🎾 Dynamics Of Ball - Vợt Impact In Tennis¶
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Dynamics Of Ball - Vợt Impact In Tennis — tài liệu 8 trang từ thư viện sách tennis.
Chủ đề chính: Racket, Racquet
Tóm tắt nội dung (trích từ tài liệu gốc): Proceedings of ACMD'02 The First Asian Conference on Multibody Dynamics 2002 July 31-August 2, 2002, Iwaki, Fukushima, Japan Dynamics of the Ball-Racket Impact in Tennis: Contact Force, Contact Time, Coefficient of Restitution, and Deformation Yoshihiko KAWAZOE Dep. of Mechanical Engineering, Saitama Institute of Technology, 1690, Okabe, Saitama, 369-0293, JAPAN E-Mail: ykawa@sit.ac.jp Keywords: Impact, Tennis, Contact Force, Contact Time, Coefficient of restitution Abstract the involvement of humans in the actual strokes. This paper has investigated the physical properties of a racket This pa
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Proceedings of ACMD'02
The First Asian Conference on Multibody Dynamics 2002
July 31-August 2, 2002, Iwaki, Fukushima, Japan
Dynamics of the Ball-Racket Impact in Tennis: Contact Force,
Contact Time, Coefficient of Restitution, and Deformation
Yoshihiko KAWAZOE
Dep. of Mechanical Engineering,
Saitama Institute of Technology,
1690, Okabe, Saitama, 369-0293,
JAPAN
E-Mail: ykawa@sit.ac.jp
Keywords: Impact, Tennis, Contact Force, Contact Time, Coefficient of restitution
Abstract the involvement of humans in the actual strokes.
This paper has investigated the physical properties of a racket This paper derives the contact forces, contact time,
and has derived the contact forces, contact time, coefficient coefficient of restitution, and deformations during impact
of restitution, and deformations during impact between a between a ball and racket. Furthermore, it predicts the
ball and racket. Furthermore, it has predicted the power or power or post- impact ball velocity with a forehand
post- impact ball velocity with a forehand groundstroke. It is groundstroke. It is based on the experimental identification of
based on the experimental identification of the dynamics of the dynamics of racket-arm system and the approximate
racket-arm system and the approximate nonlinear impact nonlinear impact analysis with a simple forehand stroke swing
analysis with a simple forehand stroke swing model. The model. The predicted results could explain the mechanism of
predicted results could explain the mechanism of impact impact between a ball and a racket with different physical
between a ball and a racket with different physical properties. It properties.
enables us to predict the various factors associated with impact 2. Prediction of Impact Forces, Contact Time, Energy
and performance of the various racket.
Loss, and Coefficient of Restitution between Ball and
Racket [3]-[7]
1. Introduction 2.1 Main Factors Associated with Impact Analysis
Traditional wooden frame tennis rackets were used for a century, Figure 1 shows schematically the test for obtaining the applied
but they have evolved through several technology force- deformation curves, where the ball is deformed between
advancements over the past 30 years. Advanced engineering two flat surfaces as shown in (a) and the ball plus strings is
technology has enabled manufacturers to discover and deformed with a racket head clamped as shown in (b). The
synthesize new materials and new designs of sport equipment. results for the ball and racket are shown in Fig.2. According to
There are rackets of all compositions, sizes, weights, shapes and the pictures of a racket being struck by a ball, it seems that the
strings tension, and very specific designs are targeted to match ball deforms only at the side, which contact to the strings.
the physical and technical levels of each user [1][2]. Assuming that a ball with concentrated mass deforms only at
However, at the current stage, the perception of performance of the side in contact with the strings [7], the curves of restoring
the tennis racket is actually based on the feel of an experienced force FB vs. ball deformation, restoring force FG vs. strings
tester or player. Since the optimum racket depends on the deformation, and the restoring force FGB vs. deformation of the
physical and technical levels of each user, there are many composed ball/strings system are obtained from Fig.2 as shown
unknowns regarding the relationship between the performance in Fig.3. These restoring characteristics are determined in order
estimated by a player and the physical properties of a tennis to satisfy a number of experimental data using the least square
racket. method. The curves of the corresponding stiffness KB, KG and
The ball-racket impact in tennis is an instantaneous KGB are derived as shown in Fig.4 by differentiation of the
phenomenon creating large deformations of ball/strings and equations of restoring force with respect to deformation. The
vibrations in the racket. The problem is further complicated by stiffness KB of a ball, KG of strings and KGB of a composed
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ball/strings system exhibit strong nonlinearity.
The measured coefficient of restitution versus the incident
velocity when a ball strikes the rigid wall is shown in Fig.5,
while the measured coefficient of restitution eBG, which is
abbreviated as COR, when a ball strikes the strings with a racket
head clamped is shown in Fig.6. Although the COR in Fig.5
decreases with increasing incident velocity, the coefficient eBG
Fig.1 Illustrated applied force- deformation test Fig.4 Stiffness vs. deformation of a ball, strings, and a
composed ball/string system assuming that a ball
deforms only at the side in contact with the strings.
with a racket head clamped is almost independent of ball
velocity and strings tension. This value of COR can be regarded
as being inherent to the materials of ball and strings, showing
the important role of strings. This feature is due to the nonlinear
restoring force characteristics of a composed ball/strings system
[4].
Since equivalent spring stiffness K of the compound system
increases as the impact velocity increases, the independence of
the damping coefficient ratio with respect to impact velocity
means that damping coefficient C is proportional to K1/2
and increases with increases in impact velocity. The energy loss
of a ball and strings due to impact can be related to the
coefficient eBG.
Fig.2 Results of a force-deformation test.
Fig.5 Measured coefficient of restitution (COR) between a ball
and a rigid wall.
The result of measured contact time, which means how long the
ball stays on the strings, with a normal racket and with a
wide-body racket (stiffer) shows that the stiffness of the racket
frame does not affect the contact time much [4]. Accordingly,
the masses of a ball and a racket as well as the nonlinear
stiffness of a ball and strings are the main factors in the deciding
of a contact time. Therefore, the contact time can be calculated
using a model assuming that a ball with a concentrated mass mB
Fig.3 Restoring forces vs. deformation of a ball, strings, and and a nonlinear spring KB, collides with the nonlinear spring KG
a Composed ball/string system assuming that a ball of strings supported by a frame without vibration, where the
deforms only at the side in contact with the strings. measured coefficient of restitution inherent to the materials of
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Fig.6. Measured COR between a ball and strings with ball-strings impact is employed as one of the sources of energy
frame clamped. loss.
The reduced mass Mr of a racket at the impact location on the
string face can be derived from the principle of the conservation
of angular momentum if the moment of inertia and the distance
between an impact location and a center of gravity are given.
Figure 7 ( super-light weighted racket 290 g) and Fig.8
(conventional weight balanced racket 370 g) show the effect of
an arm on the reduced mass on the longitudinal axis on the
racket face. There is no big difference between the reduced
masses with the arm and without it, particularly around the
center of the racket face.
The result of the experimental modal analysis [3][9] showed that
the fundamental vibration mode of a conventional type racket
supported by a hand has two nodes being similar to the mode of
a freely supported racket. The racket is assumed to be freely
suspended in terms of the performance of power.
2.2 Derivation of Approximate Impact Force and Contact
Time
400
Reduced Mass Mr[g] 350 In case the vibration of the racket frame is neglected, the
momentum equation and the coefficient of restitution eBG give
300 the post-impact velocity VB of a ball and VR of a racket at the
impact location. The impulse could be described using the
250 following equation, where mB is the mass of a ball, Mr is the
reduced mass of a racket at the hitting location, and ( VBO - VRo )
200 is the pre-impact velocity.
150
100
MH=0 [kg]
50 MH=1.0 [kg]
0 50 100 150 F ( t) dt= mB VBo - mB VB = (VBO - VRo )(1+ eBG)mB/(1+mB/Mr)
-150 -100 -50 0 (1)
Top side Center Near side [mm] Assuming the contact duration during impact to be half the
natural period of a whole system composed of mB , KGB , and
Fig7. The effect of the arm on the reduced mass of a racket at Mr , it could be obtained as
the impact locations (super-light weighted racket 290 g,
MH=1.0 kg: with arm, MH=0 kg: without arm) Tc = mB1/2/[KGB (1+ mB / Mr )]1/2 (2)
400 In order to make the analysis simpler, the equivalent force Fmean
350 can be introduced during contact time Tc , which is described as
300
Reduced Mass Mr[g] Tc F ( t ) dt = FmeanTc (3)
250 Thus, from Eq.(1), Eq.(2) and Eq.(3), the relationship between
200 Fmean and corresponding KGB against the pre-impact velocity
150 ( VBO - VRo ) is given by
100 Fmean= (VBO - VRo )(1+ eBG ) mB1/2 KGB 1/2/( 1+ mB/Mr ) 1/2
MH=0 [kg]
50 MH=1.0 [kg] (4)
0 On the other hand, from the approximated curves shown in
-150 -100 -50 0 50 100 150 Fig.3 and Fig.4, FGB can be expressed as the function of KGB
Top side Center Near side [mm] in the form
Fig.8 The effect of the arm on the reduced mass of a racket at FGB =f ( KGB ). (5)
the impact locations (conventional balanced racket 370 g, From Eq.(4) and Eq.(5), KGB and Fmean against the pre-impact
MH=1.0 kg: with arm, MH=0 kg: without arm) velocity can be obtained, accordingly TC can also be
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determined against the pre-impact velocity by using Eq.(2). experimental modal analysis [3][9] and the racket vibrations can
Figure 9 is a comparison between the measured contact times be simulated by applying the impact force-time curve to the
during actual forehand strokes [11] and the calculated ones hitting portion on the string face of the identified vibration
when a ball hits the center of the strings face of a conventional model of the racket. When the impact force Sj (2f k)
type racket, showing a good agreement between them. applies to the point j on the racket face, the amplitude Xij k of
Since the force-time curve of impact has an influence on the k-th mode component at point i is expressed as
magnitude of racket frame vibrations, it is approximated as a
half-sine pulse, which is almost similar in shape to the actual Xij k = r ij k Sj (2f k) (8)
impact force. The mathematical expression is
where r ij k denotes the residue of k-th mode between arbitrary
F (t ) = Fmax sin(t/ Tc ) (0t Tc ) (6) point i and j, and Sj (2f k) is the impact force component
of k-th frequency f k [5].
where Fmax =Fmean/2. The Fourier spectrum of Eq.(6) is Figure 11 shows the string mesh and impact location on the
represented as racket face, and Fig.12 shows the example of predicted
vibration amplitude of the racket struck by a ball at 30 m/s.
S ( f ) = 2Fmax Tccos(fTc)/ [1 - (2fTc )2] (7)
2.4 Energy Loss Due to Racket Vibrations Induced by
where f is the frequency. Impact
The energy loss due to the racket vibration induced by impact
can be derived from the amplitude distribution of the vibration
velocity and the mass distribution along the racket frame. If the
longitudinal mass distribution of racket frame is assumed to be
uniform, the energy loss E1 due to racket vibrations can be easily
derived.
2.5 Derivation of Coefficient of Restitution
Fig.9 Comparison between the measured contact times during The coefficient of restitution (COR) can be derived considering
strokes and the calculated results. the energy loss during impact. The main sources of energy
loss are E1 and E2 due to the instantaneous large deformation
Figure 10 shows the examples of the calculated shock shape of a ball and strings which is calculated by using the coefficient
during impact, where the ball strikes the center on the string face eBG. If a ball collides with a racket at rest ( VRo = 0), the
at a velocity of (a) 20 m/s and (b) 30 m/s with the racket strung energy loss E2 could be easily obtained. The coefficient of
at 55 lb, respectively. restitution er corresponds to the total energy loss E (= E1 +
2.3 Prediction of Racket Vibrations E2 ) obtained as
The vibration characteristics of a racket can be identified using
er = ( VR - VB )/ VBO = [1 - 2E ( mB + Mr )/ (mBMr VBO )]1/2
Fig.10 Calculated shock shape when a ball strikes the center on (9)
the String face at velocities of 20 m/s and 30 m/s.
3. Prediction of Rebound Power Coefficient
The post-impact ball velocity VB is represented as
VB = -VBo (er - mB/Mr) /(1+mB/Mr)+VRo (1+er) /(1+mB/Mr)
(10)
Accordingly, if the ratio of rebound velocity against the incident
velocity of a ball when a ball strikes the freely suspended racket
( VRo = 0) is defined as the rebound power coefficient e, it is
written as Eq.(11) . The rebound power coefficient is often used
to estimate the rebound power performance of a racket
experimentally in the laboratory.
e = -VB / VBO = ( er - mB/Mr ) /(1+ mB/Mr ) (11)
hen a player hits a coming ball with a pre-impact racket head
velocity VRo , the coefficient e can be expressed as
e = - ( VB - VRo ) / (VBO - VRo ) (12)
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Fig.11 String mesh (Left side) and impact location on the racket face (Right side).
Fig.12 Predicted initial amplitude of 1st mode component of racket frame vibrations.
Figure 13 is a comparison between the measured e and the Is )1/2 , where LX denotes the horizontal distance between the
predicted e when a ball hits a freely-suspended racket (about 30 player's shoulder joint and the impact location on the racket face,
m/s), showing a good agreement between them [5][6]. Ns the constant torque about the shoulder joint, and Is the
moment of inertia of arm/racket system about the shoulder joint.
4. Prediction of Post-impact Ball Velocity VB = - VBo e + VRo ( 1 + e ) (13)
The power of the racket could be estimated by the post-impact 5. Ball Control and Racket Stability
ball velocity VB when a player hits the ball [13][14]. The VB Control is simply being able to put the ball where desired, but it
can be expressed as Eq.(13). The VRo is given by LX (Ns /
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Fig.13 Comparison between the measured rebound power different physical properties. Figure 14 shows a simple forehand
coefficient e and the predicted one (VB =Vout, VBO =Vin , ground stroke swing model [12].
VRo= 0). Figure 15 shows the comparison of the predicted coefficients of
restitution er between the super-light weight racket (290 g ) and
is the most difficult to analyze. Designing equipment to conventional weight and weight balanced racket (370 g). It is
optimize control for all types of players is nearly impossible. when a player hits a coming ball with a velocity VBO of 10 m/s.
Tennis players have a wide variety of styles from smooth Figure 16 is at the longitudinal axis on the racket face. It is seen
stroking to whippy and wristy. These style differences require that er of a super-light weight racket is higher than that of a
different equipment to optimize control. However, one conventional weight and weight balanced racket at the top of the
characteristic required for control is stability. Stability refers to string face.
the ability at impact to maintain its swing path without deviation. Figure 17 shows the predicted rebound power coefficient e
Stability is also defined as the ability to resist off center hits. It is (N=56.9Nm,VBo=10m/s). It is seen that e of a conventional
desirable to maximize stability [1]. weight and weight balanced racket is higher than that of a
We can estimate the racket stability by the amount of twist or super-light weight racket anywhere on the string face.
turn about the long axis when the ball hits the strings at the Figure 18 and Fig.19 show the comparison of the predicted VB
location away from the long axis of a racket. at each hitting location on the racket face. We can see the
6. Estimation of the Performance of Tennis Rackets having difference in sweet area in terms of racket power between a
super-light weighted racket ( EOS100, 290 g) and conventional
Different Weight and Weight Balance heavier weighted racket ( PROTO-02, 370 g).
Now we can predict the various factors associated with the
tennis impact when the impact velocity or swing model and the (a) EOS100 (290 g) (b) PROTO-02 (370 g)
impact locations on the racket face are given. Furthermore we
can estimate the performance of the various rackets with Fig.15 Predicted Restitution coefficient e on the racket face
when a player hits a ball (N=56.9Nm,VBo=10m/s).
0.84
0.82
0.80
EOS100
PROTO-02
0.78
-150 -100 -50 0 50 100 150
Top side Center Near side (mm)
Fig.14 Simple forehand groundstroke swing model. Fig.16 Predicted Restitution coefficient e on the longitudinal
axis of racket face when a player hits a ball (N=56.9
Nm,VBo =10 m/s).
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38
0.6
0.5 36
VB (m/s)
0.4
34
0.3
0.2 32 EOS100
EOS100 PROTO-02
PROTO-02
0.1 50 100 150 30
0 -150 -100 -50 0 50 100 150
Top side Center Near side (mm)
-150 -100 -50 0
Fig.19 Predicted post-impact ball velocity on the longitudinal
Top side Center Near side (mm) axis of a racket face ( VBo= 10 m/s, Ns =56.9 Nm).
FIg.17 Predicted rebound power coefficient e (N= 56.9
Nm,VBo=10m/s)
Figure 20 shows the twist or turn about the long axis when the Fig.20 Twist or turn about the long axis when the ball hits the
ball hits the strings at the location away from the long axis of a strings at the location away from the long axis of a
racket. Figure 21 shows the predicted amount of the racket twist racket.
vs. distance of the impact location from the long axis, assuming
that there is no friction between the hand and the racket grip. It
is the comparison between a Super light weighted racket:
(EOS100, 290 g) and conventional heavier weighted racket
( PROTO-02, 370 g) at the topside, the center and the near side
on the racket face away from the long axis. There is no twist
about long axis at the topside away from the long axis, because
the racket turns about the location near the grip. There is big
difference in twist angles at the near side on the racket face but
there is no big difference at the topside and the center away
from long axis between the lighter racket and the heavier racket.
The conventional heavier racket seems to be desirable in
stability. However, since the hitting area for the groundstroker
is usually at the topside from the center, there is no big
disadvantage for the super-lighted weighted racket.
7. CONCLUSIONS
This paper has investigated the physical properties of a racket
and has derived the contact forces, contact time, coefficient
of restitution, and deformations during impact between a
ball and racket. Furthermore, it has predicted the power or
post- impact ball velocity with a forehand groundstroke. It is
based on the experimental identification of the dynamics of
racket-arm system and the approximate nonlinear impact
analysis with a simple forehand stroke swing model. It enables
us to predict the various factors associated with impact and
performance of the various racket.
(a) EOS100 (290g) (b) PROTO-02 (370g) ACKNOWLEDGMENTS
Fig.18 Predicted post-impact ball velocity V on the racket The author would like to thank many students in his
laboratory for their help in carrying out the study as senior
face representing sweet area in terms of power.
- 292 - Copyright � 2002 by JSME
Institute of Technology.
Angle [deg] 12 8. REFERENCES
EO S100 [1] Davis S. "Rackets science applied to golf", Prc. 5th Japan
10 International SAMPE Symposium, pp.1329-1334., 1997.
PRO TO -02 [2] Ashley S., "High-tech rackets hold court ", Mechanical
8 Engineering, ASME, pp.50-55, August (1993).
[3] Kawazoe,Y., "Dynamics and computer aided design of
6
tennis racket", Proc. Int. Sympo. on Advanced Computers
4 for Dynamics and Design'89, pp.243-248, (1989).
[4] Kawazoe,Y. (1992) Impact phenomena between racket and
2 ball During tennis stroke, Theoretical and Applied
Mechanics, Vol.41, pp.3-13.
0 [5] Kawazoe,Y., Coefficient of restitution between a ball and a
tennis racket, Theoretical and Applied Mechanics, Vol.42,
0 20 40 60 80 (1993), pp.197-208.
[6] Kawazoe,Y., Analysis of coefficient of restitution during
B [mm] a nonlinear impact between a ball and strings
considering vibration modes of racket frame, Trans.
(a) Top side on the racket face JSME, 59-562, (1993), pp.1678-1685. (in Japanese)
[7] Kawazoe,Y., Effects of String Pre-tension on Impact
Angle [deg] 12 Between Ball and Racket in Tennis, Theoretical and Applied
EO S100 Mechanics, Vol.43, (1994), pp.223-232.
10 [8] Kawazoe,Y., Computer Aided Prediction of the Vibration
PRO TO -02 and Rebound Velocity Characteristics of Tennis Rackets
with Various Physical Properties, Science and Racket
8 Sports, (1994) , pp.134 -139. E & FN SPON.
[9] Kawazoe, Y. , Experimental Identification of Hand-
6 held Tennis Racket Characteristics and Prediction of
4 Rebound Ball Velocity at Impact, Theoretical and
2 Applied Mechanics, Vol.46, (1997), pp.165-176.
0 [10] Kawazoe, Y., "Mechanism of Tennis Racket Performance
in terms of Feel", Theoretical and Applied Mechanics,
0 20 40 60 80 Vol.49, (2000), pp.11-19.
[11] Nagata,A., "Analysis of tennis movement", J. J. Sports
D Sci., 2-4, (1983), pp.245-259. (in Japanese)
[12] Kawazoe, Y. and Kanda, Y., Analysis of impact
[mm] phenomena in a tennis ball-racket system (Effects of frame
(b) Center on the racket face vibrations and optimum racket design), JSME International
Journal, Series C, Vol.40, No.1, (1997), pp.9-16.
12 [13] Kawazoe, Y. and Tomosue, R., "Sweet area prediction
of tennis rackets estimated by ball post-impact velocity
Angle [deg] 10 EO S100 (comparison between two rackets with different frame mass
PRO TO -02 distributions)", Proc. Symp. on Sports Engineering,
(1996), pp.55-59. Japan Society of Mechanical Engineers,
8 Tokyo. (in Japanese)
[14] Kawazoe, Y. and Tomosue, R., "Prediction of a sweet
6 area on a racket face in a tennis impact (Restitution
coefficient, rebound power coefficient and ball post-
4 impact velocity)", Trans. JSME, 64-623, (1998),
pp.2382-2388. (in Japanese)
2
0
0 20 40 60 80
F [mm]
(c) near side on the racket face
Fig.21 Calculated amount of the racket twist vs. distance of the
impact location from the long axis, assuming that there is
no friction between the hand and the racket grip. (Super
light weighted racket: EOS100: 290 g, Conventional
weighted racket: PROTO-02: 370 g)
students during the academic year. He would also like to thank
the International Tennis Federation (ITF) for funding the
research. This work was supported by a Grant-in-Aid for
Science Research of the Ministry of Education, Culture, Sports,
Science and Technology of Japan, and a part of this work was
also supported by the High-Tech Research Center of Saitama
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