In the results obtained by the employment of this rule, an allowance of one-third the thickness for wear, and the margin for safety is included, so that the thickness of tooth arrived at is that to be given to the actual tooth. Further, the rule supposes the breadth of the tooth to be not less than twice the height of the same, any extra breadth not affecting the result (as already explained), when the pressure falls on a corner of the tooth.
In practical application, however, the diameter of the wheel at the pitch circle is generally, or at least often a fixed quantity, as well as the amount of stress, and it will happen as a rule that taking the stress as a fixed element and arriving at the thickness of the tooth by calculation, the required diameter of wheel, or what is the same thing, its circumference, will not be such as to contain the exact number of teeth of the thickness found by the calculation, and still give the desired amount of side clearance. It is desirable, therefore, to deal with the stress upon the tooth at the pitch circle, and the diameter, radius, or circumference of the pitch circle, and its velocity, and deduce therefrom the required thickness for the teeth, and conform the pitch to the requirements as to clearance from the tooth thickness thus obtained.
To deduce the thickness of the teeth from these elements we have Robertson Buchanan’s rule, which is as follows:—
Find the amount of horse-power employed to move the wheel, and divide such horse-power by the velocity in feet per second of the pitch line of the wheel. Extract the square root of the quotient, and three-fourths of this root will be the least thickness of the tooth. To the result thus obtained, there must be added the allowance for wear of the teeth and the width of the space including the clearance which will determine the number of teeth in the wheel.
In conforming strictly to this rule the difficulty is met with that it would give fractional pitches not usually employed and difficult to measure on an existing wheel. Cast wheels kept on hand or in stock by machinists have usually the following standard:—
Beginning with an inch pitch, the pitches increase by 1⁄8 inch up to 3-inch pitch, from 3 to 4-inch pitches the increase is by 1⁄4 inch, and from 4-inch pitch and upwards the increase is by 1⁄2 inch. Now, under the rule the pitches would, with the clearance made to bear a certain proportion to the pitch, be in odd fractions of an inch.
It appears then, that, if in a calculation to obtain the necessary thickness of tooth, the diameter of the pitch circle is not an element, the rule cannot be strictly adhered to unless the diameter of the pitch circle be varied to suit the calculated thickness of tooth; or unless either the clearance, factor of safety, or amount of tooth thickness allowed for wear be varied to admit of the thickness of tooth arrived at by the calculation. But if the diameter of the pitch circle is one of the elements considered in arriving at the thickness of tooth requisite under given conditions, the pitch must, as a rule, either be in odd fractions, or else the allowance for wear, factor of safety, or amount of side clearance cannot bear a definite proportion to the pitch. But the allowance for clearance is in practice always a constant proportion of the pitch, and under these circumstances, all that can be done when the circumstances require a definite circumference of pitch circle, is to select such a pitch as will nearest meet the requirements of tooth thickness as found by calculation, while following the rule of making the clearance a constant proportion of the pitch. When following this plan gives a thinner tooth than the calculation calls for, the factor of safety and the allowance for wear are reduced. But this is of little consequence whenever more than one tooth on each wheel is in contact, because the rules provide for all the stress falling on one tooth. When, however, the number of teeth in the pinion is so small that one tooth only is in contact, it is better to select a pitch that will give a thicker rather than a thinner tooth than called for by the calculation, providing, of course, that the pitch be less than the arc of contact, so that the motion shall be continuous.
But when the pinions are shrouded, that is, have flanges at each end, the teeth are strengthened; and since the wear will continue greater than in wheels having more teeth in contact, the shrouding may be regarded as a provision against breakage in consequence of the reduction of tooth thickness resulting from wear.
In the following table is given the thickness of the tooth for a given stress at the pitch circle, calculated from Tredgold’s rule for teeth supposed to have contact when new at one corner only.
| Stress in lbs. at pitch circle. | Thickness of tooth in inches. | Actual pitches to which wheels may be made. | ||||
| 400 | .52 | 1 | 1⁄8 | to | 1 | 1⁄4 |
| 800 | .75 | 1 | 1⁄2 | „ | 1 | 5⁄8 |
| 1,200 | .90 | 1 | 7⁄8 | „ | 2 | |
| 1,600 | 1.03 | 2 | „ | 2 | 1⁄8 | |
| 2,000 | 1.15 | 2 | 1⁄4 | „ | 2 | 3⁄8 |
| 2,400 | 1.26 | 2 | 1⁄2 | „ | 2 | 5⁄8 |
| 2,800 | 1.36 | 2 | 5⁄8 | „ | 2 | 3⁄4 |
| 3,200 | 1.43 | 2 | 7⁄8 | „ | 3 | |
| 3,600 | 1.56 | 3 | 1⁄8 | „ | 3 | 1⁄4 |
| 4,000 | 1.63 | 3 | 1⁄4 | „ | 3 | 3⁄8 |
| 4,400 | 1.70 | 3 | 3⁄8 | „ | 3 | 1⁄2 |
| 4,800 | 1.78 | 3 | 1⁄2 | „ | 3 | 5⁄8 |
| 5,200 | 1.86 | 3 | 5⁄8 | „ | 3 | 3⁄4 |
| 5,600 | 1.93 | 3 | 3⁄4 | „ | 4 | |
| 6,000 | 2.00 | 4 | „ | 4 | 1⁄4 | |