Hatze
142
JOURNAL OF APPLIED BIOMECHANICS, 1993, 9, 143-148
© 1993 by Human Kinetics Publishers, Inc.
Expression for Energy Losses in the Ball
Equation 14 in the main text sums up all energy losses occurring in the ball during
and after impact. From experimental observations (to appear elsewhere) the following
approximation for llb(t) can be obtained:
(Al9)
where 'tb is a point in time between 0 and T, and ijb = iMJ) denotes the maximum of the
ball indentation 'llb at a given.impulse J. Note that f,y(t) is not symmetric about T/2, the
reason being the change in sign of the damping force f3bg(T]b)iJb of the ball during impact,
and the asymmetry between compression and expansion phase. The exact form of the
function g(T]b) is not known. On the grounds of present experimental evidence, a likely
candidate function for g(T]b) is given by
g(T]b) = (24.9 1lb)4 + (0.465 1lb)2/(cJ' - 'llb),
(A20)
where o' = 0.03 m denotes the ball radius cr reduced by the shell thickness of
3 mm. It must, however, be emphasized that more appropriate approximations for g(T]b)
could possibly be obtained in the future when more experimental results lJecome available
on the energy losses in tennis balls. When Equation' A20 and Equations 14-17 of the
main text are used, the expression for Eb(T) is found to be
Eb(T) = 13b[(o.os9i:W + co.662Eb) 4 + co.552Eb) 6Jff
(A21)
+ kb[exp(cbT]b(T)) - l)/cb - T]b(T)] + 0.44T]b(T)i]~(T),
where Eb is the normalized maximum indentation defined by Eb = ijb(v)/o'. Because the
values of Eb, T]b(T), and i]b(T) at impact end time T are all unknown, it is impossible to
decompose Equation A21 into its three components. However, experimental relations
Eb(T,J) can be obtained and are discussed in the main text. It should be noted that an
experimental function Eb(T,J) is valid only for given ball properties kb and cb which,
in turn, depend on similar string properties k.(~,s), c.(~,s). In other w'ords, the experimentally determined values kb and cb appearing in Equation 16 are themselves functions of
string properties so that, more precisely, one should write Eb = Eb{kb[k,(~,s).c,(l;;s)J,
cb[k,(/;.~),c,(l;.OJ,J); that is, the dynamic behavior of a tennis ball during impact also
depends on string properties characteristic for the impact point S(l;.s) at the string membrane and, possibly, also on the form of the impulsive force function f,y(t).
Effect of String Tension and Impact
Location on Ball Rebound Accuracy
in Static Tennis Impacts
Duane Knudson
The rebound accuracy of tennis impacts was studied by measuring the vertical
angles of approach and rebound of tennis balls projected in a vertical plane
at a clamped racket. Three identical oversized tennis rackets were strung
with nylon at 200, 267, and 334 N of tension. Ten impacts were filmed at
200 Hz for each string tension with the ball impacting the strings centrally
and 8 cm off-center. A two-way ANOVA revealed significant (p < .01) main
effects for string tension, impact location, and the interaction of string tension
and impact location. Treatment-Contrast Interactions demonstrated one significant (p < .01) difference: The decrease in rebound accuracy from a string
tension of 200 to 334 N was significantly different for central versus offcenter impacts.
The accuracy of shot placement is of primary importance in the game of
tennis, and string tension may be a key factor in determining the accuracy of
ball rebound. The selection of a string tension in order to improve shot accuracy
has been based largely on opinion because of a lack of experimental evidence
(Knudson, 199la).
Groppel (1984) hypothesized that higher string tensions improved stroke
accuracy. He theorized that the greater ball deformation on the strings (Groppel,
Shin, Thomas, & Welk, 1987) and the shorter dwell time (Brody, 1979, 1987;
Groppel, Shin, Thomas, & Welk, 1987) allowed the ball to be directed more
accurately, Brody (1987) outlined how string tension may either increase or
decrease tennis rebound accuracy based on the laws of physics.
The first study to examine stringing and rebound accuracy in tennis reported
that string type, string tension, and the interaction between the two have significant
effects on ball rebound accuracy for midsized tennis rackets (Knudson, 199la).
The more accurate rebounds observed in lower string tensions were hypothesized
to be related to their greater elastici~y (Knudson, 1991a). Studies have clearly
demonstrated that lower string tensions have greater elasticity than higher string
tensions (Baker & Wilson, 1978; Brannigan & Adali, 1981; Elliott, 1982a; Ellis,
Elliott, & Blanksby, 1978; Groppel, Shin, Thomas, & Welk, 1987).
Duane Knudson is with the Department of HPER, Baylor University, P.O. Box
97313, Waco, TX 76798-7313.
Knudson
144
String Tension and Impact Location
145
A study was needed that tested the effect of string tension on rebound
accuracy using oversized rackets because studies have shown significantly greater
ball rebound speeds for oversized rackets compared to smaller racket heads
(Elliott, Blanksby, & Ellis, 1980; Groppel, Shin, Thomas, & Welk, 1987). Impact
conditions would also be more realistic if off-center impacts were studied because
research has shown that most impacts occur off-center (Knudson, 199lb;
Ohmichi, Miyashita, & Mizumo, 1979; Plagenhoef, 1979) and off-center impacts
have been observed to have nonsignificant and significant reductions in ball
rebound velocity compared to central impacts (Elliott, 1982a, 1982b; Elliott et
al., 1980; Ellis et al., 1978). Controlled impact conditions are needed to make
comparisons across string tensions because the intraindividual variation of handheld rackets is quite large (Hatze, 1992; Knudson & White, 1989).
The purpose of this study was to examine the effect of three string tensions
and two impact locations on ball rebound accuracy in a vertical plane for an
oversized racket. If ball rebound accuracy data from different racket head size
and impact location conditions supported the hypothesis of greater rebound accuracy for lower string tensions (Knudson, 1991a), the choice of string tension to
maximize accuracy in flat shots like volleys would be more clear.
Method
Three identical Prince oversized tennis rackets were strung alternately with Prince
Synthetic Gut (nylon) strings (16 gauge) at three string tensions (200, 267, and
334 N). String tensions represented the range of typical tensions and corresponded
to the more common English units of 45, 60, and75 lb, respectively. Each racket
was marked with 3M reflective tape strips and a round wooden bead (15 mm in
diameter) that was covered with reflective tape and glued to the racket (Figure
1). Strips (5 mm wide) of tape marked the tip of the racket head and the center
of the throat, while the reflective bead marked the top, lateral geometric center
of the racket head. The rackets were secured in a horizontal position to a wooden
arm by three C clamps. Nine new tennis balls were bounce tested and projected
- from a distance of 2.29 m by a Lobster ball machine stabilized with weights and
elastic cords. Impact locations on the racket face were central and 0.08 m superior
to the geometric center of the racket face. Mean (±SD) preimpact ball velocities
were 28.9 ± 1.9 m·s- 1 at an angle of 25.4 ± 1.8° to the horizontal. The balls had
negligible rotation prior to impact (<10 rad·s- 1).
Ten central impacts for each condition were filmed at 200 fps by a Photosonics lPL camera placed 7.92 m from the center of the racket face. Camera
speed was established by internal timing lights set at 100 Hz. A mirror was
placed at a 45° angle to the racket to record impact location in the longitudinal
direction of the racket face. Data were digitized on a Numonics 1224 digitizer
interfaced to a Zenith 128 computer. Trials were digitized for ±3 frames before
and after impact for the following points: top (lateral geometric center) of the
racket head, tip of the racket head, center of the ball, mirror image of the tip of
the racket, mirror image of ball position, and the mirror image of the throat of
the racket.
~ine~~tic data were not smoothed because the greater accuracy of a small
field size limited the number of images of the ball in the field of view (Hudson
1990). Also, there were no smoothing procedures that handled very small dat~
Figure 1 - Schematic of the racket, reflective markers, impact locations, and angles
of interest. The angle of approach (0), angle of rebound (<l>), central impact location
(C), and off-center impact location (0) are illustrated.
sets, eliminated the effect of adjacent points on the smoothed data, and were
unaffected by boundary effects (D' Amico & Ferrigno, 1992; Smith, 1989;
Woltring, 1985; Wood, 1982). To minimize errors in angle and velocity calcul;tions two analysis procedures were used. First, position data for each trial were
est~blished by averaging three digitizations by one skilled operator (Groppel,
Shm, Spotts, & Hill, 1987; Looney, Smith, & Srinivasan, 1990). Second, the
angles and velocities of ball approach and rebound were determined from the
mean of two finite difference calculations between the three samples before and
after impact. The racket angle at impact was calculated from the frame prior to
impact to avoid detection of motion of the frame caused by impact.
The vertical plane angles of approach (8) and rebound (<l>) relative to the
ho?zontal were calculated to monitor the consistency of the impact conditions
(Figure 1). So rebound accuracy across trials could be compared, an accuracy
ratio (AR) representative of a tennis volley was calculated as the angle of rebound
divided by the angle of approach, both measured relative to an axis normal to
the racket face. An accurate rebound in a vertical plane was defined as an angle
of rebound close to normal to the racket face, since the study simulated a primarily
flat shot, like a volley, with a target at right angles to the racket. An AR of unity
corresponded to equal angles of approach and rebound, so the smaller the AR,
the more accurate the ball rebound. Data were analyzed by a two-way ANOVA
(2 x 3-Location x String Tension) with statistical significance accepted at the
.01 level of probability.
Knudson
14b
Impact locations measured from film were consistent. Across all trials, the mean
impact location was 0.8 ± 1.8 cm proximal to the geometric center of the racket
face. Mean impact locations in the vertical dimension were 0.08 ± 0.9 cm above
geometric center for central impacts, and 0.80 ± 1.8 cm above for the off-center
trials. Due to timing variations in the projection of the ball by the ball machine,
not all conditions had 10 observations.
Mean and mean-standard deviation accuracy ratios (AR) for all conditions
are plotted in Figure 2. A smaller AR corresponds to greater accuracy or a
rebound more in line with normal to the racket face. The unequal frequencies
and different variances (Figure 2) of the AR data required the variances in the
data to be equalized with a square-root transformation prior to ANOV A analysis
(Kirk, 1982). The two-way ANOVA revealed significant main effects for impact
location, F(l, 44) = 84.2, p < .01; string tension, F(2, 44) = 11.8, p < .01; and
the interaction of string tension and location, F(l, 44) = 6.2, p < .01.
Treatment-Contrast Interactions were calculated since the significant interaction precluded valid post hoc comparisons between means (Kirk, 1982). Treatment-Contrast Interactions demonstrated that the interaction b~tween impact
location and the difference in string tension from 200 to 334 N was significant,
F(2, 44) = 5.75, p < .01. In other words, the change in AR (slope in Figure 2)
across string tensions from 200 to 334 N was different for central versus off1.8
Central
o
Off-Center
1.7
1.6
0
1.5
ii
a:
>-
u
f!
1.4 -
"uu
<
147
cent~r impac_ts. Off-center impacts showed a greater loss in ·accuracy as string
tens10n was mcreased.
Results
0
;mmg 1ens1on ana impact Location
Discussion
~esults of the pr_esent study su~ported ~revious research demonstrating that the
~ncrea~ed elas_tic~ty of lower stnng tens10ns tends to improve rebound accuracy
m static tenms impacts (Knudson, 1991a). The oversized racket strung with
~00 N of te~sion had the most accurate rebounds (AR = 1.11 ± 0.09) in central
impacts. This was more accurate than a study of central impacts on a midsized
racket (AR = 1.50 ± 0.16) strung with nylon at a slightly higher tension of
222 N (Knudson, 1991a). It appears that oversized rackets may provide more
accurate rebounds because the oversized racket strung at 267 N was more accurate
~AR = 1.16 ± O._l) than t~e previous study (Knudson, 1991 a) of a midsized racket
I~ t~e same strmg tens10n and impact location conditions (AR = 1.52 ± 0.2).
S1m1lar to the previous st~dy ofrebound accuracy (Knudson, 199la), the present
study found that the typical standard deviation for mean ARs was 0.1, which
corresponded to a variance of± 2.5° in rebound angle for these impact conditions.
. . How these results relate to players attempting to impart large amounts of
spm is. not clear. Stroking a tennis ball with a slightly open/closed racket face
an? usmg a steep. racket path less aligned with the target, in order to generate
spm, may place different demands on the strings and the frame. Future studies
should utilize high-speed vid~ography (2000 Hz) to make higher sampling rates
~nd larger samples cost-effe_ct1_ve. Data are needed in a variety of stroking condit10ns to estabhs~ whether lt is the effect of string elasticity or the combined
effect of dwell time and racket displacement that has the greatest influence on
ball rebound accuracy when string tension is varied.
I~ was concluded that string tension, impact location, and the interaction
?f tension and location significantly affect ball rebound angle in static tennis
1mpac_ts. Resu~ts supp?rted previous research (Knudson, 199la) demonstrating
that h1~her stnng tens10ns have less accurate rebounds in a vertical plane. Offcenter impacts s~owed l~ss _accurate vertical plane rebounds and a greater loss
of accuracy as strmg tens10n mcreased. The oversized rackets in this study created
mor~ accura~e rebounds in central impacts than midsized rackets in similar string
tension and impact conditions.
1.3
References
1,2
1.1
1
1 0
1 0
20
0
2 0
2 0
2
3 0
3 0
3
String Tension (N)
Figure 2 - Mean and mean-standard deviation rebound accuracy ratios. The accuracy ratio is the angle of reflection divided by the angle of approach. A smaller
accuracy ratio corresponds to a more accurate rebound (more in line with normal
to the racket face).
Baker, J., & Wilson, B. (1978). The effect of tennis racket stiffness and string tension
on ball velocity after impact. Research Quarterly, 49, 255-259.
Brannigan, M., & Adali, S. (1981). Mathematical modelling and simulation of a tennis
racket. Medicine and Science in Sports, 13, 44-53.
Brody, H. (1979). Physics of the tennis racket. American Journal of Physics, 47, 482-487.
Brody, H. (1987). Tennis science for tennis players. Philadelphia: University of Pennsylvama Press.
D'Amico, M., & .Ferrigno, G. (1992). Comparison between the more recent techniques ·
for smoothmg and derivative assessment in biomechanics. Medical and Biological
Engineering and Computing, 30, 193-204.
Elliott, B.C. (1982a). The influence of racket flexibility and string tension on rebound
140
Knudson
velocity following dynamic impact. Research Quarterly for Exercise and Sport,
53, 277-281.
Elliott, B.C. (1982b). Tennis: The influence of grip tightness on reaction impulse and
rebound velocity_- Medicine and Science in Sports and Exercise, 14, 348-352.
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of conventional and oversized tennis rackets. Research Quarterly for Exercise and
Sport, 51, 608-615.
Ellis, R., Elliott, B., & Blanksby, B. (1978). The effect of string type and tension in jumbo
and regular sized rackets. Sports Coach, 2(4), 32-34.
Groppel, J. (1984). Tennis for advanced players. Champaign, IL: Human Kinetics.
Gtoppel, J., Shin, I., Thomas, J., & Welk, G. (1987). The effects of string type and tension
on impact in midsized and oversized tennis rackets. International Journal of Sport
Biomechanics, 3, 40-46.
Groppel, J., Shin, I., Spotts, J., & Hill, B. (1987). Effects of different string tension
patterns and racket motion on tennis racket-ball impact. International Journal of
Sport Biomechanics, 3, 142-158.
Hatze, H. (1992). Objective biomechanical determination of tennis racket properties.
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Hudson, J. (1990). Problems in data reduction: Tracking a round object. In M. Nosek, D.
Sojka, W.E. Morrison, & P. Susanka (Eds.), Proceedings of the VIIIth International
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Kirk, R.E. (1982) .. Experimental design (2nd ed.). Pacific Grove, CA: Brooks/Cole.
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in static tennis impacts. Journal of Human Movement Studies, 20, 39-47.
Knudson, D.V. (1991b). Factors affecting force loading on the hand in the tennis forehand.
Journal of Sports Medicine and Physical Fitness, 31, 527-531.
Knudson, D.V., & White, S.C. (1989). Forces on the hand in the tennis forehand drive:
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for Exercise and Sport, 61, 154-160.
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during the tennis stroke. In J. Teradus (Ed.), Science in racquet sports (pp. 89-95).
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JOURNAL OF APPLIED BIOMECHANICS, 1993, 9, 149-159
© 1993 by Human Kinetics Publishers, Inc.
A Panning Videographic Technique
to Obtain Selected Kinematic
Characteristics of the Strides
in Sprint Hurdling
John W. Chow
This paper describes a panning videographic technique for measuring stride
lengths and horizontal velocities of strides over an entire hurdle race. The
technique requires that tapes of alternate black and white sections be placed
at the inside border of the inside lane and the outside border of the outside
lane of a track as spatial reference and that a vertical reference be videotaped
when it is erected at different locations within the track. The stride lengths
and horizontal velocities obtained with the panning technique were compared
with the corresponding values that were obtained with conventional stationary
camera techniques. The results indicate that if three panning cameras are
used to cover the entire hurdle race, the average absolute errors in stride
length and horizontal velocity are 0.07 m and 0.15 m/s, respectively. Such
errors are considered acceptable for some applications. Certain properties of
the panning technique are discussed.
A single-camera panning videographic technique for obtaining the stride
lengths of a runner during a 100-m race was proposed by Chow (1987) several
years ago. The technique requires that paired background markers be placed at
10-m intervals on opposite sides of the eight-lane track. The known locations of
the markers are used as spatial reference for determining the stride lengths.
The technique was later adapted by Hay and Koh (1988) to obtain the
stride lengths during the approach run of the long and triple jumps. Instead of
using background markers, Hay and Koh (1988) painted parallel dashed lines
(6 cm wide) on each side of the runway. These lines were placed at 1.80-m
intervals, and the known locations of these lines relative to the takeoff board
aided in determining the horizontal distance from the toe of the support foot to
the front edge of the takeoff board for each support phase of the approach.
The technique has been further developed so that the horizontal velocity .
of the center of gravity (CG) during the flight phase of a stride can be determined.
Acknowledgments
This research was supported in part by a grant from the United States Tennis
Association. The support of Prince Manufacturing Inc. and the Baylor Institute of Graduate
Statistics is gratefully acknowledged.
John W. Chow is with the Department of Exercise Science, ElOl Field House,
University of Iowa, Iowa City, IA 52242.
149