DOI Page / Citation link:
https://doi.org/10.11588/diglit.25888#0073
DRAWING OF MACHINERY.
57
trundle. The trundle or lantern wheel is constructed by
inserting the extremities of a certain number of cylin-
drical pieces, called staves, into equidistant holes formed
near the circumference of two parallel plates or discs;
through the centres of which the axis of rotation passes
perpendicularly. In this kind of gearing, in which timber
is usually the material employed, it is invariably the
wheel to be driven that is made in the form of a trundle;
but such rude contrivances are now almost obsolete,
being superseded by the invention of bevil gear, which
admits of much greater accuracy, strength, and elegance.
Examples of this kind of gearing have been admitted into
this work mainly with the view of elucidating the prin-
ciples on which teeth of the more modern fashion should
be constructed.
Internal gearing, in which the pinion is situated
within the circumference of the wheel, the teeth of which
are consequently formed on the interior of the rim.
Back gearing. This form of gearing is employed to con-
vert a rotatory into a rectilinear motion, or vice versa.
In this arrangement, a straight bar, armed with teeth set
perpendicularly, is substituted for the wheel, and gears
with the pinion in the same manner as if it were a wheel
of an infinite diameter.
The tangent screw and worm wheel, in which the teeth
of the wheel are formed to engage with the threads of a
screw, which is invariably the driver.
Bevil gearing, strictly so called, which consists of toothed
wheels formed to work together in different planes, their
teeth being disposed at an angle to the planes of their
faces.
Face or crown wheels are such as have their teeth per-
pendicular to the plane of their faces, which, in the case of
trundle gearing, are parallel with the axis of the trundles.
The curves which theory prescribes, and to which it
should be the aim of practical men to approximate as nearly
as possible, in the formation of the teeth of every descrip-
tion of wheels intended to work together, are the Cycloid,
the Epicycloid, and the Involute. Of the last we have
already explained the nature, and exhibited the geometri-
cal and mechanical methods of construction; we shall now
proceed to the consideration of the two former, which,
particularly the epicycloid, have a more direct and fre-
quent application to the present subject.
Construction of the Cycloid and Epicycloid.—■
Plate XX.
The Cycloid is the curve described by a point in the
circumference of a circle which rolls, without sliding,
upon a straight line. The major axis of the cycloid is, of
course, equal to the circumference of the generating circle.
Fig. 1.—To construct the cycloid described by a point A
in a given circle A B D, tangent to the straight line
M N, supposing the circle to roll upon the straight line.
Throughout the motion of the generating circle upon
the line M N, it is obvious that its centre C moves
simply in a straight line C C5, parallel to the former.
Now, divide the circle A B D 4 into any number of
equal parts, 1, 2, 3, &c., and set off in succession from A,
upon the given straight line a number of points whose
common distance is equal to the length of one of these
parts (supposed to be reduced to a straight line). Then
through each of these points F, 2', 3', &c., draw perpen-
diculars cutting the straight line C C5; and through each
of the points 1, 2, 3, &c., in the circle, draw horizontal
lines indefinitely; by this construction we shall easily
be enabled to determine a number of points in the curve
sought.
For if we conceive the centre C of the generating circle
to have arrived at O', the point 1 will then coincide with
1' on the line M N, and the point A, being raised to an
elevation equal to the height of the first division, will be
in the horizontal passing through 1; it is still, however,
in the circumference of the given circle, whose centre has
now assumed the position C': therefore, by describing
from this last point a circle with the radius C A, its inter-
section with the line 1 A will be one point in the cycloid.
Further, supposing the circle to have rolled through a
space equal to another of the divisions, so that its centre
is at O', the point 2 will now be at 2' on M N, and the
point A will have risen to the horizontal b 2; hence a
circle described from the point O with the radius A C
will cut that line in a point A2, which is a second point in
the curve required. By a similar method any number of
additional points A3, A4, &c., may be obtained. It is
almost unnecessary to remark that the arcs A' 1', A' 2',
&c., when resolved into their equivalent straight lines,
are respectively equal to the distances A f, A 2', &c.
Another Method.—After having drawn horizontals
through all the primary divisions of the circle A B D 4,
and intersected each by a vertical line drawn through
each division on the line M N, take up in the compasses
the distance 2 b, for instance, from the point 2 to the line
A D, and set it off from d to A2; lay off similarly the
distance c 3 from e to A3, &c. These points may also be
obtained by transferring the distances 2d, 3 e, &c. from b,
to A2, from c to A3, and so on.
By this second method we have constructed the cycloid
A E, where we have supposed the generating circle to roll
upon M N from right to left. We have also introduced
into this figure a combination of two equal and similar
portions of cycloids, for the purpose of exhibiting the form
which should be given to the teeth of a rack intended to
drive a pinion.
The Epicycloid is the curve described by a point in the
circumference of a circle which rolls without sliding, upon
another circle, supposed to be at rest. The former is
called the generating, and the latter the fundamental
circle, and that part of its circumference upon which the
epicycloid rests is called the base.
When the generating circle rolls on the convex circum-
ference of the fundamental circle, the curve is termed an
exterior epicycloid; if it move within, or on the concave
circumference, it is called an interior epicycloid.
This is the curve which determines the true form of the
teeth of wheels working into each other.
H
57
trundle. The trundle or lantern wheel is constructed by
inserting the extremities of a certain number of cylin-
drical pieces, called staves, into equidistant holes formed
near the circumference of two parallel plates or discs;
through the centres of which the axis of rotation passes
perpendicularly. In this kind of gearing, in which timber
is usually the material employed, it is invariably the
wheel to be driven that is made in the form of a trundle;
but such rude contrivances are now almost obsolete,
being superseded by the invention of bevil gear, which
admits of much greater accuracy, strength, and elegance.
Examples of this kind of gearing have been admitted into
this work mainly with the view of elucidating the prin-
ciples on which teeth of the more modern fashion should
be constructed.
Internal gearing, in which the pinion is situated
within the circumference of the wheel, the teeth of which
are consequently formed on the interior of the rim.
Back gearing. This form of gearing is employed to con-
vert a rotatory into a rectilinear motion, or vice versa.
In this arrangement, a straight bar, armed with teeth set
perpendicularly, is substituted for the wheel, and gears
with the pinion in the same manner as if it were a wheel
of an infinite diameter.
The tangent screw and worm wheel, in which the teeth
of the wheel are formed to engage with the threads of a
screw, which is invariably the driver.
Bevil gearing, strictly so called, which consists of toothed
wheels formed to work together in different planes, their
teeth being disposed at an angle to the planes of their
faces.
Face or crown wheels are such as have their teeth per-
pendicular to the plane of their faces, which, in the case of
trundle gearing, are parallel with the axis of the trundles.
The curves which theory prescribes, and to which it
should be the aim of practical men to approximate as nearly
as possible, in the formation of the teeth of every descrip-
tion of wheels intended to work together, are the Cycloid,
the Epicycloid, and the Involute. Of the last we have
already explained the nature, and exhibited the geometri-
cal and mechanical methods of construction; we shall now
proceed to the consideration of the two former, which,
particularly the epicycloid, have a more direct and fre-
quent application to the present subject.
Construction of the Cycloid and Epicycloid.—■
Plate XX.
The Cycloid is the curve described by a point in the
circumference of a circle which rolls, without sliding,
upon a straight line. The major axis of the cycloid is, of
course, equal to the circumference of the generating circle.
Fig. 1.—To construct the cycloid described by a point A
in a given circle A B D, tangent to the straight line
M N, supposing the circle to roll upon the straight line.
Throughout the motion of the generating circle upon
the line M N, it is obvious that its centre C moves
simply in a straight line C C5, parallel to the former.
Now, divide the circle A B D 4 into any number of
equal parts, 1, 2, 3, &c., and set off in succession from A,
upon the given straight line a number of points whose
common distance is equal to the length of one of these
parts (supposed to be reduced to a straight line). Then
through each of these points F, 2', 3', &c., draw perpen-
diculars cutting the straight line C C5; and through each
of the points 1, 2, 3, &c., in the circle, draw horizontal
lines indefinitely; by this construction we shall easily
be enabled to determine a number of points in the curve
sought.
For if we conceive the centre C of the generating circle
to have arrived at O', the point 1 will then coincide with
1' on the line M N, and the point A, being raised to an
elevation equal to the height of the first division, will be
in the horizontal passing through 1; it is still, however,
in the circumference of the given circle, whose centre has
now assumed the position C': therefore, by describing
from this last point a circle with the radius C A, its inter-
section with the line 1 A will be one point in the cycloid.
Further, supposing the circle to have rolled through a
space equal to another of the divisions, so that its centre
is at O', the point 2 will now be at 2' on M N, and the
point A will have risen to the horizontal b 2; hence a
circle described from the point O with the radius A C
will cut that line in a point A2, which is a second point in
the curve required. By a similar method any number of
additional points A3, A4, &c., may be obtained. It is
almost unnecessary to remark that the arcs A' 1', A' 2',
&c., when resolved into their equivalent straight lines,
are respectively equal to the distances A f, A 2', &c.
Another Method.—After having drawn horizontals
through all the primary divisions of the circle A B D 4,
and intersected each by a vertical line drawn through
each division on the line M N, take up in the compasses
the distance 2 b, for instance, from the point 2 to the line
A D, and set it off from d to A2; lay off similarly the
distance c 3 from e to A3, &c. These points may also be
obtained by transferring the distances 2d, 3 e, &c. from b,
to A2, from c to A3, and so on.
By this second method we have constructed the cycloid
A E, where we have supposed the generating circle to roll
upon M N from right to left. We have also introduced
into this figure a combination of two equal and similar
portions of cycloids, for the purpose of exhibiting the form
which should be given to the teeth of a rack intended to
drive a pinion.
The Epicycloid is the curve described by a point in the
circumference of a circle which rolls without sliding, upon
another circle, supposed to be at rest. The former is
called the generating, and the latter the fundamental
circle, and that part of its circumference upon which the
epicycloid rests is called the base.
When the generating circle rolls on the convex circum-
ference of the fundamental circle, the curve is termed an
exterior epicycloid; if it move within, or on the concave
circumference, it is called an interior epicycloid.
This is the curve which determines the true form of the
teeth of wheels working into each other.
H