Three-dimensional analysis of forced vibrations of rectangular at-cut quartz plates

In the present paper, a previous technique using a hybrid method (analytical approach and numerical approach) is extended to deal with forced vibrations of rectangular AT-cut quartz plate resonators with partial electrodes. The mass loading of electrodes is considered with weak piezoelectric coupling. First rectangular acoustic waveguides with and without electrodes coating on the major faces are examined. The vibration modes of each waveguide are determined. The displacement of the resonator is defined with a linear combination of the modes. Second, the trapped-energy resonators containing a single-plated region and a double-unplated region are considered. A set of the amplitudes of the modes is derived from continuity conditions of displacement and stress at the interface between the regions, and stress-free boundary conditions at both ends of the resonator. Next, the input admittance characteristics of the resonators are calculated. Finally, the equivalent inductance of each resonance is determined from the obtained input admittance curve