Free flexural vibrations of a piezoelectric bimorph plate with periodic edge conditions

This work analyzes the vibrations of a fully-electroded annular piezoelectric bimorph plate with a free inner edge and an outer edge that is built-in with a periodicity. To this end, a variational formulation with the extensive use of Lagrange multipliers for a bimorph plate with polar orthorhombic symmetry is performed first. The mechanical displacement and the electric potential that must satisfy constraint conditions at the electrodes are expanded as the sums of powers in the thickness coordinate. The resulting piezoelectric bimorph plate equations are used along with the introduction of appropriate Lagrange multipliers to analyze the polar orthorhombic annular sectorial plates with free radial and inner circumferential edges, and an entirely built-in or free outer edge. The results are then combined to obtain the solutions for the mixed boundary value problem. The extended Hamilton's principle with the method of Lagrange multipliers is employed, followed by a Frobenius-type series expansion for solution functions. The eigensolutions are calculated from the resulting transcendental equation and compared with those obtained from an FEA to ensure the validity of the procedure.