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The dynamic behavior of squash balls

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The behavior of a squash ball constitutes an excellent case study of the dynamic behavior of rubbery materials. It is shown that the complex viscoelastic behavior of rubber can be investigated using simple drop bounce tests and compression tests. The drop tests show that the coefficient of restitution increases as the ball temperature increases. The compression tests show that as the speed of compression increases or as the ball temperature decreases, the compressive force and the energy loss both increase. These effects are due to the viscoelastic nature of the rubber and are an excellent example of the time-temperature equivalence of polymers. Compression tests were performed on balls with small holes at the base to separate the effects of the internal air pressure from the material deformation. It was found that the internal air pressure contributed about one-third to the compressive force, but contributed little to energy loss. This behavior shows that the rubber material dominates the rebound behavior and that the normal warming up process at the start of a squash game is important to raise the temperature of the rubber rather than to increase the internal air pressure.
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The dynamic behavior of squash balls
Gareth J. Lewis, J. Cris Arnold, and Iwan W. Griffiths
School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom
Received 26 February 2010; accepted 23 September 2010
The behavior of a squash ball constitutes an excellent case study of the dynamic behavior of rubbery
materials. It is shown that the complex viscoelastic behavior of rubber can be investigated using
simple drop bounce tests and compression tests. The drop tests show that the coefficient of
restitution increases as the ball temperature increases. The compression tests show that as the speed
of compression increases or as the ball temperature decreases, the compressive force and the energy
loss both increase. These effects are due to the viscoelastic nature of the rubber and are an excellent
example of the time-temperature equivalence of polymers. Compression tests were performed on
balls with small holes at the base to separate the effects of the internal air pressure from the material
deformation. It was found that the internal air pressure contributed about one-third to the
compressive force, but contributed little to energy loss. This behavior shows that the rubber material
dominates the rebound behavior and that the normal warming up process at the start of a squash
game is important to raise the temperature of the rubber rather than to increase the internal air
pressure. ©
2011 American Association of Physics Teachers.
DOI: 10.1119/1.3531971
I. INTRODUCTION
The bouncing of a ball is a familiar phenomenon that in-
volves complex dynamics and materials behavior. Changes
in the physics of the bounce can significantly change the
nature of the activity and the chance of injury in many
sports. The study of ball bouncing is amenable to simple
techniques, is relevant to many students with interests in
sports, and involves a range of physics principles that can be
explored across a wide educational spectrum.
The behavior of squash balls presents an especially inter-
esting case due to the nature of their materials and construc-
tion. Some balls rely mainly on internal air or gas pressure to
provide the dynamic behavior, with tennis balls being the
best example. There are good physics lessons here, with balls
kept cool before use to minimize gas diffusion, balls being
changed after a certain number of games as the internal pres-
sure drops, and novel nanocomposite layers in the wall of
Wilson double-core balls to reduce diffusion.
1
The behavior
of solid balls is determined solely by the ball’s material, with
examples being baseball, cricket, and field hockey balls.
Squash balls are intermediate because they are hollow, but
have thick walls and are not internally pressurized. Their
behavior is therefore partly controlled by the wall material
and partly by the air inside. The rubber used for squash balls
also produces interesting changes of the dynamic behavior
with temperature, which is the reason that squash balls need
to be warmed up before a game starts. It is also possible to
use different grades of squash ball indicated by colored
spots, which give different levels of bounce and can be
matched to the skill and fitness level of the players.
The aim of this paper is to show how the viscoelastic and
temperature dependent nature of rubber can be demonstrated
using simple experiments. The viscoelastic behavior of rub-
ber has applications in tire technology, earthquake protection
bearings, vibration damping, and seals the Challenger Space
Shuttle disaster was caused by such behavior. It is a subject
in which the structure-property relations of a material can be
related from the molecular level how polymer chain
segments move to the macrolevel the mechanical proper-
ties and then to real-world situations. Despite its impor-
tance, it is a difficult subject to convey to students. The fol-
lowing study presents an interesting, amenable, and
alternative way of demonstrating the principles of viscoelas-
tic behavior.
II. BACKGROUND
The most important feature of squash balls is that they
have low rebound resilience. The resilience of a material can
be thought of as its ability to absorb energy elastically on
loading and then to release that energy when the material is
unloaded. When a squash ball makes contact with racket
strings, a wall, or the floor of a court, some of its energy is
stored elastically in the rubber, some in the racket strings,
and some in the increased internal air pressure. Some energy
will be lost as sound, but more of the energy becomes inter-
nal thermal energy in the ball itself. This energy has two
effects—the air inside the ball becomes pressurized and the
rubber compound from which the ball is made becomes more
resilient. As a result, the ball bounces higher. The playing
temperature of the squash ball is usually around 45 ° C,
which is achieved after the ball has been warmed up by the
players.
2
This temperature is where equilibrium is reached,
and the thermal energy lost to the strings, walls, floor, and air
equals the energy gained from deformation.
The rebound resilience is defined as the ratio of the energy
remaining in the ball after an impact to the energy before
impact. It is related to the coefficient of restitution COR,
which is the ratio of speeds before and after impact. Al-
though the rebound resilience is often easier to measure, the
COR is more directly relevant to a squash game because it
governs the speed of a bounce away from a wall and the
floor. A perfectly elastic collision has COR =1 and a rebound
resilience of 100%. In this case, the ball bounces and returns
to the height at which it was dropped. A perfectly inelastic
collision has COR=0, for example, a spherical lump of soft
putty. No balls are perfectly elastic, although hard rubber
“superballs,” solid metal, and glass marbles bouncing on a
rigid and elastic surface come close.
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1 1Am. J. Phys. 79 2, February 2011 http://aapt.org/ajp © 2011 American Association of Physics Teachers
Balls used for all sports exhibit some energy loss due to
damping. Several models have been developed for the be-
havior of balls, which most commonly use combinations of
masses, springs, and dampers to account for energy losses.
35
These models can account reasonably well for the change in
behavior with speed,
3,6
but have not been used to deal with
the effects of changes in temperature.
The most common ways of assessing the properties of
balls are to measure the coefficient of restitution and the
force required to compress the ball. Measurements of the
coefficient of restitution are most commonly made by drop-
ping balls onto hard rigid surfaces and measuring the height
of rebound.
7,8
This height can then be converted into the
speed ratio. Drop tests only give fairly low impact speeds,
which means that measurements at higher speeds require
more advanced measurement systems such as high speed
video photography.
6
The measurement of the coefficient of
restitution is relevant to the impact speed seen during play,
although in squash, a range of speeds are seen, from the high
speed rebound of a hard shot into the wall compared to a
gentle drop-shot against the wall and then onto the court
floor.
9
The other common method is to determine the force re-
quired to compress the ball. The compression behavior is
important because it determines the forces and time scales of
impact. For instance, a more compressible ball will have a
slower impact and lower peak forces, which usually means
lower injury potential. However, more compressible base-
balls have a greater chance of chest injury due to the impact,
which produces resonant vibrations in the chest cavity and
vital organs.
11
A study of lacrosse balls found that more com-
pressible balls had a greater chance of passing through face
guards.
10
Although squash balls have less potential for injury,
there is major concern over impacts near the eye socket. For
this reason, the World Squash Federation specifies a maxi-
mum compression stiffness.
12
Compression tests are done at
much lower speeds than that found in an impact.
By continually measuring the force-displacement behav-
ior, the hysteresis of the ball percentage energy lost during
compression and unloading can be measured, which can
then be related to the dynamic energy losses measured by the
coefficient of restitution.
7,8
Ideally, the speed of compression
should match the impact speed, but this ideal is difficult to
achieve. Scarton et al.
11
developed a dynamic hardness mea-
sure from impact force measurements and claimed that this
measure relates more to injury potential.
The aim of our investigation was to examine the mechan-
ics of two types of squash balls using methods amenable to
most educational institutions. To explore the effects of time
scale and temperature on the energy lost during deformation
of a ball, two test methods were used. Drop tests were used
to study the effects of increasing temperature on the coeffi-
cient of restitution. We also measured the force required to
compress the squash balls as a function of speed and tem-
perature.
Previous research suggests that the elasticity of the rubber
is the determining feature of bounce height rather than the
pressure of the air in the ball.
13
This suggestion was investi-
gated by compressing balls with small holes to allow free
movement of air in and out of the ball. These results allow us
to make comparisons of ball stiffness and rebound resilience
with the World Squash Federation specifications.
12
III. MEASUREMENT OF COEFFICIENT
OF RESTITUTION
We used Dunlop international competition double yellow
dot balls and Dunlop Max blue balls. The double yellow dot
ball is used for competitions and has a low resilience. The
Max blue ball has a diameter of 44.8 mm, which is 12%
larger than the standard size of 40 mm, has a higher resil-
ience and a 40% longer time between vertical bounces than
the double yellow dot, and is designed more for beginners.
The coefficient of restitution of the squash balls was mea-
sured using a simple drop test. The collision was between the
ball and a wooden squash court floor, creating a testing en-
vironment reflecting the nature of the game. The balls were
dropped from the viewing balcony of a standard squash court
onto the court below a distance of 3.55 m. A Sony DCR-
HC62E video camera operating at 50 frames/s on a rigid
tripod was used to record the bounce of the squash ball
against a background includinga1mcalibration rule. Six of
each type of squash ball were tested at different temperatures
achieved by immersion in an electrically powered thermo-
static water bath set to various temperatures. The tempera-
ture of the ball on impact is somewhat lower due to cooling
during the drop, but by dropping balls immediately after re-
moval from the water bath, the cooling is small.
The coefficient of restitution was determined using the
relation
COR =
h
2
h
1
, 1
where h
1
is the drop height and h
2
is the bounce height. The
effects of air resistance were taken into account when calcu-
lating the coefficient of restitutions from rebound heights
using a standard drag coefficient for a sphere and was shown
to give about a 3% difference. Although this correction is not
described here, such a calculation can be used to demonstrate
the use of differential equations and can be solved either
analytically or numerically.
The vertical speeds just before impact about 8.2 m/s and
just after impact were measured from the video images. Al-
though this method is not as precise, the results were in
agreement with the coefficient of restitution calculated from
rebound heights. Figure 1 shows the results, from which it
can be seen that the coefficient of restitution is between 0.45
Fig. 1. The temperature dependence of the coefficient of restitution for two
types of squash balls.
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2 2Am. J. Phys., Vol. 79, No. 2, February 2011 Lewis, Arnold, and Griffiths
and 0.65 resilience between 20% and 35%, increases with
increasing temperature, and is higher for the Max blue balls
than for the double yellow dot balls. In a real squash game,
the ball speeds could be considerably higher. The World
Squash Federation specifications for the double yellow dot
ball states that the resilience should be between 26% and
33%, as measured in a drop test at 45 ° C. Our results give a
value of 27.1% 0.5%, which is just inside the lower limit
set by the World Squash Federation.
IV. COMPRESSION TESTS
Compression tests were performed using a Hounsfield
25 kN mechanical testing machine with balls compressed
between parallel steel plates. The plates were moved together
at a constant rate until the deformation reached 20 mm, at
which the plates were moved apart at the same rate. The
compression rates were 10, 30, 100, 300, and 1000 mm/min.
The testing machine includes a load cell above the top com-
pression plate. This cell comprises of a stiff bending beam,
which is instrumented with strain gauges and allows the
force to be recorded continuously during each test. Although
this type of equipment is standard to materials testing labo-
ratories, not all physics departments have access to this
equipment, and a simpler compression apparatus can be con-
structed using a water tank for loading and a dial gauge, as
shown in Fig. 2. Loading and unloading can be achieved
rapidly by filling and emptying tubes, and a video camera
can be used to record and then analyze a fast moving dial
gauge and the liquid level. The compression to 20 mm was
chosen as being half of the ball diameter. This compression
is realistic because the energy of compression to 20 mm is
later shown to be about 1 J, similar to the kinetic energy of
the squash ball in the drop tests.
For each test speed, two balls of each type were each
tested twice. In this way, each ball was tested twice at each
speed and variability from ball to ball could also be assessed.
The first test conducted on a ball gave a higher force than
subsequent tests due to the well-known Mullins effect,
14
whereby weak bonds formed during manufacture are broken
on the first deformation. Following the first compression,
which was discounted, the results were reproducible, with
standard deviations of the order of 5%. The Mullins effect
was not significant for the drop tests because all balls were
dropped several times before the actual measurements.
Typical force/compression curves are shown in Fig. 3 for
the double yellow dot ball at various test speeds. The two
faster tests shown are offset by 60 and 120 N to show the
behavior clearly on one graph. Figure 3 shows that the force/
compression behavior is not quite linear, with a slight up-
ward curvature. The force on unloading is smaller, giving a
hysteresis loop between loading and unloading. As the test
speed increases, the forces increase and the area of the hys-
teresis loop also increases. Figure 4 shows the force/
compression behavior for the double yellow dot balls and the
Max blue balls, both at a test speed of 100 mm/min. The data
show that the Max blue balls require a slightly higher force
to achieve the same compression, not surprising given their
larger size. There is little difference in the size of the hyster-
esis loops.
Fig. 2. Simple apparatus for measuring compression and recovery.
Fig. 3. Force/compression behavior of
yellow dot balls at different compres-
sion rates.
Fig. 4. Force/compression behavior at 100 mm/min for two types of squash
balls.
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3 3Am. J. Phys., Vol. 79, No. 2, February 2011 Lewis, Arnold, and Griffiths
There has long been considerable debate over the relative
importance of the rubber and internal air for squash balls.
12
To investigate this importance, compression tests were con-
ducted with squash balls with holes. One small hole of about
3 mm diameter was cut in each ball using a scalpel. The ball
was then compressed as described, with the hole carefully
positioned at the base of the test so that the effect of the hole
on the rubber deformation is minimized. Because the pur-
pose of the hole is to allow free passage of air in and out of
the ball, the ball was placed on a steel disk that had a central
hole linked to a groove on the base of the disk, thereby
allowing free passage of air. The force/compression behavior
of the double yellow dot balls with and without holes is
shown in Fig. 5. The data show that the initial behavior is
almost identical. Only when the compression has reached 15
mm does the hole make a significant difference, causing a
decrease in the force. The area of the hysteresis loop is about
the same size. Measurements of the area under the loading
and unloading curves show that the energy lost by a ball with
a hole is only slightly smaller than one without a hole. The
force at 20 mm compression is 82 N without a hole, com-
pared to 50 N with a hole, indicating that at this degree of
compression about 61% of the compression force of a squash
ball arises from rubber deformation, with about 39% arising
from compression of the air inside the ball. More signifi-
cantly, almost all of the energy lost during a compression
comes from deformation of the rubber.
There is a widely held view that the purpose of warming
up a squash ball is to achieve more bounce due to the inter-
nal air warming and the pressure increasing.
2
This interpre-
tation is incorrect because it is the rubber that determines
almost all of the energy loss, and it is the increase in the
rubber temperature that gives a higher bounce.
To demonstrate the effects of deformation speed and tem-
perature on the behavior of squash balls, compression tests
were also conducted with cold balls. Although it is possible
to use a low temperature chamber around the test machine,
such facilities are not that common, and hence a simpler
method was used of precooling the balls in a freezer, fol-
lowed by immediate testing at high speed. Balls were re-
turned to the freezer for at least 1 h between tests. Figure 6
shows the force/compression curves for the double yellow
dot balls at 23 and 0 °C, both at a speed of 1000 mm/min.
The data show that it takes larger forces to compress the cold
ball, and there is also a significantly larger hysteresis loop,
which would mean more energy lost during compression and
so a lower bounce.
Fig. 6. Force/compression behavior at 1000 mm/min for yellow dot balls at
23 and 0 ° C.
Fig. 7. Variation of hysteresis with rate and temperature for yellow dot balls.
Fig. 8. The variation of force at 16 mm compression with rate and tempera-
ture for double yellow dot balls.
Fig. 5. Force/compression behavior at 100 mm/min for yellow dot balls
with and without a hole.
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4 4Am. J. Phys., Vol. 79, No. 2, February 2011 Lewis, Arnold, and Griffiths
V. THE EFFECTS OF TEMPERATURE AND RATE
ON COMPRESSION TESTS
The hysteresis was determined by measuring the area be-
tween the loading and the unloading curves and dividing by
the area under the loading curve. Figure 7 shows the average
hysteresis for the double yellow dot balls plotted against the
loading rate, with the effects of holes and low temperature
also shown. We see that the hysteresis increases slightly as
the rate of compression increases and more significantly as
the temperature decreases. The hysteresis values of the balls
with holes are larger, reflecting the lower energy stored
rather than any more energy lost. The Max blue balls exhib-
ited a similar behavior.
Figure 8 shows the force at 16 mm deformation, which
also increases with the loading rate and with lower tempera-
ture. The balls with holes have considerably lower values
because the contribution from compressing the air is now
absent. Figure 9 shows the stiffness of the balls as deter-
mined from the slope of the loading curve. The stiffness
increases with loading rate and with lower temperature, but
shows little effect of the holes, indicating that the compres-
sion of the air only becomes important with more compres-
sion.
VI. ENERGY LOSS
By measuring the hysteresis loops of the compression tests
and comparing them with the rebound tests, it is possible to
investigate the effects of time scale and temperature on the
percentage of energy lost during deformation. The coefficient
of restitution values were converted to hysteresis as
Hysteresis% = 1 COR
2
100. 2
The time scale of the impacts in the bounce tests was
estimated by comparing the change in momentum from the
coefficient of restitution results with a typical average force
during impact taken to be 50 N from the compression tests.
The average hysteresis values for each test condition are
shown in Table I. We see that as the rate of deformation
increases, the hysteresis increases, and as the temperature
increases, the hysteresis decreases. This dependence is an
excellent example of the time-temperature equivalence of the
time-dependent viscoelastic mechanical behavior of poly-
mers. The mechanical properties arise from the movement of
polymer chains. More polymer chain movement occurs with
slower compression rates more time or at higher tempera-
tures more mobile chains. This mechanism of deformation
leads to the idea that slower compression tests at a low tem-
perature are equivalent to faster compression rates at a higher
temperature.
Fig. 10. The effects of temperature and rate on the hysteresis of rubber.
Fig. 9. The variation of stiffness with rate and temperature for double yel-
low dot balls.
Table I. Hysteresis determined from bounce and compression tests. The last two columns are in percent.
Conditions
Deformation
time scale ms
Hysteresis of double
yellow dot balls
Hysteresis of
Max blue balls
Bounce at 35 ° C 5.5 79.5 73.7
Bounce at 40 ° C 6 74.7 66.1
Bounce at 45 ° C 6 72.9 63.6
Bounce at 50 ° C 6 70.5 62.1
Bounce at 55 ° C 6 67.5 58.4
Compression 10 mm/min 23 ° C 240,000 21.1 21.4
Compression 30 mm/min 23 ° C 80,000 22.1 21.5
Compression 100 mm/min 23 ° C 24,000 23.3 22.4
Compression 300 mm/min 23 ° C 8,000 25.7 23.6
Compression 1000 mm/min 23 ° C 2,400 28.3 25.0
Compression 1000 mm/min 0 ° C 2,400 47.9 43.1
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5 5Am. J. Phys., Vol. 79, No. 2, February 2011 Lewis, Arnold, and Griffiths
The hysteresis of a polymer is related to its glass-transition
temperature T
g
, the region where its behavior changes from a
hard rigid glassy material below T
g
to a softer rubbery ma-
terial above T
g
.
15
Near T
g
the stiffness decreases rapidly
with temperature, and the hysteresis increases.
15
This behav-
ior occurs because some chain segments are flexible, while
some are still more immobile; moving one past another
causes energy losses.
Figure 10 shows a schematic of the effect of time scale
and temperature on hysteresis for rubber, where T
g
defined
as the peak of the curve is below room temperature. The two
curves represent different deformation rates. The faster de-
formation leads to a T
g
that is significantly higher than for
the slow rate. The behavior of squash ball rubber is indicated
by the gray circles. The playing conditions at 45 °C and fast
deformation rates give a large hysteresis. The effects of
warming are indicated by the arrow, where increasing tem-
perature leads to a decrease in hysteresis more bounce. The
much slower compression tests are also shown, which ex-
plains why the hysteresis is less for the compression tests
than for the bounce tests. A more detailed analysis of the
results shows that the hysteresis values from the rebound
tests are higher than those from the compression tests, even
accounting for the effects of loading rate and temperature.
The explanation comes from the differences between the two
types of tests. In the compression test, the ball is compressed
equally on both sides between steel plates at a constant rate.
In the rebound test compression is on just one side, onto a
slightly compliant wooden floor, and the rate of compression
varies through the bounce as the ball slows down and then
rebounds. The equivalent curves for the blue balls would be
shifted slightly to the left because the different rubber com-
pound has a slightly lower value of T
g
.
Figure 10 also shows that if compression tests are to be
used to predict the playing behavior of a squash ball, it is
better to perform the tests cold as the behavior will be closer
to the behavior of a ball during play. This point is more
clearly illustrated in Fig. 11, which shows the ball stiffness
plotted against temperature for two deformation rates. The
stiffness drops dramatically with increasing temperature near
T
g
and does so at a higher temperature for the faster rate. If
the compression tests are used to determine the injury poten-
tial of a squash ball, then compression at low temperatures
would give a better indication of the stiffness under playing
conditions than compression at room temperature, as is cur-
rently specified by the World Squash Federation.
13
VII. COMMENTS
The use of much slower compression tests needs to be
treated with caution for predicting squash ball behavior. If
such tests are to be used, it is better to use balls cooled to a
lower temperature. The use of lower temperatures means the
properties are closer to those of much faster rate tests at
higher temperatures as found under playing conditions. This
understanding suggests that the specifications of the World
Squash Federation regarding compression should be modi-
fied.
1
William D. Callister, Materials Science and Engineering: An Introduc-
tion, 7th ed. Wiley, New York, 2007, pp. 611–612.
2
Philip Yarrow and Aidan Harrison, Squash. Steps to Success, 2nd ed.
Human Kinetics, Champaign, IL, 2010, pp. ix–ixx.
3
Lloyd V. Smith and Joseph G. Duris, “Progress and challenges in numeri-
cally modelling solid sports balls with application to softballs,” J. Sports
Sci. 27 4, 353–360 2009.
4
Syed M. M. Jafri and John M. Vance, “Modeling of impact dynamics of
a tennis ball with a flat surface,” Proceedings of the Fifth International
Conference on Multibody Systems, Nonlinear Dynamics and Control,
24–28 September 2005 ASME, Long Beach, CA, 2005, pp. 399–412.
5
M. Nagurka and S. G. Huang, “A mass-spring-damper model of a bounc-
ing ball,” Int. J. Eng. Educ. 22 2, 393–401 2006.
6
L. P. Cordingley, S. R. Mitchell, and R. Jones, “Measurement and mod-
elling of hollow rubber spheres: Surface-normal impacts,” Plast. Rubber
Compos. 33 2–3, 99–106 2004.
7
Rod Cross, “The coefficient of restitution for collisions of happy balls,
unhappy balls, and tennis balls,” Am. J. Phys. 68 11, 1025–1031
2000.
8
Rod Cross, “The bounce of a ball,” Am. J. Phys. 67 3, 222–227 1999.
9
S. P. Hendee, R. M. Greenwald, and J. J. Crisco, “Static and dynamic
properties of various baseballs,” J. Appl. Biomech. 14, 390–400 1998.
10
J. J. Crisco, E. I. Drewniak, M. P. Alvarez, and D. B. Spenciner, “Physi-
cal and mechanical properties of various field lacrosse balls,” J. Appl.
Biomech. 21 4, 383–393 2005.
11
Peter A. Giacobbe, Henry A. Scarton, and Yau-Shing Lee, “Dynamic
hardness SDH of baseballs and softballs,” in International Symposium
on Safety in Baseball and Softball (STP 1313), edited by Earl E. Hoerner
and Francis A. Cosgrove American Society for Testing and Materials,
West Conshohocken, PA, 1997, pp. 47–66.
12
N. J. Bridge, “The way balls bounce,” Phys. Educ. 33, 174–181 1998.
13
“Squash specifications for courts rackets and balls” World Squash Fed-
eration, East Sussex, 2009.
14
L. Mullins, “Engineering with rubber,” Rubber Chem. Technol. 59 3,
G69–G83 1986.
15
John D. Ferry, Viscoelastic Properties of Polymers Wiley, New York,
1980, pp. 154–163.
Fig. 11. The effects of temperature and rate on the stiffness of rubber.
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6 6Am. J. Phys., Vol. 79, No. 2, February 2011 Lewis, Arnold, and Griffiths
AUTHOR QUERIES 011102AJP
#1 Au: Reference 9 was not cited in text. Please check our insertion.
NOT FOR PRINT! FOR REVIEW BY AUTHOR NOT FOR PRINT!
... A short literature review, shows that no scientific consensus exists upon inflated ball mechanics. It is almost well understood that ball elastic properties come from the gas compression inside the ball during the impact for the common inflated balls (Cross 1999;Goodwill & Haake 2001;Stronge & Ashcroft 2007) or partly from the shell and the gas (squash ball, see Lewis et al. (2011)), or uniquely from the shell (tennis-table ball, see Cross (2014)). However, the source of dissipation is still unclear and several authors have given various explanations to stand for it such as momentum flux force dissipation (Stronge & Ashcroft 2007), vibrations of the shell (Cross 2014), dissipation in the gas (Georgallas & Landry 2015) and solid friction (Pauchard & Rica 1998). ...
Article
The behaviour of sports balls during impact defines some special features of each sport. The velocity of the game, the accuracy of passes or shots, the control of the ball direction after impact, the risks of injury, are all set by the impact mechanics of the ball. For inflated sports balls, those characteristics are finely tuned by the ball inner pressure. As a consequence, inflation pressures are regulated for sports played with inflated balls. Despite a good understanding of ball elasticity, the source of energy dissipation for inflated balls remains controversial. We first give a clear view of non-dissipative impact mechanics. Second we review, analyse and estimate the different sources of energy dissipation of the multi-physics phenomena that occur during the impact. Finally, we propose several experiments to decide between gas compression, shell visco-elastic dissipation, solid friction, sound emission or shell vibrations as the major source of energy dissipation.
... For lower level or less motivated students, their interest in the experiment can be reinforced by proposing them to compare their experimental results for different sport balls with the official ball specifications [13]. On the other hand, for physics or engineering students, this experiment and the techniques used in it can be specially interesting as similar techniques are used in research in materials science [15,16,17,18,19], to study materials behaviour and hysteresis [20], or even granular materials [21]. ...
Article
A simple experiment on the determination of the coefficient of restitution of different materials is taken as the basis of an extendable work that can be done by students in an autonomous way. On the whole, the work described in this paper would involve concepts of kinematics, materials science, air drag and buoyancy, and would help students to think of physics as a whole subject instead of a set of, more or less, isolated parts. The experiment can be done either in teaching laboratories or as an autonomous work by students at home. Students' smartphones and cheap balls of different materials are the only experimental materials required to do the experiment. The proposed work also permits the students to analyse the limitations of a physical model used in the experiment by analysing the approximations considered in it, and then enhancing their critical thinking.
... Elastic hysteresis is particularly accessible to undergraduate students, and may help to prepare them for the more complicated processes of ferromagnetic and ferroelectric hysteresis. Elastic hysteresis has been studied previously for torsion of rubber tubing [43], stretched bands [44,45], bent plastic foil [46], and compressed balls and Silly Putty [47][48][49]. ...
Article
Full-text available
A simple, collapsible design for a large water balloon slingshot launcher features a fully adjustable initial velocity vector and a balanced launch platform. The design facilitates quantitative explorations of the dependence of the balloon range and time of flight on the initial speed, launch angle, and projectile mass, in an environment where quadratic air drag is important. Presented are theory and experiments that characterize this drag, and theory and experiments that characterize the nonlinear elastic energy and hysteresis of the latex tubing used in the slingshot. The experiments can be carried out with inexpensive and readily available tools and materials. The launcher provides an engaging way to teach projectile motion and elastic energy to students of a wide variety of ages.
... Un handballeur n'a pas intérêt à smasher le ballon au but pour deux raisons : la frappe est moins efficace que lancer en terme de vitesse atteinte, et il risque de se blesser. [211,212,213,214,215]. (à 70 ○ C, e = 0,4 aux grandes vitesses, et e = 0,7 aux faibles vitesses, soit 1,19 < v ′ 2 v 1 < 1,44). ...
Thesis
Full-text available
Physics tends to understand the world and to find the laws which govern it. Physics of Sports consists in observing with a physicist’s eye the phenomena which happen on the fields. These fields are wide and this study is built on several points which capture our attention : - records sports : to understand the records of speed or strength sports, we focus on muscle contraction mechanisms. We extract the dynamics of a simple gesture at Bench Press and understand it through simple mechanics laws and microscopic model of muscle contraction. - badminton trajectories : we solve the equations of motion of a particle which undergoes gravity and aerodynamic drag (proportional to the velocity sqaure, for high Reynolds numbers) and extract an analytical expression of the range. At each time the launching velocity is higher than the terminal velocity (which is the constant velocity of vertical falling of the projectile under gravity and drag), one observes a Tartaglia triangular shape of the trajectory. It is the case for most of sports balls, for fireworks, for firemen water jets, for cannonballs... For all these trajectories, the range saturates with the initial velocity : even if you hit stronger, the ball will not go further. We show several applications of this property. For exemple in sport, as there is no use to play on a field which is larger than the useful length, the size of sports fields is bound to be linked to the maximal range of the associated ball, from table tennis to golf. - balls impacts : we study soccer kicks and focus on the way of kicking the ball. The toe poke is said to be more efficient than the push pass. We perform experiments which show that the ball velocity does not depend on the shape of the impactor, but only on the velocity of kicking. We then discuss some ways to enhance the impactor velocity by using joints or elasticity of the launcher.
Article
The bouncing motion of a spherical ball following its repeated inelastic impacts with a horizontal flat surface is analyzed. The effect of air resistance on the motion of the ball is accounted for by using the quadratic drag model. The effects of inelastic impacts are accounted for by using the coefficient of restitution, which is assumed to remain constant during repeated impacts. Also presented is an extension of the analysis allowing for a velocity-dependent coefficient of restitution. Closed-form expressions are derived for the velocity, position, maximum height, duration, and dissipated energy during each cycle of motion. The decrease of successive rebound heights in the presence of air resistance is more rapid for higher values of the launch velocity, because the drag force is stronger and acts longer. Air resistance can significantly affect the value of the coefficient of restitution determined in a dropping ball test. For a given number of rebounds, the energy dissipated by inelastic impacts is greater than the energy dissipated by air resistance, if the launch velocity is sufficiently small. The opposite is true for greater values of the launch velocity. The derived formulas are applied to analyze the bouncing motion of a ping pong ball, tennis ball, handball, and a basketball.
Article
The coefficient of restitution (COR) is an important constant that represents the energy dissipation during contact between two objects. Simulation using the conventional discrete element method (DEM) involves a constant COR. This study presents a DEM simulation method that uses a parameter-dependent COR. The parameter-dependent COR was obtained from a collision incident between spherical particles and a plate surface using a drop-test apparatus. Glass and polypropylene beads of 3–6-mm diameter were used while acrylic and steel were used as the plate surfaces. The particle trajectories were captured by a high-speed camera and analyzed by an image analyzer. The COR was then correlated to a parameter-dependent COR function that depends on the material, impact velocity, and temperature. Free-fall DEM simulations using a constant COR and parameter-dependent COR were compared. The parameter-dependent COR approach obtained better agreement with experimental results than the constant-COR approach. The proposed concept could be applied for other material combinations with a wide range of operating conditions to obtain a database of parameter-dependent COR values for the simulation of solid handling applications.
Article
Zusammenfassung Racketlon umfasst als Mehrkampf‐Sportart Tischtennis, Badminton, Squash und Tennis. Bei jeder Disziplin unterscheiden sich die physikalischen Eigenschaften von Bällen und Schlägern. Verblüffenderweise erreicht der Badminton‐Federball kurz nach dem Schlag die bei Weitem höchsten Fluggeschwindigkeiten bis über 400 km/h, bremst aber schnell ab. Beim Squash beeinflusst die Temperatur des Balls das Sprungverhalten stark. Solche unterschiedlichen Eigenschaften der nacheinander gespielten Disziplinen machen Racketlon zu einer anspruchsvollen Sportart.
Chapter
The study of human motion begins with classifying the many types of locomotion and continues with descriptions of muscles in the body. The analysis of standing starts with a discussion of the overall and local stability of the body and of friction, and continues with models of walking. This includes a review of harmonic motion and pendulums. These concepts are applied to running, which is also compared to walking. The energetics and dynamics of jumping and the pole vault are followed by modeling how to throw balls, such as baseballs, and swing objects, such as golf clubs. This is then related to a more formal treatment of multisegment modeling. Collisions of the body due to falls, in contact sports such as boxing and karate, and in hitting balls are investigated. The flight of balls that are thrown or hit is analyzed, including how it is affected by drag and ball spinning. The effect of friction is probed in skiing and in the bouncing and rolling of balls.
Article
Full-text available
Measurements are presented of the force acting on ping-pong and squash balls impacting on a force plate. Both ball types are hollow and have the same diameter but deform in very different ways. Ping pong balls are relatively stiff and buckle inwards at high impact speeds, while squash balls are softer and tend to squash or flatten. The buckling process generates a large-amplitude, high-frequency oscillation of the force acting on a ping-pong ball. Squash balls are initially very stiff before they soften, with the result that the force on the ball rises to about half its maximum value in the first 20 μs. Ping-pong balls have a high coefficient of restitution (COR), while squash balls have a low COR. Results for both ball types are interpreted in terms of additional experimental observations.
Data
Full-text available
The objective of this experiment is to investigate the use of Dynamic mechanical analysis in determining the mechanical properties of materials. The ultimate aim is to set up an apparatus of DMA testing in the laboratory which is use to investigate the properties of material specimen. The material specimen used in this experiment is the Squash Ball. The damping (tan δ) effect of the specimen is to be evaluated using some dimensions alike force, acceleration and displacement at different frequencies. The behaviour of squash ball constitutes the study of dynamic nature of rubbery materials. Also, the temperature variations are measured at the different range of frequency, which can be correlated with the energy lost during damping in the specimen. To carry out the experiment some vibration setup is arranged with the help of shaker, FFT analyser, amplifier and a computer system. The whole setup for this experiment is works on the Pulse Labshop version 15 software. Also all the instruments use for carrying out the results are of Brüel & Kjaer. Keywords δ -Damping coefficient (Phase lag/angle) E΄ -Storage Modulus E -Loss Modulus ξ -Point on which the force is exerted θ -The angle of rotation T g -Glass Transition Temperature T m -Melting Temperature KE -Kinetic energy COP -Centre of Percussion FEA -finite element analysis DMA-Dynamic Mechanical Analysis COR -Coefficient of restitution dy/ dx -Small Change in x or y direction Mr.
Article
Full-text available
A perfectly happy ball is one that bounces to its original height when dropped on a massive, rigid surface. A completely unhappy ball does not bounce at all. In the former case, the coefficient of restitution (COR) is unity. In the latter case, the COR is zero. It is shown that when an unhappy ball collides with a happy ball, the COR increases from zero to unity as the stiffness of the happy ball decreases from infinity to zero. The COR is independent of the mass of each ball. The implication of reducing the COR of a tennis ball, as a possible means of slowing the serve in tennis, is also considered. It is shown that (a) the COR for a collision with a racket varies with the impact point and is a maximum at the vibration node near the center of the strings, and (b) the serve speed is reduced by only about 20% if the COR for a bounce on the court is reduced to zero.
Article
Full-text available
In this paper, the dynamics of a bouncing ball is described for several common ball types having different bounce characteristics. Results are presented for a tennis ball, a baseball, a golf ball, a superball, a steel ball bearing, a plasticene ball, and a silly putty ball. The plasticene ball was studied as an extreme case of a ball with a low coefficient of restitution (in fact zero, since the collision is totally inelastic) and the silly putty ball was studied because it has unusual elastic properties. The first three balls were studied because of their significance in the physics of sports. For each ball, a dynamic hysteresis curve is presented to show how energy is lost during and after the collision. The measurement technique is quite simple, it is suited for undergraduate laboratory experiments, and it may provide a useful method to test and approve balls for major sporting events.
Article
In this study we investigated the compressive quasi-static mechanical properties and dynamic impact behavior of baseballs. Our purpose was to determine if static testing could be used to describe dynamic ball impact properties, and to compare static and dynamic properties between traditional and modified baseballs. Average stiffness and energy loss from 19 ball models were calculated from quasi-static compression data. Dynamic impact variables were determined from force-time profiles of balls impacted into a flat stationary target at velocities from 13.4 to 40.2 m/s. Peak force increased linearly with increasing ball model stiffness. Impulse of impact increased linearly with ball mass. Coefficient of restitution (COR) decreased with increasing velocity in all balls tested, although the rate of decrease varied among the different ball models. Neither quasi-static energy loss nor hysteresis was useful in predicting dynamic energy loss (COR2). The results between traditional and modified balls varied widely in both static and dynamic tests, which is related to the large differences in mass and stiffness between the two groups. These results indicate that static parameters can be useful in predicting some dynamic impact variables, potentially reducing the complexity of testing. However, some variables, such as ball COR, could not be predicted with the static tests performed in this study.
Article
Impact research of soft materials has led to a new general dynamic hardness test method and hardness scale entitled the Scarton Dynamic Hardness (SDH) Method. Experimental impulse modal analysis testing is used to evaluate the transient or response characteristics of test materials such as sports equipment. The SDH method was used to evaluate and classify different types of sports balls by dynamic hardness and damping. A direct relationship was found between ball damping and the coefficient of restitution (COR). Sports safety applications show that a ball with a lower SDH has a reduced chance of inflicting head injury; if the SDH is too low, chest impacts may contribute to heart anomalies. Acoustical and impacter design research indicates that keeping the impacter resonances above the ball SDH will prevent their excitations. Subsequent reductions in acoustic "pinging" and enhanced performance are anticipated. The study included: baseballs, softballs, golfballs, racquetballs, marbles, ping pong, bowling, lacrosse, cricket, bocce, squash, tennis, and billiard balls.
Article
The author shows how the buildup of basic knowledge of the behavior of rubber has evolved into an understanding of how to achieve particular properties that lead to performance required for industrial applications. A discussion is presented of the behavior of rubber which shows that nonlinear stress-strain curves, creep, and hysteresis, and its properties are influenced not only by the method of fabrication but also by its previous history. This results not only in less precision in design compared with metals but also in less consistency in properties. Factors relative to strain amplification, elastic behavior and, strength and fatigue behavior are examined. The engineering applications of rubber are also discussed.
Conference Paper
A model of impact of a tennis ball with a flat surface is developed based on a planar, two-mass, linear, four degree of freedom vibration system idealization. The impact is assumed to be incident on a flat surface with friction. The incident parameters of the ball include the centre of mass translational velocity, angle of impact with the surface and the incident angular spin of the ball. The linear, piecemeal vibration model predicts the corresponding rebound parameters of the tennis ball. The model also predicts the duration of contact of the tennis ball with the flat surface, the transition of motion of the tennis ball during contact with the ground from sliding to rolling contact, and the resulting contact forces developed between the tennis ball and the flat surface. The model is computationally efficient because the governing differential equations of motion are linear and their standard solutions can be easily implemented on a personal computer. Predictions of the rebound parameters from the model are compared with experimental findings on tennis balls which are incident on a flat surface with various angles, velocities and angular spins (zero spin, topspin and backspin). For selected parameters of the two-mass model, the comparisons show excellent agreement between the model and the measurements.
Article
The bounce of a ball is a good topic for investigation at either GCSE or A-level. At King's School Canterbury pupils have experimented with both squash balls and inflatable play balls, varying the drop height, pressure and temperature and measuring the effect on bounce height, contact area and contact time. Worthwhile predictions can be made from quite simple theory and the experimental results provide ample opportunities for discussion and evaluation.
Article
A simple theory presented recently to describe the bounce of an air-filled ball predicted that the contact time should be independent of the speed of impact. Experimental results, however, disagreed. Amendments suggested here lead to a model giving good agreement with experiment.
Article
The performance of natural rubber sports balls under impact conditions is dominated by the material's behaviour under high strain rate conditions dictated by the impact velocity and ball dimensions. To design improved products, sports ball manufacturers need to better understand the physical phenomena associated with ball impact against both rigid and deformable surfaces. This understanding will provide the foundation for performance prediction and optimisation design tools as well as more appropriate product and ultimately material testing techniques. Rebound characteristics of pressurised and pressureless tennis balls and their respective rubber cores subject to normal impacts are presented for a range of incident velocities. High-speed video analysis has been used to measure coefficient of restitution, impact duration and 'whole ball' deformation to validate a surface-normal impact finite element method based predictive model as the first step towards a more comprehensive oblique impact model. Accounting for strain rate dependent stiffness and damping material properties has achieved close correlations between model predictions and observed impact behaviour. The propagation of dominant bending and hoop-strain waves through the ball during the impact is revealed to illustrate the methodology's effectiveness in predicting ball performance associated with difficult to observe impact phenomena.