Modeling perturbations induced in plate resonator characteristics due to flexural bending

In this work, the changes in frequency occurring in thickness-shear mode quartz resonators due to bending are discussed and modeled. First order strain gradients and second and third order elastic coefficients for micromachined thin plate quartz resonators are important in the calculation of the frequency shifts. These gradients and coefficients are summarized and derived using Lee´s theory. This theory is later combined with quartz/magnetostrictive layer unimorphs, where the tip deflection formulations are given. As a case study, the theoretical frequency shifts in a micromachined quartz resonator laminated with magnetostrictive thin film is calculated and later compared with experimental data. The combined model fits well with the experiments.