Fig. 67.
Now instead of rolling the intermediate describing circle r within the annular wheel p for the face curves of the teeth upon p, we may find some other circle that will give the same curve and be small enough to be rolled within the pinion q for its teeth flanks. Thus in [Fig. 67] p represents the pitch circle of the annular wheel and r the intermediate circle, and if r be rolled within p, a point on the circumference of r will trace the curve v w. But if we take the circle s, having a diameter equal to the difference between the diameter of r and that of p, and roll it within p, a point in its circumference will trace the same curve v w; hence s is a perfect substitute for r, and a portion of the curve v w may be used for the faces of the teeth on the annular wheel. The circle s being used for the pinion flanks, the wheel faces and pinion flanks will work correctly together, and as the circle s is rolled within the pinion for its flanks and within the wheel for its faces, it may be distinguished as the interior describing circle.
To prove the correctness of the construction it may be noted that with the particular diameter of intermediate describing circle used in [Fig. 65], the interior and exterior describing circles are of equal diameters; hence, as the same diameter of describing circle is used for all the faces and flanks of the pair of wheels they will obviously work correctly together, in accordance with the rules laid down for spur gearing. The radius of s in [Fig. 69] is equal to the radius of the annular wheel, less the radius of the intermediate circle, or the radius from a to c. The radius of the exterior describing circle t is the radius of the intermediate circle less the radius of the pinion, or radius c b in the figure.
Fig. 68.
Now the diameter of the intermediate circle may be determined at will, but cannot exceed that of the annular wheel or be less than the pinion. But having been selected between these two limits the interior and exterior describing circles derived from it give teeth that not only engage properly and avoid the interference shown in [Fig. 62], but that will also have an additional arc of action during the recess, as is shown in [Fig. 68], which represents the wheel and pinion shown in [Fig. 62], but produced by means of the interior and exterior describing circles. Supposing the pinion to be the driver the arc of approach will be along the thickened arc of the interior describing circle, while during the arc of recess there will be an arc of contact along the dotted portion of the exterior describing circle as in ordinary gearing. But in addition there will be an arc of recess along the dotted portion of the intermediate circle r, which arc is due to the faces of the pinion acting upon the faces as well as upon the flanks of the wheel teeth. It is obvious from this that as soon as a tooth passes the line of centres it will, during a certain period, have two points of contact, one on the arc of the exterior describing circle, and another along the arc of r, this period continuing until the addendum circle of the pinion crosses the dotted arc of the exterior describing circle at z.