Parametric instability of a circular ring subjected to moving springs

This work investigates parametric instabilities of in-plane bending vibrations of a thin elastic ring subject to forces from discrete rotating springs of arbitrary number, spacing, and orientation. Several configurations are examined, including systems with symmetric and asymmetric circumferential spring spacing, and systems with and without fixed springs. The method of multiple scales is applied to analytically identify principal and combination instability boundaries as closed-form expressions. Two different numerical approaches are used to verify the analytical results. The effects of different system parameters on the instability boundaries are studied analytically: the bending stiffness of the ring, the number of springs, and their stiffness, location, orientation and rotation speed. For several cases, well-defined properties for the occurrence or suppression of instabilities are obtained as simple relations in the system parameters.