Rayleigh’s Quotient of Linear Piezoelectricity and Its Use in the Approximate Calculation of the Natural Frequencies of Piezoelectric Continua

The paper discusses an extension to the linear theory of piezoelectricity of the Rayleigh quotient used in the analysis of the properties and the approximate calculation of the natural frequencies of elastic continua. It is shown that for a piezoelectric continuum an infinite number of equivalent expressions can be obtained which generalize the classical Rayleigh quotient. The stationarity conditions of any of these quotients under additional constraints imposed by the Gauss equation of electrostatics and the prescribed natural electrical boundary condition are shown to result in the complete set of the governing equations of the free vibration problem of a piezoelectric continuum. The general results discussed in the paper are illustrated by the approximate calculation of the natural frequencies of a piezoelectric rod by the Rayleigh–Ritz method. Unlike in the case of elastic structures, no monotonic convergence of the approximate frequencies is guaranteed for a piezoelectric continuum, the property which can be explained using the introduced Rayleigh quotients.