Preface: Paul Zarda notes that an elliptical shaped tennis racquet, with
a dual loading condition of main string tensions and cross string
tensions, will be shown to carry only a low-stress-level compressive
load and not the expected high-stress-level bending moment. This 2-D
loading condition applied to a tennis racquet is analogous to the
historical 1-D loading of a parabolic arch (carrying only axial load)
that is well documented in mechanical engineering and engineering
mechanics. As far as the author is aware of, this is the first ever
documentation and corresponding proof that an elliptical shaped tennis
racquet head, loaded using a derived formula for the main and cross
string tensions, will be shown to carry only a low-stress axial
compressive load. In order to minimize stress-levels in a tennis
racquet due to stringing, these in-racquet-derived-formula-string
tensions should be a targeted final-string-tension level for the
stringing of any elliptical or nearly-elliptical tennis racquet.
Figure
101
Figure
101 shows the head of a tennis racquet that is subjected to tensioned
strings that cause loads in the cross string direction (x-direction) and
the main string direction (y-direction). As shown in Figure 101, Wx
corresponds to a loading per unit length for the cross string tensions,
and Wy corresponds to a loading per unit length for the main string
tensions.
Paul Zarda of Sanford notes the insert shown
in Figure 101 exposes the internal forces that can develop in the frame
caused by the string tensions. M is the bending moment on the x-section,
V is the shear force on the x-section, and N is the axial force on the
x-section. It will be shown that if the in-plane x-y shape of the tennis
frame is elliptical, then the bending moment and shear force will be
identically zero. This will be shown to only occur for a specific ratio
of Wy/Wx.
Figure 102
Figure 102 shown a free body diagram of the
right half of the tennis frame. As shown in Figure 102, the symmetry
plane cut at x=0 only exposes normal forces Fx (and from symmetry about
the y=0 plane, Fx is the same for the top x-sectional cut and the bottom
x-sectional cut shown in Figure 102).
Sum of forces in the X-direction = 0:
Equation 101
Equation 102
Figure
103 is a free body diagram of the top half of the tennis frame. The
x-sectional cut on the symmetry plane y=0 exposes the axial force Fy (as
shown in the Figure). Fy is the same axial force for the left cut and
the right cut of the frame since x=0 is also a symmetry plane.
Figure 103
From Figure 103, using sum of forces in the Y-direction = 0:
Equation 103
Equation 104
Figure
104 is a free body diagram of the tennis frame in the 1st quadrant. It
shows the frame with x-sectional cuts both on the symmetry plane y=0 and
also a cut at the (x,y) position shown in the Figure. The x-sectional
cut at position (x,y) exposes the axial force N acting an angle θ as
shown in Figure 104.
Figure 104
From Figure 104, with the assumption that M
and V are 0 at the x-sectional cut at the position (x, y), then the sum
of forces in the x-direction = 0 gives
Equation 105
From Figure 104, sum of forces in the y-direction = 0 gives
Equation 106
From Figure 104, sum of moments (counterclockwise) about point A = 0 gives
Equation 107
Substituting Ncosθ from Equation 105 and Nsinθ from Equation 106 into Equation 107, one gets
Equation 108
Combining similar terms
Equation 109
Simplifying, and substituting in for
from Equation 104
Equation 110
Expanding
Equation 111
With the orange terms canceling, and rearranging
Equation 112
And dividing by the right hand side
Equation 113
Re-adjusting the first term
Equation 114
Re-writing Equation 114
Equation 115
If
the term in { brackets } in Equation 115 is unity, Equation 115 is the
equation of an ellipse, centered at (0, 0), with minor/major axis of
2a/2b, respectively, as shown in Figure 101. Hence, setting the bracket
term to unity
Equation 116
Or
Equation 117
Statement:
Equation 117, in conjunction with Equation 115, says that if the cross
string tensions per unit length, Wx, is a fraction (a/b)2 of the main
string tension per unit length, Wy, then, with a racquet head in the
form of an ellipse per Equation 115, the frame will only develop a
low-stress-level compressive force N. It is the author’s opinion that
this is the first time in history that this revealing statement and
corresponding proof has been publically offered.
Figure 105
Paul
Zarda of Orlando notes the observations made in the above statement have
impact both on stringing a tennis racquet and the performance of a
tennis racquet. Figure 105 shows an elliptical shape head of a tennis
racquet with several definitions provided associated with the stringing
of a tennis racquet. If we define, as shown in Figure 105, that Tx is
the cross string tension, Ty is the main string tension, nx is the
number of cross strings, and ny is the number of main strings, then Wx,
the cross string loading per unit length, is approximated by (using an
average spacing)
Equation 118
Similarly,
Wy, the main string loading per unit length, is approximated by (using an average spacing)
Equation 119
Substituting Equations 118 and 119 into Equation 117
Equation 120
Simplifying
Equation 121
To
somewhat quantify this formula, let’s do the calculation where the head
width 2a of a racquet is 10 inches and the head height 2b is 13 inches,
and the string pattern is 16 x 19 (ny = 16 main strings, nx = 19 cross
strings, a somewhat typical racquet). Using a reference main string
tension of Ty = 60 lbs, the above formula produces a cross string
tension Tx of 38.9 lbs. This cross string tension of 38.9 lbs assumes
equal spacing of main and cross strings (an approximation), and is
relative to the main string tension of 60 lbs. This set of tensions ( Tx
,Ty ) = (38.9, 60), in an elliptical shaped head (or nearly elliptical)
of 10 inches x 13 inches, will produce a low-stress-level axial
compressive load (see Equations 102 and 104 for representative axial
compressive loads; using these formulas, at 12 o-clock Fx = 370 lbs and
at 3 o’clock Fy = 480 lbs).
Paul Zarda of Orlando and Sanford
notes it is important to point out that Equation 121 provides the final
cross string tension Tx in a frame relative to the final main string
tension Ty. These tensions are not necessarily (or even likely) the
tensions pulled when stringing a racquet in a stringing machine. This
observation is due to the fact that, during the stringing process, a
racquet’s frame can deform, and that deformation will cause the pulled
tensions to change. This deformation is related to the flexibility of
the stringing machine and its support of the tennis racquet, the
flexibility of the tennis racquet and the stiffness of the strings
themselves. There are stringing machines that adequately support a
racquet by not allowing a tennis frame to deform during stringing; in
this case the pulled tensions are representative of the final, in-place
tensions. For flexible conditions, determining what tensions to pull
that would result in the recommended final tensions of Equation 121 is
the subject of future investigation.
In a number of stringing
machines that support a frame at 2-6 points, it is a recommended
practice to pull both the main and cross string tensions at the same
level (say 60 lbs). What is interesting is that the measured final
tensions in such a strung frame are closer to the Tx/Ty tensions
predicted above. This is due to racquet deformation (and stringing
machine deformation) and it represents the racquet trying to reach, via
deformations as best as it can, the low strain energy state implied by
equation 121.
One final observation: a ball can rebound in an
unexpected direction from a string-bed whose string tensions are
not-uniform. This effect has never been quantified and perhaps is
handled by a player in his/her swing, or perhaps it results in a ball’s
trajectory that is not completely player controllable. This is also a
topic for future investigation.
