Three-dimensional natural vibration analysis and energy considerations for a piezoelectric rectangular plate

The paper presents results of the exact three-dimensional analysis of the natural frequencies and mode shapes of a rectangular piezoelectric plate poled along the direction perpendicular to the plate middle plane. The solutions are obtained in non-dimensional form for flexural modes of vibration of a plate simply supported along the plate edges and two types of electrical boundary conditions on the plate faces, namely for the case of short and open circuit. A numerically stable algorithm is described, which allows the calculation of the natural frequencies and mode shapes of the piezoelectric plate. The details of the algorithm have been found to be especially important in the case of equal wavelengths along the two plate edges (e.g. for the lowest flexural mode of a square plate). The natural frequencies of a rectangular plate are given for a rectangular and square plate for the case of short-circuited and open-circuit plate faces, together with the through-thickness distributions of the mechanical and electric fields. Energy balance for piezoelectric continua is used to derive two equivalent formulae which extend the Rayleigh quotient used in free vibration analysis of elastic continua. These expressions are used to verify the consistency of the results and to obtain additional insight into the phenomenon of electromechanical coupling in a vibrating piezoelectric plate.