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.

Symplectic Superposition Solution of Free Vibration of Fully Clamped Orthotropic Rectangular Thin Plate on Two-Parameter Elastic Foundation

2021 ◽  
pp. 2150122
Author(s):  
Xin Su ◽  
Eburilitu Bai ◽  
Alatancang Chen
Keyword(s):  
Free Vibration ◽  
Elastic Foundation ◽  
Series Expansion ◽  
Separation Of Variables ◽  
Solution Process ◽  
Thin Plates ◽  
Symplectic Space ◽  
Superposition Method ◽  
Vibration Problem ◽  
Two Parameter

Based on the method of separation of variables in Hamiltonian system and superposition method, the series expansion solution of the free vibration problem of orthotropic rectangular thin plates (ORTPs) with four clamped edges (CCCC) on two-parameter elastic foundation is obtained. The original vibration problem is decomposed into two subproblems with two opposite sides simply supported, and the general solution of each subproblem is obtained by using the expansion of symplectic eigenvectors. Then by superposing these two general solutions, the series expansion solution of the original problem is obtained. The advantage of this method is that the solution process is carried out in symplectic space, and the validity of variable separation and symplectic eigenvectors expansion ensures the rationality of the solution process, while avoiding the presetting of the solution form. Finally, the correctness of symplectic superposition solution obtained in this paper is verified by calculating three concrete examples of fully clamped rectangular thin plates.

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.

On the symplectic superposition method for new analytic free vibration solutions of side-cracked rectangular thin plates

2020 ◽  
Vol 489 ◽  
pp. 115695
Author(s):  
Zhaoyang Hu ◽  
Yushi Yang ◽  
Chao Zhou ◽  
Xinran Zheng ◽  
Rui Li
Keyword(s):  
Free Vibration ◽  
Thin Plates ◽  
Superposition Method

A General Analytical Solution for Free Vibration of Rectangular Plates Resting on Fixed Supports and With Attached Masses

1992 ◽  
Vol 114 (2) ◽  
pp. 239-245 ◽  
Author(s):  
R. K. Singal ◽  
D. J. Gorman
Keyword(s):  
Free Vibration ◽  
Analytical Procedure ◽  
Thin Plates ◽  
Mode Shapes ◽  
Experimental Program ◽  
Space Vehicles ◽  
Superposition Method ◽  
Rectangular Plates ◽  
Solar Panels ◽  
Good Agreement

A comprehensive analytical procedure based on the superposition method is described for establishing the free vibration frequencies and mode shapes of thin plates resting on rigid point supports and with attached masses. Effects of rotary inertia of the attached masses are incorporated into the analysis and are shown to be highly significant. Results of an extensive experimental program are reported and very good agreement is demonstrated between theory and experiment. The analytical procedure has application in numerous contemporary industrial problems, in particular, in the design of solar panels for space vehicles and in the field of electronic packaging.

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