Improvements in Toothed and other Gearing.
Year: 1891

page_text: Date of Application, 24th Nov., 1891
Complete Specification Left, 22nd Aug., 1892-Accepted, 29th Oct., 1892
PROVISIONAL SPECIFICATION.
Improvements in Toothed and other Gearing.
We AUGUSTE DESCOFFE and LEONIDAS ALEXANDRE DI GIORGIO both of
Lille, France, Engineers, do hereby declare the nature of this invention to be as
follows:-
Our invention consists in the adoption in the setting out of gear teeth of homo-
logous tangential spiroidal curves of reverse curvature, whereby an uninterrupted
action is obtained without friction or shock, and consequently without loss of
work.
The spiral adopted for our mode of setting out gearing is the development on a
plane surface of a helicoidal curve drawn on the generatrix of a cylinder, the pitch
of which is equal to six times its radius, say an inclination of 43° 40′ 20″ to the
plane perpendicular to the axis, that is to say that 60° traversed by the circum-
ference of this helix around its axis correspond to a movement of translation
parallel to this axis equal to the length of a radius of the helix.
The radius of the cylinder on which the helix is set out being OB Figure 1, we
take a disc having a plane surface and a radius equal to 3 x OB = OA; then we
cause a point to advance upon this surface with a uniform rectilinear movement
from A to B and at the same time we impart to the circumference of the disc a
uniform rotary movement around its axis for the length of an arc of 60°, say
from A to D. The point will thus trace on this disc the curve AC which
represents the development on a plane surface of the helicoidal curve proposed,
which serves as a basis for the construction of the curve of the teeth according to
our system.
Our toothed wheels are chiefly distinguished from all others by the appearance
of their teeth, the very wide base and the narrow apex of which give to them the
form of a spiroidal cone so that the material can always work in compression and
never in tension. Further the height of these teeth having for their limit half the
radius of the wheel, and the number of these teeth being reduced to five, allow
of establishing multiple transmission of motion while preserving very practical
dimensions of the teeth on small diameters.
Figure 2 represents part of a wheel having five teeth being the minimum number
with the maximum height, which is half the radius of the wheel.
From the point O as a centre we describe the following circles,―
A, which is the apex of the teeth;
9, which is the base of the same teeth, Oy being
=
AO
2 and
ΔΟ
m, which is the constructive circle, Om being = 40.
The circumference A will be divided into 5 equal parts representing arcs of 72°,
and each of these arcs will be subdivided into segments of 30°, then 12º,
then 30° 72°; each of the points thus obtained is joined to the centre, thereby
forming the intersecting points bede. Then from the points A. c d we draw arcs
of a circle of the radius Am, and finally from the points of intersection of these
arcs which are the centres of the curve we draw the two sides of the tooth c A. d.
We repeat this setting out of the five arcs of a circle having the radius Am for
each of the teeth.
It will be understood that the curves thus obtained being arcs of a circle represent
only approximately parts of spirals, and that they have not the precision which is

page_text: We can obtain greater approximation by setting out the teeth with curves of
more than one centre.
For a given number n of teeth the arc occupied by each of them is equal
360°
to of the circle, and the distance between the teeth at their base is equal to
71
one-sixth of this arc.
Our mode of forming toothed gearing is applicable to all arrangements for
transmitting or converting motion, whether by spur gearing of all kinds, or by
racks, or bevel gearing, or worm wheels and worms, or chain wheels and chains, or
by other apparatus in which teeth are employed.
SPUR GEARING.
In order to construct according to our invention a spur wheel and pinion having
for example a relation of 1 to 3 and adopting the maximum height for the teeth
at a distance from the centres of 18 centimetres, the original radii are 4.5 centi-
metres for the pinion and 13.5 centimetres for the wheel; and the number of
teeth is 5 for the pinion and 15 for the wheel. The height of the teeth of the
pinion will be 3 centimetres, the outer radius 6 centimetres, the inner radius
360°
3 centimetres. The arc occupied by each tooth will be
72° of which
5
theiz. 12° is for the space between the teeth at their bases. For the wheel the
height of the teeth will be 3 centimetres; the outer radius 15 centimetres; the
inner radius 12 centimetres. The arc occupied by each tooth will be
giving
2.4°
6"
=
4° for the space between the teeth at their base.
=
360°
15
= 24º,
Under the same conditions of distance of the centres and of relation between
the original radii we can arrange spur gearing having 4 times the number or any
other desired number of teeth as will be well understood.
The contact between the teeth of the wheel and pinion takes place theoretically
without break of continuity, but as in practice one cannot calculate upon an
absolutely perfect construction it is well to adopt one of the following arrange-
ments, which will make good any imperfections which may be produced in the
construction of the gearing,-
1st. We can divide into two the width of the teeth of the same wheel so as to
form two half wheels juxtaposed, the teeth of one half being placed in the middle
of the space between the teeth of the other.
2nd. We can obtain the same result by making the teeth oblique in such
manner that one of the faces projects over and covers half the space between
the teeth.
3rd. We can also have this obliquity double and in contrary directions so as to
form a V.
RACK AND SPUR WHEEL.
As a wheel for gearing with a rack has spur teeth we construct it in the manner
above explained, but as regards the rack it should be considered as a wheel the
centre of which would be situated at an infinite distance; consequently the spiral
loses its curve and becomes a straight line connecting the base to the apex of the
tooth. The width of this tooth is the development of the arc of the original circle
comprised between two teeth of the wheel; its height equals that of the teeth of
this wheel and the space between the base of the rack teeth is equal to one-sixth of
the distance between the teeth.
For example the original radius of the wheel being 7.5 centimetres, the
development of the original circle is 47.124 centimetres, the number of teeth
being 10.

page_text: BEVEL GEARING.
For bevel gearing the setting out of the teeth is effected on the inclined outer
and inner faces of the wheels and pinions under the same conditions as for spur
gearing, the elementary rules adopted for ordinary bevel gearing being observed.
WHEEL AND WORM OR ENDLESS SCREW.
Seeing that a helix of any radius and pitch can be developed as hereinbefore
described as an equivalent spiral on a plane surface of a radius perpendicular to its
axis, the spiral can also be formed in a helicoidal curve drawn on the generatrix of
a cylinder cutting this plane at a right angle. We have thus all the elements
which enable us to construct according to our system a worm wheel and a worm or
endless screw under conditions other than those hitherto observed. For this
purpose we form on the planes perpendicular to the axes of two cylinders of any
radii, the lines of two spur wheels on our system. Then we cut out on these
cylinders as many helical threads as there are teeth by giving to these helices
half the pitch which the two spirals respectively form at the apex of the teeth,
and by preserving in the helicoidal hollows the profile of the lines of the teeth.
If on one of these cylinders thus formed we cut a section, in the direction
perpendicular to the axis, of a width suitable for the resistance to be overcome,
and we then couple it with the other cylindrical helix so that the axes of the two
cylinders are parallel, the portion of this cylinder will form the toothed wheel
which gearing exactly with the helix will act identically the same as spur gearing
with teeth formed according to our system, that is to say without friction or
shock and consequently without loss of work.
As regards chain wheels and chains we will remark that a chain gearing with
two wheels should be considered either as a wheel common to these two wheels or
as a rack likewise common to them. In the first case the setting out of the teeth
in each of the links is the same as that of the teeth of the wheels, but with the
contrary curvature. In the second case the setting out of the teeth of the links
conforms to that of the teeth of a rack as already described. In both cases the
length of the links, that is to say the length comprised between two consecutive
joints is equal to the chord which connects the axes of the teeth on their original
radius; and the inner and outer curves of the links coincide with the arcs of the
inner and outer circles of the teeth of the small wheel.
Dated this 24th dow of November 1801
J. C. MEWBURN,
55 & 56, Chancery Lane, London, Chartered Patent Agent.
COMPLETE SPECIFICATION.
Improvements in Toothed and other Gearing.
We AUGUSTE DESCOFFE and LEONIDAS ALEXANDRE DI GIORGIO, both of
Lille, France, Engineers, do hereby declare the nature of this invention and in what
manner the same is to be performed, to be particularly described and ascertained
in and by the following statement :-
Our invention consists in the adoption in the setting out of gear teeth of
homologous tangential spiroidal curves of reverse curvature, whereby an uninterrupted

page_text: action is obtained without friction or shock, and consequently without loss of work.
The spiral adopted for our mode of setting out gearing is the development on a
plane surface of a helicoidal curve drawn on the generatrix of a cylinder. If we
take a cylindrical helix the pitch of which is equal to its circumference, its develop-
ment on a plane surface is obtained as follows:-
=
The radius of the cylinder on which the helix is set out being OB, Figure 1, we
take a disc having a plane surface and a radius equal to 3x OB = OA of which
AB 2× OB; then we cause a point to advance upon this surface with a uniform
rectilinear movement from A to B and at the same time we impart to the circum-
ference of the disc a uniform rotary movement around its axis for the length of an
are equal to the radius OA, say from A to D. The point will thus trace on this
disc the curve AC which represents the development on a plane surface of the
helicoidal curve proposed. The construction of the curve of the teeth according to
our system is based upon the line of this curve, a portion of same, determined by
the rules and formula hereinafter given, being taken.
The height of the teeth beyond the radius is for the minimum number of teeth,
viz. 5 taken as equal to the development of the arc forming the base of the tooth
on the internal circle, and the space between two tooth bases is constant and equal
to th of the are comprised between two tooth axes on the internal circle of the
wheel. Under these conditions the number of degrees appertaining to the base of
each of the 5 teeth being.
64°, the length of the arc corresponding
72°
360°
5
=72°-
=
9
to this angle at the centre =
64°
180
of the mean radius = 1·117 +
1-117
Re = 1.5585 +
=
× T=1117 (1 being the internal radius Ri; that
1·117
2
1.5585 and that of the external radius
I
2.117 consequently the relation between the height of the
tooth and the mean or pitch radius Rp is
height and the external radius
=0.716717 x Rp.
1
1.5585
1.117
2.117
1.39525, and that between the
-
1-89525; therefore h
Rp
1.117
1.39525
Rp
Now as the arcs are in proportion to the number of degrees, and as these are in
relation to the number of teeth the height h for a number n of teeth h = x 139525
=
Rp
or P
27 Rp
n
5
n x 0.27905,
(P being the pitch). The relation between the
pitch and the height will be = 2π; 0.716717 x n; this relation being constant
whatever be the number of teeth n we have h
1
P
1.75333
=0.570343 × P.
Our toothed wheels are chiefly distinguished from others by the appearance of
their teeth, the very wide base and the narrow summit of which give to them the
form of an ogive so that the material can always work in compression and never in
tension. They are practically unbreakable. Further as these teeth can be given
a great height which can in some cases exceed half the radius of the wheel, and as
the number of the teeth can be reduced to 5, transmissions of motion can be
established while preserving very practical dimensions of the teeth on small
diameters.
Figure 2 represents part of a pinion having 5 teeth the height of which
gA
From the point as a centre we describe the following circles :-
OA
1.89525.
A, which is the summit of the teeth,
y, which is the base of the same teeth, Og being = OA – gA ; and

page_text: m, which is the constructive circle, Om being
=
The circumference A will be first divided into 5 equal parts representing arcs
of 72° and each of these arcs will be subdivided into segments of 32°, then 8°,
then 32° 72°; each of the points thus obtained is joined to the centre, thereby
forming the intersecting points b, c, d, e. Then from the points A, c and d we draw
arcs of a circle of the radius Am and finally from the points of intersection of these
arcs which are the centres of the curve we draw the two flanks or sides of the
tooth c A. d. We repeat this setting out of the 5 sides of a circle having the
radius Am for each of the teeth.
It will be understood that the curves thus obtained being arcs of a circle represent
only approximately parts of spirals and that they have not the precision which is
obtained by setting out mechanically as in Figure 1, but for teeth of sinall height
they are sufficiently practical for use.
We can obtain greater approximation by setting out the teeth with curves of
more than one centre as indicated in Figure 3.
Our mode of forming toothed gearing is applicable to all arrangements for
transmitting or converting motion, whether by spur gearing of all kinds, or by racks
or bevel gearing, or worm wheels and worms, or chain wheels and chains, or by
other apparatus in which teeth are employed.
SPUR GEARING.
In order to construct according to our invention a spur wheel and pinion having
for example a relation of 3 to 1 and adopting the maximum height for the teeth and
a distance of the centres of 0.16743 m. (see Figure 4). The primitive or pitch
radii being 0.0418575 m. for the pinion and 0·1255725 m. for the wheel, and the
In order to construct according to our invention a spur wheel and pinion having
for example a relation of 3 to 1 and adopting the maximum height for the teeth and
a distance of the centres of 0.16743 m. (see Figure 4). The primitive or pitch
radii being 00418575 m. for the pinion and 0.1255725 m. for the wheel, and the
number of teeth for the pinion and 5 for the wheel, we get by the application of
the formulæ hercinbefore given the following:

page_text: External radius Re: = 0.1255725 m. +
0.03 m.
2
= 0.1405725 m.
Are between the axes of two teeth
360°
15
= 24°.
Arc occupied by the space between the bases of two teeth
24°
==
= 2-666° 2°40'.
=
9
Under the same conditions of distance of the centres and of relation between the
primitive or pitch radii we can arrange spur gearing having four times the number
or any other desired number of teeth as will be well understood. Figure 5 shows
spur gearing with four times the number of teeth, and in this case the measurements
will be as follows:-
"
The contact beween the teeth of the wheel and pinion takes place theoretically
without break of continuity, but as in practice one cannot calculate upon an
absolutely perfect construction it is well to adopt one of the following arrangements
which will inake good any imperfections which may be produced in the construction
of the gearing:
1st. We can divide into two the width of the teeth of the same wheel so as to
form 2 half wheels juxtaposed as shown in Figure 6: the teeth of one half being
placed opposite the middle of the space between the teeth of the other.
2nd. We can obtain the same result by making the teeth oblique as shown in
Figure 7 in such manner that one of the faces projects over and covers half the
space between the teeth.
3rd. We can also have this obliquity double and in contrary directions so as to
form a Vas shown in Figure 8.
RACK AND SPUR WHEEL.
A rack can be considered as part of a toothed wheel the radius of which is
infinitely great and the centre situated at an infinite distance. Under these
conditions the internal, pitch and external circles become straight lines tangential to
the external, pitch and internal circles of the wheel. In like manner the curve
centres being at an infinite distance, the teeth lose their curve to become straight
lines and are thus transformed into a rack with triangular teeth the height of
which is equal to the height of the wheel with which it gears, and the base equal
to the development of the pitch P, less th for the space left between two consecutive
tooth bases.
For example (see Figure 9) the pitch radius of the pinion being 0.075 m. the
development of the pitch circle is 0-47124 m., the number of teeth being 10%; the

page_text: distance of the rack teeth will be
=
0·047124 m., of which th
10
= 0.005236 m. is for the interval between the bases of the teeth and 0.047124 m.
0-005236 m. 0.041888 m. for the body of the tooth.
BEVEL GEARING.
-
For bevel gearing the setting out of the teeth is effected on the inclined external
and internal faces of the wheels and pinions under the same conditions as for spur
gearing, the elementary rules adopted for ordinary bevel gearing being observed.
Suppose for example it be required to construct according to our invention a bevel
wheel and pinion the radii of which are as 2 to 1,-the pitch radius of the pinion
being km (see Figure 10), the height of the tooth be, the number of teeth 8, and the
generatrix of the cone oa of this pinion being equal to that of the wheel, the pitch
radius of the wheel will be 2 x kman. With these data we obtain according to
the ordinary rules the respective positions of the two wheels with their supple-
mentary cones kha and ega of the pinion and ajl and a'it of the wheel. With the
pitch radius ha of the supplementary cone of the pinion describe an arc of the
length A to B equal to that of the circumference of the circle of the radius km;
similarly with the pitch radius aj of the supplementary cone of the wheel describe
an are of the length C to D equal to that of the circumference of the circle of the
radius an; adding the half height of a tooth to each of these sectors draw the
profile of the teeth as explained for spur gearing, then apply it to the respective
external supplementary cones; proceed in the same manner for the profile of the
teeth on the internal supplementary cones pac' and big. Bevel gearing can also be
constructed with their axes out of the perpendicular or prolonged on different
planes up to an angle of 90°, but then they become like worm or screw gearing.
WHEELS AND WORM OR ENDLESS SCREW.
We establish in the usual way the axes of the worm or endless screw and of the
worm wheel at right angles by giving to the teeth of the wheel an inclination
parallel to that of the plane which is tangential at a point of the helix on the
generatrices of the cylinder, but adopting both for the helicoidal threads and for
the teeth of the wheel the profile obtained by the mode of setting out hereinbefore
described for spur gearing.
As regards chain wheels and chains we will remark that a chain gearing with two
wheels should be considered either as a pinion or wheel common to these two wheels
or as a rack likewise common to them. In the first case the setting out of the teeth
in each of the links is the same as that of the teeth of the small wheel, but with the
contrary curvature. In the second case the setting out of the teeth of the links
conforms to that of the teeth of a rack as already described. In both cases the
length of the links, that is to say the length comprised between two consecutive
joints, is equal to the cord which connects the axes of the teeth on their primitive
or pitch radius, and the inner and outer curves of the links coincide with the arcs of
the inner and outer circles of the teeth of the small wheel. Under these conditions
for a ratio of transmission m between the pinion and the wheel, the pitch radius of
the pinion being and the number of teeth being n the pitch radius of the
abowl of [360°7

page_text: from a cylindrical helix the pitch of which is equal to its circumference, which will
will be equivalent to an inclination of 45° to the plane perpendicular to the axis, we
do not limit ourselves to this precise pitch, although it must be very little departed
from. We can for example have a cylindrical helix the pitch of which is equal
to 6 times the radius of the cylinder, equivalent to an inclination of 43° 41' 20" to
the plane perpendicular to the axis. In this case the interval between the teeth
should be th of the distance between the tooth centres.
Having now particularly described and ascertained the nature of our said inven-
tion and in what manner the same is to be performed we declare that what we
claim is :-
1. Toothed and other gearing the teeth or equivalent parts of which have spiroidal
curves obtained in the manner hereinbefore set forth.
2. The construction of toothed gearing as hereinbefore described and shown in
the accompanying drawings.
Dated this 11th day of August 1892.
J. C. MEWBURN,
55 & 56, Chancery Lane, London, Chartered Patent Agent.