The Singular Nature of Rotating Circular Rings With Symmetry

Abstract

We study the S𝒪(2) symmetric deformation of circular rings. The equations of motion are based on general partial differential equations governing the elastodynamics of geometrically exact rings, which have been formulated by Dempsey [2], based on the work of Libai and Simmonds [6]. Thus, the formulation is valid for arbitrary pressure forcing, large deformations, and finite strains, although the results are tempered by a linear constitutive relation and an assumption of plane strain. With the assumption that the deformation retains S𝒪(2) symmetry, the partial differential equations are reduced to a set of coupled ordinary differential equations. Within this restricted space of solutions, we study the existence and stability of relative equilibria and discuss the effects of constant hydrostatic pressure on the dynamical response. Specifically, we find that interaction between inertial effects arising from rotational motion and the combined elastic and external pressure forces can produce unexpected behavior, including the existence of a “buckled” state which retains the symmetry, yet physically implies that material planes do not lie in the radial direction. Such a state is shown to effect the large-amplitude response of the system in a singular limit of the governing equations.