Remarks of some problems for rectangular thin plates with free edges on elastic foundations

1991 ◽  
Vol 12 (4) ◽  
pp. 415-420 ◽  
Author(s):  
Tan Jun-yu
Keyword(s):  
Thin Plates ◽  
Elastic Foundations ◽  
Free Edges

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 Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method

2013 ◽  
Vol 469 (2153) ◽  
pp. 20120681 ◽  
Author(s):  
Rui Li ◽  
Yang Zhong ◽  
Ming Li
Keyword(s):  
Finite Element Method ◽  
Symplectic Geometry ◽  
Analytic Solutions ◽  
Thin Plates ◽  
Superposition Method ◽  
The Finite Element Method ◽  
Simple Derivation ◽  
Elastic Foundations ◽  
Winkler Model ◽  
Geometry Method

Analytic bending solutions of free rectangular thin plates resting on elastic foundations, based on the Winkler model, are obtained by a new symplectic superposition method. The proposed method offers a rational elegant approach to solve the problem analytically, which was believed to be difficult to attain. By way of a rigorous but simple derivation, the governing differential equations for rectangular thin plates on elastic foundations are transferred into Hamilton canonical equations. The symplectic geometry method is then introduced to obtain analytic solutions of the plates with all edges slidingly supported, followed by the application of superposition, which yields the resultant solutions of the plates with all edges free on elastic foundations. The proposed method is capable of solving plates on elastic foundations with any other combinations of boundary conditions. Comprehensive numerical results validate the solutions by comparison with those obtained by the finite element method.

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.

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