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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tion of the two axes being the common apex of both

It has been shown that, in calculating the dimensions
of a pair of spur-wheels, if the distance of their centres,
and the ratio of their respective numbers of teeth or velo-
cities be given, their primitive circles may be found. This,
however, is not the case in regard to bevil wheels; it
is necessary, to obtain the diameter of the one, that
that of the other should be known ; and for the reason,
that otherwise it would be impossible to determine the
distance of the wheels from the common apex of the

These preliminary observations being kept in view, we
shall now describe the method of laying down the pro-
jection of a pair of bevil wheels which shall satisfy any
required conditions.

Illustrations of Bevil Gearing.—Plate XXIY.

Fig. 1.—To construct a pair of bevil wheels which
shall revolve in a given ratio to each other; the diameter
of one of the wheels, and the inclination of their axes
being given.

Having determined the diameter of the second wheel
by the rule laid down in reference to spur-wheels, at
any points m and p, taken upon each of the axes A S
and B S, raise perpendiculars, and set off the radii m n
and p o of the wheels respectively (15 inches and
71 inches), and through the points n and o draw straight
lines parallel to A S and B S ; then, from the point of
intersection E, draw perpendiculars upon the axes ; these
lines will indicate the positions of the circles forming
the bases of the cones which determine the construction of
the teeth, and which, for this reason, are sometimes called
the primitive cones, having an obvious analogy to the
primitive or pitch-circles of spur-wheels. The entire pro-
jection of these cones is obtained by setting off D C equal
to E C, and O F equal to 0 E, and joining D S, E S, and
F S ; the triangle D E S is the projection of the primitive
cone of the wheel, and the triangle E F S is that of the

In the figure now before us, the wheels are supposed to
be cut by a plane A S B, passing through their axes; this
mode of representation is adopted for the purpose of
enabling us, with greater facility, to arrive at the true
form of the exterior extremities of the teeth, and to show
the mode of deriving from them the general form. The
breadth of the wheel (If inches) is first set off from D to
H, from E to G, and from F to I; and from all these
points perpendiculars are raised upon the sides of the pri-
mitive cones, intersecting the axes in the points A, K, B,
and L; the triangles thus formed will represent four cones
forming the bounding surfaces of the extremities of the
teeth. The surface upon which their height and curvature
are determined is, strictly speaking, a sphere described
from the centre S with the radius S E; but, as the opera-
tion of obtaining an epicycloid on a spherical surface
would be difficult, and is not required in practice, it is
found sufficiently accurate to substitute the epicycloid

generated by a point in a circle of the diameter E B,
rolling upon another of the radius A E, for the teeth of
the wheel, while those of the pinion are in like manner
determined by the epicycloid described by the circle
whose diameter is A E, rolling upon another of the radius

In order, therefore, to find the form and dimensions of
the teeth, we must conceive a portion of the cones D E A
and E F B to be developed, or spread out upon a plain
surface, for which purpose let the lines A E and B E be
supposed to move parallel to themselves to any convenient
distance beyond the figure, and, with these distances as
radii, describe two circles tangent to each other; upon
these circles the operation of setting out and drawing the
curvature of the teeth is to be performed precisely as in
the case of spur-gearing. In this manner we obtain for
the height of the tooth of the pinion above the pitch-line,
the dimension E' a, which must be set off from E to a
and from F to d; and for that of the wheel the distance
E' e, which is also to be transferred to E e and D b. The
flanks E' i' and E' o', which should be a little deeper than
the faces of the teeth, are then, in like manner, indicated
upon the principal figure, and straight lines drawn through
the various points b, f, a, c, d, &c., converging to the com-
mon apex S of the primitive cones, but limited by the
lines H K, K L, and L I; these convergent lines will re-
present the contour and dimensions of the teeth.

Fig. 2.—In the case (represented in this figure), where
the angle formed by the planes D E and E F, containing
the bases of the primitive cones, is an acute instead of
an obtuse angle, the operations above detailed are fully
applicable, although in this case the common apex S of
the primitive cones, as well as those of the cones between
which the breadth of the wheel is comprised, are situated
on the other side of the base D E.

Fig. 3.—This is a sectional representation of a pair of
bevil wheels of the same dimensions as the above, but
with the planes of their primitive circles intersecting at
right angles. In this variety, which is of most frequent
occurrence in machinery, the axes of the wheels are also
of course at right angles to each other.

Fig, 4,—In this example the planes of the primitive
circles are at right angles, and the diameters of the wheels
are equal. Such wheels are termed mitre wheels, and are
employed to transfer motion from one axis to another at
right angles to it, without altering the velocity. Both
wheels of such a pair are in all respects precisely alike,
and the sides of the primitive cones form with the axes
angles of 15°; hence the sides diametrically opposite those
in contact lie in one and the same straight line.

Projections of a Bevil Wheel.—Plate XXY.

This Plate represents a specimen of a bevil wheel drawn
upon a large scale, in order to show distinctly the mode
of constructing the various projections, and of deriving
one from another. Fig. 1 is a face view, Fig. 2 an edge
view, and Fig. 3 a vertical transverse section.

We have already stated what are the principal directing
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