Natural frequency splitting and principal instability of rotating cyclic ring structures

Abstract

This work develops an analytical method to estimate the natural frequency splitting and principal instability of the rotating cyclic ring structures. An elastic model is built up under the ring-fixed frame by using the energy method. The modeling leads to a partial differential equation with time-variant coefficient. The eigenvalue is formulated to estimate the natural frequency splitting, principal instability and their relationships. The dependence of the basic parameters on the natural frequency splitting and principal instability is demonstrated. The principal instability can occur at the splitting natural frequencies but cannot at the repeating ones. A classical problem regarding the parametric instability of the rotating ring with stationary supports and the inverse problem are examined. The results verify that the natural frequency splitting does not mean unstable for the former problem, but for the latter the splitting implies unstable. Besides, the model is transformed into the support-fixed frame and thus an equivalent time-invariant model is obtained, which is solved by using the general vibration theory. The analytical method is validated through the comparisons with the results in the open literature and especially the comparisons between the results from the two types of frames.