Lecture 5: Tennis Racquet Stresses Caused by Stringing – By Dr Paul Zarda

Dr. Paul Zarda of Sanford Florida notes:  This is Lecture 5 and it is the last lecture 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.

Lecture 4 reviews the theoretical determination of loads/stresses in a tennis racquet. It also presents, based on a FE model of Wilson tennis racquet frame, loads/stresses in the frame of the racquet due to string loads.

Dr. Zarda of Orlando notes that this lecture, Lecture 5, first reviews the different kind of loads that can develop when stringing a tennis racquet. The assessment of these loads is done based on a finite element model that was developed and presented in Lecture 3.  Following that discussion, the loads that are developed in a tennis racquet (a Wilson racquet is used and is typical of almost any racquet) when it is being strung in a stringing machine are reviewed.

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. 

Figure 1 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 Lecture 4. In this case 60 lbs was chosen as the main string tension, and the cross string tension, per Equation 1 of lecture 4, was determined to be 40 lbs. For an elliptical racquet with equal string spacing, the tensions determined by Lecture 4 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 1), the bending moment is not zero but varies with position along the racquet (measure by S shown in Figure 1, 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. 

Another notable observation of Figure 1 is that the bending moment is not zero as predicted by the discussion in Lecture 4. 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 Lecture 4 not being met by the racquet shown in Figure 1, the bending moment of Figure 1 is likely at its lowest level per Lecture 4.

Dr. Zarda, Orlando and Sanford, notes that Figure 2 shows bending moment loads for the same Wilson tennis racquet under 2 additional loading conditions:  (i) main strings only, at 60 lbs, are in the racquet with no cross strings and no support of the racquet, and (ii) cross strings only, at 45 lbs, are in the racquet with no main strings and no support of the racquet. For reference, the condition of Figure 1, where the racquet is strung at 60 lbs main, 40 lbs cross, is also shown.

As Figure 2 shows, without adequate support, the main-string-only-loading condition will cause this Wilson frame to fail (the bending moment at 12 noon exceeds the failure bending moment of 372 in-lbs discussed/determined/presented in previous lectures). While the moment for the cross-string-only-loading does not show failure, it does show levels that are significantly higher than the moments shown in Figure 2 (and Figure 1) for the final strung racquet (60 main, 40 cross).

Figure 3 shows the FE-predicted bending moments during a typical stringing process. For the bending moments shown in this figure, the frame is supported at 12 and 6, and it can also have additional side supports at 10:30 – 1:30 and 7:30 – 4:30. These side supports come into play as strings are added during the stringing process. The legend shown in Figure 3 explain how and when the side supports are used or not used. The bending moment curves shown represent the bending moment in the tennis frame at various stages in the stringing process. This particular racquet is strung by first stringing the main strings (one side and then the other side), and then the cross strings from top down. The figure shows, for this 6 point support stringing machine (4 side supports and a top and bottom support), that the maximum bending moment (at any location and for any number of strings in the racquet) of approximately 210 in-lbs does not exceed the failure bending moment of  372 in-lbs (Wilson frame). This level of 210 in-lbs is almost 100% higher than the maximum bending moment (120 in-lbs) found in the final strung racquet (see Figure  1).