What is known as the Willis system for interchangeable gearing, consists of using for every pitch of the teeth a generating circle whose diameter is equal to the radius of a pinion having 12 teeth, hence the pinion will in each pitch have radial flanks, and the roots of the teeth will be more spread as the number of teeth in the wheel is increased. Twelve teeth is the least number that it is considered practicable to use; hence it is obvious that under this system all wheels of the same pitch will work correctly together.
Unless the faces of the teeth and the flanks with which they work are curves produced from the same size of generating circle, the velocity of the teeth will not be uniform. Obviously the revolutions of the wheels will be proportionate to their numbers of teeth; hence in a pair of wheels having an equal number of teeth, the revolutions will per force be equal, but the driver will not impart uniform motion to the driven wheel, but each tooth will during the path of contact move irregularly.
The velocity of a pair of wheels will be uniform at each instant of time, if a line normal to the surfaces of the curves at their point of contact passes through the point of contact of the pitch circles on the line of centres of the wheels. Thus in Fig. 41, the line A A is tangent to the teeth curves where they touch, and D at a right angle to A A, and meets it at the point of the tooth curves, hence it is normal to the point of contact, and as it meets the pitch circles on the line of centres the velocity of the wheels will be uniform.
The amount of rolling motion of the teeth one upon the other while passing through the path of contact, will be a minimum when the tooth curves are correctly formed according to the rules given. But furthermore the sliding motion will be increased in proportion as the diameter of the generating circle is increased, and the number of teeth in contact will be increased because the arc, or path, of contact is longer as the generating circle is made larger.
Thus in Fig. 42 is a pair of wheels whose tooth curves are from a generating circle equal to the radius of the wheels, hence the flanks are radial. The teeth are made of unusual depth to keep the lines in the engraving clear. Suppose V to be the driver, W the driven wheel or follower, and the direction of motion as at P, contact upon tooth A will begin at C, and while A is passing to the line of centres the path of contact will pass along the thickened line to X. During this time the whole length of face from C to R will have had contact with the length of flank from C to N, and it follows that the length of face on A that rolled on C N can only equal the length of C N, and that the amount of sliding motion must be represented by the length of R N on A, and the amount of rolling motion by the length N C. Again, during the arc of recess (marked by dots) the length of flank that will have had contact is the depth from S to Ls, and over this depth the full length of tooth face on wheel V will have swept, and as L S equals C N, the amount of rolling and of sliding motion during the arc of recess is equal to that during the arc of approach, and the action is in both cases partly a rolling and partly a sliding one. The two wheels are here shown of the same diameter, and therefore contain an equal number of teeth, hence the arcs of approach and of recess are equal in length, which will not be the case when one wheel contains more teeth than the other. Thus in Fig. 43, let A represent a segment of a pinion, and B a segment of a spur-wheel, both segments being blank with their pitch circles, the tooth height and depth being marked by arcs of circles. Let C and D represent the generating circles shown in the two respective positions on the line of centres. Let pinion A be the driver moving in the direction of P, and the arc of approach will be from E to X along the thickened arc, while the arc of recess will be as denoted by the dotted arc from X to F. The distance E X being greater than distance X F, therefore the arc of approach is longer than that of recess.