Dr. Paul Zarda of Sanford Florida has Filed a Patent for Dual Frame System

Dr. Paul Zarda of Orlando and Sanford Florida has a pending USPTO patent application that can be found here: https://www.google.com/patents/US20140274494 This patent offers a dual frame system that includes an inner frame, an outer frame, and isolators that integrate the two frames. Figure 1 depicts the basic geometry and functionality of this system. As outlined in the patent application, the system is designed so that, as a ball rebounds in the normal string-bed direction, it also rebounds in the tangential string-bed direction at the same time. This tuned system will produce added spin on the ball. The key for this system to be efficient to generate spin is directly related to the mass of the inner frame: the smaller the mass of the inner frame, the greater the amount of rebound spin of the ball. The discussion below illustrates how the mass of the inner frame can be minimized.

                           

 Dr. Paul Zarda of Orlando points out that an important feature of the spin control system is the design weight of the inner frame is to be made as small as possible.  Specifically, with reference to a tennis ball’s weight of 57.7 grams or so, the weight of inner frame (including the weight of the strings, grommets, and interface structure to the isolators, and any moving components that can move directly or indirectly with the inner frame and hence are part of its dynamic weight), should be between 20 grams and 200 grams or so, with a target weight of 30-40 grams or so. It can be shown (both thru experimental testing and simulations) that the ability of the spin control system to generate spin is inversely related to the effective dynamic mass of the inner frame (whose weight is defined above): the smaller the mass of the inner frame, the higher the amount of spin that can be achieved. In addition, the control of this inner frame effective weight (a feature of the spin control system and the inner frame), is also a claim of the patent. Controlling this weight can control the maximum amount of spin the spin control system invention can provide. Designing this effective dynamic inner frame weight to be as light as possible (compared to the ball) will allow the ball to minimize “sliding” on the string bed during impact, and thus allows the re-bounding inner frame to impart higher tangential forces to the ball, causing increased spinning of the ball during and after ball impact.

Dr. Paul Zarda notes the inner frame can have another material, instead of strings, that may cover the inner frame to provide a contact surface for the ball. The structure of the inner frame can be made from any material. A light weight, high strength, low material and manufacturing cost, is preferred. Once such candidate is a graphite composite.

The shape of the inner frame, and the stringing and string pattern of the inner frame, is an important part of the spin control system. The largest loads that the inner frame will see occur because of the string tension that is applied to the inner frame (or to any racquet frame for that matter). The ability to minimize the stresses resulting from this string tension loading will directly contribute to minimizing the weight of the inner frame and the effectiveness of the spin control system.

A basic understanding of these loads and resulting stresses is fundamental to the spin control system.  A formula for the tensioning of the strings can be derived to minimize stresses in the inner frame, and this in turn will minimize the weight of the inner frame need to accommodate those stresses.

                            

Consider, referring to Figure 2, a uniform load Wy (lbs/in) applied to an arch (like a Roman arch). For an arbitrary shape of the arch, loads will develop in the arch shown in Figure 3: an axial force N (lbs), a shear force V (lbs), and a bending moment M (lb-in).  The bending moment M causes large stresses in the structure; minimizing M will reduce stresses and hence the weight considerably.

It is well known that there is a shape for the arch that will minimize the bending moment M in the arch.  A specific parabolic shape involving L and h (see Figure 2) will cause the bending moment M (and shear V) to go to zero, thus minimizing stresses in the arch and requiring the arch to only carry, and very efficiently only carry, the axial load N. 

  

a = minor dimension

b = major dimension

Wx = cross strings load/unit length

Wy = main strings load/unit length

Tx  = Cross string tension

Ty  = Main string tension 

 nx =  # of cross strings at Tx tension

 ny =  # of main strings at Ty tension

      FIG. 4

For the double loading shown in Figure 5 (see description in Figure 4), is there a shape for that structure that will also minimize M? This structure and loading can represent a strung tennis racquet, with Wx, Wy representing cross string and main string loading, respectively.  Although not well known, and to this date still undocumented but a claim in the patent, M & V in this racquet will go to zero if the shape of the structure is a mathematical ellipse (minor axis a, major axis b, see Figure 4 and Figure 5), and the Wx, Wy string loading is not arbitrary but chosen per Equation 1 in Figure 6.

If there are Nx\Ny equally spaced cross\main strings, respectively, then, for a specified Ty main string tension, the cross string tension Tx is given by Equation 2 of Figure 6. The table of Figure 6 gives, for main string tension Ty = 60 lbs, typical cross string tensions Tx (last column) for various racquet head shapes (assuming they are elliptical). Cross string tension run about 2/3 (40 lbs) of the main string tension (60 lbs).

Stringing the inner frame based on the tension formula of Equation 2 of Figure 6, will minimize the stresses (M and V = 0) and hence will allow for minimizing the weight of the inner frame. Note that this tension formula represents the final tension in the racquet and not the tension that is actually pulled (the racquet flexing and stringing machine flexing will make those numbers different).

Dr. Zarda of Orlando makes these claims: i) The shape of the inner frame is elliptical or nearly elliptical (within 20% of an elliptical shape as measured by a maximum normal deviation normalized by the maximum dimension; note a circular shape is an ellipse and would represent minimum weight for a given area); ii) The final tensions, however they are achieved, are based on Equation 2 of Figure 6 (within 20%, including, if unequal string spacing and varying tensions apply, then average values Wx/Wy are used and compared for agreement per Equation 1 of Figure 6, and normalized by the average main string tension or by Wy, whichever applies);  iii) This applies to any strung frame, not just the inner frame presented here.

Minimizing the weight of the inner frame, subject to a specified string tension loading, will require that the inner frame be tightly engineered to remove any conservatism. Based on the discussion in the previous section, the inner frame will be elliptical in plan-view shape (and, for a specified hitting area, a circular shape would be the optimum elliptical shape for minimum weight). For the Equation 2 string loading condition, its stress field will be in a pure membrane stress field (ie, axial load only). This efficient load carrying situation will allow a minimum weight; but this loading condition will be a compressive load, and this light weight compressive loaded structure will be a strong candidate for buckling.

For a given x-sectional area of a tubular-like inner frame, simulation studies clearly show a closed x-section is significantly better than an open x-section (by a factor of 4 to 8 or so) to minimize buckling. Buckling can occur both in-plane and out-of-plane.

Racquet Head Shape b x a (inches) (FIG. 404)	Main Strings		Cross Strings
	# Main Strings	Tension Ty (lbs)	# Cross Strings	Tension Tx (lbs)
Circular Head 11 x  11	17	60	17	60
Badminton Head 9.5 x 8	16	60	19	42.5
Davis Classic 11 x 9	16	60	19	41.1
Wilson 13 x 10	16	60	19	38.9

    

FIG. 6

Dr. Zarda notes simulation studies of this inner frame indeed show that buckling is a potential failure condition. The buckling condition that was simulated was based on models of a circular inner frame with a conventional stringing pattern similar to that shown in Figure 1 (while Figure 1 shows a racquet strung, the pattern can still be applied to an inner frame). The string pattern of Figure 1 shows the main and cross strings supported at the mid-plane (z = 0) of the frame. These simulation results showed the inner frame was close to buckling for the string tension and string spacing analyzed. The modeling included the string bed modeled with a pattern and mid-plane frame support similar to that shown in Figure 6.

A more detailed description of how this reduced weight inner frame can be found in the patent application: https://www.google.com/patents/US20140274494

Dr Paul Zarda Developed a Tennis Racquet To Significantly Increase Ball’s Rotational Velocity

Dr. Paul Zarda of Orlando and Sanford Florida has developed a new tennis racquet that causes a significant increase in spin over conventional racquets. The description of the patent application can be found here: https://www.google.com/patents/US20140274494

His invention includes an inner and outer frame connected by an isolation system. The tuning of the ball’s normal rebound response to the ball’s tangential rebound response can control the spin imparted to the ball for a given tennis swing. This tuning offers a significant increase in a ball’s rotational velocity.

The basic functionality of the invention is illustrated in Figures 1 through 3. The intent of that geometry description is for illustration to highlight the functionality of Zarda’s design.

The invention presented (Dr. Paul Zarda Orlando) is a racquet capable of spin control and first illustrated in Figures 1 thru 3, where there is an inner frame 201, isolators 202, and outer frame 203. The inner frame can be any shape, but a circular shaped head (in normal-to-the-string-bed view the head shape is mathematically an exact circle) or an elliptical shaped head (in normal-to-the-string-bed view the head shape is mathematically an exact ellipse) is preferred. The isolators (see Figure 1) can be discrete (as shown) or continuous. For the discrete system, there can be any number of isolators (made of any material, including a magnetic design) and they can be at any location around the periphery of the inner and outer frame. Figure 1 shows 4 isolator locations at 12 o-clock, 3 o’clock, 6 o’clock and 9 o’clock. The inner frame 201 is strung with any string system that is used today; a stringing system that limits relative string motion is preferred. The isolator system 202 offers the structural connection between the inner and outer frames. The isolators are designed so that they are easily assembled in place or easily removed. Once removed, the inner frame can be structurally separated from the outer frame. It is intended that the inner frame would be strung separately. The spin control racquet invention provides top-spin and under-spin to the ball (if appropriately struck) by using a different design compared to the spaghetti and other racquets on the market.

Dr. Paul Zarda, Orlando, notes one unique feature of the spin control invention is an inner frame (see Figures 1 thru 3, item 201) which contains strings under tension (or another material) intended to make contact with the ball. The inner frame is connected to the outer frame using an isolation system 202 in Figures 1 thru 3 which allows movement of the inner frame relative to the outer frame upon ball impact with the strings of the inner frame.  The movement between the inner and outer frame will take place both in the XY plane (in-plane displacement) as well as out-of-plane displacement (Z-direction in Figures 1 thru 3). This relative deflection is because the isolators can offer flexibility (compliance) to the inner frame in the XY plane, as well as flexibility (compliance) to the inner frame for out-of-plane deflections.

During ball impact, except for the strings of the inner frame deflecting out-of-plane (as strings do for any conventional racquet), the inner frame moves essentially as a rigid structure. This allows the isolation system to offer overall support of the inner frame. For example, for in-plane deflections, the inner frame moves, as a rigid body, as much as 0.25 inches to 1.0 inches or more in the XY plane.  The isolators and outer frame are designed to accommodate this in-plane motion of the inner frame for any in-plane direction. Specially, the inner frame can move, in the XY plane, referring to Figure 1, in the X-direction, in the Y-direction, in a direction at 45 degrees to the X-direction and Y-direction, or in a direction at any angle Theta-Z (Theta-Z is an angular rotation direction about the Z-axis of Figure  1). 

Spin is achieved by allowing the entire string bed to move at some angle in the x-y plane and then pop back. At the same time, as the strings simultaneously are moved in the x-dir and z-dir and then re-bound, the ball is being pushed off the bed in the local z-dir of the racquet, and simultaneously being spun (about the y-dir) as it is loaded tangentially through friction. This synched motion in both directions puts the added spin on the ball while simultaneously propelling the ball off the string bed. It is this synched motion that can be achieved by choosing the appropriate isolator stiffnesses for a given swing speed and angle of contact.

The isolation system of the inner frame relative to the outer frame is to provide different stiffnesses for the isolators in the x-y plane (Kx and Ky, see Figure 3) versus the out-of-plane stiffness Kz. The Kx, Ky stiffnesses are, taken as a group, between 10 lbs/in and 1000 lbs/in and are tuned to maximize ball spin. The Kz stiffness of the entire isolation system is also tuned so that the overall out-of-plane stiffness the ball sees is between 50 lbs/in and 400 lbs/in (that stiffness includes, in series, the stiffness of the strings, the Kz isolation system, and the stiffness of the racquet).  Both the isolators and inner frame can be easily removed and replaced.  This design allows for adjustment of the Kx and Ky stiffnesses so that, no matter what the head’s motion is as it strikes the ball, the in-plane x-y stiffness the ball sees can be made the same. Hence if a racquet is swung where the motion of the head is not exactly parallel to the ground at ball impact (the racquet handle makes an angle with the ground; as in a serve) the ball will experience the same top spin. This allows the serving motion to cause significant spin, likely curing the ball in two planes.

Dr. Paul Zarda of Orlando and Sanford indicates another feature of the spin control system is the design weight of the inner frame is to be made as small as possible.  Specifically, with reference to a tennis ball’s weight of 57.7 grams or so, the weight of inner frame (including the weight of the strings, grommets, and interface structure to the isolators, and any moving components that can move directly or indirectly with the inner frame and hence are part of its dynamic weight), should be between 20 grams and 200 grams or so, with a target weight of 30-40 grams or so). It can be shown (both thru experimental testing and simulations) that the ability of the spin control system to generate spin is inversely related to the effective dynamic mass of the inner frame (whose weight is defined above): the smaller the mass of the inner frame, the higher the amount of spin that can be achieved. In addition, the control of this inner frame effective weight (a feature of the spin control system and the inner frame), is also a claim of the patent. Controlling this weight can control the maximum amount of spin the spin control system invention can provide. Designing this effective dynamic inner frame weight to be as light as possible (compared to the ball) will allow the ball to minimize “sliding” on the string bed during impact, and thus allows the re-bounding inner frame to impart higher tangential forces to the ball, causing increased spinning of the ball during and after ball impact.

The shape of the inner frame, and the stringing and string pattern of the inner frame, is an important part of the spin control system. The largest loads that the inner frame will see occur because of the string tension that is applied to the inner frame (or to any racquet frame for that matter). The ability to minimize the stresses resulting from this string tension loading will directly contribute to minimizing the weight of the inner frame and the effectiveness of the spin control system.

The isolation system is another key feature of the spin control system. The isolation system controls the motion of the inner frame relative to the outer frame by any number of methods.  In one embodiment, an isolation system (continuous isolators or a collection of discrete isolators), built of any material of known stiffness, provides a mechanical resistance to the motion of the inner frame relative to the outer frame. In other embodiments, pneumatic, hydraulic or electromagnetic means may be used to resist motion between the inner and outer frames. In another embodiment the inner frame may actually nest inside the outer frame and upon impact with the ball may move beyond the outer frame. In any of these embodiments, the material choice or design may allow stiffness that is different for different loading conditions (in-plane XY loading or out-of-plane Z-direction loading, which directions are illustrated in Figures 1 thru 3).

A key feature of the spin control system is the ability to size/tune the isolators to provide increased ball spin rates over conventional racquets. Figure 3 illustrates 4 “symbolic” isolators 302 that connect the inner frame 301 and outer frame 303 together. To help define the isolators, consider an individual isolator as a spring system as shown in Figure 3. The isolator at 12 o-clock in Figure 3 is described by its stiffness: (Kx, Ky and Kz) stiffness or (Kx-Isolator, Ky-Isolator, Kz-Isolator), or (Ktangential, Knormal, Kz), or (Ktheta, Kradial, Kz), respectively. While these springs could literally be springs, the more appropriate view of them is that the Kx, Ky and Kz springs represent the equivalent behavior of the actual mechanical isolator (like the thin walled tubes 302 of Figure 1) as it connects the inner frame 301 to the outer frame 303. For visualization purposes, the springs are shown in Figure 3 as split-in-two as they attach the inner and outer frame together.

The interpretation of the isolators 302 in Figure 3 has been illustrated as springs between the two bodies. Other interpretations can include: i) linear or non-linear static springs; ii) linear and non-linear springs that are equivalent to a linear and non-linear dynamic stiffness or compliance; iii) linear or non-linear dampers, causing energy loss between the inner and out frame; iv) any combination of these interpretations. The isolators can be designed so that each isolator is adjustable or replaceable, changing some or all of the characteristics offered here, to cause additional spin and/or accuracy control of the tennis ball during impact.

Another feature of the spin control system is the ability to easily modify the isolation system to effect the play of a racquet. The isolating system could be adjusted, replaced or supplemented to make small or large adjustments in how the racquet performs. These adjustments could take place during a match or after matches. While the adjustments could include replacing the inner-frame/string-system, it could also include removing part or all of the isolating system or replacing it with another or combining multiple isolators at different locations. The adjustments can also include some means of altering the isolating system while connected to the inner and outer frame. This could be done using some sort of tool that modifies the properties of the isolator without removing disconnecting the inner frame from the outer frame. Different isolator combinations could be designed for different playing styles, swing speeds, or talent levels.

The collection of the individual isolators of Figure 2 can be considered equivalent to the global isolator 305 shown in Figure 3. This global isolator, represented by (KGx, KGy, KGz), or (KGx-Isolator System, KGy-Isolator System, KGz-Isolator System), represents the connection of the inner frame to the outer frame (hence the collection of all the individual isolators of Figure 2). The inner frame moves, essentially, as a rigid body on the isolation system (the string system, for out-of-plane deflection, is the exception to the inner frame moving solely as a rigid body; for string bed motion out-of-plane motion, the bed acts as a spring relative to the inner frame; for in-plane motion, the string bed is very stiff for an interwoven and locked string system).

(KGx, KGy, KGz) are adjusted (by adjusting individual isolators Kx, Ky, Kz) to maximize ball spin (and control ball trajectory accuracy; see below) or optimize ball spin for a given player in a given set of conditions. String bed stiffness, measured for a collection of racquets, strings, and string tensions, ranges in stiffness from about 110/130 lbs/inch to 250 lbs/inch (string bed stiffness represents the out-of-plane stiffness a rigid tennis ball would see while center-frame Z-axis loading the bed as the racquet frame is supported).

During ball impact, for a conventional racquet, as the racquet exerts both a normal string-bed force to drive the ball over the net, and a tangential string-bed force to apply top/bottom spin to the ball, the ball is in contact with the string bed between 3-4 milliseconds to 8-9 milliseconds (with an average of 5-6 milliseconds). This contact time is primarily related to the mass of the ball, the dynamics stiffness of the ball and the dynamic stiffness of the string bed (other items can also play a role).

For a conventional racquet, the out-of-plane dynamic stiffness plays a role in determining this contact time (the softer that stiffness, the longer the contact time, and vice-versa; in addition, the ball’s inherent dynamic stiffness also plays a fundamental role). In addition, the in-plane loading for a conventional racquet, during impact between the ball and strings/racquet, is quite different than its out-of-plane loading. The tightly-spaced, interwoven string bed is very stiff in-plane as the ball and racquet/string bed are pushing against each other through the frictional contact force. For maximum ball spin, the ball must not slip on the string bed (or slipping must be minimized), and the frictional force, at least during the initial part of this contact, must adequately develop to allow the ball to transition from sliding across the string bed to rolling across the string bed (during this contact time of 5-6 milliseconds). A stiff in-plane string bed stiffness will reduce ball spin by causing the ball to slide and not roll across the string bed.

For the spin control system invention, during ball impact, under the exact same conditions discussed above for the conventional racquet, the response of the ball is entirely different. For out-of-plane ball response, the ball “sees” the out-of-plane string bed stiffness as well as the KGz stiffness of the isolation system (springs in series). If the KGz stiffness is large compared to the string bed stiffness (for example, 3 to 4 times that of the string bed stiffness), then the out-of-plane “performance/power” of the racquet will be similar to a conventional racquet with the same characteristics (assuming the overall racquet and string bed properties are matched up). If KGz is comparable to the string bed stiffness, then the overall system will be softer, and the dwell time of the ball on the string bed will increase. 

The in-plane response of this spin control system invention will also be different. The ball will see a more compliant system for the in-plane stiffness KGx and KGy of Figures 2-3. Tests/simulations have shown that if Kx and Ky are comparable to the equivalent of the out-of-plane stiffness (ball + string bed + KGz, in series), then an increase in ball spin over a non-isolated system is seen (the stiffness ratios could range from 0.1 to 10.0). An important attribute of this invention is that the stiffnesses of the discrete isolators 302 in Figure 3 can be varied, as discussed earlier, to maximize ball spin or optimize it for a given player in a given set of conditions.  This leads to a compliant in-plane string bed stiffness that will reduce the tangential force needed to take the ball from initially slipping to not slipping (ie, rolling); and a compliant, in-plane string bed can store energy during impact and return that energy to the ball’s rotational energy (thus increasing ball spin). 

Another feature of this spin control system invention is the ability to easily remove the inner frame and isolators and replace with another set of different isolators and/or different pre-strung inner frames. The simple inner frame insert allows for easy stringing of the inner frame. This “insert” design allows for automated stringing of the frame and the opportunity of patented designs of corresponding stringing machines. Inner frames of varying properties could be swapped out to offer different playing characteristics in combination with a given set of isolators. 

Dr. Paul Zarda notes the outer frame of this invention can be similar in size and shape to almost any racquet that is available today. Its weight will be less than most racquets in order that, when combined with the weight of the isolators and inner frame, the assembled weight would be comparable to racquets available today. In addition to the reduced weight restriction, the outer frame’s key properties of this invention would include: i) A design that would structurally support the isolation system; a sound structural connection that would transfer load between the inner frame and the outer frame; ii) a frame design that would allow for adequate sway space for in-plane and out-of-plane motion of the inner frame relative to the outer frame; in-plane sway space motion could be 0.2 inches or more; out-of-plane motion could be similar; iii) an outer frame design that would allow for the easy removal of the isolators, or for in-position changes of the isolators; iv) a frame, when combined with the isolators and inner frame, would result in an overall rigidity comparable to existing racquets. 

Figures 4, 5 and 6 show a designed racquet system that is functionally equivalent to the system presented and discussed in Figures 1 through 3. A description of this specific embodiment can be found in the patent application: https://www.google.com/patents/US20140274494