Geometry of vibrational stabilization and some applications

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

This paper gives a short overview of various applications of stabilization by vibration, along with the exposition of the geometrical mechanism of this phenomenon. More specifically, the following observation is described: a rapidly vibrated holonomic system can be approximated by a certain associated nonholonomic system. It turns out that effective forces in some rapidly vibrated (holonomic) systems are the constraint forces of an associated auxiliary nonholonomic constraint. In particular, we review a simple but remarkable connection between the curvature of the pursuit curve (the tractrix) on the one hand and the effective force on the pendulum with vibrating support. The latter observation is a part of a recently discovered close relationship between two standard classical problems in mechanics: (1) the pendulum whose suspension point executes fast periodic motion along a given curve, and (2) the Chaplygin skate (known also as the Prytz planimeter, or the "bicycle"). The former is holonomic, the latter is nonholonomic. The holonomy of the skate shows up in the effective motion of the pendulum. This relationship between the pendulum with a twirled pivot and the Chaplygin skate has somewhat unexpected physical manifestations, such as the drift of suspended particles in acoustic waves. Finally, a higher-dimensional example of "geodesic motion" on a vibrating surface is described.

Original languageEnglish (US)
Pages (from-to)2747-2756
Number of pages10
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume15
Issue number9
DOIs
StatePublished - Jan 1 2005

Fingerprint

Pendulum
Pendulums
Stabilization
Geometry
Curve of pursuit
Planimeters
Tractrix
Nonholonomic Constraints
Nonholonomic Systems
Bicycles
Motion
Holonomy
Periodic Motion
Nonholonomic
Pivot
Acoustic Waves
Geodesic
Mechanics
High-dimensional
Vibration

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics

Cite this

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Geometry of vibrational stabilization and some applications. / Levi, Mark.

In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, Vol. 15, No. 9, 01.01.2005, p. 2747-2756.

Research output: Contribution to journalArticle

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