The finite element analysis on frequency shift of a piezoelectric plate operating in the thickness-shear mode under biasing fields

Based on the theory for small fields superposed on finite biasing fields, the finite element model of a high frequency, double-layer plate, thickness-shear vibration pressure sensor under biasing field is set up, and a biasing field finite element method (BFFEM) is proposed. Both ends of the piezoelectric plate, vibrating in thickness-shear mode, are subjected to bending moment and axial force, excited by the ambient pressure, which induce biasing fields in the piezoelectric plate. The nonlinear biasing fields can be solved by dividing the bending moment and the axial force into a series of differential loading steps: 1) to solve the biasing fields for every loading step through the linear piezoelectricity; 2) to modify the elastic, piezoelectric and dielectric coefficients of each element in every loading step by the corresponding differential biasing fields; 3) to use the modified material coefficients as the effective material coefficients of the next loading step; 4) to repeat the above process until the resultant biasing fields are reached. Results of modal analysis and frequency shift can be obtained in the further analysis on incremental fields by BFFEM. The numerical result agrees very well with the one obtained from the linear beam theory when biasing field is small, which verifies the validity of BFFEM. As biasing fields become large, the nonlinear effect gradually generate important role in the plate vibration, and thus, the solution from BFFEM is gradually away from the one obtained through beam theory after the ambient pressure exceeds to a certain value.