Dynamic instability and parametric response of an eccentric rotating ring-shaped periodic structure

Abstract

The eccentrically rotating ring-shaped periodic structures (RPSs) are rarely spotlighted despite their pervasiveness in engineering practice, let alone consideration of time-varying eccentricity. This work investigates the dynamic instability and parametric responses of an eccentrically rotating RPS attached by additional masses. The mathematical model is derived using Hamilton's principle by incorporating the kinetic and potential energies arising from eccentric rotation and additional masses. Dynamic instabilities of free and parametric vibrations, which are caused by the time-invariant and time-varying eccentricity, respectively, are predicted by classical vibration and Floquét theories, respectively. Furthermore, the modulation feedback method is used to analyze the frequency components of the parametric responses, which are verified by the Runge-Kutta method. The results show that the instability regions involving time-varying excitations exhibit nonlinear combination characteristics. Even if the time-varying excitation is not generated by periodic units, the changes in its topological structure can still affect the distribution of parametric instability. The frequency components of the parametric responses are composed of a linear combination of the principal vibration and excitation frequencies.