The dynamic behavior of squash balls

Abstract

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.

Full text

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
共Received 26 February 2010; accepted 23 September 2010兲
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.
关DOI: 10.1119/1.3531971兴
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.1The 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 共indicated by colored
spots兲, 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 共the Challenger Space
Shuttle disaster was caused by such behavior兲. It is a subject
in which the structure-property relations of a material can be
related from the molecular level 共how polymer chain
segments move兲to the macrolevel 共the mechanical proper-
ties兲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.2This 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 共COR兲,
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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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.3–5
These models can account reasonably well for the change in
behavior with speed,3,6but 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.7,8This 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.6The 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 共percentage energy lost during
compression and unloading兲can be measured, which can
then be related to the dynamic energy losses measured by the
coefficient of restitution.7,8Ideally, 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 共a distance of 3.55 m兲. 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 = 冑h2
h1
,共1兲
where h1is the drop height and h2is 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 共about 8.2 m/s兲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 1shows 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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and 0.65 共resilience between 20% and 35%兲, 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. 3for
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 3shows 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 4shows 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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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.2This 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 7shows 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 8shows 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 9shows 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共%兲=共1 − COR2兲100. 共2兲
The time scale of the impacts in the bounce tests was
estimated by comparing the change in momentum 共from the
coefficient of restitution results兲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 共viscoelastic兲mechanical behavior of poly-
mers. The mechanical properties arise from the movement of
polymer chains. More polymer chain movement occurs with
slower compression rates 共more time兲or at higher tempera-
tures 共more mobile chains兲. 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 共ms兲
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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The hysteresis of a polymer is related to its glass-transition
temperature Tg, the region where its behavior changes from a
hard rigid glassy material 共below Tg兲to a softer rubbery ma-
terial 共above Tg兲.15 Near Tgthe 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 Tg共defined
as the peak of the curve兲is below room temperature. The two
curves represent different deformation rates. The faster de-
formation leads to a Tgthat 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 共more bounce兲. 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 Tg.
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
Tgand 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.
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6 6Am. J. Phys., Vol. 79, No. 2, February 2011 Lewis, Arnold, and Griffiths

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