A New General Solution for the Bending and Vibration of Orthotropic Rectangular Thin Plates with Four Free Edges

2010 ◽  
Vol 168-170 ◽  
pp. 1158-1162 ◽  
Author(s):  
Hong Zhang ◽  
Hai Qun Que ◽  
Huan Ding
Keyword(s):  
General Solution ◽  
Vibration Analysis ◽  
Thin Plates ◽  
Physical Parameters ◽  
Winkler Foundation ◽  
Vertical Force ◽  
Rectangular Plates ◽  
Cosine Series ◽  
Vibration Problems ◽  
Free Edges

This paper firstly introduces a new general solution constructed by double trigonometric cosine series with supplementary terms for the bending and vibration analysis of orthotropic rectangular plates with four free edges on the Winkler foundation subjected to arbitrary vertical force. The general solution, which is fourth-order continuously differentiable with less undetermined coefficients, can be used to solve the bending and vibration problems of orthotropic rectangular plates on the Winkler foundation with various physical parameters requiring no classification and superposition. This makes the bending and vibration analysis of orthotropic rectangular plates with four free edges on the Winkler foundation more unified, simplified and regulated. This paper also gives a Series of analytical example to prove that the method is feasible.

Vibrations of Completely Free Shallow Shells of Curvilinear Planform

1986 ◽  
Vol 53 (3) ◽  
pp. 647-651 ◽  
Author(s):  
Y. Narita ◽  
A. Leissa
Keyword(s):  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Rectangular Plates ◽  
Algebraic Polynomials ◽  
Shallow Shells ◽  
Zero Curvature ◽  
Vibration Problems ◽  
Arbitrary Curvature ◽  
Made In ◽  
Free Edges

A method is presented for the free vibration analysis of shallow shells having free edges of arbitrary curvilinear shape. The method of Ritz which was developed for free rectangular plates is extended to the present problem. Components of displacement are expressed as algebraic polynomials. Shells of arbitrary curvature may be treated. Results are obtained for the previously unsolved vibration problems of cylindrical, spherical and hyperbolic paraboloidal shells having free edges of circular and elliptical planform. Convergence of the method is demonstrated. Comparisons with previous solutions are made in the case of zero curvature (i.e., a flat plate). Effects of increasing curvature and ellipticity upon vibration frequencies are examined.

Free In-Plane Vibration Analysis of Rectangular Plates With Elastically Point-Supported Edges

2010 ◽  
Vol 132 (3) ◽  
Author(s):  
Jingtao Du ◽  
Zhigang Liu ◽  
Wen L. Li ◽  
Xuefeng Zhang ◽  
Wanyou Li
Keyword(s):  
Fourier Series ◽  
Vibration Analysis ◽  
Series Representation ◽  
Fourier Method ◽  
Rectangular Plates ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Plane Vibration ◽  
Auxiliary Functions ◽  
Time In Literature

In comparison with the transverse vibrations of rectangular plates, far less attention has been paid to the in-plane vibrations even though they may play an equally important role in affecting the vibrations and power flows in a built-up structure. In this paper, a generalized Fourier method is presented for the in-plane vibration analysis of rectangular plates with any number of elastic point supports along the edges. Displacement constraints or rigid point supports can be considered as the special case when the stiffnesses of the supporting springs tend to infinity. In the current solution, each of the in-plane displacement components is expressed as a 2D Fourier series plus four auxiliary functions in the form of the product of a polynomial times a Fourier cosine series. These auxiliary functions are introduced to ensure and improve the convergence of the Fourier series solution by eliminating all the discontinuities potentially associated with the original displacements and their partial derivatives along the edges when they are periodically extended onto the entire x-y plane as mathematically implied by the Fourier series representation. This analytical solution is exact in the sense that it explicitly satisfies, to any specified accuracy, both the governing equations and the boundary conditions. Numerical examples are given about the in-plane modes of rectangular plates with different edge supports. It appears that these modal data are presented for the first time in literature, and may be used as a benchmark to evaluate other solution methodologies. Some subtleties are discussed about corner support arrangements.

Vibration Cosine Solutions of Free Orthotropic Rectangular Plates on the Elastic Half-Space

2011 ◽  
Vol 291-294 ◽  
pp. 2094-2097
Author(s):  
Chun Ling Wang ◽  
Huan Ding ◽  
Hai Xia Zhang
Keyword(s):  
General Solution ◽  
Rectangular Plate ◽  
Half Space ◽  
Analytic Solutions ◽  
Elastic Half Space ◽  
Integral Representations ◽  
Rectangular Plates ◽  
Practical Applications ◽  
Cosine Series ◽  
Orthotropic Rectangular Plate

In this paper, the analytic solutions of steady vibration of free orthotropic rectangular plate loaded with vertical steady loading on the elastic half-space was given by combining the general solution of double trigonometrically cosine series with supplementary terms with dynamic integral representations for displacements of the elastic half-space loaded with arbitrary vertical steady loading. This solution not only is four-order derivative, but also has less undetermined coefficients. It can be used to solve the problems of bending and steady vibration of orthotropic rectangular plates on the elastic half-space without be classified and be superimposed. This causes this kind of things, bending and steady vibration of orthotropic rectangular plates with four free edges on the elastic half-space, unionization, simplification and systematization. When the material is isotropic, the solutions are turned into analytic solutions of steady vibration of free rectangular plate on the elastic half-space. At last some computational examples are presented and the results are coincided with those in literatures. Then the method in this paper will be of important practical applications.

Analytical free vibration solutions of fully free orthotropic rectangular thin plates on two-parameter elastic foundations by the symplectic superposition method

2020 ◽  
pp. 107754632096782
Author(s):  
Xin Su ◽  
Eburilitu Bai
Keyword(s):  
Free Vibration ◽  
Fundamental Solutions ◽  
Thin Plates ◽  
Superposition Method ◽  
Elastic Foundations ◽  
Vibration Problem ◽  
Separation Variable ◽  
Vibration Problems ◽  
Two Parameter ◽  
Free Edges

The free vibration of orthotropic rectangular thin plates with four free edges on two-parameter elastic foundations is studied by the symplectic superposition method. Firstly, by analyzing the boundary conditions, the original vibration problem is converted into two sub-vibration problems of the plates slidingly clamped at two opposite edges. Based on slidingly clamped at two opposite edges, the fundamental solutions of these two sub-vibration problems are respectively derived by the separation variable method of the corresponding Hamiltonian system, and then the symplectic superposition solution of the original vibration problem is obtained by superimposing the fundamental solutions of the two sub-problems. Finally, the symplectic superposition solution obtained in this study is verified by calculating the frequencies and mode functions of several concrete rectangular thin plates with four free edges.

scholarly journals Vibration Analysis of Rectangular Plates with Free Corners Using Spline-Based Differential Quadrature

2004 ◽  
Vol 11 (2) ◽  
pp. 119-128 ◽  
Author(s):  
Hongzhi Zhong ◽  
Qiang Guo
Keyword(s):  
Vibration Analysis ◽  
Differential Quadrature Method ◽  
Free Vibrations ◽  
Differential Quadrature ◽  
Rectangular Plates ◽  
Cardinal Spline ◽  
Approximation Of Derivatives ◽  
Weighting Coefficients ◽  
Spline Interpolants ◽  
Free Edges

A spline-based differential quadrature method (SDQM) is elaborated and applied to the vibration analysis of rectangular plates with free edges. The sextic B-spline functions are used to construct the pertaining cardinal spline interpolants. Thus, explicit expressions of weighting coefficients for approximation of derivatives are obtained. Free vibrations of rectangular plates with free edges, which pose considerable challenge to the conventional differential quadrature, are dealt with and the results are in excellent agreement with those in the literature, highlighting the effectiveness and potential of the spline-based differential quadrature.

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