A frequency shift based on thermoelastic dissipation of solid torus micro- or nano-ring model

Abstract

The analysis of thermoelastic damping (TED) is an important issue in designing precise MEMS vibrating resonators. Especially, the frequency shift (FS) in the approximated dissipation model needs to be considered in terms of accuracy and sensitivity. In this paper, a formula is developed to investigate the FS due to a 3-dimensional (3D) TED in a circular cross-sectional ring. The temperature profile of the existing TED model for a torus ring is summarized in the form of a thermal moment. And then, the moment is then integrated across the cross-sectional area to form an equation of motion and expressed as an eigenfrequency with thermal effect according to Young’s modulus. The difference with the existing isothermal frequency is then summarized in the form of a ratio to develop an expression of explicit function for FS. The convergence of the model is examined by numerically analyzing the results at various temperatures for Si, which is known as a common semiconductor material. Additionally, a practical analysis is performed with vibration mode number and aspect ratio as independent variables. The trend of results suggests the expected conditions for FS to be considered, especially for high temperatures, high vibration mode numbers, and high aspect ratios.