Parametric instability of spinning elastic rings excited by fluctuating space-fixed stiffnesses

Abstract

This study investigates the vibration of rotating elastic rings that are dynamically excited by an arbitrary number of space-fixed discrete stiffnesses with periodically fluctuating stiffnesses. The rotating, elastic ring is modeled using thin-ring theory with radial and tangential deformations. Primary and combination instability regions are determined in closed-form using the method of multiple scales. The ratio of peak-to-peak fluctuation to average discrete stiffness is used as the perturbation parameter, so the resulting perturbation analysis is not limited to small mean values of discrete stiffnesses. The natural frequencies and vibration modes are determined by discretizing the governing equations using Galerkin's method. Results are demonstrated for compliant gear applications. The perturbation results are validated by direct numerical integration of the equations of motion and Floquet theory. The bandwidths of the instability regions correlate with the fractional strain energy stored in the discrete stiffnesses. For rings with multiple discrete stiffnesses, the phase differences between them can eliminate large amplitude response under certain conditions.