Study on the equivalent circuit and coupled vibration for the longitudinally polarized piezoelectric ceramic hollow cylinders

Longitudinally polarized piezoelectric ceramic disks with central holes have been the most commonly used elements in underwater acoustics and ultrasonics. For very thin piezoelectric ceramic disks, its vibration can be regarded as one-dimensional thickness extensional vibration, or plane radial vibration, and the coupling between these two different vibrational modes is neglected. However, for practical piezoelectric ceramic disks with central holes, the geometrical dimensions do not satisfy the requirements for the thin ring, and the coupling must be considered. In this paper, the coupled vibration of the longitudinally polarized piezoelectric ceramic disks with central holes is studied using an approximate analytical method. When the mechanical coupling coefficient is introduced and the shearing strain is ignored, the coupled vibration of the piezoelectric ceramic hollow cylinder is divided into two equivalent vibrations. One is the equivalent longitudinal extensional vibration, and the other is the equivalent radial vibration. These two equivalent vibrations are not independent; they are correlated together by the mechanical coupling coefficient. The equivalent circuit for the piezoelectric ceramic hollow cylinders in coupled vibration is derived, and the resonance frequency equations are obtained. Based on the coupled frequency equations, the longitudinal and radial resonance frequencies can be obtained when the dimensions and the material parameters are given. Compared with one-dimensional theory, the computed resonance frequencies in this paper are in good agreement with the measured results. Compared with the numerical methods, the analytic method presented in this paper is simple in computing the longitudinal and radial resonance frequencies and in analyzing the coupled vibrational modes of the piezoelectric ceramic disks with central holes.