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A theoretical analysis of billiard ball dynamics under cushion impacts

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journal contribution
posted on 02.07.2014 by Senthan Mathavan, Michael Jackson, Robert M. Parkin
The last two decades have seen a growing interest in research related to billiards. There have been a number of projects aimed at developing training systems, robots, and computer simulations for billiards. Determination of billiard ball trajectories is important for all of these systems. The ball’s collision with a cushion is often encountered in billiards and it drastically changes the ball trajectory, especially when the ball has spin. This work predicts ball bounce angles and bounce speeds for the ball’s collision with a cushion, under the assumption of insignificant cushion deformation. Differential equations are derived for the ball dynamics during the impact and these equations are solved numerically. The numerical solutions together with previous experimental work by the authors predict that for the ball–cushion collision, the values of the coefficient of restitution and the sliding coefficient of friction are 0.98 and 0.14, respectively. A comparison of the numerical and experimental results indicates that the limiting normal velocity under which the rigid cushion assumption is valid is 2.5 m/s. A number of plots that show the rebound characteristics for given ball velocity–spin conditions are also provided. The plots quantify various phenomena that have hitherto only been described in the billiards literature.

History

School

  • Mechanical, Electrical and Manufacturing Engineering

Published in

PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART C-JOURNAL OF MECHANICAL ENGINEERING SCIENCE

Volume

224

Issue

C9

Pages

1863 - 1873 (11)

Citation

MATHAVAN, S., JACKSON, M.R. and PARKIN, R.M., 2010. A theoretical analysis of billiard ball dynamics under cushion impacts. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 224 (9), pp. 1863 - 1873.

Publisher

Sage Publications / © The Authors

Version

SMUR (Submitted Manuscript Under Review)

Publication date

2010

Notes

This article was published in the journal, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science [Sage Publications on behalf of IMechE / © The Authors]. The definitive version is available at: http://dx.doi.org/10.1243/09544062JMES1964

ISSN

0954-4062

Language

en

Exports