Three-dimensional free vibration of multi-layered piezoelectric plates through approximate and exact analyses

Abstract

This article presents two models that have the aim of analysing three-dimensional freely vibrating plates made of an arbitrary combination of structural and/or piezoelectric layers. The first model is derived from a displacement-based variational statement, and it investigates the possibility of approaching exact three-dimensional results at any degree of accuracy. This model has been developed as if the plates were virtually made of a single layer, and it is herein referred to as the approximate analysis model. The second model is based on solving the set of three-dimensional linear equations coupling the relevant mechanical and electric quantities, and it therefore provides exact results. The latter model is derived from the transfer-matrix technique which, having shown numerical instability in the multiphysics problem being dealt with, was then successfully modified to provide exact and reliable results. Excellent agreement has been obtained between the models, and this shows how the exact approach here designed is stably able to overcome ill conditioning problems, while the first model, having been validated by the exact results, could be applied to effectively investigate multiphysics problems for general boundary conditions, and for cross- and/or angle-ply laminates, at any level of required accuracy.

Benchmark solutions for the free vibration of layered piezoelectric plates based on a variational formulation

“…This is important because using only Chebyshev polynomial nodes makes it possible to minimize uniformly the error due to the Lagrange interpolation. A comparison with the 3D analytical solutions (Heyliger and Saravanos, 1995; Messina and Carrera, 2015) is presented. As it turned out, the proposed SaS formulation provides from 12 to 14 right digits for the first six natural frequencies corresponding to half-wave numbers r = s = 1 (see, e.g.…”

“…The most popular state space approach was extensively utilized for solving the dynamic problems of simply supported electroelastic plates by Chen et al (1998), Ding et al (1999), Chen and Ding (2002), Deü and Benjeddou (2005), and Zhong and Yu (2006) and magneto-electro-elastic plates by Pan and Heyliger (2002), Chen et al (2005), and Chen et al (2007a, 2007b). Messina and Carrera (2015) proposed to utilize the transfer matrix method to solve the ordinary differential equations in terms of the displacements and electric potential derived from the system of partial differential equations through the separating variable procedure. The dynamic response of laminated piezoelectric plates through Taylor series expansions in the thickness direction was studied by Gao et al (1998), Vel et al (2004), and Baillargeon and Vel (2005).…”

A new semi-analytical solution method for free vibration analysis of composite rectangular plates with general edge constraints coupled with single piezoelectric layer

“…Using Fourier series expansion, they transferred the 2D electromechanical equations of motion to a generalized eigenvalue problem and obtained the natural frequencies as well as mode shapes. Messina (2002), Messina and Carrera (2015), and Messina and Carrera (2016) analyzed 3D free vibration of multilayered piezo-composite plates using an expansion of displacement-based variational statement with any degree of precision in a unique formulation. Kulikov and Plotnikova (2017) implemented the sampling surfaces method for analyzing the 3D vibration of multilayered simply supported piezoelectric plate.…”

Three-dimensional analysis of freely vibrating multilayered piezoelectric plates through adaptive global piecewise-smooth functions

“…Based on the above-mentioned references, Messina and Carrera (2014) investigated an approximate methodology aimed at approaching the three-dimensional dynamics of freely vibrating piezoelectric plates using an expansion level allowing any degree of accuracy in a unique formulation for single- or multilayer piezoelectric plates. Such an objective was pursued by expanding the unknown essential functions (displacement and electric potential) on both classical smooth bases (on the middle plane) and global piecewise-smooth function (GPSF) series (Messina, 2002) (through the thickness of the plate).…”

“…The approximate model introduced by Messina and Carrera (2014) was also successfully compared with an exact model. The exact model was developed in the same article where numerical instabilities were challenged and solved by appropriately scaling the electric potential.…”