Exact solutions for the free in-plane vibration of rectangular plates with two opposite edges simply supported

Gorman, Daniel

doi:10.1016/j.jsv.2005.10.023

Cited by 67 publications

(33 citation statements)

References 4 publications

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“…Park [10] derived the exact frequency equations for the FIV of the clamped circular plate by using the separation of the variables. Gorman [11] obtained the exact solutions for the FIV of rectangular plates with two opposite edges simply supported, the other opposite edges being both clamped or both free. In Gorman's elegant work, only one quarter of the rectangular plate was analyzed, and it was shown that by this approach, the interpretation of the computed mode shapes with mode family separation becomes a much more manageable task, the probability of missing an eigenvalue can be greatly reduced, and the problem of repeated eigenvalues can be avoided.…”

Section: Introductionmentioning

confidence: 99%

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Exact solutions for the free in-plane vibrations of rectangular plates

Xing

Liu

2009

International Journal of Mechanical Sciences

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Section: Introductionmentioning

confidence: 99%

“…conditions including clamped condition, free condition and two distinct types of simple support boundary conditions that are denoted by the symbols SS1 and SS2 [11].…”

Section: Introductionmentioning

confidence: 99%

Exact solutions for the free in-plane vibrations of rectangular plates

Xing

Liu

2009

International Journal of Mechanical Sciences

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“…(48) is accomplished in an analytical manner by solving a number theory problem. It is well-known that the exact solution for the natural frequency of an all-round simply supported (S 2 ) element is available [17,19,43]. The natural frequency ωmn for the case can be expressed analytically in the following nondimensionalised form where η = a/b andm andn are the number of half-waves in the x and y direction respectively.…”

Section: The Wittrick-williams Algorithm Enhancement and Mode Shape Cmentioning

confidence: 99%

A spectral dynamic stiffness method for free vibration analysis of plane elastodynamic problems

Xiang

Banerjee

2017

Mechanical Systems and Signal Processing

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Citation: Liu, X. and Banerjee, J. R. (2017). A spectral dynamic stiffness method for free vibration analysis of plane elastodynamic problems. Mechanical Systems and Signal Processing, 87, doi: 10.1016/j.ymssp.2016.10.017 This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent AbstractA highly efficient and accurate analytical spectral dynamic stiffness (SDS) method for modal analysis of plane elastodynamic problems based on both plane stress and plane strain assumptions is presented in this paper. First, the general solution satisfying the governing differential equation exactly is derived by applying two types of one-dimensional modified Fourier series. Then the SDS matrix for an element is formulated symbolically using the exact general solution. The SDS matrices are assembled directly in a similar way to that of the finite element method, demonstrating the method's capability to model complex structures. Any arbitrary boundary conditions are represented accurately in the form of the modified Fourier series. The Wittrick-Williams algorithm is then used as the solution technique where the mode count problem (J 0 ) of a fully-clamped element is resolved. The proposed method gives highly accurate solutions with remarkable computational efficiency, covering low, medium and high frequency ranges. The method is applied to both plane stress and plane strain problems with simple as well as complex geometries. All results from the theory in this paper are accurate up to the last figures quoted to serve as benchmarks.

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“…(9) and (10) above are obtained, the classical method to carry out an exact buckling analysis of a plate consists of (i) solving the system of differential equations in Navier or Lèvy-type closed form in an exact manner, (ii) applying particular boundary conditions on the edges and finally (iii) obtaining the stability equation by eliminating the integration constants [41,42,43,44]. This method, although extremely useful for analysing an individual plate, it lacks generality and cannot be easily applied to complex structures assembled from plates for which researchers usually resort to approximate methods such as the FEM.…”

Section: Dynamic Stiffness Formulationmentioning

confidence: 99%

Buckling of composite plate assemblies using higher order shear deformation theory—An exact method of solution

Fazzolari

Banerjee

Boscolo

2013

Thin-Walled Structures

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Citation: Fazzolari, F. A., Banerjee, J. R. and Boscolo, M. (2013). Buckling of composite plate assemblies using higher order shear deformation theory-An exact method of solution. Thin-Walled Structures, 71, pp. 18-34. doi: 10.1016/j.tws.2013 This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent AbstractAn exact dynamic stiffness element based on higher order shear deformation theory and extensive use of symbolic algebra is developed for the first time to carry out a buckling analysis of composite plate assemblies. The principle of minimum potential energy is applied to derive the governing differential equations and natural boundary conditions. Then by imposing the geometric boundary conditions in algebraic form the dynamic stiffness matrix, which includes contributions from both stiffness and initial pre-stress terms, is developed. The Wittrick-Williams algorithm is used as solution technique to compute the critical buckling loads and mode shapes for a range of laminated composite plates including stiffened plates. The effects of significant parameters such as thickness-to-length ratio, orthotropy ratio, number of layers, lay-up and stacking sequence and boundary conditions on the critical buckling loads and mode shapes are investigated. The accuracy of the method is demonstrated by comparing results whenever possible with those available in the literature.

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Exact solutions for the free in-plane vibration of rectangular plates with two opposite edges simply supported

Cited by 67 publications

References 4 publications

Exact solutions for the free in-plane vibrations of rectangular plates

Exact solutions for the free in-plane vibrations of rectangular plates

A spectral dynamic stiffness method for free vibration analysis of plane elastodynamic problems

Buckling of composite plate assemblies using higher order shear deformation theory—An exact method of solution

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