Multimode thermoelastic dissipation.

In this paper, we investigate thermoelastic dissipation (TED) in systems whose thermal response is characterized by multiple time constants. Zener [Phys. Rev. 52, 230 (1937)] analyzed TED in a cantilever with the assumption that heat transfer is one dimensional. He showed that a single thermal mode was dominant and arrived at a formula for quantifying the quality factor of a resonating cantilever. In this paper, we present a formulation of thermoelastic damping based on entropy generation that accounts for heat transfer in three dimensions and still enables analytical closed form solutions for energy loss estimation in a variety of resonating structures. We apply this solution technique for estimation of quality factor in bulk mode, torsional, and flexural resonators. We show that the thermoelastic damping limited quality factor in bulk mode resonators with resonator frequency much larger than the eigenfrequencies of the dominant thermal modes is inversely proportional to the frequency of the resonator unlike in flexural mode resonators where the quality factor is directly proportional to the resonant frequency. Purely torsional resonators are not limited by TED as the deformation is isochoric. We show that it is possible to express the quality factor obtained by full three-dimensional analyses as a weighted sum of Zener formula based modal quality factors. We analytically estimate the quality factor of a cantilever and a fixed-fixed beam and corroborate it with data to show that the assumption of a single dominant thermal mode, which is valid in one-dimensional analysis, is violated. The analytical formulation described in this paper permits estimation of energy lost due to heat transfer in orthogonal directions. It is found that the entropy generated due to heat transfer along the beam becomes significant in beams with aspect ratio (length/width) below 20. [ABSTRACT FROM AUTHOR]