A Mathematical Model of a Gymnast on the Horizontal Bar

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Abstract

The purpose of the study was to develop and validate a predictive mathematical model of hinged, three-segment, rigid body, planar motion as representative of a gymnast performing on the horizontal bar. The three-segment model consisted of the upper limbs, head and trunk, and lower limbs hinged at the "shoulder" and "hips". The gymnast was assumed to rotate about a fixed axis of rotation and a constant friction moment proportional to the normal force was assumed to act between the bar and the gymnast.

Equations of motion were developed through application of Newtonian mechanics to the three-segment model. The resulting system of equations was reduced to a single equation of motion which was solved numerically predicting the angular motion of the first segment (upper limbs), given the angular motion of the "shoulder" and "hip" joints. The predicted path of the total body center of gravity was also calculated.

Separate computer programs were developed to predict and validate the model against known movements and to predict the path of a theoretical movement. The measured initial conditions and the continuing kinematic constraints of hip and shoulder joint motion for the Validation trials were validation trials were derived from angular data smoothed using a digital filter. The predictive program utilized a Fourier series as representative of the theoretical shoulder and hip joint motions. conducted using four experienced gymnasts executing a beat swing, ordinary grip giant circle, reverse grip giant circle and glide kip up. The subjects were filmed at 200 frames per second using a Locam camera.

Evaluation of the reduced data showed that the absolute mean difference between the measured and predicted angle of the upper limbs with respect to a vertical for 16 trials was 12.98 degrees over a mean simulation interval of 1.42 seconds. The absolute mean difference between the measured and predicted positions of the centers of gravity was found to be 6.21 inches.

The model was concluded to be successful based on comparison with similar human movement simulations.

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