scholarly journals New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Xinran Zheng ◽  
Mingqi Huang ◽  
Dongqi An ◽  
Chao Zhou ◽  
Rui Li
Keyword(s):  
Hamiltonian System ◽  
Free Vibration ◽  
Eigenvalue Problems ◽  
Analytic Solutions ◽  
Symplectic Space ◽  
Superposition Method ◽  
Mathematical Solution ◽  
Simply Supported ◽  
Solution Methods ◽  
Symplectic Eigenvalue

AbstractNew analytic bending, buckling, and free vibration solutions of rectangular nanoplates with combinations of clamped and simply supported edges are obtained by an up-to-date symplectic superposition method. The problems are reformulated in the Hamiltonian system and symplectic space, where the mathematical solution framework involves the construction of symplectic eigenvalue problems and symplectic eigen expansion. The analytic symplectic solutions are derived for several elaborated fundamental subproblems, the superposition of which yields the final analytic solutions. Besides Lévy-type solutions, non-Lévy-type solutions are also obtained, which cannot be achieved by conventional analytic methods. Comprehensive numerical results can provide benchmarks for other solution methods.

On New Analytic Free Vibration Solutions of Doubly Curved Shallow Shells by the Symplectic Superposition Method Within the Hamiltonian-System Framework

2020 ◽  
Vol 143 (1) ◽  
Author(s):  
Rui Li ◽  
Chao Zhou ◽  
Xinran Zheng
Keyword(s):  
Hamiltonian System ◽  
Free Vibration ◽  
Original Problem ◽  
Analytic Solutions ◽  
Solution Procedure ◽  
Symplectic Space ◽  
Lagrangian System ◽  
Superposition Method ◽  
Shallow Shells ◽  
Doubly Curved

Abstract This study presents a first attempt to explore new analytic free vibration solutions of doubly curved shallow shells by the symplectic superposition method, with focus on non-Lévy-type shells that are hard to tackle by classical analytic methods due to the intractable boundary-value problems of high-order partial differential equations. Compared with the conventional Lagrangian-system-based expression to be solved in the Euclidean space, the present description of the problems is within the Hamiltonian system, with the solution procedure implemented in the symplectic space, incorporating formulation of a symplectic eigenvalue problem and symplectic eigen expansion. Specifically, an original problem is first converted into two subproblems, which are solved by the above strategy to yield the symplectic solutions. The analytic frequency and mode shape solutions are then obtained by the requirement of the equivalence between the original problem and the superposition of subproblems. Comprehensive results for representative non-Lévy-type shells are tabulated or plotted, all of which are well validated by satisfactory agreement with the numerical finite element method. Due to the strictness of mathematical derivation and accuracy of solution, the developed method provides a solid approach for seeking more analytic solutions.

scholarly journals New Analytic Free Vibration Solutions of Rectangular Thick Plates With a Free Corner by the Symplectic Superposition Method

2018 ◽  
Vol 140 (3) ◽  
Author(s):  
Rui Li ◽  
Pengcheng Wang ◽  
Bo Wang ◽  
Chunyu Zhao ◽  
Yewang Su
Keyword(s):  
Free Vibration ◽  
Analytic Solutions ◽  
The Other ◽  
Superposition Method ◽  
Element Analysis ◽  
Thick Plates ◽  
Simply Supported ◽  
Vibrating Plates ◽  
First Time ◽  
Mathematical Derivation

Seeking analytic free vibration solutions of rectangular thick plates without two parallel simply supported edges is of significance for an insight into the performances of related engineering devices and structures as well as their rapid design. A challenging set of problems concern the vibrating plates with a free corner, i.e., those with two adjacent edges free and the other two edges clamped or simply supported or one of them clamped and the other one simply supported. The main difficulty in solving one of such problems is to find a solution meeting both the boundary conditions at each edge and the condition at the free corner, which is unattainable using a conventional analytic method. In this paper, for the first time, we extend a novel symplectic superposition method to free vibration of rectangular thick plates with a free corner. The analytic frequency and mode shape solutions are both obtained and presented via comprehensive numerical and graphic results. The rigorousness in mathematical derivation and rationality of the method (without any predetermination for the solutions) guarantee the validity of our analytic solutions, which themselves are also validated by the reported results and refined finite element analysis.

Accurate Analytical-Type Solutions for the Free Vibration of Simply-Supported Parallelogram Plates

1991 ◽  
Vol 58 (1) ◽  
pp. 203-208 ◽  
Author(s):  
D. J. Gorman
Keyword(s):  
Free Vibration ◽  
Equal Length ◽  
Superposition Method ◽  
Vibration Modes ◽  
Type Solution ◽  
Simply Supported ◽  
Wide Range ◽  
Plate Geometry ◽  
Special Case ◽  
Comprehensive Study

A comprehensive study of the free vibration of simply-supported parallelogram plates is conducted. Solutions are obtained by utilizing the superposition method and by taking advantage of symmetry inherent in the problem. Toward this end a new alternating Le´vy-type solution is introduced. Verification tests are conducted by comparing computed eigenvalues with those of rhombic plates in the special case where all plate edges are of equal length. Eigenvalues are stored for eight vibration modes and for a wide range of plate geometry.

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.

Free vibration of simply supported and clamped elliptical plates

1992 ◽  
Vol 158 (2) ◽  
pp. 383-386 ◽  
Author(s):  
K.L. Prasad ◽  
A. Venkateshwar Rao ◽  
B. Nageswara Rao
Keyword(s):  
Free Vibration ◽  
Simply Supported

Backward Error Analysis for Eigenproblems Involving Conjugate Symplectic Matrices

2015 ◽  
Vol 5 (4) ◽  
pp. 312-326 ◽  
Author(s):  
Wei-wei Xu ◽  
Wen Li ◽  
Xiao-qing Jin
Keyword(s):  
Optimal Control Problems ◽  
Eigenvalue Problems ◽  
Backward Error Analysis ◽  
Linear Quadratic ◽  
Backward Error ◽  
Linear Quadratic Optimal Control ◽  
Quadratic Optimal Control ◽  
Symplectic Eigenvalue ◽  
Backward Errors ◽  
Symplectic Matrices

AbstractConjugate symplectic eigenvalue problems arise in solving discrete linear-quadratic optimal control problems and discrete algebraic Riccati equations. In this article, backward errors of approximate pairs of conjugate symplectic matrices are obtained from their properties. Several numerical examples are given to illustrate the results.

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