Dynamic analysis of functionally graded plate integrated with two piezoelectric layers, based on a three-dimensional elasticity solution

2009 ◽  
Vol 223 (6) ◽  
pp. 1297-1309 ◽  
Author(s):  
M Shakeri ◽  
S N Sadeghi ◽  
M Javanbakht ◽  
H Hatamikian
Keyword(s):  
Differential Equations ◽  
Dynamic Analysis ◽  
Series Expansion ◽  
Three Dimensional ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Simply Supported ◽  
Fg Plates ◽  
Piezoelectric Layers ◽  
Three Dimensional Elasticity

In this article, dynamic analysis of functionally graded (FG) plates integrated with two piezoelectric layers has been carried out. The rectangular plate is simply supported at four edges and exposed to dynamic excitation. Three-dimensional elasticity equations have been considered. Using a series expansion of mechanical and electrical displacements, the partial differential equations have been reduced to ordinary differential equations (ODEs) with variable coefficients. The solution of the resulting system of ODEs has been carried out using the Galerkin method. The final result is obtained by taking just one term in the series expansion. The Newmark method has been used to move forward in the time domain for a dynamic solution. Finally, numerical results have been presented for a simply supported rectangular FG plate integrated with two piezoelectric layers. In some cases, results have been compared to previously published works.

Three-dimensional elasticity solution for laminated cross-ply panel under localized moment

2007 ◽  
Vol 221 (8) ◽  
pp. 859-866
Author(s):  
A Alibeigloo ◽  
M Shakeri
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Three Dimensional ◽  
Axial Direction ◽  
Variable Coefficients ◽  
Cylindrical Panel ◽  
Fourier Series Expansion ◽  
Simply Supported ◽  
Elasticity Solutions ◽  
Three Dimensional Elasticity

Three-dimensional elasticity solutions have been presented for thick laminated crossply circular cylindrical panel. The panel is under localized patch moment in axial direction and is simply supported at all edges with finite length. Ordinary differential equations with variable coefficients are obtained by means of Fourier series expansion for displacement field and loading in the circumferential and axial directions. Resulting ordinary differential equations are solved using Taylor series. Numerical results are presented for (0/90°) and (0/90/0°) lay-up, and compared with the results for simple form of loading published in literatures.

A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels integrated with piezoelectric layers

2014 ◽  
Vol 21 (4) ◽  
pp. 571-587 ◽  
Author(s):  
Hamid Reza Saeidi Marzangoo ◽  
Mostafa Jalal
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Simply Supported ◽  
Piezoelectric Layers ◽  
Curved Panels

AbstractFree vibration analysis of functionally graded (FG) curved panels integrated with piezoelectric layers under various boundary conditions is studied. A panel with two opposite edges is simply supported, and arbitrary boundary conditions at the other edges are considered. Two different models of material property variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential law distribution of the material properties through the thickness are considered. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. For the simply supported boundary conditions, closed-form solution is given by making use of the Fourier series expansion, and applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free-end conditions. Natural frequencies of the hybrid curved panels are presented by solving the eigenfrequency equation, which can be obtained by using edges boundary conditions in this state equation. The results obtained for only FGM shell is verified by comparing the natural frequencies with the results obtained in the literature.

Dynamic analysis of functionally graded shell with piezoelectric layers based on elasticity

2009 ◽  
Vol 223 (9) ◽  
pp. 2039-2047 ◽  
Author(s):  
M Javanbakht ◽  
M Shakeri ◽  
S N Sadeghi
Keyword(s):  
Differential Equations ◽  
Piezoelectric Layer ◽  
External Surface ◽  
Internal Surface ◽  
Longitudinal Direction ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Galerkin Finite Element ◽  
Piezoelectric Layers ◽  
Functionally Graded Shell

A study on the elasticity solution of the functionally graded (FG) shell with two piezoelectric layers is presented. In this article, the structure is finitely long, simply supported, and FG with two piezoelectric layers under pressure and electrostatic excitation. The equations of equilibrium, which are coupled partial differential equations, are reduced to ordinary differential equations (o.d.e.) with variable coefficients by means of trigonometric function expansion in the longitudinal direction. The resulting o.d.e. are solved by the Galerkin finite-element method and the Newmark method. Numerical results are presented for a FG cylindrical shell with a piezoelectric layer as an actuator in the external surface and a piezoelectric layer as a sensor in the internal surface.

Elasticity solution for thick laminated anisotropic cylindrical panels under dynamic load

2002 ◽  
Vol 216 (3) ◽  
pp. 315-322 ◽  
Author(s):  
A Alibiglu ◽  
M Shakeri ◽  
M R Eslami
Keyword(s):  
Differential Equations ◽  
Dynamic Load ◽  
Three Dimensional ◽  
Variable Coefficients ◽  
Cylindrical Panel ◽  
Great Length ◽  
Elasticity Equations ◽  
Shell Panel ◽  
Asymmetric Load ◽  
Three Dimensional Elasticity

The dynamic response of an axisymmetric arbitrary laminated anisotropic cylindrical panel subjected to asymmetric load is studied on the basis of three-dimensional elasticity equations. The shell panel has a great length and is simply supported at both edges. The highly coupled partial differential equations (PDEs) are reduced to ordinary differential equations (ODEs) with variable coefficients by means of trigonometric function expansion in circumferential directions. The resulting OPEs are solved by Galerkin's finite element method. Numerical examples are presented for 45°/-45° and 45°/-45°/45° laminations under dynamic load. Finally, the results are compared with published results.

Three-dimensional elasticity solution of simply supported functionally graded rectangular plates with internal elastic line supports

2009 ◽  
Vol 44 (4) ◽  
pp. 249-261 ◽  
Author(s):  
Y P Xu ◽  
D Zhou
Keyword(s):  
Boundary Conditions ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Three Dimensional ◽  
Lower Surface ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Elastic Line ◽  
Simply Supported ◽  
Three Dimensional Elasticity

This paper studies the stress and displacement distributions of simply supported functionally graded rectangular plates with internal elastic line supports. The Young's modulus is graded through the thickness following the exponential law and the Poisson's ratio is kept constant. On the basis of three-dimensional elasticity theory, the solutions of displacements and stresses of the plate under static loads, which exactly satisfy the governing differential equations and the simply supported boundary conditions at four edges of the plate, are analytically derived. The reaction forces of the internal elastic line supports are regarded as the unknown external forces acting on the lower surface of the plate. The unknown coefficients in the solutions are then determined by the boundary conditions on the upper and lower surfaces of the plate. Convergence and comparison studies demonstrate the correctness and effectiveness of the proposed method. The effect of variations in Young's modulus on the displacements and stresses of rectangular plates and the effect of internal elastic line supports on the mechanical properties of plates are investigated.

scholarly journals Three-Dimensional Piezothermoelastic Stress of a Finite Functionally Graded Cylindrical Shell with Piezoelectric Layer

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Yong-gang Kang ◽  
Zhong-qi Wang ◽  
Gongnan Xie
Keyword(s):  
Cylindrical Shell ◽  
Fourier Series ◽  
Material Properties ◽  
Piezoelectric Layer ◽  
Radial Direction ◽  
Electric Displacement ◽  
Three Dimensional ◽  
Functionally Graded ◽  
Simply Supported ◽  
Piezoelectric Layers

Three-dimensional piezothermoelastic solutions for a finite functionally graded cylindrical shell with piezoelectric layer are carried out in this paper. The cylindrical shell is simply supported at four end edges and is subjected to axisymmetric thermomechanical loads. The piezoelectric layers are polarized along radial direction as a sensor. The material properties are assumed to be temperature independent and radially dependent but are assumed to be homogeneous in each layer; the variables are expanded in Fourier series to satisfy the boundary conditions and multilayer approach is used. Numerical results of mullite/molybdenum functionally graded cylindrical shell are presented; the temperature change, stresses, electric potential, and electric displacement distributions are given and briefly discussed.

scholarly journals Exponential matrix method for the solution of exact 3D equilibrium equations for free vibrations of functionally graded plates and shells

2017 ◽  
Vol 21 (1) ◽  
pp. 77-114 ◽  
Author(s):  
Salvatore Brischetto
Keyword(s):  
Differential Equations ◽  
Functionally Graded Material ◽  
Material Properties ◽  
Matrix Method ◽  
Three Dimensional ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Equilibrium Equations ◽  
Graded Material ◽  
Exponential Matrix

The present paper analyzes the convergence of the exponential matrix method in the solution of three-dimensional equilibrium equations for the free vibration analysis of functionally graded material structures. The three-dimensional equilibrium equations are written in general orthogonal curvilinear coordinates for one-layered and sandwich plates and shells embedding functionally graded material layers. The resulting system of second-order differential equations is reduced to a system of first-order differential equations redoubling the variables. This system is exactly solved using the exponential matrix method and harmonic displacement components. In the case of functionally graded material plates, the differential equations have variable coefficients because of the material properties which depend on the thickness coordinate z. For functionally graded material shells, the differential equations have variable coefficients because of both changing material properties and curvature terms. Several mathematical layers M can be introduced to approximate the curvature terms and the variable functionally graded material properties to obtain differential equations with constant coefficients. The exponential matrix is applied to solve the resulting system of partial differential equations with constant coefficients, where the used expansion has a very fast convergence ratio. The present work investigates the convergence of the proposed method related to the order N used for the expansion of the exponential matrix and to the number of mathematical layers M used for the approximation of curvature shell terms and variable functionally graded material properties. Both N and M values are analyzed for different geometries, thickness ratios, materials, functionally graded material laws, lamination sequences, imposed half-wave numbers, frequency orders, and vibration modes.

A Study on Free Vibrational Response of Functionally Graded Nanocomposite Sandwich Plates Reinforced by Randomly Oriented Straight Carbon Nanotubes

2016 ◽  
Vol 880 ◽  
pp. 77-82
Author(s):  
Vahid Tahouneh
Keyword(s):  
Carbon Nanotube ◽  
Three Dimensional ◽  
Functionally Graded ◽  
Sandwich Plates ◽  
Rectangular Plates ◽  
Reinforced Composites ◽  
Technical Literature ◽  
Simply Supported ◽  
Vibrational Response ◽  
Three Dimensional Elasticity

This paper is motivated by the lack of studies in the technical literature concerning to the three dimensional vibration analysis of thick laminated rectangular plates with continuously graded carbon nanotube-reinforced (CGCNTR) sheets. The formulations are based on the three-dimensional elasticity theory. The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The structure is supported by an elastic foundation with Winkler’s (normal) and Pasternak’s (shear) coefficients. The material properties of the functionally graded carbon nanotube reinforced composites are graded along the thickness and estimated through Mori-Tanaka method.

Sign in / Sign up

Export Citation Format

Share Document