Free and parametric vibrations of an elastic ring structure induced by rotating internal and external time-varying excitations

Ring structures can generate significant vibration due to the internal and external excitations. A mathematical model of a thin ring subjected to time-varying springs and forces is established by using Hamilton's principle, where the stiffnesses and geometric nonlinearity caused by external excitations are introduced. The external excitations, which usually act as a forcing term for forced vibrations, can be found in the stiffness coefficients. Thus, the external excitation is confirmed to be an primary factor in changing vibration characteristics. The natural frequencies of the ring subjected to identical equally-spaced external excitations split when the external excitations count N and wavenumber n satisfy 2n/N = integer. The parametric vibrations induced by time-varying internal and external excitations are analyzed and classified into seven cases in terms of the parameter combinations. The instability areas of typical cases with the multi-frequency excitations are obtained based on Floquet theory. The influences of primary parameters such as the time-varying stiffness and force, orientation angle, and damping on the instabilities are studied. The relationships between natural frequency splitting and parametric instability are revealed and compared with the existing results in the open literature. The results in this study lay a foundation for predicting the parametric instabilities of similar structures.

Automated summary

This study examines the vibration characteristics of a thin ring subjected to internal and external excitations, and reveals that external excitations are the primary factor in changing vibration characteristics. By analyzing and classifying parameter combinations, the study identifies seven cases of parametric vibrations. Based on Floquet theory, the instability areas of multi-frequency excitations are obtained, and the impact of primary parameters on instabilities is studied. The study also uncovers the relationship between natural frequency splitting and parametric instability, and compares the findings with existing results in the literature.