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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.
1. 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.
The 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.
FIG. 1 is a top plan view of a tennis racket head having an overall rectangular shape according to a preferred embodiment of the present invention;
FIG. 2 is a graph showing deflection of a racket element upon application of string tension forces;
FIG. 3 is a graph of the deflection according to FIG. 2 but with the vertical axis exaggerated to better illustrated the quantity of deflection;
FIG. 4 is a top plan view of a tennis racket according to an embodiment of the present invention which establishes the dimensions described in the specification;
FIGS. 5a-5j are schematic illustrations in cross section of a plurality of embodiments of string grommets as used in preferred embodiments of the present tennis racket;
FIG. 6 is a schematic diagram of a beam supported at its ends and along which is applied regularly spaced forces to demonstrate forces on a side of a strung racket;
FIG. 7 is a schematic illustration of deflection of the beam of FIG. 6 when subjected to the stringing forces;
FIG. 8 is a cross section through a hollow cylindrical tennis racket frame showing stringing forces thereon;
FIG. 9 is a cross section through a tennis racket frame according to the principles of the present invention;
FIG. 10 is a plan view of a tennis racket according to one embodiment of the present invention;
FIG. 11 is a schematic representation of a tennis racket according to another embodiment of the invention;
FIG. 12 is an enlarged fragmentary view of a corner of a rectangular tennis racket showing string forces;
FIG. 13 is a schematic illustration of a beam to which is applied a force;
FIG. 14 is a cross sectional view through a side of a tennis racket frame in a preferred embodiment;
FIG. 15 is an enlarged fragmentary view of a corner of a rectangular tennis racket of a preferred embodiment;
FIG. 16 is a schematic representation of a sandwich of materials used in making a tennis racket according to a preferred embodiment;
FIG. 17 is a schematic representation of a tennis racket manufactured as described;
FIGS. 18a and 18b are a plan view and side view, respectively, of a tennis racket face according to a preferred embodiment;
FIGS. 19a, 19b and 19c are a plan view, side view and end view of a handle portion of a tennis racket according to a preferred embodiment;
FIG. 20 is a schematic representation of a force measuring apparatus used in the present method;
FIG. 21 is a perspective view of a testing apparatus as used in the present method;
FIG. 22 is a close up top view of the measuring apparatus;
FIG. 23 is a diagram of a frame prototype manufactured to demonstrate a proof of concept;
FIG. 24 is a perspective view of the frame prototype of FIG. 23 undergoing testing;
FIG. 25 is another view of the frame prototype undergoing testing;
FIG. 26 is a top perspective view of a racket constructed according to the teachings of the present invention; and
FIG. 27 is a view of test equipment for testing the present tennis racket.
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.
The 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.
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 FIG. 1.
In particular, FIG. 1 shows a tennis racket 20 having a face or frame 22 made up of substantially straight sides 24 and substantially straight ends 26. The frame 22 has rounded corners 28 and a neck 30 that extends to a handle (not shown in this view). Main strings 32 extend between the ends 26 and cross strings 34 extend between the sides 24.
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 FIG. 2. A curve 36 is plotted on the graph to illustrate the deflection of the beam along the length of the frame beam.
With reference to FIG. 3, for viewing ease, the plot of the deflection curve 36 is redrawn with a 4 to 1 aspect ratio.
With the stated values of the relevant parameters, the beam, representing the long side 24, in FIG. 1, of the racket face, is seen to deflect a maximum distance of about 0.5 inch. This maximum deflection of course occurs at the center of the beam (x=6.5 inch), and the deflection distances continuously decrease in value away from the center. The horizontal string 34 at the center of the racket face therefore decreases in length by about 1 inch (0.5 inch from the bending of each side 24).
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/s^{2}), 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=l1^{2}/l2^{2}.
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.
In 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 FIG. 4. Let w be the initial constant distance between the top and bottom (long) sides 24 of the face 22, and let 1 be the distance between left and right (short) sides or ends 26 of the face 22. Let x denote the distance from the left side of the face, 0<x<1. Assume we start the stringing process with the leftmost vertical string, at a distance x=x1. This string is attached at the bottom side and pulled through the top locking grommet with a chosen tension t1, and then locked into place. This string will pull the upper racket side down a distance d1(x). The lower side will be pulled up the same distance, so that the length of the string after locking will be w−2d1(x1). When the second (adjoining) string section is similarly attached at x=x2 with a chosen tension t2, the upper racket side will be pulled down a further distance d2(x). This will decrease the length of the first string section to w−2d1(x1)−2d2(x1), and so the tension in this string section will be reduced to t1−2kd2(x1), where k is the spring constant for the chosen string
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 FIGS. 5a-5j. Each of the schematic illustrations of the locking grommets show a section through a right side beam 24 or 26 of a racket face, so that the racket face is to the left of the segment. Each segment contains a single hole 40 through which a string 32 or 34, depicted as a right-facing arrow, passes. For this illustration it is assumed that the string 32 or 34 is attached to the opposite (left) side of the face, and that the desired tension is applied to the string by pulling the string to the right through the depicted hole 40 in the direction of the arrow. The illustrations provide methods of locking the string within the hole 40 in the beam at this chosen tension using the locking grommet.
In FIG. 5a, the hole 40 is conical and the string 32 or 34 is locked in place by inserting a conical plug 42, shown in FIG. 5b, into the hole 40. This simple locking grommet mechanism uses the friction between the string 32 or 34 and the plug 42 and hole 40 to restrain the string. In FIG. 5c, the conical hole 40 is threaded as indicated by heavy lines 44 and the string 32 or 34 passes through a center hole 48 in a compressible threaded conical screw 50. The screw 50 is of a material that compresses when tightened into the conical hole 40 so as to decrease the diameter of the central hole 48. The screw 50 includes means for threadably tightening the screw into the hole 40, such as a slot or shaped recess that receives a screwdriver, for example. When the screw 50 is turned into the hole 40, it compresses onto the string 32 or 34 and locks the string into place. The conical threaded screw 50 is shown in cross section in FIG. 5d with the central hole 48 in an open, string receiving position. An outer surface 52 of the conical screw 50 is threaded with a threading corresponding to the threading 44 in the conical hole 40 of FIG. 5c. It is understood that the two part illustration of FIG. 5d shows the two halves of a single conical screw.
In a further conical-hole 40 example of a locking grommet, shown in FIG. 5e, the string 32 or 34 passes under a cylindrical bar 54 that resides in the hole 40 to the left (with respect to the drawing) of a stopping block 56 (shown as a black rectangle). The stopping block 56 is secured in place in the conical hole 40 by being anchored into the hole wall 40 although other anchoring means are also contemplated. The string 32 and 34 can be freely pulled to the right as the cylinder 54 rotates, but when the pulling force on the string is released, the cylinder 54 is wedged into the conical hole 40 and holds the string in place.
The drawing in FIG. 5f depicts a locking grommet with a hole 58 in the racket frame having a stepped configuration and includes a pair of elements 60 (shown as shaded bars that are angled toward one another) mounted at the step in the hole 58. The elements 60 that are forced together by spring mechanisms shown schematically at 62 that for example are positioned between the racket body and each element 60 to bear against the elements and press them against the string. Two or more such elements 60 may be provided at the hole 58. The friction between the elements 60 and the string 32 or 34 prevents the string from moving back to the left, but the string can be easily pulled to the right. The string is therefore locked in position at the desired tension by the locking grommet.
The locking grommet of FIG. 5g depicts the use of a tightening screw 64 to lock the string 32 or 34 into place. The string passes through a central hole 66 in an external block 68 (as shown in black) that is affixed to the frame of the racket at the hole 70. The screw 64 is threaded into cooperating threads in a bore 72 in the block that extends transverly to the hole 66 through which the string extends. The screw 64 is turned down into the bore 72 until it bears against the string 32 or 34, forcing the string against an opposite will of the central hole 66 to lock the string in place. The external block 68 is affixed to the frame of the racket so as to not come loose during use.
Referring to FIG. 5h, a similar threaded screw 64 is threadably received into a bore 72 in a portion 74 of the frame as defined by a recess 76. By tightening the screw 64 into the bore 72, the screw 64 presses the string 32 or 34 against an inside wall of the hole 70 so as to secure the string at the desired tension and position. The recess 76 and/or the screw 64 are configured to permit the screw to be tightened and loosened by a user's fingers or by a tool. Thus, this embodiment locks the string position by a locking grommet built into the frame itself.
In FIG. 5i is shown a locking grommet embodiment that makes use of a compressible elastic element 78 through which the string 32 or 34 passes as indicated by the broken line. The compressible element 78 is disposed in a conical shaped sleeve or cone 82 formed in the racket frame. The compressible element is compressed by screwing a threaded nut 80 onto the threaded cone 82. The threaded cone 82 is disposed in a recess 84 in the frame. Rotating the nut 80 to the left causes the cone 82 to compress the element 78 onto the string and thus hold it in place. In FIG. 5i, the element 78 may release the string for tension adjustment by screwing the threaded nut 80 to the right on the threaded cone 82. Rotating the nut 80 to the left again locks the string at the desired tension by the compression of the elastic element 78 onto the string and to hold it in place.
A similar concept is shown in FIG. 5j, wherein the element 78 is compressed by screwing a threaded cone 86 over a threaded cylinder 88. Rotating the cone 86 to the left causes the cone 86 to compress the element 78 onto the string and thus hold it in place.
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.
General 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.
BULK PROPERTIES OF MATERIALS | ||||
MATERIAL | E/10{circumflex over ( )}6 (psi) | p (pci) | E/p10{circumflex over ( )}6 | |
Aluminum | 10 | 0.10 | 100 | |
Carbon Fiber | 22 | 0.06 | 367 | |
Titanium | 27 | 0.17 | 159 | |
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 FIG. 6. The beam 90 is a simple model of a side 24 or 26 of the face of a rectangular racket with supports 92 near each end. The strings attached to the side exert an approximately uniform force across the top of the beam 90, of magnitude F=NT, where T is the tension in each string and N is the number of strings. This force causes the beam 90 to deflect downward, a maximum distance D, as shown in FIG. 6.
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 FIG. 6.
As shown in FIG. 7, the beam 90 deflects or bends under the applied force. The deflection distance D is determined by the applied force F, the beam length L, the material Young's Modulus E, and the area moment of inertia I of the beam cross-section according to the following equation:
I is defined by the geometry of a cross-sectional slice of the beam according to the equation:
I=∫dAy^{2}. (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 FIG. 8, with inner radius r1 and outer radius r2, y is the vertical distance between the horizontal neutral axis (as indicated by a dashed line) and the area elements in the material between the circles (as indicated by a shaded area). In this case.
The vertical arrow 98 in the FIG. 8 represents the force exerted by the string tension.
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 FIG. 8. Such a tubular beam is much stronger that a solid rod of the same material and equal weight, because I for the tube, Eq. 9, is much larger than 1 for the rod, I=πr^{4}/4, where r is the radius of the rod.
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.
The 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 FIG. 8. As we noted above, this structure is, however, not strong enough to form a preferred light-weight rectangular racket face embodiment. In this section we will teach how a much stronger frame can be constructed using carbon fiber sandwiches. Such a sandwich consists of two parallel relatively thin carbon fiber plates separated by a light filler material.
The cross-section of a beam 100 made out of such a sandwich is illustrated in FIG. 9. The neutral axis 102 is indicated by the dashed line, and the vertical arrow 104 represents the force exerted by the string tension. Comparison of FIGS. 8 and 9 shows that the material in the tubular structure has some material far from the neutral axis, but also some material close to the neutral axis. The sandwich structure, on the other hand, has all the material (apart from the light filler material) far from the neutral axis.
The dimensions given in FIG. 9 are d, the overall height, h, the height of the filler material 108 between the plates 106, and b, the width of the sandwich. The thickness of the carbon fiber plates 106 is (d−h)/2 each. The area moment of inertia of this sandwich cross-section is defined by equation
I_{S}=b(d^{3}−h^{3})/12=b(d−h)(d^{2}+dh+h^{2})/12=A_{S}(d^{2}+dh+h^{2})/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
I_{A}=π(r_{2}^{4}r_{1}^{4})/4=π(r_{2}^{2}−r_{1}^{2})(r_{2}^{2}+r_{1}^{2})/4=A_{A}(r_{2}^{2}+r_{1}^{2})/4, (Eq. 13)
where A_{A}=π(r_{2}^{2}−r_{1}^{2}) 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 FIG. 9, can be combined to construct a racket 110, as shown in FIG. 10. The face corners 112 consist of sandwich beams butted together and held together with a suitable epoxy. This simple construction is, however, not strong enough to withstand the stresses at the corners caused by the string tensions. Reinforcement at the corners is necessary, and that adds to the weight and top-heaviness of the racket.
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 FIG. 11. The corners 116 are stronger, but still not strong enough without using carbon fiber plates that are relatively thick and wide and therefore rather heavy.
To understand why the corners must be strengthened, consider the corner 116 illustrated in FIG. 12. This is a front view of the racket face, with the parallel carbon fiber plates 106 shown as shaded elements, and the strings 118 are represented by the vertical arrows. The tensions in these strings 118 exert strong stresses on the corners 116 because the carbon fiber plates 106 are long and thin. We will estimate these stresses below.
With reference to FIG. 13, to estimate the corner stress, we will model the upper plate as a cantilever beam 120 rigidly attached to a solid support 122 at the left end. If a force F is applied on the top of the beam 120 at a distance L from the support, and the beam 120 has width w and thickness t, the resultant stress at the support is given by the equation
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
σ_{max}=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
σ_{rupt}=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 FIG. 11 tends to pull the plate downward, and it is difficult for the weak filler material 126 between the plates to withstand this force. Stronger and heavier filler must be used to prevent the force from crushing it, resulting again in unwanted additional frame weight.
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.
Although 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 FIG. 9. Such a table cross-section is illustrated in FIG. 14. This section has three carbon fiber outer plates 130, 132 and 134, instead of the two plates 106 shown in FIG. 9. The upper (table top) plate 134 is similar to the upper sandwich plate 106, but the bottom sandwich plate 106 is replaced by two side (table leg) plates 130 and 132. The neutral axis 136 is indicated by the dashed line, and the vertical arrow 138 represents the force exerted by the string tension. The volume 140 between the table legs 130 and 132 consists of light filler material as in the sandwich. This table section will have a smaller area moment of inertia per pound than the sandwich section because it has more carbon fiber material placed closer to the neutral axis. It is therefore necessary for the table beam 134 to be heavier than the sandwich beam in order to have equal strength, but this small additional weight is more than made up for by the fact that the racket constructed from these table beams requires no corner or filler reinforcement. We will establish these facts below.
The dimensions given in FIG. 14 are d, the width of the upper (table top) carbon fiber plate 134, b, the height of the side (leg) plates 130 and 132, h, the width of the filler material 140 between the plates, and c, the thickness of the upper plate 134. The thickness of the side plates 130 and 132 is (b−h)/2 each.
The total area of the carbon fiber sections in FIG. 14 is A_{T}=2ab+cd, and the area moment of inertia of this cross-section is defined by equation
I_{T}=d(3b^{2}c+6bc^{2}+4c^{3})/12l+ab^{3}/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
I_{T}≈db^{2}c/4+ab^{3}/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 I_{T }(Eq. 19) will be less than the sandwich moment I_{S }(Eq. 12). However, the dimensions a, b, c, d, can be chosen such that I_{T }is at least 90% as large as I_{S }(and therefore, given Eq. 15, over 80% larger than the annulus moment I_{A}). 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 FIG. 12 above. The corresponding table corner 142 is illustrated in FIG. 15. We will first estimate the stress induced by the tensions in the vertical strings using the cantilever of FIG. 13 to model the plates. For the sandwich corner 116, the plate is thin (t˜0.1 inch) and wide (w˜0.75 inch), whereas for the table corner, the plate 134 is fat (t˜0.75 inch) and narrow (w˜0.1″). This makes a large difference in the stress σ_{max }at the support.
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
σ_{max}=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 FIG. 14 that, since the top carbon fiber plate 134 rests on the side plates 130 and 132, the string tensions 138 applied to it do not exert significant force on the filler material 140. Essentially all of the applied force is resisted by the side plates 130 and 132. Laboratory verification of this fact will be reported in the following section.
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 FIG. 16. It consists of top and bottom carbon fiber plates 150 and 152, separated by light filler material 154. The filler material 154 may be a fiber or non-fibrous material, an expanded foam material, or other materials made of natural or synthetic materials. These plates will become the side (table leg) plates 130 and 132 of the table in FIG. 14.
The racket frame can be cut out of this sandwich 148 in a single piece 156, as illustrated in FIG. 17. Then one attaches a long thin carbon fiber strip, of thickness equal to the dimension c in FIG. 14, and width equal to the thickness of the above sandwich (the dimension d in FIG. 14). This strip can be attached to the outer perimeter of the racket face in FIG. 17, using suitable epoxy. For added strength, this strip can have upper and lower lips, rendering it into a c-shape. Such a strip can be fitted around the outer perimeter of the racket face and bonded with epoxy. The lips or extensions extend onto the top and bottom plates 150 and 152, adding not only strength but a finished look. Finally, stringing holes can be drilled in appropriate places around the face perimeter 158, grommets, or grommet strips, can be inserted into these holes, and the carbon fiber handle stem 160 can be supplemented with a light material and then strapped. The racket would then be ready for stringing.
The shape of the cutout in FIG. 17 can be generalized to include the improvements discussed previously. The corners 162 can be rounded, to improve the racket strength and appearance. The face sides 164 can be bowed outward so that a purely rectangular face will obtain after stringing. The side strips can be grooved or otherwise altered to accommodate grommets and strings. This ease of construction is a major advantage of the present table construction.
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.
To 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 FIG. 1 illustrates this pattern. The gaps between the first and last strings and the inside frame sides are 1 inch in the vertical direction and 1.5 inches in the horizontal direction.
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 FIG. 14), large enough so that the sides and corners of the racket are strong enough to withstand string tensions in the usual range 55-65 lbs, and small enough so that the 14 oz. weight limit is not exceeded. Once these table dimensions are determined, the amount of inward bowing of the face sides, caused by the string tensions, must be calculated or measured. If the maximum bowing distance exceeds the 0.23 inch limit derived above (following Eq. 6), then the racket must be constructed with outward side bowing in order to compensate for this.
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=2×10^{7 }psi, and the density is ρ=0.96 oz/in^{3}.
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 I_{T}=0.0083 in^{4}, 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 FIGS. 18a and 18b (for the racket face 166) and FIGS. 19a, 19b and 19c (for the throat 168 and handle 170). In addition to the features specified above, these drawings incorporate some other properties. Since the entire frame is cut out of a carbon fiber sandwich of thickness d=0.75 inch, all of its elements (face 166, throat 168, handle stem 170) will have this thickness. The inside corners are rounded with radius 1 inch, and the outside corners are rounded with radius 1.625 inches. The throat sides have width b=0.625 inch. The throat tapers into a handle stem 0.625 inch wide. All of the other dimensions are specified in the drawings. A number of holes are depicted in the drawings. These serve to lighten the racket in areas where there is sufficient strength to accommodate them. For example, holes 174 are shown at the corners 176 where the throat 168 attaches (see FIG. 18a). Holes 178 and 180 are aligned along the center of the handle 170 in the top view of FIG. 19a and holes 182 are transverse to the holes 180 as seen in the side view of FIG. 19b. The holes may be filled with a light weight material or left as openings in the racket frame. Handle shaping elements 184 are provided on the handle portion 170 to provide a comfortable grip.
FIGS. 18a and 18b show a tennis ball 172 in phantom for a sense of scale.
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 FIGS. 19a, 19b and 19c as the handle shaping elements to enlarge the handle into the desired size (width 1.25 inches and height 1.14 inches), and shape (an octagonal shape is standard and is used here).
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 FIG. 18b. Grommets and preferably locking grommets as described above or other types of locking grommets are to be inserted into these holes, and then the racket can be strung. Finally, an end cap 186 and strapping can be attached to the handle.
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 (FIG. 14), with the chosen dimensions (Eq. 22), is strong enough to hold the inserted grommets and withstand the force from the string tensions, we have devised the apparatus illustrated in FIG. 20. A section of the table beam 200 (shown in grey) is held in place by a pair of solid stops 202 (shown in black). A racket string 204 is threaded through grommets 206 in two holes in the beam 200, and attached to a load cell 208 (shown as a shaded element to the right). The load cell 208 is attached to a block 210 (shown in black) that can be moved backward by rotating an inserted threaded bolt 212. This backward movement exerts a force F on the strings, and the magnitude of the force is indicated on a display unit 214 (shown as a shaded block in the lower center) wired to the load cell.
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 FIGS. 21 and 22.
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 FIG. 23. The central part 214 of the upper part of the replica 212 has the same cross-section dimensions (Eq. 22) as the table cross-section shown in FIG. 14. The inside replica length is 4.5 inch, and the outside length is 5.75 inch. (The inside length of the racket face drawn in FIG. 18a is 12 inches.) The corner curvatures of the replica and the real racket are the same.
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 FIGS. 24 and 25. We continually increased the applied force F until one of the upper corners in the replica began to rupture. (One of the corners was made thicker than the other to insure that the other corner ruptured first.) We observed such a rupture when the applied force reached 2250 lbs. Since this force is well above the 1390 lb equivalent of the 1140 lb force applied by the strings on our real racket, we conclude that this real rectangular racket has corners that are more than sufficiently strong to withstand the encountered stresses. This confirms the correctness of our theoretical estimates of our corner strength and gives us the confidence to proceed with the construction of the racket.
A photograph of a produced racket 230, constructed according to the above specifications, is shown in FIG. 26. The measured weight (W), center of mass (COM) and weight moment of inertia, about the axis perpendicular to the racket handle and parallel to the racket face (MOI), located 6 inches from the end of the handle, are as follows
W=14 oz
COM=9.87″
MOI=2400 oz.in^{2}. (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 FIG. 27.
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.
FIXED FRAME VELOCITY RATIO COMPARISONS | ||||||
Rectangular | ||||||
Racket | Oval Racket | |||||
Speed | center | 3.5″ down | % decr | center | 3.5″ down | % decr |
46.5 mph | 0.771 | 0.767 | 0.52 | 0.811 | 0.726 | 10.48 |
65.3 mph | 0.667 | 0.662 | 0.75 | 0.725 | 0.663 | 8.55 |
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.
FREE FRAME COR CHANGE COMPARISONS | |||
Speed = 65.3 mph | Rect Racket | Oval Racket | |
Equal weight, MOI, COM | 1.19% increase | 7.45% decrease | |
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.