Again, a wheel having 40 teeth of 8 diametral pitch would have a pitch circle diameter of 5 inches, because 40 ÷ 8 = 5, and its full diameter would be 51⁄4 inches, because the diametral pitch is 1⁄8, and this multiplied by 2 gives 1⁄4, which added to the pitch circle diameter of 5 inches makes 51⁄4 inches, which is therefore the diameter of the addendum, or, in other words, the full diameter of the wheel.
Suppose now that a pair of wheels require to have pitch circles of 5 and 8 inches diameter respectively, and that the arc pitch requires to be, say, as near as may be 4⁄10 inch; to find a suitable pitch and the number of teeth by the diametral pitch system we proceed as follows:
In the following table are given various arc pitches, and the corresponding diametral pitch.
| Diametral Pitch. | Arc Pitch. | Arc Pitch. | Diametral Pitch. | ||||
| Inch. | |||||||
| 2 | 1 | .57 | 1 | .75 | 1 | .79 | |
| 2 | .25 | 1 | .39 | 1 | .5 | 2 | .09 |
| 2 | .5 | 1 | .25 | 1 | .4375 | 2 | .18 |
| 2 | .75 | 1 | .14 | 1 | .375 | 2 | .28 |
| 3 | 1 | .04 | 1 | .3125 | 2 | .39 | |
| 3 | .5 | .890 | 1 | .25 | 2 | .51 | |
| 4 | .785 | 1 | .1875 | 2 | .65 | ||
| 5 | .628 | 1 | .125 | 2 | .79 | ||
| 6 | .523 | 1 | .0625 | 2 | .96 | ||
| 7 | .448 | 1 | .0000 | 3 | .14 | ||
| 8 | .392 | 0 | .9375 | 3 | .35 | ||
| 9 | .350 | 0 | .875 | 3 | .59 | ||
| 10 | .314 | 0 | .8125 | 3 | .86 | ||
| 11 | .280 | 0 | .75 | 4 | .19 | ||
| 12 | .261 | 0 | .6875 | 4 | .57 | ||
| 14 | .224 | 0 | .625 | 5 | .03 | ||
| 16 | .196 | 0 | .5625 | 5 | .58 | ||
| 18 | .174 | 0 | .5 | 6 | .28 | ||
| 20 | .157 | 0 | .4375 | 7 | .18 | ||
| 22 | .143 | 0 | .375 | 8 | .38 | ||
| 24 | .130 | 0 | .3125 | 10 | .00 | ||
| 26 | .120 | 0 | .25 | 12 | .56 | ||
From this table we find that the nearest diametral pitch that will correspond to an arc pitch of 4⁄10 inch is a diametral pitch of 8, which equals an arc pitch of .392, hence we multiply the pitch circles (5 and 8,) by 8, and obtain 40 and 64 as the number of teeth, the arc pitch being .392 of an inch. To find the number of teeth and pitch by the arc pitch and circumference of the pitch circle, we should require to find the circumference of the pitch circle, and divide this by the nearest arc pitch that would divide the circumference without leaving a remainder, which would entail more calculating than by the diametral pitch system.
The designation of pitch by the diametral pitch system is, however, not applied in practice to coarse pitches, nor to gears in which the teeth are cast upon the wheels, pattern makers generally preferring to make the pitch to some measurement that accords with the divisions of the ordinary measuring rule.
Fig. 11.
Of two gear-wheels that which impels the other is termed the driver, and that which receives motion from the other is termed the driven wheel or follower; hence in a single pair of wheels in gear together, one is the driver and the other the driven wheel or follower. But if there are three wheels in gear together, the middle one will be the follower when spoken of with reference to the first or prime mover, and the driver, when mentioned with reference to the third wheel, which will be a follower. A series of more than two wheels in gear together is termed a train of wheels or of gearing. When the wheels in a train are in gear continuously, so that each wheel, save the first and last, both receives and imparts motion, it is a simple train, the first wheel being the driver, and the last the follower, the others being termed intermediate wheels. Each of these intermediates is a follower with reference to the wheel that drives it, and a driver to the one that it drives. But the velocity of all the wheels in the train is the same in fact per second (or in a given space of time), although the revolutions in that space of time may vary; hence a simple train of wheels transmits motion without influencing its velocity. To alter the velocity (which is always taken at a point on the pitch circle) the gearing must be compounded, as in [Fig. 11], in which a, b, c, e are four wheels in gear, b and c being compounded, that is, so held together on the shaft d that both make an equal number of revolutions in a given time. Hence the velocity of c will be less than that of b in proportion as the diameter, circumference, radius, or number of teeth in c, varies from the diameter, radius, circumference, or number of teeth (all the wheels being supposed to have teeth of the same pitch) in b, although the rotations of b and c are equal. It is most convenient, and therefore usual, to take the number of teeth, but if the teeth on c (and therefore those on e also) were of different pitch from those on b, the radius or diameters of the wheels must be taken instead of the pitch, when the velocities of the various wheels are to be computed. It is obvious that the compounded pair of wheels will diminish the velocity when the driver of the compounded pair (as c in the figure) is of less radius than the follower b, and conversely that the velocity will be increased when the driver is of greater radius than the follower of the compound pair.