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

Xing, Yufeng; Liu, B.

doi:10.1016/j.ijmecsci.2008.12.009

Cited by 87 publications

(40 citation statements)

References 11 publications

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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%

“…Much later, Gorman [18] carried out a thorough investigation for exact solutions of simply supported plates by using Levy-type solutions. Xing and Liu [19][20][21] provided closed-form exact solutions for all possible cases of simply supported plates by using the Rayleigh quotient method. The classical dynamic stiffness method [22][23][24][25], first developed for plates in the 70s [22], can be applied to plate assemblies but restricted to cases with two opposite plate edges simply supported.…”

Section: Introductionmentioning

confidence: 99%

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

confidence: 99%

Section: Introductionmentioning

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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“…(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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“…Their energy can be generated by various kinds of in-plane forces including the tangential stresses on the flow structural interface, 5 and when reaching the structural material and geometric discontinuities, it can also be transformed into the energy of flexural waves which are directly responsible for structural sound radiation. 6 Many of the recent studies [7][8][9][10][11][12][13][14][15][16][17][18][19] on the free and forced responses of in-plane wave in finite plates have been motivated by this recognition. Although brief reviews of the progress in this area over the last two decades have been given by Dozio, 16 and by Liu and Xing,17 of particular interest is the series of papers by Gorman 5,10,12,13,15 on the semianalytical solution of the free in-plane vibration in finite plates, and by Xing and Liu [17][18][19] on exact solution for inplane vibrations of plates.…”

Section: Introductionmentioning

confidence: 99%

Waveguide characteristics of coupled in-plane waves

Lü

Qiu

2012

The Journal of the Acoustical Society of America

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In-plane waves in a waveguide made from a thin plate are described by a superposition of a set of orthogonal functions that satisfy the edge conditions of the waveguide. Due to the Poisson and shear effects, the displacement components of the in-plane waves along the two in-plane orthogonal coordinates are coupled and this coupling affects the propagation and spatial properties of the waveguide modes. The orthogonal functions and their associated wavenumbers represent the characteristics of the uncoupled modes of the waveguide where the above mentioned couplings are ignored. This study demonstrates that the characteristics of the waveguide modes are determined by the couplings of the uncoupled mode pairs, which become significant when the pairs satisfy the conditions of spatial coincidence. At some frequencies, certain waveguide modes can be determined by a single pair of uncoupled modes. For this case, the analytical solution for the waveguide modes exists and provides both a qualitative and quantitative interpretation of the characteristics of the waveguide mode.

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

Cited by 87 publications

References 11 publications

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

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

Waveguide characteristics of coupled in-plane waves

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