Nonlinear Vibrations of Magnetically Levitated Rotating Rings

A geometrically exact model of a magnetically levitated ro- tating ring subject to radial electromagnetic (EM) forces and un- dergoing three-dimensional (3D) motions is proposed. The equa- tions of motion are derived in the context of a total Lagrangian formulation assuming 3D finite kinematics. The specialization to extensional-flexural motions of the special Cosserat theory of curved rods is illustrated. The nonlinear partial differen- tial equations governing the motion of the ring are numerically solved via a finite element approach and the effects of the angu- lar speed and the EM interaction forces on the nonlinear modal properties of the ring are investigated. The nonlinear equilib- rium of the ring is first evaluated at different angular speeds and the associated eigenproblem is then solved by linearizing the equations about each prestressed equilibrium state. Free nonlin- ear motions involving the mth mode together with its companion mode and other interacting modes are investigated to study the effects of geometric and constitutive nonlinearities on stability.