Fig. 29.
In this illustration the diameter of the rolling or describing circle q, being less than the radius of the wheels a a or b b, the flanks of the teeth are curves, and the two wheels being of the same diameter, the teeth on the two are of the same shape. But the principles governing the proper formation of the curve remain the same whatever be the conditions. Thus in [Fig. 28] are segments of a pair of wheels of equal diameter, but the describing, rolling, or curve-generating circle is equal in diameter to the radius of the wheels. Motion is supposed to have occurred in the direction of the arrows, and the tracing point to have moved from n to m. During this motion it will have marked a curve y m, a portion of the y end serving for the face of a tooth on one wheel, and also the line k x, a continuation of which serves for the flank of a tooth on the other wheel. In [Fig. 29] the pitch circles only of the wheels are marked, a a being twice the diameter of b b, and the curve-generating circle being equal in diameter to the radius of wheel b b. Motion is assumed to have occurred until the pencil point, starting from n, had arrived at o, marking curves suitable for the face of the teeth on one wheel and for the flanks of the other as before, and the contact of tooth upon tooth still, at every point in the path of the teeth, occurring at some point of the arc n o. Thus when the point had proceeded as far as point m it will have marked the curve y and the radial line x, and when the point had arrived at o, it will have prolonged m y into o g and x into o f, while in either position the point is marking both lines. The velocities of the wheels remain the same notwithstanding their different diameters, for the arc n g must obviously (if the wheels rotate without slip by friction of their surfaces while the curves are traced) be equal in length to the arc n f or the arc n o.
Fig. 30.
In [Fig. 30] a a and b b are the pitch circles of two wheels as before, and c c the pitch circle of an annular or internal gear, and d is the rolling or describing circle. When the describing point arrived at m, it will have marked the curve y for the face of a tooth on a a, the curve x for the flank of a tooth on b b, and the curve e for the face of a tooth on the internal wheel c c. Motion being continued m y will be prolonged to o g, while simultaneously x will be extended into o f and e into h v, the velocity of all the wheels being uniform and equal. Thus the arcs n v, n f, and n g, are of equal length.
Fig. 31.