Two Methods for Determining Impact Time in the Bouncing Ball System

Many physical systems can be modelled relatively simply and accurately by a ball bouncing off a surface; hence, the dynamics of this system have been studied by mathematicians and physicists for several decades. If the surface that the ball is rebounding from is fixed, the problem is simple, and can be analyzed as a damped harmonic oscillator. However, if the surface itself is vibrating, very complex behavior can arise. There are two cases to consider: elastic collisions (where no energy is lost due to impact), and inelastic collisions. Both cases have been studied quite extensively, and chaotic behavior has been shown to arise for various parameter values in both circumstances. Holmes (1982) gives a good introduction to the complexities. The bouncing ball problem has a number of significant applications, from the field of granular flow physics to robotics (through analogies to a person walking). In particular, it is important to be able to model a person walking over uneven or vibrating terrain if robotic technology is to be able to eventually learn to “walk.” Therefore, the bouncing ball on a vibrating platform is an important problem to study and understand, even though much work has been done on the problem already. In our research, however, we were unable to find a modern simulator that effectively models the behavior of the ball and platform. At least two older programs exist; one accompanies a text called An Experimental Approach to Nonlinear Dynamics and Chaos, and is on a 3.5” floppy diskette that only runs on old Apple computers (Tufillaro et al., 1992). A similar program is available at Thomas (2000) that will plot phase diagrams for the bouncing ball program, but the code is poorly-maintained and not as extensive as the original Bouncing Ball software. Therefore, we endeavor to create a