The tennis racket has developed since the origins of Lawn Tennis in the 1870s. This study investigated how the tennis racket developed from 1874 to 2017, using measurements and material classifications for 525 samples. Racket measurements covered geometric, inertial and dynamic properties, and the number of strings. Rackets predating 1970 were mainly wooden, and typically characterised by head areas below 0.05 m2, masses over 350 g and natural frequencies below 120 Hz. Rackets from the 1970s were made from wood, metal and fibre–polymer composites, with most postdating 1980 made from fibre–polymer composites with a larger head, lower mass and higher natural frequency than their predecessors. Principal component analysis was used to reduce the dimensionality of the number of variables. Principal component one (PCA1) accounted for 35% of the variance in the measured racket properties, and was found to be significantly affected by material. Head width was best correlated with principal component one (r = 0.897, p < 0.001), followed by head length (r = 0.841, p < 0.001) and natural frequency (r = 0.813, p < 0.001). Early rackets were constrained by the limitations of wood, and the move to composites, which began in the 1970s, allowed this observed increase in head size and natural frequency. As material development has been a major driver of racket design in the past, we propose that new materials and manufacturing techniques, like additively manufactured composites, could further improve the tennis racket. The measurement techniques described here can be used to monitor developments in racket design.
Tennis racket properties are of interest to sports engineers and designers as it allows them to evaluate performance, review trends and compare designs. This study explored mathematical models that correlated to the mass moments of inertia of a tennis racket, both about an axis through the butt and about the longitudinal axis, using its dimensions, mass and centre of mass location. The models were tested on 416 rackets, dating from 1874 to 2017. Results showed that moments of inertia about the butt and longitudinal axis can be estimated to within − 4 to 5% and − 11 to 12% of measured values, respectively, using the proposed models on original rackets. When rackets were customised, with 30 g of additional mass, moment of inertia about the butt could be estimated within 6%, but the model for moment of inertia about the longitudinal axis was less accurate (largest error at 25%). A Stepwise Linear Regression model indicated that racket mass and then centre of mass location had the largest effect on moment of inertia about the handle, with head width having the largest effect on moment of inertia about the longitudinal axis.Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Abstract:Tennis rackets have advanced significantly since the invention of the game in 1874, including innovations in both shape and materials. Advances in these design parameters have implications for racket performance, especially swing speed. This study tested one hundred rackets, spanning brands and eras, using simple, portable instruments in order to pilot protocols and make recommendations for streamlining testing procedures for tennis rackets. A wide range of properties were measured and documented for each racket. We suggest that since Transverse and Lateral Moment of Inertia are well correlated, measuring both is not necessary when processing a large number of rackets. In addition, it is also possible to predict the Transverse Moment of Inertia well from models that use simple dimension and mass measurements, which may be preferable in larger studies. Exploring the use of more complex modelling will allow us to better understand the impact of tennis racket design on performance in the future.
The International Tennis Federation (ITF) is responsible for protecting the nature of tennis. The ITF uses computational models to predict how trends in equipment parameters could affect the games future. The current ball-racket impact model is limited to non-spinning, on-axis, normal ball impact simulations. The aim of this project was to develop a model of oblique, spinning, on-and off-axis ball-racket impacts. Large scale test data (n > 1000) was collected using an impact rig and calibrated highspeed cameras. Impacts for a range of realistic velocities, spin rates and impact locations were collected, measured using automated image processing algorithms to digitise ball centroids. An established spin measurement method was improved to correct for perspective errors associated with the proximity of the cameras to the test volume. The automated algorithms were validated with experimental data and manual methods. Multi-variate polynomial models to predict the lateral and vertical components of rebound velocities and rebound spin rate were trained and validated using a curve fitting toolbox and 'n-fold and leave one out cross-validation' method. Second order models best fit the training data, with the low predictive errors. Root-mean-squared errors were calculated using a test dataset, independent of the training data. These were 0.57 m•s-1 for the lateral rebound velocity model, 0.48 m•s-1 for the vertical rebound velocity model and 30.5 rad•s-1 for the rebound spin rate model. Variance was partially explained by experimentally established inherent variability of the ball and stringbed. Model output confidence was established by simulating small changes in model inputs. The simulated lateral and vertical components of rebound velocity, but not the simulated spin rate, were an order of magnitude greater than the measurement precision. The new models were combined with ball aerodynamics and ball-to-surface impact models to simulate tennis court trajectories for oblique, spinning, on-and off-axis ballracket impacts. Increasing stringbed stiffness or the lateral offset of impact location were found to decrease rebound velocity and increase rebound angle-markedly so for a 60 mm lateral offset. Increasing lateral offset also increased the rebound spin rate.
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