On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates

Lim, C.W.; Li, Chaofeng; Xiang, Yang; Yao, Weian

doi:10.1016/j.ijengsci.2008.08.003

Cited by 140 publications

(36 citation statements)

References 15 publications

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“…Gorman and Ding proposed a semi-inverse superposition method based on the use of Lévy-type single series to obtain the free vibration solutions of rectangular thin plates [13]. Lim et al [14,15] developed the analytic solutions for free vibration of Lévy-type rectangular thin plates by the symplectic elasticity approach, which was originated by Zhong's group [16,17] and has been applied in many research fields [18], including elasticity [17,19], symplectic numerical methods [20,21], fracture mechanics [22,23], piezoelectricity [24], functionally graded effects [25], magneto-electro-elasticity [26]. Hu et al [27] further developed the method to obtain the vibration solutions of rectangular orthotropic plates with combinations of clamped and simply supported edges.…”

Section: Edgesmentioning

confidence: 99%

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Analytic solutions for the free vibration of rectangular thin plates with two adjacent corners point-supported

Wang

2015

Arch Appl Mech

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Analytic free vibration solutions of rectangular thin plates with two adjacent corners point-supported are obtained for the first time by using an up-to-date symplectic superposition method, which demonstrates the ability to yield solutions via step-by-step derivations from the governing equations, without predetermining any forms of solutions. The essential difference between the current method and the other ones is that the solution procedure is conducted in the symplectic space rather than in the classical Euclidean space. This extends the scope of application of eigen expansion and paves the way for more analytic solutions of similar problems. Comprehensive highly accurate numerical results are presented to show the validity of the current solutions and to provide the benchmarks for future comparison and verification of other methods.

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Section: Edgesmentioning

confidence: 99%

“…into Eqs. (14) and (15), and then their summation, i.e., w (x, y) = w 1 (x, y) + w 2 (x, y) (20) For the sake of convenience, the same number of four sets of constants N is taken in the calculation; that is, m, n = 1, 2, 3, . .…”

Section: Symplectic Superposition For Analytic Free Vibration Solutiomentioning

confidence: 99%

Analytic solutions for the free vibration of rectangular thin plates with two adjacent corners point-supported

Wang

2015

Arch Appl Mech

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“…During the past two decades, the symplectic method has been introduced into various fields of engineering applications successfully. In elasticity, a great amount of symplectic techniques can be found in literature including the work on free vibration, thin plate bending, and circular cylindrical problems [9][10][11]. Moreover, the symplectic elasticity approach has been recently extended to problems with complex material properties and structures.…”

Section: Introductionmentioning

confidence: 99%

The Symplectic Method in Polar Coordinates for Linear Viscoelastic Materials

Zhang¹,

Bai²

2015

TOMSJ

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Abstract:In this study, the symplectic method is applied to a two-dimensional annular-sector viscoelastic domain under the polar coordinate system. By applying variable separation approach, all fundamental solutions are derived in analytical form. Furthermore, using the method of variable substitution, lateral conditions are transformed into finding a particular solution for the governing equations, and this particular solution is derived with the use of eigensolution expansion. In the numerical example, the boundary condition problem is discussed in detail to analyze the stress response of viscoelastic solids.

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“…Lim et al [45] gave analytic solutions for bending of a rectangular thin plate supported only at its four corners. Lim et al [46] obtained exact frequency equations for Lévy-type thin plates. Zhong & Li [47] performed exact bending analysis for fully clamped rectangular thin plates subjected to arbitrary loads.…”

Section: Introductionmentioning

confidence: 99%

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

Zhong

2013

Proc. R. Soc. A.

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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.

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On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates

Cited by 140 publications

References 15 publications

Analytic solutions for the free vibration of rectangular thin plates with two adjacent corners point-supported

Analytic solutions for the free vibration of rectangular thin plates with two adjacent corners point-supported

The Symplectic Method in Polar Coordinates for Linear Viscoelastic Materials

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

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