Single-phase-lag thermoelastic damping in out-of-plane vibrating micro-ring resonators

Abstract

Thermoelastic damping (TED) has been identified as a primary energy dissipation mechanism in micro-resonators operating under vacuum conditions. The precise prediction based on the explicit TED model is crucial for optimizing micro-resonators with high quality factor. In this work, a novel two-dimensional single-phase-lag (2D-SPL) TED model is developed for the micro-ring resonator exhibiting out-of-plane vibration. The proposed model addresses the limitations of classical Fourier-based formulations, enabling more reliable TED predictions under extreme conditions. Initially, in the context of the SPL non-Fourier model, the governing equation of thermoelasticity is derived incorporating heat conduction in both transverse and circumferential directions. Subsequently, the function of fluctuation temperature is resolved using the Galerkin approach. Finally, an analytical 2D-SPL TED model is achieved through the energy-definition method. As emphases, the impacts of the equilibrium temperature, geometric dimensions, vibration frequencies, and mode orders on TED are investigated systematically. The results demonstrate that the SPL non-Fourier effect significantly affects TED at low equilibrium temperatures, high frequencies, and high-order modes. Additionally, the discrepancies among the one- and two-dimensional TED spectra become increasingly pronounced for low ratios of radius to thickness, higher frequencies, and larger mode orders.