0.5

1 cm

58

ENGINEER AND MACHINIST’S DRAWING-BOOK.

and it is necessary that they should be fitted to the

shaft in two distinct pieces. The part B is very ac-

curately fitted to the body of the eccentric A; the joints

being rebated to prevent lateral displacement; and the

parts are held immovably together by round steel pins

a, a, passing through the lines of junction. A catch or

projection b is cast upon one side of the boss of the ec-

centric, which is carried round with the crank-shaft by

either of two similar catches affixed to the latter; the

position of these, as well as the length of the part b, being

so adjusted that, when, in the revolution of the shaft, one

of its catches abuts against one end of the part b, the ec-

centric shall then be in its proper position for effecting the

forward action of the engine; and similarly, when the

opposite catch is in contact with the other extremity of

the projection on the eccentric, the reversing or backward

motion of the engine is produced.

Fig. 9. To describe the involute of a circle.

The involute is the curve described by the development

or resolution of the successive parts of another given curve

into their equivalent straight lines. The mode of its

generation is as follows:—Suppose an inelastic thread to

be accurately applied to the outline or circumference of

the curve o, 1, 2, 8-9; if we cause the extremity o to

move by unwinding the thread gradually from off the

circumference, the curve 1', 2', S',-9', described by that

point, will be the involute of the given curve. This

definition obviously suggests a mode of drawing the curve

mechanically. The application of this curve in machinery

is of not unfrequent occurrence; it is the form given to

many cams or wypers for various purposes; it is also the

curve proper to be given to the teeth of a pinion driving

a rack.

The geometrical construction of the involute of a circle

is as follows:—Describe a circle from the centre C, with

the radius C A, and, commencing at o, divide its circum-

ference into any number of equal parts, (which must be

so small that each may be looked upon as a straight

line). Then draw indefinitely through each of the points

1, 2, 3, &c., tangents to the given circle, and upon the

first, set off from 1 to 1' the length of the arc o 1; upon

the second, from 2 to 2', twice the length of the arc o 1,

in other words the arc o 2 reduced approximately to a

straight line, and so on. The curve which passes through

all the points thus obtained will be the involute of the

circle o A 9.

Supposing it is required, by means of an involute cam

or wyper to raise a point A to A', a circle described

from the centre C, and passing through A', will intersect

the curve at a point 5', such that the portion o 5' will

represent the length sufficient to produce the required

effect.

The involute may also be described mechanically thus:

let A (Fig. 178, annexed), be the circle ot which the

involute is required, and let R be a straight ruler, at

whose extremity a pin p is fixed with the point resting

upon the initial point q of the curve; then by rolling the

straight ruler upon the circumference, so that the point

at which it touches the circle may move gradually from q

towards r, the curve traced by the pin p will be the

involute required.

Fig. 10. To produce the involute of a circle so as to

form a spiral.

If we continue or prolong the involute of the circle a e

f i indefinitely, the curve will then assume the form and

title of a spiral. This curve is such that, after the first

revolution round the axis, the generating point a having

arrived at b, all straight lines such as b c, c d, l m, &c., are

equal to a b, and consequently to the circumference of the

primary circle or nucleus. This curve may be described

by means of a succession of circular arcs, whose centres are

taken successively upon the points of contact of tangents

drawn from various points e, f i, &c., taken upon the cir-

cumference of the primary circle. Thus, from the point of

contact e, with the radius e g, equal to the arc a e reduced

to a straight line, draw the arc a g, from f with the radius

h f, equal to the arc a e f, describe the arc g h from i the

arc h k, and so on. Also, from a, with the radius a b

draw the arc b l; and, with the radius a c, the arc c m, &c.

Classification of Gearing.

Combinations of toothed gearing, as is well known, are

employed for the transmission of motive power with a

determinate velocity. To produce a regular, smooth, and

equable motion by means of toothed gearing, and thus to

prevent shocks which might be injurious and destructive

to the machinery, the form of the teeth must be such that

the wheels shall work together in precisely the same man-

ner as if they were impelled merely by the friction of the

circumferences of their pitch circles. To attain this object,

the form of the teeth must be subjected, as we shall now

proceed to point out, to certain invariable laws depen-

dent upon the relative diameters of the gearing pairs.

Toothed gearing may be divided into two great classes,

spur and bevel wheels; in the former, the axes of the

driving and driven wheels are parallel to each other;

in the latter, they may be situated at any angle. Of

these classes there are numerous varieties, distinguished

by appropriate names.

Thus we have spur gearing, strictly so called, consisting

of wheels of which the teeth are disposed at the circumfer-

ence and converge towards the centre (see Fig. 1, Plate

XXI.) Trundle gear, in which a spur or face wheel moves a

ENGINEER AND MACHINIST’S DRAWING-BOOK.

and it is necessary that they should be fitted to the

shaft in two distinct pieces. The part B is very ac-

curately fitted to the body of the eccentric A; the joints

being rebated to prevent lateral displacement; and the

parts are held immovably together by round steel pins

a, a, passing through the lines of junction. A catch or

projection b is cast upon one side of the boss of the ec-

centric, which is carried round with the crank-shaft by

either of two similar catches affixed to the latter; the

position of these, as well as the length of the part b, being

so adjusted that, when, in the revolution of the shaft, one

of its catches abuts against one end of the part b, the ec-

centric shall then be in its proper position for effecting the

forward action of the engine; and similarly, when the

opposite catch is in contact with the other extremity of

the projection on the eccentric, the reversing or backward

motion of the engine is produced.

Fig. 9. To describe the involute of a circle.

The involute is the curve described by the development

or resolution of the successive parts of another given curve

into their equivalent straight lines. The mode of its

generation is as follows:—Suppose an inelastic thread to

be accurately applied to the outline or circumference of

the curve o, 1, 2, 8-9; if we cause the extremity o to

move by unwinding the thread gradually from off the

circumference, the curve 1', 2', S',-9', described by that

point, will be the involute of the given curve. This

definition obviously suggests a mode of drawing the curve

mechanically. The application of this curve in machinery

is of not unfrequent occurrence; it is the form given to

many cams or wypers for various purposes; it is also the

curve proper to be given to the teeth of a pinion driving

a rack.

The geometrical construction of the involute of a circle

is as follows:—Describe a circle from the centre C, with

the radius C A, and, commencing at o, divide its circum-

ference into any number of equal parts, (which must be

so small that each may be looked upon as a straight

line). Then draw indefinitely through each of the points

1, 2, 3, &c., tangents to the given circle, and upon the

first, set off from 1 to 1' the length of the arc o 1; upon

the second, from 2 to 2', twice the length of the arc o 1,

in other words the arc o 2 reduced approximately to a

straight line, and so on. The curve which passes through

all the points thus obtained will be the involute of the

circle o A 9.

Supposing it is required, by means of an involute cam

or wyper to raise a point A to A', a circle described

from the centre C, and passing through A', will intersect

the curve at a point 5', such that the portion o 5' will

represent the length sufficient to produce the required

effect.

The involute may also be described mechanically thus:

let A (Fig. 178, annexed), be the circle ot which the

involute is required, and let R be a straight ruler, at

whose extremity a pin p is fixed with the point resting

upon the initial point q of the curve; then by rolling the

straight ruler upon the circumference, so that the point

at which it touches the circle may move gradually from q

towards r, the curve traced by the pin p will be the

involute required.

Fig. 10. To produce the involute of a circle so as to

form a spiral.

If we continue or prolong the involute of the circle a e

f i indefinitely, the curve will then assume the form and

title of a spiral. This curve is such that, after the first

revolution round the axis, the generating point a having

arrived at b, all straight lines such as b c, c d, l m, &c., are

equal to a b, and consequently to the circumference of the

primary circle or nucleus. This curve may be described

by means of a succession of circular arcs, whose centres are

taken successively upon the points of contact of tangents

drawn from various points e, f i, &c., taken upon the cir-

cumference of the primary circle. Thus, from the point of

contact e, with the radius e g, equal to the arc a e reduced

to a straight line, draw the arc a g, from f with the radius

h f, equal to the arc a e f, describe the arc g h from i the

arc h k, and so on. Also, from a, with the radius a b

draw the arc b l; and, with the radius a c, the arc c m, &c.

Classification of Gearing.

Combinations of toothed gearing, as is well known, are

employed for the transmission of motive power with a

determinate velocity. To produce a regular, smooth, and

equable motion by means of toothed gearing, and thus to

prevent shocks which might be injurious and destructive

to the machinery, the form of the teeth must be such that

the wheels shall work together in precisely the same man-

ner as if they were impelled merely by the friction of the

circumferences of their pitch circles. To attain this object,

the form of the teeth must be subjected, as we shall now

proceed to point out, to certain invariable laws depen-

dent upon the relative diameters of the gearing pairs.

Toothed gearing may be divided into two great classes,

spur and bevel wheels; in the former, the axes of the

driving and driven wheels are parallel to each other;

in the latter, they may be situated at any angle. Of

these classes there are numerous varieties, distinguished

by appropriate names.

Thus we have spur gearing, strictly so called, consisting

of wheels of which the teeth are disposed at the circumfer-

ence and converge towards the centre (see Fig. 1, Plate

XXI.) Trundle gear, in which a spur or face wheel moves a