0.5

1 cm

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-