Exact Electroelastic Field of a Functionally Graded Piezoelectric Cantilever Beam Subjected to Pure Body Force Loading

2010 ◽  
Author(s):  
A. A. Emami ◽  
R. Hashemi ◽  
M. H. Kargarnovin ◽  
R. Naghdabadi
Keyword(s):  
Differential Equations ◽  
Body Force ◽  
Functionally Graded ◽  
The Body ◽  
Cantilever Beams ◽  
Exponential Distributions ◽  
Partial Differential ◽  
Numerical Studies ◽  
Piezoelectric Cantilever ◽  
Functionally Graded Piezoelectric

The electroelastic response of functionally graded piezoelectric cantilever beams which includes the effect of body force is presented in this paper. The material properties such as elastic compliance, piezoelectric and dielectric impermeability are assumed to be graded with different indices in the thickness direction according to exponential distributions. Systems of fourth order inhomogeneous partial differential equations (PDEs) which are satisfied by the stress and induction functions and involve the body force terms are derived. Spectral forms for electrical and mechanical variables in the x-axis are employed to convert the partial differential governing equations and the associated boundary conditions into sets of ordinary differential equations, and the resulting equations are solved in a closed form manner. Subsequently, in numerical studies, the effects of the material property graded indices are examined upon the electroelastic response of FGP cantilever beams under pure body force loadings.

scholarly journals A Multi-Parameter Perturbation Solution for Functionally Graded Piezoelectric Cantilever Beams under Combined Loads

Materials ◽  
2018 ◽  
Vol 11 (7) ◽  
pp. 1222 ◽  
Author(s):  
Yongsheng Lian ◽  
Xiaoting He ◽  
Sijie Shi ◽  
Xue Li ◽  
Zhixin Yang ◽  
...  
Keyword(s):  
Cantilever Beam ◽  
Functionally Graded ◽  
Airy Stress Function ◽  
Parameter Perturbation ◽  
Combined Loads ◽  
Piezoelectric Cantilever ◽  
Basic Equations ◽  
Perturbation Parameters ◽  
Functionally Graded Piezoelectric ◽  
Parameter Perturbation Method

In this study, we use a multi-parameter perturbation method to solve the problem of a functionally graded piezoelectric cantilever beam under combined loads, in which three piezoelectric coefficients are selected as the perturbation parameters. First, we derive the two basic equations concerning the Airy stress function and electric potential function. By expanding the unknown Airy stress function and electric potential function with respect to three perturbation parameters, the two basic equations were decoupled, thus obtaining the corresponding multi-parameter perturbation solution under boundary conditions. From the solution obtained, we can see clearly how the piezoelectric effects influence the behavior of the functionally graded piezoelectric cantilever beam. Based on a numerical example, the variations of the elastic stresses and displacements as well as the electric displacements of the cantilever beam under different gradient exponents were shown. The results indicate that if the pure functionally graded cantilever beam without a piezoelectric effect is regarded as an unperturbed system, the functionally graded piezoelectric cantilever beam can be looked upon as a perturbed system, thus opening the possibilities for perturbation solving. Besides, the proposed multi-parameter perturbation method provides a new idea for solving similar nonlinear differential equations.

Curvature Approximation Effects in the Free Vibration Analysis of Functionally Graded Shells

2016 ◽  
Vol 08 (06) ◽  
pp. 1650079 ◽  
Author(s):  
Salvatore Brischetto
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Curvilinear Coordinates ◽  
Equilibrium Equations ◽  
Partial Differential ◽  
Exact Geometry

The present work investigates the effects of the curvature terms in the three-dimensional (3D) equilibrium equations used for the free vibration analysis of functionally graded material (FGM) structures. The 3D equilibrium equations have been written in general orthogonal curvilinear coordinates which are valid for spherical shells. They automatically degenerate in those for cylindrical shells and plates considering one of the two radii of curvature and both radii of curvature equal to infinite, respectively. The approximation of curvature terms in the 3D equilibrium equations has been evaluated by means of frequency analyses. Results obtained via 3D equilibrium equations with exact geometry have been compared with those calculated via 3D equilibrium equations written with the approximation of the curvature terms. The effects of the curvature approximations depend on the thickness and curvature of the structures, on the materials, lamination sequences and FGM laws, on the frequency orders and vibration modes. The resulting system of second order partial differential equations has been reduced into a system of first order partial differential equations redoubling the variables. Therefore, the exponential matrix method has been employed using a layer wise approach. The final 3D equations have been solved in exact form considering harmonic displacement components and simply supported structures. The approximation of the curvature terms has been introduced in the 3D equilibrium shell equations. For numerical reasons, interlaminar continuity conditions and the top and bottom boundary and loading conditions have been written including the exact geometry. The introduction of curvature approximations only in the equilibrium equations is sufficient to obtain an exhaustive qualitative analysis of the importance of curvature terms in the free vibration problems for FGM structures.

Nonlinear Vibration of Functionally Graded Material Cylindrical Shell Based on Reddy’s Third-Order Plates and Shells Theory

2012 ◽  
Vol 625 ◽  
pp. 18-24 ◽  
Author(s):  
Lu Dong ◽  
Yu Xin Hao ◽  
Jian Hua Wang ◽  
Li Yang
Keyword(s):  
Cylindrical Shell ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Vibration ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Third Order ◽  
Partial Differential ◽  
Graded Material ◽  
Plates And Shells

In this paper, an analysis on nonlinear dynamics of a simply supported functionally graded material (FGM) cylindrical shell subjected to the different excitation in thermal environment. Material properties of cylindrical shell are assumed to be temperature-dependent. Based on the Reddy’s third-order plates and shells theory[1], the nonlinear governing partial differential equations of motion for the FGM cylindrical shell are derived by using Hamilton’s principle. Galerkin’s method is utilized to transform the partial differential equations into a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined parametric and external excitation. The effects played by different excitation and system initial conditions on the nonlinear vibration of the cylindrical shell are studied. In addition, the Runge–Kutta method is used to find out the nonlinear dynamic responses of the FGM cylindrical shell.

An Analytical Solution for Nonlinear Vibrations Analysis of Functionally Graded Plate Using Modified Lindstedt–Poincare Method

2020 ◽  
Vol 12 (01) ◽  
pp. 2050003 ◽  
Author(s):  
S. Hashemi ◽  
A. A. Jafari
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Rectangular Plate ◽  
Volume Fraction ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Published Data ◽  
Power Law Distribution ◽  
Partial Differential

In this research, the nonlinear free vibrations analysis of functionally graded (FG) rectangular plate which simply supported all edges are investigated analytically using modified Lindstedt–Poincare (MLP) method for the first time. For this purpose, with the aid of von Karman nonlinearity strain-displacement relations, the partial differential equations of motion are developed based on first-order shear deformation theory (FSDT). Afterward, by applying Galerkin method, the nonlinear partial differential equations are transformed into the time-dependent nonlinear ordinary differential equations. The nonlinear equation of motion is then solved analytically by MLP method to determine the nonlinear frequencies of the FG rectangular plate. The material properties are assumed to be graded through the direction of plate thickness according to power law distribution. The effects of some system parameters such as vibration amplitude, volume fraction index and aspect ratio on the nonlinear to linear frequency ratio are discussed in detail. To validate the analysis, the results of this paper are compared with both the published data and numerical method, and good agreements are found.

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