The Modified Fourier-Ritz Approach for the Free Vibration of Functionally Graded Cylindrical, Conical, Spherical Panels and Shells of Revolution with General Boundary Condition

et al. 2018

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A series solution for the transverse vibration of Mindlin rectangular plates with elastic point supports around the edges is studied. The series solution for the problem is obtained using improved Fourier series method, in which the vibration displacements and the cross-sectional rotations of the midplane are represented by a double Fourier cosine series and four supplementary functions. The supplementary functions are expressed as the combination of trigonometric functions and a single cosine series expansion and are introduced to remove the potential discontinuities associated with the original admissible functions along the edges when they are viewed as periodic functions defined over the entire x-y plane. This series solution is approximately accurate in the sense that it explicitly satisfies, to any specified accuracy, both the governing equations and the boundary conditions. The convergence, accuracy, stability, and efficiency of the proposed method have been examined through a series of numerical examples. Some numerical examples about the nondimensional frequency and mode shapes of Mindlin rectangular plates with different point-supported edge conditions are given.

Section: Point-supported Edge Conditions the Rectangularmentioning

confidence: 98%

A Series Solution for the Vibration of Mindlin Rectangular Plates with Elastic Point Supports around the Edges

et al. 2018

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“…However, the conventional Fourier series just adapt to a few of simple boundary conditions due to the convergence problem along the boundary conditions. Recently, a modified Fourier series technique proposed by Li [50] has been widely applied in the vibration problems of plates and shells subject to different boundary conditions by the Ritz method, e.g., [42,[51][52][53][54][55][56][57][58][59][60]. In this technique, each displacement of the structure under study is written in the form of a conventional cosine Fourier series and several supplementary terms.…”

Section: Shock and Vibrationmentioning

confidence: 99%

An Accurate Solution Method for the Static and Vibration Analysis of Functionally Graded Reissner‐Mindlin Rectangular Plate with General Boundary Conditions

Liu

et al. 2018

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This paper presents an accurate solution method for the static and vibration analysis of functionally graded Reissner-Mindlin plate with general boundary conditions on the basis of the improved Fourier series method. In the theoretical formulations, the governing equations and the general elastic boundary equations are obtained by using Hamilton’s principle. The components of admissible displacement functions are expanded as an improved Fourier series form which contains a 2D Fourier cosine series and auxiliary function in the form of 1D series. The major role of the auxiliary function is to remove the potential discontinuities of the displacement function and its derivatives at the edges and ensure and accelerate the convergence of the series representation. The characteristic equations are easily obtained via substituting admissible displacement functions into governing equations and the general elastic boundary equations. Several examples are made to show the excellent accuracy and convergence of the current solutions. The results of this paper may serve as benchmark data for future research in related field.

“…Kiani [23] dealt with vibration characteristics of carbon nanotube-reinforced composite spherical panels according to Hamilton's principle and Ritz method. Li et al [24] carried out the free vibration of four-parameter FG moderately thick spherical panels with general boundary conditions in modified Fourier-Ritz approach. Pang et al [25] proposed a unified vibration analysis approach on the basis of variational operation for FG shell of revolution with general boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

A Semianalytical Approach for Free Vibration Characteristics of Functionally Graded Spherical Shell Based on First‐Order Shear Deformation Theory

Gao

Cui

et al. 2019

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This paper describes a unified solution to investigate free vibration solutions of functionally graded (FG) spherical shell with general boundary restraints. The analytical model is established based on the first-order shear deformation theory, and the material varies uniformly along the thickness of FG spherical shell which is divided into several sections along the meridian direction. The displacement functions along circumferential and axial direction are, respectively, composed by Fourier series and Jacobi polynomial regardless of boundary restraints. The boundary restraints of FG spherical shell can be easily simulated according to penalty method of spring stiffness technique, and the vibration solutions are obtained by Rayleigh–Ritz method. To verify the reliability and accuracy of the present solutions, the convergence and numerical verification have been conducted about different boundary parameters, Jacobi parameter, etc. The results obtained by the present method closely agree with those obtained from the published literatures, experiments, and finite element method (FEM). The impacts of geometric dimensions and boundary conditions on the vibration characteristics of FG spherical shell structure are also presented.