🎾 The Double Pendulum In Tennis¶
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The Double Pendulum In Tennis — tài liệu 24 trang từ thư viện sách tennis.
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Tóm tắt nội dung (trích từ tài liệu gốc): The Double Pendulum in Tennis about:reader?url=http://twu.tennis-warehouse.com/learning_center/doub... twu.tennis-warehouse.com The Double Pendulum in Tennis 27-34 ph�t 1. INTRODUCTION Swinging a racquet is a process that is not easy to describe in words since the actions of all the various body segments are quite complicated. However, if we focus just on the forearm and the racquet then the task is a little easier, especially since these two segments by themselves act like a double pendulum. A double pendulum is just two single pendulums joined end to end. To illustrate the point, consider th
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The Double Pendulum in Tennis
27-34 ph�t
1. INTRODUCTION
Swinging a racquet is a process that is not easy to describe in
words since the actions of all the various body segments are quite
complicated. However, if we focus just on the forearm and the
racquet then the task is a little easier, especially since these two
segments by themselves act like a double pendulum. A double
pendulum is just two single pendulums joined end to end. To
illustrate the point, consider the two film clips shown below where
the action of the arm and racquet in a serve is compared with a
simple mechanical double pendulum. Both were filmed in slow
motion, at 300 fps, to observe the action more clearly. The
pendulum version is upside-down, but the actions are very similar.
Pendulum Motion of the Serve
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0:00 / 0:01
Movie Screen 1. -- Double Pendulum Motion in the serve.
(Note: Movies are best viewed frame by frame using keyboard
arrow keys or movie controls.)
Double Pendulum Demonstration
Choose Movie
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0:00 / 0:04
Movie Screen 2. -- The difference in racquet and forearm speed
between a light "racquet" and a heavy racquet.
(Note: Movies are best viewed frame-by-frame using keyboard
arrow keys or movie controls.)
The server struck the ball at 100 mph so the racquet swung much
faster than the mechanical pendulum. The mechanical pendulum
was constructed using a sawn-off baseball bat as a forearm and a
long wood dowel as a racquet, connected by an artificial wrist joint
consisting of two eye hooks held together loosely with a small bolt.
There is no wrist or elbow action in the mechanical pendulum, and
gravity alone caused the pendulum to rotate. Even so, the
similarity with a real serve is remarkable. With a few extra
modifications to the pendulum, it could be made to behave just like
a real serve. It needs only a bit of elbow action and a bit of wrist
action to behave in a more realistic manner.
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Several things stand out. One is that the forearm slows down while
the racquet speeds up. Another is that the racquet rotates through
a larger angle than the forearm. The forearm and racquet start out
at right angles and end up almost in line by the time the racquet
strikes the ball. In a real serve, the forearm and racquet remain at
right angles for about 80% of the swing time. In the pendulum
swing, the dowel drops straight down before it starts to swing
around, because there is no wrist to lock the dowel at right angles
to the forearm. We could ask several questions about this. Is it
actually necessary to lock the wrist at the start of a serve? Does
the wrist rotate the racquet at the end of the swing or is it the other
way around? That is, does the racquet rotate the wrist? Would it
help if the forearm didn't slow down? These sorts of questions are
easily answered by looking at the mechanics of a pendulum,
especially when we add some artificial elbow and wrist action.
The actions of a double pendulum have been studied for many
years, especially in relation to the swing of a golf club. The physics
of swinging a tennis racquet has not been studied in nearly as
much detail. Part of the problem is that there are many different
ways of swinging a racquet. A golfer has a simpler task since the
ball just sits there waiting to be struck and the golfer needs only to
hit the ball toward the hole. Except when serving, a tennis player
needs to chase down the ball and then hit it from a range of
different heights to various parts of the court, either using a
forehand or a backhand or a volley or a half-volley or a smash.
Golfers only ever hit the ball with backspin, but tennis players can
hit the ball with topspin or backspin or side spin or any other spin
they like.
In tennis, the wrist is not just a passive hinge, like it is in a
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mechanical double pendulum, but plays an active role in
controlling the racquet. Similarly, the elbow joint is not passive, but
plays an active role in controlling motion of the forearm. The
beauty and the complication of this arrangement is that motion of
the forearm affects motion of the racquet, and motion of the
racquet affects motion of the forearm, so the two cannot be
controlled separately or independently. The results are sometimes
unexpected but are consistent with the way that a double
pendulum behaves.
If the wrist is completely limp, then the player has no direct control
over the racquet and the racquet will just swing where it wants to
swing. In general, the player rotates the forearm at a controlled
speed, using muscles in the upper arm, and uses the wrist as an
extra aid to control exactly where the racquet needs to go. If the
wrist is locked then the forearm and the racquet behave as a
single, solid object and both rotate at the same speed. That is, if
the forearm rotates through say 30 degrees, then so does the
racquet. The rotation speeds (measured in degrees per second) of
the forearm and the racquet are the same even though the actual
speed of the racquet (measured in mph or feet per second) is
greater than the actual speed of the forearm.
The complex interaction between the forearm and the racquet is
nicely described in terms of the physics of a double pendulum.
Players don't worry about that since they know from experience
what happens to the racquet when they swing their arm or use
their wrist to position the racquet. However, they may not be aware
of some of the fine details. For example, a surprising feature is that
the forearm slows down rapidly just before a player strikes the ball.
The forearm speeds up at the start of the swing but the main
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object of the exercise is to swing the racquet rather than the
forearm. The best way to do that is to allow the forearm to slow
down just before striking the ball so that the forearm transfers its
energy to the racquet. In fact, there is a whole chain of events
involved in swinging a racquet. Before the forearm slows down,
the upper arm slows down to transfer its energy to the forearm.
And before that, the upper torso slows down to transfer its energy
to the upper arm.
It is difficult for a player to predict how the swing will change if the
player uses a different racquet or the racquet is modified in some
way by adding weight to the tip or the handle. But the changes can
be predicted using a double pendulum calculation. Similarly, one
can predict what will happen if the player uses more wrist action or
deliberately swings the forearm faster.
In this article we will consider only a forehand and a serve, the
object of the exercise being to see what insights can be gained by
treating the forearm and the racquet as a double pendulum. The
particular advantage of this approach is that it is easy to calculate
the effect of different wrist and elbow actions. It is much harder to
measure the effects. I admit that the calculations ignore several
real effects, but the basic double pendulum action is obvious from
the video film.
2. WALKING AND RUNNING
As an aside, it is worth mentioning that the action of a double
pendulum can be seen in many different human activities, not just
in swinging a racquet or a golf club. The best known examples are
walking and running. The upper leg is swung by muscles around
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the hip and the lower leg is swung by muscles around the knee.
Walking and running involve two double pendulums working in
unison. Success requires that each segment be swung at the right
time, at the right speed, and by the right amount. Movie Screen 3
shows the double pendulum action involved in walking.
Double Pendulum Motion While Walking
0:00 / 0:05
Movie Screen 3. -- Double pendulum motion while walking.
(Note: Movies are best viewed frame by frame using keyboard
arrow keys or movie controls.)
3. SERVE DATA
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Figure 1 -- Positions of racquet and forearm viewed from the side.
The serve data in from the video in Movie Screen 1 provides a
good basis for further calculations, although a direct comparison
with a double pendulum is not straightforward. The main problem
is that the serve action occurs in all three dimensions, whereas a
double pendulum normally swings in just one plane. Another
problem is that the upper arm, forearm and racquet together form
a triple rather than a double pendulum. In effect, the upper arm
and forearm form one double pendulum, while the forearm and
racquet form another double pendulum. The serve action is shown
in Figs. 1 and 2 and can be divided into three stages. In Stage 1,
the upper arm reaches maximum speed. In Stage 2, the forearm
reaches maximum speed. In Stage 3, the racquet reaches
maximum speed.
Stage 1: t = 0 to t = 0.05 s. The upper arm swings from an
approximately horizontal position to a vertical position while the
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forearm remains horizontal and remains locked at right angles to
the upper arm. The racquet remains locked at right angles to the
forearm. Since the forearm has not yet started to rotate in the
vertical plane, this stage of the serve action is not part of the
double pendulum involving the forearm and the racquet.
Figure 2 -- Positions of racquet, forearm and upper arm viewed
from the rear.
Stage 2: t = 0.05 to t = 0.103 s. The upper arm remains
approximately vertical while the forearm swings from a horizontal
to a vertical position. The racquet remains locked at right angles to
the forearm so the racquet and the forearm both rotate at the
same speed. This is the start of the double pendulum motion,
analogous to the start of the swing of a golf club where the club is
locked at right angles to the arms. Since the forearm and the
racquet both rotate through 90 degrees in 0.053 seconds, the
average rotation speed of each segment is 90/0.053 = 1700
degrees/sec or 4.72 rev/sec or 283 rpm.
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Stage 3: t = 0.103 to t = 0.123 s. The forearm rotates forward
about 10 degrees while the racquet rotates through 90 degrees.
This is the final stage of the double pendulum action where the
wrist unlocks to allow the racquet to rotate as fast as possible. The
average rotation speed of the forearm is 10/0.02 = 500 deg/s = 1.4
rev/s = 83 rpm and the average rotation speed of the racquet is
90/0.02 = 4500 deg/s = 12.5 rev/s = 750 rpm. Just before
impacting with the ball, the racquet was actually rotating at close to
6000 deg/s = 1000 rpm.
Because of the three-dimensional nature of the serve, it is difficult
to track accurately the rotation speed of the forearm and the
racquet through the entire swing. However, we can use the
average speeds of the forearm and the racquet to construct a
reasonable two-dimensional model of the process. Two models
could be considered. One starting at t = 0 and ending at t = 0.103
s would describe the double pendulum action of the upper arm and
forearm. During this time, the racquet is simply an extra mass in
the hand, locked at right angles to the forearm. The other model,
starting at t = 0.05 s and ending at t = 0.123 s, is the one we will
consider in more detail. During this time the upper arm does not
rotate by any significant amount but the elbow moves upward and
forward due to motion of the whole body.
One extra piece of information needed to analyze the serve is the
elbow speed, shown in Fig. 3. The upper arm and the elbow
rotated in a vertical plane. The elbow was rising rapidly at t = 0 and
slowed almost to a stop by the time the player struck the ball.
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Figure 3 -- (a) Path of elbow in video serve (b) Elbow speed in the
vertical and horizontal directions. The horizontal speed is negative
since the elbow moves from right to left in the video.
4. WRIST ACTION
When a golfer swings a club, the wrist normally relaxes half way
through the stroke. In a golf swing, the club head swings around
so fast there is very little a player can do with the wrist near the
end of the swing to change the swing speed. In effect, the club
rotates the wrist, not the other way around.
A tennis racquet has a smaller swing weight than a golf club and
can be swung with only one arm. The effect of the wrist on the
swing speed might therefore be more significant. We can estimate
the effect with a simple calculation. Suppose that a player takes
0.1 sec to accelerate the racquet from 0 to 600 rpm in a serve. 600
revolutions in a minute corresponds to 10 revolutions in one
second or one revolution in 0.1 seconds. That is the maximum
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rotation speed of the racquet. Most pros swing their racquet a bit
faster than that in a first serve. The numbers here are typical of a
second serve. If it takes 0.1 s to get up to that speed then the
torque on the racquet is about 30 Nm or about 22 ft-lbs. Not all of
that torque is supplied by the wrist, but some of it might be.
The maximum torque that can be exerted by the wrist can be
estimated from a simple experiment. A player can easily hold a
racquet in a horizontal position with one hand. If a 1 lb weight is
added near the tip of the racquet, say 2 ft from the wrist, then it is
harder to hold the racquet in a horizontal position since the extra
weight exerts a torque of 2 ft-lbs on the wrist. The wrist needs to
exert a torque of 2 ft-lbs on the racquet to stop it rotating. At most,
a player might be able to support a 10 lb weight in this way, so the
maximum torque that can be exerted by the wrist is about 20 ft-lbs.
Holding a 10 lb weight like this would put a real strain on the wrist,
but a player might be able to exert a torque of this magnitude for a
split second without straining the wrist too much. In that case,
almost all of the torque needed to swing a racquet in a serve could
be provided by wrist action. The question is, does a player actually
use the wrist in this way or is sufficient torque provided just by
rotation of the arm and the body?
Watching a player doesn't answer the question. If the hand is
rotating about an axis through the wrist then that doesn't mean the
player is using the wrist to rotate the racquet. It might be the other
way around. The racquet might be rotating so fast that it causes
the hand to rotate. The forearm can also cause the hand to rotate.
If you relax your wrist, you can rotate your hand back and forth just
by shaking your forearm back and forth. Or you can use your wrist
to speed up motion of the hand or to stop your hand flapping
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around when you shake your forearm. If the forearm is swinging
rapidly and suddenly slows down then the hand and the racquet
will be flung forward. That is how a double pendulum works and it
is how players generate extra speed when swinging a racquet or a
golf club.
In theory, it should be possible to estimate the amount of positive
or negative wrist action in terms of measured rotation speeds of
the racquet and the forearm, but that is difficult to do. As far as I
know, nobody has ever done that. An alternative and simpler
approach is to examine the mechanics of swinging a racquet to
determine how much wrist action is actually needed.
5. LIMP VS LOCKED WRISTS
Figure 4 -- At the start of a racquet stroke the wrist can remain
limp as in the middle diagram or it can be locked as in the diagram
at the right.
Consider the start of a swing where the racquet and the forearm
are both at rest or almost at rest. A typical situation is shown in
Fig. 4 where the racquet and forearm are at right angles and
where the forearm starts to swing forward. Two possible results
are shown. One where the wrist remains limp, and the other where
the wrist is "locked" as if the racquet and forearm are bolted
together. If the wrist remains limp and acts as a simple hinge then
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the wrist acts only to pull the racquet forward and there is no
rotation of the racquet at the start of the swing. If the wrist is
locked then the racquet and the forearm both rotate together as a
single unit.
When the wrist remains limp there is only one force acting on the
racquet and it acts in line with the racquet since that is the
direction that the wrist is moving at the start of the swing. If the
forearm rotates in a circular arc then the wrist also rotates in a
circular arc so the force on the racquet will change direction and
then the racquet will start to rotate to follow the path of the wrist.
That is what happened to the wood dowel in the video at the start
of the swing. That sort of behavior can be seen in some forehand
strokes where the forearm rotates faster than the racquet at the
start of the swing. It doesn't mean that the wrist is completely limp.
It means that the wrist is not working hard enough to keep the
racquet locked at a fixed angle to the forearm. As seen below, both
Federer and Djokovic do that, so it is not necessarily a bad thing.
Federer and Djokovic "Floppy Wrist"
Choose Movie
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0:00 / 0:06
Movie Screen 4. -- Federer and Djokovic both display the "floppy
wrist" at the beginning of the forward motion of the swing.
(Note: Movies are best viewed frame-by-frame using keyboard
arrow keys or movie controls.)
(Reference: Courtesy of John Yandell's 'High Speed Stroke
Archives' at Tennisplayer.net).
6. TORQUES AND COUPLES
Figure 5 shows four different situations where forces can be
applied to the handle of a racquet to rotate the racquet. If the force
is applied in a direction along the handle, as in Fig. 5(a) then the
racquet will accelerate in a straight line in the direction of the force
but the racquet won't rotate. If the racquet is already rotating when
a force is applied along the handle in this manner, then the rotation
speed won't change since there is no torque on the racquet. In
order to apply a torque, the force needs to be directed at an angle
to the handle, as in Fig. 5(b). That is what happens soon after the
start of the swing when the wrist remains relaxed. The racquet
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starts to rotate because the wrist starts pulling the handle at an
angle to the racquet.
Figure 5 -- Forces used to rotate a racquet.
If the wrist alone is used to rotate a racquet, the hand exerts a
force along the whole length of the hand. The effect of the hand
can be regarded as two separate forces on the handle, as shown
in Fig. 5(c). Rotation of the hand exerts a downward force near the
first finger and an upward force near the little finger. The two
forces are almost equal and opposite but not quite. Two equal and
opposite forces acting like this are known as a couple. The
magnitude of the couple is given by C = Fd where F is the value of
each force and d is the distance between the two forces. The
torque acting on the racquet is equal to C. If each force is say 30
lb and d = 4 inch = 1/3 ft then the torque on the racquet is 10 ft-lbs.
More generally, a player will rotate a racquet by exerting a force at
an angle to the handle, as in Fig. 5(b), and also by deliberately
rotating the hand to apply a couple, as shown in Fig. 5(d). The
total torque is then FD + C where D is the perpendicular distance
from the center of mass (CM) of the racquet to the line of action of
the force. If the acceleration of the racquet is measured and the
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swing weight of the racquet is measured then the total torque can
be calculated, but the couple cannot be determined separately by
this calculation.
7. DOUBLE PENDULUM SERVE CALCULATIONS
The equations used to describe a double pendulum are too
complicated to describe in this article, but can be found in
American Journal of Physics 79, May 2011, 330-339 and
elsewhere. Solutions of these equations were obtained by
modelling the forearm as a uniform rod of mass 1.5 kg and length
0.3 m, swung by applying a couple C1 to the forearm. The racquet
was swung by applying a couple C2 to the racquet. Different swing
styles were modeled by varying the torques applied to the forearm
and the racquet.
Figure 6 -- Positions of the forearm and racquet at intervals of
0.01 s for a serve calculated for a forearm couple C1 = 44 ft-lb and
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for wrist couples (a) C2 = 15 ft-lb and (b) C2 = 20 ft-lb.
Figure 6 shows two calculations, each for a 300 gram racquet
swung by applying a constant forearm couple C1 = 44 ft-lb during
the whole serve. The two calculations correspond to two different,
constant wrist couples, either 15 ft-lb or 20 ft-lb. The elbow speed
was taken from the measured speed shown in Fig. 3. The racquet
and the forearm were both assumed to be at rest, not rotating, at
the start of the serve. The racquet shown in Fig. 6(a) swung
around into a vertical position after 0.094 seconds and was then
rotating at 875 rpm. In Fig. 6(b), the racquet swung around into a
vertical position after 0.095 seconds and was then rotating at 760
rpm.
At first sight, it might seem like serve (a) is better than serve (b)
since the racquet was swinging faster, despite the fact that the
wrist couple was actually smaller in (a) than (b). This is a very
surprising result. Less wrist action can actually generate a higher
serve speed. However, serve (a) is not good since the forearm is
inclined too far forward when the racquet arrives in a vertical
position. Serve (b) is better since the forearm is inclined only 10
degrees forward, as it was in the first video.
Figure 7 shows the rotation speed of the racquet and the forearm
for the two serves in Fig. 6. The serve with the smaller wrist couple
allowed the forearm to rotate at a higher speed before the speed
started to drop. The bigger drop in forearm speed then generated
a larger racquet speed. If a large wrist couple is applied to the
racquet early in the serve then it acts back on the forearm and
reduces the rotation speed of the forearm. The end result is that
the racquet doesn't rotate as fast, but the racquet lines up correctly
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when the racquet is just slightly in front of the elbow.
Figure 7 -- Rotation speeds of the forearm and racquet for the
serves shown in Fig. 6.
There are many different ways of applying wrist and elbow couples
in a serve. A large couple could be applied for the first part of the
serve and then it could drop to a smaller value or even zero for the
remainder of the serve. Or the elbow couple could decrease to
zero while the wrist couple increases with time. One such serve is
shown in Fig. 8, as an attempt to reproduce the serve in the video
film.
Figure 8(a) shows the position of the forearm and racquet at
intervals of 0.01 seconds. Figure 8(b) shows the rotation speed of
the arm and the racquet vs time and it also shows the couple C1
applied to the forearm and the couple C2 applied to the racquet.
The two couples were chosen to give reasonable agreement with
the serve in the video film. C1 was held constant at 44 ft-lbs for
0.05 seconds and was then allowed to decrease to zero. C2 was
allowed to decrease in time at the start of the serve so that the
racquet would remain locked at right angles to the forearm. The
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wrist was then allowed to relax until t = 0.035 s, at which time the
wrist couple was increased to 19 ft-lb.
For the calculations in Fig. 8, the elbow velocity was taken as that
in Fig. 3 (but starting at t = 0.05 s in Fig. 3), and the arm and
racquet were assumed to be rotating at 13 rad/s at the start of the
swing. The resulting serve is very similar to the video serve. The
racquet is correctly aligned at t = 0.073 s, is rotating at 880 rpm at
that time, and the forearm has slowed almost to a stop at the
impact time.
Figure 8 -- Calculations for serve shown in video. The rotation
speeds are shown in rpm. C1 is the couple applied to the forearm
by the elbow and C2 is the couple applied to the racquet by the
wrist. The elbow is relaxed when C1 = 0. The wrist is relaxed when
C2 = 0.
One thing is clear from Figs. 6 to 8, and that is the server must use
the wrist in an active manner. It is possible to relax the wrist briefly
in the middle of the serve action, as shown in Fig. 8, but the wrist
must be used near the end of the serve to apply a positive couple
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to the racquet. In that respect, a serve is different from a golf swing
where the golfer can relax the wrists after unlocking the wrists. On
the other hand, the elbow can relax near the end of a serve, as
shown in Fig. 8(b). By relaxing the elbow just before impacting the
ball, the forearm slows down more rapidly and transfers its
rotational energy to the racquet.
8. A TYPICAL FOREHAND
A forehand in tennis is usually struck with topspin by swinging the
racquet upward as well as forward. In order to apply the double
pendulum model, we need to ignore vertical motion of the racquet
and assume that both the forearm and the racquet swing in a
horizontal plane (or perhaps in some other plane inclined to the
horizontal). The vertical speed component of the outgoing ball is
relatively small unless the player attempts a topspin lob over the
opponent's head. Forward motion of the racquet usually
commences when the forearm is pointing in a direction
approximately toward the back fence and the racquet is about 50
degrees further around. Players don't always commence forward
motion of the racquet from that position, but these values are
typical.
In a tennis forehand, the ball is normally struck in front of the body
with the forearm extended forward while the racquet is aligned
nearly parallel to the net at the point of impact. That way, the
outgoing ball heads toward the net. This type of swing differs from
one in golf or baseball where the wrists remain locked with the
forearms at right angles to the club or the bat. Soon after the start
of a golf or baseball swing, the wrists relax and the striking
implement then rotates rapidly to align with the arms at the point of
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impact.
A realistic forehand can be modeled by assuming that the couples
C1 and C2 remain constant in time. If the wrist remains locked then
the forearm and the racquet swing at the same rotational speed at
the beginning of the swing, and C2 decreases with time. If C2
remains constant in time then the forearm swings faster than the
racquet at the beginning of the stroke, while the racquet swings
faster than the forearm toward the end of the stroke. Both types of
forehand are relatively common and can be viewed on YouTube
and at www.tennisplayer.net.
Calculations for a medium pace forehand are shown in Figs. 9 and
10 for a racquet of length 70 cm, mass 300 g, swing weight 310
kg.cm2 and with a balance point (center of mass) 35 cm from the
butt end of the handle. The mass of the forearm was taken as 1.5
kg, and the mass of the hand was taken as 0.5 kg. The racquet
was swung with C1 = 25 Nm and C2 = 2.5 Nm, resulting in good
alignment of the racquet at the nominal impact point.
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Figure 9 -- View from above showing the positions of the elbow,
wrist, forearm and racquet at 0.05 s intervals for a 300 g racquet
swung in a horizontal plane to impact the ball. The elbow swings
forward due to rotation about the shoulder and forward motion of
the player.
Figure 10 -- Angular velocities of the forearm and the racquet for
the forehand shown in Fig. 9. After 0.2 seconds, the racquet
speeds up rapidly and the forearm slows down.
The angular velocities of the forearm and the racquet are shown in
Fig. 10. Despite the fact that C2 was held constant, rotational
energy transferred from the arm to the racquet during the swing,
with the result that 1 decreased to a minimum and 2 increased
to a maximum near the end of the swing. Acceleration and
deceleration of the elbow simulates the effect of a triple pendulum
whereby rotational energy in the upper arm is transferred to the
forearm in an analogous manner. In an efficient tennis (or golf or
baseball) swing, energy is first transferred from the upper arm to
the forearm and is subsequently transferred from the forearm to
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the racquet after a short time delay.
9. CONCLUSION
In this article we have just scratched the surface in terms of
analyzing different tennis strokes and in understanding the subtle
interactions between the player and the racquet. We did not
consider what might happen if racquet properties are changed and
we considered only one forehand stroke. Nevertheless, we have
shown that tennis strokes have many similarities to a mechanical
double pendulum and have explained how motion of the forearm
affects the motion of the racquet, and vice versa, at least in simple
situations where the arm and racquet rotate in the same plane.
Slow motion video is a powerful tool in analyzing player strokes. It
also needs to be combined with a clear understanding of what is
being observed and what is happening if players and coaches are
to reap maximum benefits by using this tool.
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