Numerical Evaluation of Impact Characteristics of Cue in Billiards

Abstract

Impact characteristics of a cue in billiards are evaluated numerically by developing a simple method. It is assumed that the cue and ball are elastic and rigid, respectively. The displacement of the cue is evaluated by the finite element method and that of the ball is calculated by solving the equation of motion. The time variation of impact force is estimated by equating these displacements at contact point. The accuracy of the analysis is checked by performing a detailed finite element analysis and an experiment, in which a freely supported cue is hit with a swinging ball. It is shown that (1) the impact force and the ball velocity increase with increase in the stiffness of the shaft, (2) the contact period decreases with increase in the stress wave velocity in the cue, (3) with increase in the stiffness of the tap, the impact force rises rapidly, reaches a large maximum value and falls in a short time (4) the ratio of slip to rotation of a ball during the contact period increases with increase in the stiffness of the shaft and the tap and (5) a peak appears in the time variation of impact force for a soft shaft due to the metal screw in the middle of the shaft.