Bevel and Mitre Wheels must be regarded as two cones rolling upon each other, and the teeth are drawn upon the same principle as those of spur wheels, the maximum pitch diameter being always taken as the diameter of bevel and mitre wheels.

Form of Teeth of Wheels.-The following simple method of forming the teeth of wheels gives good results. Teeth thus formed and wheels made to the following proportions work accurately and smoothly together, wear uniformly, maintain their shape, and make very little noise in working. The utmost strength being given to the roots of the teeth, the liability to breakage and wear and tear is reduced to a minimum, and all wheels of the same pitch work properly together.

When the flank-or side of the tooth below the pitch line-is curved, the radius of the flank equals the pitch of the tooth, and the point from which this radius is struck is part of the pitch in depth below the pitch line, as shown at Fig. 92.

The radius of the point or face of the tooth,—or that portion of the tooth above the pitch circle,-equals the pitch for wheels with less than 21 teeth, and the pitch for wheels with upwards of 20 teeth. The point from which each radius is struck, is part of the pitch in depth, below the pitch line; the radius of the curve at the root of the tooth is the pitch. The flank of the tooth may also be made flat or parallel, and joined to the rim with a curve at the root of the tooth having a radius of the pitch, for wheels with more than 20 teeth; but for wheels with flat flanks with less than 21 teeth, the flanks should radiate towards the wheel roots of the teeth should join the rim with a small curve. Proportions of the Teeth of Iron and Steel Wheel-Gearing.— The author's method of drawing the teeth of wheel-gearing is clearly shown in Fig. 92. The pitch is divided into 15 equal parts, and the teeth are formed to the following proportions:

centre, and the

From the pitch line of the wheel to the top of the tooth = 5 parts. From the pitch line of the wheel to the bottom of the tooth = 6 parts.

Thickness of the tooth at the pitch line = 7 parts.

Space between the teeth at the pitch line = 8 parts.
Thickness of the rim = 7 parts.

Depth of feather or rib under the rim 8 parts.

Thickness of feather or rib under the rim

Thickness of the arm=7 parts.

= 7 parts,

Thickness of the feather or rib on the arm = 4 parts.

Depth of the feather or rib on the arm = 3 parts.

Diameter of the boss

Depth of the boss =

twice the diameter of the shaft.

times the width of face of the wheel. Depth of the feather or rib round the boss 8 parts. Thickness of the feather or rib round the boss Radius of curve at the root of the tooth = 2 parts, as shown in Fig. 92. Radius of the point or face of the tooth of wheels with upwards of 20

= 7 parts.

teeth = II parts. Radius of the point or face of the tooth of wheels with less than 21 teeth = 9 parts. Point below the pitch line of the wheel, from which the radius of the point or face of the tooth is struck = 1 part.

Breadth of the arm at the rim = 1 the pitch of the teeth, when the wheel face does not exceed 2 times the pitch in width; and 2 times the pitch for widths of face from 2 to 3 times the pitch; and 3 times the pitch for width of face equal to 4 times the pitch.

Breadth of the arm at the boss, should be increased by tapering the arm down from the rim to the boss, at the rate of inch per foot, on each side of the arm. The tendency of the strain being to twist the arm, the power acts with the greatest effect near the boss.

The teeth of wheels above 4 inches pitch may be shorter than given above

=

say 4 parts from the pitch line to the top of the teeth, and 5 parts from the pitch line to the bottom of the tooth = a total depth of tooth of 18 the pitch. The Maximum Working-Speed of Toothed-Wheels at the pitch-line, in feet per minute, consistent with freedom from excessive wear and tear, isCast-Iron Wheels, with straight teeth

1,800 feet 2,100,,

2,200

2,300

2,400

270

Fig. 93 shows a form of tooth used for crab-wheels, called knuckle gear.

Fig. 94 shows the way to project a pair of bevel-wheels, with their shafts at right angles.

The delineation of a worm and worm-wheel is shown in Fig. 95.

Fig. 96 shows the way to project a pair of angle wheels, or bevel wheels, with their shafts at an angle of 65°.

Number of Arms.-Wheels under 2 feet diameter should have 4 arms; wheels from 2 to 7 feet 6 inches diameter, 6 arms; wheels from 8 to 12 feet, 8 arms; and wheels from 13 to 16 feet diameter, 10 arms.

Width of Face.-The least width of face necessary to resist the full transverse strain on the tooth is 1 times the pitch, but for the sake of durability the width should not be less than 2 times the pitch; 24 times the pitch is the usual width. The following are good proportions :

Pitch of wheel, in inches

Width of face of wheel, in inches

1111111112 2 2 2 3 3 3 4

222 3 3 3 4 4 5 5 6 7 8 9 10 11 12 15

Mortice Wheels.-The wood teeth of a mortice wheel are made thicker than the teeth of its iron fellow, to compensate for the difference in strength of the material; consequently the thickness of the iron tooth has to be reduced, and the length of tooth is also reduced so as to be in proportion to the thickness. A wood-cog is shown in Fig. 99.

Thickness of wood cog = 9 parts, or of the pitch.
Thickness of tooth of iron wheel or fellow = 6 parts.
From the pitch line to the top of tooth = 4 parts.

Cog.

From the pitch line to the bottom of the tooth = 5 parts. Fig. 99.-Wood Thickness of the rim the pitch of the teeth multiplied by 12. Width of face of wheel same as for spur and bevel wheels given above. Width of mortice or shank of wood coginch narrower than the face of the tooth. Methods of fixing wood-cogs are shown in Figs. 97 & 98. Thickness of metal at each end of mortice = 6 parts.

No clearance is required: the wood cogs should be trimmed to fit accurately between the iron teeth.

When a pair of wheels of large diameter and quick speed work together, the larger one should have wood teeth, and the smaller one iron teeth. Wood teeth wear out sooner, but are not more liable to break than iron teeth. Beech, oak, and maple are sometimes used, but horn-beam, ironwood, and crab-tree are the best woods, for making the cogs. When working they should be smeared with a mixture of soft soap and plumbago. Worm-Wheels.-When the shafts are at right angles, the action of a worm and worm-wheel is similar to that of a rack and pinion, and the formation of the teeth at the section at the centre of a worm-wheel, should be the same as those of a spur-wheel of the same diameter, and the section of the thread of the worm should be the same as the teeth of a rack of the same pitch of tooth. Each revolution of the worm, turns the worm-wheel, to the extent of one tooth with a single thread worm, and 2 teeth with a double thread worm. The teeth of worm-wheels are made shorter than spur wheels. The amount the teeth are angled or skewed is equal to the pitch of the teeth. A worm and worm-wheel are shown in Fig. 95. Thickness of tooth = 7 parts, or of the pitch.

Space between the teeth

=

8 parts.

From the pitch line to the top of the tooth =

4 parts.

From the pitch line to the bottom of the tooth = 51⁄2 parts.
Radius of the point or face of the tooth

Flank of tooth, straight and flat.

= 9 parts.

Width of face of tooth = 1 times the pitch.

External diameter of worm = 4 times the pitch.

Pitch of Small Wheels.-The pitch of change wheels and other small wheels, is reckoned on the diameter of the pitch circle of the wheel instead of the circumference, and it is called the pitch per inch.

To find the number of teeth, in a wheel of a given diameter and pitch per inch:

Multiply the diameter of the pitch circle in inches, by the given pitch per inch.

To find the diameter of the pitch circle, to contain a given number of teeth of a given pitch per inch :

Divide the number of teeth by the required pitch per inch.

TABLE 18.-PITCH PER INCH IN DIAMETER AND CIRCULAR PITCH

The pitch per inch in diameter (Table 18), bears the same ratio to the circular pitch, as the diameter to the circumference, a diametral pitch of 1 inch, corresponds with a circular pitch of 3'1416 inches; hence to find the circular pitch divide 3:1416 by the given diametral pitch, and to find the diametral pitch divide 3'1416 by the given circular pitch. The outside diameter of a wheel-over the top of the teeth-is found by adding two parts of the diametral pitch to the pitch diameter, for instance a wheel of 48 teeth, 8 per inch pitch, is 6 + 3ths = 64 inches diameter outside. The depth of tooth of these small wheels is usually ths the pitch. Angular and Circumferential Velocity of Wheels.-The angular velocity of a revolving body, is the velocity of a point at a unit's distance from the centre of motion, or the angle swept through in a second by a line perpendicular to the axis of motion, the angle being expressed in circular measure. Every point of a revolving wheel has a different velocity in proportion to its distance from the centre of motion, for instance in a revolving pulley, the boss will make the same number of revolutions as the rim, but the angular velocity of the rim will be greater than that of the boss.

To find the circumferential velocity of a wheel:-Multiply the circumference in feet by the number of revolutions per minute, the product will give the space passed through by any point of the circumference in feet per minute, which divided by 60 will give the velocity in feet per second.

To find the angular velocity of a wheel, or the number of revolutions made in a given time:-Divide the circumferential velocity per second, (found by the last rule) by the circumference in feet, the result will give the angular velocity, which multiplied by 60 will give the number of revolutions per minute.

The Centre of Gyration is a point in a revolving body in which the momentum, or energy of the moving mass, is supposed to be concentrated.

The radius of gyration of a fly-wheel (including arms and rim) and of gearing may be assumed in practice as the radius of the inside of the rim. To find the amount of force, to apply at the radius of a wheel, to cause it to make a certain number of revolutions, in a given number of seconds, Rule: multiply the number of revolutions by the weight of the wheel in lbs., and multiply the product by the square of the distance in feet from the centre of motion to the centre of gyration, and call the result A. Then multiply the constant number 153'5 by the number of seconds during which the force is applied, and multiply the product by the radius in feet on which the force acts, and call the result B.; lastly, divide the result A. by the result B., which will give the required force in lbs.

The Radius of Gyration of a solid wheel of uniform thickness, or of a circular plate, or of a solid cylinder of any length, revolving on its axis, is to the radius of the object multiplied by '7071.