Modelling of a Tennis Racquet String-bed

Abstract

This research is conducted within the field of sports engineering by focusing on vibrations in a tennis racquet. In previous studies, experimental work identifies characteristics of different string materials, but the tests are on single strings rather than the full string-bed. The aim of this research is to theoretically model and experimentally evaluate the string-bed. The specific focus is on the mechanical properties of the string-bed and gaining knowledge about its complete response and the presence of damping. The string-bed has been previously modelled in several ways usually by simple elastic systems and with some recent attempts to include non-linearity inside a partial differential equation. The author believes there is a research gap in how the string-bed is represented and so the desire is to further this area with a model that included string-bed properties.
In the experimental work of this research, the frame of the racquet was fixed so that only the strings vibrated from the impact of a tennis ball and the vibrations were captured by a high-speed camera. In the theoretical work, a model was created for predicting the frequency of a string-bed. The model is a discretized circular membrane which was modified for elliptic geometries. The nature of the damping found from the string-bed responses was analysed using a single degree-of-freedom system for non-linear effects. The discretized circular membrane model is modified for elliptical geometries using work by Troesch and this agrees well with a heuristic modification, also of the discretized circular membrane. This provides a means of predicting the string-bed’s frequency using string-bed properties, rather than membrane features, such as, distance between the strings, the number of crossing points, and the elliptical shape which is common to most tennis racquet frames. The discretized and modified circular membrane model is also used to calculate the string tension. In practice, the value of tension is used as a metric for predicting how a string-bed behaves. Previous studies have tested the change in string tension and show the drop to be surprisingly high. The discretized model’s frequency equation was transposed and the empirical frequency, found using fast Fourier transform analysis, was used to calculate the corresponding reduction in string tension.
The damping characteristics of different materials were found from the “free” vibrations of the string-bed’s responses. To interpret the type of damping present, a simple, single degree-of-freedom, oscillatory system comprising a lumped mass was modelled to simulate different damping types. The velocity component was varied to create non-linear effects, and, also, the stiffness component to which the string tension contributes. This showed that the damping is non-linear and is related to the string tension.

Date of Award	8 Oct 2025
Original language	English
Awarding Institution	University of Bath
Supervisor	Nigel Johnston (Supervisor) & Glen Mullineux (Supervisor)