Spring damper model of an impact between a tennis ball and racket

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Abstract
A model has been derived that determines the ball, stringbed and racket frame motion for an impact between a tennis ball and racket. This paper describes the model and methods used to verify it experimentally. The model incorporated parameters such as racket mass, moment of inertia and ball stiffness. The work was conducted to produce a tool that could be used to identify the importance of each of the parameters on the impact. The ball was modelled as simple spring and damper in parallel while the stringbed was modelled as a spring in series with the ball. The values of these spring and damper parameters were determined experimentally. It was assumed that the racket frame was a rigid body to simplify the analysis. Two different methods of supporting the racket were modelled, and the balls were always projected perpendicular to the string plane. Firstly, the racket was head-clamped and all the balls impacted at the geometric string centre of the racket. Good agreement was found between the experiment and model data for both ball rebound velocity and maximum stringbed deflection during impact. The second method of support involved freely suspending the racket at the tip on a small pin. Three different impact positions along the longitudinal axis of the racket were tested. Good agreement between the experiment and model data was found for the ball rebound velocity for impacts at the geometric string centre of the racket. The model over-predicted the experimental racket rebound velocity (generally by less than 5 per cent) for all impact positions. A qualitative analysis assigned this small difference to the assumption that the frame was a rigid body and therefore vibration losses were not accounted for.

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