On the symplectic superposition method for analytic free vibration solutions of right triangular plates

Yang, Yushi; An, Dongqi; Xu, Houlin; Wang, Bo; Li, Rui

doi:10.1007/s00419-020-01763-7

Cited by 10 publications

(2 citation statements)

References 45 publications

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“…If the geometries of the plate are changing, we need to construct different symplectic superposition systems and derive new formulas. The exact solution procedure can be seen in "On the symplectic superposition method for analytic free vibration solutions of right triangular plates" [31] Circular plates have been studied using the symplectic method, the details can be found at "Natural vibration of circular and annular thin plates by Hamiltonian approach" [32] and "The dynamic behavior of circular plates under impact loads" [33]. We are currently conducting some research on the use of symplectic superposition for irregularly shaped plates (plates with a rectangular central cutout and L-shaped plates) "New analytic free vibration solutions of plates with a rectangular central cutout by symplectic superposition" "New analytic free vibration solutions of L-shaped moderately thick plates by symplectic superposition" which are being submitted for publication.…”

Section: Introductionmentioning

confidence: 99%

Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method

Yang,

Xu,

Chu

et al. 2024

Mathematics

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The buckling behavior of rectangular thin plates, which are supported at their four corner points with four edges free, is a matter of great concern in the field of plate and shell mechanics. Nevertheless, the complexities arising from the boundary conditions and governing equations present a formidable obstacle to the attainment of analytical solutions for these problems. Despite the availability of various approximate/numerical methods for addressing these challenges, the literature lacks accurate analytic solutions. In this study, we employ the symplectic superposition method, a recently developed method, to effectively analyze the buckling problem of rectangular thin plates analytically. These plates have four supported corners and four free edges. To achieve this, the problem is divided into two sub-problems and solve them separately using variable separation and symplectic eigen expansion, leading to analytical solutions. Finally, we obtain the resolution to the initial issue by superposing the sub-problems. The current solution method can be regarded as a logical, analytical, and rational approach as it begins with the basic governing equation and is systematically derived without assuming the forms of the solutions. To examine various aspect ratios and in-plane load ratios of rectangular thin plates, which are supported at their four corner points with four edges free, we provide numerical examples that demonstrate the buckling loads and typical buckling mode shapes.

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Section: Introductionmentioning

confidence: 99%

Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method

Yang,

Xu,

Chu

et al. 2024

Mathematics

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“…The simplistic superposition method was used for analytic free vibration solutions of right triangular plates [22]. The problem was solved by dividing it into three subproblems that were solved by the symplectic techniques via imposing the variable separation on the Hamiltonian-system-based governing equation and symplectic eigen expansion.…”

Section: Introductionmentioning

confidence: 99%

Bending of Orthotropic Scalene Triangle Plates: Finite Difference Modeling

Akhmediev,

Beketova,

Nuguzhinov

et al. 2023

Preprint

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The object of study is a transversely bent triangular plate made of orthotropic material fixed along the edges of the plate under the action of a uniformly distributed load. Fourth-order differential equilibrium equations with variable orthotropy parameters were used. The equations were approximated by finite differences for a grid of scalene triangles. This type of mesh accurately describes the boundary contour of the triangular-shaped plates. The boundary conditions for the mesh consider the orthotropy of the plate material. Seven standard finite-difference equations have been developed considering the boundary conditions along the three edges of the plate and the presence of three angles of an irregularly shaped triangle. A finite difference matrix is obtained. The matrix structure makes calculating a triangular plate at different base angles possible. The boundary conditions were varied in the form of rigid or hinged support of triangular plates. The calculation method considers the orthotropic parameters of the material in two mutually perpendicular planes. The adaptation of the numerical method to the calculation of orthotropic plates of arbitrary shape is described. Relationships are given for determining the rigidity characteristics of orthotropic materials. An algorithm for simple engineering calculation of triangular orthotropic plates is proposed, which makes it possible to perform accurate calculations in variant design.

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On the symplectic superposition method for new analytic bending, buckling, and free vibration solutions of rectangular nanoplates with all edges free

et al. 2020

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On the symplectic superposition method for analytic free vibration solutions of right triangular plates

Cited by 10 publications

References 45 publications

Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method

Analytic Solution for Buckling Problem of Rectangular Thin Plates Supported by Four Corners with Four Edges Free Based on the Symplectic Superposition Method

Bending of Orthotropic Scalene Triangle Plates: Finite Difference Modeling

On the symplectic superposition method for new analytic bending, buckling, and free vibration solutions of rectangular nanoplates with all edges free

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