EP0104930B1

EP0104930B1 - Frame for sports racket

Info

Publication number: EP0104930B1
Application number: EP83305749A
Authority: EP
Inventors: Tsai Chen Soong
Original assignee: Individual
Current assignee: Individual
Priority date: 1982-09-27
Filing date: 1983-09-27
Publication date: 1989-04-26
Also published as: EP0104930A1; DE3379712D1

Description

Frames for sports rackets, and particularly for tennis rackets, present an engineering challenge. They must be strong enough to withstand enormous loads, be as nearly rigid as possible, and yet use only a few ounces of material. For example, a conventional tennis racket weighs approximately 355 to 411 grams; and its center of gravity is in the vicinity of its throat, which makes the weight attributed to the frame extending around the ball-hitting region from 170 to 199 grams. This low weight of material must sustain a tremendous string load of up to 36 kilograms per string and a ball-hitting load of 46 kilograms or more, repeated for perhaps 40,000 shots without a failure. Understandably, sports racket frames have not yet fully met such a challenge.
Steel frame rackets are known to too flexible or "whippy". Since steel is heavy, its walls have to be made thin to remain light in weight, giving its frame section insufficient moment of inertia for resisting bending and torsion loads.
Frame sections formed of aluminum alloy can have thicker walls and be more rigid, but they tend to permanently deform due to lower yield strength. (Aluminum Company of America) (Alcoa) heat-treatable 60-T6 series or 70 series improve the strength of aluminum considerably, but not enough to eliminate frame problems.
Graphite and composite materials, although expensive, have produced frame strips of very high strength-to-weight ratios that increase possible alternatives.
Frame strips presently used in metal rackets fall into two categories-those which are tubular with an oval or rectangular cross section and those which are of I-beam configuration with solid or tubular flanges. For the latter, the tubular flange on both ends of the web provides torsional and bending rigidity resisting ball impact; and the thick web provides a bearing seat supporting string holes. Although quite popular, an I-beam section has the inherent problem of a marginal bending movement of inertia to resist the pulling load from the strings in the plane of the string surface, since most of the sectional mass is along the web of the I-beam of the frame to provide a solid seating for the strings. For example, the moment-of-inertia ratio between the axis perpendicular to the web and the axis coinciding with the web for the Head Edge racket frame section is 7.6 to 1.0.
For a tubular frame of rectangular cross-section, the disparity between moments of inertia along the two principal axes of the cross-section is not as drastic as for I-beam type frames, but even these are usually narrowed in the middle of the section to provide the necessary string support. US-A-1937787 shows a range of tubular frames which are all relatively narrow and would lack sufficient moment of inertia for the game played today. However, wider sections would lead to an unwieldy and overweight construction.
Graphite rackets also follow the general geometry of metal tubing frames, and they too have a narrow neck at the middle of the cross-section where the string hole is bored through the frame strip.
I have thoroughly studied the problems of sports racket frames, and tennis racket frames in particular, and have used the finite-element structural mechanical analysis method to study the loads imposed on a tennis racket from the strings and from the impact of the ball. Through such analysis, I have discovered a better cross-sectional shape and overall configuration for a racket frame having several important advantages. My analysis not only revealed the weaknesses of conventional racket frames, but showed that frames having the improved shape according to the invention can be made stronger and more rigid without increasing weight, even though still using existing materials.
Another important advantage of my improved frame section is a longer free vibrational length for the strings, which substantially improves the performance of the string network. By keeping the free vibrational length of the strings to a maximum within the overall size limitations of a particular racket frame and by making the frame stronger and more rigid, my invention adds considerably to the performance of racket frames.
According to the invention there is provided a sports racket having a ball-hitting region defined by string network (3) surrounded and supported by a frame having a nose region and a throat region where the majority of longitudinal strings are anchored, and a middle region comprising two lateral side regions of substantial lengths, measured along the axis of the racket overthe ball-hitting region, where the majority of cross strings are anchored, the frame having a hollow structural form whose cross-section perpendicular to the plane of the string network (3) is symmetrical about the said plane and comprises four sides:

an outer peripheral side (16, 17);
an inner peripheral side (5); and
two opposed supporting sides (14, 15) extending on opposite sides of the plane of the string network (3) and connecting the said outer peripheral side (16, 17) to the said inner peripheral side (5);
the frame thus having
an outer face corresponding to the outer peripheral side (16, 17) of the section;
an inner face corresponding to the inner peripheral side (5) of the section and having openings (8) provided for the passage of strings (3) such as to allow the strings (3) to vibrate without touching the inner face of the frame; and
a front and back face corresponding to the said opposed supporting sides (14, 15) of the section;
characterised in that the said cross-section is approximately a trapezoid with straight or curved sides, the distance (18) measured from the inboard edge of said inner peripheral side (5) to the inboard edge of said outer peripheral side (16, 17) being at least 0.64 cm, the length of said inner peripheral side (5) being not less than the length of said outer peripheral side (16, 17); and that the front and back faces have a plurality of weight-saving openings (12, 13) so that for the majority of these openings the volume of said opening is not less than the volume of the material between two adjacent openings.

The inner face, formed to provide clearance openings around the strings, may be formed as a separate perforated strip secured to the front and back faces. Not only the sides of the lateral side regions have a string clearance depth of at least 0.64 cms, but this string clearance depth may also extend to a nose region between the lateral side regions.
The clearance of the supporting sides from the strings must be sufficient to accommodate vibration of the strings on average impact with the ball without the strings touching the support region of the frame, and this feature may be expressed as the clearance of the support region from the strings allows the strings to vibrate freely within an angle of at least 5° on either side of the plane of the string network.
Examples of the invention will now be described with reference to the accompanying drawings in which:

Figure 1 is a plan view of a tennis racket made according to my invention with variable frame strip dimensions and labelled to identify regions of the racket and nodal points used in my analysis;
Figure 2 is a cross-sectional shape of racket frames made according to my invention and subjected to stress and stability analyses;
Figure 3 is a perspective view of a preferred embodiment of a tennis racket, with surface openings omitted, made according to my invention;
Figures 4A and 4B are plans of alternative embodiments;
Figure 5 is a frame cross section taken from US-A-3,899,172 as typical of prior art hollow tubular I-beam type tennis racket frames;
Figure 6 is a partially schematic, cross-sectional view of a tennis racket frame according to my invention and labeled to show measurements used in analysis and explanation;
Figure 7-10 are graphic displays of forces acting on the numbered nodal points of a preferred tennis racket made according to my invention and illustrated in Figure 1;
Figure 11 is a graphic display of lateral deflection of the racket of Figure 1 compared with prior art rackets;
Figure 12 is the cross-sectional shape of an alternative racket frame according to my invention; and
Figure 13 is a fragmentary plan view of a preferred embodiment of a racket according to my invention with a wider frame strip section along its lateral sides.

Detailed description

My discovery of a better racket frame came about from several factors. First, I have been analyzing and working on tennis rackets for several years; and my work on the dynamics of racket strings, as explained in US-A-4,333,650, has led to considerable knowledge about string loads and forces involved in hitting a ball.
Added to this is my knowledge of structural mechanics, giving me insight into structures best suited to withstand stresses involved in tennis racket frames. From these I was able to devise an improved cross-sectional shape for a tennis racket frame as represented by the section of Figure 2.
By using analytical methods I was able to calculate the effectiveness of the section of Figure 2 compared to the prior art section of Figure 5. The analysis shows that the section of Figure 2 and alternative structures as shown in section in Figure 12 substantially improve over the prior art as explained below.
Generally, my improved frame anchors the strings at the outer perimeter of the frame strip and forms a support region of the frame extending inward from the outer perimeter toward the ball-hitting region. Providing the support or mechanical strength for the frame section in regions formed inwardly from the outer perimeter anchorage adds a small but significant extra length to the nominal string length and thus enlarges the free vibrational area of the string network for the same size racket head.

Analysis by FEM Method (Finite-Element Method of Structural Analysis

To study and compare different tennis racket frame strip sections under actual stringing load and ball impact load, I performed a finite-element structural mechanical analysis (FEM). For this I used a conventionally shaped racket head approximately elliptical in its playing area and having a curved throat piece assumed to be the same as the frame strip.
Measured from the neutral axis of the frame strip, the major and minor radii of the ellipse are 16.33 cm and 14.05 cm, respectively. The two lateral sides converge to the handle, and the analysis assumes that the end of the grip towards the shank region provides a fixed-end support to the racket. Since the racket and the load are symmetric with respect to the longitudinal axis of the racket, only one-half of the racket needs to be meshed.
Figure 1 shows the mesh of the analyzed racket. There are 34 beam elements in the analysis, which contains 35 nodes; and each node has six degrees of freedom, three translations, and three rotations. Nodes 1 and 29 are nodes to maintain symmetry with the right half of the racket. The throat piece is joined rigidly with the side frame at node 20. There could be another beam element to join the two parts at node 21 to 22, or from 23 to 24, to make the frame more rigid. But this additional reinforcement will not affect appreciably the stress at nodes 29 and 35. So the additional beam is omitted, and the calculated result to estimate stress and deflection of the racket should be on the safe side.

Applied loads from stringing and ball impact

For the ball-hitting load, each node except nodes 1 and 29 in the elliptical circumference are loaded with a force of 987 g in the z direction of Figure 1. The sum is 45.4 kg at the center of the network. This static load is equivalent to a tennis ball having a weight of 57.9 g travelling at 129 Km/h and being stopped within 0.0046 seconds in a constant deceleration. If a frame can sustain this static load for an indefinite time, it should be able to sustain a transient load with a peak load of much greater magnitude. So, this 45 kg sustained load may taken as a realistic field load on a racketto repeatedly sustain a volley at a 160 km/h ball speed.
For the inplane string load, each node from node 2 to 9 and from 17 to 27 bears a longitudinal string, and each node from 5 to 21 bears a lateral string load. This produces 16 longitudinal and 16 lateral string loads, and the string force at each node is 36 kg.

Analyzed frame strip section of Figure 2

Figure 2 shows a preferred cross-sectional shape made according to the invention for frame strips analyzed and compared with a prior art frame as explained below. As is apparent from the wider frame section of Figure 2, there can be differences in shapes, sizes, and wall thicknesses; and such differences can be affected by manufacturing methods, materials, head sizes, and racket weights. The invention is by no means limited to the geometry analyzed here as examples. Governing principles in selecting such alternatives remain the same and are explained below.
Section 1 of Figure 1 has a string hole or grommet seat 2 located at the outer perimeter where string 3 enters the frame and leads into the network. The width 4 of seat 2 is as short as possible, about 0.5 cm or less. There is ample opening or cutout at the inner peripheral side 5 to let the string vibrate without interference. For the analysis, the height 6 in the section of Figure 2 is taken as about 2 to 2.5 cm but it can be reduced when stronger material than the say, Aluminum Company of America (Alcoa) 6061-T6 is used. The width 7, designated as d, is 2.5 cm for section 1 of Figure 2 used in the analysis but can vary from 1.14 to 3.05 cm depending on objectives. The distance 18 is then 1.14-0.5=0.64 cm as the minimum. In the analysis, the height 6 is taken as 2.5 cm and d is varied from 1.14 to 2.54 cm. For widths 7 less than 1.14 cms the design will not yield enough effective string length increase to benefit the performance. Widths greater than 3 cm will make the frame strip too bulky.
The string clearance opening 8 can, for example, be round, oval, or rectangular in shape; and each string can have its own opening, or use an enlarged opening to accommodate several strings, so that in between holes 8, there is ample material to form a web to connect the upper side 9, 14 and lower side 10, 15. The material removed from opening 8 can be added to the web between the neighboring openings, so that, in the analysis, the wall thickness 11 is made the same as the sides 9,14 and 10, 15, whose thickness in section 1_-is preferably about 0.14 cm for aluminum, for example.
If the material is very strong, such as graphite, and the upper and lower sides 9,14 and 10, 15 are stiff enough, inner peripheral side 5 can form a continuous angle section with sides 9 and 10; and no web is needed for connecting the two sides at the inner perimeter. To keep a frame weight within accepted limits and still accommodate a frame having a 2.5 cm width 7 as shown in Figure 2, weight-reducing openings 12 and 13 are formed in supporting sides, 9, 14 and 10, 15 respectively. Although openings 8, 12 and 13 are all illustrated in section 1 for convenience, in actual practice, they may be staggered or spaced along the length of a frame strip so that they do not all lie on a single section, for evenly distributing the material and strength along the frame strip length.
In analysis, for simplicity, the removed material of the openings 8, 12 and 13 is assumed to have the same volume as the remaining material in the walls. Then a uniform wall thickness of 0.064 cm, can be assumed, for use in the analysis with the local opening assumed as being eliminated. This greatly reduces the complexity of the analysis. This "smeared average" method of dealing with local irregularity in wall thickness is well accepted in structural analysis. This is true especially for estimating local structural instability to which a thin-walled web connecting two strong, parallel flanges is often vulnerable.
Openings 12 and 3 can be round, oval, or rectangular in shape, with the remaining web extending between side regions 9 and 16 and between 10 and 17. Openings 12 and 13 can also be shaped by triangles, leaving panels between openings inclined as in a truss assembly. Then the frame will have its outer and inner peripheral sides supported by a plane truss on each side of the string plane. This can be structurally more rigid.
Openings 12 and 13 reduce weight, as well as reduce air resistance when the racket is swung. This is necessary when the section width 7 is more than 1.8 cm.
In the analysis, the side regions 9 and 10 in Figure 2 are 0.64 cm high and 0.14 cm thick, and side regions 16 and 17 are 0.5 cm high and 0.14 cm thick. These are continuous flanges providing major bending rigidity to resist moments due to the string and ball impact loads. They also provide necessary mass to guard against damage when the racket hits the ground.
The thickness of string anchorage wall 2 at the outer perimeter of the frame section can be 0.089 cm for an aluminum section. Especially around the nose of the racket, a plastic cushion strip can be provided to resist court-scuffing damage. Side regions 14 and 15 can be inwardly curved or recessed along their outer surfaces to reduce damage when the racket hits the ground.
Due to the well balanced mass distribution, the inventive sections have extremely high ratios of strength to weight for torsion and bending in the two principal axes, which pass through the cross section's center of gravity and are parallel or perpendicular to the plane of the string network. These values were rigorously calculated and are reported next. Foamed polyurethane integral stuffing used to fill the internal space of the frame strip for damping purposes is an option, but its affect on strength and weight is not included. Although the sections of Figure 2 have the desired strength-to-weight ratios, changes are possible; and the invention is not limited to the geometry of specifications illustrated as sample sections here.
Any variation in sectional shapes for frames according to the invention preferably keeps the string clearance depth distance 18 to a maximum. This string clearance depth is measured along a perpendicular to the frame section in the plane of the string network from the inner perimeter 5 outward to the point where a string 3 or grommet clears the inside of the outer perimeter anchorage region 2. In other words, the outermost point where a string 3 can vibrate free from intereference with the anchorage region is preferably located as close to the outer perimeter 2 of the frame as possible, and vibrational clearance is preferably provided for the strings from the point inward toward the ball-hitting region. The importance and extent of vibrational clearance for strings 3 is explained more fully below.
A racket having a generally conventional shape and made with a frame strip having a cross-sectional shape such as shown in Figure 2 is illustrated in Figure 3, with surface openings omitted. The cross-sectional shape of the frame strip used in the racket of Figures 4A, 4B can be formed as an extrusion or draw in which string openings 8 and surface openings, are bored, or it can be formed as an open channel extrusion to which an inner perimeter wall with preformed openings 8 is secured. Graphite frames can be made by different applicable types of construction.
Figure 4A illustrates a frame with circular apertures formed in the upper and lower side regions 9 and 10. Those at the centre are slightly larger than those near the nose or shaft. Figure 4B shows triangular apertures in the regions 9 and 10, making a lattice pattern.

Analysis of frame sections

To determine the physical properties of different sections, I carried out rigorous analysis based on structural mechanics for the sections shown in Figure 2 for a prior art section of Figure 5.
Torsion Rigidity: Torsional rigidity of a one-cell box with variable wall thickness, as shown in Figure 2, is given by the following equation:

where 8 is the angle of twist per unit length, T is the torque applied, G is the shear modulus of the material, Jet; is the effective polar moment of inertia, A_o is the area bounded by the center line of the box, L, is the length of a particular segment, and t, is its wall thickness with i as the subscript index of that particular segment. There are eight segments of different wall thickness in the sections of Figure 2.
The shear stress at web 5 which is vulnerable to local instability is given by:
Figure 5 shows a prior art drawn aluminum frame strip section presently used in the AMF Company's Head Edge medium-sized head racket. This particular section, as detailed in US-A-3,899,172 issued August 1975, was said to have a very high strength-to-weight ratio. In the disclosure, the strength ratio of I/A, which is the moment of inertia to the cross-sectionaal area ratio, was said to range from 0.33 cm² to 0.37 cm². For comparison purposes, I enlarged Figure 2 of the patent fourteen times and calculated its geometrical properties. It turned out to have an area A=₀.₇₂ cm², l_y/A=0.34 cm² and l_z/A=0.044 cm² which, excluding h/A, agreed with the claims.
The effective polar moment of inertia, as related to St. Venant torsion of two-tubes-connected-by-a-web type section, can be found from the following formula:
where L₂ is the length of the web, t₂ is the web's thickness, A_o is the area bounded by the centerline of the tubular hole, t₁ and L₁ are the wall thickness and the circumferential length of the tubular hole, respectively. With the measured quantites substituted into the above equation, we have for the prior art section:
The maximum shear stress at the web occurs at a point on the outer boundary of the web on the y-axis, as shown in Figure 5. With the applied torque designed at T, the shear stress is:
The moment of inertia about the y and z axes for the inventive section and for the prior art section of Figure 5 can be obtained by the usual method. Table 1 shows the section properties where d is the width 7 of the section in Figure 2 varied from 1.2 cm to 2.5 cm.
Table 2 is the strength-to-area ratio calculated from Table 1, and Table 3 is the ratio of comparison of strength-to-area ratio based on Table 2, with the strength ratio of the prior art section of Figure 5 taken as the base for comparison.
Table 3 shows that the inventive frame strip is far superior to the prior art frame strip in all respects. Consider the inventive section having a width of 1.5 cm. Its cross section is 14.5% lighter than the prior art section. With unit weight of Aluminum Company of America's Alcoa 61 S-T6 taken at 2.71 g/cm³ for a frame strip length of 117 cm the saving in weight of a complete racket is about 33.2 g, which is about 9.4% of the total weight.
In addition, as clear from Table 3, the inventive racket is 84% more stiff than the prior art racket in resisting ball impact load. This makes the returning ball fly back faster. The inventive section is also 657% more stiff in resisting inplane load. This not only makes the racket extremely strong against permanent deformation during stringing, but also helps to make the racket more rigid in resisting the ball load. When the string network tightens to resist the penetration of the ball, it not only bulges out to contain the ball, but each string has to pull inward toward the centre of the net. A racket having a stiffer in plane rigidity, which is represented by its I_z value, will make the net hard to be pulled inward toward its center, hence a stiffer frame allows the network to store more energy and impart its larger stored energy to the rebounding ball. This a major advantage of the invention. Indeed, the LJA value of 0.21 corresponding to d=1.14 cm in Table 2 may be taken as practicable or preferred minimum ratio to achieve the benefit.
The inventive racket is also 530% more stiff in torsion. This rigidity reduces the "whippy" feeling of a racket, which affects play accuracy and reduces the strain energy loss to the frame.
The inventive racket also increases the free vibration area of the string network by increasing the free vibration length of its strings. Since the strings are anchored at the outer perimeter region of the frame and the support region, which includes the inner perimeter of the frame, does not interfere with free vibration of the strings, the strings have a free vibration length that extends within the frame section to the region of the string anchorage at the outer perimeter.
This is illustrated schematically in Figure 6 where the dimension L_s applied to all the strings of the network defines a net area commonly called the "string area" or "playing area" of a racket. The smaller dimension L_b of Figure 6, when applied to the racket strings, defines the net area that a ball can actually touch. This is an area bounded by the frame and the throat minus an outer band width equal to the radius of the ball.
The free vibrational length of the strings is shown by the longer dimension L, extending for the full length of each string between the points where the string clears its anchorage at the outer perimeter of the racket frame. This dimension L_f applied over all the strings of the network gives a larger free vibration area than the conventional "string area" based on the dimension L. for prior art rackets.
Applying this to the inventive section having a value d=1.14 cm in Table 1 where the strip width is about the same width as a conventional extrusion, the increase in the free vibrational area is about 10%. An over-sized head for a conventional racket has only about 23% more playing area than a medium-sized head racket. So, applying even a narrow form of the inventive racket frame to a medium size racket head to increase the free vibration area of the string network by 10%, when accompanied by a frame section that is 87% and 372% stronger respectively in bending stiffnesses and 363% stiffer in torsion as shown in Table 3 and 19% lighter in weight as shown in Table 1, produces a substantial improvement over the prior art. Also, a medium-sized head racket having an inventive frame strip with a width of 1.93 cm has a string network with a free vibration area equal to a conventional over-sized head racket. The resulting medium-sized racket head is half an inch narrower in its overall width than an over-sized racket head and does not look as large, even though it performs at least as well.

Loads on the racket

Ordinarily by comparison of the principal moment of inertia about the three axes and the strength-to-weight ratio of the inventive section with prior art sections, a merit comparison could be established and there would be no need to analyze stress from actual loads on the racket. However, since the inventive section improves its strength-to-weight ratio by distributing the mass away from its center to increase the moment of inertia while leaving the interior open to admit the vibrating string without interference, some segments of the wall of the section have to be thinner than the prior art walls. Consequently, I have studied the critical stress cases to show that the inventive section is indeed adequate to resist such particular failure modes.
Based on the finite-element method applied on the racket as shown in Figure 1, results of the loading of the racket frame from a 36 kg string load case and a 45 kg ball load are obtained and shown in Figures 7 to 10.
Figures 7 and 8 respectively depict the bending moment at each nodal section about the local z-axis and the axial force. The shear force can be obtained from the equilibrium of moments at the two ends of an element. The shear is quite small, however, and is neglected. From Figures 7 and 8, it is clear that the stringing load on the frame is maximum at node 1, with a magnitude of 2.76105 cm . g for the 36 kg string tension system. The axial force is compressive and is almost uniform at about 318 kg from node 1 to 20 at the throat bracket.
Based on Table 1 properties of sections, the bending stress is maximum at the outer perimeter of a section with _Cz as the distance from the neutral z-axis. The stress is M_zC_z/l_z, where the c_z/l_z= value of the prior art section and of the inventive section with a width d=2.5 cm are respectively 13.5 cm-³ and 2.1 cm-³. The maximum bending stress for the two sections are also in that ratio, which is a ratio of six to one in favor of the inventive section. Since the cross-sectional areas A of each section are almost equal, the axial compressive stress is almost the same.
To investigate local instability of the inventive section at its inner perimeter due to the combined bending an axial force, I obtained the combined stress from the following (using c_z=1.046 for inner-periphery):
at node 1. From a classical buckling equation ("Theory of Elastic Stability", by Timoshenko and Gere, Second Edition, page 366), for a thin plate supported by strong parallel flanges and compressed uniformly along the flange direction at the ends, the critical stress the web can sustain is:
For the inventive section, E=0.7×10⁶ for aluminium, h=0.064 cm for web thickness, and b=2.24 cm for web height, the critical stress allowed is 3267 kg/cm² compared with the actual stress of 835 kg/cm² from the 36 kg string force system, the local instability of the thin web is of no concern at all. On the other hand, the combined compressive bending stress at node 1 for the prior art section is more than 3500 kg/cm².
When the contour geometry of the racket based on the neutral axis line of the racket frame is fixed, the difference in the frame strip properties do not appreciably change the loadings of the frame from the ball or string load. This means that loadings due to external force on the inventive racket and on the prior art racket are approximately the same, but stress and displacement are different.
Furthermore, the loads on the sections are linearly proportional to the applied loads. For example, if the bending moment acting at node 1 from the 36 kg string load system is 277 cm - kg then the moment becomes 346 cm . kg when the string load system is increased to 45 kg each, with all the other things remaining the same. The load from the ball impact similarly increases the bending moment. Therefore, the information revealed in Figures 7 to 10 affords a very useful loading reference for a tennis racket of conventional size and shape.
Figures 9 and 10 show loads on the frame strip at different node points from the impact of the ball. The ball impact produces no axial force along the longitudinal axis of a section, but it roduces two bending moments. One is a twisting or torque moment, M,,, about the longitudinal axis of the section. The twist in the shank region beyond node 20 can be reduced by stiffening the throat piece. The maximum twisting torque is at node 19, which is about 150 cm . kg in magnitude.
The maximum bending about the local y-axis from Figure 10 occurs at the handle node 35 where My is 703 cm . kg. The material distance to inertia ratios, cylly, for the prior art section and for the inventive section with a width of d=2.5 cm are respectively 3.88 cm-³ and 2.78 cm-³. Consequently, their maximum stresses are also in that ratio.
Therefore, the maximum material stress of the inventive section is only 71 % of the prior art section, regardless of the actual size of the moment. For the 703 cm-kg bending moment, the stresses are 2716 and 1949 kg/cm² respectively, in favor of the inventive section.
Figure 11 shows the lateral deflection at node 1, at the nose of an aluminum racket, for a ball impact load of 45 kg. The prior art section deflects twice as much as the inventive section at different section widths d. Stiffer material can reduce the deflection, but the ratios remain, and the deflection is proportional to different impact forces. For determining relative merits, comparisons between strength-to-weight ratios and magnitudes of stress and displacement for the inventive section and the prior art section are more important than absolute magnitudes, per se.
The inventive frame section shapes shown in Figure 2 and subject to the foregoing analysis, apply basically to medium and large size racket heads and frames made of metal, graphite, and other high strnegth to area ratio materials. Variations from the shapes shown in Figure 2 are possible and practical for these materials.
Frame 60 of Figure 12 is similar in overall shape to frame section 1 of Figure 2. Its anchorage web 62 is also at its outer perimeter and supports strings 3. Openings 64 formed in supporting sides 63 have edges 65 that are formed to bend inward as illustrated. This helps strengthen sides 63 around opening 64.
Instead of an integral inner peripheral side, section 60 has an inner peripheral side 66 formed as a .separate strip perforated with openings 67 having inturned edges 68 as illustrated and securely attached to the inner edges 69 of sides 63. Wall 66 and side edges 69 can be secured together by welding, for example. Such construction allows perforations 64 and 67 to be die shaped, to have inturned edges 65 and 68 for greater strength and smooth outer surfaces. As with other preferred embodiments, string 3 can vibrate clear of supporting sides 63 and inner peripheral side 66.
Racket frame sections are not necessarily uniform throughout the length of the frame and can vary in width and shape. Frame sections according to the invention can accommodate this and can be shaped to accommodate the loads encountered at different regions of a frame. For example, greater widths, thicknesses, and strengths are appropriate in the throat, shank, and lateral side regions and thinner widths, thicknesses, and strengths in the nose region of a racket.
Also, it is especially important for transverse strings of the string network to have maximum free vibrational length so that the string clearance depth of the frame section should be at a maximum along lateral side regions of the frame where transverse strings are anchored. Maximum string clearance depth is not so necessary for longitudinal strings anchored in the nose region of the racket. The elliptical shape of conventional rackets makes longitudinal strings longer than transverse strings, anyway.
Greater width of the racket frame strip in the lateral side regions is also preferred for'the advantage of increasing the moment of inertia of the racket about its longitudinal axis to counteract shots made off the longitudinal axis of the racket. Racket head 80 of Figure 13 is formed of a frame strip 81 that is wider in lateral side regions 82 than in nose region 83 for accomplishing both objectives. The greater width of frame strip 81 in lateral side regions 82 not only increases the moment of inertia against a twisting moment, but also allows a greater-string clearance depth. The inner perimeter 84 of frame strip 81 preferably has the same elliptical shape as a convention racket head, and the widening of frame strip 81 in lateral side regions 82 is formed to increase the distance between the outer perimeter region 85 where the transverse strings are anchored. This increases the free vibrational length of the transverse strings and makes them more effective components of the vibrating string network.
Widening of frame strip 81 in lateral side regions 82 preferably sufficient to exceed the width of frame strip 81 in nose region 83 by at least 0.3 cm and preferably by about 0.9 cm. Such widening also preferably increases the string clearance depth by the same amounts to increase the free vibrational length of the transverse strings while also increasing the moment of inertia of the racket about its longitudinal axis.
String clearance depth the inventive racket is measured perpendicular to the frame strip and in the plane of the string network. This distance extends from the inner perimeter of the racket frame along the string plane in a direction perpendicular to the frame strip to the point where the strings clear and depart inwardly from their anchorage at the outer perimeter of the racket. Support regions of the racket frame section extending inward from the string anchorage at the outer perimeter clear the strings perimeter clear the strings by a sufficient margin to allow their free vibration under normal playing conditions. Then the strings, instead of vibrating only within the area enclosed by the inner perimeter of the racket frame, vibrate through their entire length including their string clerance depth within the frame to the region where they contact their anchorage at the frame's outer perimeter.
The clearance of the support region from the strings is preferably sufficient to allow the strings to vibrate freely within an angle of at least 5° on either side of the plane of the string network. This means that the support regions of the frame, including the inner perimeter, preferably clear the strings by an angle of 5° on either side of the plane of the string network extending inward from the string anchorage region. Such a 5° clearance angle is adequate to accommodate string deflection in response to a normal ball impact load. An 7° clearance angle on either side of the plane of the string network is preferred for accommodating the most severe ball impact forces for a racket can be expected to encounter.
Within practical weight requirements that limit the cross-sectional area of the frame of up to about 0.723 cm² for aluminum alloy materials and up to about 1.143 cm² for graphite or other composite materials of similar specific weight, the inventive cross-sectional shape for a racket frame preferably has an inertia to area ratio about its z-axis (h/A) of between 0.7 to 1.23 cm² and about its y-axis (h/A) of between 0.39 to 0.65 cm²for a section having a height from 1.65 to 2.29 cm and a width of from 1.54 to 2.16 cm and a wall thickness of from 0.13 to 0.20 cm. Comparing this with the section of US-A-3,899,172, which has an h/A value of 0.044 cm² and a l_y/A value of 0.34 cm² as representative of the state-of-the-art for an aluminum alloy frame strip having a cross-sectional area of 0.72 cm² the inventive section is much superior in its strength to area ratios.
Racket frames made according to my invention enlarge and maximize the free vibrational area of the string network and thus clearly improve racket performance. My frames are also stronger, stiffer, and better able to withstand string load without being heavier. They are less likely to be deformed under stringing or ball impact load, are less whippy, and provide a larger sweep spot playing area.

Claims

1. A sports racket having a ball-hitting region defined by string network (3) surrounded and supported by a frame having a nose region and a throat region where the majority of longitudinal strings are anchored, and a middle region comprising two lateral side regions of substantial lengths, measured along the axis of the racket over the ball-hitting region, where the majority of cross strings are anchored, the frame having a hollow structural form whose cross-section perpendicular to the plane of the string network (3) is symmetrical about the said plane and comprises four sides:

an outer peripheral side (16, 17);

an inner peripheral side (5); and

two opposed supporting sides (14, 15) extending on opposite sides of the plane of the string network (3) and connecting the said outer peripheral side (16,17) to the said inner peripheral side (5); the frame thus having

an outer face corresponding to the outer peripheral side (16, 17) of the section;

an inner face corresponding to the inner peripheral side (5) of the section and having openings (8) provided for the passage of strings (3) such as to allow the strings (3) to vibrate without touching the inner face df the frame; and

a front and back face corresponding to the said opposed supporting sides (14, 15) of the section;
characterised in that the said cross-section is approximately a trapezoid with straight or curved sides, the distance (18) measured from the inboard edge of said inner peripheral side (5) to the inboard edge of said outer peripheral side (16, 17) being at least 0.64 cm, the length of said inner peripheral side (5) being not less than the length of said outer peripheral side (16, 17); and that the front and back faces have a plurality of weight-saving openings (12, 13) so that for the majority of these openings the volume of said opening is not less than the volume of the material between two adjacent openings.

2. A racket as claimed in Claim 1 or Claim 2, wherein said openings of said front and back faces are of a triangular shape.

3. A-racket as claimed in Claim 1 or Claim 2, wherein said openings of said front and back faces are of a round, rectangular or oval shape.

. 4. A racket as claimed in Claim 2, wherein said triangular openings leave panels between them inclined as in a truss assembly.

5. A racket as claimed in any of Claims 1 to 4, where the width of the frame (front and back faces) in the lateral side- regions is greater than that in the nose. region.

6. A racket as claimed in any of Claims 1 to 5, wherein the weight-saving openings (12, 13) are located only in the lateral side regions.