Thermoelastic damping in micro-scale T-, U- and Z-shaped frame structures using wave propagation approach

Studying the thermo-mechanical coupling behavior and establishing the thermoelastic damping (TED) model are essential for improving the quality factor of the microelectromechanical resonator. In this work, an analytical TED model for planar frame resonators is first developed using the wave propagation approach. The method is illustrated for typical T-, U- and Z-shaped frame structures and can be extended to more complex frame structures. The frame is viewed as a set of beam segments interconnected by rigid joints, each following the Euler-Bernoulli beam theory. Based on the wave propagation approach, the wave transmission and reflection properties at discontinuities are determined taking into account the flexural and longitudinal motions. The displacement fields of the frame structure are established for both free and forced vibrations. The coupled temperature field of the frame is obtained by solving for the temperature variation arising from the flexural and longitudinal vibrations separately. Finally, the expression of TED is achieved by its energy definition. The validity of the present TED model is confirmed by comparison with finite element (FE) simulations. Numerical results indicate that the TED behavior of frame structures is too complicated to be predicted by previous TED models of simple structures. The often-overlooked longitudinal motion will cause a dramatic TED reduction around specific frequencies. The overall TED is determined by all segments together, and the influence of a segment is reflected by its energy share. Moreover, the effects of different boundary conditions and geometric parameters on TED are evaluated. Practical conclusions to guide the design of high Q-factor resonators are drawn based on the TED behavior of frame structures as summarized from the numerical study.