# Numerical simulations of the frictional collisions of solid balls on a rough surface

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## Abstract

Three-dimensional simulations of the frictional collision between solid balls moving on a rough surface are analyzed in this paper. The analysis is performed in the context of pool and snooker, two popular games in the pocket billiards family. Accurate simulations of ball motion in billiard games are useful for television broadcasts, training systems and any robotic game playing systems. Studying solid ball collisions in a three-dimensional space requires careful consideration of the different phenomena involved in ball motion such as rolling, sliding and ball spin about a general axis. A set of differential equations are derived describing ball dynamics during collisions. In the absence of explicit analytical solutions to the differential equations, a numerical procedure is performed to determine post-collision ball velocities and spins after collision. In addition, the paper also presents a methodology to analyze the curved, slip trajectories of balls immediately after impact. The results presented here, when compared with some experimental shots, show that the percentile errors in post-collision velocities are reduced by the proposed method. The prediction accuracies for ball travel direction are increased twofold by the proposed impact simulation algorithm.

## Keywords

Impulse with friction Frictional impact Solid balls Billiards Snooker Pool Massè## List of symbols

*e*Coefficient of restitution between the balls

*F*Force

*I*Moment of inertia of the balls

*M*Mass of the balls

*N*Number of iterations

*P*Accumulated impulse at any time during impact

*P*_{I}^{c}Accumulated impulse at the termination of compression

*P*_{I}^{f}The final accumulated value of impulse

*R*Radius of the balls

*s*Slip speed between the balls

*s′*Slip speed between the cue ball and the table

*s″*Slip speed between the object ball and the table

*V*_{0}Incident speed of the ball

*W*Work done due to impulse force

*x*Ball position on the table along the X-axis

*y*Ball position on the table along the X-axis

*z*Ball position on the table along the X-axis

*ΔP*Impulse during a time of

*Δt**θ*The angle that the common normal of the ball–cushion contact point makes with the horizontal

*μ*_{bb}Coefficient of sliding friction between the balls

*μ*_{s}Coefficient of sliding friction between the ball and cushion

*Φ*Direction of slip between the balls

*Φ′*Direction of slip between the cue ball and the table

*Φ″*Direction of slip between the object ball and the table

*ω*Angular speed of the ball

*ω*_{0}^{T}Topspin of the ball at incidence

*ω*_{0}^{S}Sidespin of the ball at incidence

- \(\dot{\omega }_{r}\)
Resistance of the table surface to ball sidespin

## Superscripts

- C
Cue ball

- O
Object ball

## Subscripts

- G
Parameters measured about ball centroid

- I
Along normal impulse between the balls

*n*Iteration number of numerical simulation

- N
Common normal at the point of contact between the balls and the table (for forces)

*N*_{f}Terminal iteration number of numerical simulation

- R
Along the surface at the point of contact between the balls and the table (for frictional forces)

- S
At the end of the slipping phase of a ball

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