Lapping of ultra-precision ball surfaces. Part I. Concentric V-groove lapping system

A general closed-form analytical solution is derived for the lapping tracks with its kinematics for the concentric V-groove lapping system. Based on the kinematics, the theoretical prediction in the prediction of the ball-spin angle β is in very good agreement with the experimental results from the observation of the contact track, where a small slip ratio occurs at the contact points. The lapping tracks on the ball surface for the three contact points are fixed circles, and their lengths of the lapping tracks are linearly proportional to |sin(θ1+β)|, |sin(θ2+β)|, and |cos β|, respectively. To understand the generation mechanism of spherical surface, the ball is enforced to rotate a certain degree of δ with respect to its previous orientation when it enters the lap again. Results show that the lapped area ratio, Lr, is significantly influenced by the angular speed ratio, wb/Ωb, and δ. When the angular speed ratio is integer and δ is small enough or randomized, the lapped area ratios at the contact points of A, B, and C are linearly proportional to [sin θ+sin(θ+2β)], [sin θ+sin(θ−2β)], and (1+cos 2β), respectively. In practice, if the orientation is randomized as the ball enters the lap again, then the distribution of the lapping tracks are dense after many cycles, and the larger the lapping length in each cycle, the smaller is the number of cycles required to achieve the maximum lapped area ratio. In the geometry design of ball lapping, the V-groove half-angle should be larger than 45°, but to prevent the splash of abrasives, it should be less than 75°.