Note : Les descriptions sont présentées dans la langue officielle dans laquelle elles ont été soumises.
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TITLE
STRING LOAD APPORTIONED RACKET
~ACKGROUND OF THE IN~ENTION
This invention involves several discoveries
reached by experience, experimentation, and supportive
analysis to improve significantly on the string network
of tennis rackets, racquetball rackets, and other sport
rackets. The effort generally is directed toward deter-
mining optimum string parameters and arrangements to make
a string network that is more effective, efFicient, and
responsive in applying hitting force to a ball.
The invention not only recognizes that longer
strings have important advantages, but it recognizes why
longer strings work better and how they can be arranged
to produce improved results. I:t includes several sug-
gesti:ons for extending longitudinal strings into the throat
or shank region of a racket to have a substantially longer
strung length, and it proposes several different arrange-
ments for fanning out, guiding, and anchoring longer
longitudinal strings
The invention also recognizes that longer longi-
tudinal strings should be strung with a higher tension
than shorter transverse strings, and the invention determines
both the reasons for and the extent of the higher tension
for the longer strings to achieve a significantly better
working relationship within the string network.
Investigation of the dynamics of string mechanics
by using experimentation, mathematical analysis, and play
experience has produced considerable verified information
3Q on truly effective lengths and tensions for the longitudinal
and transverse strings to work effectively together. The
information reveals that the necessarily shorter but
equally tensioned transverse strings in prior art rackets
bear much more than half of the load in hitting a ball.
35 This not only wastes the superior capacity of the longi-
tudinal strings to bear the ball-hitting load, but also
1 ~7~
contributes to twisting torque and shock delivered to
the player's arm from shots hit off center.
The invention not only recognizes the advantages
of longer long;tudinal strings strung at higher tension
than the shorter trans-vers~e stringsl ~ut also quantifies
an approximate functioning relationship that balances
the greater tension and length of the longitudinal strings
with the lesser tension and length of the transverse strings
effectively to apportion more of the ball~hitting load
to the longitudinal strings, This gives a string network
a higher coefficient of restitution imparting a higher
velocity to a rebounding ball, spreads the higher coeffi-
cient of restitution throughout a wider network area,
reduces losses from stretching and rubbing strings and
1~ deforming the ball, and lessens torque shock to the arm
of the player. Tennis rackets strung according to the
invention have been made, tested, and used in play to
verify measureable clata, confirm analysis, and establish
subjectively that the invention produces better controll
higher velocity returns, and a lively and shock-free feel
in shot making.
SUMMARY OF THE INVENTION
My discoveries about functionally interrelating
string lengths and tensions for tuning string networks
to improved performance in hitting balls applies to tennis
and other sports rackets. These have a hand grip joined
to a frame supporting a string network that extends through-
out a ball-hitting region spaced from the grip, and the
frame has a shank region extending from the grip and
flaring outward in a throat region and extending around
a generally oval ball-hitting region spanned by transverse
and longitudinal strings.
I have found that at least a central plurality
of the longitudinal strings, and preferably all the longi-
tudinal strings, should have a strung length at least30% longer than the transverse strings. A preferred way
of accomplishing this is to extend the longitudinal strings
into the throat or shank region of the frame and possibly
~lt7~
as far as the region of the grip. These longer longitudinal
strings can either fan outward across the ball-hitting
region or be approximately parallel in the ball-hitting
region and ~uided in the throat region to angle toward
the shank region.
I have also ~ound that the longer longitudinal
strings should be strung with at least 30~ more tension
than the trans~verse strings. This not only tunes ~he
longer and shorter stri-ngs to operate harmoniously, but
it also converts more of the ball-~itting force to initial
string tension and reduces losses that occur from ball
deformation, string stretchin~, and interstring friction.
I have also discovered an important functional
relationship between the l~nger strung length and greater
tension of the longer longitudinal strings and the shorter
strung length and lesser tension of the transYerse strings.
String lengths and tensions selected according to this
relationship effectively apportion to the longer longi-
tudinal strings from approximately half to substantially
more than half of the string force that decelerates a
ball penetrating the string network as the ball is hit.
I`n other words, the greater length and tension of the
longitudinal strings is selected relative to the lesser
length and tension of the transverse strings to place
nearly half or considerably more than half of the ball-
hitting load on the longitudinal strings in contrast to
prior art rackets that place substantially more than half
of the ball-hitting load on the transverse strings.
This relationship significantly improves over
a conventional string network in several ways. The longer
longitudinal strings bear more of the load in hitting
a ball and have a greater influence on the shot; and
since the longer longitudinal strings have a greater
- capacity to store and return energy to the ball than the
shorter transverse strings 3 this alone produces consider-
able improvement. Longer strings stretch less than shorter
strings in deforming as a ball penetrates the string network
so that longer strings lose less energy in string stretching
~7
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and interstring friction. The higher strung tension of
the longer strings also provides more of the ball resisting
force and further reduces the need for string stretching.
The longer, tauter strings stop a ball with less force
and more deformation to reduce ball deformation and the
energy loss that entails. Moreover, longitudinal strings
anchored nearer the longitudinal axis of the racket
are geometrically more suited to bearing the ball-hitting
load than the transverse strings anchored at the sides
of the frame and transmitting more twisting shock to the
pla~yer from off center hits. Advantages related to these
include a more responsive sweet spot area, a higher coef-
~icient of restitution of the string network3 more control
and velocity for shots, and less vibration.
DRAWINGS
Figures l and 2 are respective plan view~ of
alternative preferred embodiments of rackets strung
according to my invention;
Figures 3-5 are respective plan~ side elevation,
and end elevation views of a schematic racket model for
analyzing string networks according to my invention;
Figures 6 and 7 are graphic diagrams of string
forces involved in hitting a ball respectively with a
prior art racket and with a racket strung according to
my invention; and
Figures 8 and 9 are scale schematic diagrams
comparing experimentally determined coefficient of restitu-
tion areas using representative frames strung according
to my invention on the left and according to the prior
art on the right.
DETAILED DESCRIPTION
GENERALLY
Most of the recent improvements in tennis rackets
have involved frame and racket structure, rather than
string network. Considerable work has been done on the
size and location of the sweet spot, more properly called
the center of percussion, where the impact of the ball
is least felt by the player. This is affected by the
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geometry, shape, size, rigidity, and weight distribution
of the frame, including the throat and handle, and only
to a lesser extent by string tensions and lengths.
Except for a few changes in string network size,
string materials 7 and varia61y spaced strings, string
networks have not been varied. The present state of the
art of racket making universally applies the same tension
to transverse strings and longitudinal strings, even though
racket frames provide a generally oval ball-hitting region
so that longitudinal strings have a longer average length
than transverse strings.
RACKET AND BALL MECHANrCS
Unders~anding the invention requires a general
understanding of racket and ball mechanics. When
1~ a ball and string network collide, the kinetic energy
carried by the ball due to its velocity relative to the
racket is divided into three ~arts. The first part is
spent on hending the frame, the second is consumed in
flattening the ball, and the third is spent on penetrating
the string network, which increases the string tension
and dents the net. Among the three parts, the energy
spent on bending the frame is almost a total loss. The
ball contacts the network for only two to three thousandths
of a second, and the frame is still bent when the ball
rebounds from the network; so that energy stored in the
bent frame is not returned quickly enbugh to add to the
rebound of the ball The energy spent in deforming the
ball, due to the final impact force between the network
and the ball is at least partially lost, because the ball
is still partially deformed when it rebounds from the
string network so that some of the energy spent in de-
forming the ball is not recovered in rebounding.
The energy losses from frame bending and ball
deforming can be seen clearly from high speed photographs
and are generally recognized as an unfavorable part of
racket mechanics Improvements in tennis balls to retain
a high internal pressure and use of high strength materials
such as composite, metal, and boron reinforced synthetics
--6--
to make racket frames light but rigid are both efforts
to reduce these losses of dynamic energy.
The third part involving the energ~ stored in
the string network as the ball penetrates it on impact
5 and the reaction of the string network in returning kinetic
energy to the rebounding ball is known to be important;
and different string materials and tensions have explored
this. However, apart from a few suggestions that were
never adopted in the art, string networks have been limited
10 to the oval ball-hitting region and have used transverse
and longitudinal strings arranged at right angles to each
other, ~ormed of t~e same material, and strung with the
same tension.
STRING MECHANICS
The tension that develops in a string on impact
with the ball consists of two components--an initial strung
tension To and an additional tension AE(x/Ll)2 from stretch-
ing or elongating the string, where A is the string's
cross sectional area and E is its Young's modulus, x is
20 the ball penetration, and Ll is the half length of the
string.
These two components combine to form a retarding
force that resists the advance of the ball while storing
up the diminishing kinetic energy of the ball. A differ-
25 ential equation describing this dynamic e~uilibrium takenfrom Timoshenko, "Vibration Problems in Engineering",
D. VanNostrand Co., New York, p. 116, is:
F = m ~ = _ ~1OXL ~ AE(x/Ll) ] (1)
where F is the force acting on the ball from the string,
and the negative sign indicates that it is a decelerating
force.
It is important to recognize that the initial
35 tension To term is much larger than the stretching term
and is linearly proportional to the ball penetration
distance x, Initial string tension thus acts much like
~ ~'7~ ~ ~ 8
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a linear spring in receiving and storing the kinetic energy
of the ball. The stretching term AE is small since it
is proportional to the cube of x/Ll which is very small
when ball penetration x is small. However, when the
relative speed of the ball is high and its penetration
is large, the stretching term AE becomes increasingly
significant.
My invention recognizes the fact that a longer
string with a large Ll reduces the influence of the stret-
ching term AE and indirectly increases the contributionof initial tension To~ both of which benefit the perform-
ance of the network. Repeated stretching and unstretching
of a string cause hysteresis loss from molecular friction
within the string, and string stretching also causes
rubbing, wear, and ~riction loss as strings move against
one another. This suggests that the stretching term AE
sh~uld be kept as small as possible, and that long strings
are the best way to achieve this.
~hen string length increases, the initial strung
2~ tension To should also be increased so that the To/Ll
term is not reduced. This results in a longer, tauter
string with a high tension resistance to penetration of
a hall and a much smaller portion of ball resistance
derived from string stretching. Also, from the vibrational
point of vie~, string tension should increase proportion-
all~ with increases in string length so both strings
vibrate at the same frequency.
Since the length of the transverse or cross
strings is limited by the width of the racket frame, only
the longitudinal strings can be made longer to take advan-
tage of higher tension resistance. Longer longitudinal
strings can be extended into the throat, shank, and even
into the handle to provide a substantially longer strung
length than the transverse strings.
My previous applications suggest several anchor-
age and guidance arrangements for extending longitudinal
~trings into the shank or grip region of a racket, and
many other possibilities are probably workable. The two
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most preferred arrangements are schematically shown in
FIGS. 1 and 2.
Longitudinal strings 15 of a preferred racket
10 of FIG. 1 fan outward across the ball-hitting region
3 from an anchorage 16 in shank 11. Anchorage 16 can be
positioned anywhere from throat 12 to grip 13, depending
on the length and tension desired for strings 15.
The other preferred racket 20 of FIG. 2 has
longitudinal strings 25 that either extend axially parallel
10~ or diverge slightly across the ball-hitting region from
a throat piece guide 22 having guide elements 24 that
angle the strings between their anchorage 26 in shank
21 and their course across the ball~hitting region.
Again) anchorage 26 can 6e positioned along shank 21 or
3 Wi thin grip 23.
The embodiment of FIG. 2 looks more conventional
and might be better received, but its throat guide 22
produces some friction loss~ The embodiment of FIG. 1
is preferred not only far reducing friction, but for the
~3 additional advantage of reducing twisting torque from
off center hits. Throat guide 22 can also provide an
anchorage for longitudinal strings extending somewhat
deeper into the throat region than is ordinary. The
tendency of different string lengths and tensions to
;~5 produce a desired performance is explained more fully
below.
Both the embodiments of FIGS. 1 and 2 arrange
the longer longitudinal strings 15 or 25 to bear more
of the ball-hitting load than the transverse strings 17
or 27, and thus reduce the twisting torque from off center
- hits. But the fan out arrangement of FIG. 1 spaces the
longitudinal strings closer together in the central region
where most balls are hit and disposes strings 15 within
a closer average distance from the racket axis to keep
~5 twisting torque to a min;mum. This relieves the so-called
tennis elbow caused by repeated twisting movement of the
play-er's arm from ball-hitting shock.
~ 1't'9;~
MATHEMATICAL ANALYSIS
The practical possibility of longer longitudinal
strings strung at much higher tensions raises the issue
of the optimum relationship between longer and shorter
strings. This required mathematical analysis deriving
a more realistic dynamic equation and using a more real
istic mathematical model to determine the effect of changing
string parameters on the load distribution to different
strings.
FIGS. 3-5 show a mathematical model simplifying
and approximating the action of a central longitudinal
string 30 and a central transverse string 31 perpendicular
to each other and elastically supported by other strings
in the network to be deformed as shown when hltting a
ball. The string width 2b adjacent the ball simulates
the string portion that conforms with the flattened surface
of the ball when the ball penetrates into the string
network, The overall string lengths LO are divided into
subscript portions to account for different lengths of
string depressed by different amounts. The broken lines
32 and 33 simulate the elastic support from other strings
supporting the two string system shown in solid lines,
and the penetration d of the ball into the string network
in the area of contact also dents the elastic supporting
strings 32 and 33 by d/2.
Dynamic equations based on the model of FIGS.
3-5 as explained below approximate more closely the complex
realities of the interaction between longitudinal and
transverse strings. These equations aid in determining
appropriate values for string lengths and tensions to
achieve optimum string network response.
The elastically supported, two string network
of FIGS. 3-5 resists the force represented by the mass
m of the ball traveling at an initial relative velocity
~O in decelerating the ball as the two strings share the
load. With r representing the percentage of the load
borne by the longitudinal string, and with subscripts
c and L referring respectively to parameters of the cross
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--1 o--
string 31 and the longitudinal string 30, a more involved
analysis arrives at the following equations to describe
the dynamic equilibriums of the two strings under various
string lengths and tensions:
r mV02 - (pTo)~d2 ~ q~Lcl4 = O (2a)
tl - r)mVO - (PTO~cd ~ (A2Eq)Ccl4 = (2b)
The parameters p and q are given as:
p = (3/2) ~ L1-bl + l/(L2-b~] (3a)
q = (27/32) [l/(Ll-b~ + l/(L2-b~2/Lo ~3b)
For the cross string, L2=Ll and Lo=2Ll.
The maximum penetration d is found ~or the
longitudinal str;ng from:
r r ~
d = / (pTo~L ~ /1 2rmVO2~AEq~L ~ (4a)
L / _
~J (AEq)L ~/ (PTo)L
which bears a percentage r of the ball-hitting load, and
is found for the transverse string from:
r
(P o~c f I 2(,1-r)mV02(AEq)~ .~
d ~ (4b)
~ ~AEq~c~ pTo)C2
which bears a percentage l-r of the ball-hitting load.
The string force resisting the advance of the
ball increases with penetration of the ball into the string
network and reaches its maximum value when the ball is
stopped. At that instant, the deceleration i5 maximum,
and the force Fo is greatest. This maximum force, rFO
on the long string and (l-r~FO on the transverse string,
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which determines the final deformation of the ball, is
given respectively by:
r Fo = ~PTO~L d + (AEq)L d3 (5a)
(l-r)FO = ~PTo~c d + (AEq~ d (5b)
where Fo~ which is equal to the mass times the deceleration,
is the combined force on the ball from the two string
system, r is the load percentage borne by the long string
30~ l-r is the load percentage borne by the cross string
31, and d is the maximum penetration 6y the ball. For
the same penetration, a smaller Fo will deform the ball
less and hence is preferable.
It is quite clear From the mechanics of the
string network that the consistent practice oF the prior
art in stringing the longitudinal and transverse strings
with the same tension has forced the shorter transverse
strings to bear a much larger portion of the ball-hitting
load. The above equations give an approximation of the
load disparity between the two strings and show the tendency
of present rackets to overburden the transverse strings.
For example, the Prince racket with its ov~r sized head
and relatively long 11 inch transverse strings working
with 13 inch longitudinal strings apportions 57% of the
load to the transverse strings and only 43% to the longi-
tudinal strings when both strings are strung at the recom-
mended tension of 72 pounds. The corresponding loadd;stribution for the Dunlop Volley II is 56/~ on the trans-
verse strings and 44% on the longitudinal strings. The
preponderance of the ball-hitting load on the transverse
strings is substantially more than half for all rackets
3~ presently being sold.
Calculations using the above equations to approxi-
mate a realistic example comparing conventional stringing
with longer and tauter longitudinal strings balanced with
transverse strings according to the invention help clarify
the importance of the inventive improvement. Longitudinal
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string force in a conventionally strung prior art racket
having equal tension on longitudinal and transverse strings
as shown in FIG. 6 is compared with a racket having longer
and tauter longitudinal strings balanced with the trans-
verse strings according to the invention as shown in FIG.7. The comparison assumes a tennis ball traveling at
a velocity of 50 miles per hour and hitting a stationary
racket and string network. It also assumes that four
transverse strings and four longitudinal strings are in
contact with the ball and provide the -force required for
stopping the ball~
The previous equations used with these assumptions
show that the mass of the ball impacting on the contacted
strings penetrates the network to a distance of 20.5
millimeters for both the prior art and the inventive
rackets using the indicated string lengths. Th;s makes
the duration of ball contact and control o-f the shot about
equal for each racket. Both transverse and longitudinal
strings in the prior art racket are tensioned at 50 pounds,
and the inventive racket tensions the transverse strings
at 50 pounds and the longitudinal strings at 93 pounds.
The graphs of FIGS. 6 and 7 plot the impact
force against the penetration of the ball into the string
networks and divide the ball-resisting force into the
portion attributable to initial string tension To and
the portion attributable to stretching of the string AE
as previously explained. The results clearly show that
longer strings at higher tensions allocate a much smaller
portion of the ball-stopping force to string stretching.
The results also show that the maximum impact force at
the end of the ball penetration is higher for the prior
art racket than for a racket strung according to the inven-
tion. Since the ball penetration is the same for both
string networks~ shot control is the same, and the lesser
maximum force for the inventive network means a more
efficient rebound. Both of these differences represent
significant qualitative advantages for the inventive
network.
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Reducing the force involved in stretching strings
reduces losses that necessarily occur from internal friction
as a string stretches and from interstring friction as
strings rub together. It also reduces string wear and
fatigue so that the network lasts longer. Reducing the
maximum force required to stop the ball wastes less energy
in ball deformation and means a springier~ more responsive
string network that is more effective in returning energy
to the rebounding ball.
Of course, a real racket has a much more complex
string net~ork than assumed in these calculations and
includes a large number of perpendicular string systems
~f different lengths and actual tensions. However, the
tendenc~ shown by the calculated comparison should and
does prove true when applied to real racket string networks.
TEST VERIFICAT~ON
Test measurements haYe compared string networks
strung acc~rding to the invention with conventionally
strung string networks for two of the best tennis rackets
in the current market Because the invention involves
improved performance from an optimally strung network
and not an improved shape or conFiguration of racket or
frame, the frames of the two best rackets available were
chosen for comparison of stringing efficiency. One is
the "Volley II" made by the Dunlop Company as a medium
size head racket. The trade magazine "Tennis World" has
a special feature report in the April 1980 issue praising
this racket as excellent. The other racket is the famed
"Prince Classic"~ an over size head racket made by Prince
3~ Manufacturing Company according to U.S. Patent No. 3,~99,765.
Since the relevant comparison involves differences
in string networks and not differences in frame structure
or weight distribution that effect the overall performance
of the racket, the tests were made by clamping the periphery
of the racket frame in a horizontal position leaving the
string network free, dropping a ~ennis ball down from
a ~ixed height of 49 2 inches, and accurately measuring
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the height of the rebound of the ball from the string
network. The rebound height was measured by an "Instar"
video camera that recorded on magnetic tape and allowed
playback on a television to stop the frame showing maximum
rebound height. The tests were conducted by Dr. William
Parzygnat, who has a PhD in Mechanical Engineering from
Cornell University and works for the Xerox Corporation.
Both the Dunlop Volley II and the Prince Classic
rackets were first tes~ed with a new nylon string network
with uniformly tensioned strings at factory-recommended
values of 62 pounds tension for the Volley II and 72
pounds tension for the Prince Classic.
The ball drop tests were made on each racket at
dif~erent points in the ball-hitting region, and the ball
rebound heights were accurately recorded and measured to
establish the coefficient of restitution, which ls the
rebound height divided by the drop height. The results of
these tests are drawn in scale and schematically shown in
the right hand portions of FIGS. 8 and 9.
Then an identical racket frame was strung with
longer longitudinal strings and with string lengths and
tensions selected according to calculations. The network
strung according to the invention used longitudinal strings
anchored in the shank near the grip and fanned out across
the ball-hitting region as shown in FIG. lo To establish
string lengths and tensions in these rackets, calculations
assumed a relative ball velocity of 50 miles per hour with
the ball contacting four transverse strings and four
longitudinal strings as previously described. With the
ball's weight established at 0.103 pounds (46.7 gm), the
mass shared by one transverse string and one longitudinal
string is calculated to be 0.0008 lb.-sec.2/~t.
-15
For the regular Volley II racket strung with
nylon strings having an AE of 2260 pounds and with both
strings tensioned at the ~actory-recommended 62 pounds,
equations 4a and 4b indicate a ball penetration oF 0.69
inches or 1.76 centimeters. These same equations suggest
that the same racket frame strung according to the inven-
tion to achieve the same ball penetration and thereby
the same impact duration and shot-making control should
tension the 9 inch transverse nylon strings at 42 pounds
and use 18 inch longitudinal s~rings strung at 100 pounds
tension on a "Kevlar" string having an ~E of 13,000 pounds.
This makes the long strings twice as long as the transverse
strings and more than twice as taut and substantially
changes the load apportionment be~ween the transverse
and longitudinal strings. The original factory-strung
Volley II racket apportions 56~ of the ball-hitting load
to the transverse strings and only 44% to the longitudinal
strings, while the inventive string network apportions
59% of the load to the longitudlnal strings and only 41%
to the transverse strings.
Ball drop tests were then made on the Volley
II racket strung according to the invention to record
and measure the rebound height and the coefficient of
restitution at different points in the string network,
and the results of these measurements are plotted in
scale on the left side of FIG. 80 The test results show
a significant improvement.
The inventive string network achieves a 0.76
maximum coefficient of restitution that is higher than
any coefficient of restitution attained with conventional
stringing for the same racket. The region of the highest
coef-Ficient of restitution from 0.74 to 0.75 for conven-
tional stringing is only 9.0 square inches in the centerof the network and is enlarged to 34.4 square inches in
the inventive network, an increase by a factor of 3.82.
An outer region having a smaller coefficient of restitution
of 0.72 to 0.73 for the conventionally strung racket
amounting to 23.4 square inches was enlarged in the inven-
~ 3
-16-
tive net~ork to 51.5 square inches for an increase by
a factor of 2.2. These tests clearly show that the invention
subst~ntially improves over the conventional by making
the string network generally more lively and efficient
in rebounding a ball and by greatly enlarging the most
effective areas of the network.
In the test comparison of t.he Prince racket
as illustrated in FIG. 9~ calculations suggested that
instead of 11 inch nylon transverse strings and 13 inch
nylon longitudinal strings both strung at the recommended
72 pounds, the transverse strings should be tensioned
at 45 pounds and the longitudinal strings should be ex-
tended to 18 inches to an anchorage 1 inch away from the
handle grip and should be formed of Kevlar to withstand
a higher tension of 100 pounds. This changed the load-
bearing ratio from the original stringing placing 57%
oF the load on the transverse strings and 43% on the
lonyitudinal strings to the inventive stringing that
apportions 58% of the load on the longitudinal strings
and 42% on the transverse strings.
~ all drop tests were repeated to measure the
rebound height and coefficient of restitution of the
inventive network as plotted on the left side of FIG.
9, The results show that the invention enlarged the
central region with the highest coefficient of restitution
of 0.76 from the original 7.1 square inches to 44.3 square
inches for an increase by a factor of almost 6.3. The
outer area having a coefficient of restitution of 0.74
to 0.75 ~tso enlarged from the original 38.7 square inches
to 65.8 square inches for an increase by a factor of
1.7. This improvement represents an enormous increase
in the area of highest rebound responsiveness and shows
the clear superiority of the inventive network.
The coefficient of restitution values obtained
in these tests represent only the comparative efficiencies
of the string networks in rebounding the ball, because
the racket frames were constrained during the tests and
-17-
not involved in the interaction. In tests of a Prince
racket held at its handle when a ball hits the string
network as reported in U.S. Patent No. 3,999,765, the
coefficients of restitution were in the neighborhood of
~.3 to 0.4.
Racket performance depends not only on the string
network, but also on frame configuration~ material, and
weight distribution. So the improvement the invention
achieves in the string network may not result in a directly
proportional improvement in overall racket performance.
On the other hand, the inventive improvement in the network
stringing can be applied to existing rackets without addi-
tional cost~ and the drop tests establish that the invention
makes a more efficient string network with better ball-
rebounding ability that undoubtedly improves a racket'soverall performance. Rackets strung according to the
invention have been used extensively by experienced players
who have compared them with conventionally strung rackets
and reported a sub~ective impression confirming the test
results. Rackets strung according to the invention are
lively and responsive, feel definitely "playable", and
make well controlled and powerful shots.
The calculations and comparisons between con-
ventional string networks and the inventive string network
suggest another reason why the inventive network makes
a racket superior. Longer and tauter strings are able
to absorb the energy of the ball with less force applied
to the ball and consequently reduce deformation of the
ball. This increases the ball's rebound speed, because
less energy is lost in deforming the ball and more energy
stored in the strings is returned to the ball as kinetic
energy.
Considering the Volley II racket as an example,
calculations with equations 5a and 5b show that the con-
ventional string sys-tem stopped the ball with a final
load or peak force of 62 pounds from the two strings.
This seemingly large force lasts only for a brief duration,
- 18 -
because the total contact time between the ball and the
network is only two to three thousandths of a second. In
comparison with the inventive string network, the
transverse strings at 4~ pounds tension contributed 23
pounds toward stopping the ball, and the longer
longitudinal strings at 100 pounds tension contributed
33.8 pounds in a load-bearing ratio of 4:6. The maximum
string force applied to the ball is 56.~ pounds, which is
about 92% of the peak force from the conventionally strung
racket. This reduction in the maximum impact force reduces
the ball deformation and increases the rebound velocity.
Test results have also confirmed the shock
reduction capability of rackets strung according to the
invention. Ayain, using as an example the Dunlop Volley
II strung according to ~he invention as explained above,
comparative test play by several professionals and
experienced amateurs verifies that this racket is
remarkedly shock free and suppresses vibration better than
all other known rackets, including oversized rackets and
graphite frame rackets. This can particularly benefit
players who wish to avoid tennis elbow and want a racket
that vibrates the least.
PRACTICAL LIMITS
Although longitudinal strings can extend all the
way to the proximal end of the grip, calculations show
that such long strings would require very high tensions
exceeding the capacity of present string materials and
racket frames. Nylon tennis racket strings cannot
withstand tension more than about 90 pounds, and the upper
limit for 17 gauge Kevlar is about 100 pounds. If more
tension resistant string material is developed and
stronger frame materials are available, then longitudinal
strings can be lengthened into the handle to take full
advantage of the invention.
Within the present limits for string and frame
materials, a string network can be structured to emphasize
-19-
either control or power. High string tension and moderate
string length emphasize power and make the ball and network
contact brief, which reduces control. Conversely, ex-
ceptionally long strings with moderately high tension
increase the duration of ball and network contact to
inlprove control and reduce shock at the expense of hitting
power. The invention improves the network performance
so that control, power, and shock recluction can all be
enhanced; and the calculations aid in preselecting ways
of emphasizing one of these characteristics.
Information developed by the invention suggests
that for conventionally strung prior art rackets such
as the Volley II or the Prince, simply uniformly increasing
the tension of all the longitudinal strings in direct
proportion to their slightly greater length will make
the network too stiFf on the sides and will reduce the
size of the sweet spot. So to take advantage of the
improvements produced by ~he invention~ conventional
longitudinal strings must be lengthened at least a little
relative to the transverse strings. Both calculations
and experience show that the longitudinal strings should
be at least 30% longer than the transverse strings to
achieve a worthwhile improvement. The longitudinal strings
should also be strung with at least 30~ more tension than
the tranverse strings, and the functional relationship
between the longitudinal and transverse s~rings should
be predetermined to place about half or more of the ball-
hitting load on the longitudinal strings.
To achieve the 30~ minimum excess in length
and tension for the longitudinal strings compared to the
transverse strings requires lengthening the longitudinal
strings by at least an inch or two for conventional rackets
such as the Prince or Volle~ Ir. This can be done by
converting the oval frame to an egg shape with the blunt
end outward and the more pointed end toward the grip
or by a modified throat piece that provides a string anchorage
closer to the grip.
- 2~ -
For example, a Prince racket with transverse
strings strung at 70 pounds can have longitudinal strings
fanning out from a throat piece one inch behind the present
throat piece, and the greater length of these strings can
be tensioned to the 90 pound limit of nylon to increase the
ball-hitting load on the longitudinal strings from 43% to
47~. Field tests have shown that this 30% increase in the
tension of the longitudinal strings over the cross strings
makes a superior racket that is more playable, more
responsive, and smooth; maintains the same control with
added power to the center hits; and vibrates much less from
off center 'nits.
Longitudinal strings tensioned at less than 30%
more than the cross string tension do not produce a
significant improvement if the corresponding length change
is not appropriate. Also, longitudinal strings with at
least 50% more tension than the transverse strings are
clearly desirable, and this generally requires extending
the longitudinal strings well into the throat or shank
region of the racket. To take full advantage of the
invention's possibilities for improvement, it is best to
lengthen and tighten the longitudinal strings enough to
apportion at least 50% and up to 65~ of the ball-hitting
load on the central longitudinal strings. The string
network can also be varied to fit the styles of different
players by emphasizing either power hitting or control and
reduced shock.
STRING LENGTH LIMITATIONS
The International Tennis Confederation (ITC) in July
1981 adopted new rules restricting the length of tennis
racket strings. As quoted from the Encyclopedia Brittanica
Book of the Year for 1982, the new rules say: "The hitting
-- 21 --
surfllco Qf tho rack~t mw~ consist of strings alternataly
intorlac~d or bondod wh~ro the frame including the handle
shall not b~ long~r ~chan 3~ inches (81 cm), or string
surfaco ~xc~d 15~ 2 x 11~1/2 inches (39.4 x 29.9 cm)..O.
As spplicd to my lnvention, this rule may limit
th~ ngth of longitudinal ~rings to lS 1/2 inches so
that thoy c~nno c bo str~g to an anchorage sd~ acant the
h~dlo without violat~ng ~he rul~ by being overly lon~.
Tho ~nchorago ~or the longitudin~l strings must then be
po~ition~d aw~ from ~h~ Rrip and into the shank rogion
to a point wh~ro the long~qt longltudinal strings will
not ~xc~d 15-1/2 inches.
Anoth~r ~ff~ct of this rul~ is that if th~ longi-
tudinsl ~trlng~ aro fanned o~t ~cro~s the string n~twork
~5 I pr~for and if ~h~ rack~t is as wid~ as pormitt~d,
at l~a~t th~ two out~rmo~ longi'cudlnal ~trings that extend
along oppo~lte ~ido~ of ~he n~tworX cannot be 30~ longer
than tho longo~t tra.n~r~o string. For example, the
Princo Cla~ic rack~ wi ~ch a width of 10. 75 inches and
a maximum longitudinsl ~trinR l~ngth o 15-1/2 inches
with annod out longitudinnl strings has two lon~itudinal
string~ on oa.ch ~id~ of the rack~t that ~r~ only 14~ and
24~ long~r thsn tho longa~t tran~verse string. Longi-
tudinal strings str~mg parallel wlth each other, ins~ead
of being fannod out acro~s th~ string network 9 can also
produco sido string~ thQt are shortar than desirable7
To deal appropriately with longitudinal strings
strtm~ along tho sid~ of the network with lengths that
do not ~xce~d 1:h~ l~ngth of the longest transvflrse string
by at l~ast 30~ roquirss different tensioning. O~erly
tonsionlng side longitudinal ~trings tha~c are inadequately
long can msk~ the sldo~ of the string ~etwork ~oo stiff
and offacti~ely nasrow th~ high performance bnll-hitting
r~ gi on .
It ls possiblo to string difforent lengths of
strlngs ~t dlfforant tension~ and u~e three or more tcnsions
- 22 ~
on ~trlngs wlthin th~ lntarlaclng network. This cnn allow
sid~ longitudinal strings of lntormedista length to be
giv~n interm~dlato tonsions botween the lower tension
applied to tr~v~r~o strlngs ~d thc higher tension
appllod to contral longitudinal strings. This would be
espoclally d~sirablo wh~n two or more side longitudinal
strin~s ha~o ~ intormediato longth and when the centr~
longieudinal ~trin~ are ~trung with a tension exceeding
tho transvorso ~tring ~on~lon by 50~ or more. S~eral
interm~diato string~ at an interm~diato tension would
th~n flll a rolativ~ly wido ten~lon gap between the
central longitud~nal str1 ngs and the transverse strings.
If only one or two sidc longitudinal strings
fall~ slightly short of the desired goal of a~ least 30
greater l~n~th than th~ longa~t tr~nsv~rse string, an
acc~ptabl~ solution wo~ d be to string the side longi-
tudlnal ~erings at th~ ~m~ t~n~ion as the other longi-
eudinal s~rings, particularly if this tcnsion is only
30 to 40% moro than the transverso strin~ tension. As
larger numbors of sido longitudinal strings fall short
of th~ 30~ longor goal and as maximum and minimum tensions
respectively for central longitudinal strings and transverse
string~ diffor from ~ach other by substantially mors than
the minimum Roal of 30~, ~t is better to use an intermediate
t~nsion to sm~o~h the tension transition from the two
extro~es.
Although the changa in the rules reduces some
of tho option~ for kueping the longitudin~l strings ade-
quat~ly long within the geometry of the short~r ~engt}l
limits now permitted by the rulss, the general objective
remains tho same. This ls to keep the longitudinal strings
as long as po~sibl~ and serung at tuitably higher tensions
so as to bear about hal f or more of the ball-hitting load.
Side longitudinal strings may require tensioning at higher or
intermediate tensions may be used for side longitudi`nal strings,
but these circumstances do not change the general objective.
