5 _
20 X 25 X 30
56
560
5X8X14
70X75X80
Example: To cut 48 threads per inch. Lead-screw 2
threads per inch.
Solution: A = .20 =„2X2X5 ^20x25x30^
48 480 5x8x12 60x75x80
for Screw-cutting on Lathes. 55
{in) The Cutting of Left-liand Threads.
So far, it has been implicitly taken for granted that only
right-hand threads had to be cut ; it can, however, happen,
though not often, that a left-hand thread has to be cut. For
this purpose, the leadscrew must rotate in an opposite direction
to the lathe-spindle. This is obtained by connecting up an
idle wheel at will. In double transmission, a fifth wheel (idle),
chosen at will, may also be introduced.
A number of lathes have been constructed of late which
render the connecting-up of an intermediate wheel un-
necessary. With these lathes, all that is required is to shift
the reverse-plate at the headstock which reverses the move-
ment of the pinions which drive the change-wheels, thus
causing these wheels and the leadscrew to rotate in an opposite
direction. This is a decided improvement, as there is not
much space to spare when five or six wheels are on the shear.
With a double compound train generally the larger number
are only small wheels, but with four wheels, however, every
proportion is possible, so that the placing of a fifth wheel can
sometimes be very troublesome.
56
Tfie Calculation of Change- Wheels
CHAPTER III.
THREADS AND THEIR CONSTRUCTION.
{a) Forms of Thread.
There are different forms of thread, a few of which are
illustrated in Figs. 12-15.
Fig. 12 shows the Vee thread in its general form, which is
constructed in different types, and is most often met with.
Fig. 13 illustrates the square or flat thread, the section of
which is either a square or a right-angle, and which is much
in use for larger diameters and coarser threads. In Fig. 14,
the trapezium thread is seen, the section of which is a trape-
FiGS. 12, 13, 14, 15.
zium, much in vogue for the leadscrews of lathes, the worm
being also a trapezium thread. Fig. 15 is the round thread,
formed by the intersection of semicircles.
Very little need be said with reference to the last three
types, for which it is impossible to speak of any one system,
the form of the section being dependent on circumstances,
and determined by each individual at will.
Different varieties, however, exist of the Vee thread.
{b) Types of Threads.
The type chiefly employed is certainly the Whitworth
system ; Fig. 16 shows the construction.
The depth of the Whitworth thread is equal to 0*64 of
for Screw-cutting on Lathes.
57
the pitch, the sides of the thread forming an angle of 55°
with top and bottom rounded through \ of the line h,
drawn perpendicular from the apex of the triangle to its
base, the radius of rounding being equivalent to o* 143 h.
Not only is the sectional form of the Whitworth thread
definitely fixed, but also the number of threads per inch for
Fig. 16.
h = 0.96. S
all diameters up to and including 6 inches, and this has been
fixed at from 20-2| threads per inch.
The sectional form is precisely similar for the finest as well
as the coarsest threads, and it is for this reason that the exact
dimensions and strength of the thread are determined by the
simple determination of the outside diameter.
Table I.— Whitworth Thread.
Diameter at
Bottom.
Diameter of
Thread.
No. of
Threads
Diameter at
Bottom.
Diameter of
Thread.
No. of
Threads
per inch.
in.
mm.
in.
nun.
in.
mm.
in.
mm.
i
6-35
•18
4-72
20
'1
34-92
116
29-46
6
A
7 '94
•24
6-09
18
li
38- 1
1-29
32-68
6
I
9-52
-29
736
16
If
41-27
»-37
35-28
5
i.
ti'll
•34
8-64
14
I|
44-45
1-49
37-84
5
h
12-70
•39
9-91
12
15
47*62
'*59
40-38
4i
I
15-87
•SI
12-92
"
2
50-82
1-71
43-43
Ah
J
19-05
-62
15*74
ID
2i
57*15
1-93
49-02
4
\
22-22
•73
1854
9
2i 63-5
218
55-37
4
I
25-4
•84
21-33
8
2 J 69-85
2-38
6045
3i
14
28 -57
•94
23-87
7
3 76-2
2-63
66-80
3i
«i
31-75
I 07
26*92
7
58 The Calculation of Change- Wheels
Table I. gives the various dimensions of the Whitworth
thread.
A Whitworth thread of certain dimensions can also be cut
on a considerably larger outside diameter, the exact strength
of the thread being fixed by simply determining which
dimension of the Whitworth system is required.
Table I. gives not only the outside diameter, but also the
diameter at bottom of thread, so that the height of the thread
can be arrived at by subtracting the latter from the former,
and dividing the difference by two.
When cutting threads on the lathe, which deviate in
diameter from this system, it is necessary to know the depth
of the thread both for cutting inside and outside threads.
The depth of the thread can also be arrived at by a simple
calculation.
For this purpose, just look at Fig. i6. By drawing a
perpendicular from the apex of the triangle, a right-angled
triangle is formed, the smallest angle of which is equal to
5S°4-2 = 27°3o'.
Tang. 27° 30' =0-52. Therefore, if the long side of the
right-angle = i, then the short side = 0*52, and the base of
the triangle of 55° = 1-04.
This base is, however, equal to S, i.e. the pitch.
Whence it follows that ^ : S = i : i '04, or 0'96 : i.
The real depth of the thread is, however, only | h. So
that the ratio between the depth of the thread and the
pitch is equal to f ^ : S = (0-96 X f ) : i = 0-64 : i. | A thus
equals 0*64 S.
If we take the outside diameter D, the diameter at the
bottom of the thread d, and the pitch S, then, ^ = D —
2 X o • 64 S, or </ = D — I • 28 S.
The gas thread universally adopted by the pipe trade, given
in Table II., is also according to the Whitworth system, and in
1903 was also adopted as the standard thread for pipes and
fittings for gas, water, and steam by the Association of
German Engineers, the Association of German Plumbers, the
Association of the German Central Heating Industry, and
the Union of German Pipe Manufacturers.
for Screw-cutting on Lathes.
59
On the other hand, in the autumn of 1898, an attempt was
made by a number of influential associations of Continental
engineers, assembled in congress at Zurich, and including,
amongst others, the Association of German Engineers, the
Table II, — Whitworth Screwing Thread.
Nominal
Internal
Diameter
of Pipe.
External
Diameter
of Pipe.
Diameter
at Bottom
of Thread.
Nominal
Internal
Diameter
of Pipe.
External
Diameter
of Pipe.
Diameter
at ISottom
of Thread.
■d
rt
Is
r- I.
■oJL
6
2
in.
mm.
in.
mm.
in.
mm.
in.
mm.
in.
mm.
in.
mm.
\
3-17
•382
9-71
-336
8-55
28
li
38-1
1-882
47-81
I -76s
44-85
i
6-35
-S18
13-15
-451
11-44
19
If
41-27
2-02
51-33
1-904
48-37
i
952
•656
16-67
•589
14-95
19
ij
44*45
2-047
52
1-93
49-03
4
12-7
-826
20-97
•734
18-64
14
2
50-8
2*347
59-61
2-23
56-65
1
15-87
-902
22-91
•81
20-59
14
2i
57-15
2-587
65-72
2-47
62-76
i
19-05
1-04
26-44
-949
24- II
14
2i
63-5
3-
76 23
2-882
73-27
\
22*22
1-089
30-2
1-097
27-87
14
2|
69-85
3-247
82-47
3-13
79-51
I
25-4
1-309
33-24
1*192
30-28
II
3
76-2
3-485
88-51
3-368
85-51
li
28-57
1-492
37-89
1-375
34-93
II
3i
88-9
3-912
99-36
3-795
96-39
li
31-75
1-65
41-91
1-533
38-95
II
4
loi 6
4-339
100-2
4-223
107-26
If
34-92
1-745
44-32
1-628
41-36
II
Table III.— S. I. Thread.
Diam.
Pitch.
Diameter
at Uottom
of Thread.
Diam.
Pitch.
Diameter
at Uottom
of Thread.
Diam.
Htch.
Diameter
at Bottom
of Thread.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
6
I
4-7
20
2-5
16-75
48
5
415
7
I
5-7
22
2-5
18-75
52
5
455
8
1-25
6-37
24
3
20-1
56
5-5
48-85
9
I-2S
7-37
27
3
23- 1
60
5-5
52-85
10
I 50
8-05
30
35
25-45
64
6
56 -02
II
1-50
9-05
33
3-5
28-45
68
6
6o*o2
12
'•75
9-72
36
4
30-8
72
6-5
63-55
14
2
11-4
39
4
33-8
76
6-5
67-55
16
2
13-4
42
4-5
36-15
80
7
7009
18
2-5
14-75
45
4-5
39- »5
6o
The Calculation of Change-Wheels
Swiss 'Association of Machine-Tool Makers, the Society for
the Encouragement of National Industries, etc., to replace the
Whitworth system, which is based on the English system of
measurements, by a metric thread, and it was unanimously
decided to adopt the S. I. thread (" Syst^me International "),
as per Table III.
Owing to the universal application of the Whitworth
thread, the innovation makes but little headway, though,
especially of late years, this system is being more and more
used on the Continent, especially by the Automobile Industry,
for threads cut on the lathe.
The construction and form of the S. I. thread is given in
Figs. 17 and 1 8.
The apex is an angle of 60*^. The section is consequently
an equilateral triangle.
Hence it follows that the perpendicular h, dropped from
the apex to the base, is equivalent to
\/(s'-(|J)=/-= = o-866S.
The truncation equals \ h, so that the thread has a height
of O • 75 //, or o • 6495 S.
for Screw-cutting on Lathes.
6i
Whilst the Whitworth thread bears not only at the sides
but also at the bottom, the S. I. thread, on the contrary, has
a play at the bottom of, at the most, ^^ h, equivalent to the
half truncation, the rounding of the thread is equal to the
■*"> I
•\A/%^tvVVM!
wmmmMmmm.
! I
-d ^, p p,
Fig. i8.
play, the radius of the rounding in this case being y\j h. The
rounding and play amount, as is generally accepted, to at
least .}^ h. Loevve strikes an average for this, and fixes the
play and rounding at ^ h.
The outside diameter of the male-screw is thus smaller
than the diameter at bottom of the thread in the nut, and
Fig, 19.
vice versd, the diameter at bottom of the thread of the male-
screw is smaller than the outside diameter of thread in the nut.
62
The Calculation of Change-Wheels
If we take the play a, then the actual depth of the thread
of both male-screw and nut equals 0"j^h -V a. If we fix the
play at its maximum, equals ^^ h, then the height equals
0*0625 h + 0*75 h. = 0-8125 h, or 0-703625 S = ~o-7S.
The Lowenherz thread (Table IV.) is in general use up to
Table IV. — Lowenherz Thread,
Oiam.
Pitch.
Diameter
at Bottom
of Thread.
Diam.
Pitch.
Diameter
at Bottom
of Thread.
Diam.
Pitch.
Diameter
at Bottom
of Thread.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
I
0-25
0-625
2-6
o"45
1-925
5-5
0-9
4-15
1-2
0*25
0-825
3
o'5
2-25
6
I
4-5
»'4
0-3
0-95
3*5
0-6
2-6
7
I-I
5-35
1-7
0-3S
I -175
4
0-7
2-95
8
1-2
6-2
2
0-4
1-4
4-5
0-75
3-375
9
13
7-05
2-3
0-4
1-7
5
0-8
3-8
10
1-4
79
a diameter of 10 mm. for instruments of every description,
especially in Germany and Switzerland, and in screw works,
the screws are almost exclusively made by this system.
The construction of the Lowenherz thread is shown in
Fig. 19. The apex is 53° 8'.
1
""k — r'Tr"^"
Fig. 20.
This angle results from h = S. The thread is truncated
flat on the outside diameter and at bottom with a \ truncation,
so that the real depth of the thread is = 0*75 //.
The Sellers thread (Table V.) is an American thread, con-
structed as per Fig. 20.
for Screw-cutting on Lathes.
Tahle V. — Sellers Thread.
63
Diameter,
inch.
Number of
Threads
per inch.
Diameter,
inch.
Number of
Threads
per inch.
Diameter,
inch.
Number of
Threads
per inch.
k
40
14
7
3i
3i
A
24
li
7
3i
3i
i
20
13
6
3l
3
A
18
'i
6
4
3
1
16
I|
5i
4i
A
h
14
If
5
4i
2|
\
13
IS
5
4l
2|
A
12
2
4i
5
2i
1
II
2i
4i
5i
2^
1
10
2^
4
5i
2i
i
9
2i
4
5i
2i
I
8
3
3i
6
2i
The apex is an angle of 60°, so that the perpendicular ^1
dropped from the apex to the base, is again = o*866S. The
thread is flat-faced at bottom and on the top with \ trunca-
tion, consequently
/ = I /j, and o*75 X o- 866 = 0-6495 S.
The thread which resembles the S. I. thread very much
has, however, no play and is divided according to English
measurements.
Fig. 21.
Although largely displaced by the Sellers thread, the sharp
V thread .still exists and is u.sed in America. (See Table VI.)
The section is an equilateral triangle not truncated.
64
The Calculation of Change- Wheels
The B. A. S. (British Association Standard), as per Table
VII., is an English thread much used in England for screws of
small diameter, especially for electric fittings. The apex,
Fig. 21, is an angle of 47^". The thread is truncated, and
Table VI.— Sharp V Thread.
Diameter.
Number of
Threads
per inch.
Diameter.
Number of
Threads
per inch.
Diameter.
Number of
Threads
per inch.
Diameter.
Number of
Threads
per inch.
inch.
inch.
inch.
inch.
i
20
H
10
15
5
22
4
h
18
I
9
a
4i
3
3i
§
16
\l
9
2
4i
3!^
3i
h
14
«
2h
4i
3i
3i
h
12
li
7
H
4i
31
3J
h
12
H
7
2|
4i
3i
3i
I
II
li
6
2i
4
3i
3i
\\
II
li
6
2t
4
31
3
f
10
i|
5
2|
4
3i
4
3
3
top and bottom are rounded, leaving the depth of the thread
equal to 0*6 S.
In addition to the foregoing, the Delisle, Sauvage, Acme,
and Thury systems are to be met with.
The total number of thread systems exceeds fifty, but only
the seven most used have been treated of here.
Table VII.— B.A.S. Thread.
j
Number O I
2
3
4
5
6
7
8 9
10
12
14
19
Diameter. g
mm.
S-3
4-7
41
3-64
3-2
2-8
2-5
2'2
1-9
1-7
i'3
I
0-79
Pitch. ,
mm.
0-9
o-8i
0-73
0-66
0-S9
0-53
048
0-430-39
1
1
o-3SjO'28o-23o-i9
1 i
for Screw-cutting on Lathes.
65
{c) Screw-aitting Tools.
A tool used for screw-cutting must first and foremost be
perfectly true. It is not to be looked upon as an ordinary
tool, nor may it be ground on a stone which does not run true.
When cutting deep threads, whether they be V or square,
it is always advisable to use separate tools for roughing and
finishing.
The cutting angle must be about 70°, whilst the tool must
not be pointed or semi-circular, but flattened at the edge
(Figs. 22 and 23), as otherwise the angle will not be true, and.
Fig. 22.
Fig. 23.
at the same time, it will be impossible to grind the tool
accurately. The tool must not only stand on its edge in the
angle B, Fig. 22, but the sides A A must also have clearance.
The angle in which the thread lies on the work has also to be
taken into consideration, and the line A B, Fig. 24, must run
Fig. 24.
Sh
g>
Fig. 26.
Fig. 25.
at the same angle. Suppose that a i in. pitch has to be cut
on a diameter of 2 in. Then, imagine C D, in Fig. 26, to be the
angle at which the thread lies on the work, the line A B of
F
66
The Calculation of Change-Wheels
the tool, Fig. 25, must thus run parallel to the line C D in
Fig. 26. This is still more evident in the case of square
threads with a coarse pitch, Fig. 27. In this case, the clearance
Fig. 27.
on the sides of the tool must be different. The diameter of
the thread on the top, as also the angle of the thread there
is indicated in Fig. 28, that at bottom of the thread in
Fig. 29, a and b being the circumference,
c and c the pitch, which is the same for both,
and there are consequently two angles.
The hypotenuses d and e show the angle of
the thread at top and bottom. If the
clearance of the tool is correct on the top, it
will be incorrect when at the bottom. The
steeper the pitch, the more noticeable this
will be. The tool must have more clearance
on the right-hand side for bottom than at
the top, but less on the left-hand side.
The tool must consequently be ground in
such a manner that the right-hand side will
have enough clearance at bottom of the
thread, whilst the clearance for the left-
hand side must concur with the angle at
the top, that is to say, for a right-hand thread, as in Fig. 27 ; for
left-hand threads or for internal threads the opposite conditions
will exist in regard to angles. The tool must accordingly be
tf-i
rC-A
Figs. 28 and 29.
for Screw-cutting on Lathes,
ground as indicated in Fig. 27, A B being the slope of the
right-hand side of the tool, A C on the left-hand side. The
upper cutting surface of the tool must run square on the line
A D. When cutting an inside right-hand thread, everything
is reversed, what is right-handed becoming left.
For a Vee thread, the tool must be ground in accordance
with the angle of the system of the thread. It need scarcely
be said that this must not be left only to eye or the rough
estimate of the operator. A gauge should be provided, as
Fig. 30.
per Fig. 30, giving the precise angle. And yet, notwithstand-
ing that it is far more difficult for a workman to judge an
ajigle with the eye than to guess a certain length, and no one
would ever think of permitting an operator to estimate a
certain length without using his rule, it is an exception when
the operator is provided with a suitable angle gauge.
// is utterly impossible that a thread can be true when the
operator lias judged tlie angle of tJie tool with his naked eye.
Fig. 31.
This gauge fulfils a second, and not less important,
purpose. Even though the tool be ground to the precise
angle, it is still possible to cut a wrong thread, for the tool
must be so placed in the holder that an imaginary line drawn
perpendicularly from the apex of the triangle to the imaginary
base, must also fall perpendicularly on the side of the cylinder
on which the thread is to be cut. Not having this gauge,
F 2
68
The Calculation of Change-Wheels
the operator judges with his eye the position in which he
thinks the tool should be placed. But the most experienced
workman can make a mistake, it is not possible for tlie thread
to be true. If the tool has been placed with the utmost care
in the position which might reasonably be supposed to be
correct, and this is afterwards checked with an angle gauge,
Fig. 32.
it will almost invariably be found that the position is incorrect.
The reason is that the two lines forming the angle are usually
very short in proportion to the other lines of the tool, the eyes
being consequently deceived.
In Fig. 31, at A, is shown the manner of gauging the angle
to which a lathe centre should be turned ; at B, the angle to
Fig. 33.
which a screw thread-cutting tool should be ground ; and at
C, the correctness of the angle of a screw thread already cut.
In Fig. 32, the shaft with a screw thread is supposed to be
held between the centres of a lathe. By applying the gauge
as shown at D or E, the thread tool can be set at right angles
for Screw-cutting on LatJies. 69
to the shaft, and then fastened in place by the bolts in the
tool post, thereby avoiding imperfect or leaning threads.
In Fig. 33, at F and G, the manner of setting the tool for
cutting internal threads is illustrated.
{d) Cutting t/ie Thread.
As previously stated, it is always advisable to begin
cutting a thread that has anything like a deep curve with a
roughing tool which is at a cutting point and which need not
be ground precisely to the angle.
The thread should afterwards be gone over with a
finishing tool. When engaged in cutting shallow threads, the
tool can cut on both sides at the same time, and it can be
put exactly on the direction of the shaft. With deeper
threads, i.e. quick pitches, this is no longer possible. Cutting
with both sides of the tool at the same time causes it to snap,
the thread is rough, and very often it is impossible to continue
working ; the tool should, therefore, work but one side at a
time, should frequently be set slightly in a parallel direction
to that of the shaft, and directly there is any play between
the tool and the thread, it must again be set square on the
direction of the shaft. Each time that the tool has gone
completely over the thread, it should be withdrawn and again
set in the original position at the commencement, though
increased with the amount cut at one passage.
For this purpose a graduated collar is provided to the
feed screw by means of which the traverse movement can be
read, and by which the tool can be set in the exact position
every time.
The operator formerly got out of the difficulty by marking
the position of the screw spindle with a piece of chalk.
On lathes of up-to-date construction, the graduated collar
is now always to be found on the screw spindle.
A very practical construction is shown in Fig. 34.
Advantage is here taken of the movement of the two half-
nuts when opening and closing, to withdraw the cutting tool
from the curve, and vice versd, back again to the exact
70
Tfie Calculation of Change-Wheels
position, so that instead of having to carry out various opera-
tions at the end of the thread, a simple movement of a
handle is all that is required.
The construction is as follows : Over the two half-nuts
which move under the can-iage in the same direction as the
cross-slide, and are opened and closed by a double right- and
left-hand screw, is placed a n-shaped slide fixed on knobs of
the upper portion of the half-nuts. The screw spindle of the
cross-slide fits in the upper portion of this slide on the one
side by a turned up edge, and on the other by lock-nuts.
The screw spindle must consequently follow the movement of
the slide. Holes are drilled right through the projecting
Fig. 34.
parts of the half-nuts, and the slide. A steel pin fits closely
into these holes. Oblong holes, in which the pin has play, are
bored in the carriage for same.
Before beginning to cut internal or external threads, the
pin is set in the foremost or hindmost nut, so that the half-nut
through which the pin is placed is coupled with the slide in
which the screw spindle fits, and consequently they must
follow tjic movement of the half-nut in question together
with the cross-slide and tool. It is worked as follows : As
soon as the tool has arrived at the end of the thread, the
half-nuts of the lead-screw are opened and by this means the
tool is withdrawn from the thread. The carriage is then
for Screw-cutting on Lathes.
7»
returned by hand by means of the pinion, the tool set so
much farther in with the screw spindle as it is desired to cut
deeper, and the half-nuts are closed again. This causes the
tool to resume its original position, only cutting the material
so much deeper as it has been set farther in by hand. If no
Fig. 35.
thread is to be cut, the connection between the slide and half-
nut is broken by withdrawing the locking-pin, and the slide is
coupled to the cross-slide by inserting the pin in the hole
bored through both slides.
When screw cutting, this arrangement results in a decided
I
X.
-^.
i
-^ —
Figs. 36-39.
saving of time, besides preventing the possibility of mistakes
arising from inserting the tool either too far or not far
enough in.
There should be an outlet for the tool at the end of the
72 The Calculation of Change-Wheels
thread. If the diameter is sufficiently large to permit of it,
an entire circular groove should be turned, Fig. 35. If, for
some reason or other, a circular groove is not possible, a
suitable outlet, as per Figs. 36-39, must be drilled for vee or
square threads. Before commencing cutting, the tool should
be so fixed that it will arrive just at these holes.
It was formerly the custom to return the carriage when
the tool had gone over the thread, by reversing the move-
ment of the lathe. But with the present-day construction
of the lathe, by which it is possible to return the carriage
quickly by hand by means of the handle, the half-nuts are
opened and the carriage returned by hand. If the thread being
cut is of the same pitch as, or an aliquot part of the pitch of
the leadscrew, the half-nuts can be dropped into engagement
at any point of the leadscrew without any difficulty, the tool
always returning to its precise position in the thread. This
is, however, not so when the number of threads per inch are
uneven or broken, and other means must be adopted to ensure
the tool returning to its precise position in the groove. Con-
sequently, when starting to cut the thread, a stop, or marking
line, is placed on the bed, the half-nut closed and a chalk line
drawn on top of the leadscrew, and another chalk line at the
front side of the chuck-plate. When the tool has gone over
the thread and the carriage has been returned by hand as far
as the stop or the line, the head spindle is turned round till
both chalk lines are again in their original position, the nuts
closed, and the tool is once more in its precise place in the
path which has just been cut.
This comparatively troublesome and primitive manner of
working is done away with, if the carriage is provided with a
thread indicator as shown in Fig. 40.
The following is the principle of this attachment : A small
worm-wheel runs on the leadscrew, and by means of a pinion
gearing, causes an indicator to move on a circular index-
plate. All that is now necessary is to note the position of
the indicator at the starting point, after which, the half-nuts
can be closed, and the tool will come precisely in the path
each time the indicator resumes its original position.
for Screw-cutting an Lathes.
73
{e) The Cutting of Datable or Multiple Threaded Screws.
The cutting of double or multiple threaded screws causes
a good deal of trouble, as, in addition to exercising ordinary
care that the thread cut is true, another most important point
has to be taken into consideration, viz. that the setting of the
tool is also exactly equidistant. The manner of working is
similar to that for a single thread, btit care should be taken as
far as possible that when cutting a double thread the spindle
wheel is divisible by two, and for a treble thread by three.
After the first incision has been made to the required
depth, the tool must be shifted exactly to the centre between
Fig. 40.
two threads for a double, and to one-third of the intermediate
space for a treble thread. The distance the tool is to be
shifted should, however, never be measured off, as this can
never be exact, but must be obtained by mechanical means,
either by turning the work-piece while the leadscrew is
stationary, or by turning the leadscrew while the work-piece
remains stationary. If a double thread has to be cut, one
of the teeth of the spindle wheel coming between two teeth is
marked with chalk, as also the two teeth which the tooth in
question engages. After this the spindle wheel is bisected and
this tooth is also chalked ; the spindle wheel is then released
74 The Calculation of Change- Wheels
from the wheel it engages, the spindle is given half a turn by
hand, so that the opposite tooth comes between the two
marked teeth, and the two wheels are once more engaged.
If the spindle wheel is not divisible by two, then this must be
found on the wheel on the leadscrew, but the pitch of the
thread to be cut must in this case be taken into consideration.
For example, — A double threaded screw of 4 threads per
3 inches is to be cut on a 'lathe with a leadscrew of 2 threads
per inch.
The fraction is -I- = = •
i 2 50
The spindle wheel is, however, not divisible by 2, and as
the factor 3, which is indivisible by 2, will invariably be found
in that wheel, 4 wheels are used so that the factor 3 can be
placed in the intermediate wheel.
75 _ ^00 ^ ^°
"50 ~ 50 X 80 ■
If there is any reason, for instance, with heavy lathes not to
turn the spindle but to shift the carriage by turning the lead-
screw, this is accomplished as follows for the above example : —
Pitch = f in. The carriage must thus be shifted | -H 2 =
I in. ; the leadscrew has a pitch of \ in., and so must make
f -r- i = I revolution ; the wheel of the leadscrew has 80
teeth, and consequently 80 X | = 60 teeth must be moved.
If the same pitch is to be cut on this lathe but for a three-
thread, then the first-mentioned wheels, '^- are the best to
50
use ; the wheel with 75 teeth can be divided into three, and
25 teeth turned each time.
If it is desired to move the carriage, this must be moved
I -T- 3 = |-in., the leadscrew make \ revolution, and the wheel
with 50 teeth be moved 50 X i = 25 teeth.
For example. — To cut a pitch of \\ in. Double threaded
screw. Leadscrew i in. pitch.
Solution: ii^ 15 ^100x60
I 8 40 X 80
Joy Screw-cuiiirig on Lathes. 75
For a double threaded screw, the spindle wheel is divisible
by 2.
I J -^ 2 = IjI in. The leadscrew must thus make |^ -f- I
= j-| revolution.
j-jl X 80 = 75. The wheel on the leadscrew must thus be
moved 75 teeth.
Example. — To cut 6 threads per 15 in., three-threaded
screw. Leadscrew ^ inch pitch.
Solution: 11= ii_^ = 75_X_8o^
i 6 30 X 40
For a three-thread, the spindle wheel can be divided into
3 X 25 teeth.
The carriage must be shifted ig5 _i, ^ _ 5 {^^^ ^q that the
leadscrew must make | -i- i = ^^ revolutions.
The wheel with 30 teeth is placed on the leadscrew, and
30 X y^ = 50 teeth are moved = 50 -^ 30 = i revolution and
20 teeth.
(/) The Cutting of very Coarse Thread.
When cutting coarse thread, a difficulty may possibly
occur which will require careful consideration. When the
thread to be cut is considerably coarser than that of the lead-
screw, the movement of the leadscrew must be appreciably
quickened. There is, however, a limit to this, and that is the
resistance offered by the teeth of the gear-wheels. If the
pitch is too coarse, these will break off. The extent to which
the pitch may be increased depends, naturally, entirely on
the strength of the wheels supplied with the lathe. Generally
speaking, the pitch may safely be a four-fold of the leadscrew,
anything exceeding this being attended with considerable
danger and the off-chance of the teeth breaking.
In order to permit thread to be cut which is many times
coarser than that of the leadscrew, a gearing can be attached
to the fast headstock, as illustrated in Fig. 2.
The wheel 15 can be set in connection with the small gear-
wheel of the double back gearing. If then the lathe runs with
1^
Tlie Calculation of Change-Wheels
double back gear, the ratio of speed between the cone-pulley
and lathe spindle will be i : 8, that between the wheel 1 5 and
the cone-pulley pinion 2:1, and the wheel 10 will complete
4 revolutions to i of the spindle. In the case of a thread
which is four times coarser than the leadscrew, there is a ratio
DiACHAn0fTwm6eJMS 90S-906
tuusTRATiMs me/iM/r Spamoenvtia
tXAHPU-smrriMe 6c/inOMEMO0rL£M>-
scKeir\ati'iOfv.Coujnit t wrmtuiae aifi
in '9 Hou cm 2 riiuAos Fe/t/tKH
fir^CoufVi'i ■■ 4 ■ - -
Fig. 41.
of I : I between the change-wheels, whilst for a thread eight
times as coarse, there is only a ratio of i : 2. Taking as above
that the teeth are strong enough for a ratio of i : 4, and that
the leadscrew has a ^ in. pitch, then a 4X4Xi =8 in. pitch
may easily be cut in this manner.
for Screiv- cut ting on LatJies.
77
{g) The Hendey-Norton System.
One of the newest designs for screw-cutting is that of the
Hendey-Norton system, which, by means of a train of gears
placed under and at the side of the headstock, renders it
possible to cut a number of threads of different pitches
Fig. 42.
without the necessity of fixing different change-wheels.
Change-wheels, as they have up till now been understood
in connection with the lathe, have been entirely superseded.
On a lathe provided with the Hendey-Norton system, it is no
longer necessary to fix up or take off change-wheels, the
various wheels being simply and solely geared up in the space
78
The Calculation of Change-Wlieels
formed between the spindle and the leadscrew by the shifting
of handles. The calculation of change-wheels is consequently
a thing of the past.
But, in this work which treats of the whole question of
screw-cutting in an abridged form, a description of this
system, which will certainly come more and more to the front
in the struggle for economical tools, and has already been very
largely adopted, must not be missing.
vCONE GEARS
\ .SLIP GEAR SHATT
CONEGE,i>R I"-
SHAf
TUMBLfR, AND KNOB
Fig. 43.
Arrangement of wheels in a Lodge and Shipley lathe, the fast headstock
being removed.
On a lathe of this description, screw-cutting has been
reduced to its simplest possible form. A clever workman
may, it is true, be quite capable of calculating the wheels
required to cut a certain thread quickly, and can possibly
reckon it out in his head, but even so, the actual fixing up
of the wheels seriously interferes with the steady progress of
for Screw-cutting on Lathes. 79
the work, whilst the difficulty is at once doubled whenever
turning, drilling, and thread-cutting have to be performed
periodically, as, with so many lathes, the attendant circum-
stances are such that it cannot be arranged for all at the
same time.
The lathes under discussion are constructed in such a
manner that a great variety of threads can be cut without
requiring the fixing up or taking off of a single wheel.
In the earlier constructions of this type of lathe, there was
invariably one great drawback, viz. that the number of
pitches which could be cut was comparatively small (10-12
pitches), but this number has now been extended to from
40 to 44 different pitches.
The foregoing illustration (Fig. 43) shows the complete
arrangement of the wheels.
This gives a clear view of the bed, the fast headstock
having been removed for the purpose.
The arrangement of the wheels consists of two separate
groups of wheels. The first group (9-1 1 wheels) is placed
under the headstock, the second being in a closed box
attached to one side of the lathe.
The action performed by a workman in gearing up the
wheels for the cutting of different pitches is extremely simple,
so that after a brief explanation it is sufficiently clear even to
a novice, and it can be executed so quickly that not more
than from 10-20 seconds are required to change the wheels for
another pitch than that for which they were geared up.
An index plate is affixed to the gear-box, which is given
on page 80 in its exact size.
A handle with pointer is placed under the plate. This
pointer can be moved over the entire length of the index plate
and set in the middle of either of the four divisions of the
plate. This handle is connected with the wheel indicated in
Fig. 41, by the number 862, which accordingly moves this
wheel with it ; whilst under the holes in the headstock the
numbers 1-8 or i-i i appear, according to the dimensions of
the lathe.
If, for example, it is required to cut 5^ threads per inch
8o
The Calculation oj Change-Wheels
Thds.
Knob.
Thdf.
Knob.
Thds.
Knob.
Thds.
Knob.
I8
2
9
2
4i
■ 2
2
I
19
3
9J
3
4-.'
3
2i
2
20
4
10
4
5
4
2i
4
22
5
II
5
Si
5
24'
5
23
6
IlJ
6
5^:
6
2?
6
24
7
12
7
6
7
3
7
26
8
13
8
6J
8
3}
8
28
9
14
9
7
9
3*
9
30
10
15
10
7i
10
3?
10
32
II
16
II
16
II
4
II
FEEDS.
8ot
3 40
40 to 20
20 to 10
10 to 5
Index Plate.
the pointer is placed by means of the handle in the middle of
that division in which the number in question appears under
the letters Thds. (Threads), in this particular case, in the
second division on the right hand side. On the same line on
which the number 5^ appears, the figure 6 is to be found.
The handle on the headstock is now placed in the hole above
the figure 6, and the wheels are then geared up for cutting
the desired thread. For all other threads appearing on the
index plate, the procedure is identical. The topmost handle
957 is placed in the highest or lowest position, according as it
is desired to cut left or right hand thread.
We will now proceed to give a detailed description of the
construction of this gearing.
Wheel 968 (see Fig. 42) is fast on the lathe spindle and
engages wheel 922 (Fig. 41) whenever right-hand thread is to
becut In this case wheel 923 is idle. For a left-hand thread,
wheel 968 engages 923, and wheel 922 is caused to rotate by
wheel 923, so that the direction of movement is just the
reverse to that in the first case. Both wheels run loose on
studs fastened in plate 920, and are shifted by the middle
for Screw-cutting on Lathes. 8i
handle. Wheel 922 engages wheel 955 which is fixed on
shaft 952, which is consequently brought into motion. This
same shaft 952 imparts motion to wheel 959, which, by means
of a keyway, can be moved in a transverse direction by the
handle under the fast headstock. Wheel 959 engages 961,
which can be geared up, by means of the handle already
referred to, with all the different wheels 651-659 under the
fast headstock, which wheels are all fixed on shaft 662 ; wheel
961 consequently imparting motion to the shaft. Wheels 6^
and ^"j are also keyed to shaft 662. Wheel 862 (Fig. 41)
movable by a keyway, is mounted on the leadscrew. Con-
sequently the motion of shaft 662, to which the gear-wheels
are keyed, is transmitted to wheel 862 by one of the wheels
(^ or 66^, vii two sets of double wheels 905 and 906, both of
which sets are identical.
This train of gears can be seen in the detailed drawing*
Fig. 41, to the left of the side view of the fast headstock. It
should be noted that wheels 905 and 906 are coupled, but that
each set is independent of the other, and can consequently
rotate at different speeds ; this is, moreover, apparent with the
whole train of gears, seeing that, whilst wheels 6^6 and 66"]
also coupled, and each engages one of the sets 905 and 906,
the latter obtain various speeds. This train of gears gives
four different speeds between shaft 662 and the leadscrew.
Wheel 666 engages 905 and 906 on the right. Wheel 667
engages 905 and 906 on the left.
By moving wheel 862 on the leadscrew (this wheel is
also to be seen in the illustration. Fig. 43), and by changing
handle 964, which turns on shaft 662 and to which at the
same time the two sets of wheels 905 and 906 are keyed,
wheel 862 can be placed in four different positions, i, 2, 3,
and 4. (See detailed drawing, Fig. 41.)
Wheel ^y = 906 and wheel 666 = 905 = 862. The
proportion of 667 to 906 = i : i, of 666 to 906 = 2 : i, so that
if wheel 862 engages 905 on the right, the speed of shaft 662
is doubled, seeing that ^ ^ ^ =2.
1x2
If wheel 862 engages 906 on the right, the motion of the
G
82
The Calculation of Change-Wheels
shaft is transmitted without any variation, and wheel 906 on
the right simply serves as an idle wheel. If 862 engages 905
to the left, there is a double reduction in speed ; if 862 engages
906 on the left, the diminution is four times as great. Con-
sequently, if the handle on the fast headstock is set in opening
No. 9:
With the pointer in column i, 3J pitches per inch
»» >» 2, 7 ,, „
»» >» 4» 28 „ ,,
will be cut.
In this manner, with 1 1 wheels on shaft 662, 44 different
pitches can be cut.
Fig. 44.
Fig. 45,
The swing plate of the fast headstock is further so con-
structed that, by setting up one wheel, the speed of the lead-
screw can once more be doubled, or by removing the same
wheel, it can be reduced to half as slow again, so that all the
threads appearing on the index table can now be cut, with
double or half the number per inch. The reserve hole in the
swing plate can be clearly seen in Fig. 41, close to 923.
In the foregoing illustrations. Fig. 44 gives the combina-
tion for fine threads. Fig. 45 for coarse threads, whilst Fig. 46
shows the position of the wheel 955.
for Screw-cutting on Lathes.
83
The usual gearing is : Wheel 968 engages 922, and 922
engages 955, the wheel on shaft 952. For fine threads, 968
engages 922, and 922 engages 923, consequently 923 engages
924. which is a double wheel with 925, the proportion between
them being i : 2. Finally, 923 engages 955. Wheel 955
does not engage 922, but is moved a wheel's width to one side.
(See Fig. 46.)
posmaiforfmt/faiD C£A/i w//£//
CVrr/MS EXTKA TH/fEADS.
T/XDOTTEO LW£S SMOW THE
POS/r/<M/ wMEAf carr/A/e the
ttr-VifJUi THHCAOS OF IHOIX
Fig. 46.
For coarse threads, 968 engages 922, and 922 engages 925,
consequently 924 engages 923 and 923 engages 955. From
a careful consideration of these two combinations for fine and
coarse threads, it will be seen that wheels 924 and 925 on
the one side, and wheel 923 on the other side, are mutually
interchanged for the two cases.
So far it has only been multiples or fractions of an inch,
or both, which could be cut in this manner. Should it, how-
ever, be necessary to deviate herefrom for any special pitch,
other threads than those of the English system can also be
cut by a certain proportion between the two wheels 924 and
925.
LONDON :
PKINTED BY WILLIAM CLOWKS AND SONS, LIMITED, GREAT WINDMILL STREET, W,
ANti DUKB STKKKl, STAMFORIl STREET, S.B^
THE SOUTHWARK
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PATTERNS CAN BE SUPPLIED TO DRAWINGS.
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(NOT TOYS)
Correct to Scale, Manufactured throughout on the Premises.
REPAIRS to all kinds of Models (Engines, LocomotlTCS,
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OUR SPECIALITIES ARE
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MARINE ENGINES, &c., TO ALL SCALES
Illustrated Catalogues of Specialities, 2s. each.
INVENTORS' IDEAS & DESIGNS worked out and advised upon
by Expert Mechanicians. Estimates given.
Personal Inspection Cordially Invited.
3
The Phosphor -Bronze Go.
LIMITED
87 SUMNER STREET, SOUTHWARK
Sole Makers of the following ALLOYS:
PHOSPHOR BRONZE ("Cog Wheel Brand" and
" Vulcan "). Ingots, Castings, Rolled Plates, Strip, Bars and Rods
"DURO METAL" {^'^^r^) ALLOYS A, B & C.
A Hard Bronze for Roll Bearings, Wagon Brasses, &c.
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(" Cog Wheel " Brand). The best qualities made.
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Brand). Ingots and Castings.
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PLASTIC METAL ("Vulcan" Brand).
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.Superior to any White Brass.
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CASTINGS in Bronze, Gun Metal, Brass & Aluminium
Branch Foundry :
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4