comparison of a finite element model of a tennis racket to experimental data
TRANSCRIPT
ORIGINAL ARTICLE
Comparison of a finite element model of a tennis racketto experimental data
Tom Allen • Steve Haake • Simon Goodwill
Published online: 5 December 2009
� International Sports Engineering Association 2009
Abstract Modern tennis rackets are manufactured from
composite materials with high stiffness-to-weight ratios. In
this paper, a finite element (FE) model was constructed to
simulate an impact of a tennis ball on a freely suspended
racket. The FE model was in good agreement with experi-
mental data collected in a laboratory. The model showed
racket stiffness to have no influence on the rebound char-
acteristics of the ball, when simulating oblique spinning
impacts at the geometric stringbed centre. The rebound
velocity and topspin of the ball increased with the resultant
impact velocity. It is likely that the maximum speed at
which a player can swing a racket will increase as the
moment of inertia (swingweight) decreases. Therefore, a
player has the capacity to hit the ball faster, and with more
topspin, when using a racket with a low swingweight.
Keywords Ball � Finite element � High-speed video �Impact � Racket � Spin � Tennis
1 Introduction
Tennis racket materials have changed over the years, from
wood to aluminium alloy to fibre composites [1–3] and
these developments have changed the way in which the
game is played. Advances in racket technology, especially
developments in materials, have allowed players to hit
shots faster and with greater accuracy [4], effectively
increasing the speed of the game [5]. Manufacturers
began experimenting with composite materials in 1970s
[2, 3], mainly due to their high stiffness-to-weight ratios,
in comparison to metals. Currently, the majority of
rackets are manufactured from composite lay-ups as this
allows materials to be precisely placed for desired stiff-
ness and mass distributions. The reduction in the mass
and the increase in the structural stiffness of tennis
rackets, dating from 1870s to 2007, have allowed serve
speeds to increase by approximately 17.5% and the
reaction time available to the receiver to reduce by
approximately 15% [6]. Finite element (FE) techniques
have been used by previous authors to further the scien-
tific understanding of tennis equipment [7–11]. An earlier
FE model by Allen et al. [7] was successfully validated as
a good approximation of a head-clamped tennis racket for
oblique spinning impacts. The frame of a head-clamped
racket can essentially be thought of as infinitely heavy as
it cannot be displaced during an impact with a ball. Head-
clamped rackets are used for analysing the effect of
stringbed properties, such as string type and tension [12].
However, rigidly clamping a racket by the head is clearly
not a good representation of how it will be supported
during play, especially as the frame of the racket cannot
be displaced or deformed.
Brody [13] demonstrated that the frequency response of
a freely suspended tennis racket is similar to that of a
handheld tennis racket. Therefore, freely suspending a
racket is currently the best representation of how it will be
supported during an actual tennis shot. Previous authors
have found that, for impacts normal to the face on the long
axis of a freely suspended racket, the rebound velocity of
the ball is dependent on impact location and is lowest at the
tip and highest in an area near the throat [4, 14, 15].
T. Allen (&) � S. Haake � S. Goodwill
Faculty of Health and Wellbeing, Sports Engineering Research
Group, Centre for Sport and Exercise Science,
Sheffield Hallam University, Sheffield, UK
e-mail: [email protected]
Sports Eng (2010) 12:87–98
DOI 10.1007/s12283-009-0032-5
Goodwill and Haake [15] produced spring damper models
for normal impact on rigid and flexible rackets. The flex-
ible racket model showed closer agreement with the
experimental data than the rigid racket model. The rigid
racket model overpredicted the rebound velocity of the
ball, for impacts offset from the geometric stringbed centre
(GSC) along the longitudinal axis. In addition, by using
only impacts normal to the face on the long axis of the
stringbed, the models by Goodwill and Haake [15] were
less representative of a typical elite player’s shot [16]; thus
they went onto investigate oblique spinning impacts.
Goodwill and Haake [17] analysed the oblique impact of
a tennis ball with no inbound spin on a freely suspended
racket. The inbound angle was set at 36� to the stringbed
normal and the inbound velocity was in the range from 15
to 40 m s-1. All of the impacts were reported to be at the
GSC, as this was stated to be where players typically hit the
ball during play. The rebound velocity and topspin of
the ball both increased with the inbound velocity, whilst
the rebound angle remained essentially constant. Analysis
of tennis shots from elite players has highlighted that the
ball can have spin rates of around 300–550 rad s-1 prior to
impact with the racket [18, 19]. Therefore, the impacts
would have been more representative of a typical tennis
shot from an elite player if the balls had been projected
onto the racket with initial spin.
The aim of this paper is to produce and validate an FE
model of a freely suspended tennis racket. The freely
suspended racket model will be an extension of the head-
clamped racket model produced by Allen et al. [7] in
ANSYS/LS-Dyna 10.0. ANSYS/LS-Dyna is an Explicit FE
solver which can be applied to a variety of different impact
scenarios, including sporting applications [11, 20–22]. The
frame of the freely suspended racket will have the capacity
to displace and deform during an impact with the ball.
Initially, the freely suspended racket model will be vali-
dated for impacts normal to the face at a variety of loca-
tions on the stringbed. The aim of simulating impacts
normal to the face at a variety of locations will be to
provide a rigorous validation of the model. Following the
comparison with experimental data for impacts normal to
the face, the model will be validated for oblique spinning
impacts close to the GSC. An oblique spinning impact
close to the GSC is a good representation of a typical tennis
shot from an elite player [16].
2 FE model
2.1 Description of the model
The model ultimately consisted of three parts: (1) the
frame, (2) stringbed of the racket and (3) the ball.
2.1.1 Frame
The racket modelled had an overall length of 0.68 m and a
head size of 0.35 9 0.27 m (Fig. 1). These dimensions are
representative of a modern tennis racket frame [6]. The
freely suspended racket model was based on the head-
clamped racket model published by Allen et al. [7], with
some major modifications to the frame of the racket, as
detailed below. (1) No constraints were applied to the
frame in the freely suspended racket model. Having no
constraints allowed the frame of the racket to displace
during an impact. (2) The rigid body material model
(MAT_RIGID) was changed to a linear elastic material
model (MAT_ELASTIC) to enable deformation of the
racket frame to be simulated. A linear material model was
considered to be adequate due to the relatively small
deformations of a racket during an impact with a ball.
(3) The frame geometry was also separated into three parts,
i.e. the handle, throat and head (Fig. 1). With this model,
the mass distribution of the racket can thus be adjusted by
changing the shell thicknesses and densities of the handle,
throat and head sections. International Tennis Federation
(ITF) branded rackets (carbon-fibre construction) were
used for the laboratory-based validation experiments.
Therefore, the mass and mass distribution of the racket in
the FE model was set to correspond to the ITF branded
racket, as shown in Table 1. The mass, balance point and
mass moment of inertia (swingweight) of the ITF racket
were taken from Goodwill [23]. The polar moment of
inertia (twistweight) of the ITF racket was obtained
experimentally using bifilar suspension theory [24].
2.1.2 Stringbed
The most complex task of the FE model was simulating the
interwoven stringbed. A load of 150 N was applied to a
rigid cylinder attached to both ends of every string during
the dynamic relaxation phase of the simulation. The pur-
pose of the applied load was to produce an initial contact
Fig. 1 Finite element model racket geometry with three separate
sections
88 T. Allen et al.
force at the intercepts of the individual strings in the
interwoven stringbed. The convergence tolerance for
dynamic relaxation was 0.06. The ends of the strings were
tied to the frame during the transient phase of the simula-
tion to produce a strung racket. The tied contact between
the strings and racket was set to initiate at a simulation time
of 0.00135, 0.00015 s before the ball impacted the string-
bed. Full details of the method used to define stringbed
tension are given in Allen et al. [7]. A linear elastic
material model (MAT_ELASTIC) was used for the strings,
with a Young’s modulus of 7.2 GN m-2, a density of
1,100 kg m-3, and a Poisson’s ratio of 0.3 [9]. A coeffi-
cient of friction (COF) of 0.4 was defined between the ball
and stringbed [25]. Previous work, using the FE model of a
stringbed published by Allen et al. [9], indicated that
increasing the COF between the ball and strings from 0.4 to
0.6 has little influence on the rebound characteristics of the
ball [26]. Reducing the COF to 0.2 caused the model to
overpredict the rebound topspin of the ball in comparison
to experimental data.
2.1.3 Ball
The ball consisted of a pressurised rubber core and felt
cover; full details of the ball model and its validation
can be found in Allen et al. [10]. The initial velocity
and spin of the ball were defined using INITIAL_
VELOCITY_GENERATION.
2.2 Details of the simulations
Table 2 shows the details of the material models and ele-
ments which were used for the main parts of the model.
The FE model acted as a base for the geometry and mass of
the racket. Two versions were created from the base model
to encompass the large range of values of racket stiffness
typically found (Table 3). Previous authors have also used
the natural frequency of a tennis racket to determine the
required structural stiffness for the frame in an analytical
model [15, 23]. The natural frequencies of tennis rackets
dating from 1870s to 2007 are within the range of 70–
190 Hz [6]. The natural frequency of the ITF racket used in
the laboratory experiment was 134 Hz [23]. Modal analysis
was undertaken on the FE model of the racket frame, for
different values of effective modulus. An effective modu-
lus of 20 GN m-2 resulted in a natural frequency of
135 Hz, which is within 1% of the value of 134 Hz for the
ITF racket. The reason for using two values of natural
frequency was to determine the effect of racket stiffness on
the rebound characteristics of the ball. Using two values of
effective modulus to produce rackets with natural fre-
quencies that bracket the ITF racket justifies the use of an
isotropic material model for simulating an anisotropic
composite lay-up.
3 Experimental methods
Tennis balls were projected from a modified pitching
machine (BOLA) onto the freely suspended ITF racket,
using the impact rig detailed in Choppin [27] (Fig. 2a). The
bespoke impact rig was specifically made for analysing the
impact of a tennis ball on a racket. The balls were projected
onto an initially stationary racket, as the impacts were all
conducted in the laboratory frame of reference [4]. The
racket was supported at the tip, from a small horizontal pin,
with its longitudinal axis vertical. The pin was located
underneath the tip of the frame between the two central
Table 1 Racket mass
distribution in the finite element
model
Part Mass
(kg)
Balance point
from butt (m)
Mass moment of inertia
about butt (kg m2)
Polar moment of
inertia (kg m2)
Handle 0.098 N/A 0.00098 0.000021
Throat 0.090 N/A 0.00622 0.000125
Head 0.162 N/A 0.04331 0.001446
Complete racket FE model 0.348 0.324 0.05111 0.001592
ITF racket 0.348 0.325 0.05337 0.001550
Table 2 Material model, type of elements and number of elements for the main parts of the finite element model
Part Material model Type of elements Number of elements
Ball (felt cover) Foam (MAT_LOW-DENSITY_FOAM) Reduced integration 8-node brick (SOLID164) 21,600
Ball (rubber core) Hyperelastic (MAT_OGDEN_RUBBER) Reduced integration 8-node brick (SOLID164) 21,600
Strings Linear elastic (MAT_ELASTIC) Reduced integration 8-node brick (SOLID164) 29,891
Racket frame Linear elastic (MAT_ELASTIC) Belytschko-Tsay formulation shell (SHELL163) 27,410
Comparison of a finite element model of a tennis racket to experimental data 89
main strings. Using a pin to support a racket is a technique
which had been used by previous authors [14, 15, 23].
Three identical rackets were used for the experimental
testing, all strung at 289 N (65 lbs).
Non-spinning impacts normal to the face were simulated
at four different impact positions on the stringbed, labelled:
centre, off-centre, tip and throat (Fig. 3; Table 4). The
inbound velocity of the balls in the impacts normal to the
face was in the range from 10 to 40 m s-1. Oblique spin-
ning impacts were simulated at nominal inbound speeds of
20 and 30 m s-1 and a nominal angle of 25� to the z axis,
on a plane parallel to the x and z axes (refer to axes on
Fig. 2a). Changing the frame of reference from the court to
the laboratory means that the ball should have backspin
prior to impact to represent a topspin shot [12] (Fig. 4).
The inbound spin of the oblique impacts was in the range
from 100 rad s-1 of topspin to 500 rad s-1 of backspin.
The nominal impact location of the oblique impacts was
the centre of the stringbed, although the impacts were
slightly offset from the long axis of the racket to com-
pensate for the horizontal displacement of the ball whilst it
remains in contact with the strings [17]. The impacts were
captured using two synchronised Phantom V4.3 high-speed
video cameras, operating at 1,900 fps and an exposure time
of 0.2 ms. The two cameras were positioned on separate
sides of the impact rig to provide three dimensional (3D)
coordinates of the ball and racket, as detailed by Choppin
[27] (Fig. 2b).
The impacts were recorded as bitmap images and ana-
lysed using Richimas v3 image analysis software. A
detailed description of Richimas v3 is given in Goodwill
and Haake [12]. The 2D positions of the ball were obtained
manually from each camera using Richimas. The pairs of
2D coordinates obtained using Richimas were converted
into global 3D coordinates (camera frame of reference)
using a freely available MATLAB Toolbox which was
developed by Bouguet [28]. The 3D calibration was
undertaken using a checkerboard, as developed by Zhang
[29] and applied to tennis impact testing by Choppin [27].
To measure the impact position on the stringbed, a trans-
formation matrix was used to convert the 3D coordinates of
the ball into the racket frame of reference, with the origin at
the GSC (Fig. 5). The origin was located at the GSC by
obtaining the global 3D coordinates of three white markers
at known locations on the frame of the racket (Fig. 5).
Impact was assumed to initiate at the first instance when
the ball’s perpendicular distance (Z) from the stringbed (XY
plane at Z = 0) was less than its radius (33 mm). The
horizontal (x) and vertical (y) impact positions (relative to
the GSC) were obtained from the position of the ball at the
first point of contact with the stringbed. For full details of
the method used to reconstruct tennis ball to racket impacts
in 3D using two high-speed video cameras, refer to
Choppin [27].
The spin of the balls from the oblique impacts was
assumed to be about the y axis, which is top/back spin
relative to the racket. Ball spin was calculated in 2D using
Table 3 Natural frequencies of the two racket frame models with
different values of effective modulus
Racket Effective
modulus
(GN m-2)
Poisson’s
ratio
Natural
frequency
(Hz)
FE model: low structural stiffness 10 0.3 96
FE model: high structural stiffness 70 0.3 253
ITF carbon-fibre racket N/A N/A 134
The natural frequency of the ITF Carbon-fibre racket was taken from
Goodwill [23]
Fig. 2 a Impact rig used for simulating impacts on a freely
suspended tennis racket. b Relative camera positions for measuring
the trajectory of a tennis ball in 3D (Modified from Choppin [27])
Fig. 3 Impact positions on the stringbed for the validation of the
freely suspended racket model for impacts normal to the face
Table 4 Impact locations for the impacts normal to the face on a
freely suspended racket (mean ± SD)
Impact location Horizontal distance
from the stringbed
centre (mm)
Vertical distance from
the stringbed centre (mm)
(? = towards tip)
Centre 13 ± 7 8 ± 7
Off-centre 31 ± 10 4 ± 7
Throat 18 ± 8 -55 ± 16
Tip 13 ± 11 49 ± 7
90 T. Allen et al.
markings, which were drawn on the felt (Fig. 5). The
process involved using Richimas v3 to obtain the coordi-
nates of the geometric ball centre (GBC) see point A on
Fig. 6 and a marker, which is the intercept of lines on the
ball, see point B on Fig. 6. The radius of the ball and
the distance (X) were then used to obtain the angle h. The
process was repeated to obtain four angles, before and after
impact. The spin of the ball was calculated from the gra-
dient of the angle time data. To ensure the highest possible
accuracy with this method, the spin was calculated inde-
pendently from both cameras, and the mean value was used
to compare with the model. The root mean squared error
(RMSE) between the between the spin measured from the
two cameras was 21 rad s-1 before impact and 11 rad s-1
after impact. A difference of 21 rad s-1 equates to 21% at
100 rad s-1 and 4% at 500 rad s-1.
Table 4 shows the calculated impact locations for the
impacts normal to the face. The RMSE between the
resultant and z velocities (normal to stringbed) for all
the impacts normal to the face in the laboratory-based
experiment was 0.008 m s-1 for inbound and 0.04 m s-1
for rebound. As the RMSE is very low the impacts were
considered to be normal to the face of the racket and the
resultant velocities were analysed against the FE models.
Table 5 shows the calculated velocities and angles before
impact and the impact locations on the stringbed for the
oblique impacts.
A repeatability study was undertaken to assess the level
of human error in the manual tracking method. An impact
with low, medium and high inbound spin was selected and
analysed ten times (Table 6). The impacts had a nominal
inbound velocity of 20 m s-1. The uncertainties in the
measured values are similar to those reported by Goodwill
and Haake [17].
FE simulations with the initial conditions shown in
Table 7 were undertaken to correspond to the laboratory-
based experimental data. The impact positions on the
stringbed were identical to the mean values in Table 4 for
the impacts normal to the face and Table 5 for the oblique
impacts. The initial conditions of each impact and the
Fig. 4 Diagram to show that
the ball should impact the racket
with backspin in the laboratory
frame of reference to represent a
topspin shot
Fig. 5 Racket position showing throat and side markers and axis
coordinate system. B is an intercept of the markings on the ball
Fig. 6 Diagram to show the method used for calculating the top/back
spin of a tennis ball, by calculating the change in h over time (T). B is
an intercept of the markings on the ball and A is the GSC. h was
calculated using trigonometry from the radius of the ball (R) and the
horizontal distance between A and B(X)
Comparison of a finite element model of a tennis racket to experimental data 91
material properties of the racket were set using the tennis
design tool (TDT). The TDT is a parametric modelling
programme which was produced in Visual Basic 2005 [26].
4 Results
The aim of this investigation was to compare an FE model
against experimental data with the intention of determining
the effect of tennis racket structural stiffness. Two racket
models were produced to bracket the ITF racket in terms of
structural stiffness. The stiffness of the racket was mea-
sured from the fundamental frequency. Results from the
simulations showed the following key findings. Firstly, for
impacts normal to the face, the structural stiffness of the
racket frame only affected the rebound velocity of the ball
for impacts in the throat region. Secondly, for oblique
spinning impacts at the GSC, the structural stiffness of the
racket frame did not affect the rebound velocity, angle nor
spin of the ball. In addition, the rebound velocity, angle and
spin of the ball all decreased as the inbound backspin
increased; whilst, the rebound velocity, angle and spin of
the ball all increased with the inbound velocity of the ball.
These results will be explained in detail below.
Figure 7 shows a comparison of the FE model with the
experimental data for the impacts normal to the face. The
results are expressed as the resultant velocity of the ball
after an impact with the racket. There are four nominal
impact positions on the stringbed. There are two sets of
data from the FE model corresponding to different racket
frame stiffnesses. The rebound velocity of the ball was
slightly lower for the off-centre impacts in comparison to
those at the centre. The rebound velocity was lowest for the
tip impacts and highest for those at the throat, in agreement
with Goodwill and Haake [14, 15] and Kanda et al. [30].
Figure 7c shows that four of the tip impacts, which had an
inbound velocity below 20 m s-1, had a larger rebound
velocity than expected from the trend of the rest of the data
(see highlighted data points). Three of these four impacts
were closer to the GSC in comparison to the mean impact
location, in both the vertical and horizontal directions. The
remaining impact had an offset distance from the long axis
of the stringbed which was less than the mean value.
Raising the effective modulus of the racket frame in the
FE model increased the rebound velocity of the ball for the
throat impacts whilst having a negligible effect on those at
the other locations, in agreement with Goodwill and Haake
[15]. The discrepancy between the two models increased
with inbound velocity, which can be accounted for by
energy losses due to racket frame vibrations [15]. The
model with the lower structural stiffness will deform more,
particularly at high impact speeds, resulting in a decrease
in the rebound velocity of the ball. The FE model of the
racket with the effective modulus of 10 GPa, was in rela-
tively good agreement with the experimental data for all
four of the impact locations on the stringbed. The model
with the higher effective modulus of 70 GPa slightly
overpredicted the rebound velocity of the ball for the throat
and off-centre impacts when the inbound velocity was
greater than 20 and 25 m�s-1, respectively. The overpre-
diction of rebound velocity for the off-centre impacts is
likely to be due to the model underpredicting the defor-
mation of the racket in torsion.
Figure 8 shows a comparison of the FE model with the
experimental data, for the oblique impacts at the two
inbound velocities. As with the impacts normal to the face,
there are two sets of data for the FE model corresponding
Table 5 Inbound velocities, angles and impact locations for the
oblique impacts on a freely suspended racket (mean ± SD)
Nominal inbound velocity (m s-1) 20 30
Nominal inbound angle (�) 25 25
Calculated inbound velocity (m s-1) 18.0 ± 0.5 28.0 ± 0.4
Calculated inbound angle (�) 23.7 ± 1.3 22.9 ± 0.9
Horizontal distance from the stringbed
centre (mm)
9 ± 16 15 ± 10
Vertical distance from the stringbed centre
(mm) (? = towards tip)
9 ± 12 8 ± 11
Table 6 Results of a repeatability test for impacts with low medium and high inbound spin
Low spin (-5 rad s-1) Medium spin (252 rad s-1) High spin (530 rad s-1)
Resultant inbound velocity (m s-1) 0.1 (0.4%) 0.1 (0.6%) 0.1 (0.5%)
Resultant rebound velocity (m s-1) 0.1 (1.0%) 0.1 (0.8%) 0.1 (1.5%)
Inbound angle (�) 0.3 (1.4%) 0.5 (2.0%) 0.4 (1.4%)
Rebound angle (�) 0.3 (0.9%) 0.5 (3.1%) 0.9 (18.1%)
Inbound spin (rad s-1) 9 (176%) 8 (3.2%) 21 (3.9%)
Rebound spin (rad s-1) 9 (9%) 4 (38.5%) 8 (18.9%)
Impact distance from long axis (mm) 1 (56%) 2 (17.4%) 2 (13.0%)
Impact distance from short axis (mm) 1 (9%) 1 (19.8%) 1 (2.7%)
(value) = SD as a percentage of the mean
92 T. Allen et al.
to different racket stiffnesses. All of the impacts were close
to the GSC. The rebound speed of the ball decreased with
increasing inbound backspin and was lower for the impacts
at 18 m s-1 than those at 28 m s-1. Three of the impacts in
Fig. 8b had a rebound speed which was lower than
expected from the trend of the rest of the data (see high-
lighted data points). All three of these impacts had: (1) an
inbound speed which was lower than the mean speed and
(2) an impact location which was further from the GSC in
comparison to the mean impact location, in both the hori-
zontal (X) and vertical (Y) directions. Goodwill and Haake
[17] state that the rebound characteristics of the ball are
highly dependent on the impact position, as a result of the
non-uniformity of the stringbed. The resultant rebound
speeds obtained from the two FE models were in good
agreement with the experimental data for both inbound
speeds. There was only a very small difference in the
rebound speeds obtained from the two FE models of dif-
ferent racket stiffnesses. The negligible effect of racket
frame stiffness on the rebound speed of the ball was in
agreement with the results obtained for the impacts normal
to the face at the centre of the stringbed.
Figure 9 shows that the rebound angle of the balls
(relative to the racket normal) decreased with increasing
inbound backspin. The rebound angles were virtually
identical for both inbound velocities when the balls had a
negligible amount of inbound spin, in agreement with
Goodwill and Haake [17]. However, the rebound angle
Table 7 Initial conditions used in the FE model to simulate an impact between a tennis ball and freely suspended racket
Inbound velocity
(m s-1)
Inbound
angle (�)
Inbound backspin
(rad s-1)
Number of impact
positions
Impacts normal to the face 10, 20, 30 and 40 0 0 4
Low velocity oblique impacts 18 23.7 0, 200 and 400 1
High velocity oblique impacts 28 22.9 0, 200 and 400 1
Fig. 7 Ball rebound velocity
for impacts normal to the face
on a freely suspended racket.
a Centre, b off-centre, c tip
and d throat
Comparison of a finite element model of a tennis racket to experimental data 93
decreased more with increasing inbound backspin when the
inbound velocity of the balls was 18 m s-1, in comparison
to 28 m s-1.
As with rebound speed, there was very little difference
in the results obtained from the two FE models. The FE
models were both in relatively good agreement with the
experimental data, although the models slightly underpre-
dicted the rebound angle of the ball by a few degrees, for
both inbound velocities. Goodwill and Haake [17] found
rebound angle to increase with string tension. Therefore, it
is likely that the FE model underpredicted rebound angle
because the structural stiffness of the stringbed was too
low.
Figure 10 shows that the rebound spin of the balls
decreased with increasing inbound backspin. The rebound
spin was lower for the inbound velocity of 18 m s-1, and it
decreased more with inbound backspin. As with rebound
velocity and angle, there was very little difference in the
results obtained from the two FE models. The FE models
were in good agreement with the experimental data for
inbound backspins which were lower than approximately
200 rad s-1. At higher inbound backspins, the models
slightly underpredicted the rebound spin of the balls.
5 In-depth analysis of an oblique spinning impact
An investigation was then undertaken to ascertain how and
why the rebound properties of the ball changed with
inbound spin. An impact at 28 m s-1 and 23�, with
200 rad s-1 of backspin was selected for analysis. These
values of inbound velocity and backspin were considered
to correspond well with those employed in play [16]; these
impacts were also in good agreement with the trend of the
experimental data. Figure 11 shows how the horizontal and
vertical forces acting on the ball and its horizontal velocity
and spin change throughout the impact. The horizontal and
vertical planes are defined as being parallel and normal to
the face of the stringbed, respectively. The vertical force
shows a nonlinear rising portion because (1) the internal
pressure of a ball increases with deformation [11] and (2)
the tangential stiffness of the stringbed increases with
Fig. 8 Ball rebound velocity
for oblique impacts on a freely
suspended racket. a 18 m s-1
and 24�, b 28 m s-1 and 23�
Fig. 9 Ball rebound angle for
oblique impacts on a freely
suspended racket. a 18 m s-1
and 24�, b 28 m s-1 and 23�
94 T. Allen et al.
contact area and displacement [14]. The initial horizontal
force was negative which means that the force was acting
in the opposite direction to the horizontal motion of the
ball. The negative horizontal force caused a decrease in the
horizontal velocity of the ball and an increase in its topspin.
At approximately 2.25 ms the ball reached its minimum
horizontal velocity and maximum topspin. The contacting
region of the ball then ‘gripped’ the stringbed and the ball
deformed forwards storing energy [31]. Approximately
0.25 ms later the ball lost its ‘grip’ with the stringbed and
the horizontal force reversed sign. The reverse in the sign
of the horizontal force caused an increase in the horizontal
velocity of the ball and a decrease in its topspin. This
reversal of the horizontal force occurs when the spin rate of
the ball exceeds that associated with rolling; this is com-
monly referred to as ‘over-spinning’. The horizontal force
acting on the ball then converged towards zero. Once the
horizontal force equalled zero, there was no further change
in the horizontal velocity or spin of the ball, which implied
that the ball was rolling off the stringbed. There was no
horizontal force at the end of the impact despite a non-zero
vertical force because the ball was rolling.
A further investigation was undertaken with the FE
model to substantiate the interesting findings in Fig. 11.
This can be referred to in three parts: (1) How does
changing the COF between the ball and stringbed influence
the results? (Fig. 12) (2) Does the structural stiffness of the
stringbed (Fig. 13) or (3) the ball, alter the findings further
(Fig. 14)? The impact conditions were identical to those
detailed above. The horizontal force reversed sign and the
ball was rolling off the stringbed for all the impacts, in
agreement with the results shown in Fig. 11.
Figure 12 shows the effect of altering the COF
between the ball and stringbed. Values of 0.2 and 0.6
Fig. 10 Ball rebound spin for
oblique impacts on a freely
suspended racket. a 18 m s-1
and 24�, b 28 m s-1 and 23�
Fig. 11 a Force, b horizontal velocity and c spin obtained from the FE model, throughout an impact at the centre of a freely suspended racket
with an inbound velocity of 28 m s-1, angle of 23� and with 200 rad s-1 of backspin (70 GPa/253 Hz)
Comparison of a finite element model of a tennis racket to experimental data 95
were used for the COF in the model. The COF had no
noticeable effect on the vertical force. The initial hori-
zontal force was larger when the COF was 0.6, causing
the ball to ‘overspin’ earlier in the impact. The hori-
zontal force was also larger when it reversed sign,
resulting in the ball leaving the stringbed at a slightly
higher horizontal velocity and with very slightly less
topspin.
Figure 13 illustrates the effect of changing the structural
stiffness of the stringbed. The Young’s modulus of the
strings was decreased and increased by 25% to produced
values of 5.4 and 9.0 GPa, respectively. The vertical force
was larger on the rising portion for the high stiffness
stringbed. The initial horizontal force was also very slightly
larger for the stiffer stringbed and it reversed sign earlier in
the impact. The horizontal force was very similar in
magnitude when it reversed sign for both stringbeds.
Therefore, the ball rebound from both stringbeds with a
very similar horizontal velocity and topspin, in agreement
with Goodwill et al. [12].
Fig. 12 a Force, b horizontal velocity and c spin obtained from two FE models with different values of ball-to-string COF
Fig. 13 a Force, b horizontal velocity and c spin obtained from two FE models with different stringbed stiffness. The low and high stiffness
strings had a Young’s modulus of 5.4 and 9.0 GPa, respectively
Fig. 14 a Force, b horizontal velocity and c spin obtained from two FE models with different stiffness balls. The stiffness of the low stiffness
ball was 20% lower than the original model and the stiffness of the high stiffness ball was 20% higher than the original model
96 T. Allen et al.
Figure 14 shows the effect of changing the structural
stiffness of the ball. The modulus of the rubber core was
increased by 20% to produce a ball with high structural
stiffness and decreased by 20% to produce a ball with low
structural stiffness. The maximum vertical force was very
slightly higher for the stiffer ball. The horizontal force was
also very slightly larger for the stiffer ball when it reversed
sign. Overall, the different between the horizontal and
vertical forces acting on the two balls was marginal;
therefore both balls rebounded from the stringbed with the
same horizontal velocity and topspin.
6 Discussion
When simulating impacts normal to the face, the FE model
was in relatively good agreement with the experimental
data for all four of the impact locations used in this
investigation. The FE model was in better agreement with
the experimental data when the effective modulus of the
racket was 10 GPa as opposed to 70 GPa, due to the nat-
ural frequency of the 96 Hz racket being closer to that of
the ITF racket. Increasing the structural stiffness of the
racket resulted in an increase in the rebound velocity of the
ball for impacts at the throat, in agreement with Goodwill
and Haake [15] and Kanda et al. [30]. The structural
stiffness of the racket had only a very marginal effect on
the rebound velocity of the ball at the other impact loca-
tions. The small effect of two very different values of
effective modulus provides justification for the use of an
isotropic material model to simulate a composite lay-up.
When simulating oblique spinning impacts, the two
racket models of different stiffness were both in good
agreement with the experimental data, in terms of the
rebound velocity of the ball. The two models were also in
relatively good agreement with the experimental data for
rebound angle and spin; although, they did slightly un-
derpredict the rebound angle of the ball for the entire range
of inbound backspins. The models also underpredicted the
rebound spin of the ball for inbound backspins greater than
approximately 200 rad s-1. However, it was difficult to
precisely determine the accuracy of the FE model due to
the uncertainty in experimentally measuring both inbound
and rebound spin (*20 rad s-1). The stiffness of the
racket frame had very little influence on the rebound
characteristics of the ball. The difference between the two
models was much lower than the scatter in the experi-
mental data. This agreed with the results obtained for
impacts normal to the face close to the GSC. The GSC
corresponds to a node point of the racket and hence
exhibits very low vibration of the fundamental mode. It is
likely that racket stiffness will have a greater influence on
the rebound characteristics of the ball for impacts away
from the GSC, particularly in the throat region, as found
with the impacts normal to the face. The results indicate
that the ball was ‘over-spinning’ at around the mid-point of
the impact. Goodwill and Haake [12] also found the ball to
be ‘over-spinning’ when performing experimental impacts
on a head-clamped tennis racket. The results also indicate
that the ball was rolling off the stringbed at the end of the
impact. Therefore, adjusting the COF between the ball and
stringbed did not have a large effect on the rebound char-
acteristics of the ball.
This investigation has indicated that the structural
stiffness of a tennis racket does not have an influence on
the rebound characteristics of the ball when simulating a
typical groundstroke from an elite player, i.e. an oblique
spinning impact at the GSC. However, the rebound
velocity and topspin of the ball both increased with the
inbound velocity of the ball. Therefore, for a set inbound
ball velocity, a player can increase the resultant impact
velocity of a shot by simply swinging their racket faster.
Mitchell et al. [32] have shown that during a serve, swing
speed increases as racket swingweight decreases; therefore,
the effect of the mass of the racket on the impact should
also be considered. For a constant swing speed, less
momentum will be transferred to the ball if a player uses a
lighter racket [6, 33]. Haake et al. [6] analysed the effect of
racket mass in the range from 0.29 to 0.36 kg on serve
speeds. They concluded that the ball is launched faster as
the mass of the racket decreases. Miller [33] states that
decreasing racket mass below 0.25 kg would be counter-
productive, as the transfer of momentum to the ball would
be less effective. If swing speed also increases with
decreasing swingweight for groundstokes, as would be
expected, then it is likely that a player will be able to hit the
ball faster and with more topspin when using a lighter
racket. This indicates that the use of composite materials in
tennis rackets has indeed increased the speed of the game.
Further research into the effect of racket mass on oblique
spinning impacts and the relationship between swing speed
and swingweight should be undertaken to strengthen this
hypothesis.
7 Conclusion
An FE model of a freely suspended tennis racket was
compared against experimental data for both normal to the
face and oblique impacts. When simulating impacts normal
to the face, the FE model of the racket with the natural
frequency of 96 Hz had the best agreement with the
experimental data. The results from the FE model showed
that the stiffness of the racket had no notable effect on the
rebound characteristics of the ball for oblique impacts at
the GSC. The structural stiffness of the racket had no effect
Comparison of a finite element model of a tennis racket to experimental data 97
on the rebound characteristics of the ball because the GSC
corresponds to a node point of the racket. The results from
the FE simulations indicated that the ball was ‘over-spin-
ning’ during the oblique impacts. It would have been very
difficult to measure ‘over-spinning’ using a conventional
laboratory-based experiment alone. This is the first FE
model with the capability to accurately simulate oblique
spinning impacts, at different locations on the stringbed of
a freely suspended racket. The model can now be used to
determine the influence of different racket parameters, such
as mass and swingweight, on the game of tennis.
Acknowledgments The authors would like to thank Prince for
sponsoring the project. They would also like to thank Terry Senior,
Simon Choppin and John Kelley.
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