| The design and analysis of quartz crystal resonators in the fundamental thickness-shear mode have been extensively studied with many methods including the simple model based on finite plates for the vibration frequency and Mindlin plate theory for couplings of the fundamental thickness-shear and spurious modes. These methods are widely used in the design process for the optimal determination of crystal blanks and electrode configuration. In order to study the overtone vibrations of quartz crystal resonators, Mindlin plate theory is used in the form of the third-order equations with selected modes to obtain the dispersion and frequency spectra in the vicinity of the third-order thickness-shear mode. In our earlier studies, a set of correction factors have been suggested for the Mindlin plate equations to be accurate at the cut-off frequency at the third-order thickness-shear mode. By checking the accuracy at the cut-off frequencies and of the dispersion relations, the third-order plate equations of selected modes are chosen for the calculation. The coupling of modes and effect of electrodes for the third-order overtone vibrations at the thickness-shear mode will be used for the design of quartz crystal resonators of overtone types. |