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, Y.

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

SN - 0950-1207

VL - 462

SP - 3253

EP - 3275

JO - PROC. R. SOC. - A.

JF - PROC. R. SOC. - A.

IS - 2075

ER -