The piezoelectrically forced vibrations of AT-cut quartz strip resonators

The one-dimensional vibration equations of crystal strips of AT-cut quartz with narrow width and finite length are given by Lee and Wang (J. Appl. Phys., 75(12):7681, 1994). The mechanical vibrations of such strips have been studied by neglecting the coupling due to piezoelectric effect. In the present study, the one-dimensional equations are further expanded to include the piezoelectric effect sets of coupled equations. By neglecting the components of high-order displacements and electric potentials, particular attention is given to the thickness-shear vibrations of the strips. Analytical solutions of the free vibrations, including components of thickness-shear, flexural, width-shear, width-stretch, width-flexural, and the zeroth- and first-order electric potential, are obtained. The dispersion relations, frequency spectra, and mode shapes of the free vibrations are given as functions of the ratios of length to thickness and width to thickness. The forced vibrations are studied with these solutions, and forced response and capacitance ratios are obtained. Comparisons are made with known experimental results, A quartz strip resonator is studied with this seven modes model. The effect of the length of the electrodes, the width of the crystal, and the forcing frequency on the resonance frequency, vibration mode shapes, and capacitance ratios are obtained.