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.
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.
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
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