TENNIS RACKET AND METHOD
A tennis racket is provided that has a rectangular ball striking face with rounded corners. The racket is formed of a carbon fiber sandwich with light material mounted therebetween. A strip of carbon fiber is attached to the outer edge spanning the two carbon fiber top and bottom sheets. The rectangular frame is formed with outwardly bowed sides so as to be pulled into a straight sided configuration by the string tension. Locking grommets of various constructions are provided to hold the strings at predetermined tensions. Testing equipment and methods are provided for testing tennis rackets.
This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61,436,259, filed Jan. 26, 2011, which is incorporated herein by reference.
BACKGROUND OF THE INVENTION1. Field of the Invention
The present invention relates generally to a tennis racket and method of making and stringing a tennis racket.
2. Description of the Related Art
In U.S. Pat. No. 6,344,006 B1 and U.S. Pat. No. 7,081,056 B2, R. Brandt taught the advantages of tennis rackets with strings of equal length in the vertical and in the horizontal directions, or, more generally, equal string frequencies of vibration. In the present patent document, these ideas will be further developed and improved, and construction methods for manufacturing and testing such rackets will be presented.
SUMMARY OF THE INVENTIONThe present invention provides a tennis racket and method of making a tennis racket the provides a structure by which the strings of a tennis racket may be provided with equal string lengths and with desired string vibration frequencies when striking a tennis ball to provide improved ball striking performance. In a preferred embodiment, the tennis racket includes grommets to establish and maintain the desired string length and string vibration frequencies. In a further development, a layered structure is provided for the racket structure, preferably in the form of a carbon fiber material sandwich. The racket head of one embodiment includes outwardly bowed sides configured to be drawn into a straight configuration when subject to string tension forces. The string or strings used in the present racket may have varying densities to achieve the desired string vibration frequencies. The present invention also provides a method for testing racket structures.
In further detail, a number of possible structures and methods for achieving the desired string lengths and frequencies are achieved by the use of grommets at the string mounting openings of the racket. The grommets are structural elements at the sides of racket faces through which the strings pass. The present invention provides locking grommets for holding the strings and algorithms for setting the tensions using the grommets. The present invention also provides a plurality of types of grommets, formed for example of Teflon or other low friction material, of elastic materials, or having an adjustable height, which help achieve the desired string lengths and frequencies. A configuration of the racket head is provided that quantitatively establishes how far the sides of the racket face are bowed outward before stringing in order that the string lengths will become equal after stringing.
In a further development, the present invention provides methods for manufacturing rackets according to the principles of the present invention. In a preferred embodiment the entire racket body is fabricated out of a carbon fiber sandwich. This construction provides superior strength and simplicity compared to other racket constructions. The dimensions of the carbon fiber racket embodiment are determined so as to be optimized for racket strength to weight ratio. The present invention also provides methods for testing the strength and performance of the rackets constructed according to the invention, as well as other racket constructions. The strength of the racket corners and the strength of the grommet string holders is tested. The performance of the rackets is tested by laboratory measurements. The laboratory testing provides data that confirms the performance advantages of the present racket constructions over other constructions.
In general, the tennis racket structures and procedures according to the present invention apply to rectangular-shaped tennis rackets, although other shapes may be encompassed within the scope of the invention. Included in various embodiments of the present invention is (1) a derived outward bowing profile that compensates for inward bowing, and the determination of when outward bowing is necessary for optimal performance; (2) a means to achieve equal horizontal and vertical string frequencies by using different string tensions and/or string mass densities; (3) use of locking grommets that maintain different tensions on different strings, enabling strings to vibrate with equal frequency without equal length, and enabling the adjustment or replacement of individual strings; (4) a derivation of an algorithm for determining initial string tensions that provide equal string frequencies; (5) a specification of a sandwich-constructed embodiment of rectangular rackets that are approximately twice as strong per pound as conventional rackets fabricated from tubular elements; (6) a specification of a superior table-constructed embodiment of rectangular rackets that are nearly as strong per pound as the sandwich-constructed embodiment on the face sides, and have much stronger corners, provide better protection for filler material, and allow for unprecedented ease of construction; (7) a use of a cantilever model to analytically estimate the stresses on racket corners; (8) designs and implementations of devices that test the strength of racket side beams and corners.
The present invention will be described by reference to the drawings. A first consideration is the vibration frequencies of the strings upon striking a tennis ball. General considerations for the string frequencies include the following.
String FrequenciesThe calculation of the performance of a given tennis racket is a complicated problem in significant part because all of the strings interact with each other during an impact with a tennis ball. Such calculations were reported in U.S. Pat. No. 5,672,809 and U.S. Pat. No. 6,344,006, and the main conclusion was that a rectangular face racket, with strings of equal length in each direction, provides the optimal performance. The reasons for this are the following.
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- a) Strings of equal length (and tension and density) vibrate with the same frequency. They therefore respond in unison to an impacting ball, thus providing the maximal rebound force.
- b) When a ball impacts two parallel strings of unequal length, it experiences different forces from each string, resulting in an unintended rebound direction. When parallel strings all have the same length, this error is eliminated.
- c) When a ball impacts a string away from the string center, it experiences unequal forces from each side of the depressed string. The shorter the string length, the larger is this effect. For a rectangular faced racket, each string has the same (maximal) length, and so this source of error is minimized.
- d) For a given racket weight, the rectangular faced racket has a much larger moment of inertia about the longitudinal central axis than a conventional racket. An impact away from this axis will therefore cause the racket to rotate away from an impacting ball much less than for a conventional racket. The resultant angular error and speed diminution are therefore greatly reduced.
- e) Because a rectangular racket has less angular rotation than a conventional one, physical injuries to a player's arm (commonly referred to as “tennis elbow”) due to off center hits are reduced.
The major benefits of equal string lengths obtain if the lengths are not exactly equal, but the string vibration frequencies are equal. The frequency of a vibrating string with fixed ends is given by the equation
where l is the string length, T is the tension on the string, and m is the linear mass density of the string. For conventional rackets, T and m are constant, but the string lengths vary considerably and therefore so do the vibration frequencies. With rectangular-faced rackets, the values of T, m, and l (in each direction) can be all constant, and so the vibration frequencies f can be all constant. That is the simplest possibility, and innovative ways to accomplish this will be taught below. We will afterwards consider more general possibilities.
The strings on a tennis racket are of course not free between the sides of the frame. The strings cross over and under each other many times along their lengths, and so the performance of a given racket must be determined by complicated calculations on a computer, as was described in U.S. Pat. No. 6,344,006 B1. It was shown in this patent that rackets with strings of equal fundamental frequencies (as calculated using Eq. 1) provide the optimal performance.
When we speak of a racket with a rectangular face, it is understood that what is essential is for the opposite sides of the face where the strings are attached to be parallel. This leaves us with the freedom to round the corners of the face frame in the sections beyond where the stringing holes are placed. Such rounded corners make the racket stronger and better looking. Such a racket is illustrated in
In particular,
The simplest way to obtain equal string frequencies in each direction is to have the values of l, T, and m all constant. We start first with a perfectly rectangular face before stringing and assume for the moment that the face remains essentially rectangular after stringing. As the racket is strung, the chosen tension T is applied to each string, but as the stringing proceeds, the addition of each new string at tension T slightly changes the tensions imparted to the previous strings. After all the strings are attached, the string tensions equilibrate and the resultant tension T′ existing on the strings is different from the applied tension T. A competent stringer will know how to choose the applied tensions such that the resultant tension is the desired one.
With conventional grommets through which the string extends at the racket sides and ends, the tension equalization is imperfect. The friction on the string within each grommet and on the outer face sides between grommets impedes the tension equalization. In addition, the forces applied on the strings from impacts with a ball will, at least temporarily, redistribute the tensions.
These difficulties can be avoided with the use of Teflon, or Teflon coated, grommets and side strips. Teflon is extremely slippery, having the lowest coefficient of friction (about 0.05 against polished steel) among practical solids. Its use will insure that the tension equalization obtains quickly and nearly perfectly during stringing and during impacts with balls. Other materials having a low coefficient of friction may be used as well to provide equalization of string tension across the face of the racket.
In reality, if the racket face is perfectly rectangular (apart from the corners) before stringing, the applied tensions in the strings will cause the face sides to bow inwards. This will cause all of the strings lengths to shorten, more at the center of the sides and less near the ends of the sides. The result will be a racket with unequal string lengths and reduced tension on the strings.
To illustrate this effect, consider a beam of length l, supported at each end, subjected to the forces from n uniformly spaced attached strings of tensions T. If E is the Young's Modulus of the beam material and I is the area moment of the beam cross-section, then the deflection y(x) of the beam at distance x from the beginning of the beam is given by the equation
Using l=13 inches, T=60 lbs., n=19 strings, and typical values of E and I for a carbon fiber tube, this function is plotted in
With reference to
With the stated values of the relevant parameters, the beam, representing the long side 24, in
The deflections of actual racket face sides 24 are more complicated than this because these sides 24 are not rigidly supported beams and because the forces applied by the strings are not uniform. These deflections are not given by simple analytical expressions such as Eq. 2 above, but they can readily be evaluated by computer for any given racket geometry and material, and they can be directly measured. The realistic side bending calculations are complicated by the fact that the applied forces are discrete and are different at each string location, even after the string tensions have equilibrated to a constant value, because of the geometry of the deflected sides. The equations to be solved are therefore highly coupled and non-linear because the string tensions determine the amount of side bending, and the amount of side bending determines the string tensions. Representative results will be given below.
As an example of these results, consider the 13 inches long parallel sides of a rectangular carbon fiber racket face 22, with 19 string holes, the first of which is 2 inches from the corner 28, and the rest are spaced 0.5 inch apart. Assuming conventional values for the Young's Modulus of the fiber and the cross-section of the face, each of the face sides would bow inward a maxim distance at the center of the beam of 0.21 inch if the strings each carried 60 lbs. of tension. However, if the racket were strung by applying 60 lbs. of tension to each string, because of the bowing, the resultant string tensions would be reduced to only 51.7 lbs., and the maximum deflection distance for each side 24 would be only 0.18 inch. In order for the strings to carry the desired 60 lbs. of tension, the applied stringing tensions would have to be 69.6 lbs. These calculations are original and encompassed within the scope of the present invention.
Because of the bowing of the racket sides 24 described above, if we start with a rectangular racket face 22, stringing the racket will result in a constant tension (after equalization, preferably aided by Teflon grommets and side straps), but not in strings of equal length, and consequently not in strings with equal frequencies of vibration. In order to achieve strings of equal length, the present invention constructs the racket face so that its face sides bow outward by an appropriate (tension dependent) amount before stringing. Then, after stringing, the string tensions will bring the sides into a nearly rectangular configuration. The details of this construction will be given herein below for a preferred embodiment. These calculations are original and encompassed within the scope of the present invention.
In order to determine the amount of induced inward bowing that is significant enough to warrant the use of compensating outward bowing, we use Eq. 1 to relate the change in string frequency Δf to the change in string length Δl, which leads to equation:
Δf/f=−Δl/l. (Eq. 3)
Using the typical values l=10 inches, T=60 lbs., mg=0.014 oz/in (g=32 ft/s2), a typical frequency is f=670/s. A typical impact time is t0=0.004 s, and so the string vibrates about ft0=2.7 times, and undergoes a phase change of about 2πft0=17 rad, during the impact. In order that strings of different lengths Δl interact coherently with an impacting ball, we require that the phase change difference be less than π/4, i.e. 2πΔft0<π/4, or Δf/f<⅛ ft0≈0.046. (If the phase change difference were greater than π/4, the adjacent strings would move in opposite directions near the end of the impact time, thus reducing the rebound force on the ball.) Eq. 3 then gives Δl<0.46 inch. This implies that the compensating outward bowing should be used whenever the inward bowing is more than Δ½≈0.23 inch on each side 24.
In addition to the above use of outward bowing to establish equal string lengths, there is another way to achieve equal string frequencies when the face sides bow inward. The idea is to use string with variable mass density m. If bowing causes one string length to decrease from 1 to l1 and another string length to decrease from 1 to l2, these two strings will nevertheless vibrate at the same frequency if the density m1 of the l1 string and the density m2 of the l2 string are chosen in the ratio m1/m2=l12/l22.
Above is described how tennis rackets with rectangular faces have stings of equal length and frequencies in the horizontal and in the vertical direction. An even more uniform behavior will obtain if, furthermore, the frequencies in the horizontal and in the vertical direction are the same. One way to accomplish this is to have the face not just rectangular, but square. Then, if the horizontal and vertical tensions are equal, the horizontal and vertical frequencies will also be equal. Most players would, however, prefer the rectangular shaped face because it would give them a larger hitting area under current ITF (International Tennis Federation) Rules limiting racket dimensions.
Another, more practical way to achieve equal frequencies in the horizontal and vertical directions is to choose the horizontal string lengths lh and tensions Th, and the vertical string lengths lv and tensions Tv in the ratio Th/Tv=(lv/lh)2. Then, according to Eq. 1, the horizontal and vertical frequencies will be equal. For this arrangement to be practical, the horizontal and vertical string lengths could not differ by more than 10% since string tensions are usually chosen in the 55 lb-65 lb range and √(55/65)=0.92. A suitable example is lh=11 inches and lv=12 inches.
Variable TensionIn this section we will teach how to obtain strings with the same frequencies even if the string tensions are not equal. The idea is to construct the rectangular racket such that the tension in each string can be set separately. This can be achieved by using locking grommets that lock their attached strings in place. These grommets allow a string to be pulled through in one direction (outward from the face), but not in the opposite (inward) direction. The (initial) tensions of each string can then be set as desired, and there will be no equalization of the tensions on different strings because there will be no slippage of the strings through the grommets.
We will teach how to construct such locking grommets below. We will first explain how they can be used to implement the goal of equal frequencies. To this end, picture the rectangular racket with its face 22 to the left and its handle 38 to the right, as shown in
Proceeding in this way by sequentially attaching the N string sections, locking the i′th string section at position xi with initial tension ti, i=1, 2, . . . , N, the sides will end up bowed inward by the total distance d(x)=Σdi(x), and the tension on the i′th string section will end up at Ti=ti−2kd(xi). After the perpendicular strings are similarly attached, the strings will attain their final lengths wi and tensions Ti whose values will depend on all of the chosen initial tensions t1, . . . , tN.
The goal is to choose the applied string tensions t1, . . . , tN such that the final string frequencies fi, according to the formula
are all equal (fi=constant). For any given rectangular racket (geometry and material) and string, the N equations (fi=constant) can be solved by computer for the N applied string tensions t1, . . . , tN. We will do this for our preferred embodiments in the following sections.
There are further advantages to using our locking grommets. For example, the locking grommets can be used to implement desired tension differences at different areas of the racket face 22. The areas of the face near the sides are, even for rectangular faces, areas of lower performance. This can be partially made up for by choosing the string tensions in these areas to be less than the tension on the strings near the center of the face. This will provide more uniform power across the entire racket face.
Another possible application of the locking grommets is to incorporate some elasticity within the grommet itself. This will provide additional options for controlling the string lengths, tensions, and frequencies.
Another important advantage of the locking grommets is the ability the grommets provide to compensate for decreases in string tensions that result from hitting with the racket. Each impact between a ball and the racket strings tends to lengthen and weaken the strings and reduce their tensions. The simple pulling of a string through a locking grommet can compensate for this by shortening the string section and increasing its tension. The need for restringing will therefore be significantly reduced. (A simple hand-held tension gauge can be used in series with the pulled string to tell when the desired string tension is achieved.)
Likewise, a broken string can be easily replaced on the rackets that are strung with separate individual strings, without the need for a total restringing. This will result is a significant savings in player time and expense, and compensate for the additional time needed to string the racket.
Now a number of locking grommets embodiments will be described. These are illustrated in
In
In a further conical-hole 40 example of a locking grommet, shown in
The drawing in
The locking grommet of
Referring to
In
A similar concept is shown in
There are many other possible embodiments of locking grommets that may be used within the scope of the present invention, as will be apparent to those of skill in the art that achieve the goal of maintaining individual string tensions.
Frame EmbodimentsGeneral considerations of frame construction are considered herein. In a preceding portion of the present specification, the performance advantages of the rectangular tennis racket face were reviewed and amplified. In this section the construction of such rackets is described. Conventional racket construction techniques are inadequate for a rectangular racket unless the racket is too heavy to be playable. The problem is that the square corners, even if rounded, are areas where large stresses are concentrated when the racket is strung. In order for the racket to withstand these stresses, a relatively large amount of reinforcing material must be incorporated into the corners, making the racket heavy and unbalanced.
The two properties that determine a racket's strength are materials and geometry. The main material used to fabricate essentially all contemporary rackets is carbon fiber. The reason is that carbon fiber has the largest strength to weight ratio of any practical material. The bulk measure of the strength of a material is its Young's modulus E. This modulus is defined as the ratio of stress (applied force per unit area, F/A) to strain (elongation or compression per unit length δ1/1) according to the equation:
F/A=E δ1/1. (Eq. 5)
The bulk measure of weight of a material is its density ρ, the ratio of weight W to volume V according to the equation:
W=ρV. (Eq. 6)
The following table lists E and ρ for aluminum, carbon fiber, and titanium.
The Young's Modulus values show that Carbon Fiber is stronger than Aluminum but weaker than Titanium, but it is much less dense than these metals so that its strength to weight ratio is much larger. This is why Carbon Fiber is the racket material of choice.
To understand the role of geometry in racket strength, consider a supported uniform beam 90 of a material of length L as shown in
Due to the deflection, the upper part of the beam is compressed, and the lower part of the beam is stretched. There is therefore a plane 94 through the beam 90, called the neutral axes, whose length remains unchanged after the deflection. This plane is indicated by the dashed line 94 through the center of the beam in
As shown in
I is defined by the geometry of a cross-sectional slice of the beam according to the equation:
I=∫dAy2. (Eq. 8)
The integration is over the area of the cross-section containing material, with y being the perpendicular distance between the neutral axis and the area element dA.
For example, if the cross-section is a circular annulus 96 as shown in
The vertical arrow 98 in the
The area of the cross-section of the uniform beam is defined by equation.
A=∫dA, (Eq. 10)
and the weight is defined by
W=ρAL. (Eq. 11)
The goal of the racket construction is to achieve adequate strength, so that the deflection D is relatively small, without requiring a relatively large area A, so that the weight W is also relatively small. For a given material (given E and ρ), and fixed weight (fixed A and L), the goal is therefore to choose the geometry of the cross-section such that I is as large as practically possible. According to its definition (Eq. 8), I becomes larger when the racket material is placed as far as possible from the neutral axis of the face side (beam).
This is the reason that conventional rackets are constructed out of tubular elements. A thin-walled tube has much of its material lying relatively far from the neutral axis. This is illustrated by the annulus cross-section in
This construction works well for a conventional oval shaped racket. For a rectangular racket, however, this construction is not ideal. The large stresses at the corners of the racket face necessitate relatively thick tube walls for stability. In the following sections we will teach alternative construction designs that are better suited for the present rackets, and have further advantages. These designs will achieve adequate strength without requiring unacceptable weight.
Sandwich EmbodimentThe tubular carbon fiber racket frame is strong because much of the material is far from the neutral axis, as indicated for the circular annulus of
The cross-section of a beam 100 made out of such a sandwich is illustrated in
The dimensions given in
IS=b(d3−h3)/12=b(d−h)(d2+dh+h2)/12=AS(d2+dh+h2)/12, (Eq. 12)
where AS=b(d−h) is the area of the carbon fiber material in the sandwich cross-section. We will show that this moment is about twice as large as the annulus moment with the same area
IA=π(r24r14)/4=π(r22−r12)(r22+r12)/4=AA(r22+r12)/4, (Eq. 13)
where AA=π(r22−r12) is the area of the carbon fiber material in the annulus cross-section.
If the cross sections have equal area (AS=AA), so that the beams have equal weight (Eq. 11), the ratio of these moments is defined by equation
Using the fact that the carbon fiber material is thin, so that h≈d and r1≈r2=d2/2, in terms of the diameter d2 of the outer circle, this reduces to
Thus, if the cross-sections are of similar height, so that d2≈d, we have shown that IS is about twice as large as IA, so that the sandwich beam is about twice as strong as the annulus beam.
It follows that constructing the rectangular racket frame out of sandwich beams instead of tubes, the frame will be stronger and require less weight at the corners. The construction is, however, not optimal. The problem is again the corners. Sandwich beams, with cross-sections as shown in
A better method of construction is to fabricate the entire racket face 114 out of a single sandwich beam. Such an embodiment, with rounded corners 116, is shown in
To understand why the corners must be strengthened, consider the corner 116 illustrated in
With reference to
With 19 strings at 60 lbs tension each, F=19×60 lbs=1140 lbs. With the side length of 12″, we take L to be the average distance 6″. We take w=0.75″, a typical value of a racket face width. If the racket is to weigh at most 14 oz, then the beam thickness can be at most about t=0.1″. Using these values, Eq. 16 gives
σmax=5,500,000 psi. (Eq. 17)
This value is large because t must be small in order to have an acceptable racket weight.
The rupture stress of carbon fiber is about
σrupt=820,000 psi. (Eq. 18)
Since the applied stress is much greater than the rupture stress, we conclude that this sandwich embodiment of the rectangular racket must be significantly strengthened in order for the corner not to rupture. This is difficult to do without significantly increasing the weight of the racket.
The cantilever model is, of course, a simplification, and it does not incorporate the strengthening effects of the rounded corners and parallel plates. Also, one can use stronger fiber and a larger width, more complicated geometry, and more optimal fiber lay-up patterns. We have, however, used realistic finite element computer calculations to evaluate the relevant stresses, and the results substantiate the conclusions drawn from the model, Given these large estimates of the corner stress, the difficulty described here is obviously serious.
There is another problem with the above embodiment. The tension in the string attached to the top plate 124 in
In the next section we will describe a preferred embodiment of the present rectangular racket that avoids all of the above difficulties. The result will be a strong yet light racket that possesses all of performance advantages, including those described U.S. Pat. No. 6,344,006 B1 and U.S. Pat. No. 7,081,056 B1.
A Preferred EmbodimentAlthough the above sandwich embodiment material is about twice as strong per pound as conventional tubular material, we established that it possessed two significant properties that largely negated this material strength. It gave rise to weak corners unless relatively heavy reinforcement is added, and it exposed the filler material to damaging compressional forces unless relatively heavy resistive material is incorporated. In this section we will teach how to construct a modified sandwich structure that solves these problems, has other advantages, and enables the fabrication of a rectangular racket that is both strong and light.
The idea is to use beams in which the cross-section has a “table” structure instead of the sandwich structure illustrated in
The dimensions given in
The total area of the carbon fiber sections in
IT=d(3b2c+6bc2+4c3)/12l+ab3/6. (Eq. 19)
The first term in (Eq. 19) is the contribution from the top and the second term is the contribution from the legs. Since c will always be much less than b, a good approximation is
IT≈db2c/4+ab3/6. (Eq. 20)
We proceed to explain why this embodiment does not possess the difficulties (weak corners, exposed filler) of the previous one, and provides a strong and light racket. We consider first the beam strength, as quantified in the beam deflection distance (Eq. 7). For the same force F, length L, and modulus E, the larger the moment, the stronger the beam. As already explained, for a given cross-section area (and therefore a given beam weight), the table moment IT (Eq. 19) will be less than the sandwich moment IS (Eq. 12). However, the dimensions a, b, c, d, can be chosen such that IT is at least 90% as large as IS (and therefore, given Eq. 15, over 80% larger than the annulus moment IA). It follows that if we choose the cross-sectional area of the table beam to be only slightly larger than that of the sandwich beam, the strengths of the two beams will be identical. Below we will provide examples of table beam dimensions that yield strong and light rectangular rackets.
We will next show that the rectangular racket corners of frames that use table sections are much stronger than those of frames that use sandwich sections. The sandwich corner 116 is illustrated in
Substitution of the sandwich values (F=1040 lbs, L=6 inches, t=0.1 inch, w=0.75 inch) into the expression (Eq. 16) for the stress at the support gave the very large stress of 5,500,000 psi, which is much larger than the carbon fiber rupture stress of 820,000 psi. On the other hand, substitution of the sandwich values (F=1040 lbs, L=6 inches, t=0.75 inch, w=0.1 inch) into this expression gives the much smaller stress
σmax=730,000 psi. (Eq. 21)
This is significantly less than the rupture stress, confirming that the table corner is much stronger than the sandwich corner, and strong enough to withstand the stress induced by the string tensions.
The cantilever model does not take into account the strengthening effects of the rounded corners and parallel plates. We have accurately calculated the corner stress on rackets with rounded corners fabricated from sandwich and table beams, using finite element analysis techniques. The results are stress values that are less than the above estimates. The range in calculated stresses for the sandwich corner is 1,700,000-2,400,000 psi for the sandwich corner, and 530,000-650,000 psi for the table corner. The size of these ranges arises from use of different dimensions, lay-up patterns, and epoxies. With any of these possibilities, the superiority of the table construction is apparent.
Use of the table elements thus enables the construction of rectangular rackets that have corners that are strong enough to withstand the applied forces, without the need for any reinforcement. These elements thus eliminate the weak-corner problem associated with the sandwich elements. They also eliminate the filler-crushing problem associated with the sandwich elements. It is clear from the table cross-section shown in
Rectangular rackets fabricated from the present carbon fiber table elements thus possess strong sides, strong corners, and protected filler material. That is why they are the preferred embodiment for a rectangular racket. There is, in addition, another major advantage to this fabrication. It is particularly easy to construct the rackets out of a single carbon fiber sandwich. Such a sandwich 148 is illustrated in
The racket frame can be cut out of this sandwich 148 in a single piece 156, as illustrated in
The shape of the cutout in
The preferred rectangular racket embodiment described in this section has been based on theoretical calculations. In the following section, we will provide the details of how to actually construct and test specific examples of these rackets. Dimensions will be given, materials will be specified, photographs of constructed examples will be shown, and strength and performance data will be presented.
Examples and Tests of a Preferred EmbodimentTo fabricate a rectangular racket that exemplifies a preferred embodiment, we begin by choosing the desired features. Most contemporary rackets have 19 short strings and 16 long strings, and we will comply with this choice. According to the performance calculations reported in the Brandt U.S. Pat. No. 6,344,006 B1, string separations of 0.5 inch are close to optimal, and so we will also make that choice. We choose inside face dimensions of 9.5 inches×12 inches to accommodate this string pattern. The racket face shown in
Most contemporary tennis rackets weigh between 10 and 14 oz., and have a center of mass located within the throat area, and we will also incorporate these constraints. We must therefore choose the table dimensions b, h, d, and c (defined in
For the plates comprising the table beams, we have chosen commercially available carbon fiber products. The table legs 130 and 132 are fabricated out of carbon fiber plates with a 45° lay-up pattern. The tabletops 134 are fabricated out of carbon fiber plates with a unidirectional lay-up pattern. The Young's Modulus of this material is about E=2×107 psi, and the density is ρ=0.96 oz/in3.
With these values, and any chosen values of the table dimensions b, h, d, and c, substituted into the moment equation Eq. 19, we can use Eq. 7 to estimate the racket side deflections, we can use Eq. 16 to estimate the racket corner stresses, and we can calculate the weight of the racket face. We can then confirm these results by performing a finite element analysis. These calculations enable us to choose table dimensions that yield rackets that are sufficiently strong and light.
These calculations imply that a suitable choice of the table dimensions for the 12 inch face side is as follows:
b=0.625 inch
h=0.55 inch
d=0.75 inch
c=0.0625 inch (Eq. 22)
The table top thickness is thus c=0.0625 inch and the table leg thickness is (d−h)/2=0.1 inch. With these values, the area moment of inertia of the beam cross-section is IT=0.0083 in4, and, with 60 lb tensions, the maximum deflection of a 12 inches side is 0.155 inch. For the 9.5 inches face sides, b can be reduced to 0.5 inch.
The 12 inches side of the racket face has been designed to withstand the 1140 pounds of force arising from the 19 attached strings at 60 lbs. tension each. The throat and handle of the racket need not be this strong. The forces on these elements arise during the brief impact times (about 0.004 sec), during which the racket strikes a ball. This force is at most about 250 lbs, and lasts for such a short time that the throat dimensions can be significantly less than those in Eq. 22.
The racket is preferably strung with separate string sections for each of the main strings and cross strings. It is also contemplated that the string may be a single continuous string or may be of several string segments.
The remaining elements of the racket design are not significant for performance and can be chosen to be consistent with current conventions. The final racket design is illustrated in
The racket handle stem 170, cut out of the sandwich, has width 0.625 inch and height 0.75 inch. Light material, such as balsa wood, is attached to this stem, as shown in
After the frame is cut out of the carbon fiber sandwich and the side (table top) plate is attached around the perimeter, string holes are to be drilled as indicated in
The string tensions are preferably set so be equal for each main string and each cross string. In a preferred embodiment, the string tensions on both the main strings and cross strings are equal to one another. The string vibration frequencies of the main strings are preferably equal and the string vibration frequencies of the cross strings are preferably equal to one another. In one embodiment, the vibration frequencies of both the main strings and the cross strings are equal. In some calculations, the variables ti and li refer to the tension and length, respectively, of each main string, and the variables sj and kj refer to the tension and length, respectively, of each cross string. The linear density of the strings are designated mj for the density of the main strings and m′j for the densities of the cross strings.
The stated dimensions in the above drawings have been chosen with the aid of theoretical calculations of strength and performance. Before proceeding to fabricate the racket, it is wise to confirm these calculations with laboratory strength measurements. We have devised appropriate strength testing protocols that we teach below.
To confirm that the table construction (
When the display 214 reads a value F, the tension in each of the two strings 204 is F/2. In order for the beam to withstand string tensions of 60 lbs, it must withstand an exerted force of 120 lbs. We tested the table beam, with dimensions given in Eq. 22, in this way and we found that it withstood applied forces of over 200 lbs. This confirms the beam strength calculations. Photographs of the testing apparatus are set forth in
Having experimentally confirmed that the tested length of table beam, with the specified dimensions, is as strong as predicted by our calculations, we proceed to describe how to similarly test the strength of our racket corners. The method we have devised to do this utilizes a small replica 212 of a racket face side and corners, as illustrated in
The force on the long side of the real racket face arising from 60 lbs. of tension on each of 19 strings is 1140 lbs. The equivalent force F on the replica 212, the force that produces the same stress at the corners, when applied to the center of the replica, is 1390 lbs. (This force is greater because the effective lever arm is less.) To confirm that the replica can withstand a force of this magnitude, we inserted the replica 212 into a hydraulic press in series with a load cell that measures the applied force. Photographs of this testing apparatus are given in
A photograph of a produced racket 230, constructed according to the above specifications, is shown in
W=14 oz
COM=9.87″
MOI=2400 oz.in2. (Eq. 3.18)
The racket of the present invention may have a weight with string of less than 14 oz. It is also possible to construct a racket according to the present invention with a weight of less than 12 oz., or even less than 10 oz.
We have measured the performance of a similar prototype racket 232 which has slightly different dimensions and is heavier with a different CF by impacting the racket with tennis balls 234 propelled from a cannon at various impact speeds and various locations on the racket face. The racket face 232 was rigidly clamped to a solid surface 236 perpendicular to the cannon direction. The incident balls 234 were without spin and struck the racket 232 at right angles. The incident and rebound ball speeds were accurately measured using light gates. (The racket does not move during these impacts. The transformation of these data to the game situation in which the racket is essentially a free body during the impact will be given afterwards.) For comparison, the same type of measurements were made on a conventional oval racket. A photograph of the setup is shown in
The quantity that characterizes the performance of a fixed racket at any point on the face is the velocity ratio at that point, the ratio of the rebound speed to the incident speed. In comparing the ratios between the rectangular and oval rackets, it is important to compare the differences between the ratios at the racket centers (the point of best performance for a fixed racket) and at a given distance away from the center, because the performances at the racket centers can be adjusted by changing the string tensions. It is also important to compare the differences at the same incident speed, because the ratios always decrease as the incident speed increases.
The following table exhibits velocity ratio data that is typical of our measurements. The impact speeds are given in the first column. For the rectangular and oval rackets, the velocity ratios are compared at the center and at the point 3.5″ below the center. The ratio is seen to decrease by 0.52% and 0.75% for the rectangular racket, and by 10.48% and 8.55% for the oval racket. The off-center performance of the rectangular racket is seen to be far better.
To transform these data into data that characterize the performance of the rackets in game situations, in which the racket is essentially a free body during the impact, we replace the velocity ratios with coefficients of restitution (CORs). The COR between a tennis ball and a tennis racket is the ratio of the relative rebound speed to the relative incident speed. This quantity, which reduces to the velocity ratio when the racket is fixed, is what determines the performance of a swung racket. The rebound ball linear and angular velocities are completely determined by the COR at the impact point, along with the details of the racket stroke (linear velocity, angular velocity), the racket kinematics (weight, COM, MOIs), and the incident ball properties (weight, linear velocity, and angular velocity).
The following table exhibits changes in the COR data that results from the above velocity ratio data for the 65.3 mph impacts. We have assumed that the rectangular and oval rackets have equal weights, MOIs, and COMs. The actual larger MOIs of the rectangular rackets will further improve their superiority over the oval rackets.
For the fixed frame rackets, the best performance is at the center of the faces. For the free rackets, the point of best performance is shifted towards the handle because that is the direction towards the COM of the racket. For the rectangular racket, the effect of this is to replace the 0.75% decrease in performance by a 1.19% improvement in performance 3.5″ down from the center. For the rectangular racket, the effect of this is to replace the 8.55% decrease in performance by a 7.45% decrease in performance 3.5″ down from the center.
The COR differences exhibited in the above table give rise to significant differences in the trajectory of a struck tennis ball. The velocity and spin of a hit ball is determined by the ball-racket COR together with all the details of the stroke, the incident ball, and the kinematics. For typical values of these quantities, the difference in the hit ball speed at the center of the rectangular racket and 3.5 inches down is only about 0.15 mph. Assuming typical values for the ball friction coefficient and drag coefficient, the resultant difference in the hit ball trajectory is less than 4 inches overall. The corresponding difference for the oval racket is 6.5 mph, and the resultant difference in the hit ball trajectory is more than 12.5 feet. The superiority of the rectangular racket is clear
We have presented specific preferred embodiments of the rectangular racket concept, but there are many other possible embodiments of our inventions as will be understood by those of skill in the art.
Although other modifications and changes may be suggested by those skilled in the art, it is the intention of the inventors to embody within the patent warranted hereon all changes and modifications as reasonably and properly come within the scope of their contribution to the art.
Claims
1. A sports racket, comprising:
- a racket body including: a racket head having a substantially rectangular face shape, said substantially rectangular face shape including rounded corners, said racket head defining a plurality of string mounting positions in a predetermined arrangement on the four opposing sides of said rectangular face shape; a racket handle affixed to said racket head; and
- at least one string mounted at said string mounting positions to define main string portions and cross string portions, said main string portions each being of substantially equal length and substantially equal tension, said cross string portions each being of substantially equal length and substantially equal tension.
2. A sports racket as claimed in claim 1, wherein said racket body with said at least one string weighs less than 14 ounces.
3. A sports racket as claimed in claim 1, wherein said racket body with said at least one string weighs less than 12 ounces.
4. A sports racket as claimed in claim 1, wherein said racket body with said at least one string weighs less than 10 ounces.
5. A sports racket as claimed in claim 1, wherein said main string portions and said cross string portions are of substantially equal string tensions when strung in said racket head.
6. A sports racket as claimed in claim 1, wherein said rectangular racket head is formed with a shape having outwardly bowed sides such that when strung the racket head sides are drawn into substantially straight sides by the string tension.
7. A sports racket, comprising:
- a racket body including: a racket head having a substantially rectangular face shape, said substantially rectangular face shape including rounded corners, said racket head defining a plurality of string mounting positions in a predetermined arrangement on four opposing sides of said rectangular face shape; a racket handle affixed to said racket head; and
- at least one string mounted at said plurality of string mounting locations to define main string portions and cross string portions, said main string portions each having a main string fundamental vibration frequency where each main string frequency is substantially the same, said cross string portions each having a cross string fundamental vibration frequency where each cross string frequency is substantially the same.
8. A sports racket as claimed in claim 7, wherein said racket body and string weighs less than 14 ounces.
9. A sports racket as claimed in claim 7, wherein said racket body and string weighs less than 12 ounces.
10. A sports racket as claimed in claim 7, wherein said racket body and string weighs less than 10 ounces.
11. A sports racket as claimed in claim 7, wherein said at least one string has a substantially constant linear string density, and wherein (√ti)/li=constant and (√sj)/kj=constant, where ti is a string tension of a main string portion, li is a string length of the main string portion, sj is a string tension of a cross string portion, and kj is a string length of the cross string portion.
12. A sports racket as claimed in claim 7, wherein said at least one string has a variable linear string density, and wherein that (√ti/mi)/li=constant and (√sj/m′j)/kj=constant, where ti is a string tension of a main string portion, mi is a string density of the main string portion, li is a string length of the main string portion, sj is a string tension of a cross string portion, m′j is a string density of the cross string portion, and kj is a string length of the cross string portion.
13. A sports racket as claimed in claim 7, wherein (√ti/mi)/li=(√sj/m′j)/kj.
14. A sports racket as claimed in claim 7, further comprising:
- lockable grommets mounted at least some of said string mounting locations to hold said string portions in place.
15. A sports racket as claimed in claim 7, further comprising:
- a lockable grommet at one end of each of said main string portions and at one end of each of said cross string portions to hold said string portions in place.
16. A sports racket as claimed in claim 7, further comprising:
- a lockable grommet at each end of each of said main strings and at each end of each of said cross strings to hold said strings in place.
17. A sports racket as claimed in claim 14, wherein said lockable grommets hold said string portions at said string mounting locations to maintain a predetermined tension on predetermined ones of the string portions.
18. A sports racket as claimed in claim 17, wherein said lockable grommets on said cross string portions maintain tensions on said cross string portions so that (√sj)/kj=constant, where sj is the tension on the cross string portions and kj is the length of the cross string portions, and
- wherein said lockable grommets on said main string portions maintains tension on said main string portions so that (√ti)/li=constant, where ti is the tension on the main string portions and li is the length of the main string portions.
19. A sports racket as claimed in claim 17, wherein said lockable grommets on said cross string portions maintain tensions on the cross string portions such that (√sj/m′j)/kj=constant, where sj is the tension on the cross string portions, m′j is the mass density of the cross string portions, and kj is the length of the cross string portions; and
- wherein said lockable grommets on said main string portions maintain tensions on the main string portions such that (√ti/mi)/li=constant, where ti is the tension on the main string portions, mi is the mass density of the main string portions, and li is the length of the main string portions.
20. A sports racket, comprising:
- a racket head including an open frame defining an arrangement of string holes at opposing sides of said open frame, said string holes including substantially conical hole portions;
- a handle affixed to said racket head;
- at least one string disposed through said string holes and extending across said open frame to define a racket face; and
- a plurality of lockable grommets disposed in ones of said string holes, said grommets including substantially conical grommet portions inserted into said substantially conical hole portions, said lockable grommets engaging and restraining said at least one string under tension by frictional engagement between said substantially conical hole portions and said substantially conical grommet portions and said string.
21. A sports racket, comprising;
- a racket head including an open frame defining an arrangement of string holes at opposing sides of said open frame, said string holes including substantially conical hole portions, said substantially conical hole portions being threaded;
- a handle affixed to said racket head;
- at least one string disposed through said string holes and extending across said open frame to define a racket face; and
- a plurality of lockable grommets disposed in ones of said string holes, said grommets including threaded substantially conical grommet portions inserted into said threaded substantially conical hole portions, said lockable grommets engaging and restraining said at least one string under tension by threaded engagement between said threaded substantially conical hole portions and said threaded substantially conical grommet portions and said string.
22. A sports racket, comprising:
- a racket head including an open frame defining an arrangement of string holes at opposing sides of said open frame, said string holes including substantially conical hole portions;
- a handle affixed to said racket head;
- at least one string disposed through said string holes and extending across said open frame to define a racket face; and
- a plurality of lockable grommets disposed in ones of said string holes, said grommets each including a substantially cylindrical bar disposed in said substantially conical hole portions, said grommets each including a stopping block in said substantially conical hole, said substantially cylindrical bar being wedged in said substantially conical hole to hold said at least one string in place under tension, said substantially cylindrical bar permitting said at least one string to be pulled through said grommet to increase tension on the string but preventing movement of the string in an opposite direction.
23. A sports racket, comprising:
- a racket head including an open frame defining an arrangement of string holes at opposing sides of said open frame;
- a handle affixed to said racket head;
- at least one string disposed through said string holes and extending across said open frame to define a racket face; and
- a plurality of lockable grommets disposed in ones of said string holes, said grommets including first and second locking portions and a biasing member, said biasing member biasing said first and second locking portions toward one another to engage said at least one string to hold said at least one string under tension, said first and second locking members permitting said at least one string to move through said locking portions in a direction to increase tension on the string but preventing movement of the string in an opposite direction.
24. A sports racket, comprising:
- a racket head including an open frame defining an arrangement of string holes at opposing sides of said open frame;
- a handle affixed to said racket head;
- at least one string disposed through said string holes and extending across said open frame to define a racket face; and
- a plurality of lockable grommets disposed at ones of said string holes, said grommets including a string passageway and a threaded bore opening onto said string passageway and a threaded member threadably engaged in said threaded bore, said threaded member bearing against said at least one string in said string passageway to secure said at least one string in said string passageway.
25. A sports racket as claimed in claim 24, further comprising:
- an external block defining said string passageway and said threaded bore and into which said threaded member is engaged.
26. A sports racket as claimed in claim 24, further comprising:
- a portion of said racket head defining said threaded bore and into which said threaded member is engaged.
27. A sports racket, comprising:
- a racket head including an open frame defining an arrangement of string holes at opposing sides of said open frame;
- a handle affixed to said racket head;
- at least one string disposed through said string holes and extending across said open frame to define a racket face; and
- a plurality of lockable grommets disposed at ones of said string holes, said grommets each including a compressible elastic string gripping element engaging said string, said grommets each including a threaded cone engaged with said gripping element, said grommets each including a threaded nut threadably engaged on said threaded cone so as to compress said gripping element and thereby secure the string in position.
28. A sports racket, comprising:
- a racket head including an open frame defining an arrangement of string holes at opposing sides of said open frame;
- a handle affixed to said racket head;
- at least one string disposed through said string holes and extending across said open frame to define a racket face; and
- a plurality of lockable grommets disposed at ones of said string holes, said grommets each including a compressible elastic string gripping element engaging said string, said string gripping elements being held in threaded nuts, and threaded portions being provided at said string holes onto which said threaded nuts are threadably engaged so as to hold said string under tension in said string holes.
29. A sports racket, comprising:
- a racket frame;
- a racket handle connected to said racket frame,
- said racket frame and said racket handle being formed of a carbon fiber composite materials and having an sandwich structure, said sandwich structure including first and second opposed carbon fiber sheets between which is disposed a light core material.
30. A sports racket as claimed in claim 29, wherein said racket frame and said racket handle is in a single piece and has a structure characterized by having been cut from a single sheet of sandwich material.
31. A sports racket as claimed in claim 30, wherein said sports racket has a moment of inertia of a face cross section at least twice as large as an annular cross section of a same area and similar dimensions.
32. A sports racket as claimed in claim 29, further comprising:
- a carbon fiber strip affixed to an outer perimeter face of at least portions of said sports racket.
33. A sports racket as claimed in claim 32, wherein said carbon fiber strip is of a “C” shape and has first and second portions overlapping first and second outer layers of the sandwich structure, respectively.
34. A sports racket as claimed in claim 32, further comprising:
- a plurality of layers of Kevlar material applied to said carbon fiber strip.
35. A sports racket as claimed in claim 32, wherein an area moment of inertia of a face cross section is at least 1.8 times as large as a moment of inertia of an annulus of a same area.
36. A sports racket as claimed in claim 32, wherein said racket weighs less than 14 ounces.
37. A sports racket as claimed in claim 32, wherein said racket weighs less than 12 ounces.
38. A sports racket as claimed in claim 32, wherein said racket weighs less than 10 ounces.
39. A sports racket as claimed in claim 32, wherein said racket frame has a inner face length is between 11 inches and 15 inches, said racket frame having a inner face width of between 9 inches and 11 inches, said racket frame has a face side width of between 0.25 inches and 0.75 inches, said racket frame has a face side depth is between 0.5 inches and 1 inch, and said racket frame has a sandwich plate thickness is between 0.025 inches and 0.15 inches.
40. A sports racket as claimed in claim 39, wherein stress induced in side portions of the racket frame by string tensions of less than 70 pounds is less than 80 percent of a rupture force of the frame face material.
41. A sports racket as claimed in claim 32, wherein said racket frame is of a substantially rectangular face shape, said substantially rectangular face shape including rounded corners, said racket head defining a plurality of string mounting positions in a predetermined arrangement on four opposing sides of said rectangular face shape; at least one string mounted at said plurality of string mounting locations to define main string portions and cross string portions, said main string portions each having a main string fundamental vibration frequency, said cross string portions each having a cross string fundamental vibration frequency, said main string fundamental vibration frequency being substantially equal to said cross string fundamental vibration frequency.
- a racket handle affixed to said racket head; and
42. A sports racket as claimed in claim 39, further comprising:
- lockable grommets mounted so as to hold at least one string in said racket frame.
43. A sports racket as claimed in claim 42, wherein said lockable grommets maintain a predetermined string tension on at least one string strung in said racket frame.
44. A sports racket as claimed in claim 42, wherein said lockable grommets maintain predetermined tension on said at least one string, said predetermined tension being such that main string portions of the at least one string have substantially equal fundamental vibration frequencies, said predetermined tension being such that cross string portions of the at least one string have substantially equal fundamental vibration frequencies.
45. A sports racket as claimed in claim 44, wherein said fundamental vibration frequencies of said main string portions is substantially equal to said fundamental vibration frequencies of said cross string portions.
46. A sports racket as claimed in claim 29, wherein said sports racket weighs less than 14 ounces.
47. A sports racket as claimed in claim 29, wherein said sports racket weighs less than 12 ounces.
48. A sports racket as claimed in claim 29, wherein said sports racket weighs less than 10 ounces.
49. A method for providing a sports racket, comprising the steps of:
- forming a racket frame so that said frame has a substantially rectangular shape;
- stringing the racket frame with at least one string to provide main string portions and cross string portions extending between opposing sides of said substantially rectangular frame;
- tensioning the main string portions so that each of the main string portions is of substantially equal length and substantially equal tension; and
- tensioning the cross string portions so that each of the cross string portions is of substantially equal length and substantially equal tension.
50. A method as claimed in claim 49, wherein said step of forming includes forming the substantially rectangular racket frame with outwardly bowed sides such that said steps of tensioning the main string portions and tensioning the cross string portions exerts sufficient force on the bowed sides to draw the sides into substantially straight sides under string tension.
51. A method as claimed in claim 49, wherein said step of stringing applies initial tension to the main string portions and the cross string portions; and
- wherein said tensioning steps apply final tension to the main string portions and the cross string portions.
52. A method as claimed in claim 49, wherein said tensioning steps applied substantially the same tension to the main string portions and the cross string portions.
53. A method as claimed in claim 49, wherein said steps of tensioning include the sub-steps of locking the string portions in lockable grommets to maintain tension on the string portions.
54. A method as claimed in claim 53, wherein said substep of locking includes locking the string portions in the lockable grommets by wedging force.
55. A method as claimed in claim 53, wherein said sub-step of locking includes locking the string portions in the lockable grommets by threadably engaging a first threaded member with a second threaded member.
Type: Application
Filed: Jan 18, 2012
Publication Date: Aug 23, 2012
Inventor: Richard A. Brandt (New York, NY)
Application Number: 13/352,853
International Classification: A63B 49/02 (20060101); A63B 51/00 (20060101);