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Bouncing ball on a vibrating periodic surface.

Avishai Halev1, Daniel M Harris1

  • 1Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA.

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Summary
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A bouncing ball on a vibrating surface can become unstable, leading to horizontal motion. Breaking surface symmetry can direct this motion, enabling controlled "walking" states.

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Area of Science:

  • Physics
  • Nonlinear Dynamics
  • Classical Mechanics

Background:

  • Investigates the complex dynamics of a partially elastic ball interacting with a vertically vibrated sinusoidal surface.
  • Builds upon previous work by McBennett and Harris (Chaos 26, 093105 (2016)) on chaotic systems.

Purpose of the Study:

  • To demonstrate the instability of simple periodic vertical bouncing on a sinusoidal surface.
  • To explore the resulting period doubling cascade and horizontal motion.
  • To investigate the possibility of periodic
  • walking
  • states and directional control.

Main Methods:

  • Numerical simulations and theoretical analysis of a bouncing ball model.
  • Analysis of the critical curvature leading to instability.
  • Parameter space exploration for different surface topographies and ball properties.

Main Results:

  • Simple periodic bouncing becomes unstable above a critical surface curvature.
  • Instability leads to a period doubling cascade and persistent horizontal motion.
  • Periodic "walking" states are observed, and directional motion is induced by breaking symmetry.

Conclusions:

  • The study reveals a transition from vertical to horizontal motion in a vibrated bouncing ball system.
  • Periodic "walking" states are achievable under specific conditions.
  • Asymmetric surface topography provides a mechanism for controlling the direction of horizontal motion.