Accurate coupled vibration analysis of a piezoelectric array element by the superposition method

Abstract

In this paper, the superposition method is extended to obtain analytical solutions for the coupled vibration of a piezoelectric slender bar in a configuration corresponding to a typical ultrasonic linear array transducer element. The problem can be described mathematically by three partial differential equations with electrical and mechanical boundary conditions. To solve this, the vibrations in lateral and thickness directions are referred to as two building blocks. In each building block, the expressions of displacements and electric potential are assumed first based on their symmetry properties and then the induced dynamic responses, such as in-plane stress and electric displacements, are calculated. Finally, the vibration responses of the two building blocks are superimposed to satisfy the boundary conditions using Fourier series expansions. Electrical impedance and mode shapes, represented by the spatial distribution of displacements and electric potential, are calculated analytically and compared with the results of the finite element method. An excellent agreement is observed. The method can be applied to design and optimize piezoelectric array transducers for various applications.