0.5

1 cm

62

ENGINEER AND MACHINIST’S DRAWING-BOOK.

wheel. This may be most advantageously effected by

taking for the curvature of the teeth of the pinion, the

epicycloid A d described by the point A in the circle

M A N, rolling over the circle BAD. It will be observed

that, as in the preceding examples, the tooth E of the

pinion begins its action upon the tooth F of the wheel at

the point of contact of their respective primitive circles,

and that it is unnecessary that it should be continued be-

yond the point c, because the succeeding tooth H will then

have been brought into action upon G; consequently the

teeth of the wheel might be bounded by a circle passing

through the point e. It is, however, one of the practical

advantages which this species of gearing has over wheels

working externally, that the surfaces of contact of the

wheel and pinion admit of being more easily increased;

and by making the teeth somewhat longer than simple

necessity demands, the strain may be diffused over two or

more teeth at the same time. The flanks of the teeth

of the wheel are formed by radii drawn to the centre 0,

and their points are rounded off* to enable them to enter

freely into the spaces of the pinion.

Projections of a Spur Wheel and Pinion in Gear.—

Plate XXIII.

The student who has paid due attention to the preced-

ing illustrations, and to the practical instructions given

in reference to Plate XVI., will now be in a condition to

lay down most fully and accurately the projections of any

pair of spur wheels intended to work together. In the

example now before us, we assume that the distance O 0'

of the centres of the wheels, and the number of teeth in

each, are known, and that each wheel is qualified to be

the driver. The distance between the axes is 2 feet

4f inches, the number of teeth in the wheel 54, and in the

pinion 36, giving a velocity-ratio of 3 to 2.

In the first place, find the diameters of the pitch

circles ABC and C D E by the rule already laid down,

which in this case is thus stated :—The sum of the teeth

of the two wheels, 90, is to the distance of their centres

2875 inches, as the number of teeth in the wheel, 54, is to

its radius unknown; the result of this calculation gives

as the radius of the wheel 17 25 inches, making that of

the pinion 2875 — 17-25 = 11-5 inches. The circles

having been laid down, they are to be divided into the

required number of equal parts, according to the number

of teeth they are respectively to contain; and the mode of

effecting this with neatness and despatch has already been

pointed out in our notice of Plate XVI. If we regard

the wheel as driven by the pinion, the true curvature of

the teeth of the latter will obviously be the epicycloid

generated by a point in the circle 0 G C rolling upon the

pitch circle C D E of the pinion.

It may be useful in this place to point out the mode

usually adopted in the pattern-shop for setting out the

teeth of wheels. As the workman cannot be expected to

* Strictly speaking, the curve of these parts is an epicycloid,

generated by a circle having the diameter A C, but as a very small

portion only is required, it is sufficiently accurate in practice to

employ circular arcs, as shown in the figure.

be in possession of that accurate mathematical knowledge

which would enable him to determine the requisite forms

by purely theoretical means, he has recourse to the mecha-

nical method of drawing the outline of the teeth by means

of templates. The method of drawing the epicycloid

mechanically has already been described, and illustrated

by Fig. 4, Plate XX. We shall now proceed to show in

full detail the application of that process in such a case

as that now before us.

The pitch of the teeth being given, and the ratio of the

angular velocities, the diameters of the wheels are deter-

mined to the nearest number of teeth which correspond

to these conditions. Four slips of thick veneer are then

provided; on two of these, small arcs d d of the pitch

circles are drawn, and on the other two, similar arcs of

circles whose diameters are equal to the radii of the

wheels. The edges of the pieces towards which these arcs

are convex, are worked accurately to the arc-lines; and

thus prepared, the first pair B. representing segmental

portions of the wheels at their pitch circumferences, are

fixed by small screws upon the faces of two pieces of clean

hardwood board C C. This completes the pair of templates,

as represented by the annexed cut (Fig. 179).

179.)

c c

0 0

The operation of describing the form of the teeth is thus

conducted: The thickness ac of the tooth (Fig. 180)

lug. iso.

being set oft" upon the pitch circle d d, the segment D,

{Ftg 18l0 of which the diameter is equal to the radius

of the opposite wheel, is then applied convexly

against the edge on d, d, as shown edgewise

in Fig. 181. This segment has a tracing

point inserted obliquely into its edge at p,

which being brought to coincide with a, in

the pitch line d d, and the segment made to

roll towards the right, the edges meantime

being lightly pressed together, the tracing

point describes the curve a t upon the surface

C C. To describe the side c t, the point p is

made to coincide with c, and the segment D is rolled

towards the left, till the curves intersect at t. To pre-

ENGINEER AND MACHINIST’S DRAWING-BOOK.

wheel. This may be most advantageously effected by

taking for the curvature of the teeth of the pinion, the

epicycloid A d described by the point A in the circle

M A N, rolling over the circle BAD. It will be observed

that, as in the preceding examples, the tooth E of the

pinion begins its action upon the tooth F of the wheel at

the point of contact of their respective primitive circles,

and that it is unnecessary that it should be continued be-

yond the point c, because the succeeding tooth H will then

have been brought into action upon G; consequently the

teeth of the wheel might be bounded by a circle passing

through the point e. It is, however, one of the practical

advantages which this species of gearing has over wheels

working externally, that the surfaces of contact of the

wheel and pinion admit of being more easily increased;

and by making the teeth somewhat longer than simple

necessity demands, the strain may be diffused over two or

more teeth at the same time. The flanks of the teeth

of the wheel are formed by radii drawn to the centre 0,

and their points are rounded off* to enable them to enter

freely into the spaces of the pinion.

Projections of a Spur Wheel and Pinion in Gear.—

Plate XXIII.

The student who has paid due attention to the preced-

ing illustrations, and to the practical instructions given

in reference to Plate XVI., will now be in a condition to

lay down most fully and accurately the projections of any

pair of spur wheels intended to work together. In the

example now before us, we assume that the distance O 0'

of the centres of the wheels, and the number of teeth in

each, are known, and that each wheel is qualified to be

the driver. The distance between the axes is 2 feet

4f inches, the number of teeth in the wheel 54, and in the

pinion 36, giving a velocity-ratio of 3 to 2.

In the first place, find the diameters of the pitch

circles ABC and C D E by the rule already laid down,

which in this case is thus stated :—The sum of the teeth

of the two wheels, 90, is to the distance of their centres

2875 inches, as the number of teeth in the wheel, 54, is to

its radius unknown; the result of this calculation gives

as the radius of the wheel 17 25 inches, making that of

the pinion 2875 — 17-25 = 11-5 inches. The circles

having been laid down, they are to be divided into the

required number of equal parts, according to the number

of teeth they are respectively to contain; and the mode of

effecting this with neatness and despatch has already been

pointed out in our notice of Plate XVI. If we regard

the wheel as driven by the pinion, the true curvature of

the teeth of the latter will obviously be the epicycloid

generated by a point in the circle 0 G C rolling upon the

pitch circle C D E of the pinion.

It may be useful in this place to point out the mode

usually adopted in the pattern-shop for setting out the

teeth of wheels. As the workman cannot be expected to

* Strictly speaking, the curve of these parts is an epicycloid,

generated by a circle having the diameter A C, but as a very small

portion only is required, it is sufficiently accurate in practice to

employ circular arcs, as shown in the figure.

be in possession of that accurate mathematical knowledge

which would enable him to determine the requisite forms

by purely theoretical means, he has recourse to the mecha-

nical method of drawing the outline of the teeth by means

of templates. The method of drawing the epicycloid

mechanically has already been described, and illustrated

by Fig. 4, Plate XX. We shall now proceed to show in

full detail the application of that process in such a case

as that now before us.

The pitch of the teeth being given, and the ratio of the

angular velocities, the diameters of the wheels are deter-

mined to the nearest number of teeth which correspond

to these conditions. Four slips of thick veneer are then

provided; on two of these, small arcs d d of the pitch

circles are drawn, and on the other two, similar arcs of

circles whose diameters are equal to the radii of the

wheels. The edges of the pieces towards which these arcs

are convex, are worked accurately to the arc-lines; and

thus prepared, the first pair B. representing segmental

portions of the wheels at their pitch circumferences, are

fixed by small screws upon the faces of two pieces of clean

hardwood board C C. This completes the pair of templates,

as represented by the annexed cut (Fig. 179).

179.)

c c

0 0

The operation of describing the form of the teeth is thus

conducted: The thickness ac of the tooth (Fig. 180)

lug. iso.

being set oft" upon the pitch circle d d, the segment D,

{Ftg 18l0 of which the diameter is equal to the radius

of the opposite wheel, is then applied convexly

against the edge on d, d, as shown edgewise

in Fig. 181. This segment has a tracing

point inserted obliquely into its edge at p,

which being brought to coincide with a, in

the pitch line d d, and the segment made to

roll towards the right, the edges meantime

being lightly pressed together, the tracing

point describes the curve a t upon the surface

C C. To describe the side c t, the point p is

made to coincide with c, and the segment D is rolled

towards the left, till the curves intersect at t. To pre-