### Abstract

Following parts I and II of this series, the geometry of steady states for a general convex axisymmetric rigid body spinning on a horizontal table is analysed. A general relationship between the pedal curve of the cross-section of the body and the height of its centre-of-mass above the table is obtained which allows for a straightforward determination of static equilibria. It is shown, in particular, that there exist convex axisymmetric bodies having arbitrarily many static equilibria. Four basic categories of non-isolated fixed-point branches (i.e. steady states) are identified in the general case. Depending on the geometry of the spinning body and its dynamical properties (i.e. position of centre-of-mass and inertia tensor), these elementary branches are differently interconnected in the six-dimensional system phase space and form a complex global structure. The geometry of such structures is analysed and topologically distinct classes of configurations are identified. Detailed analysis is presented for a spheroid with displaced centre-of-mass and for the tippe-top. In particular, it is shown that the fixed-point structure of the flip-symmetric spheroid, discussed in part I, represents a degenerate configuration whose degeneracy is destroyed by breaking the symmetry. For the spheroid, there are in general nine distinct classes of fixed-point structures and for the tippe-top there are three such structures. Bifurcations between these classes are identified in the parameter space of the system.

Original language | English |
---|---|

Pages (from-to) | 371-390 |

Number of pages | 20 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 462 |

Issue number | 2066 |

DOIs | |

Publication status | Published - 2006 |

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### Keywords

- Dynamical systems
- Non-isolated fixed-points
- Rigid body dynamics
- Spinning bodies

### ASJC Scopus subject areas

- General

### Cite this

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

*462*(2066), 371-390. https://doi.org/10.1098/rspa.2005.1586

**Dynamics of an axisymmetric body spinning on a horizontal surface. III. Geometry of steady state structures for convex bodies.** / Branicki, M.; Moffatt, H. K.; Shimomura, Yutaka.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 462, no. 2066, pp. 371-390. https://doi.org/10.1098/rspa.2005.1586

}

TY - JOUR

T1 - Dynamics of an axisymmetric body spinning on a horizontal surface. III. Geometry of steady state structures for convex bodies

AU - Branicki, M.

AU - Moffatt, H. K.

AU - Shimomura, Yutaka

PY - 2006

Y1 - 2006

N2 - Following parts I and II of this series, the geometry of steady states for a general convex axisymmetric rigid body spinning on a horizontal table is analysed. A general relationship between the pedal curve of the cross-section of the body and the height of its centre-of-mass above the table is obtained which allows for a straightforward determination of static equilibria. It is shown, in particular, that there exist convex axisymmetric bodies having arbitrarily many static equilibria. Four basic categories of non-isolated fixed-point branches (i.e. steady states) are identified in the general case. Depending on the geometry of the spinning body and its dynamical properties (i.e. position of centre-of-mass and inertia tensor), these elementary branches are differently interconnected in the six-dimensional system phase space and form a complex global structure. The geometry of such structures is analysed and topologically distinct classes of configurations are identified. Detailed analysis is presented for a spheroid with displaced centre-of-mass and for the tippe-top. In particular, it is shown that the fixed-point structure of the flip-symmetric spheroid, discussed in part I, represents a degenerate configuration whose degeneracy is destroyed by breaking the symmetry. For the spheroid, there are in general nine distinct classes of fixed-point structures and for the tippe-top there are three such structures. Bifurcations between these classes are identified in the parameter space of the system.

AB - Following parts I and II of this series, the geometry of steady states for a general convex axisymmetric rigid body spinning on a horizontal table is analysed. A general relationship between the pedal curve of the cross-section of the body and the height of its centre-of-mass above the table is obtained which allows for a straightforward determination of static equilibria. It is shown, in particular, that there exist convex axisymmetric bodies having arbitrarily many static equilibria. Four basic categories of non-isolated fixed-point branches (i.e. steady states) are identified in the general case. Depending on the geometry of the spinning body and its dynamical properties (i.e. position of centre-of-mass and inertia tensor), these elementary branches are differently interconnected in the six-dimensional system phase space and form a complex global structure. The geometry of such structures is analysed and topologically distinct classes of configurations are identified. Detailed analysis is presented for a spheroid with displaced centre-of-mass and for the tippe-top. In particular, it is shown that the fixed-point structure of the flip-symmetric spheroid, discussed in part I, represents a degenerate configuration whose degeneracy is destroyed by breaking the symmetry. For the spheroid, there are in general nine distinct classes of fixed-point structures and for the tippe-top there are three such structures. Bifurcations between these classes are identified in the parameter space of the system.

KW - Dynamical systems

KW - Non-isolated fixed-points

KW - Rigid body dynamics

KW - Spinning bodies

UR - http://www.scopus.com/inward/record.url?scp=33750541164&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750541164&partnerID=8YFLogxK

U2 - 10.1098/rspa.2005.1586

DO - 10.1098/rspa.2005.1586

M3 - Article

AN - SCOPUS:33750541164

VL - 462

SP - 371

EP - 390

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 - 2066

ER -