0.5

1 cm

64

ENGINEER AND MACHINIST’S DRAWING-BOOK.

tion of the two axes being the common apex of both

cones.

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

cones.

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

pinion.

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

EB.

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

ENGINEER AND MACHINIST’S DRAWING-BOOK.

tion of the two axes being the common apex of both

cones.

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

cones.

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

pinion.

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

EB.

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