Comparative analysis of the electroelastic thickness vibrations of layers with curved boundaries

Abstract

The three-dimensional equations of electroelasticity in Cartesian, cylindrical, and spherical coordinates are represented in Hamiltonian form with respect to the thickness coordinate. The boundary-value problem with a harmonic potential difference and zero mechanical load given on the boundaries is solved numerically. The amplitude-frequency characteristics and natural frequencies are compared. The resonant and antiresonant frequencies of the current and the dynamic electromechanical coupling coefficient are determined Keywords: piezoelectric layer, sphere, plate with transverse polarization, electroelastic thickness vibrations, amplitude-frequency characteristics, resonant and antiresonant frequencies Introduction. Piezoelectric elements (plates, cylinders, spheres, etc.) are widely used in various devices [1,3,9]. The study of their dynamic characteristics is of scientific and applied importance. The exact analytic solution describing the thickness vibrations of a plate-like vibrator is well known [1, 6, etc.], whereas the analytic solutions for cylindrical and spherical vibrators with arbitrary boundary conditions are formal because they require evaluating integrals of special functions. When curvature is small, it is expedient to seek solutions in the form of ordinary power series of linear functions of the radial coordinate. This approach was followed to solve many problems of wave propagation and vibration in elastic and piezoelectric cylindrical and spherical bodies [6, 10 11]. The electroelastic vibrations of cylinders were studied using the variational difference and finite-element methods [6, etc.]. The method of lines and spline functions (for approximation in one coordinate) were used in [7,8] to study the vibrations of electroelastic cylinders. The problems addressed in these papers were partially solved earlier, but no comparison with their solutions was made. As for the accuracy of the solutions, no particulars were given in [7,8], just generalities.The present paper outlines a unified approach to the solution of the problem for a plate, a cylinder, and a sphere. We will reduce the three-dimensional equations of electroelasticity to an operator Hamiltonian system with respect to the thickness coordinate and analyze the properties of this system. In the case of a harmonic electric potential, we will numerically determine the amplitude-frequency characteristics, resonant and antiresonant frequencies, and dynamic electromechanical-coupling coefficient. The results will be subject to comparative analysis.1. Problem Formulation. To study the harmonic electroelastic vibrations of a piezoelectric plate, cylinder, and sphere, we will start with the equations of mechanical vibrations [6]

The approach to the calculation of harmonic oscillations of electroelastic cylinders

“…Copyright (c) 2016 Безверхий О. І., Григор'єва Л. О. В роботі [10] просторову систему рівнянь електропружності для радіально поляризованого циліндра шляхом розкладу розв'язку в ряди по поздовжній та окружній координаті зведено до набору систем рівнянь типу Гамільтона. В статті [11] використовуються рівняння гамільтонового типу по товщинній координаті для аналізу одномірних гармонічних коливань кулі, циліндру і плоского шару. Закладена в цій статті ідея узагальнюється у цій роботі для осесиметричної задачі електропружності для циліндру.…”