The fifth-order overtone vibrations of quartz crystal plates with corrected higher-order mindlin plate equations

Higher-order overtone resonators have been widely used in various electronic products for their higher vibration frequencies, which are in the much-needed frequency range beyond the reach of the fundamental mode. However, the existing designs of higher-order overtone resonators and further improvement for meeting more precise requirements are largely based on empirical approaches. As an analytical effort, we have derived the corrected fifth-order Mindlin plate equations with the consideration of electric potential and overtone displacements. The elimination and truncation of the infinite two-dimensional equations has been done to ensure the exact cut-off frequencies of the fundamental, the third-order overtone, and fifth-order overtone thickness-shear modes in comparison with the three-dimensional equations. The frequency spectra are plotted in the vicinity of overtone thickness-shear modes for analysis of couplings and interactions with spurious modes, and the optimal design of quartz crystal blanks for overtone vibrations has been suggested. The equations, solutions, and method will be important in design of the higher-order overtone thickness-shear vibration resonators.