2014 STLE Annual Meeting & Exhibition
May 18-22, 2014
Disney’s Contemporary Resort
Lake Buena Vista, Florida, USA
EXPERIMENTAL COEFFICIENT OF FRICTION FOR THE IMPACT OF A
TENNIS BALL
Track or Catagory
Control Id: 1834053 and Current Topic: Lubrication Fundamentals
Authors and Institutions
Cermik, Ozdes1; Ghaednia, Hamid1; Marghitu, Dan1
1. Mechanical Engineering, Auburn University, Auburn, AL, United States.
INTRODUCTION
The impact of a tennis ball with a surface has been mostly studied for the
normal impact. Garwin [1] and Brody [2] have studied the physics of the oblique
impact. Brody [2] studied the oblique impact of a tennis ball on tennis courts. He
used Newtonian mechanics with the assumption of constant coefficient of
restitution, and the tennis ball was considered rigid. However, it has been shown
that the coefficient of restitution reduces with the increase of the initial velocity for
the impacts with a rigid surface, [3; 4; 5]. The increase of the initial velocity
before the impact reduces the coefficient of restitution for the impact of a tennis
ball with a clamped tennis racket, [3; 4]. The coefficient of restitution of a tennis
ball is higher for the oblique impacts than for the normal impacts as stated in, [6;
7; 8].
Cross [9] studied the theoretical oblique impact of a tennis ball with the
strings to analyze the influence of changing the coefficient of friction. He used a
similar approach as Brody. The coefficient of friction was obtained from sliding
experiments not from a ball-racket impact situation. His model shows the
coefficient of friction affects the rebound characteristics of the tennis ball. The
coefficient of sliding friction below around 0.3 is critical and small decrease
causes large change in the rebound angle of the tennis ball.
In this study, an experimental set up was built in order to measure the
coefficient of restitution and the coefficient of friction between the tennis ball and
the racket. A robotic arm has been used to drop the tennis ball vertically from
different heights in the range of 0.025 m to 1.031 m on a fixed racket.
0∘ , 15∘ , 30∘ , 45∘ , 57∘ initial impact angles have been used and tested in order to
determine the effects on the coefficient of restitution and the coefficient of friction.
The motion of the ball bas been recorded with a high-speed camera with 10 000
frames/second and analyzed image processing method. The impact interval was
divided into compression and restitution phases. For each phase an expression
for contact force was determined.
EXPERIMENTAL RESULTS
The center of the ball was tracked by Image processing method. The
position of the center of the ball has been calculated before, during and after the
impact in the global [𝑖! , 𝑗! , 𝑘! ] and local coordinates [i, j, k].
Experimental results for the Coefficients of Restitution and Friction
The coefficient of restitution has been calculated as the ratio of the normal
velocity after and the normal velocity before the impact for different initial
velocities and different impact angles. The coefficient of restitution is constant for
different impact angles β and has a value of 𝑒 = 0.88 as seen in Fig 1a.
1
0.5
0.9
0.4
Coefficient of Friction
Coefficient of Restitution
The coefficient of friction is calculated from the tangential and the normal
velocity before and after the impacts. The coefficient of friction is the ratio
between the tangential and the normal impulses. Figure 1b shows the variation of
the coefficient of friction with respect to the angle of the racket. The results show
that the coefficient of friction increases as the impact angle increases.
0.8
0.7
0.6
0.5
0
Experiments
Average
10
20
30
Impact angle,
40
(a)
50
60
Experiments
Average
0.3
0.2
0.1
0
0
10
20
30
Impact angle,
40
50
60
(b)
Fig. 1. (a) Averaged coefficient of restitution for different impact angles β. (b) Variation of the
coefficient of friction for different impact angles β.
Contact Force Coefficients
In order to verify the theory for the oblique impacts, the contact force
coefficients, k and b need to be calculated. The results for the normal impact has
been used in order to find the contact force coefficients. The mean absolute
errors of the displacement, and the error of the velocity after the impact have
been taken into consideration. This process has been done for 15 experiments
with β = 0∘ (normal impact). The sum of the errors between the experiments and
the theory for the displacement and velocity has been calculated. The minimum
error has been selected for the calculation of k and b. The contact force
coefficients have been calculated by averaging all of the calculated coefficients in
each case.
Oblique Impact
The normal and tangential components of the displacement were
compared for the theory and experiments. The final experimental velocity was
also compared with the theory. The error for both normal and tangential
component of the velocity and the displacement in all cases is less than 10%.
YC (m)
0.02
0.01
0
−0.01
−0.01
β=45
before
impact
0.02
Experiments
Theory
o
impact
−0.005
0.01
XC (m)
0.03
after impact
0
β=45
before
impact
o
Experiments
Theory
impact
after impact
−0.01
0
0.005
Time (sec)
(a)
0.01
0.015
−0.02
−0.01
−0.005
0
0.005
Time (sec)
0.01
0.015
(b)
Fig. 2. Comparison between the theory and the experiments for normal and tangential
components of the displacement. (a) The normal component for β=45∘ . (b) The tangential
component for β=45∘
The errors between the theory and the experiment for the normal and tangential
components of the velocity are 1.9% and 6.2% respectively when impact angle
𝛽 = 45∘ .
CONCLUSION
Experimental results show that the coefficient of restitution is constant for
different impact angles and different low velocities. The coefficient of friction is
increasing when the impact angle increases; however, it is constant for different
velocities for the same impact angle within our initial velocity range.
Normal impact experiments have been used in order to obtain the contact
force coefficients. The coefficients are used for the simulation of the oblique
impacts. The theory has been compared with the experiments for β=
15∘ , 30∘ , 45∘ , 57∘ . Both the displacement and final velocity are analyzed. The
simulation results are in a good agreement with the experimental results.
KEYWORDS
Friction: Friction Test Methods, Coefficient of Friction, Digital Image Processing,
Impact
REFERENCES
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1969;37(1):88–92.
[2] Brody H. That’s how the ball bounces. The Physics Teacher. 1984;(22):494–497.
[3] Cross R. Dynamic properties of tennis balls. Sports Engineering. 1999;2(1):23–34.
[4] Haake SJ, Carre MJ, Goodwill SR. The dynamic impact characteristics of tennis balls
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[8] Cross R. Measurements of the Horizontal coefficient of restitution for a superball and
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[9] Cross R. Effects of friction between the ball and strings in tennis. Sports Engineering.
2000;3(2):85–97.