1139
Modelling of tennis ball impacts on a rigid surface
S R Goodwill* and S J Haake
Sports Engineering Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK
Abstract: A viscoelastic model of a tennis ball impact at normal incidence on a rigid surface is
presented in this study. The ball model has three discrete elements that account for the structural
stiffness, material damping and momentum flux loading. Experiments using a force platform are
performed to determine the force that acts on the ball during impact, for a range of ball inbound
velocities. The inbound and rebound velocities of the ball are measured using speed gates. The contact
time and coefficient of restitution for the impact are also determined in these experiments. The model
parameters are determined such that the values of the coefficient of restitution and contact time that
are calculated by the model are consistent with those values determined experimentally. The model
can be used to calculate the force that acts on the ball during impact. Generally, the force–time plots
calculated by the model were consistent with those determined experimentally. Furthermore, the
model can be used to calculate the three components of the force that acts on the ball during impact.
It is shown that the main component of the force during the first 0.6 ms of impact is that due to
momentum flux loading. This is approximately equal in magnitude for each ball type and explains
why the total force acting on each ball is very similar during this period.
Keywords:
tennis ball, impacts, modelling, rigid surface
NOTATION
AC
AK , a
cB
cM
COR
dCONT
FM
kB
kBð0Þ
kCAL
kCONST
mB
M 1, M 2
t
TC
VB
VB0
xB , x_ B , x€B
damping area constant
constants used to define the ball stiffness
dashpot damper (material damping)
dashpot damper (momentum flux
component)
coefficient of restitution
diameter of the contact area
momentum flux force acting on the ball
spring stiffness
ball stiffness at zero displacement
force platform calibration factor
prebuckle spring stiffness
ball mass
mass of sections 1 and 2 respectively
time
contact time
ball inbound velocity
ball rebound velocity
displacement, velocity and acceleration of
mass mB respectively
The MS was received on 16 May 2003 and was accepted after revision
for publication on 15 June 2004.
* Corresponding author: Sports Engineering Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield
S1 3JD, UK. E-mail: s.r.goodwill@sheffield.ac.uk
C07703
# IMechE 2004
d_ B1 , d_ B2
Dt
rarea
1
velocity of sections 1 and 2 respectively
time step
mass per unit surface area
INTRODUCTION
The modelling of a tennis ball impact on a surface/
racket has been attempted by many researchers over the
last 30 years [1–4]. The motivation for this has generally
been to gain an understanding of the influence of ball
construction on the impact. This is valuable for both the
governing body who set the rules of the game and the
ball/racket manufacturers who produce a product that
abides by these rules.
By understanding the impact mechanism, a racket
manufacturer can optimize their product by choosing
appropriate materials for each component of the racket.
One such manufacturer was Prince who, in collaboration with Howard Head, introduced the oversize tennis
racket into the game [5]. Since then, the use of carbon
fibre has enabled manufacturers to produce lighter and
stiffer tennis rackets, which has coincided with a marked
increase in the ball speeds measured on the men’s
professional circuit. Furthermore, many commentators
believe that modern rackets have enabled the serve to be
hit with increased speed and accuracy so that even the
Downloaded from pic.sagepub.com at PENNSYLVANIA
STATE
UNIV
on February
20, Vol.
2016
Proc.
Instn
Mech.
Engrs
218 Part C:
J. Mechanical Engineering Science
1140
S R GOODWILL AND S J HAAKE
fastest player reaction times are sometimes too slow to
allow proper return of service. It is generally agreed that
some part of this change can be assigned to the
improved training, athleticism and physique of modern
players. However, the International Tennis Federation
(ITF) has still been criticized for not imposing some
control on the ‘power’ of a tennis racket.
This has motivated the ITF to embark on a research
project that is aimed at advancing their existing knowledge of the mechanisms involved in the impact between
a tennis ball and racket. This will be achieved using both
experimental investigations and theoretical modelling
techniques. The impact between a tennis ball and tennis
racket involves a complex interaction between many
components. Furthermore, a detailed understanding of
each of these parameters is required in order to
construct the overall model of the impact between the
ball and racket. Therefore, this overall model needs to
be developed in several discrete stages. The logical first
stage of this study involves the development of a model
of a much simpler impact in which a tennis ball lands
normal to a rigid surface. This model can be used to
obtain an understanding of the physical properties of
several different types of tennis ball. Furthermore, it will
form a first approximation of the ball for a model of an
impact between a ball and racket.
The aim of this study is to develop a model of an
impact between a ball and rigid surface that can be used
to predict the dynamic response of the ball. This model
must be capable of calculating the force that acts on the
ball and the motion of the ball centre of mass during
impact and also the velocity of the ball after impact.
The objectives of this study are:
(a) to measure experimentally the dynamic response of
a tennis ball for an impact with a rigid surface;
(b) to derive a physically realistic model of a tennis ball
impact on a rigid surface;
(c) to assess the accuracy of the model by comparing
experimentally obtained data with the data calculated by the model.
2
EXPERIMENTAL PROCEDURE
The equipment used in this experiment is shown in
Fig. 1. The balls were projected at a piezoelectric force
platform using an air cannon. The speed gates were used
to determine the inbound and rebound velocity of the
ball, defined as VB and VB0 respectively. These values
were used to calculate the coefficient of restitution which
is defined as the ratio of the rebound to inbound ball
velocity. The ball was propelled at inbound speeds of
between 14 and 30 m/s (31.3 and 67.1 mile/h), perpendicular to the surface of the force platform. The
magnitude of the ball deformation that is associated
Proc. Instn Mech. Engrs Vol. 218 Part C:
with these inbound speeds is equivalent to that
measured for typical ball–racket impacts.
Detailed construction details for this force platform
are given in references [3] and [4]. A 610 probe was used
to connect the piezoelectric platform to a charge
amplifier to increase the time constant of the circuit,
as explained in more depth in reference [3]. The voltage
output from the charge amplifier was sampled using an
analogue-to-digital converter (ADC) and laptop PC, at
a rate of approximately 67 kHz.
The force platform output a voltage V which was
sampled by the ADC and PC to give a voltage–time
plot. The same equipment was used in reference [3], and
it was verified that the output voltage was linearly
proportional to the force. However, the calibration
factor relating these two parameters was not constant
for all impacts. Consequently, the calibration factor,
kCAL, has to be determined for each impact. This was
obtained using the assumption that the impulse applied
to the
to the ball, mðVB0 VB Þ, was proportional
Ð
integral of the voltage–time signal, 0TCV dt (where TC
is the contact time for the impact). The ADC sampled at
a sufficiently high rate for the trapezium rule to be used
to integrate the voltage data with negligible error.
The sampled data obtained from the force platform
were used to determine the force that acted on the ball
during impact. These data were then numerically
integrated to calculate the velocity and displacement of
the ball centre of mass during impact. The data were
used to determine the contact time for the impact, the
contact time being defined as the period when a force
acts on the ball.
Four different ball types were used in this study, and
the physical properties of these balls are given in
Table 1. Ball 1 has an approximate internal air pressure
of 103 kPa (15 lbf/in2) above atmospheric. The oversize
ball (ball 3) has a similar internal air pressure but is
approximately 8 per cent larger in diameter. Ball 2 is
pressureless and contains air at atmospheric pressure.
Ball 4 is identical to ball 1 but with the internal air
pressure released by piercing it with a 1 mm needle at
intervals along the seam. The needle was removed
when a pressure gauge attached to it measured atmospheric air pressure inside the ball. It was assumed that
air was not allowed to escape from the interior during
impact.
Table 1 shows the measures recorded for the balls
studied using approval tests specified by the International Tennis Federation [6]. Balls are dropped from
254 cm (100 in) and their rebound is recorded. For
approval, balls must rebound between 134.6 and
147.3 cm. In a second test, the balls are deformed
between two flat plates at 2.54 mm/s to a load of 80 N,
held there for 5 s, increased to a compression of 25.4 mm
and then returned to 80 N. The load is held again for
10 s and reduced to zero. The forward and return
deformations are recorded at 80 N on the compression
Downloaded Engineering
from pic.sagepub.com
at PENNSYLVANIA STATE UNIV on February 20, 2016
J. Mechanical
Science
C07703 # IMechE 2004
MODELLING OF TENNIS BALL IMPACTS ON A RIGID SURFACE
1141
Fig. 1 Layout of experiment showing a ball projected at a force platform using an air cannon
Table 1 Physical properties of the four types of tennis ball studied. The tests are specified by rules of tennis laid down by the
International Tennis Federation [6]
Ball type
Mass (g)
Ball rebound
from 254 cm (cm)
Forward deformation
at 80 N (mm)
Return deformation
at 80 N (mm)
1.
2.
3.
4.
57.7
57.1
56.7
57.5
138.9
142.2
137.4
103.8
6.2
6.4
6.8
8.4
8.6
9.6
9.3
15.2
Pressurized
Pressureless
Oversize
Punctured
and return phases of the test. Balls 1 to 3 passed all
approval tests, but ball 4 failed all tests.
3
MODELLING THE IMPACT
3.1
Previous modelling attempts
kB ¼ mB
There are many modelling techniques that could be used
to model the impact between a tennis ball and rigid
surface. In this study, a viscoelastic model was chosen
owing to the relative versatility of this method; the
stiffness and damping properties of the ball can be
defined using any function of the ball deformation and
velocity.
The most recent study which has used a viscoelastic
model to simulate the impact between a tennis ball and
rigid surface is described in reference [7]. In that paper, a
simple Kelvin–Voigt model was used to simulate the
impact, as shown in Fig. 2a. This model contains a
spring in parallel with a dashpot damper. The mass mB
is equal to the mass of the ball. In this model, the values
of kB and cB represent the linear stiffness and damping
of the ball respectively, and therefore the governing
equation for this system is
mB x€B þ cB x_ B þ kB xB ¼ 0
C07703
# IMechE 2004
where xB is the displacement of mass mB and is
referenced to the rigid surface.
In reference [7] it was assumed that the values of the
stiffness and damping coefficients, kB and cB respectively, remained constant throughout the impact. The
analytical equations used to define parameters kB and cB
are
ð1Þ
p2
TC2
ð2Þ
and
cB ¼ 2mB
lnðCORÞ
TC
ð3Þ
where TC and COR are the contact time and coefficient
of restitution for the impact respectively.
The values of kB and cB were calculated analytically
using experimentally determined values of contact time
and coefficient of restitution for the impact. The values
of kB and cB were determined for a range of ball impact
velocities and several different ball types. It was found
that the stiffness parameter increased as the ball impact
velocity was increased. This implies that the structural
stiffness of the ball increases with an increase in the ball
deformation, which is consistent with reference [8]. In
that study, several tennis balls were quasi-statically
compressed between two flat plates, and it was found
Downloaded from pic.sagepub.com at PENNSYLVANIA
STATE
UNIV
on February
20, Vol.
2016
Proc.
Instn
Mech.
Engrs
218 Part C:
J. Mechanical Engineering Science
1142
S R GOODWILL AND S J HAAKE
Fig. 2
(a) Kelvin–Voigt viscoelastic model of a tennis ball impact on a rigid surface and (b) typical
comparison between the force calculated by the viscoelastic model and the data obtained
experimentally using the force platform (ball inbound velocity 28 m/s)
that the structural stiffness of the balls increased with
increasing ball deformation.
The term cB x_ B in equation (1) represents the rate
dependent force resulting from the dynamic deformation of the material. This force leads to hysteresis losses
in the system. The magnitude of cB x_ B varies throughout
impact and is clearly dependent on the instantaneous
velocity of the ball centre of mass, x_ B , and the damping
coefficient, cB. In reference [7] it was found that cB
increased with increasing ball impact velocity. This
increase in cB is due to the larger volume of rubber being
deformed at higher ball speeds.
This viscoelastic model can be used to calculate the
force that acts on the ball during impact. A typical
comparison between these calculated data and those
obtained experimentally for impacts on a force platform
is given in Fig. 2b. It can be seen that the correlation
between the two sets of data is poor, especially during
the restitution (rebound) phase of impact. Furthermore,
using the model, a tensile force is calculated during the
final 0.5 ms of impact, as illustrated by the negative force
shown in Fig. 2b. This is due to the assumption that the
damping coefficient is constant throughout impact,
which is clearly not realistic.
The main weaknesses of this modelling technique are
the assumptions that the stiffness and damping coefficients are constant throughout impact. In the following
section, the simple Kelvin–Voigt model described above
will be developed with the aim of improving the physical
similarity between the model and the actual impact
mechanism. The objective is for this advanced model to
be a suitable first-order approximation for the ball
component in a more complex modelled impact (e.g.
ball impact on a racket).
Proc. Instn Mech. Engrs Vol. 218 Part C:
3.2
Improved viscoelastic model
A viscoelastic model similar to that described above will
be used as the foundation of the advanced model
derived in this study. The weaknesses that have been
identified in the previous section will be addressed in this
advanced model. In brief, a spring will be used to
simulate the structural stiffness of the ball and a dashpot
damper will be used to simulate the hysteresis loss in the
ball. The methods used to define these parameters are
discussed individually.
3.2.1 Spring parameter kB (structural stiffness)
The spring parameter kB is used to simulate the
structural stiffness of the ball, and therefore the function
that defines this parameter should be representative of
the stiffness of the ball. The relationship between kB and
the ball centre-of-mass displacement xB cannot be
measured directly. Previous authors [9, 10] have successfully used a power law function to relate these two
parameters for other sports ball impacts. A suggested
form of the relationship between the spring stiffness kB
and ball centre-of-mass displacement xB is
kB ¼ kBð0Þ þ AK xaB
ð4Þ
The parameter kBð0Þ corresponds to the stiffness of the
ball at zero displacement. The parameters AK and a are
constants that are used to define the relationship
between the stiffness kB and the ball centre-of-mass
displacement xB. The method used to obtain the values
of the parameters kB(0), AK, and a for each ball type is
discussed in a later section.
Downloaded Engineering
from pic.sagepub.com
at PENNSYLVANIA STATE UNIV on February 20, 2016
J. Mechanical
Science
C07703 # IMechE 2004
MODELLING OF TENNIS BALL IMPACTS ON A RIGID SURFACE
1143
Fig. 3 (a) Illustration of the ball centre-of-mass displacement xB and contact length dCONT at time t and
(b) relationship between xB and dCONT
3.2.2
Dashpot damping parameter cB
(material damping)
3.2.3 Momentum flux contribution
In reference [7] it was found that the value of the
damping parameter cB increased with the ball impact
velocity, and this was attributed to the increase in the
volume of rubber being deformed at higher speeds.
Thus, in this advancement of the model, it is assumed
that the damping parameter is a function of the volume
of rubber being deformed. In this model, a relationship
between these two parameters needs to be determined.
Figure 3a illustrates the ball centre-of-mass displacement xB and the contact length dCONT (diameter of the
contact area) at time t. In reference [8], a relationship
was obtained between these two variables for the
compression phase of impact. It was found that this
relationship was independent of the ball type and ball
impact velocity. This relationship is illustrated in Fig. 3b
and defined by
dCONT ¼ 2:776105 x4B þ 1:746104 x3B 453x2B
þ 7:66xB
ð5Þ
It is assumed that the relationship defined in equation
(5) can be used for both the compression and restitution
phases of impact. As mentioned previously, the damping
parameter cB should be a function of the volume of
rubber being deformed, which is proportional to the
contact area. Also, it is assumed that the section of the
ball that is in contact with the surface is effectively a flat
disc, as illustrated in Fig. 3a. The parameter cB can
therefore be defined as
cB ¼
Bp
ðdCONT Þ2
4
ð6Þ
where B is a constant. Let AC ¼ Bðp=4Þ and therefore
cB ¼ AC ðdCONT Þ2
ð7Þ
where AC is defined as the damping area constant.
C07703
# IMechE 2004
A viscoelastic model, such as that shown in Fig. 2,
typically consists of a point mass attached to a series of
springs and dampers to simulate the impact. Clearly, a
tennis ball is a highly deformable hollow ball with a
relatively complex mass distribution, and is physically
very different to a mass attached to a spring. In
reference [11], a table tennis ball was used to illustrate
the mechanism of a hollow ball bouncing on a flat
surface. In that study, analytical equations were developed for the individual components that make up the
stiffness of the ball, i.e. the shell stiffness and internal air
pressure. This model also accounted for the contribution
made to the reaction force that arose from the
instantaneous change in momentum of material being
brought to rest on the surface during the compression
phase of impact. In references [11] to [13] this force was
defined as the momentum flux component, and, to be
consistent with that work, this term will be used in the
remainder of this study. In these papers it was shown
that the total force acting on a ball during impact is
composed of two components: one due to the quasistatic structural stiffness (including air pressure) and the
other due to the momentum flux. In those papers,
material damping was not considered, so this rate
dependent force did not contribute to the total force
acting on the ball.
The concept of momentum flux can be illustrated by
considering the impact between a thin-walled spherical
membrane (no structural stiffness or internal air
pressure) and a rigid surface. The initial velocity of
this hollow sphere is VB. At the start of the impact, the
material in the initial contact region rapidly decelerates
from VB to zero. A reaction force, numerically equal to
the rate of change in momentum, acts on the surface.
During the compression phase of impact, the sphere can
be considered as two separate sections: section 1
continues to move towards the surface while section 2
is in contact with the surface, as shown in Fig. 4. For
simplicity it is assumed that section 2 is flat and
stationary and therefore remains in contact with the
Downloaded from pic.sagepub.com at PENNSYLVANIA
STATE
UNIV
on February
20, Vol.
2016
Proc.
Instn
Mech.
Engrs
218 Part C:
J. Mechanical Engineering Science
1144
S R GOODWILL AND S J HAAKE
Fig. 4 Definition of the contact length dCONT, ball COM displacement xB and ball deformation rate d_ B
during impact
surface during impact. Furthermore, no energy is stored
in section 2 (this assumption will be discussed later). It is
also assumed that section 1 is undeformed, and therefore all points on this section move towards the surface
with the same velocity d_ B1 . When a segment of section 1
impacts on the surface it is assumed that the velocity
changes from d_ B1 to zero, and the size (and mass) of
section 2 increases. The masses of sections 1 and 2 are
defined as M1 and M2 respectively, and the total mass of
the ball is defined as mB. By definition, the term
‘momentum flux’ implies a ‘flow’ or ‘transportation’ of
momentum. During the compression phase of the
impact, the ‘flow’ of mass into section 2 is defined as
M_ 2 . The force FM acting on the ball at time t is
numerically equal to the rate of change in momentum
and is defined using
ðFM Þt ¼ ðM_ 2 d_ B1 Þt
ð8Þ
Following the notation in reference [11], it can be stated
that the force ðFM Þt is due to momentum flux. In the
above analysis, only the compression phase of the
impact has been accounted for. Furthermore, given
the assumption of zero structural stiffness, the hollow
sphere would eventually flatten against the surface. It
was assumed that, upon impact, material in contact with
the surface (section 2) remains flat and stationary, and
no energy is stored. Therefore, the hollow membrane
will not rebound from the surface. In this way, the
momentum flux concept represents a dissipative term
during the restitution phase of the impact.
In reality, a tennis ball is very different from a
spherical membrane. However, the force due to
momentum flux will nevertheless contribute to the total
force acting on the tennis ball during the compression
phase of the impact. Furthermore, the force can be
calculated using equation (8). This force, ðFM Þt , acts
perpendicular to the surface of the tennis ball, and thus
compresses the material (rubber core and felt). In the
example above, it was assumed that the energy stored in
Proc. Instn Mech. Engrs Vol. 218 Part C:
this deformation is not recovered as kinetic energy
during the restitution phase. However, this cannot be
assumed for the impact of a tennis ball on a rigid
surface, and therefore the motion of section 2 is
investigated below.
Initially it is assumed that a tennis ball is made from a
perfectly elastic material, such as that used to make a
perfect superball. When this superball bounces on a
rigid floor, all of the initial kinetic energy is stored in the
deformation of the ball. Most of this potential energy is
returned to the ball during the restitution phase of the
impact, and the ball rebounds with a velocity close to
that of the inbound velocity. This perfect superball was
studied numerically using an elastic finite element (FE)
model created in Ansys/LS-DYNA version 8.0. The
solid superball was modelled using 70 304 eight-node
brick elements. The ball was modelled as an Ogden
rubber with quasi-static material properties representative of the rubber used in a tennis ball. These properties
were measured experimentally from samples of rubber
supplied by a ball manufacturer. The solid ball model
had a diameter of 47 mm and a mass of 58 g (equal to
that of a tennis ball). The ball model was projected at a
rigid surface at a velocity of 5.00 m/s, and rebounded at
4.96 m/s, with a contact time of 1.9 ms. A small amount
of energy was stored in the oscillations of the rebounding ball. However, not all perfectly elastic objects
rebound in such an efficient manner. Figure 5 shows
two orientations of a perfectly elastic rubber block
ð565630 mmÞ modelled in Ansys/LS-DYNA. Intuitively, it would be expected that the rebound velocity of
the block would be approximately 5 m/s for both
orientations. When the block was dropped on its end
(orientation A) it rebounded at 4.50 m/s, and the contact
time was 0.65 ms. When the block was dropped on its
side (orientation B) it rebounded at 3.05 m/s and the
contact time was 0.1 ms. In the latter case, approximately 50 per cent of the initial kinetic energy is stored
in the vibration of the block after impact. Also, the
simulation has illustrated that the contact time for the
Downloaded Engineering
from pic.sagepub.com
at PENNSYLVANIA STATE UNIV on February 20, 2016
J. Mechanical
Science
C07703 # IMechE 2004
MODELLING OF TENNIS BALL IMPACTS ON A RIGID SURFACE
Fig. 5
Finite element model of a rubber block impacting on a rigid surface in two different orientations
impact is considerably shorter for the block dropped in
orientation B. This is because the block is effectively
stiffer in this orientation, and thus it is subjected to larger
contact forces. Furthermore, immediately after initial
impact, the edges of block B momentarily rebound off
the surface, but the majority of the block remains in
contact. This leads to a non-uniform compression of the
block, and thus a non-uniform distribution of the contact
forces. These forces cause transverse deformations in the
block which lead to a significant proportion of the initial
kinetic energy being stored in vibration of the block after
impact. Conversely, in orientation A the contact force
distribution is effectively uniform owing to the geometry
of the block, and therefore negligible transverse vibrations are excited.
Referring back to Fig. 4, the motion of section 2 of a
tennis ball will be analogous to that of a rubber block
landing in orientation B. The only main difference is
that the tennis ball material is not perfectly elastic and
dissipates a large fraction of its energy during impact.
The FE analysis has illustrated that the contact time for
a block in orientation B is much less than the duration
of the compression phase of the impact between a tennis
ball and rigid surface which lasts approximately 2 ms.
Therefore, during this compression phase, section 2 of
the tennis ball will rebound off the surface at a fraction
of its inbound velocity. This velocity will be even lower
than that predicted by FE because of the inelastic nature
of the tennis ball material. Section 1 (and the internal air
pressure) inherently prevents section 2 from rebounding
freely, and the velocity of section 2 will decelerate to
zero. This initializes an oscillation of section 2 which will
eventually damp out owing to the large hysteresis energy
losses in the tennis ball material. Therefore, it can be
assumed that no energy is recovered from section 2.
Consequently, there will be no component representing
the momentum flux term during the restitution phase of
impact.
It is assumed that section 2 remains approximately
flat throughout impact. Therefore, M2 can be determined from the contact length, dCONT, and the mass per
unit surface area, rarea. The relationship between dCONT
C07703
1145
# IMechE 2004
and xB is given by equation (5). It is assumed that the
tennis ball shell is inextensible and therefore the value of
rarea remains constant throughout impact. For a
standard-size tennis ball, with an effective radius of
29.5 mm and a mass of 57 g, the value of rarea is
5.212 kg/m2. Equation (8) can be solved to determine the
momentum flux component of the total force for a unit
time interval Dt:
ðFM Þt ¼
brarea p½ðdCONTðtÞ Þ2 ðdCONTðtDtÞ Þ2 c _
ðdB1 Þt
4Dt
ð9Þ
The velocity of the ball centre of mass x_ B is different to
velocity d_ B1 , as noted in reference [11]. In the model, the
bottom section of the ball is stationary, and therefore
the momentum of the ball is due solely to the motion of
the top section. Consequently, it can be assumed that
the relationship between x_ B and d_ B1 is
mB
_
ðdB1 Þt ¼
ð10Þ
x_ B
M1
t
Substituting equation (10) into equation (9) gives
ðFM Þt ¼
mB frarea p½ðdCONTðtÞ Þ2 ðdCONTðtDtÞ Þ2 g
ðx_ B Þt
4DtðM1 Þt
ð11Þ
which is an equation of the form
ðFM Þt ¼ ðcM x_ B Þt
ð12Þ
where
ðcM Þt ¼
mB frarea p½ðdCONTðtÞ Þ2 ðdCONTðtDtÞ Þ2 g
ð13Þ
4DtðM1 Þt
It is important to stress that equations (12) and (13) only
apply during the compression phase of impact, and,
during the restitution phase, cM is equal to zero.
The advanced model, which includes a second dashpot damper, cM, is shown in Fig. 6. The equation for
Downloaded from pic.sagepub.com at PENNSYLVANIA
STATE
UNIV
on February
20, Vol.
2016
Proc.
Instn
Mech.
Engrs
218 Part C:
J. Mechanical Engineering Science
1146
S R GOODWILL AND S J HAAKE
Fig. 6 Illustration of the viscoelastic model including a
dashpot damper to simulate the force due to the
momentum flux
the momentum flux force [equation (12)] is analogous
to the equation of motion for a dashpot damper with
a damping coefficient defined as cM. However, this
damper does not represent material hysteresis losses,
unlike damper cB. Instead, damper cM is simply a neat
method of defining the momentum flux component.
prior to this point is due to the high structural stiffness
of the ball before buckling occurs.
Analysis of the published experimental data has
shown that the ball wall generally buckles after
approximately 0.2 ms. Therefore, the second minor
modification of the model involves the assumption
that the spring should be assigned a high, constant
stiffness kCONST during the first 0.2 ms of impact. After
0.2 ms, it is assumed that the spring stiffness changes
to a lower value which is proportional to ball
COM displacement, as defined by equation (4), i.e.
kBð0Þ þ AK xaB . The stiffness kCONST represents the
stiffness of the ball structure prior to buckling. The
value of kCONST cannot be determined either analytically or numerically. In practice a value of kCONST equal
to 80 kN/m gave a good correlation between the force
calculated by the model and the published experimentally obtained data. This will be confirmed later in this
study.
It should be noted that the two minor modifications
discussed here do not strongly influence the overall
model solution. However, the inclusion of these
modifications acts to improve the physical similarity
between the model and the actual impact.
3.2.5 Model summary
3.2.4
Minor modifications to the model
The values of the parameters kB , cB and cM have been
fully defined in this section, except for two minor
modifications which will be explained here. In this
model, the values of the three components of the force
are assumed to be defined by simple, continuous
functions. However, experimental studies conducted in
references [5] and [8] have identified several discontinuities in the force measured by the force platform. For
example, in reference [5] it was shown that the ball
experiences a very low load during the initial period of
impact. This relatively low load was assigned to the
compression of the cloth on the ball, which has a much
lower stiffness than that of the ball. Published experimental data show that this low load generally occurs for
ball centre-of-mass displacements xB below 2 mm. The
model can easily be modified to simulate this negligible
force that occurs for these displacements by enforcing
the assumption that the force acting on the ball was
equal to zero for xB < 2 mm. This is the first minor
modification that is made to the model.
Published experimental data show that the force
acting on the ball rises sharply after the cloth has
compressed ðxB & 2 mmÞ. This rise is followed by a
sudden dip which is assigned to the ball wall buckling at
a critical point [5]. Furthermore, it was concluded that
the high load to which the ball is subjected immediately
Proc. Instn Mech. Engrs Vol. 218 Part C:
The governing equation for the viscoelastic model
illustrated in Fig. 6 is
mB x€B þ ðcB þ cM Þx_ B þ kB xB ¼ 0
ð14Þ
The terms kB and cB can be considered to represent the
structural stiffness and material damping of the section
of the ball. The term cM accounts for the momentum
flux component acting on the ball. For convenience, cM
is grouped with cB in equation (14). However, it should
be noted that these coefficients represent very different
mechanisms and are therefore not combined into a
single component. The momentum flux component only
acts during the compression phase because the energy is
dissipated and not recovered in the restitution phase.
The aim of this study is to determine a model of an
impact between a tennis ball and rigid surface. This
model must be capable of predicting the force that acts
on the ball during impact, the ball velocity after impact
and the contact time. This requires the model to be
solved using a method that allows these two parameters
to be calculated. Clearly, the model solution will require
the actual values of the parameters kB , cB and cM to be
known. However, initially, an algebraic method will be
described to illustrate the solution procedure. After this,
the method used to obtain the parameters kB , cB and cM,
for several ball types, will be described.
Downloaded Engineering
from pic.sagepub.com
at PENNSYLVANIA STATE UNIV on February 20, 2016
J. Mechanical
Science
C07703 # IMechE 2004
MODELLING OF TENNIS BALL IMPACTS ON A RIGID SURFACE
3.3
Table 2
Numerical solution for the model
The parameters cB , cM and kB are all functions of xB,
and therefore equation (14) cannot be solved analytically. Instead, a numerical solution is required which
involves the calculation of the parameters xB , x_ B and x€B
at a finite number of time intervals. A numerical
solution of equation (14) can easily be determined using
the finite difference method with a time step Dt of
0.01 ms.
Assuming that the velocity x_ B does not change during
this small time step, the finite difference form of
equation (13) at time t is
"
#
ðxB ÞtþDt 2ðxB Þt þ ðxB ÞtDt
mB
ðDtÞ2
ðxB Þt þ ðxB ÞtDt
þ ½ðcB Þt þ ðcM Þt Dt
þ ðkB Þt ðxB Þt ¼ 0
ð15Þ
which, rearranged, gives the displacement of xB at time
t þ Dt as
2
Dt
ðxB ÞtþDt ¼ ðkB Þt ðxB Þt þ ðcB Þt þ ðcM Þt
mB
ðxB Þt þ ðxB ÞtDt
6
Dt
2ðxB Þt þ ðxB ÞtDt
ð16Þ
Equation (16) can be used to determine xB for time steps
of Dt using the two boundary conditions
ðxB Þt¼0 ¼ 0
and
ðxB Þt¼ Dt ¼ VB0 Dt
A simple routine was written to calculate the displacement, xB, velocity, x_ B , and acceleration, x€B , of the ball
centre of mass at each time interval until impact ceased.
The end of impact was defined as the point at which the
force acting on the ball returned to zero.
3.4
Determining the model parameters kB , cB and cM
The model solution requires the parameters cB , cM and
kB to be known. More specifically, it requires the
functions that are used to define these parameters to be
known. To recap, these functions are
kB ¼ kBð0Þ þ AK xaB
cB ¼ AC ðdCONT Þ
ð7Þ
2
mB frarea p½ðdCONTðtÞ Þ ðdCONTðtDtÞ Þ g
4DtðM1 Þt
ð13Þ
The unknown parameters in the above equations are
C07703
# IMechE 2004
Spring parameters kBð0Þ , AK and a and damping
coefficient AC for four ball types
Ball type
kBð0Þ (kN/m)
AK (kN/m2)
a
AC (kN s/m3)
1.
2.
3.
4.
21
23
21
16
16 000
12 500
3600
60 000
1.65
1.70
1.30
2.00
3.5
4.0
3.2
5.8
Pressurized
Pressureless
Oversize
Punctured
kBð0Þ , AK , a and AC, all the others having been quantitatively defined previously. These unknown parameters
are determined using experimental data obtained for an
impact between a ball and a rigid surface. The two
experimentally measured variables that are utilized to
determine the unknown parameters are the contact time
and the ball rebound velocity.
The contact time calculated by the model is solely a
function of the spring stiffness, kB, and is not dependent
on the damping coefficient, cB. This characteristic was
utilized in reference [7] in the development of a simple
viscoelastic model of a tennis ball impact on a rigid
surface. In that model it was assumed that the spring
stiffness kB was constant during impact, as mentioned
previously. The spring stiffness was calculated analytically using the empirical data for the contact time. This
ensured that the contact time measured experimentally
was equal to that calculated by the model. The spring
stiffness was obtained for several ball types and a range
of ball impact velocities, including very low velocities. In
reference [7] a plot of the relationship between spring
stiffness and ball impact velocity is presented for each
ball type. These plots could be extrapolated to predict
the spring stiffness kB for an infinitely small impact
velocity. In the present study it is assumed that this
extrapolated stiffness is equivalent to the spring stiffness
at zero displacement, defined here as kBð0Þ . Thus, using
the data from reference [7], the values of kBð0Þ , for each
ball type, could be determined directly. The four ball
types used in reference [7] were the same as those used in
the present study, and relevant values of kBð0Þ for these
four ball types are shown in Table 2.
The other two parameters that are required to define
the spring stiffness kB are AK and a. The pair of values
AK and a and the damping parameter AC were obtained
for each ball type using an iterative procedure that will
be explained thoroughly in the section below. The
procedure adopted to determine AK , a and AC was split
into two discrete stages.
ð4Þ
2
2
ðcM Þt ¼
1147
3.4.1 Stage one
In this model, the contact time calculated by the model
is solely a function of the spring parameters kBð0Þ , AK
and a. The value of kBð0Þ had been previously defined
and therefore the aim of this first stage was to determine
a pair of values for AK and a that would result in a
Downloaded from pic.sagepub.com at PENNSYLVANIA
STATE
UNIV
on February
20, Vol.
2016
Proc.
Instn
Mech.
Engrs
218 Part C:
J. Mechanical Engineering Science
1148
S R GOODWILL AND S J HAAKE
Fig. 7 Typical examples of a comparison between the model and experimental data for (a) contact time and
(b) coefficient of restitution for the impact
contact time calculated by the model that was equal
to that measured experimentally for all ball impact
velocities.
The stiffness parameters AK and a were initially
assigned arbitrary values of 10 000 kN/m2 and 1.3
respectively. The model was then used to calculate the
contact times for ball impact velocities ranging from 10
to 30 m/s. The data calculated by the model were plotted
on a graph along with the experimental data, similar to
the curve labelled ‘initial attempt’ in Fig. 7a. The
coefficients of a second-order polynomial trend line for
the model data were calculated using least-squares
regression analysis. This polynomial was used to
quantify the sum of the (squared) differences between
the ‘initial attempt’ curve and the experimental data. A
Visual Basic Script Macro was written that utilized the
Goal Seek function to facilitate the iterative process of
finding the appropriate values of AK and a that
minimized the sum of the (squared) differences. The
Macro was run repeatedly until the solution converged
on a pair of values for the two parameters. This
converged solution was achieved when further iterations
of the model did not reduce the sum of the differences. A
typical converged solution for the model is shown in
Fig. 7a.
3.4.2
used to obtain the converged solution. A typical
solution is shown in Fig. 7b.
3.5
Summary
The two stages required to obtain the values of AK , a
and AC were conducted for each of the four ball types,
and the values of these parameters are shown in Table 2.
This procedure produced plots similar to those in Fig. 7
for each ball type. Typically, the correlation between the
converged model solution and the experimental data
was within 5 per cent for all ball types. This observation
confirms that the model can confidently be used to
predict the contact time and coefficient of restitution for
the impact.
Figure 8 gives an illustration of the relationship
between the spring stiffness kB and the ball centre-ofmass displacement xB. These relationships were calculated using equation (4) and the parameters given in
Table 2. This figure shows that ball 1 (standard
pressurized) is the stiffest ball and ball 3 (oversize
Stage two
The second stage of the modelling procedure involved
the determination of the damping parameter AC. This
involved a similar procedure to that used to determine
the spring parameters AK and a. The values of AK and a
that were determined from the first stage were input into
the model, and AC was initially assigned a value of
3.5 kN/m3. The model was used to calculate the
coefficient of restitution for a range of ball impact
velocities. As before, a Visual Basic Script Macro was
Proc. Instn Mech. Engrs Vol. 218 Part C:
Fig. 8 Illustration of the spring stiffness parameter kB for the
four ball types
Downloaded Engineering
from pic.sagepub.com
at PENNSYLVANIA STATE UNIV on February 20, 2016
J. Mechanical
Science
C07703 # IMechE 2004
MODELLING OF TENNIS BALL IMPACTS ON A RIGID SURFACE
pressurized) is the next stiffest. The pressureless ball
(ball 2) has a relatively high stiffness at small ball centreof-mass displacements, similar to that of the two
pressurized balls. However, at larger displacements,
the stiffness of this ball was relatively low compared
with the two pressurized balls. The punctured ball (ball
4) has a relatively low stiffness at small ball centre-ofmass displacements. However, at larger displacements,
the stiffness of this ball is similar to that of the
pressureless ball.
4
COMPARISON OF MODEL AND
EXPERIMENTAL DATA
It has been shown that the model can be used to
calculate the coefficient of restitution and contact time
for the impact. These calculated values correlate very
closely with the data obtained experimentally. However,
it should be remembered that the model parameters
were determined using these experimental data which
partially explains this high correlation.
A more appropriate method for validating the
accuracy of the model involves a comparison of the
force-time and force-displacement plots calculated by
the model and those determined experimentally. (The
displacement referred to here is that of the ball centre of
mass.) Typical force–time and force–displacement plots
are shown in Fig. 9 for the four ball types and two
different impact velocities. Both the experimentally
obtained data and the data calculated by the model
are presented.
The experimental data in Figs 9e to h show that the
ball initially experiences a very low load which is due to
the compression of the cloth on the ball, as mentioned
previously. The force calculated by the model correlates
very closely with the experimental data during this initial
phase of the impact. The model has been defined such
that it calculates a zero loading on the ball for ball COM
displacements of less than 2 mm during the compression
phase. For displacements greater than 2 mm, the
structural stiffness of the ball is momentarily defined
as the value kCONST. This high constant value simulates
the high structural stiffness of the shell of the tennis ball
prior to buckling. The plots in Fig. 9 confirm that the
chosen value of kCONST is of the correct order of
magnitude for all the ball types and inbound velocities.
After a time t ¼ 0.2 ms, it is assumed that the ball wall
buckles, and it is at this point that the modelled spring
stiffness changes from the high value ðkCONST Þ to a
lower value that is proportional to ball COM displacement ðkBð0Þ þ AK xaB Þ. In the remaining part of the
impact, the model and experimental force–displacement
(and force–time) plots exhibit a very close correlation,
with the two sets of results never differing by more than
approximately 10 per cent.
C07703
# IMechE 2004
1149
It can be seen that, during impact, the experimental
data often exhibit oscillations that are not evident in the
model data. This can be attributed to the fact that the
impact between a tennis ball and rigid surface is very
complex and difficult to simulate using a simple onedegree-of-freedom (1DOF) model. A tennis ball is a
multi-DOF object that deforms considerably during
impact. It is being modelled as a 1DOF system capable
of simulating a first-order approximation of the
structural stiffness of the object, but it may not be
able to model the higher-order modes of vibration which
will clearly have some influence on the experimentally
measured force. For example, towards the end of the
restitution phase, the measured force acting on ball 4
exhibits a local peak as illustrated in Figs 9d and h. This
also occurs in the other unpressurized ball (ball 2) and is
attributed to the ball ‘flipping out’ immediately prior to
the end of impact. This is analogous, but opposite in
direction, to the buckling of the ball during compression. This hypothesis was subsequently confirmed using
high-speed cinematography [15]. This mechanism is only
evident for the two unpressurized balls and has been
attributed to the fact that these two balls have a lower
structural stiffness owing to the lack of internal
pressurization.
5
CONTRIBUTION OF EACH COMPONENT
OF THE MODEL
The viscoelastic model shown in Fig. 6 is composed of
three components:
(a) structural stiffness (spring parameter kB),
(b) material damping (dashpot parameter cB),
(c) momentum flux (dashpot parameter cM).
Typical contributions for each of these components are
illustrated in Figs 10a and b for ball 1 (standard
pressurized) and ball 4 (punctured) respectively.
In the model it was assumed that the momentum flux
component is proportional to the change in momentum
of the flattened section of the ball. This will be highest in
the initial stage of impact, thus explaining the initial
steep rise in the momentum flux component. Also, it
should be noted that a large fraction of the initial model
force ðt < 0.6 msÞ is due to this momentum flux
component. Figure 10 compares the momentum flux
contributions for the stiffest and least stiff balls
respectively and shows that the two magnitudes are
very similar. Furthermore, during this initial period
ðt < 0.6 msÞ the experimentally measured loads for these
two balls are very similar. This is an interesting
observation because it has previously been concluded
that the high initial loading is due to the high structural
stiffness of the ball [5]. However, the model can be used
to show that the dominant component of the total force,
Downloaded from pic.sagepub.com at PENNSYLVANIA
STATE
UNIV
on February
20, Vol.
2016
Proc.
Instn
Mech.
Engrs
218 Part C:
J. Mechanical Engineering Science
1150
S R GOODWILL AND S J HAAKE
Fig. 9 (a)–(d) Force–time and (e)–(h) force–displacement plots for the four ball types. The data obtained
experimentally and those calculated by the model are both presented, for two different ball impact
velocities
Proc. Instn Mech. Engrs Vol. 218 Part C:
Downloaded Engineering
from pic.sagepub.com
at PENNSYLVANIA STATE UNIV on February 20, 2016
J. Mechanical
Science
C07703 # IMechE 2004
MODELLING OF TENNIS BALL IMPACTS ON A RIGID SURFACE
1151
Fig. 10 Typical comparison of experimental and model force–time data, showing the contribution of each
component of the total model force (ball impact velocity 20 m/s): (a) pressurized ball; (b) punctured
ball
during this period, is that due to the momentum flux
component.
Figures 10a and b both show that the principal
component of the total model force at maximum COM
displacement (or maximum total force) is the structural
stiffness component. It is therefore not surprising that
the structurally stiffer ball 1 exhibits a higher maximum
total model force, compared with the less stiff ball 4, at
this point.
6
DISCUSSION
6.1
Summary of the model
The model described in this paper requires a number of
parameters to be determined for each ball type. The
methods used to obtain these model parameters have
been thoroughly described in a previous section of this
paper and are summarized here:
1. kCONST—arbitrarily defined constant equal to
80 kN/m for all ball types. This represents the
structural stiffness of the ball prior to buckling.
2. kBð0Þ —ball stiffness for very small deformations. This
can be obtained using experimental data collected for
a simple drop test (in which the ball deformation is
very low).
3. AK , a and AC—parameters of the equations defining
the stiffness and damping of the ball model.
Experimental values of the coefficient of restitution
(COR) and contact time TC are obtained for an
impact between a ball and a force platform for a
range of ball impact velocities. A convergence
C07703
# IMechE 2004
technique is used to obtain model parameters that
generate a model solution calculating values of COR
and TC comparable with those determined experimentally.
6.2
Application of the model
The main aim of this study was to develop a model of an
impact between a ball and a rigid surface that can be
used to predict the dynamic response of the ball. Ball
manufacturers can use this model to predict the effect of
ball construction on, for example, the peak force acting
on the ball during impact.
The experimental and theoretical results obtained in
this study are also of benefit to the International Tennis
Federation. Currently, there is no ITF ruling to regulate
the dynamic properties of a tennis ball at realistic ball–
racket impact speeds. The only test that regulates the
coefficient of restitution for a tennis ball is conducted at
a relatively low speed. In reference [7] it was shown that
all standard-production balls perform very similarly at
this low speed impact but often act distinctly differently
at higher speeds. Clearly, the current ITF ruling does
not regulate the properties of the balls at these higher
speeds. A potential consequence of this is that ball
manufacturers may develop a new tennis ball that
conforms to the current specifications (at low speeds)
but acts undesirably at higher speeds. Therefore, the
ITF should consider the implementation of a new ruling
that will regulate the performance of the balls at these
higher speeds to ensure that the nature of the game is
not adversely affected. The experimental and theoretical
Downloaded from pic.sagepub.com at PENNSYLVANIA
STATE
UNIV
on February
20, Vol.
2016
Proc.
Instn
Mech.
Engrs
218 Part C:
J. Mechanical Engineering Science
1152
S R GOODWILL AND S J HAAKE
results obtained in this study will enable the ITF to
increase their understanding of the impact mechanism
and aid them in the definition of a proposal for a new
ruling for the dynamic properties of tennis balls.
7
CONCLUSIONS
A viscoelastic model of a tennis ball impacting
perpendicular to a rigid surface has been developed in
this study. One of the main features of this model is that
each constituent represents an actual component that is
present in the physical impact mechanism. This model
supersedes simpler models published by other authors
[7, 16]. In these studies it was assumed that the physical
properties of the ball (e.g. stiffness) were constant
throughout impact, and the magnitude was dependent
on the ball impact velocity.
In the model derived in this study, the physical
properties of the ball have been defined using realistic
functions of the ball centre-of-mass displacement.
Therefore, this model can be used as a first-order
approximation of the ball in a model of a more complex
impact. Indeed, the authors intend to advance this work
by modelling an impact between a tennis ball and
racket.
In this study, the force acting on the ball during
impact has been determined experimentally using a force
platform. This force was also calculated using the
derived model. The experimental and model force–time
plots (and force–displacement plots) exhibit a very good
correlation, especially for internally pressurized balls.
For other ball types there are supplementary features of
the impact mechanism that are not simulated by the
model, and therefore a weaker correlation is obtained.
However, it should be noted that, for the majority of the
impact, the model and experiment correlate to within 10
per cent. Therefore, this model is a significant improvement on previous attempts.
It has been shown that the model derived in this study
can be used to determine the magnitude of each of the
components of the total force acting on the ball during
impact. Simple experimental techniques are not capable
of performing this task. The three discrete components
of the total force that acts on the ball are the structural
stiffness, the material damping and the momentum flux.
The third component listed here has not been included
by other authors, but it has been shown to be a
significant constituent of the total force acting on a
tennis ball during the compression phase of impact.
The model has been used to elucidate the differences
between the experimentally determined force–time and
force–displacement plots for the different ball types.
Analysis of the individual contribution of each element
of the model has shown that the main component of the
force during the first 0.6 ms of impact is the momentum
Proc. Instn Mech. Engrs Vol. 218 Part C:
flux. In the model, this is relatively independent of the
ball stiffness, and therefore the magnitude of the force is
similar for all the balls. At the maximum ball centre-ofmass displacement during impact, the magnitude of the
modelled force is primarily a function of the structural
stiffness. This finding supports the experimental data
which show that the stiffest ball exhibits a considerably
larger force at this point, compared with the least stiff
ball.
ACKNOWLEDGEMENTS
Grateful thanks go to the International Tennis Federation for the financial and practical support given to this
project. The affiliation of Dr S. J. Haake as Technical
Consultant to the ITF is also acknowledged. The
authors would like to express thanks to the Engineering
and Physical Sciences Research Council (EPSRC) for
co-funding this project.
REFERENCES
1 Kawazoe, Y. Coefficient of restitution between a ball and a
tennis racket. Theoret. Appl. Mechanics, 1993, 42, 197–208.
2 Cross, R. Impact of a ball with a bat or racket. Am. J.
Physics, 1999, 67(8), 692–702.
3 Thomson, A. Dynamic characteristics of tennis balls and
the determination of ‘player feel’. MEng thesis, University
of Sheffield, 2000.
4 Cross, R. Dynamic properties of tennis balls. Sports Engng,
1999, 2(1), 23–33.
5 Head, H. Tennis racket. US Pat. 3,999,756, 1976.
6 International Tennis Federation, Rules of Tennis, 2002
(Wilton, Wright and Son Limited, London).
7 Haake, S. J., Carre, M. J. and Goodwill, S. R. The
dynamic impact characteristics of tennis balls with tennis
rackets. J. Sports Sci., 2003, 21, 839–850.
8 Goodwill, S. R. Modelling of tennis ball impacts on tennis
rackets. PhD thesis, University of Sheffield, 2002.
9 Carré, M. J. The dynamics of cricket ball impacts and the
effect of pitch construction. PhD thesis, University of
Sheffield, 2000.
10 Lieberman, B. B. and Johnson, S. H. An analytical model
for ball–barrier impact. Part 1: models for normal impact.
In Science and Golf II: Proceedings of the World Scientific
Congress of Golf (Eds A. J. Cochran and M. R. Farally),
1994, pp. 309–314 (E&FN Spon, London).
11 Hubbard, M. and Stronge, W. J. Bounce of hollow balls on
flat surfaces. Sports Engng, 2001, 4(2), 49–61.
12 Percival, A. L. The impact and rebound of a football. The
Manchester Association of Engineers, Session 1976–77,
Vol. 5, pp. 17–28.
13 Johnson, W., Reid, S. W. and Trembaczowski-Ryder, R. E.
Impact, rebound and flight of a well-inflated pellicle as
exemplified in association football. The Manchester
Association of Engineers, Session 1972–73, Vol. 5, pp. 1–
25.
Downloaded Engineering
from pic.sagepub.com
at PENNSYLVANIA STATE UNIV on February 20, 2016
J. Mechanical
Science
C07703 # IMechE 2004
MODELLING OF TENNIS BALL IMPACTS ON A RIGID SURFACE
14 Howard, A. Sports surface characterisation. MEng thesis,
University of Sheffield, 2000.
15 Taylor, D. The dynamic properties of tennis balls on
impact with a racket and their relationship to the subject of
‘player feel’. MEng thesis, University of Sheffield, 2002.
C07703
# IMechE 2004
1153
16 Dignall, R. J. and Haake, S. J. Analytical modelling of the
impact of tennis balls on court surfaces. In Tennis Science
and Technology, 2000, pp. 155–162 (Blackwell Science,
Oxford).
Downloaded from pic.sagepub.com at PENNSYLVANIA
STATE
UNIV
on February
20, Vol.
2016
Proc.
Instn
Mech.
Engrs
218 Part C:
J. Mechanical Engineering Science