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Horizontal stability of a bouncing ball.

Brendan G McBennett1, Daniel M Harris1

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

Chaos (Woodbury, N.Y.)
|October 27, 2016
PubMed
Summary
This summary is machine-generated.

A bouncing ball on a vibrating parabolic surface becomes unstable. It shifts to lateral motion, leading to complex periodic and chaotic bouncing patterns as the surface steepens.

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Area of Science:

  • Physics
  • Nonlinear Dynamics
  • Classical Mechanics

Background:

  • Understanding the dynamics of bouncing objects is crucial in various fields.
  • Investigating the stability of periodic motion under external forcing.

Purpose of the Study:

  • To analyze the stability of a partially elastic ball bouncing on a vertically vibrated concave parabolic surface.
  • To identify transitions from simple vertical motion to complex bouncing behaviors.

Main Methods:

  • Two-dimensional simulation of a bouncing ball system.
  • Analysis of stability for periodic solutions under horizontal perturbations.
  • Characterization of emergent motion patterns.

Main Results:

  • Simple vertical bouncing becomes unstable to horizontal perturbations above a critical parabolic coefficient.
  • A new periodic solution emerges with persistent lateral motion over the parabola's vertex.
  • Further increases in parabolic steepness lead to complex periodic and chaotic bouncing states.

Conclusions:

  • The stability of bouncing ball dynamics is highly sensitive to surface geometry and external vibrations.
  • Concave parabolic surfaces can induce complex nonlinear behaviors, including chaos.
  • Lateral motion is a key characteristic of unstable bouncing states on these surfaces.