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

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.

Comparison of a finite element model of a tennis racket to experimental data

“…The pin was located underneath the tip of the frame between the two central main strings. Using a pin to support a racket is a technique which had been used by previous authors [14,15,23]. Three identical rackets were used for the experimental testing, all strung at 289 N (65 lbs).…”

“…Therefore, freely suspending a racket is currently the best representation of how it will be supported during an actual tennis shot. Previous authors have found that, for impacts normal to the face on the long axis of a freely suspended racket, the rebound velocity of the ball is dependent on impact location and is lowest at the tip and highest in an area near the throat [4,14,15]. Goodwill and Haake [15] produced spring damper models for normal impact on rigid and flexible rackets.…”

Viscoelastic impact characterisation of solid sports balls used in the Irish sport of Hurling

“…Over the course of testing the sliotar cores, a difference was observed between these two measurements for ball types that exhibited lateral expansion. Previous publications' statements of equivalence [30][31][32] between COM displacement and diameter compression methods were shown in the present study as being acceptable, not for the reasons given in the original papers but rather due to the absence of lateral expansion in the ball impacts examined. As the lateral expansion occurred parallel to the impact plate and hence was imperceptible to the axial load-cell, it augmented the diameter compression due to volumetric spreading of ball material without affecting the COM displacement as derived from the load-cell force data.…”

“…Numerous methods for quantifying deformation are featured in the literature: diameter compression, the reduction of diameter of the ball normal to the impact plate; lateral expansion, the increase in ball diameter parallel to the impact plate; and centre of mass (COM) dis-placement, calculated from the double time integral of the force divided by ball mass. Many publications assume, either explicitly or implicitly, that diameter compression and COM displacement are equivalent [30][31][32], neglecting the effect (if any) of lateral expansion. The studies that have considered lateral expansion do not refer to COM displacement [33,34].…”

Simulation of the impact response of a sliotar core with linear and non-linear contact models

“…More specifically, the Kelvine-Voigt model predicts that contact forces at the beginning of the impact are discontinuous (due to the damping term being proportional to velocity alone), that coefficients of restitution do not vary impact velocity (which is inaccurate, as COR has been shown experimentally to decrease with impact speed) and that small attractive force terms appear directly prior to the separation of the bodies (when from physical reasoning, if the body has expanded back to its original dimensions, no net force should act upon it). Such deficiencies were also observed by Haake et al (2003) and Haake (2001,2004) [10,17,18]. None-theless,comparing the Hunte-Crossley model predictions to an analogous Kelvine-Voigt model can facilitate understanding of the impact process.…”