NONLINEAR PARAMETRIC RESONANCE OF SPINNING RINGS

Non-linear vibration of rotating thin circular rings under parametric excitation is analyzed. First, a geometrical discretization is perform ed by applying an energy principle. The resulting dynamical model invo lves 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 determi ned by applying the method of multiple time scales. It is first shown that only combination parametric resonance of the additive type is pos sible for the system examined. For this case, the existence and stabil ity properties of the constant solutions of the averaged equations-cor responding to trivial or quasi-periodic motions of the original system -are investigated. Then, emphasis is placed on understanding the relat ion between the response of the slightly damped and the undamped syste m, as well as the transition from a rotating to a stationary state. Fi nally, a numerical study of the original dynamical system with small d amping is performed, demonstrating the existence and coexistence of a quasi-periodic response with subharmonic, chaotic and unbounded motion s. (C) 1995 Academic Press Limited