Lapping of ultra-precision ball surfaces. Part II. Eccentric V-groove lapping system

A general closed-form analytical solution is derived for the kinematics of the eccentric V-groove lapping system. Since, the ball-spin angular speed with its angle, continuously varies with time for the eccentric lapping system and its oscillation increases with increasing eccentricity, lapping tracks are able to be distributed over the whole ball surface without the interrupted process, therefore, the efficiency of ball lapping becomes higher. Consequently, an approach is proposed to linearize the equation of motion in evaluating the lapping tracks, where the spin vector is assumed to be constant in each time step. Results show that the lapped area ratio is significantly influenced by the spin angular speed and its increment rate. For the spin angle linearly varied with time, it is interesting to find that the lapped area ratio is significantly influenced by the ratio of the total frequency to the increment rate of the spin angle, wt/wc. With increasing wt/|wc|, the lapped area ratio increases quickly to a saturated value close to one except wt/|wc|∈N. For a triangular periodic function of the spin angle with frequency, w, and the amplitude, a, the lapped area ratio varies with wt/w very quickly for wt/w less than two, and as wt/w is larger than eight, it approaches to a saturated value except wt/w∈N. Since, the spin angular speed with its angle continuously varies with time for the eccentric lapping system, the lapped areas are complementary at the contact points of A and B. The total lapped area ratio is higher than 87% for a slip ratio less than 0.5. Hence, it is possible to lap all the surface of a ball by changing the slip ratio during the lapping process. The total lapped area increases to a saturated value with increasing eccentricity. The eccentricity is suggested to be 2Rb for θ=45° and Rg/Rb=5. This result is in very good agreement with the Zhang's experiment [B. Zhang, N. Umehara, K. Kato, Effect of the eccentricity between the driving shaft and the guide ring on the behavior of magnetic fluid grinding of ceramic balls, J. JSPE 61 (4) (1995) 586–590 (in Japanese)].