Size-dependent nonlinear free vibration of multilayer functionally graded graphene nanocomposite Timoshenko microbeam under different boundary conditions

The aim of this article is to find an exact analytical solution for free vibration characteristics of thin functionally graded rectangular plates with different boundary conditions. The governing equations of motion are obtained based on the classical plate theory. Using an analytical method, three partial differential equations of motion are reformulated into two new decoupled equations. Based on the Navier solution, a closed-form solution is presented for natural frequencies of functionally graded simply supported rectangular plates. Then, considering Levy-type solution, natural frequencies of functionally graded plates are presented for various boundary conditions. Three mode shapes of a functionally graded rectangular plate are also presented for different boundary conditions. In addition, the effects of aspect ratio, thickness—length ratio, power law index, and boundary conditions on the vibration characteristics of functionally graded rectangular plates are discussed in details. Finally, it has been shown that the effects of in-plane displacements on natural frequencies of functionally graded plates under different boundary conditions have been studied.

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3-D Free Vibration Analysis of Functionally Graded Thick Circular and Annular Plates on Elastic Foundation

ASME 2010 10th Biennial Conference on Engineering Systems Design and Analysis, Volume 2 ◽

10.1115/esda2010-24340 ◽

2010 ◽

Author(s):

Vahid Tajeddini ◽

Abdolreza Ohadi ◽

Mojtaba Sadighi

Keyword(s):

Free Vibration ◽

Boundary Conditions ◽

Vibration Analysis ◽

Ritz Method ◽

Three Dimensional ◽

Free Vibration Analysis ◽

Small Strain ◽

Functionally Graded ◽

Different Boundary Conditions ◽

Fg Plates

This paper describes a study of three-dimensional free vibration analysis of thick circular and annular functionally graded (FG) plates resting on Pasternak foundation. The formulation is based on the linear, small strain and exact elasticity theory. Plates with different boundary conditions are considered and the material properties of the FG plate are assumed to vary continuously through the thickness according to power law. The kinematic and the potential energy of the plate-foundation system are formulated and the polynomial-Ritz method is used to solve the eigenvalue problem. Convergence and comparison studies are done to demonstrate the correctness and accuracy of the present method. With respect to geometric parameters, elastic coefficients of foundation and different boundary conditions some new results are reported which maybe used as a benchmark solution for future researches.

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Dynamic buckling of functionally graded multilayer graphene nanocomposite annular plate under different boundary conditions in thermal environment

Engineering With Computers ◽

10.1007/s00366-020-01169-7 ◽

2020 ◽

Author(s):

Farshid Allahkarami

Keyword(s):

Boundary Conditions ◽

Annular Plate ◽

Thermal Environment ◽

Dynamic Buckling ◽

Functionally Graded ◽

Multilayer Graphene ◽

Different Boundary Conditions ◽

Graphene Nanocomposite

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Free Vibration Analysis of Functionally Graded Spherical Torus Structure with Uniform Variable Thickness along Axial Direction

Shock and Vibration ◽

10.1155/2019/2803841 ◽

2019 ◽

Vol 2019 ◽

pp. 1-22

Author(s):

Cong Gao ◽

Xuhong Miao ◽

Lin Lu ◽

Ruidong Huo ◽

Qiaolin Hu ◽

...

Keyword(s):

Free Vibration ◽

Boundary Conditions ◽

Variable Thickness ◽

Domain Decomposition Method ◽

Ritz Method ◽

Axial Direction ◽

Functionally Graded ◽

Spherical Torus ◽

Different Boundary Conditions ◽

Vibration Responses

Based on the Ritz method, this paper focused on the free vibration of functionally graded (FG) spherical torus with uniform variable thickness along axial direction under different boundary conditions. The first-order shear deformation theory (FSDT) is employed to formulate the analytical model. The method involves partitioning of the spherical torus structure into proper shell segments in order to satisfy the computing requirement of high-order vibration responses according to the domain decomposition method. The two adjacent segments are connected by using the penalty method, where penalty parameters are defined by the artificial springs; the continuity condition and different boundary conditions can be obtained by assigning the appropriate values of springs. The displacement functions’ components are double mixed series, in which Fourier series and unified Jacobi polynomials, respectively, represent displacement function along circumferential direction and axial direction. Then the Ritz method is used to obtain final solutions. The numerical results obtained by the proposed method show great agreement with previously published literatures and those from the finite element program ABAQUS. The effects of boundary conditions and geometric parameters on the vibration responses of the structure are also presented. The most novelty of this paper is to generalize the selection of admissible displacement functions by using Jacobi polynomial.

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