Fig. 49.
It has been shown, when referring to [Figs. 42] and [44], when treating of the amount of sliding and of rolling motion, that the smaller the diameter of rolling circle in proportion to that of pitch circle, the longer the acting length of flank and the more the amount of rolling motion; and it follows that the teeth would also preserve their original and true shape better. But the wear of the teeth, and the alteration of tooth form by reason of that wear, will, in any event, be greater upon the pinion than upon the wheel, and can only be equal when the two wheels are of equal diameter, in which case the tooth curves will be alike on both wheels, and the acting depths of flank will be equal, as shown in [Fig. 47], the flanks being radial, and the acting depths of flank being shown at j k. In [Fig. 48] is shown a pair of wheels with a generating circle, g and g′, of one quarter the diameter of the base circle or pitch diameter, and the acting length of flank is shown at l m. The wear of the teeth would, therefore, in this latter case, cause it in time to assume the form shown in [Fig. 49]. But it is to be noted that while the acting depth of flank has been increased the arcs of contact have been diminished, and that in [Fig. 47] there are two teeth in contact, while in [Fig. 48] there is but one, hence the pressure upon each tooth is less in proportion as the diameter of the generating circle is increased. If a train of wheels are to be constructed, or if the wheels are to be capable of interchanging with other combinations of wheels of the same pitch, the diameter of the generating circle must be equal to the smallest wheel or pinion, which is, under the Willis system, a pinion of 12 teeth; under the Pratt and Whitney, and Brown and Sharpe systems, a pinion of 15 teeth.
But if a pair or a particular train of gears are to be constructed, then a diameter of generating circle may be selected that is considered most suitable to the particular conditions; as, for example, it may be equal to the radius of the smallest wheel giving it radial flanks, or less than that radius giving parallel or spread flanks. But in any event, in order to transmit continuous motion, the diameter of generating circle must be such as to give arcs of action that are equal to the pitch, so that each pair of teeth will come into action before the preceding pair have gone out of action.
It may now be pointed out that the degrees of angle that the teeth move through always exceeds the number of degrees of angle contained in the paths of contact, or, in other words, exceeds the degrees contained in the arcs of approach and recess combined.
Fig. 50.
In [Fig. 50], for example, are a wheel a and pinion b, the teeth on the wheel being extended to a point. Suppose that the wheel a is the driver, and contact will begin between the two teeth d and f on the dotted arc. Now suppose tooth d to have moved to position c, and f will have been moved to position h. The degrees of angle the pinion has been moved through are therefore denoted by i, whereas the degrees of angle the arcs of contact contain are therefore denoted by j.
The degrees of angle that the wheel a has moved through are obviously denoted by e, because the point of tooth d has during the arcs of contact moved from position d to position c. The degrees of angle contained in its path of contact are denoted by k, and are less than e, hence, in the case of teeth terminating in a point as tooth d, the excess of angle of action over path of contact is as many degrees as are contained in one-half the thickness of the tooth, while when the points of the teeth are cut off, the excess is the number of degrees contained in the distance between the corner and the side of the tooth as marked on a tooth at p.