Dynamic modelling and quality factor evaluation of hemispherical shell resonators

A novel strategy is proposed for the dynamic modelling and quality factor evaluation of the hemispherical shell resonators (HSR) considering the coupling of hemispherical shell and support rod. The artificial spring technique is introduced to realize the coupling connection of substructures and simulate the arbitrary boundary conditions. Hamilton's principle with the Rayleigh-Ritz method is employed to establish the equation of motion of the HSR, and the natural frequencies of the HSR are obtained by solving the eigenvalue problem of the equation of motion. The thermal energy method is applied to evaluate the thermoelastic damping (TED) of the HSR. The model of anchor loss is based on a separation and transfer method, in which the HSR and its substrate are first separated for analysis and then the stress from the clamped region of the support rod is transferred to the substrate for determining the vibration displacement across the clamped region. The energy losses to the substrate are formulated as the integral of the product of the stress and the vibration displacement across the clamped region per cycle of vibration. The results calculated from the present theoretical method are compared with those from the published literature and the finite element method (FEM), which validates the correctness and feasibility of the present theoretical model. The effects of the boundary conditions and geometrical parameters on the vibration behaviors of the HSR are analyzed, and the influences of the boundary conditions, geometrical parameters and material properties of the HSR on the quality factors related to the TED and anchor loss are investigated. The present model can be used to optimize the design of the HSR with a high quality factor.