Effects of Different String Tensi~n Patterns and Racket Motion on

Transcription

INTERNATIONAL JOURNAL OF SPORT BIOMECHANICS, 1987, 3, 142-158
Effects of Different String
Tensi~nPatterns and Racket Motion
on Tennis Racket-Ball Impact
Jack L. Groppel, In-Sik Shin,
Julie Spotts, and Barbara Hill
Much study has been done to examine the various aspects of tennis racket
performance including racket materials, shape, balance, and flexibility, but
recently the string in the frame has come to interest scientists. The purpose
of this investigation was to determine the effects of varying string tension
patterns on the mechanical behavior of racket, string, and ball during and
immediately after impact. Two separate experiments were conducted; one
examined 12 rackets strung differently with a string lock system by filming
the impacts at 3,500 frames per second, and the other quantitatively evaluated the forehand drive results of a skilled player who experimented with the
rackets. Differences were found in the various impacts examined, and these
differences were attributed to the various characteristics of the string, racket, and ball. In addition, larger coefficients of restitution than those found
in previous literature were also calculated.
Much engineering research has been conducted to determine optimal tennis
racket flexibility and string tension for different size rackets made from various
types of materials. Numerous variables have been considered in order to assess
the mechanical performance of the racket and string system along with their effect on postimpact ball velocity. Among the variables developed in the numerous
tennis racket studies, the coefficient of restitution (COR) has been frequently utilized (Baker & Putnam, 1979; Baker & Wilson, 1978; Brody, 1979; Ellis, Elliott, & Blanksby, 1978). The COR has been defined simply as the ratio of rebound
Jack L. Groppel is with the Department of Physical Education at the University
of Illinois. In-Sik Shin is an assistant professor at Seoul National University in South Korea.
Julie Spotts is a bioengineering student at the University of Illinois. Barbara Hill is a physical
education student at the University of Delaware.
Direct all correspondence to Jack L. Groppel, University of Illinois, Department
of Physical Education, 906 So. Goodwin, Urbana, IL 61801.
TENNIS RACKET-BALL IMPACT
143
ball speed immediately after impact over preimpact ball speed. Ellis et al. (1978)
found that for two different string types and tensions, a mean COR of an oversized aluminum racket was 0.48, which was higher than that of a regular-sized
aluminum racket (0.40). Furthermore, Baker and Wilson (1978) reported that
string tensions above 50 lbs did not result in higher COR, whereas Groppel, Shin,
Thomas, and Welk (1987) found that the COR corresponding to 40 lbs of string
tension was the highest.
Although the simply defined COR has been utilized to evaluate racket performance, most of the investigations studying COR have only been concerned
with ball motion. Even though Baker and Putnam (1979) showed the time history of some parts of the racket, they did not include string motion in their study.
In addition, when investigating duration of ball contact with the string in oversized rackets, Groppel et al. (1987) found that the duration of contact fell in a
range between 5.2 and 8.6 msec. This range of values was much higher than
the result of a study done by Baker and Putnam (1979), in which ball contact
duration in conventional rackets was found to be approximately 4 msec.
In an attempt to improve ball-racket interaction, a string-lock system was
developed that utilizes small wedges to secure a string to each hole. This new
stringing method provided several advantages including individualized adjustments
of each string, accurate tension control, and simplicity of tension change. This
stringing method was also thought to provide the possibility for fine-tuning the
racket for individual players.
The purpose of this study was to examine the effects of individual stringing
on string and frame behavior during impact. Several space and time variables
were selected as dependent variables for this investigation. These dependent variables included (a) maximum displacement of the geometric center of the racket
head from the resting position, (b) maximum displacement of the peak string surface from the resting position, (c) deflection of the peak string surface relative
to the geometric center of the racket head, while the ball was in contact with
the string, (d) impact duration from the beginning of ball contact with the string
until its departure, (e) the coefficient of restitution between the ball and racket,
and (f) the slope determined by the ratio of maximum deflection of peak string
surface from the geometric center of the racket head and the time interval from
beginning of ball contact with the string to maximum deflection of the string.
The following terms are defined to depict how the dependent variables
were examined. The geometric center of the racket head (GCH) was the midpoint of two lines: the long axis that separates the racket head into right and left
halves, and the axis perpendicular to the long axis that separates the racket into
upper and lower halves. The peak string surface (PKS) was the point where the
string was pushed farthest from its resting position during impact.
Methods
Two different experiments were conducted to investigate string and frame behavior, as well as the stringing effect upon the coefficient of restitution between
the ball and racket. Twelve identical HEAD midsized aluminum rackets, strung
line by line using the string-lock system, were used for this investigation. The
configurations of the string tensions are shown in Figure 1.
GROPPEL, SHIN, SPOTTS, A N D HILL
General Description of Rackets Utilized
Racket # I
I
--FJ
L
65 Ibs
1- t
=
Key
all tensions equal
on crosses (tuned
to same musical
note; same vibrational mode).
= all tensions equal
on mains.
Racket 82
>-.
= tension change of
3 Ibs for each
succeeding string.
Racket #3
Racket #6
Racket #7
+
Racket #10
65
Racket #11
Racket #4
50
+
11121b
per string
Racket #8
Racket # I 2
Racket #5
Racket #9
Figure 1
- String tension patterns for the 12 rackets examined.
+ 1 112 Ib per
string beginning
at 50 Ibs
145
Experiment I
Each racket was clamped vertically at the grip by a large vise. An Appollo Wizard Ball Machine was used to project new Dunlop tennis balls at the geometric
center of the racket head in a vertical plane perpendicular to the vertical racket
face. The nozzle of the ball machine was located 0.90 m from the racket face.
The vise and ball machine were stabilized by being fixed on a wooden platform,
thus ensuring less variability in the ball hitting position on the racket face.
The racket frame was marked at four points in order to determine the
geometric center of the racket head. These points were the center of the top, bottom, and two sides of the frame head. In addition, the string cross points were
painted flourescent red and green to serve as a reference when digitizing to determine peak string deflection. The hitting position on the racket face was checked
by shooting several balls against a carbon paper-covered board at the level of
the racket. This enabled accurate placement of the racket in the vise by lining
up the image of the ball with the geometric center of the racket head. The position of ball contact on the strings was checked on film by looking at the string
face reflected off a mirror, located 45" from the ball's moving direction and the
plane of the string face.
A 16-mm Hycam was set at a speed of 3,500 fps and placed 6 m from the
clamped racket. The optical axis of the Hycam's 70-mm lens had an angle of
8.7" from the string plane, which made it possible to film the entire region of
the string face during the ball contact duration. In addition, a 1,000-Hz external
timing light generator was connected to the camera to mark a timing light on
the film every 1 msec. Kodak Videonews 7250 reversible color film was used.
Six 500-watt flood lights were used to produce a total of 3,000 watts, a light intensity sufficient to cause the aperture of the Hycam to open to its maximum of 3.3.
Experiment 2
To estimate the coefficient of restitution during the impact of a forehand drive
hit with slight topspin, the investigators had a former NCAA Division I varsity
tennis player serve as the subject. After a sufficient warm-up, each of the 12 rackets
was given to the subject sequentially, starting with the racket marked #l. The
numerical order of the rackets was randomly determined, and the subject was
requested to hit one forehand drive directly back toward a ball machine located
on the opposite side of the court. This procedure was repeated three times for
each racket. However, when the subject did not feel satisfied with a certain stroke
(i.e., because of an off-center contact) the trial was repeated in order to obtain
consistent results. This subjective evaluation seemed important since players often
select a tennis racket based on how it feels during a stroke (Groppel, 1984).
Two 16-mm Locam cameras located at right angles to each other were
used to film the motions of the racket and ball during the stroke. One was positioned overhead while the other was located to the side of the ball's direction
of flight. The film speeds for the Locams were set at 100 fps.
Data Analysis
The motions of the frame and string were described by five points, including
the four points on the frame head and one point on the most deflected position
146
GROPPEL, SHIN, SPOTTS, AND HILL
of the string face. Digitizing of points included all frames, starting with the frame
just before ball contact and ending with the frame just after ball departure. These
data were smoothed with a Butterworth recursive digital filter set at a cutoff frequency of 350 Hz.
This cutoff frequency was determined by assuming that both string and
racket motions were sinusoidal. Since the ball contact time has been reported
at approximately 5 msec, the frequency of the string motion during impact was
assumed to be 100 Hz. Since the highest possible frequency of the digitized data
could be 1,750 Hz, the cutoff frequency was determined at midharmonics frequency between the possible signal and the highest noise frequencies.
After the digitized data were smoothed, GCH and PKS were determined.
The ball was moving in a horizontal direction that coincided with the X-ordinate
of the film taken during Experiment 1. However, one camera used in Experiment 2 captured the ball moving in the vertical direction, which became the Yordinate on the digitizing tablet.
Because of perspective error, the scaling factor at the middle of the string
face was one, whereas that for the front edge of the frame was less than one and
that for the rear edge was greater than one by the same amount. Therefore the
position of the geometric center (GCH) was corrected and determined using the
following equation:
GCH = (x [front side] - x [rear side])/2,
where x (front side) and x (rear side) denote the horizontal coordinates of the
points marked on the frame head at the level of the geometric center.
The coefficient of restitution was obtained by analyzing two consecutive
frames just prior to ball contact and two consecutive frames just after impact,
excluding the frame if it showed any contact between the ball and strings. Thus,
a total of four film frames were involved in the calculation. Because so few fdm
frames were available before and after impact (due to limited field width), data
smoothing was not conducted and raw data were used in the calculation. The digitized points for a film frame were the front edge of the ball, tip of the racket
head, and upper boundary of the grip for the second experiment.
It was necessary to determine the motion of the GCH just prior to and
immediately after impact to estimate the coefficient of restitution. In Experiment
1 the position of the GCH was determined as explained previously. To determine
the GCH in Experiment 2, the investigators used the positions of the racket tip
marker and the upper boundary of the grip. The position of the GCH of the swung
racket was estimated using the ratio of the real distance from the tip of the racket
to the upper boundary of the grip and the projected distance between the same
markers.
The coefficient of restitution was determined by dividing the relative speed
of the ball and the GCH after impact by the same relative speed before impact.
When the relative speeds were determined, it was possible to estimate the value
by calculating the ratio of the relative displacement between the ball and GCH.
Because the frame rate was constant, the time terms were cancelled out. Thus
the coefficient of restitution (COR) used in this study is shown by the following
equation:
COR =
y (R), -
(B),
mb-
(Rib
where represents the displacement of the racket (R) at the GCH and ball (B)
for the two consecutive frames while a and b imply after and before impact, respectively. It was noted that a small error in either of the position data would magnify
the error in the COR. Therefore the digitizing was repeated three times by three
investigators.
Results and Discussion
String and Frame Behavior
The string and frame behavior is depicted in Figures 2-13. Each figure depicts
the changes in the peak of string surface (PKS) and geometric center of the racket head (GCH) versus the time of ball contact. The third curve in each figure
represents the change in position between PKS and GCH, and consequently is
labeled deflection of PKS and GCH.
The basic description of string and frame motion can be generated from
these graphs. First, when the ball made contact with the strings, a deformation
of both the ball and string occurred. Although the strings deformed, the frame
did not move until the shock wave created by the impact traveled along the strings
to the frame. This explains the delay of increase in the GCH curve. Once the
shock wave reached the frame of the racket head, the center of the frame also
started to deflect. The strings continued to stretch until the momentum change
FRAME and STRING MOTION
E
0
y
RACKET No. 1
LEGEND :
PEAK OF STRING SURFACE
41
I :t...,bw.
.$7;.>.=
(PKS)
GEOMETRIC CENTER 0. HEAD '"CH'
DEFLECTION OF PKS FROM GCH
BALL CONTACT AND DEPARTURE
TIME (msec)
Figure 2
- Frame and string motion for racket 1.
RACKET No. 2
LEGEND :
1-
--
Q--Q
3
(PKS)
GEOMETRIC CENTER OF HEAD (GCH)
- - - -+ , +
2
1
0
TIME (msec)
-
RACKET No. 3
(PKS)
TIME (msec)
Figures 3 and 4
- Frame and string motion for rackets 2 and 3.
RACKET No. 4
-
LEGEND :
-ct
(PKS)
DEFLECTION OF PKS FROM GGH
$?&%
3
A
-
2
1
0
1
3
2
4
TIME (msec)
I
4t--
RACKET No. 5
LEGEND :
-H- PEAK
OF STRING SURFACE
(PKS)
3
2
1
0
TIME (msec)
Figures 5 and 6 - Frame and string motion for rackets 4 and 5.
5
-
RACKET No. 6
1
3
2
4
5
TIME (msec)
RACKET No. 7
+--+-
(PKS)
kT<&3i;;.
1
2
3
4
TIME (msec)
Figures 7 and 8 - Frame and string motion for rackets 6 and 7.
5
-
RACKET No. 8
(PKS)
1
3
2
4
5
TlME (msec)
I
RACKET No. 9
LEGEND :
++-
(PKS)
1
2
3
4
TIME (msec)
Wgures 9 and 10 - Frame and string motion for rackets 8 and 9.
5
RACKET No. 10
+-+-
(PKS)
TlME (msec)
RACKET No. 1 1
+A
PEAK
OF STRING SURFACE
(PKS)
!,:?KiSBALL
4$ CONTACT
1
AND DEPARTURE
2
3
4
TlME (msec)
Figures 11 and 12
- Frame and string motion for rackets 10 and 11.
5
4l
RACKET No. 12
LEGEND :
^
E
0
1
I ...
.....
t:::.....s,
(PKS)
GEOMETRIC CENTER OF HEAD (GCHi
TIME (msec)
Figure 13 - Frame and string motion for racket 12.
of the ball was equal to the impulsive force in the stretched strings (impulse =
A momentum). Furthermore, the impact caused the frame to bend, as shown in
Figures 2-13. This was due to the impulsive force applied from the ball to the
string, in addition to the reaction of the string pushing off the ball and pulling
the racket frame. Six variables were chosen to investigate the string and frame
behavior: max displacement of GCH,max displacement of PKS, relative deflection of PKS from GCH,impact duration, time to max deflection of PKS from
GCH, and slope of tangent line to deflection of PKS from GCH curve. The values
for these variables for all 12 rackets are given in Table 1.
The mean value for the maximum displacement of GCH was found to be
2.34 cm. In any given static situation, the bending deflection of the racket frame
could be a function of the load, the position of the load, and frame stifmess. Therefore the difference between rackets in displacement of GCH should be related
to preimpact ball velocity, ball contact position, and representative string surface tension.
The displacement of PKS was calculated as a position in space out of the
plane of the racket over time. Thus the PKS just before impact should have been
the same as the GCH, or zero. The mean value for PKS was 2.54 cm. The mean
value for the relative deflection of the PKS from GCH was 1.93 cm.
Since the digitizing was started just one frame before impact and stopped
at one frame after impact, the duration of impact was adjusted by one timed frame.
154
Table 1
Frame I String Space-Time Relationship During Impact
Racket #
Max
Max
peak
geometric string
Relative
center
surface deflection of
(~m)
(cm)
peaklframe
(GCH)
(PKS)
(cm)
Slope of
Time to
tangent line
max
to incline of
Impact
deflection
deflection
duration peaklframe peaklframe
(rnsec)
(msec)
(CMlmsec)
Racket
wlpeak
value
9
1
11
2, 12
13,1112
4
Racket
wlrninimum
value
8
7
1
7,8
5, 7
3
*average value
The correction factor of one frame was selected because of the sampling rate
of the film. It is possible the actual impact and departure of the ball into and away
from the strings may have occurred between film frames. It was anticipated that
any error due to improper determination of contact duration would be reduced.
The mean impact time for the 12 rackets was 3.98 msec.
The time to maximum deflection of PKS from GCH dictates the time it
took from impact for the strings to reach their maximum deflection relative to
the GCH. This variable had to be adjusted for the same reason as for the impact
duration, but the correction factor was only 112 of a timed frame because the
time it took for strings to return to zero was omitted. The mean time for rnax
deflection of PKS from GCH was 1.78 msec.
The last variable observed, the slope of the tangent line to the deflection
of PKS from GCH curve, indicates the change of deflection versus time. The
155
larger or steeper this slope, the faster the deflection occurs. The mean value for
the 12 rackets tested was 0.88 cmlmsec.
In order to determine if any general trends in the values of the variables
for any one racket emerged, the investigators rank-ordered the 12 rackets from
high to low in each variable. Table 2 depicts this rank ordering. Rackets 2, 9,
11, and 12 were ranked first or near first for almost every variable, whereas rackets
7 and 8 were ranked last or close to last for several variables. Because of the
varying lengths of strings across a tennis racket, when the ball makes contact
and causes a wave to travel along the strings, they may behave differently. The
complexity of string harmonics, however, is beyond the scope of this paper. In
order to fully explain string and racket behavior, more research needs to be conducted in this area.
Table 2
Rank Order of Rackets From High to Low for Each Variable
Max
Max
Peak
Relative
geometric string
deflection of
center
surface PKS from GCH
Impact
duration
Time to
Slope of tangent
max
line to deflection
deflection of PKS from GCH curve
Coefficient of Restitution
The values of the calculated coefficients of restitution for the 12 rackets from
the top and side views during the forehand drive are given in Table 3. The range
of the COR was 0.09 with the limits of 0.75 to 0.84. These values were somewhat larger than the findings of previous researchers: 0.57-0.66 by Elliott (1982),
0.44 by Watanabe, Ikegami, and Miyashita (1979), and 0.48 by Ellis et al. (1978).
It is possible these investigators did not fully account for the motion of the racket
just after impact. Without considering this backward motion after impact, the
Table 3
Coefficient of Restitution During Forehand Drive
Top view
Racket #
Top view
Side view
Side view
M
COR calculations would be underestimated, therefore explaining the higher values
obtained in this investigation. (The rank-ordering of the rackets from high to low
values of CORs is shown in Table 5 .) Racket 11 is close to first, as it was for
the six other variables.
The coefficient of restitution for the fixed racket condition was determined
by the same method used for the moving racket. The mean value of the COR
for the fixed rackets was 0.58 (Table 4), which was far less than that of the swung
rackets (range from 0.78 to 0.84), as shown in Table 5. The time of the ballracket impact in the two different experiments may help explain this result. The
mean impact time in the fixed-racket experiment was 3.98 msec. Although contact duration could not be measured in Experiment 2 when examining the forehand drive (due to the low frame rate), other researchers have noted that this
time is longer during an actual stroke (e.g., Plagenhoef, 1971). If that informa-
Table 4
Coefficient of Restitution for Fixed Racket Testing
Racket#l
COR
2
3
4
5
6
7
8
9
10
11
12
M
524 .565 511 .436 .536 .655 .705 .555 .535 .521 559 .631 .578
157
Table 5
Rank Order of Racket Coefficient of Restitution From High to Low
Coefficient of restitution during forehand
TOP
Side
TOP
Side
Coefficient of
restitutionfixed racket
tion holds true for Experiment 2, it seems possible the subject added a considerable amount of force to the impact. In Experiment 1 there could be no external
force involved because the racket was fixed. In Experiment 2, however, the racket
was swung forward to drive the ball. In a tennis stroke executed by a skilled player,
the racket has been seen to accelerate up to the point of contact (Groppel, 1984).
Therefore an external force may be applied that was neglected in the COR calculation. Further study is recommended to examine this phenomenon.
References
AMF HEAD Division (1981). The new HEAD string-lock system is here. Headlines,
1(1):1-4.
s
and ball responses during impact
Baker, J.A., & Putnam, C.A. (1979). T e ~ i racket
under damped and freestanding conditions. Research Quarterly, 50(2): 164-170.
Baker, J.A.W., & Wilson, B.D. (1978). Effect of racket stiffness and string tension on
ball velocity after impact. Research Quarterly, 49(3): 255-259.
Brody, H. (1979). Physics of the tennis racket. American Jouml of Physics, 47(6):482487.
Elliott, B.C. (1982). The influence of tennis racket flexibility and string tensions on
rebound velocity following a dynamic impact. Research Quarterlyfor fiercise and
Sport, 53(4):277-281.
Ellis, R., Elliott, B.C., & Blanksby, B. (1978). The effect of string type and tension
in jumbo and regular sized tennis rackets. Sports Coach, 2:32-34.
Groppel, J.L. (1984). Tennis for advanced players: And those who would like to be.
Champaign, IL: Human Kinetics.
158
Groppel, J.L., Shin, I., Thomas, J., & Welk, G. (1987). The effects of string tension
on impact in midsized and oversized tennis rackets. International Journal ofSport
Biomechanics, 3(1):40-46.
Plagenhoef, S. (1971). Patterns of human motion: A cinematographical analysis. Englewood
Cliffs, NJ: Prentice-Hall.
Watanabe, T., Ikegami, Y., & Miyashita, M. (1979). Tennis: The effects of grip firmness
on ball velocity after impact. Medicine and Science in Sports, 11(4):359-361.

Effects of Different String Tensi~n Patterns and Racket Motion on

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