It is not improbable, however, that the contact of tooth upon tooth extends in cast gears across at least two-thirds of the breadth of the tooth, in which case the rules for ascertaining the strength of cut teeth of equal thickness may be employed, substituting 2⁄3rds of the actual tooth breadth as the breadth for the purposes of the calculation.
If instead of supposing all the strain to fall upon one tooth and calculating the necessary strength of the teeth upon that basis (as is necessary in interchangeable gearing, because these conditions may exist in the case of the smallest pinion that can be used in pitch), the actual working condition of each separate application of gears be considered, it will appear that with a given diameter of pitch circle, all other things being equal, the arc of contact will remain constant whatever the pitch of the teeth, or in other words is independent of the pitch, and it follows that when the thickness of iron necessary to withstand (with the allowances for wear and factor of safety) the given stress under the given velocity has been determined, it may be disposed in a coarse pitch that will give one tooth always in contact, or a finer pitch that will give two or more teeth always in contact, the strength in proportion to the duty remaining the same in both cases.
In this case the expense of producing the wheel patterns or in trimming the teeth is to be considered, because if there are a train of wheels the finer pitch would obviously involve the construction and dressing to shape of a much greater number of teeth on each wheel in the train, thus increasing the labor. When, however, it is required to reduce the pinion to a minimum diameter, it is obvious that this may be accomplished by selecting the finer pitch, because the finer the pitch, the less the diameter of the wheel may be. Thus with a given diameter of pitch circle it is possible to select a pitch so fine that motion from one wheel may be communicated to another, whatever the diameter of the pitch circle may be, the limit being bounded by the practicability of casting or producing teeth of the necessary fineness of pitch. The durability of a wheel having a fine pitch is greater for two reasons: first, because the metal nearest the cast surface of cast iron is stronger than the internal metal, and the finer pitch would have more of this surface to withstand the wear; and second, because in a wheel of a given width there would be two points, or twice the area of metal, to withstand the abrasion, it being remembered that the point of contact is a line which partly rolls and partly slides along the depth of the tooth as the wheel rotates, and that with two teeth in contact on each wheel there are two of such lines. There is also less sliding or rubbing action of the teeth, but this is offset by the fact that there are more teeth in contact, and that there are therefore a greater number of teeth simultaneously rubbing or sliding one upon the other.
But when we deal with the number of teeth the circumstances are altered; thus with teeth of epicycloidal form it is manifestly impossible to communicate constant motion with a driving wheel having but one tooth, or to receive motion on a follower having but one tooth. The number of teeth must always be such that there is at all times a tooth of each wheel within the arc of action, or in contact, so that one pair of teeth may come into contact before the contact of the preceding teeth has ceased.
In the construction of wheels designed to transmit power as well as simple motion, as is the case with the wheels employed in machine work, however, it is not considered desirable to employ wheels containing a less number of teeth than 12. The diameter of the wheel bearing such a relation to the pitch that both wheels containing the same number of teeth (12), the motion will be communicated from one to the other continuously.
It is obvious that as the number of teeth in one of the wheels (of a pair in gear) is increased the number of teeth in the other may be (within certain limits) diminished, and still be capable of transmitting continuous motion. Thus a pinion containing, say 8 teeth, may be capable of receiving continuous motion from a rack in continuous motion, while it would not be capable of receiving continuous motion from a pinion having 4 teeth; and as the requirements of machine construction often call for the transmission of motion from one pinion to another of equal diameters, and as small as possible, 12 teeth are the smallest number it is considered desirable for a pinion to contain, except it be in the case of an internal wheel, in which the arc of contact is greater in proportion to the diameters than in spur-wheels, and continuous motion can therefore be transmitted either with coarser pitches or smaller diameters of pinion.
For convenience in calculating the pitch diameter at pitch circle, or pitch diameter as it is termed, and the number of teeth of wheels, the following rules and table extracted from the Cincinnati Artisan and arranged from a table by D. A. Clarke, are given. The first column gives the pitch, the following nine columns give the pitch diameters of wheels for each pitch from 1 tooth to 9. By multiplying these numbers by 10 we have the pitch diameters from 10 to 90 teeth, increasing by tens; by multiplying by 100 we likewise have the pitch diameters from 100 to 900, increasing by hundreds.
TABLE FOR DETERMINING THE RELATION BETWEEN
PITCH DIAMETER, PITCH, AND NUMBER OF TEETH
IN GEAR-WHEELS.
| Pitch. | Number of Teeth. | |||||||||
| 1. | 2. | 3. | 4. | 5. | 6. | 7. | 8. | 9. | ||
| 1 | .3183 | .6366 | .9549 | 1.2732 | 1.5915 | 1.9099 | 2.2282 | 2.5465 | 2.8648 | |
| 1 | 1⁄8 | .3581 | .7162 | 1.0743 | 1.4324 | 1.7905 | 2.1486 | 2.5067 | 2.8648 | 3.2229 |
| 1 | 1⁄4 | .3979 | .7958 | 1.1937 | 1.5915 | 1.9894 | 2.3873 | 2.7852 | 3.1831 | 3.5810 |
| 1 | 3⁄8 | .4377 | .8753 | 1.3130 | 1.7507 | 2.1884 | 2.6260 | 3.0637 | 3.5014 | 3.9391 |
| 1 | 1⁄2 | .4775 | .9549 | 1.4324 | 1.9099 | 2.3873 | 2.8648 | 3.3422 | 3.8197 | 4.2971 |
| 1 | 5⁄8 | .5173 | 1.0345 | 1.5517 | 2.0690 | 2.5862 | 3.1035 | 3.6207 | 4.1380 | 4.6552 |
| 1 | 3⁄4 | .5570 | 1.1141 | 1.6711 | 2.2282 | 2.7852 | 3.3422 | 3.8993 | 4.4563 | 5.0134 |
| 1 | 7⁄8 | .5968 | 1.1937 | 1.7905 | 2.3873 | 2.9841 | 3.5810 | 4.1778 | 4.7746 | 5.3714 |
| 2 | .6366 | 1.2732 | 1.9099 | 2.5465 | 3.1831 | 3.8197 | 4.4563 | 5.0929 | 5.7296 | |
| 2 | 1⁄8 | .6764 | 1.3528 | 2.0292 | 2.7056 | 3.3820 | 4.0584 | 4.7348 | 5.4112 | 6.0877 |
| 2 | 1⁄4 | .7162 | 1.4324 | 2.1486 | 2.8648 | 3.5810 | 4.2972 | 5.0134 | 5.7296 | 6.4457 |
| 2 | 3⁄8 | .7560 | 1.5120 | 2.2679 | 3.0239 | 3.7799 | 4.5359 | 5.2919 | 6.0479 | 6.8038 |
| 2 | 1⁄2 | .7958 | 1.5915 | 2.3873 | 3.1831 | 3.9789 | 4.7746 | 5.5704 | 6.3662 | 7.1619 |
| 2 | 5⁄8 | .8355 | 1.6711 | 2.5067 | 3.3422 | 4.1778 | 5.0133 | 5.8499 | 6.6845 | 7.5200 |
| 2 | 3⁄4 | .8753 | 1.7507 | 2.6260 | 3.5014 | 4.3767 | 5.2521 | 6.1274 | 7.0028 | 7.8781 |
| 2 | 7⁄8 | .9151 | 1.8303 | 2.7454 | 3.6605 | 4.5757 | 5.4908 | 6.4059 | 7.3211 | 8.2362 |
| 3 | .9549 | 1.9099 | 2.8648 | 3.8197 | 4.7746 | 5.7296 | 6.6845 | 7.6394 | 8.5943 | |
| 3 | 1⁄4 | 1.0345 | 2.0690 | 3.1035 | 4.1380 | 5.1725 | 6.2070 | 7.2415 | 8.2760 | 9.3105 |
| 3 | 1⁄2 | 1.1141 | 2.2282 | 3.3422 | 4.4563 | 5.5704 | 6.6845 | 7.7986 | 8.9126 | 10.0268 |
| 3 | 3⁄4 | 1.1937 | 2.3873 | 3.5810 | 4.7746 | 5.9683 | 7.1619 | 8.3556 | 9.5493 | 10.7429 |
| 4 | 1.2732 | 2.5465 | 3.8197 | 5.0929 | 6.3662 | 7.6394 | 8.9127 | 10.1839 | 11.4591 | |
| 4 | 1⁄2 | 1.4324 | 2.8648 | 4.2972 | 5.7296 | 7.1619 | 8.5943 | 10.0267 | 11.4591 | 12.8915 |
| 5 | 1.5915 | 3.1831 | 4.7746 | 6.3662 | 7.9577 | 9.5493 | 11.1408 | 12.7324 | 14.3240 | |
| 5 | 1⁄2 | 1.7507 | 3.5014 | 5.2521 | 7.0028 | 8.7535 | 10.5042 | 12.2549 | 14.0056 | 15.7563 |
| 6 | 1.9099 | 3.8196 | 5.7295 | 7.6394 | 9.5493 | 11.4591 | 13.3690 | 15.2788 | 17.1887 | |