Dynamics of an axisymmetric body spinning on a horizontal surface. IV. Stability of steady spin states and the 'rising egg' phenomenon for convex axisymmetric bodies

M. Branicki, Yutaka Shimomura

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Following part III in the series, the linear stability of previously identified steady states is analysed for a general convex axisymmetric body spinning on the horizontal plane in order to determine spin orientations leading to the 'rising egg' phenomenon. The viscous friction law is assumed between the body and the plane, which is linear in the velocity of the point of contact and allows for analytical treatment of the problem. In the analysis, the emphasis is put on the relationship between the geometrical structure of interconnected structures of non-isolated fixed-points, representing the steady-spin states in the system phase space, and their stability properties. It is shown that the rising egg phenomenon, discussed initially in part I for the flip-symmetric geometry of a uniform spheroid, occurs in a much broader class of spinning axisymmetric bodies. It is also shown that for some geometries, the steady spin configurations of minimum potential energy are always stable, contrary to the flip-symmetric case, so that even a rapid spin does not cause the centre-of-mass to rise. Particular attention is focused on a spheroid with displaced centre-of-mass and the tippe-top.

Original languageEnglish
Pages (from-to)3253-3275
Number of pages23
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume462
Issue number2075
DOIs
Publication statusPublished - 2006 Nov 8

Fingerprint

axisymmetric bodies
eggs
metal spinning
Horizontal
Geometry
spheroids
Flip
Barycentre
Potential energy
center of mass
Friction
Linear Stability
geometry
Phase Space
friction
Fixed point
potential energy
Contact
Configuration
Series

Keywords

  • Dynamical systems
  • Non-isolated fixed points
  • Rigid-body dynamics
  • Rising egg phenomenon
  • Spinning bodies
  • Stability analysis

ASJC Scopus subject areas

  • General

Cite this

@article{796d49e831b047098b00c00a0c529067,
title = "Dynamics of an axisymmetric body spinning on a horizontal surface. IV. Stability of steady spin states and the 'rising egg' phenomenon for convex axisymmetric bodies",
abstract = "Following part III in the series, the linear stability of previously identified steady states is analysed for a general convex axisymmetric body spinning on the horizontal plane in order to determine spin orientations leading to the 'rising egg' phenomenon. The viscous friction law is assumed between the body and the plane, which is linear in the velocity of the point of contact and allows for analytical treatment of the problem. In the analysis, the emphasis is put on the relationship between the geometrical structure of interconnected structures of non-isolated fixed-points, representing the steady-spin states in the system phase space, and their stability properties. It is shown that the rising egg phenomenon, discussed initially in part I for the flip-symmetric geometry of a uniform spheroid, occurs in a much broader class of spinning axisymmetric bodies. It is also shown that for some geometries, the steady spin configurations of minimum potential energy are always stable, contrary to the flip-symmetric case, so that even a rapid spin does not cause the centre-of-mass to rise. Particular attention is focused on a spheroid with displaced centre-of-mass and the tippe-top.",
keywords = "Dynamical systems, Non-isolated fixed points, Rigid-body dynamics, Rising egg phenomenon, Spinning bodies, Stability analysis",
author = "M. Branicki and Yutaka Shimomura",
year = "2006",
month = "11",
day = "8",
doi = "10.1098/rspa.2006.1727",
language = "English",
volume = "462",
pages = "3253--3275",
journal = "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",
issn = "0080-4630",
publisher = "Royal Society of London",
number = "2075",

}

TY - JOUR

T1 - Dynamics of an axisymmetric body spinning on a horizontal surface. IV. Stability of steady spin states and the 'rising egg' phenomenon for convex axisymmetric bodies

AU - Branicki, M.

AU - Shimomura, Yutaka

PY - 2006/11/8

Y1 - 2006/11/8

N2 - Following part III in the series, the linear stability of previously identified steady states is analysed for a general convex axisymmetric body spinning on the horizontal plane in order to determine spin orientations leading to the 'rising egg' phenomenon. The viscous friction law is assumed between the body and the plane, which is linear in the velocity of the point of contact and allows for analytical treatment of the problem. In the analysis, the emphasis is put on the relationship between the geometrical structure of interconnected structures of non-isolated fixed-points, representing the steady-spin states in the system phase space, and their stability properties. It is shown that the rising egg phenomenon, discussed initially in part I for the flip-symmetric geometry of a uniform spheroid, occurs in a much broader class of spinning axisymmetric bodies. It is also shown that for some geometries, the steady spin configurations of minimum potential energy are always stable, contrary to the flip-symmetric case, so that even a rapid spin does not cause the centre-of-mass to rise. Particular attention is focused on a spheroid with displaced centre-of-mass and the tippe-top.

AB - Following part III in the series, the linear stability of previously identified steady states is analysed for a general convex axisymmetric body spinning on the horizontal plane in order to determine spin orientations leading to the 'rising egg' phenomenon. The viscous friction law is assumed between the body and the plane, which is linear in the velocity of the point of contact and allows for analytical treatment of the problem. In the analysis, the emphasis is put on the relationship between the geometrical structure of interconnected structures of non-isolated fixed-points, representing the steady-spin states in the system phase space, and their stability properties. It is shown that the rising egg phenomenon, discussed initially in part I for the flip-symmetric geometry of a uniform spheroid, occurs in a much broader class of spinning axisymmetric bodies. It is also shown that for some geometries, the steady spin configurations of minimum potential energy are always stable, contrary to the flip-symmetric case, so that even a rapid spin does not cause the centre-of-mass to rise. Particular attention is focused on a spheroid with displaced centre-of-mass and the tippe-top.

KW - Dynamical systems

KW - Non-isolated fixed points

KW - Rigid-body dynamics

KW - Rising egg phenomenon

KW - Spinning bodies

KW - Stability analysis

UR - http://www.scopus.com/inward/record.url?scp=33750562357&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750562357&partnerID=8YFLogxK

U2 - 10.1098/rspa.2006.1727

DO - 10.1098/rspa.2006.1727

M3 - Article

AN - SCOPUS:33750562357

VL - 462

SP - 3253

EP - 3275

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2075

ER -