0.5

1 cm

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being superseded by the invention of bevil gear, which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