Free vibrations of axially polarized piezoceramic hollow cylinders of finite length

Abstract

The problem of the free axisymmetric vibrations of longitudinally polarized piezoceramic hollow cylinders is solved by a numerical analytic method. The spline-collocation method with respect to the longitudinal coordinate is used to reduce the original problem of electroelasticity to an eigenvalue boundary-value problem for ordinary differential equations with respect to the radial coordinate. This problem is solved by the stable discrete-orthogonalization and incremental search methods. Numerical results are presented and the natural frequencies of the cylinders are analyzed for a wide range of their geometric characteristics Keywords: free vibrations, hollow piezoceramic cylinder, spline-approximation, boundary conditions Introduction. Piezoceramic materials have significant advantages over natural piezoelectrics (quartz, tourmaline, Rochelle salt, etc.) such as good moldability, low cost, high sensitivity, and high thermal stability. Modern piezoceramic elements and devices are solid-state and can be made in three-dimensional, two-dimensional, and integrated forms. They show high noise immunity, low intrinsic noise, and high radiation resistance. This is why piezoceramic materials are widely used in various fields of science and engineering, subject to intensive development, and their properties are of huge and ever-increasing interest.The widespread use of piezoceramic elements and devices is due to the tendency to better describe real processes in piezoceramic structural materials and to reveal and study three-dimensional effects occurring in thick-walled elements. Despite the great number of relevant publications, there are only few studies on the vibrations of piezoceramic cylinders of finite length based on the three-dimensional theory of elasticity [1-3, 9, 10].Recent trends in computational mathematics, mathematical physics, and mechanics are toward the use of spline-functions. This is due to the following advantages of spline-approximations: (i) stability of splines against local perturbations (the local behavior of a spline at a point does not affect its overall behavior, unlike, for example, polynomial approximation); (ii) good convergence of spline-interpolation (unlike polynomial interpolation); and (iii) simplicity and convenience of numerical implementation of spline algorithms. When used in various variational, projective, and other discrete-continuous methods, spline-functions yield much better results than classical polynomials do, simplify the numerical implementation of these methods, and improve the accuracy of the solution. Noteworthy are the publications [5][6][7][8], which use spline-approximation to study the mechanical behavior of various plates and shells.In the present paper, we study the free vibrations of a longitudinally polarized hollow piezoceramic cylinder. The lateral surfaces of the cylinder are free from external loads. The cylinders are considered either hinged or clamped.1. Governing Equations. The vibrations of piezoelectric bodies are described by the equations of elasti...

The approach to the calculation of harmonic oscillations of electroelastic cylinders

“…Раніше для дослідження усталених коливань п'єзокерамічних циліндрів застосовувався розклад розв'язку в ряди [1] або варіаційно-різницевий метод (ВРМ) [2], [3]. В роботі [4] для визначення власних частот поздовжньо поляризованих циліндрів за допомогою сплайн-апроксимацій по поздовжній координаті отримано серію крайових задач на власні значення по радіальній координаті. В [5] наводяться чисельні та експериментальні результати щодо вимушених коливань п'єзокерамічного циліндра з закріпленим торцем.…”

“…Коефіціенти системи (7) співпадають з коефіціентами рівнянь (4). Очевидним плюсом системи (7) є присутність в розв'язуючих функціях напружень та електричної індукції, що значно спрощує постановку граничних умов, в тому числі імпедансних.…”

Double Legendre polynomial quadrature-free method for axisymmetric vibration of functionally graded piezoelectric circular plates

“…Homogeneous and inhomogeneous materials with piezoelectric effect are of particular engineering interest, and their free vibration characteristics were studied by the extensive worldwide scholars. The free vibration problem of piezoelectric structure was often solved by analytical method (Ding et al, 2021; Daemi and Eipakchi, 2019; Li et al, 2012; Li et al (2011); Wang and Lin, 2021; Razavi et al (2016)), semi-analytical method (Nie and Zhong, 2007; Santos et al, 2006), Rayleigh–Ritz method (Lee and Kwak, 2015), differential quadrature method (Alibeigloo and Simintan, 2011; Bisheh et al, 2019; Sharma and Parashar, 2016), the Fourier series method (Su et al, 2016; Zhang et al, 2018), finite element method (Carrera and Boscolo, 2007; Lezgy-Nazargah et al, 2013), numerical analysis method (Grigorenko et al, 2010), meshless method (Jan et al, 2013; Kwak et al, 2022), electromechanical equivalent circuit method (Wang et al (2022)), and so on. Li et al (2011) developed 3-D analytical solution for the axisymmetric problem of FGPM circular plates under tension and bending.…”