Non-linear parametric resonance of spinning rings

Abstract

Abstract Non-linear vibration of rotating thin circular rings under parametric excitation is analyzed. First, a geometrical discretization is performed by applying an energy principle. The resulting dynamical model involves two degrees of freedom, representing the vibration amplitudes of two in-plane flexural modes with the same circumferential wavenumber. These modes are coupled through gyroscopic and non-linear terms, while the parametric excitation originates by small periodic perturbations of the spin speed of the ring. Then, approximate solutions are determined by applying the method of multiple time scales. It is first shown that only combination parametric resonance of the additive type is possible for the system examined. For this case, the existence and stability properties of the constant solutions of the averaged equations—corresponding to trivial or quasi-periodic motions of the original system—are investigated. Then, emphasis is placed on understanding the relation between the response of the slightly damped and the undamped system, as well as the transition from a rotating to a stationary state. Finally, a numerical study of the original dynamical system with small damping is performed, demonstrating the existence and coexistence of a quasi-periodic response with subharmonic, chaotic and unbounded motions.