Analytical solutions for thermo-elastic damping of rotational ring resonators incorporating thermal relaxations and elastic small scales

Abstract

At the micro and nanoscales, thermo-elastic damping is regarded as one of the most crucial factors for promoting energy dissipation in vibrating structures. The thermo-elastic damping of a thin rotational ring resonator is explored in this paper based on three phase lag heat conduction model for thermal relaxations and nonclassical elasticity for small scale effects. Solving the heat conduction equation for radial direction heat flow and zero heat flux boundary conditions yields the temperature spread across ring cross-section. The equation takes into account only the rotating narrow ring’s in-plane vibrations. For the frequency dependent thermo-elastic damping, the combined nonlocal thermo-elasto-dynamic equations are solved. The present model is validated with the experimental and finite element data. The thermo-elastic damping of rotating rings has been found to be over predicted by classical elasticity. For a particular small scale factor, as rotational speed rises by 80%, the equivalent thermo-elastic damping of the ring resonator reduces by 43%. In addition, the effects of cross-sectional shape, nonlocal scale factor, rotating speed, modenumber, ambient temperature, and substance on thermo-elastic damping are thoroughly explored in this work. The results presented here are useful to build and construct future nanoscale resonant sensors and radio-frequency devices.