Free vibrations of a piezoelectric layer of hexagonal (6mm) class

An asymptotic method due to "Achenbach" is used to analyze the free vibrations of a piezoelectric layer of hexagonal (6 mm) class. In this method, the displacement components, the electric potential and the frequency are expressed as power series of the dimensionless wavenumber ε{lunate} = 27π × Layer Thickness/Wavelength. Substituting the expansions of field variables and the frequency in the field equations of piezoelectricity and in the boundary conditions, a system of coupled, second order, inhomogeneous, ordinary differential equations with thickness variable as the independent variable is obtained by collecting the terms of same order ε{lunate}n. Integration of such systems of differential equations yields the various terms in the series expansions for the field variables and the frequency, for all modes and in the whole range of frequencies, in a range of the dimensionless wavenumber 0 < ε{lunate} < ε{lunate}* < 1 where ε{lunate}* increases as more terms are retained in the expansions. The frequency coefficients reduce to the corresponding ones in the elastic case as a limit. The exact frequency equation in the case of plane strain is obtained and analyzed numerically. The results thus obtained are compared with those obtained in the asymptotic method. The results fairly agree upto three decimal places. © 1983.