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, Y.
PY - 2006/1/1
Y1 - 2006/1/1
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.
UR - http://www.scopus.com/inward/record.url?partnerID=yv4JPVwI&eid=2-s2.0-33750541164&md5=cf3ab4b3313d34b328a956590a8d25d2
U2 - 10.1098/rspa.2005.1586
DO - 10.1098/rspa.2005.1586
M3 - Article
AN - SCOPUS:33750541164
SN - 1364-5021
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
IS - 2066
ER -