A computer simulation model of tennis racket/ball impacts

Abstract

A forward dynamics computer simulation for replicating tennis racket/ball impacts is described consisting of two rigid segments coupled with two degrees of rotational freedom for the racket frame, nine equally spaced point masses connected by 24 visco-elastic springs for the string-bed and a point mass visco-elastic ball model. The first and second modal responses both in and perpendicular to the racket string-bed plane have been reproduced for two contrasting racket frames, each strung at a high and a low tension. Ball/string-bed normal impact simulations of real impacts at nine locations on each string-bed and six different initial ball velocities resulted in <3% RMS error in rebound velocity (over the 16-27 m/s range observed). The RMS difference between simulated and measured oblique impact rebound angles across nine impact locations was 1°. Thus careful measurement of ball and racket characteristics to configure the model parameters enables researchers to accurately introduce ball impact at different locations and subsequent modal response of the tennis racket to rigid body simulations of tennis strokes without punitive computational cost.

Subject-specific computer simulation model for determining elbow loading in one-handed tennis backhand groundstrokes

“…A four-segment angle-driven subject-specific computer simulation model of the torso and arm is connected to a previously developed two-segment racket model (Glynn et al, 2011) to simulate one-handed backhand groundstrokes. The resulting model is matched to two contrasting trials by an elite tennis player.…”

“…The restoring force F w in each massless spring-damper was a function of the stiffness k w and damping c w coefficients, the displacement x and velocity x  of the wobbling segment relative to the fixed rigid component (Equation (1)), based upon the work of Pain and Challis (2001b). The upper-limb model was attached to a 13 degree of freedom forward dynamics, equipment-specific computer simulation model of tennis racket/ball impacts (Glynn, King and Mitchell, 2011). The racket frame was represented using two rigid bodies connected by a frictionless pin joint with two linear massless torsional spring-dampers to resist motion in and out of the racket plane and model the fundamental modes of vibration.…”

“…A point mass representation of the tennis ball allowed normal and oblique impacts to occur at one of the nine specified locations on the stringbed (see Glynn et al, 2011 for further details). The human hand was represented by a rigid segment connecting the wrist joint centre to the proximal end of the metacarpal of the middle finger which was in turn fixed to a rigid, massless cylinder encasing the racket handle.…”

“…Equipment-specific parameters for the tennis racket frame, stringbed and tennis ball were determined from independent experimental tests (Glynn et al, 2011) and were then fixed. The inertia parameters of the racket frame parts were determined from the results of pendulum oscillation and knife-edge balance experiments.…”

“…More useful models have also incorporated subject-specific parameters to enable simulations to be compared with performances by the subject to provide a quantitative evaluation of the model accuracy King et al, 2006). In particular for tennis Glynn, King and Mitchell (2011) developed and evaluated an equipment-specific computer simulation model of ball/racket impacts and demonstrated that the model was of a sufficient complexity to accurately represent the responses of the ball, stringbed and racket frame in terms of outbound ball velocity and racket frame linear acceleration. Only when a model has been evaluated in this way can the researcher be confident in its capabilities and then use it to address the pertinent research questions (Hubbard, 1993).…”