comparison of a finite element model of a tennis racket to experimental data

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)


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


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


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


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)


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


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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