Nonlinear dynamics of a ball bouncing on a vibrating ferrite rod

This paper explores the dynamics of a ball bouncing on a ferrite rod in an alternating magnetic field. Using linear magnetostriction theory, we derive the equation for the rod's vibration and analyze the influence of frequency doubling effect on the ball's motion. We analytically determined the probability distribution function (PDF) of the ball's rebound velocity and validated our analysis with numerical simulations and experimental results. In both simulations and experiments, the analytically deduced PDF accurately predicts the rebound velocity distribution with high velocities, but shows minor discrepancies in low velocities due to the nonlinear relationship between the incoming and rebounding relative velocity. Additionally, simulation results also show that the ball will eventually enter a "locking region" and undergoes periodic motion.