### 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 language | English (US) |
---|---|

Pages (from-to) | 2747-2756 |

Number of pages | 10 |

Journal | International Journal of Bifurcation and Chaos in Applied Sciences and Engineering |

Volume | 15 |

Issue number | 9 |

DOIs | |

State | Published - Jan 1 2005 |

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### 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.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Geometry of vibrational stabilization and some applications

AU - Levi, Mark

PY - 2005/1/1

Y1 - 2005/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=27544476546&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27544476546&partnerID=8YFLogxK

U2 - 10.1142/S0218127405013745

DO - 10.1142/S0218127405013745

M3 - Article

AN - SCOPUS:27544476546

VL - 15

SP - 2747

EP - 2756

JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

SN - 0218-1274

IS - 9

ER -