Fractal dynamics of a bouncing ball on a accelerating lift tabletop with both constrained to vertical motion

Abstract

"The bouncing ball problem has proved to be an important phenomenon in engineering applications involving vibro-transportation and vibratory feeder systems. In this paper, the dynamics of a bouncing ball falling consecutively on an accelerating lift tabletop is studied. Using simulation, it is established that the dynamic interaction of accelerating lift tabletop constrained to one-dimensional motion on which the ball is bouncing is fractal. The acceleration of the lift table top was varied gradually as a percentage of acceleration due to gravity over one thousand steps while the number of bounces-off made by the bouncing ball before the lift table top covered a fall distance of 10.000 m was recorded graphically. Similarly, every lift tabletop acceleration has the set of bounce-off height of the bouncing ball recorded graphically, and taken to be one third of height of fall. The number of bounce off drastically dropped to about zero when the acceleration of the lift tabletop was 40% of acceleration due to gravity. The graphical presentation of the ball bounce off height has normal distribution shape with fractal detail. This study showed that two objects, initially at different heights, falling under gravity, maintain separating heights for the period of their fall. The equation governing the dynamics of the bouncing ball and the lift tabletop are of quadratic type but the ball bounce off height graphical results contain fractal details. "