In [Fig. 31] is shown the case of a rack and pinion; a a is the pitch line of the rack, b b that of the pinion, a b at a right angle to a a, the line of centres, and d the generating circle. The wheel and rack are shown with teeth n on one side simply for clearness of illustration. The pencil point n will, on arriving at m, have traced the flank curve x and the curve y for the face of the rack teeth.
Fig. 32.
It has been supposed that the three circles rotated together by the frictional contact of their perimeters on the line of centres, but the circumstances will remain the same if the wheels remain at rest while the generating or describing circle is rolled around them. Thus in [Fig. 32] are two segments of wheels as before, c representing the centre of a tooth on a a, and d representing the centre of a tooth on b b. Now suppose that a generating or rolling circle be placed with its pencil point at e, and that it then be rolled around a a until it had reached the position marked 1, then it will have marked the curve from e to n, a part of this curve serving for the face of tooth c. Now let the rolling circle be placed within the pitch circle a a and its pencil point n be set to e, then, on being rolled to position 2, it will have marked the flank of tooth c. For the other wheel suppose the rolling wheel or circle to have started from f and rolled to the line of centres as in the cut, it will have traced the curve forming the face of the tooth d. For the flank of d the rolling circle or wheel is placed within b b, its tracing point set at f on the pitch circle, and on being rolled to position 3 it will have marked the flank curve. The curves thus produced will be precisely the same as those produced by rotating all three wheels about their axes, as in our previous demonstrations.
The curves both for the faces and for the flanks thus obtained will vary in their curvature with every variation in either the diameter of the generating circle or of the base or pitch circle of the wheel. Thus it will be observable to the eye that the face curve of tooth c is more curved than that of d, and also that the flank curve of d is more spread at the root than is that for c, which has in this case resulted from the difference between the diameter of the wheels a a and b b. But the curves obtained by a given diameter of rolling circle on a given diameter of pitch circle will be correct for any pitch of teeth that can be used upon wheels having that diameter of pitch circle. Thus, suppose we have a curve obtained by rolling a wheel of 20 inches circumference on a pitch circle of 40 inches circumference—now a wheel of 40 inches in circumference may contain 20 teeth of 2 inch arc pitch, or 10 teeth of 4 inch arc pitch, or 8 teeth of 5 inch arc pitch, and the curve may be used for either of those pitches.
Fig. 33.
If we trace the path of contact of each tooth, from the moment it takes until it leaves contact with a tooth upon the other wheel, we shall find that contact begins at the point where the flank of the tooth on the wheel that drives or imparts motion to the other wheel, meets the face of the tooth on the driven wheel, which will always be where the point of the driven tooth cuts or meets the generating or rolling circle of the driving tooth. Thus in [Fig. 33] are represented segments of two spur-wheels marked respectively the driver and the driven, their generating circles being marked at g and g′, and x x representing the line of centres. Tooth a is shown in the position in which it commences its contact with tooth b at b. Secondly, we shall find that as these two teeth approach the line of centres x, the point of contact between them moves or takes place along the thickened arc or curve c x, or along the path of the generating circle g.