Armengaud, Jacques Eugène; Leblanc, César Nicolas   [Hrsg.]; Armengaud, Jacques Eugène   [Hrsg.]; Armengaud, Charles   [Hrsg.]
The engineer and machinist's drawing-book: a complete course of instruction for the practical engineer: comprising linear drawing - projections - eccentric curves - the various forms of gearing - reciprocating machinery - sketching and drawing from the machine - projection of shadows - tinting and colouring - and perspective. Illustrated by numerous engravings on wood and steel. Including select details, and complete machines. Forming a progressive series of lessons in drawing, and examples of approved construction — Glasgow, 1855

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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

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
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