Sports Eng (2013) 16:181–188
DOI 10.1007/s12283-013-0118-y
ORIGINAL ARTICLE
Effect of string bed pattern on ball spin generation from a tennis
racket
Alexander Nicolaides • Nathan Elliott •
John Kelley • Mauro Pinaffo • Tom Allen
Published online: 8 May 2013
Ó International Sports Engineering Association 2013
Abstract Topspin has become a vital component of
modern day tennis. Ball-to-string bed and inter-string
friction coefficients can affect topspin generation from a
racket. The aim of this research was to determine the effect
of string bed pattern on topspin generation. Tennis balls
were projected onto nine head-clamped rackets with different string bed patterns. The balls were fired at 24 m/s, at
an angle of 26° to the string bed normal with a backspin
rate of 218 rad/s and outbound velocity, spin and angle
were measured. Outbound velocity was shown to be
independent of string bed pattern. Outbound angle
increased with the number of cross strings, while outbound
topspin decreased. In the most extreme case, decreasing the
number of cross strings from 19 to 13 increased rebound
topspin from 117 to 170 rad/s.
Keywords Impact Velocity High-speed video Mechanics Angle
1 Introduction
Topspin is a vital component of modern day tennis. A
number of studies have measured topspin rates for
A. Nicolaides N. Elliott J. Kelley T. Allen
Centre for Sports Engineering Research, Sheffield Hallam
University, Sheffield, UK
M. Pinaffo
Prince Sports, Venice, Italy
T. Allen (&)
Department of Engineering and Maths, Sheffield Hallam
University, Sheffield, UK
e-mail: [email protected]
groundstrokes delivered by elite players [1–3]. Goodwill
et al. [2] measured a maximum spin rate of 398 rad/s for
males during match play, while Kelley et al. [3] reported
299 rad/s for females. Choppin et al. [1] reported a maximum of 220 rad/s during simulated play. A ball struck with
topspin has a downward (Magnus) force acting on it during
flight, causing it to drop faster [4, 5]. Increasing the topspin
applied to a ball increases the maximum speed at which the
ball will still impact within the court boundary. Developments in rackets and strings are believed to have been
influential in increasing topspin rates in modern day tennis.
A number of studies have investigated the effect of racket
[6] and string bed [7–14] parameters on topspin generation.
Goodwill and Haake [7] showed rebound topspin to be
almost twice as high for a ball projected onto a headclamped ‘spaghetti racket’, in comparison to a conventional racket. The spaghetti racket was a novel invention
patented in 1977 [15], where the strings were not interlaced. Smaller rebound angles (relative to string bed normal) were also reported for the spaghetti racket. Lower
frictional forces between contacting strings enable the main
strings to deform laterally (in plane with the string bed)
more easily in a spaghetti racket. Stored elastic energy,
combined with lower inter-string friction, can cause the
strings to return or ‘spring back’ while the ball is in contact. The returning movements of the strings were reported
to simultaneously increase the topspin and decrease the
transverse (in plane with the string bed) velocity of the ball.
The International Tennis Federation introduced a new
rule in 1978 stating that strings must be alternately interlaced and the spaghetti racket was subsequently banned
[16]. The International Tennis Federation continues to
monitor innovations as manufacturers strive to produce
equipment which will give their players a significant
advantage in terms of spin generation. The frictional
182
properties of tennis strings have been a popular topic
among researchers investigating ball spin generation.
Allen et al. [10] used a finite element model to show
rebound topspin can increase as ball-to-string coefficient of
friction decreases, for an inbound angle of 40°. Ball-tostring friction was shown to have no effect at an angle of
20°, which corresponds approximately to a groundstroke
from an elite player [1]. Inter-string friction can also affect
the rebound topspin of the ball, for both simulated impacts
on a head-clamped racket [11] and actual player strikes
[12]. When inter-string friction coefficients are low the
main strings can displace laterally and return more easily.
If the strings return while the ball is in contact they can act
to increase its topspin, while decreasing its transverse
velocity. The effect is comparable to the spaghetti racket.
Reducing inter-string contact forces, by reducing interstring friction [11, 12] or having non-interlacing strings [7],
can cause an increase in rebound topspin. Changing the
string pattern could also influence inter-string contact forces and affect rebound topspin. Groppel et al. [17] investigated the effect of string bed pattern on impacts normal to
the face of the racket. To the best of our knowledge, no
peer reviewed studies have been published on the effect of
string bed pattern on spin generation, although there are
some detailed communications online [13, 14].
There are currently few regulations regarding the string
bed pattern, the rules state that the string bed must not be
less dense at the centre than in any other region but there
are no restrictions on the number of strings [16]. The main
limitations associated with a reduced number of strings are:
(i) increased risk of string failure and (ii) maintaining
contact between the ball and string bed. The aim of this
study was to determine the effect of string bed pattern on
spin generation.
2 Methods
Tennis balls were projected onto nine head-clamped rackets with a range of string bed patterns. The inbound and
rebound properties of the ball were measured using a highspeed video camera.
2.1 Materials
Nine prototype rackets (patent pending) with the same head
size and shape were used in this research. The length of the
racket head was 0.35 m and the width was 0.25 m (taken
from the inside edge of the frame). Each racket had a
different string bed pattern, as detailed in Table 1. Some of
the string beds had the same number of cross and main
strings but different string bed densities. Each string bed
had either, (i) higher density of main strings at the centre,
A. Nicolaides et al.
Table 1 String beds used in study
String
bed
Main
strings
Cross
strings
Total number
of strings
Description
1
12
13
25
Equally spaced
2
12
13
25
Dense in the centre cross
3
12
13
25
Dense in the centre main
4
12
13
25
Dense in the centre main
and cross
5
16
12
28
Dense in the centre main
6
16
12
28
Dense in the centre main
and cross
7
12
19
31
Dense in the centre cross
8
12
19
31
Dense in the centre main
and cross
9
16
19
35
Standard
(ii) higher density of cross strings at the centre, (iii) higher
density of main and cross strings at the centre or (iv) an
equally spaced uniform pattern. A standard string bed
pattern, with 16 main and 19 cross strings, was also
included.
A central impact region, with a radius of 58 mm
(Fig. 1), was used to quantify differences between string
beds with the same number of main and cross strings but
different densities (string spacing). The central region
corresponds approximately to where an elite player will
strike the ball during a ground stroke [1]. The number of
main strings, cross strings and intersections within the
central region for each string bed are shown in Table 2.
Rackets were strung at a tension of 245 N 24 h prior to
testing with polyester string (Beast XP, Prince). Strings
lose tension continuously over time, but the loss is quickest
over the first 30 min, it then stabilises. The International
Tennis Federation approved balls (NX Tour, Prince) were
opened 24 h before testing to let the pressure settle. Each
ball was used for a maximum of 20 impacts.
2.2 Experimental methods
A ball pitching machine (BOLA pitching machine, Stuart
Williams, UK) fitted with a tapered barrel was used to
project balls at a mean angle (h1) of 26.4° (standard
deviation ± 0.81) onto the racket face (Fig. 2). The ball
was targeted at the central region of the string bed (Fig. 1).
To ensure consistent impact location the exit end of the
barrel was positioned close (resultant distance of *0.4 m)
to the racket face. Individual impacts were visually
inspected in the high-speed video footage to ensure impact
location was close to the centre of the string bed. The total
number of impacts across all nine rackets was 315 and the
number per racket fell between 29 and 39.
Effect of string bed pattern
183
Fig. 1 Diagram to show the
central region on the string bed.
A visual comparison can be
made between two different
string bed patterns, 12 9 13
(left) and 16 9 12 (right)
Table 2 Number of strings and intersections in the central region
String bed
Main strings
Cross strings
Intersections
1
6
6
32
2
7
6
36
3
8
6
40
4
8
7
45
5
10
5
46
6
10
6
48
7
6
10
52
8
8
10
64
9
10
10
78
Freely suspended rackets are often used when investigating normal impacts on the long axis, as the frequency
response is comparable to when hand held [18–20]. As the
number of variables increase, such as for an oblique spinning impact, it becomes increasingly difficult to obtain
repeatable results using a freely suspended racket. Rebound
spin from an oblique impact on a freely suspended racket is
particularly sensitive to impact location [6, 8]. The aim of
this research was to further our understanding of ball/string
interactions.
Fully constraining the racket reduces the sensitivity of
ball rebound to impact location, while allowing ball/string
interactions to be observed more easily. The racket was
fully constrained so any differences in ball rebound could
be attributed to the ball/string interaction [8]. The main
limitations associated with using a fully constrained racket
rather than a realistically supported racket are: (i) the
racket cannot recoil and (ii) higher ball/string bed contact
forces—for the same inbound ball velocity.
A high-speed video camera (Phantom V4.3, Vision
Research, Wayne, USA)—operating at 2,200 Hz at a resolution of 544 9 480 and exposure time of 80 ls—filmed
each impact (Camera 1 in Fig. 2b). The camera’s focal axis
lay perpendicular to the plane of the ball trajectory. The
bespoke impact rig was designed to allow a mirror to be
positioned beneath the string bed at an angle of 45°
(Fig. 2). A second camera (Phantom V4.3, Vision
Research, Wayne, USA) positioned with the focal axis
parallel to the cross strings, filmed ball and string interactions by viewing the mirror (Camera 2 in Fig. 2a). The
second camera filmed at 3,000 Hz at a resolution of
400 9 400 and an exposure time of 90 ls.
2.3 Data analysis
A fully automated in-house software package was used to
obtain the inbound and rebound velocity (V), angle (h) and
spin (x) of the ball. The automated calibration process
utilised the known diameter of the ball. The software was
validated against manual digitisation for the inbound and
rebound ball trajectories for ten similar ball-on-racket
impacts. Each trajectory was manually digitised five times
and the mean was compared to the value obtained from the
software. The mean absolute differences, between the
manual and software results across all twenty trajectories,
for velocity, angle and spin are shown in Table 3. As the
results obtained using the two methods are comparable the
software was a suitable replacement for the manual digitisation process.
The mean and standard deviation of the measured
inbound conditions across all impacts are shown in Table 4.
The inbound conditions correspond approximately to a low
speed groundstroke [1]. The balls were projected with
backspin as this corresponds to the spin direction before
impact—assuming the racket face is angled forward, rising
and travelling significantly faster than the ball—when
transferring from the court to the laboratory frame of
reference where the racket is stationary [21, 22] (Fig. 3).
184
A. Nicolaides et al.
Fig. 2 Diagram to show the
experimental set up using the
bespoke rig and silver faced
mirror to view ball and string
interactions. View from the
direction of a camera 1 and
b camera 2
Table 3 Mean absolute difference between digitisation results from
automated software and a manual process
Mean absolute difference
Speed (m/s)
0.20
Angle (°)
0.31
Spin rate (rad/s)
4.27
The results are from 10 impacts and include both inbound and outbound results. The manual process was repeated 5 times for each
impact and the means were compared to the results obtained using the
software
It was important to ensure any differences in ball
rebound were due to the string bed, rather than the inbound
conditions. A one-way analysis of variance (ANOVA) test
was performed on the inbound conditions between string
beds to determine if there were any significant differences
in the velocity, angle or spin between the nine groups of
impacts (string beds). There were no significant differences
in the inbound velocity, angle or spin between the nine
string beds (p [ 0.1 in all cases).
3 Results
Table 4 Inbound ball conditions for all impacts
Mean
Standard deviation
Velocity (m/s)
Angle (°)
Backspin (rad/s)
24.4
26.4
218
0.61
0.81
9.22
There were slight variations (standard deviation in Table 4)
in the inbound velocity, angle and spin between individual
impacts, due to inconsistencies in the pitching machine.
Figure 4 shows the results for the normal and transverse
rebound velocity of the ball. The results indicate that
normal rebound velocity was independent of the number of
cross strings, main strings and intersections (in the central
region). Transverse velocity was independent of the number of main strings, while it increased with the number of
cross strings and interceptions. Figure 5 shows the results
for rebound angle. The results indicate that rebound angle
was independent of the number of main strings. Rebound
Effect of string bed pattern
185
the number of main strings. Rebound topspin decreased as
the number of cross strings and intersections increased.
Figure 7 shows the relationship between rebound topspin
and rebound angle. Rebound angles tended to decrease as
rebound topspins increased.
4 Discussion
Fig. 3 Diagram to show that the ball should impact the racket with
backspin in the laboratory frame of reference to represent a topspin
shot where the racket face is angled forward, rising and travelling
significantly faster than the ball
angle increased with the number of cross strings and
intersections.
Figure 6 shows the results for rebound topspin. The
results indicate that rebound topspin was independent of
Ball spin generation from a fully constrained racket was
dependent on the string bed pattern. The number of main
strings in the central region did not affect rebound spin.
Decreasing the number of cross strings or the number of
intersections in the central region increased the rebound
spin of the ball, while maintaining normal velocity.
Rebound angles tended to decrease as topspin increased.
Figure 8 shows ball and string interactions (second
camera) for racket 5, which had the standard 16 main
strings but only 12 cross strings. Frame-by-frame inspection of the video footage indicated that the ball was in
contact with the string bed for all the images displayed
(based on observed ball deformation against the strings).
The sequence clearly shows at least three of the main
strings were deforming laterally and then returning or
‘springing back’. Provided the strings maintained contact
with the ball as they returned, it is likely they would have
acted to increase its topspin while decreasing its transverse
velocity. The mechanisms highlighted in Fig. 8 are comparable with those reported by Goodwill and Haake [7] for
a spaghetti racket and Haake et al. [11] and Kawazoe and
Okimoto [12] for a conventional racket with lubricated
strings. The findings were also in agreement with unpublished communications on the effect of string bed pattern
on spin generation [13].
Figure 9 shows ball and string interactions for an impact
on each of the rackets. The images were taken at the
observed point of maximum lateral string deformation and
hence maximum stored energy. The images indicate that
Fig. 4 Effect of the number of a main strings b cross strings and c interceptions on the normal and transverse rebound velocity (V2) of the ball.
The data points correspond to the mean for each racket and the error bars equate to one standard deviation either side
186
A. Nicolaides et al.
Fig. 5 Effect of the number of a main strings b cross strings and c interceptions on the rebound angle (h2) of the ball. The data points correspond
to the mean for each racket and the error bars equate to one standard deviation either side
Fig. 6 Effect of the number of a main strings b cross strings and c interceptions on the rebound topspin (x2) of the ball. The data points
correspond to the mean for each racket and the error bars equate to one standard deviation either side
Fig. 7 Relationship between rebound angle and topspin for all
rackets. The data points correspond to the mean for each racket
lateral deformation of the main strings was dependent on
string bed pattern. In general, lateral string deformations
can be observed to increase as the number of cross strings
and intersections decreased. It is therefore likely that the
higher rebound topspins, and lower transverse velocities,
Fig. 8 High-speed video images of a ball impacting on racket 5,
which had 16 main and 12 cross strings. The time step between
images was 0.67 ms and the timeframe from image 1 to 9 was 5.3 ms.
The strings can clearly be seen to deflect laterally and then return
during impact. Inbound velocity = 24.9 m/s, angle = 26.3° and
spin = 231.1 rad/s
Effect of string bed pattern
187
Rebound angle increased with the number of cross strings,
while rebound topspin decreased. Normal rebound velocity
was essentially independent of the string bed pattern. Highspeed images indicated lateral main string deformations
increased as the number of intersections between the main
strings and cross strings decreased. It is likely that the main
strings acted to increase the topspin, and decrease the
transverse velocity, of the ball as they returned during the
restitution phase of the impact.
Acknowledgments The authors would like to thank Mr Terry
Senior for designing and developing the test rig and Prince Sports for
providing equipment for testing. The authors also like to thank
Dr Simon Choppin and Dr Heather Driscoll for proof reading the
manuscript.
Fig. 9 High-speed video impacts showing observed maximum lateral
string movement for each racket. The numbers correspond to the
racket number, with the number of interceptions increasing from top
left to bottom right. Mean and standard deviation for inbound
velocity, angle and spin were 24.2 ± 0.4 m/s, 26.5 ± 0.3° and
221 ± 7 rad/s
were due to greater lateral deformation of the main strings.
Ball/string interactions are complex however, and it is
likely that individual string displacements will be dependent on the specific impact location and hence number of
strings in contact with the ball.
The results indicate that reducing the number of cross
strings in a tennis racket can substantially increase spin
generation, while maintaining normal velocity. The mean
topspin increased by as much as 45 % when the number of
cross strings was reduced to 13 (string bed 1) from the
‘standard’ 19 (string bed 9). The gains in topspin obtainable
by changing the string bed pattern appear to be approximately half those reported by Goodwill and Haake [7] for a
spaghetti racket. It is not possible, however, to compare the
results directly due to differences in impact conditions. It is
likely that simultaneously reducing the number of cross
strings and inter-string coefficient of friction would allow
the main strings to displace laterally and return more easily,
resulting in higher rebound spin rates. The mechanisms by
which the main strings displace laterally, store elastic
energy and return are likely to be dependent on string tension, material, diameter and the impact conditions.
5 Conclusion
Impacting tennis balls on different string bed patterns has
shown that the number of cross strings can affect the
rebound angle and spin of the ball, for inbound conditions
which correspond approximately to a groundstroke.
References
1. Choppin SB, Goodwill SR, Haake SJ (2011) Impact characteristics of the ball and racket during play at the Wimbledon qualifying tournament. Sports Eng 13:163–170
2. Goodwill S, Capel-Davies J, Haake S, Miller S (2007) Ball spin
generation of elite players during match play. Tennis Sci Technol
3(1):349–356 International Tennis Federation, London
3. Kelley J, Goodwill S, Capel-Davies J, Haake S (2008) Ball spin
generation at the 2007 Wimbledon qualifying tournament. Eng
Sport 7(1):571–578 Springer, France
4. Goodwill SR, Chin SB, Haake SJ (2004) Wind tunnel testing of
spinning and non-spinning tennis balls. J Wind Eng Ind Aerodyn
92:935–958
5. Mehta R, Pallis J (2001) The aerodynamics of a tennis ball.
Sports Eng 4:1–13
6. Allen T, Haake SJ, Goodwill SR (2011) Effect of tennis racket
parameters on a simulated groundstroke. J Sports Sci 29:311–325
7. Goodwill SR, Haake SJ (2002) Why were ‘spaghetti string’
rackets banned in the game of tennis? In: Ujihashi S, Haake S
(eds) 4th International conference on sports engineering. Blackwell Science, Kyoto, pp 231–237
8. Goodwill SR, Haake SJ (2004) Ball spin generation for oblique
impacts with a tennis racket. Soc Exp Mech 44:195–206
9. Cross R (2000) Effects of friction between the ball and strings in
tennis. Sports Eng 3:85–97
10. Allen TB, Haake SJ, Goodwill SR (2010) Effect of friction on
tennis ball impacts. Proc Inst Mech Eng P J Sports Eng Technol
224:229–236
11. Haake SJ, Allen TB, Jones A, Spurr J, Goodwill S (2012) Effect
of inter-string friction on tennis ball rebound. Proc Inst Mech Eng
J J Eng Tribol 226:626–635
12. Kawazoe Y, Okimoto K (2008) Tennis top spin comparison
between new, used and lubricated used strings by high speed
video analysis with impact simulation. Theor Appl Mech Jpn
57:511–522
13. Lindsey C (2011) Spin and string pattern. The tennis warehouse.
http://twu.tennis-warehouse.com/learning_center/stringpattern.php.
Accessed 18 Jan 2013
14. Cross R, Lindsey C (2013) Spin and string patterns old, new, and
illegal. The tennis warehouse. http://twu.tennis-warehouse.com/
learning_center/spinpatterns.php. Accessed 18 Jan 2013
15. Fischer W (1977) Tennis racket. US Pat 4(273):331
16. ITF Technical Department. International Tennis Federation.
www.itftennis.com/technical. Accessed 02 Oct 2012
188
17. Groppel JL, Shin IS, Spotts J, Hill B (1987) Effects of different
string tension patterns and racket motion on tennis racket-ball
impact. Int J Sport Biomech 3:142–158
18. Brody H (1987) Models of tennis racket impacts (Modeles
d’impacts sur raquette de tennis). Int J Sport Biomech 3:293–296
19. Cross R (1998) The sweet spots of a tennis racket. Sports Eng
1:63–78
20. Kawazoe Y (1997) Experimental identification of a hand-held
tennis racket and prediction of rebound ball velocity in an impact.
Theoret Appl Mech 46:177–188
A. Nicolaides et al.
21. Allen T, Goodwill SR, Haake SJ (2009) Comparison of a finite
element model of a tennis racket to experimental data. Sports Eng
12:87–98
22. Cross R, Lindsey C (2005) Racquets, strings, balls, courts, spin and
bounce. Technical tennis. Racquet Tech Publishing, California