Two-dimensional analysis of thickness-shear and flexural vibrations in rectangular AT-cut quartz plates using a one-dimensional finite element method

The two-dimensional coupled vibrations of thickness-shear and flexure in miniature rectangular AT-cut quartz resonators are theoretically studied. The analysis is based on Mindlin´s plate equations. To calculate the mode chart, a one-dimensional finite element technique is used. It is shown that the theoretical curves match the experimental data well. It is found that most spurious resonances in strip type miniature resonators belong to the two-dimensional flexural family. The numerical method presented is well suited to determine the points at which these flexural modes will coincide with the fundamental main thickness-shear mode in the miniature resonators elongated along the X axis