Dr. Paul Zarda of Orlando notes: This is Lecture 4 in a series of
lectures to help understand stresses that develop in a tennis racquet
when the racquet is being strung on a stringing machine.
Lecture 1 developed a basic understanding of two different kinds of loads or stresses that a tennis racquet will carry when it is being strung.
Lecture 2 reviewed typical loads that a tennis racquet can withstand. It also reviewed actual Instron failure loads of some typical racquet frames.
Lecture 3 reviewed the development of numerical models, or simulation models, called finite element (FE) models, and how such practices can be used to assess loads and stresses developed in a tennis racquet due to a typical stringing of the racquet.
This lecture, Lecture 4, reviews a theoretical basis for predicting loads in a tennis racquet frame. It then uses that theoretical background to look at real loads in a strung tennis racquet.
Dr. Zarda of Sanford notes that the shape of the tennis racquet frame, and the stringing and string pattern of the frame, is important in determining the loads and stresses in the frame. The largest loads that the frame will see occur because of the string tension that is applied to the frame. A basic understanding of these loads and resulting stresses will now be discussed. A formula for the tensioning of the strings can be derived to minimize stresses in the frame.
Lecture 1 developed a basic understanding of two different kinds of loads or stresses that a tennis racquet will carry when it is being strung.
Lecture 2 reviewed typical loads that a tennis racquet can withstand. It also reviewed actual Instron failure loads of some typical racquet frames.
Lecture 3 reviewed the development of numerical models, or simulation models, called finite element (FE) models, and how such practices can be used to assess loads and stresses developed in a tennis racquet due to a typical stringing of the racquet.
This lecture, Lecture 4, reviews a theoretical basis for predicting loads in a tennis racquet frame. It then uses that theoretical background to look at real loads in a strung tennis racquet.
Dr. Zarda of Sanford notes that the shape of the tennis racquet frame, and the stringing and string pattern of the frame, is important in determining the loads and stresses in the frame. The largest loads that the frame will see occur because of the string tension that is applied to the frame. A basic understanding of these loads and resulting stresses will now be discussed. A formula for the tensioning of the strings can be derived to minimize stresses in the frame.
Consider, referring to Figure 1, 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 as shown in the Figure 1
insert: 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 in the frame.
It is well known that there is a shape for this arch that will minimize the bending moment M in the arch. This shape is a parabolic shape involving L and h (see Figure 1) and such a shape 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. Such a loading situation (axial load only) will minimize the forces/stresses in the structure.
It is well known that there is a shape for this arch that will minimize the bending moment M in the arch. This shape is a parabolic shape involving L and h (see Figure 1) and such a shape 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. Such a loading situation (axial load only) will minimize the forces/stresses in the structure.
For the double loading
shown in Figure 2 (see description in Figure 2), is there a shape for
that structure that will also minimize M? Yes, there is. 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, but now documented in a recent patent
application, M & V will go to zero if the shape of the structure is a
mathematical ellipse (minor axis a, major axis b, see Figure 2), and
the Wx, Wy string loading is not arbitrary but chosen per Equation 1 in
Figure 2. 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 2. Stringing the
tennis racquet frame based on the tension formula of Equation 2 of
Figure 2, will minimize the stresses (M and V will go to zero) for the
tennis frame.
Figure 3 shows the bending moment that can
developed in a strung frame. These results are based on a finite element
(FE) model of a Wilson tennis racquet that is discussed in Lectures 2
and 3. Here the tensions chosen are based on the formula presented in
Figure 2. In this case 60 lbs was chosen as the main string tension, and
the cross string tension, per Equation 1 of Figure 2, was determined to
be 40 lbs. For an elliptical racquet with equal string spacing, the
tensions determined by Equation 1 and 2 of Figure 2 would produce a
bending moment M close to zero. In this case, because of non-equal
spacing of the Wilson frame (FE model is shown in Figure 3), the bending
moment is not zero but varies with position along the racquet (measure
by S shown in Figure 3, where S=0 is at 12 noon, and S=18 inches is at 6
o-clock). Note that the maximum bending moment is around 120 in-lbs and
occurs at about 3 o-clock (S=9 inches). This bending moment is
significantly less than the failure bending moment of 372 in-lbs
discussed in Lectures 2 and 3. Also note that there is no discussion
here of how this racquet was strung. It is assumed, for this discussion,
that the strings are “dropped in” and their final in-racquet tensions
are 60 lbs for the main and 40 lbs for the cross. While this final
strung condition of a tennis racquet is difficult or even impossible to
achieve in a practical way using a stringing machine, it does offer the
reader an understanding of the minimal load/stress level that one might
achieve.
In conclusion, Dr. Zarda, Orlando and Sanford, notes another notable observation of Figure 3 is that the bending moment is not zero as predicted by the discussion of Figure 2. This is caused by the fact that the shape of the racquet frame head is not a perfect ellipse and because the string spacing is not uniform. Even with the conditions implied in Figure 2 not being met by the racquet shown in Figure 3, the bending moment of Figure 3 is likely at its lowest level by using Equation 2 of Figure 2.
The next and final lecture will look at loads/stresses that are caused using an actual stringing machine.
In conclusion, Dr. Zarda, Orlando and Sanford, notes another notable observation of Figure 3 is that the bending moment is not zero as predicted by the discussion of Figure 2. This is caused by the fact that the shape of the racquet frame head is not a perfect ellipse and because the string spacing is not uniform. Even with the conditions implied in Figure 2 not being met by the racquet shown in Figure 3, the bending moment of Figure 3 is likely at its lowest level by using Equation 2 of Figure 2.
The next and final lecture will look at loads/stresses that are caused using an actual stringing machine.
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