DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.25888#0074
58
ENGINEER AND MACHINIST’S DRAWING-BOOK.
Fig. 2.—To construct the epicycloid described by a
point A in the circle A B D, in its motion round the
fixed circle MAN.
The operations requisite for describing this curve geo-
metrically are precisely analogous to those we have already
explained in reference to the cycloid, which may be re-
garded as an epicycloid generated upon a fundamental
circle of infinite diameter. After having divided into any
convenient number of equal parts the circle A B D, whose
centre C obviously travels in a circular arc B C C3, con-
centric with the fundamental circle, set off upon the circle
M A N a succession of points 1' 2' 3' &c., whose common
distance is equal to the length of one of the divisions on
the generating circle. Then, through each of the points
1, 2, 3, &c., describe circular arcs from the centre 0, and
from the same centre draw radii passing through each of
the points I', 2', 3', &c., producing them till they cut the
circle B C C3. If now we conceive the generating circle
to have moved till its centre C is at C', the point 1 will
coincide with 1', which will now be the point of contact of
the two circles; and the point A will be in the arc which
passes through 1 ; but it is still a point in the circle
BAD; if, therefore, from the centre C', with the radius
A C, a circle be described cutting the arc c i, the point of
intersection A' will be a point in the curve required. In
the same manner any number of additional points may be
obtained by tracing a line through which the epicycloid
will be constructed.
As in the case of the cycloid, this curve may be drawn
by a simpler method. The point A' may be determined
by taking the distance from 1 to c, the point of intersec-
tion of the radius C' O with the arc c i, and setting it off
on the same arc from i to A'; or, if thought preferable,
the distance 1 i may be set off from c to A'; the distance
2 b may, in like manner, be set off from d to A2, and so
on. The second epicycloid A E in our figure has been
thus described, as well as the two smaller portions towards
the left, which are so disposed as to indicate the form of
the exterior parts of the teeth of a wheel intended to work
into another.
Fig. 3 represents the interior epicycloid generated by
a circle A B D rolling upon the concave circumference of
the circle MAN. The geometrical construction neces-
sary for obtaining this curve is identical with that already
described. First divide the generating circle into equal
parts 1, 2, 3, &c., and through the points of division de-
scribe circles concentric with MAN; then draw radii
from the centre O to the points F, 2', 3', &c., which cor-
respond to the divisions on the generating circle; and,
regarding the points of intersection of these radii with the
circle B C C3 as the several positions of the centre C, de-
scribe arcs of circles from these points with the radius
A C. The intersections of these arcs with the circles
passing through the divisions of the generating circle will
give so many points A', A2, A3, &c., in the curve required.
Into this figure we have introduced another epicycloid
in all respects identical with the former, but constructed
by the simpler, and in many respects better method which
we have already sufficiently detailed. At M are also
shown two similar portions of the curve in combination,
indicating the true form of the tooth of an internal spur-
wheel driving a pinion.
Fig. 4 In the case of the generating circle BAD being
greater than the fundamental circle MAN, the second
method of describing the epicycloid A A4 is always prefer-
able ; because, by the first, the arcs which are employed to
indicate the various points in the curve, intersect each
other at angles so acute as to leave considerable room for
inaccuracy. Therefore, applying the preceding instruc-
tions to the case now before us, divide the circle A B D
into equal parts, and mark off corresponding divisions
upon the circle A M N; then draw the radii Cl', C2', C3',
&c., producing them till they cut the circular arcs, de-
scribed from the centre C through each of the points
1, 2, 3, &c. If now the distance A a, for instance, be
transferred, upon the same arc from b to A4, a point in
the required curve will be obtained, and so on for the
other points. In the present instance the entire curve is
represented in the figure.
When the radius of the generating circle C E N is half
that of the fundamental circle, A M N (the former rolling
on the concave circumference of the latter), the resulting
epicycloid becomes a straight line. For, if we make the
arc c N, for example, equal to c N, and conceive the
centre o of the small circle to be at the point o'; then,
to find the new position of the point N, we must describe
from the centre o' with the radius o N, a circle which
will cut that drawn from C with the radius C c, at a
point rv\ which in every position of the generating
circle will be found to be in the diameter C N. This
result is also obtained, and may be demonstrated, by the
second method. Thus, if a circular arc be described from
C through the point e (which is such that the arc e N is
equal to e N), then by drawing the radius C 6, it will be
obvious that its intersection with the circular arc is in
the generating circle C E N ; whence it follows that the
point N must coincide with n2 on the radius C N. By a
similar construction, the same will be found true in every
position of the generating circle. We shall shortly have
occasion to exemplify the practical value of this remark-
able property of circles bearing the above relation to each
other.
The method of drawing the epicycloid mechanically is
shown at Fig. 4, and is exceedingly simple and obvious.
A segment E D G of the generating circle is formed of
thin wood or any suitable material, and a similar seg-
ment of the fundamental circle is also prepared and
fixed upon a plain surface. If now the former be rolled
upon the latter, a point fixed at G will describe the
curve required. A similar method is applicable to the
cycloid and to the interior epicycloid.
Propositions respecting Toothed Wheels.
In every pair of wheels intended to work in gear with
each other, it is, of course, indispensably requisite that
the teeth should be all equal, and placed at equal dis-
ENGINEER AND MACHINIST’S DRAWING-BOOK.
Fig. 2.—To construct the epicycloid described by a
point A in the circle A B D, in its motion round the
fixed circle MAN.
The operations requisite for describing this curve geo-
metrically are precisely analogous to those we have already
explained in reference to the cycloid, which may be re-
garded as an epicycloid generated upon a fundamental
circle of infinite diameter. After having divided into any
convenient number of equal parts the circle A B D, whose
centre C obviously travels in a circular arc B C C3, con-
centric with the fundamental circle, set off upon the circle
M A N a succession of points 1' 2' 3' &c., whose common
distance is equal to the length of one of the divisions on
the generating circle. Then, through each of the points
1, 2, 3, &c., describe circular arcs from the centre 0, and
from the same centre draw radii passing through each of
the points I', 2', 3', &c., producing them till they cut the
circle B C C3. If now we conceive the generating circle
to have moved till its centre C is at C', the point 1 will
coincide with 1', which will now be the point of contact of
the two circles; and the point A will be in the arc which
passes through 1 ; but it is still a point in the circle
BAD; if, therefore, from the centre C', with the radius
A C, a circle be described cutting the arc c i, the point of
intersection A' will be a point in the curve required. In
the same manner any number of additional points may be
obtained by tracing a line through which the epicycloid
will be constructed.
As in the case of the cycloid, this curve may be drawn
by a simpler method. The point A' may be determined
by taking the distance from 1 to c, the point of intersec-
tion of the radius C' O with the arc c i, and setting it off
on the same arc from i to A'; or, if thought preferable,
the distance 1 i may be set off from c to A'; the distance
2 b may, in like manner, be set off from d to A2, and so
on. The second epicycloid A E in our figure has been
thus described, as well as the two smaller portions towards
the left, which are so disposed as to indicate the form of
the exterior parts of the teeth of a wheel intended to work
into another.
Fig. 3 represents the interior epicycloid generated by
a circle A B D rolling upon the concave circumference of
the circle MAN. The geometrical construction neces-
sary for obtaining this curve is identical with that already
described. First divide the generating circle into equal
parts 1, 2, 3, &c., and through the points of division de-
scribe circles concentric with MAN; then draw radii
from the centre O to the points F, 2', 3', &c., which cor-
respond to the divisions on the generating circle; and,
regarding the points of intersection of these radii with the
circle B C C3 as the several positions of the centre C, de-
scribe arcs of circles from these points with the radius
A C. The intersections of these arcs with the circles
passing through the divisions of the generating circle will
give so many points A', A2, A3, &c., in the curve required.
Into this figure we have introduced another epicycloid
in all respects identical with the former, but constructed
by the simpler, and in many respects better method which
we have already sufficiently detailed. At M are also
shown two similar portions of the curve in combination,
indicating the true form of the tooth of an internal spur-
wheel driving a pinion.
Fig. 4 In the case of the generating circle BAD being
greater than the fundamental circle MAN, the second
method of describing the epicycloid A A4 is always prefer-
able ; because, by the first, the arcs which are employed to
indicate the various points in the curve, intersect each
other at angles so acute as to leave considerable room for
inaccuracy. Therefore, applying the preceding instruc-
tions to the case now before us, divide the circle A B D
into equal parts, and mark off corresponding divisions
upon the circle A M N; then draw the radii Cl', C2', C3',
&c., producing them till they cut the circular arcs, de-
scribed from the centre C through each of the points
1, 2, 3, &c. If now the distance A a, for instance, be
transferred, upon the same arc from b to A4, a point in
the required curve will be obtained, and so on for the
other points. In the present instance the entire curve is
represented in the figure.
When the radius of the generating circle C E N is half
that of the fundamental circle, A M N (the former rolling
on the concave circumference of the latter), the resulting
epicycloid becomes a straight line. For, if we make the
arc c N, for example, equal to c N, and conceive the
centre o of the small circle to be at the point o'; then,
to find the new position of the point N, we must describe
from the centre o' with the radius o N, a circle which
will cut that drawn from C with the radius C c, at a
point rv\ which in every position of the generating
circle will be found to be in the diameter C N. This
result is also obtained, and may be demonstrated, by the
second method. Thus, if a circular arc be described from
C through the point e (which is such that the arc e N is
equal to e N), then by drawing the radius C 6, it will be
obvious that its intersection with the circular arc is in
the generating circle C E N ; whence it follows that the
point N must coincide with n2 on the radius C N. By a
similar construction, the same will be found true in every
position of the generating circle. We shall shortly have
occasion to exemplify the practical value of this remark-
able property of circles bearing the above relation to each
other.
The method of drawing the epicycloid mechanically is
shown at Fig. 4, and is exceedingly simple and obvious.
A segment E D G of the generating circle is formed of
thin wood or any suitable material, and a similar seg-
ment of the fundamental circle is also prepared and
fixed upon a plain surface. If now the former be rolled
upon the latter, a point fixed at G will describe the
curve required. A similar method is applicable to the
cycloid and to the interior epicycloid.
Propositions respecting Toothed Wheels.
In every pair of wheels intended to work in gear with
each other, it is, of course, indispensably requisite that
the teeth should be all equal, and placed at equal dis-