🎾 Vợt Control - Part 2¶
Giới Thiệu¶
Vợt Control - Part 2 — tài liệu 28 trang từ thư viện sách tennis.
Chủ đề chính: Racquet
Tóm tắt nội dung (trích từ tài liệu gốc): Racquet Contribution To Shot Control about:reader?url=http://twu.tennis-warehouse.com/learning_center/contr... twu.tennis-warehouse.com Racquet Contribution To Shot Control 26-33 ph�t 1. Introduction What does it mean to say that a racquet has great control? How much does the racquet, all by itself, independent of the player, influence the control over one's shot? To determine this, we built a mechanical racquet holder that allowed the racquet to twist and rotate upon impact. Balls were fired from a ball machine with zero spin at 40 mph (17.9 m/s) and perpendicular to the strings in all 3 dime
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Racquet Contribution To Shot Control
26-33 ph�t
1. Introduction
What does it mean to say that a racquet has great control? How
much does the racquet, all by itself, independent of the player,
influence the control over one's shot? To determine this, we built a
mechanical racquet holder that allowed the racquet to twist and
rotate upon impact. Balls were fired from a ball machine with zero
spin at 40 mph (17.9 m/s) and perpendicular to the strings in all 3
dimensions (x, y, and z). But given the inherent inconsistency of
the ball machine, there were small variations in these parameters.
Shots were fired to impact above and below the long longitudinal
axis near the tip, throat, and center of the stringbed. The racquet
was stationary prior to all impacts to also eliminate the influence of
the speed, angle, and tilt of the swing.
Normally the incident speed, spin, and angle dominate the
rebound characteristics and drown out and hide any contribution of
the racquet. The goal of this procedure was to eliminate the
incident determinants as much as possible so we could see the
racquet's influence in isolation. But even on perfectly perpendicular
bounces, there is also significant randomness that must be
factored in. A ball dropped straight onto the floor will bounce in
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unpredictable directions and spins. That is due to both small
variations in the surface and because balls are not perfectly
spherical.
Our intuition tells us that when the ball hits above or below the
long axis, the racquet will twist backwards, sending the ball up or
down. Similarly, as the impact location varies along the length of
the racquet, then the racquet will rotate backwards about the wrist,
sending the ball off in the direction of the tip. And it is also
generally intuited that the more mass at the hitting location, the
less the racquet will twist and rotate and the shot will be closer to
its intended path. Are our intuitions correct?
2. Experimental Setup
Three cameras were used to gather data -- one at the side, top,
and behind the racquet. The side camera was used to analyze the
vertical (upward/downward) launch angles, spin
(topspin/backspin), and velocity (perpendicular and vertical
tangential). The top camera recorded the horizontal (side-to-side)
angles, sidespin, and velocity (perpendicular and horizontal
tangential). The back camera was used to determine the precise
impact locations. The impact coordinates were indicated as
distance in cm from the x axis (along the longitudinal racquet axis)
and the perpendicular y axis. The origin of the back camera
coordinate axis was at the tip of the racquet. For the top and side
cameras, the x axis was perpendicular to the stringbed and the y
axis tangential to the stringbed. The coordinates for the top and
side cameras were calibrated in the z direction in order to correct
for apparent ball distance and velocity. Figure 1 shows the three
camera views.
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a. Back Camera
b. Side Camera
c. Top Camera
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Figure 1 -- Back camera (a), side camera (b), and top camera (c).
Nominal topspin, backspin and sidespin = 0 and nominal vertical
and horizontal impact angles = 90 degrees. Videos shot at 300 fps.
The back camera measured impact location, the side measured
top/back spin and vertical incident and rebound angles, and the
top measured sidespin and horizontal angles.
Upon impact, the racquet twists (Figure 2) and rotates (Figure 3).
In our experiment "Control Part 1" we showed that for any given
incident speed, spin, and angle, a greater effective mass at the
impact point would limit the twisting and rotation caused by those
inputs and thus also limit the effect on the outgoing rebound. In
this experiment we have a slightly different interest -- do the
properties of the racquet set a baseline rebound speed, angle, and
spin independent of the inputs, and if so, is the magnitude of these
dependent on the impact location? In other words, does each
impact location have a built-in bias as to the speed, spin, and
direction of the rebound?
Figure 2 -- Twisting caused by impacts above (left) and below
(right) the long axis (view from the side camera).
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Figure 3 -- Backward rotation caused by all impacts (view from
the top camera). Rotation occurs about an axis at the end of the
handle.
3. Results
3.1 Vertical Rebound Results
The actual incident parameters differed slightly from nominal and
they also differed slightly by location (whether the impact was
above or below the long axis). The statistics are presented in Table
1.
Table 1
Actual Incident Variables
Incident Variable Impact Mean Standard
Location Deviation
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Table 1
Actual Incident Variables
Incident Variable Impact Mean Standard
Location Deviation
1 (angle degrees) Above long 0.61 � 0.65
axis
1 (angle degrees) Below long -0.23 � 0.5
axis
1 (angle degrees) All impacts 0.1 � 0.7
1 (spin rpm) Above long 119 � 126
1 (spin rpm) axis 88 � 73
1 (spin rpm) 101 � 99
Below long
axis
All impacts
vx1 (perpendicular Above long 18.9 � 0.6
speed m/s) axis
vx1 (perpendicular Below long 18.6 � 0.28
speed m/s) axis
vx1 (perpendicular All impacts 18.5 � 0.2.2
speed m/s)
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Table 1
Actual Incident Variables
Incident Variable Impact Mean Standard
Location Deviation
vy1 (tangential speed Above long 0.2 � 0.21
m/s) axis
vy1 (tangential speed Below long -0.08 � 0.16
m/s) axis
vy1 (tangential speed All impacts 0.036 � 0.23
m/s)
R1 (rotational speed Above long 0.42 � 0.44
0.31 � 0.25
m/s) axis 0.35 � 0.34
R1 (rotational speed Below long
m/s) axis
R1 (rotational speed All impacts
m/s)
Vc (contact velocity Above long 0.62 � 0.53
m/s) axis
Vc (contact velocity Below long 0.23 � 0.34
m/s) axis
Vc (contact velocity All impacts 0.39 � 0.47
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Table 1
Actual Incident Variables
Incident Variable Impact Mean Standard
m/s) Location Deviation
The resulting non-nominal impact geometry is shown in Figure 4.
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Figure 4 -- Incident and rebound geometry for impacts in the
vertical plane. The actual incident parameters varied slightly from
nominal. Above the long axis the average incident angle was
slightly upward (0.59�) and below the long axis it was slightly
downward (-0.24�).
These incident variances from nominal shown in Table 1 are small,
but we may be looking for small effects that can get lost in the
jumble. The most significant variable in Table 1 that complicates
our study is contact velocity (Vc) because it affects both rebound
angle and spin. Contact velocity is the sum of the rotational and
linear speeds tangent to the strings -- Vc = R1 + vy1, where R is
the radius of the ball (0.033 m), 1 the angular velocity, and vy1
the tangential speed to the strings. The direction and magnitude of
Vc determine the direction and duration of the friction force. Vc was
positive (upward) for all impacts. That is because the positive,
counter-clockwise R1 was so much greater than vy1 in all cases,
so Vc was positive no matter the incident angle. Thus, for example,
the incident ball could be angled downward, but the contact
velocity would still be upward. Friction acts opposite the direction
of Vc, so it acted downward for all impacts, regardless of incident
angle. Figure 5 shows this.
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Figure 5 -- The direction and magnitude of Vc determine the
direction and duration of the friction force (F). Vc is the sum of
tangential component of the incident velocity (vy1) and the
rotational speed (R1). R1 was larger than vy1 in all cases,
indicated by the longer vector.
Although Vc was small, it was not zero. If any inherent speed, spin
and angle biases exist in the racquet, they will interact with the
incident parameters to affect their results. In theory, given the
same incident variables but with opposite signs, one would expect
the top and bottom halves of the racquet to produce rebounds with
speed, angle, and spin having the same magnitudes but with
opposite signs. We would expect that to be the built-in rebound
bias of the racquet. So, if the rebound in the bottom half is
downward, the rebound in the top half would be upward. If the
incident variables are not zero at each impact location, and in fact,
a bit different at each location, it becomes more difficult to decipher
inherent bias from incident causality.
Figure 6 shows the location of all 148 impacts. The red dots
rebounded downward (negative) from the horizontal and the blue
dots upward (positive). Fewer than 20 shots rebounded positively.
Their appearance and properties do not correlate strongly with any
incident variable or impact location and may be, in a large part,
random.
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Figure 6 -- Impact locations color coded to show downward (red)
and upward (blue) vertical rebounds as seen from the side camera
(Figure 1b).
Rebound angle -- Figure 6 tells us that the rebound direction (off a
stationary racquet) was predominately downward for virtually all
impact locations. This does not fit our ideal of the top-half and
bottom-half rebounds being symmetrical but opposite versions of
each other. Why not? Part of the reason is the incident variables
were not symmetrical and opposite either, and consequently, nor
were Vc and friction. Friction acted downward in almost all
impacts, so it acted in the opposite direction of the racquet twist in
the top and in the same direction as the twist in the bottom. In the
bottom half of the racquet, Vc was smaller because its component
linear and rotational speeds were in opposite directions canceling
each other, and thus the duration of friction was less. In the top
half, linear and rotational speeds were in the same direction and
added together, so Vc was larger and friction lasted longer.
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Therefore, you might think the change in the rebound angle during
impact would not only be downward but also be greater in the top
half than in the bottom half of the racquet. Just the opposite is true.
The average change in angle direction during impact is 1.32� for
the top and 5.3� for the bottom (change in angle = 2-1).
Figure 7 shows this difference. Change in angle is used instead of
the resulting angle (2) because it shows the net resulting direction
and magnitude of all forces acting tangent to the racquet face,
regardless of what the absolute value of that change might be (i.e.,
positive or negative).
Figure 7 -- The change in angle during impact is greater on the
bottom half (red dots) than on the top half (blue dots).
This highlights how racquet biases affect the rebound result -- In
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the top half, friction lasted longer but the upward-facing twist acted
to push the ball in the opposite direction of friction, decreasing the
downward result. In the bottom half, friction duration was shorter
but the downward-facing twist acted in the same direction as
friction, increasing the downward result. So in the top, the biases
tended to decrease the incident result and in the bottom they tend
to augment it, making it appear that the rebounds in the two halves
were more dissimilar than they are.
If the incident parameters are reasonably close to nominal or if
they are equal but opposite in values, and if we assume the biases
will be the same but opposite in each half, then the biases can be
approximately calculated. When the racquet twists backward in the
top half, the rebound angle will be influenced upward. On the
bottom it will be influenced downward. The amount of influence will
not be by the full amount of the twist angle but by the average of
the angle during impact. We will call that amount the impact bias,
. will be the same magnitude but opposite sign on each half of
the racquet and it will occur during every impact.
The change in angle in each half of the racquet during impact is
given by
(1a) t = 2t - 1t
and
(1b) b = 2b - 1b
where 2 is the rebound angle from perpendicular and 1 the
incident angle, and the subscripts t and b are top and bottom
respectively. Adding the bias angle to the incident angle gives us
what we might call the "effective impact angle." Adding to
equations 1a and 1b gives us our effective change in angle during
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impact.
(2a) et = 2t - 1t + t
and eb = 2b - 1b + b
(2b)
where et and eb are the effective angle changes for the top
and bottom, and t is negative and b is positive but of equal
magnitude. If we set eq. 2a and 2b equal and solve for , we get
(3) = (t - b) / 2
For example, using the average t of 1.32� and b of 5.3 we get
= 1.32 - 5.3 /2 = -1.99�. So, for the top, the total effective change
in angle will be 1.32 + 1.99 = 3.31 and for the bottom it will be 5.3 -
1.99 = 3.31.
In other words, for the top half, friction undoes the 1.99 backwards
degrees and adds another 1.32 degrees for a total effective
change in angle due to friction of 3.31�. And for the bottom half, we
subtract the added 1.99� bias angle first and then add the 5.3� due
to friction for a total effective change in angle of 3.31�. (Note: if we
change our reference frame to "upward/downward" terminology,
we would take the negative of these values if the direction of the
angle is downward, as it is in most of the impacts in this
experiment.)
That result is the average result for all impacts in the top and
bottom halves of the racquet, no matter their location. In reality the
magnitude of will change depending on the distance of the
impact location from the longitudinal axis. So, we grouped impacts
into 1 cm wide intervals on either side of the long axis and took the
average of for each group. The result for average in each
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interval is displayed in Figure 8.
Figure 8 -- Average effective impact angle bias over 1 cm
intervals from 1 to 8 cm from longitudinal axis. These values are
plus/minus values. On the top the bias would be negative and on
the bottom positive.
This analysis demonstrates how, at first glance, the results of
Figure 6 disobeyed the theoretical and experimental expectations
that the rebounds on opposite halves of the racquet are the same
but opposite reflections of each other when the incident
parameters are also the same and opposite. This inherent bias is
hidden but always contributing to the results of every impact. This
bias is dependent on the impact location (as shown in Figure 8).
We will encounter this location dependency of rebound angle in
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more detail in the discussion below concerning rebound speed.
The two are intertwined.
Rebound speed -- Rebound speed is determined by the racquet's
apparent coefficient of restitution (ACOR or "power potential").
ACOR is defined as the ratio of the rebound speed perpendicular
to the stringbed (vx2) to the incident perpendicular speed (vx1)
(Equation 4):
(4) ACOR = eA = vx2 / vx1
where eA is the symbol for ACOR used in formulas. Figure 9
illustrates these relationships.
Figure 9 -- ACOR is the ratio of vx2 and vx1.
16 trong 28 ACOR is usually about 0.5 in the center of the throat area, 0.4 in
the center of the stringbed and about 0.1-0.2 at the tip and on the
periphery. ACOR is effectively independent of incident velocity
over the range of typical tennis shots. When inserted into the
following formula, it is used to predict shot speeds:
(5) vx2 = eA(vx1 + Vx1) + Vx1
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where Vx1 is the perpendicular speed of the racquet. In our
experiment, the speed of the racquet is 0, so the formula reduces
to vx2 = eA(vx1). So, for example, if the incident velocity is 40 mph
and hits a stationary racquet in the middle of the strings with
ACOR = 0.4, then the out-going rebound will be 16 mph (16 = 0.4
x 48).
Figure 10 is a plot of ACOR vs impact distance from the long
central axis and color-coded by distance from the racquet butt. It is
apparent that ACOR depends on both the lengthwise and
widthwise impact location.
Figure 10 -- ACOR vs x and y axis impact locations. Red dots are
impacts at throat, blue dots in middle between 3 and 9 o'clock, and
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green dots at tip.
Rebound speed depends on location, and since the rebound angle
depends on the speed, it too will will depend on impact location.
Figure 11 shows why. The rebound angle (2) is the angle between
the horizontal (vx2) and vertical (vy2) components of the rebound
velocity (v2). vx2 will almost always be much larger than vy2 and
will therefore be most important in determining the rebound angle
(2). You can see from Figure 12 that if vy2 stayed nearly the same
length and vx2 were shorter, then 2 would increase. So, a smaller
ACOR will mean a larger rebound angle from perpendicular and a
larger ACOR will mean a smaller angle. If ACOR = 0, then the ball
would drop vertically at 90 degrees. So, in this angular respect,
more power means more control.
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Figure 11 -- Rebound angle (2) is the angle between the
horizontal (vx2) and vertical (vy2) components of the rebound
velocity (v2).
Figure 12 shows the dependency of the change in vertical rebound
angle on impact location (the change in angle was first graphed
against ACOR and then the fitted curve was graphed against
distance from long axis). On both the top and bottom half of the
racquet the downward change in rebound angle increases as
impact moves toward the periphery. It also shows that for impacts
at any given distance from the center axis, the change in rebound
angle will be greater for impacts further from the butt end of the
racquet.
Figure 12 -- Change in Rebound angle vs x and y axis impact
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locations. Red dots are impacts at throat, blue dots in middle
between 3 and 9 o'clock, and green dots at tip. Negative change in
angle refers to the down direction.
Rebound spin -- The effect of impact location on spin is a bit more
difficult to assess. Measured over all the impacts, the incident ball
had a counter-clockwise spin of 101 � 99 rpm (counter-clockwise
spin is positive and clockwise spin is negative). This meant that
the relative motion at the contact point to the strings was always
upward and thus there was a friction force downward for both
upper and lower halves of the racquet. Nonetheless, the results on
each half are different. As Figure 13 shows, on the bottom half of
the racquet, the further you go from the center axis, the more the
change in spin is counter-clockwise (less negative, more positive).
On the top half, the further from the center, the more the change in
spin is clockwise (less positive, more negative). Each half has a
built in spin bias. In general terms, the bottom half is topspin
biased and the top half is backspin biased. These biases increase
from the center to the periphery of the stringbed.
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Figure 13 -- Change in spin vs impact distance from long axis. On
the bottom half of the racquet (red dots), the further you go from
the center axis, the more the change in spin is counter-clockwise
(topspin here). On the top half (blue dots), the further from the
center, the more the change in spin is clockwise (backspin here).
It seems clear that for the vertical rebound a tennis racquet is
biased by impact location for speed, angle, and spin even though it
is sometimes not obvious.
3.2 Horizontal Rebound Results
As in the vertical plane, the incident angles varied from nominal in
the horizontal plane. Table 2 shows the actual incident parameters.
Table 2
Actual Incident Horizontal Variables
Incident Variable Impact Mean Standard
Location Deviation
1 (angle degrees) Throat half -0.93 � 1.0
1 (angle degrees) Tip half 0.3 � 0.52
1 (angle degrees) All impacts -0.09 � 0.92
1 (spin rpm) Throat half 136 � 51
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Table 2
Actual Incident Horizontal Variables
Incident Variable Impact Mean Standard
Location Deviation
1 (spin rpm) Tip half 139 � 43
1 (spin rpm) All impacts 138 � 45
vx1 (perpendicular Throat half -18.5 � 0.33
speed m/s)
Tip half -18.6 � 3.8
vx1 (perpendicular
speed m/s) All impacts -18.6 � 3.1
vx1 (perpendicular
speed m/s)
vy1 (tangential speed Throat half 0.3 � 0.33
m/s) Tip half -0.11 � 0.17
All impacts 0.024 � 0.3
vy1 (tangential speed
m/s)
vy1 (tangential speed
m/s)
R1 (rotational speed Throat half 0.47 � 0.17
m/s)
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Table 2
Actual Incident Horizontal Variables
Incident Variable Impact Mean Standard
Location Deviation
R1 (rotational speed Tip half 0.48 � 0.15
m/s)
R1 (rotational speed All impacts 0.47 � 0.16
m/s)
Vc (contact velocity Throat half 0.77 � 0.4
m/s)
Vc (contact velocity Tip half 0.37 � 0.23
m/s)
Vc (contact velocity All impacts 0.5 � 0.35
m/s)
Horizontal rebound angle -- Given these near nominal average
incident parameters (more so in the tip than throat half), the
geometry of the rebounds from each half of the racquet are
presented in Figure 14.
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Figure 14 -- Horizontal rebound as seen from above for impacts
near the throat and tip. The throat-half was considered to be 40-50
cm from the butt and the tip-half greater than 50 cm.
For impacts near the throat, vy1 and 1 are in the same direction
at the contact point, adding together as a contact velocity toward
the tip (R� = tangential rotational velocity, where R is the radius
of the ball). The friction force acts opposite towards the throat. This
force acts to reverse the spin and to slow the ball parallel to the
strings, causing the rebound angle (2) to be closer to the
perpendicular than was the incident angle (1). Yet, in spite of this
force, the bounce angle is actually greater than the incident angle
(in most cases). Some other factor is influencing the bounce
direction besides the friction force. Intuition suggests that factor is
the backward rotation of the racquet launching the ball at a slightly
greater angle.
For impacts near the tip, vy1 and R� are in opposite directions.
vy1 is toward the throat and R�1 toward the tip. The net contact
velocity (Vc = R� + vy1) is toward the tip because the velocity of
the sidespin is greater than the parallel velocity to the strings of the
incident ball. Friction acts toward the throat, slowing and/or
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reversing the spin and increasing the rebound angle away from the
perpendicular. If the rotation of the racquet backward has any
effect on rebound direction, it should be in the opposite direction of
the actual rebound toward the tip.
Figure 15 shows the same 148 impacts as in Figure 6, but this
time the color-coding is blue for rebounds to the right (in the
direction of the tip) and red for rebounds to the left (in the direction
of the throat).
25 trong 28 Figure 15 -- Impact locations color coded to show left (red) and
right (blue) rebounds as seen from the top camera (as seen in
Figure 1c).
All rebounds are directed to the center, no matter the impact
location. This is not too surprising given that in both halves of the
racquet the ball is incident toward the center, although by less than
1� on average. But what of the intuition that because the racquet
rotates horizontally backwards about an axis at the end of the butt,
we would expect all impacts to have a rebound bias in the
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direction of the tip (direction of the rotation), not to the center in
one half and toward the tip in the other? Nor would we expect the
rebound angles to be greater than the incident angles because
friction acts to slow the ball and cause a steeper rebound -- i.e.,
closer to the perpendicular.
The degree to which the rebounds are directed to center depends
on the impact distance from the longitudinal axis. This is shown in
Figure 16. This graph looks a lot like Figure 10 -- the angle
depending both on the distance from the center axis and butt axis.
Figure 16 -- Horizontal rebound angle vs distance from the
longitudinal axis.
The reason Figure 16 looks like Figure 10 is that both the
horizontal and vertical rebound angles share a leg -- the
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perpendicular vx2 component of the rebound velocity v2. And
because that component leg is dependent on distance (y) from the
long axis, both horizontal and vertical rebounds are also
dependent on y.
This is all somewhat puzzling. If there is indeed bias in horizontal
rebound angle in the direction of the tip, then this bias should
affect all impacts independent of impact location. Why, then, do
impacts in the throat rebound as if this bias exists and impacts in
the tip act in the opposite direction to that bias? And why, in the
throat half, does this suposed bias appear even though friction is
pushing in the opposite direction? It is true that, depending on the
direction, magnitude, and duration of the friction force, it can
supercede the action of the bias, as it did with the vertical
rebounds. So how do we coax out a horizontal bias if it exists?
With the present data set, we were unable to satisfactorily explain
these results. Several candidate theories present themsleves: that
our intuition about tip-directed rebounds is wrong and perhaps
even upside-down, that there is friction reversal, that a "drum
effect" (bounces toward the middle from all locations on racquet) is
involved, and that there is an "edge effect" involved (variations in
stiffness of adjacent strings). Exploration of these possibilites will
have to be left to "Control: Part 3."
5. Conclusion
It seems clear that the properties of each racquet set some sort of
innate behavior in terms of speed, spin, and angle. These
behaviors, in turn, are dependent on impact location. This inherent
behavior was easier to discern when examining rebounds in the
vertical plane. That is because the top and bottom halves of the
27 trong 28 2:50 CH, 07/03/2021
Racquet Contribution To Shot Control about:reader?url=http://twu.tennis-warehouse.com/learning_center/contr...
racquet behave the same but with opposite signs. In the horizontal
plane it is different. The hypothesized rotational bias of rebound
angle being in the direction of the tip and increasing as impacts
are closer to the tip did not materialize. Further studies must be
done on horizontal plane rebounds from a racquet.
28 trong 28 2:50 CH, 07/03/2021