On the stability of magnetically levitated rotating rings

The dynamic stability of rotating elastic circular rings subject to magnetic levitation and radially recentering magnetic forces is studied. A geometrically exact model of elastic rings deforming in space is formulated in the context of the special Cosserat theory of curved rods. A Lagrangian description of the motion is obtained with respect to the rotating frame. The equations of motion are transformed into a set of ODEs according to a Faedo–Galerkin discretization. The effects of the magnetic and gyroscopic forces on the equilibrium states of the ring are investigated together with the loss of stability of the low-frequency modes. Circular elastic rings with open and closed thin-walled cross sections (i.e., L-shaped and boxed cross sections) are considered. The loss of stability occurring at critical angular speeds where the critical modes become unstable is proved to depend on the ring stiffness and cross-sectional symmetry/asymmetry properties.