COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Numerical Analysis of Impact between Cue and Ball in Billiard
Shinsuke Shimamura 1*, Shigeru Aoki 2
1
2
Graduate,School of Engineering. Sci. Eng., Toyo University, Kujirai 2100, Kawagoe, Saitama350-8585 Japan
Toyo University Dept.of Comp. Sci. Eng., Toyo University, Kujirai 2100, Kawagoe, Saitama350-8585 Japan
Email: windy-scene@hotmail.co.jp, saoki@eng.toyo.ac.jp
Abstract A finite element analysis and an experiment on the impact between a cue and a ball in billiard are carried out.
It is assumed that the cue is elastic and the ball is rigid. The displacement of the cue is calculated by FEM and that of
the ball is estimated by solving the equation of motion. The contact force is determined by equating the sum of these
displacements to the impact velocity times duration. An experiment, in which a freely supported cue is hit with a
swinging ball, is performed to examine the accuracy of the analysis. It is shown that (1) the computational results on
the time variation of contact force agree well with the experimental ones, (2) the contact period is reduced by mounting
the hard plastic collar near the tip of the cue, and (3) the contact force rises steeply with the mounted hard plastic collar.
Key words: Impact, Billiard, Cue, Ball, FEM
INTRODUCTION
The billiard is a popular game widely played in the world. The cue in billiard consists of the tap, collar and shaft as
shown in Fig. 1, and they are generally made of leather, plastic and wood, respectively. While many researches have
been reported for optimal design of the racket in tennis and the club in golf, few researches for the cue in billiard have
been published[1], and the shape, size and materials of their parts are usually determined from experience.
It is inferred that the time variations of the impact force between a cue and a ball, the rotation of the ball and the slip
between the ball and the table during and immediately after the impact depend on the design of the cue. It is therefore
important to make a clear understanding of their dependence for the design of the cue.
The objective of the present research is to make clear the effect of Young’s moduli of the collar of a cue on the time
variations of the impact force, the rotation and the slip of a ball by using a simple analysis.
Figure 1: Head part of Cue
SIMPLE ANALYSIS FOR IMPACT BETWEEN CUE AND BALL
The validity of the assumptions, which are made to make the analysis simple, are examined in this chapter by
considering a simple impact between a cue and a ball.
⎯ 1145 ⎯
1. Experiment Fig. 2 shows the outline of the experiment. A ball, which is hung with two thin threads like a pendulum
as shown in Fig. 3, is released from a prescribed angle ϕ = 30° and hit the center of the tap of a cue sustained
horizontally with soft sponge. The impact speed v is estimated to be 3 m/s. The output of a strain gage mounted on the
collar of the cue is recorded in a oscilloscope via a bridge box and strain meter. The experimental result is shown as a
solid curve in Fig. 4
Figure 2: Outline of the experiment
Figure 3: Photo of cue and ball in a simple impact experiment
Figure 4: Experimental and analytical result
2. Analysis Let and x(t) represent the displacements in the horizontal (axial) direction of the cue and the ball,
respectively, as shown in Fig. 5. The following equation is obtained:
(1)
and x(t) depend on the time variation of the impact force
. Eq. (1) is solved for
by using the
The
finite difference method with the finite element method etc. The strain at the location of the strain gage is calculated
from the obtained
.
⎯ 1146 ⎯
Figure 5: Experimental and analytical
3. Comparison of the results by various methods The results, which are obtained with various methods, are
compared in Fig. 4, where the abscissa t represents the time from the beginning of the impact. The finite element mesh
is shown in Fig. 6.
The dotted curve corresponds to the results calculated by assuming that the ball is assumed to be rigid and the impact
force is a point force. It is found that the dotted curve is in good agreement with the experimental result. Based on this
result, the method used for the dotted curve is employed hereafter.
Figure 6: FEM model and result of deformation (t = 0.00098sec)
Figure 7: Effect of collar hardness on time variation of contact force
⎯ 1147 ⎯
SIMPLE ANALYSIS FOR IMPACT BETWEEN CUE AND BALL
The analysis in the previous chapter is extended to the impact of a cue against a ball on a table. The frictional force
between a ball and a table is taken into account, and the contact point is not the center of a ball (α = 0), but the upper
part of a ball (α = 26.5°). An analysis is carried out for two different Young’s models of the collar. The analytical
results on the time variations of contact force and the displacement of a cue are shown in Figs. 7 and 8, respectively. It
is found from these results that (1) the displacement of a cue decreases, (2) the contact force rises steeply and (3) the
contact period is reduced with increase in the hardness of the collar.
Figure 8: Effect of collar hardness on displacement in x direction of cue
CONCLUSION
An analysis was carried out for the impact between a cue and a ball in the billiard to achieve a better understanding of
the effect of the collar hardness of the cue. The accuracy of the analysis is checked by performing a simple impact test.
It is found in this study that (1) the displacement of a cue becomes small, (2) the contact force rises steeply and (3) the
contact period is reduced with increase in the collar hardness of a cue.
REFERENCES
1. Toda S. Mechanics. Iwanami Shoten, Tokyo, Japan, 1994 (in Japanese).
⎯ 1148 ⎯