A semi-analytical method for calculating resonator energy loss into plate substrates

This paper develops a new semi-analytical technique for estimating the energy loss in a resonator mounted on a plate substrate. Resonator energy loss due to interaction with its substrate (so called “anchor loss”) has been widely studied, but most treatments have assumed a semi-infinite medium due to the small dimensions of the resonator compared to the substrate thickness. Recent high quality factor resonators, though, typically have resonant frequencies from a few kilohertz up to several hundred kilohertz and in these cases, due to the wavelengths involved, the substrate is more accurately treated as a plate. A procedure is developed whereby modal analysis is performed on a finite element model that is coupled to analytical expressions for the plate Lamb waves. This approach is distinguished from prior semi-analytical approaches by the absence of an applied load. Thus, a more fundamental treatment of dynamics of the coupled system is possible.