Thermoelastic damping in high frequency resonators using higher-order shear deformation theories

Abstract

Thermoelastic damping (TED) in high frequency oscillators shifts their natural frequencies and attenuates the vibration amplitude. The amount of damping depends upon the temperature of the environment in which the device is operating. Here, we model the oscillator as a simply-supported beam of rectangular cross-section undergoing infinitesimal deformations. We consider three (the Levinson, the Timoshenko and a sinusoidal) through-the-thickness shear strain distributions, both translational and rotational inertia, and heat conduction to analytically delineate their effects on the shift in frequencies and the attenuation of their amplitudes for the lowest three vibration modes. The shear deformation distribution function multiplied by the Poisson ratio appears in the volumetric strain. The TED in the Timoshenko, the Levinson and a beam with sinusoidal variation of shear stresses is related to that in the Euler beam to recover the famous Lifshitz–Roukes formula. Various results are presented to qualitatively and quantitatively illustrate the dependence of the TED upon the lowest three frequencies and the resonator thickness. The analytical solutions presented here can be used as benchmark solutions for checking the numerical solutions.