Lecture 3 - Tennis Racquet Stresses Caused by Stringing

Dr. Paul Zarda of Orlando and Sanford Florida notes:  This is Lecture 3 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.

This lecture, Lecture 3, shows an overview of a finite element model of a tennis racquet. A finite element model is a simulation technique that can be used to assess the loads a tennis racquet might see and whether or not those loads might cause the tennis racquet to fail. The loads that are predicted with the finite element model can be compared to the actual failure loads discussed in Lecture 2 in order to assess if the tennis racquet might withstand a specific loading.

Dr. Paul Zarda of Orlando and Sanford notes that Figure 1 shows a typical tennis racquet that has been strung in the top left image. This is a Wilson tennis racquet that has a stringing pattern of 16 x 19, which means it has 16 main strings and 19 cross strings. The bottom right image of Figure 1 is a finite element model of that very tennis racquet. A description of finite element techniques can be found on the internet and will only be discussed here in an overview way.

Dr. Zarda of Orlando points out that Figure 2 shows more details of the racquet where both the frame and the strings are modeled as beam elements. A beam element is a work horse finite element that can accurately model structures that are beam like: a beam-like structure is a structure that is long in one direction compare to the other two (8 to 1 or more). The tennis racquet meets those conditions. Loads can be applied to this finite element beam model and the stresses in the tennis frame can be determined.

Figure 3 shows two different types of loads that were applied to the tennis racquet: a load in the 3-9 direction forcing the two sides of the frame together, and a load in the 12-6 direction forcing the top of the racquet toward the yoke. These loads can be both applied to the finite element model as well as to the actual Wilson tennis racquet. The resulting deflections for both of these loading conditions (test and FE simulation) can be compared. If they are in disagreement, attributes (for example, x-sectional properties) of the FE model (the simulation model) can be adjusted to correlate the simulation and test.

Figure 4 shows the correlated results of that simulation for the two loading conditions presented in Figure 3. The loading condition for the 3-9 load case is presented here. The deformed shape of the loading is exaggerated in the left image of Figure 4, and the un-exaggerated deformed shape of the simulation is shown in the image at the right of Figure 4. These results match the tested racquet.

This “tuned” FE model can now be used (future lectures) to predict the loads/stresses in the tennis racquet frame when the racquet is loaded during a typical stringing process.