The teeth of a mangle-wheel are, however, most commonly formed by pins projecting from the face of the disc as in Fig. 210. In this manner the pitch circles for the inner and outer wheels coincide, and therefore the velocity ratio is the same within and without, also the space through which the pinion moves in shifting is reduced.
This space may be still further reduced by arranging the teeth as in Fig. 211, that is, by placing the spur-wheel within the annular or internal one; but at the same time the difference of the two velocity ratios is increased.
If it be required that the velocity ratio vary, then the pitch lines of the mangle-wheel must no longer be concentric.
Thus in Fig. 212 the groove _k_ _l_ is directed to the centre of the mangle-wheel, and therefore the pinion will proceed during this portion of its path without giving any motion to the wheel, and in the other lines of teeth the pitch radius varies, hence the angular velocity ratio will vary.
In Figs. 209, 210, and 211 the curves of the teeth are readily obtained by employing the same describing circle for the whole of them. But when the form Fig. 212 is adopted, the shape of the teeth requires some consideration.
Every tooth of such a mangle-wheel may be considered as formed of two ordinary teeth set back to back, the pitch line passing through the middle. The outer half, therefore, appropriated to the action of the pinion on the outside of the wheel, resembles that portion of an ordinary spur-wheel tooth that lies beyond its pitch line, and the inner half which receives the inside action of the pinion resembles the half of an annular wheel that lies within the pitch circle. But the consequence of this arrangement is, that in both positions the action of the driving teeth must be confined to the approach of its teeth to the line of centres, and consequently these teeth must be wholly within their pitch line.
To obtain the forms of the teeth, therefore, take any convenient describing circle, and employ it to describe the teeth of the pinion by rolling within its pitch circle, and to describe the teeth of the wheel by rolling within and without its pitch circle, and the pinion will then work truly with the teeth of the wheel in both positions. The tooth at each extremity of the series must be a circular one, whose centre lies on the pitch line and whose diameter is equal to half the pitch.