to leave this fillet have the advantage of wearing longer than when brought up to a corner. Single curve or involute gears are, it is stated, the only gears that can be run at varying distances of axes, and transmit unvarying angular velocity. This peculiarity makes involute gears specially valuable for driving rolls, etc., the distance of whose axes

If, in any bevel gear, the teeth were sufficiently prolonged toward the apex, they would become infinitely small; that is, the teeth would all end in a point, or vanish at O.

We can also consider a bevel gear as beginning at the apex, and becoming larger and larger as we go away from the apex. Hence, as the bevel-gear teeth are tapering from end to end, we may say, that a bevel gear has a number of pitches, and pitch circles, or diameters.

In speaking of the pitch of a bevel gear, we mean always the pitch at the largest pitch circle, or at the largest pitch diameter as at BD, Fig. 324.

bevel gears, the gear OBQ, being The outer surface of a tooth as at The distance mm' is usually called

Fig. 324 is a section of three twice as large as the two others. mm' is called the face of the tooth. the length of the face of the tooth, though the real length is the distance

it occupies upon the line Oi. The outer part of a tooth at mn is called its large end, and the inner part m'n' the small end.

Having decided upon the pitch and the numbers of teeth

1. Draw centre lines of shafts AOB and COD at right angles.

For 24 teeth

2. Parallel to AOB, draw lines ab, and cd, each distant from AOB equal to half the largest pitch diameter of one gear. 4 pitch, this half largest pitch diameter is 3 in.

3. Parallel to COD, draw lines ef and gh distant equal to half the largest pitch diameter of the other gear. 12 teeth 4 pitch, this half largest pitch diameter is 1 in. 4. At the intersection of these four lines, draw Oi, Oj, these lines give the size and shape of the pitch cones. "Cone pitch lines."

from COD For a gear

Ok, and Ol;

We call them

5. Perpendicular to the cone pitch lines and through the intersection

of lines ab, cd, ef, and gh, draw lines mn, op, qr. The lines uv are drawn to show that another gear can be drawn from the same diagram.

6 Upon the lines mn, op, qr, the addenda and depth of the teeth are laid off, these lines passing through the largest pitch circles of the gears. Lay off the addendum, it being in these gears in. This gives distance mn, op, qr, and uv equal to the working depth of the teeth, which in these gears is in.

The addendum is measured perpendicularly from the cone pitch

lines at kr.

7. Draw lines Om, On, Oo, Op, Oq, Or. These lines give the height of teeth above the cone pitch lines as they approach O, and would vanish entirely at O.

It is quite as well never to have the length of teeth, or face mm',

longer than one-third the apex distance mO, nor more than two and one-half times the circular pitch.

8. Having decided upon the length of face, draw limiting lines m'n' perpendicular to iO, q'r' perpendicular to kO, and so on.

The distance between the cone pitch lines at the inner ends of the teeth m'n' and q'r', is called the inner or smaller pitch diameter; and the circle at these points is called the smallest pitch circle. We now have the outline of a section of the gears through their axes.

The distance mr is the whole diameter of the pinion. The distance qo is the whole diameter of the gear.

In practice these diameters can be obtained by measuring the drawing. The diameter of the pinion is 3'45 in., and of the gear 6°22 in. We can find the angles also by measuring the drawing with a protractor. In the absence of a protractor, templets can be cut to the drawing.

In turning the blanks to the correct angle, place one arm of the protractor or templet against the wheel boss and test the angle.

Bevel Gears (Cutting).—When axes are at right angles, the sum of angles of edge in the two gears equals 90°, and the sums of angle of edge and face in each gear are alike.

The angles of axes remaining the same, all pairs of bevel gears of the same ratio have the same angle of edge; all pairs of same ratio and of same numbers of teeth, have the same angles of both edges and faces independent of the pitch. Thus, in all pairs of bevel gears having one gear twice as large as the other, with axes at right angles, the angle of edge of large gear is 63° 26', and the angle of edge of the small gear 26° 34'.

In all pairs of bevel gears with axes at right angles, one gear having 24 teeth, and the other gear having 12 teeth, the angle of face of small gear is 59° 11'.

Data for Cutting Bevel Gears-(see table of data, p. 289).

1. The pitch and the numbers of the teeth the same as for spur gears. 2. The data for the cutter, as to its form: sometimes two cutters are needed for a pair of bevel gears.

3. The whole depth of the tooth spaces both at the outside and inside ends; D" +ƒ at the outside, and D"" +ƒ at the inside.

4. The thickness of the teeth at the outside, and at the inside; t and t'.

5. The height of the teeth above the pitch lines at the outside and inside s and s'.

6. The cutting angles, or the angles that the path of the cutter makes with the axes of the gears. In Fig. 325 the cutting angle for the gear CD is AOp, and the cutting angle for the pinion is BOo.

The form of the teeth in one of these gears differs so much from that in the other gear that two cutters are required. In determining these cutters, we do not have to develop the forms of the gear teeth, we need merely measure the lines Ac and Bc, Fig. 325, and calculate the cutter forms, as if these distances were the radii of the pitch circles of the gears to be cut.

Twice the length Ac in inches multiplied by the diametral pitch equals the number of teeth for which to select a cutter for the 24-tooth gear; this number is about 54, which calls for a No. 3 bevel-gear cutter in the list of bevel-gear cutters (see p. 283).

Twice Bc multiplied by 8 equals about 13 which indicates a No. 8 bevel-gear cutter for the pinion.

This method of selecting cutters is based upon the idea of shaping the teeth as nearly right as practicable at the large end, and then filing the small ends where the cutter has not rounded them over enough. There are several things that affect the shape of the teeth, so that the choice of cutters is not always so simple a matter as the taking the lines Ac and Bc as radii.

In cutting a bevel gear in the ordinary gear-cutting machine, the finished spaces are not always of the same form as the cutter might be expected to make, because of the changes in the positions of the cutter and of the gear blank in order to cut the teeth of the right thickness at both ends.

D

Que

B

The cutter must be thin enough to pass through the small end of the spaces so that the large end has to be cut to the right width by adjusting either the cutter or the blank sideways, then rotating the blank and cutting twice around. Thus, in Fig. 326, a gear and a cutter are set to have a space widened at the large end e', and the last chip to be cut off by the right side of the cutter, the cutter having been moved to the left and the blank rotated in the direction of the arrow. In a universal milling machine the same result would be attained by moving the blank to the right and rotating it in the direction of the arrow. It should be remembered that, in setting to finish the side of a tooth, the tooth and the cutter are first separated sideways, and the blank is then rotated by indexing the spindle to bring the large end of the tooth up against the cutter.

FIG. 325.-Bevel gear. Diagram for
dimensions.

This tends not only to cut the spaces wider at the large pitch circle, but also to cut off still more at the face of the tooth; that is, the teeth may be cut rather thin at the face and left rather thick at the root.

This tendency is greater as a cutting angle, BOo, Fig. 325, is smaller,

or as a bevel gear approaches a spur gear, because when the cutting angle is small the blank must be rotated through a greater arc in order to set to cut the right thickness at the outer pitch circle. Different workmen prefer different ways to compromise in the cutting of a bevel gear. When a blank is rotated in adjusting to finish the large end of the teeth there need not be much filing of the small end, if the cutter is right, for a pitch circle of the radius Bc, Fig. 325, which for our example is a No. 8 cutter, but the tooth faces may be rather thin at the large ends. This compromise is preferred by nearly all workmen; because it does not require much filing of the teeth.

A second approximation in cutting with a rotary cutter is to widen the spaces at the large end by swinging either the index spindle or the

cutter-slide carriage, so as to pass the cutter through on an angle with the blank sideways, called the "side-angle," and not rotate the blank at all to widen the spaces. This side-angle method is employed in automatic mitre gearcutting machines, and is available in the manufacture of mitre gears in large quantities, because with the proper relative thickness of cutter the tooth thickness comes right by merely adjusting for the side angle, but for cutting a few gears it is not so much liked, because in adjusting for the side angle the central setting for the cutter is usually lost, and has to be found by guiding into the central slot already cut.

If the side-angle mechanism pivots about a line that passes very near the small end of the tooth to be cut, the central setting of the cutter may not be lost. With this method a gear must be cut at least twice round. In widening the spaces at the large end, the teeth are narrowed practically the same amount at the root as at the face, so that this side-angle method requires a wider cutter at e e', Fig. 326, than the first or rotative method. The amount of filing required to correct the form of the teeth at the small end is about the same as in the first method.

A third approximate method consists in cutting the teeth right at the large end by going round at least twice, and then to trim the teeth at the small end and towards the large end with another cutter, going round at least four times in all. This method requires skill, and is necessarily a little slow, but it contains possibilities for considerable accuracy.

A fourth method is to have a cutter fully as thick as the spaces at the small end, cut rather deeper than the regular depth at the large end, and go only once round. This is a quick method, but more inaccurate