61

System of a Rack Driving a Trundle.

Fig. 1, Plate XXII.—Required to construct the form

of the teeth of a rack intended to communicate uniform

motion to a trundle pinion.

The primitive circle, A B D, of the trundle pinion is, in

the example now before us, divided into eight equal parts;

the points of division being taken as the centres of the

staves, their outlines are described with a radius equal

to a quarter of the distance between two contiguous

divisions of the primitive circle; by this arrangement the

space intervening between any two adjacent staves is

equal to the diameter of the staves themselves (these

measurements being, in all cases, taken upon the primi-

tive circle of the trundle). In order that these spaces

may correspond upon the rack, set off repeatedly from A

on the line MN, the distance A1, equal to one-fourth of that

between the centres of the staves; the points A, 4', G, &c.,

will be the centres of the spaces, and the points E, F, H,

&c., those of the teeth. Now, if we suppose that the staves

of the trundle have no sensible thickness, but were repre-

sented simply by their axes, the true curvature of the

teeth would be a cycloid generated by a point in the

circle A B D rolling upon the straight line M N; this

hypothesis being, for the moment, regarded as true, the

form of the tooth would be that represented at M a G,

consisting of two curves M a, a G, the construction of

which we have already explained, observing that the arc

A B, resolved into a straight line, measures the thickness

M G of the tooth, which would impel uniformly the axis

of one of the staves.

If now, curves be drawn tangents to a series of circles

drawn from any number of points b, c, d, e, &c., in the

curves M A and A G, with the radius A 1, these lines

will determine the true form of tooth suitable for pro-

pelling the staves of the trundle. We have, then, only to

transfer this form of tooth to all the other divisions

F, E, I, &c., of the rack, which we have supposed to be

moved from left to right, as indicated by the arrow.

It is to be remarked that, to diminish the friction, it

is necessary in practice, to make the thickness of the

teeth a little less than the width of the spaces ; from

one-twelfth to one-fifteenth of the pitch being usually

allowed for this purpose. For the sake of simplifying our

investigations, we have not adhered rigidly to the truth

in this respect in the present, and in most of the subse-

quent illustrations ; but the necessary modifications may

be easily made.

To form the bottoms of the spaces, it is only necessary

to describe, with the radius A1, semicircles tangents to the

curves already drawn, taking the precaution to fix their

centres a little below the points M, G, 4', A, &c., to allow

the staves to pass without contact.

All that now remains is, to cut off from the point of

the tooth, that portion which does not come into action ;

and, to aid in estimating the quantity which may be re-

moved, we shall make the following observation :—

It is of essential importance, in a pair of wheels work-

ing together, that the driving tooth should not commence

its action upon the driven tooth until their point of contact

reaches the line joining the centres of the wheels; in other

words, in the example now under consideration, the tooth

I should not quit contact with the stave B until the

next tooth E commences its action, that is, until the

centre of the stave A is upon the perpendicular drawn

from the centre of the trundle upon the line M N. In this

position, which is that represented in the figure, the point

of contact of the tooth I with the stave B is at o upon the

line A B. This point, then, indicates the height at which

a horizontal line may be drawn limiting the points of the

teeth. It is better in practice, however, to allow a little

more height, in order to obviate the liability to injurious

shocks, by allowing both teeth to be in action for a brief

period.

The Form of Teeth in a Pinion driving a Trundle

Wheel Externally.

Fig. 2.—In this case it is obvious that the teeth of the

pinion, of which the primitive circle is A M N, will be

formed on the same principles as in the preceding example,

observing, however, that the generating circle of the epi-

cycloids M a, a b, is the primitive circle A B D of the

trundle wheel.

System composed of an Internal Spur-Wheel

driving a Pinion.

Fig. 3.—The form of the teeth of the driving wheel is in

this instance determined by the epicycloid described by a

point in the circle AEG, rolling on the concave circum-

ference of the primitive circle MAN. The points of the

teeth are to be cut off by a circle drawn from the centre

of the internal wheel and passing through the point E,

which is indicated, as before, by the contact of the curve

with the flank of the driven tooth.

The wheel being supposed to be invariably the driver,

the curved portion of the teeth of the pinion may be very

small. This curvature is a part of an epicycloid generated

by a point in the circle MAN rolling upon BAD.

System composed of an Internal Wheel driven

by a Pinion.

Fig. 4.—This problem involves a circumstance which has

not hitherto come under consideration, and which demands,

consequently, a different mode of treatment to that em-

ployed in the preceding cases. The epicycloidal curve A a,

generated by a point in the circle having the diameter AO,

the radius of the circle MAN, and which rolls upon the

circle BAD, cannot be developed upon the flank A 6,

the line described by the same point in the same circle in

rolling upon the concave circumference MAN; and for

this obvious reason, that that curve is situated without

the circle BAD, while the flank, on the contrary, is

within it. It becomes necessary, therefore, in order that

the pinion may drive the wheel uniformly according to

the required conditions, to form the teeth so that they

shall act always upon one single point in those of the