Revealing the mechanism causing stepwise maximum bounce height changes in a bouncing ball system

The bouncing ball system is a simple mechanical collision system that has been extensively studied for several decades. It is a fundamental problem in impact dynamics. We studied a traditional bouncing ball system numerically and experimentally and discovered novel bifurcation structures where the maximum height of the bouncing ball with respect to the stationary state increases stepwise nonsmoothly when we increase the frequency of the oscillating table continuously even though the bouncing ball is in chaotic states [Karube et al., Chaos 30, 103111 (2020)]. We attempt to reveal the trick causing the stepwise changes of the maximum heights of the bouncing ball. We focus on the time interval for the ball to take off and land on the oscillating table at which the ball takes the maximum height. Let this time interval be denoted by t-interval. In addition, let the oscillation frequency of the table be denoted by f. The stepwise increases in the maximum heights of the bouncing ball in the stationary states occur when the multiplication of the t-interval and f coincides with integer values. This is the mechanism causing the nonsmooth maximum heights. Furthermore, results that are qualitatively consistent with the numerical ones are verified in the actual bouncing ball system using table tennis ball equipment.