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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lines and dimensions necessary to be given or assumed in
laying down the projections of any bevil wheel, namely:
the axis 0 S of the primitive cone, the diameter A B of
its base, the angle A S O which the side of the cone makes
with the axis, and the straight lines A 0, D O', perpen-
dicular to AS, and representing the sides of two cones
between which the breadth of the wheel (or length of the
teeth) is comprised. These lines having been laid down
according to the required conditions, the first operation
is to divide the primitive circle, described with the radius
A C, into a number of equal parts corresponding to the
number of teeth, or pitch of the wheel. Then upon the
section, Fig. 3, draw with the radius O A or O B sup-
posed to move parallel to itself outside the figure a small
portion of a circle, upon which construct the outlines of
a tooth M, and of the rim of the wheel, with the same
proportions and after the same manner as we have
announced and explained in reference to spur wheels;
set off from A and B the points a, d, and /,—denoting
respectively the distances from the pitch line to the
points and roots of the teeth, and to the inside of the
rim, and join these points to the vertex S of the primi-
tive cone, terminating the lines of junction at the lines
D O', E O'; the figure abed will represent the lateral
form of a tooth, and the figure c df e a section of the rim
of the wheel, by the aid of which the face view (Fig. I)
may easily be constructed.

The points a, b, c, d, and e, having been projected upon
the vertical diameter A' B', describe from the centre O' a
series of circles passing through the points thus obtained,
and draw any radius, as C' L, passing through the centre
of a tooth. On either side of the point L set off the dis-
tances L 1c, L l making up the thickness of the tooth M at
the point, and indicate, in like manner, upon the circles
passing through the points B' and d', its thickness at the
pitch-line and root; then draw radii through the points
i, l, k, g, m, &c., terminating them respectively at the
circles forming the projections of the corresponding parts
at the inner extremity of the teeth; these radial lines
will represent the rectilinear edges of all the teeth. The
curvilinear outlines may be delineated by arcs of circles,
tangents to the radii g C and i C, and passing through the
points obtained by the intersections of the radii and the
various concentric circles. The radii of these circular arcs
may in general, as in the case of spur wheels, be taken
equal to the pitch, and their centres upon the interior and
exterior pitch-circles; thus the points g and i, n and o,
for example, are the centres for the arcs passing through
the corresponding points in the next adjacent teeth, and
vice versa.

The drawing of the teeth in the edge view (Fig. 2), and of
such portions of them as are visible in the section (Fig. 3)
will present no difficulty to the attentive student, and is
sufficiently explained by inspection of the lines of projec-
tion which we have partially introduced into the plate for
this purpose. We have only to remark, that in the construc-
tion of these views every point in the principal figure from
which they are derived is situated upon the projection of

the circle drawn from the centre C' and passing through
that point. Thus the points g and i, for example, situated
upon the exterior pitch-circle will be determined in Fig. 2
by the intersection of their lines of projection with the
base A B of the primitive cone; and the points k and l
will be upon the straight line passing through a a (Fig. 3),
and so on. Farther, as the lateral edges of all the teeth
in Fig. 1 are radii of circles drawn from the centre C, so,
in Fig. 2, they are represented by lines drawn through
the various points found as above for the outer extremi-
ties of the teeth, and converging towards the common
apex S; while the centre-lines of the exterior and interior
extremities themselves all tend to the points O and O'
respectively. This circumstance will suggest a mode of
materially simplifying the operation of drawing the edge
view of the teeth when the wheels are small or executed to
a small scale; and in all cases it affords a means of testing
the accuracy of the operations, if the method of projecting
numerous points be adopted. Finally, having represented
the projection of the rim, arms, and eye, according to the
dimensions indicated, the drawing of this subject will be

Projection of a Bevil Wheel and Pinion in Gear.—
Plate XXYI.

This plate contains two views of a pair of bevil
wheels working into each other, the number of their
teeth, and consequently the ratio of their velocities, being
as 11 to 7.

Let A S and S B be the axes of the two wheels, and
D E and E F the projections of their pitch-circles. Then
regarding Fig. 1, in the first instance, as a section made
by a plane in which the lines A S and B S are situated,
proceed to represent the form of a tooth X and Y in each
wheel. Then construct upon Fig. 2 the face view of the
wheel, substituting appropriate circular arcs for the spheri-
cal epicycloids which give the exact curvature of the teeth;
the details of this operation are laid down with sufficient
minuteness in the Plate which last occupied our attention,
as also the mode of deriving from Fig. 2 the edge view of
the wheel in Fig. 1. And, since the face view of the
pinion is not given in either of the figures, it is necessary
to lay down a portion of it on any convenient part of the
paper, taking the centre, of course, upon the line A S;
this we have shown in dotted lines on the Plate, together
with the lines of projection; the teeth being disposed sym-
metrically on each side of the axis, a quadrant of the
pinion is all that is required to be thus represented. The
position of the teeth of the pinion in Fig. 2 with respect
to the axis A S being precisely the same as in Fig. 1, it is
unnecessary to draw a second plan in order to obtain the
edge view, but simply to copy that already constructed in
Fig. 1.

The forms and proportions of the teeth in the wheels
here represented differ slightly from those embodied in
the preceding example; but they are such as are ap-
proved and adopted in practice by many mill-wrights.

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