Parametric vibration of an elastic structure with stationary and rotating rings subjected to traveling loads

This work examines the parametric vibration of an elastic dual-ring structure. An analytical model is developed through Hamilton theory, where stationary supports and rotating loads between stationary and rotating rings are included. Analytical results are obtained by using Galerkin′s and multi-scale methods. The results imply that there exist four types of excitations and that vibration can occur in a coupled, uncoupled, principal, or combination manner. The unstable boundaries are derived as closed-form expressions of basic parameters. Influences of the ratio of the wavenumber to the support number, the stiffness ratio of the rotating supports to ring bending, and their speed ratios are also identified. All analytical results are validated by numerical calculations via Floquet theory. This study suggests applications in electric motors to suppress parametric vibration or piezoelectric energy harvesters to arouse such vibration to improve the energy efficiency.