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Free vibration of rings is presented via wave approach theoretically. Firstly, based on the solutions of out-of-plane vibration, propagation, reflection, and coordination matrices are derived for the case of a fixed boundary at inner surface and a free boundary at outer surface. Then, assembling these matrices, characteristic equation of natural frequency is obtained. Wave approach is employed to study the free vibration of these ring structures. Natural frequencies calculated by wave approach are compared with those obtained by classical method and Finite Element Method (FEM). Afterwards natural frequencies of four type boundaries are calculated. Transverse vibration transmissibility of rings propagating from outer to inner and from inner to outer is investigated. Finally, the effects of structural and material parameters on free vibration are discussed in detail.

Planetary gears are one part of the whole transmission chain, and the dynamics and vibration characteristics of them are strongly coupled with external rotors. In this paper, to demonstrate the interaction between multistage planetary gears and external rotors as well as investigate the lateral-torsional coupling characteristics of them, a coupling model of a two-stage planetary gear rotor system is proposed. In such a model, the two-stage planetary gear subsystem is established as a lumped-parameter model and the external rotor subsystem is established as a finite element model. The vibration mode distribution properties and lateral-torsional coupling characteristics are both analyzed by modal strain energy. Three different conditions are considered: uncoupled, partially coupled, and fully coupled. The results indicate that the coupling among multiple subsystems and the lateral-torsional coupling mainly exist in the low-mode region. Natural frequencies dominated by the two-stage planetary gear subsystem are sensitive to coupled conditions, whereas natural frequencies dominated by the input rotor subsystem are remarkably insensitive to coupled conditions. Furthermore, the natural frequency of the first torsional mode can be obtained only in the fully coupled condition. Experiments are implemented to obtain natural frequencies, and the experiment results validate the numerical results. [ABSTRACT FROM AUTHOR]

Radial vibration of the circular plate is presented using wave propagation approach and classical method containing Bessel solution and Hankel solution for calculating the natural frequency theoretically. In cylindrical coordinate system, in order to obtain natural frequency, propagation and reflection matrices are deduced at the boundaries of free-free, fixed-fixed, and fixed-free using wave propagation approach. Furthermore, radial phononic crystal is constructed by connecting two materials periodically for the analysis of band phenomenon. Also, Finite Element Simulation (FEM) is adopted to verify the theoretical results. Finally, the radial and piezoelectric effects on the band are also discussed. [ABSTRACT FROM AUTHOR]