Mechanical analysis of racket and ball during impact.

Several investigators have demonstrated experimentally that ball-rebound velocity after an eccentric impact against a tennis racket remains unchanged for two extreme conditions of grip firmness, i.e., when the grip is firmly clamped and when it is allowed to stand freely on its butt. The present study utilized a simple mathematical model from classical impact theory to provide analytical support for their experimental findings. It was shown that the functional relationship between the approach and rebound velocity of the ball is dependent on five dimensionless numbers: 1) the ratio of the ball to racket mass; 2) the ratio of the radius of gyration about the racket pivot to the distance of the geometric center of the racket head to the pivot; 3) the coefficient of restitution between the ball and the racket string ensemble; 4) the ratio of the distances of the center of mass and the center of the strings to the pivot; and 5) the ratio of the grip length to the distance from the pivot to the center of the strings. Because the mass and length ratios are very small numbers for tennis, the rebound-to-approach velocity of the ball is principally a function of the coefficient of restitution, which is practically independent of the conditions of grip firmness. Using published data generated from other experiments, analytical estimates were obtained for the values of the coefficient of restitution between a tennis ball and a racket strung to typical tensions for various rebound-to-approach velocity ratios. These estimates were validated directly by an independent experiment.