3D billiard dynamics with finite rotation, contact, collision and slip friction

The popular sports entertainment, billiards, is rich in mechanical issues such as 3D finite rotation, collision, contact and friction, evoking a research interest in rigid-body dynamics, nonlinear CAE and computational mechanics. However, the 3D nonlinear behavior of a rigid ball has so far hardly received the attention of scholars to exercise the mechanical modeling skill. The represent study focuses, therefore, on 3D nonlinear billiard dynamics and attempts to precisely predict the ball behavior in finite rotational motion with collision, contact and slip friction. In dependence upon ball situations, 4 different models are proposed. They are namely, rotation model, strike model, collision model and reflection mode. The rotation model is an elementary model and describes the 3D ball motion after a cue strike and subject to table friction only. The strike model is useful to study the effects of cue striking. For simplicity, the strike points are limited to the ball center, 12, 3, 6 and 9 o'clock' in the ball projection. The collision model ignores the ball-to-ball friction and the law of conservation of momentum holds to predict the velocities of two balls after collision. The reflection model simulates the ball-to-cushion contact in bank shots. Incidence angle, translational and rotational velocities, cushion elasticity and frictional properties may be variable in a parameter study. In all these 4 models, the ball is assumed to be in contact to the table surface. Masse or jump shots are excluded in modeling. The equations of ball motion are time-integrated by forward Euler method. The models are verified and validated in numerical examples including optimization and parameter studies. All computed results well agree with the hustler’s experiences in game practice. The present research work will contribute to develop a skill-up program of professional hustlers.