An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges

Du, Jingtao; Li, Wen L.; Jin, Guoyong; Yang, Tiejun; Liu, Zhigang

doi:10.1016/j.jsv.2007.06.011

Cited by 129 publications

(98 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The notations k (corresponding to x axis) and j (corresponding to y axis) of (k, j) denote symmetric (taking '0') or antisymmetric (taking '1') deformation with respect to the corresponding axes. [18] c Fourier series based analytical method [32] d Ritz method [23] Similar comparison can be made for the free inplane vibration of completely free isotropic plates (ν = 0.3) between the current SDSM results and those available in the literature [18,23,31,32]. All SDSM results have six significant digit accuracy which are more accurate than those available in the literature (the latter methods give no more than three or four significant figures).…”

Section: Applications To Plane Stress Problemsmentioning

confidence: 52%

“…parently, these zero frequencies were missed by other investigators [18,31,32]. The first non-rigid body inplane mode shapes of a square isotropic plate is shown in Fig.…”

Section: Applications To Plane Stress Problemsmentioning

confidence: 92%

“…Table 3 compares the first eight natural frequencies computed by the SDSM results with those obtained by other methods wherever available. The comparative methods include dynamic stiffness method based on the superposition method (DSM) [31], Gorman's superposition method (GSM) [18], Fourier series based analytical method (FSA) [32] and the Ritz method [23]. The current SDSM results, which are accurate up to the last figures quoted with six significant figures, are intended to serve as benchmark solutions.…”

Section: Applications To Plane Stress Problemsmentioning

confidence: 99%

“…The (k, j) notation is the same as that given in Table 3. a Kantorovich method [29] b Forced response analysis 1 [28] c Forced response analysis 2 [27] d finite element solution using NASTRAN [28] e Fourier series based analytical method [32] Figure 4: The first eight non-zero free inplane vibration mode shapes of a completely free square (a/b = 1) isotropic plate. The (k, j) notation and the meaning of colour of mesh is the same as those in Table 3.…”

Section: Applications To Plane Stress Problemsmentioning

confidence: 99%

“…Nefovska-Danilovic et al [31] developed the dynamic stiffness method for isotropic rectangular plates based on Gorman's superposition method. Du et al [32,33] used a Fourier series based analytical method to examine the free inplane vibration of plates with different boundary conditions. More recently, Papkov [34] provided the lower and upper bounds of natural frequencies for the free inplane vibration of completely free and fully clamped plates by an analytical method which makes use of the asymptotic behaviour of quasi-regular infinite systems.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Xiang

Banerjee

2017

Mechanical Systems and Signal Processing

View full text Add to dashboard Cite

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.

show abstract

Section: Applications To Plane Stress Problemsmentioning

confidence: 52%

“…parently, these zero frequencies were missed by other investigators [18,31,32]. The first non-rigid body inplane mode shapes of a square isotropic plate is shown in Fig.…”

Section: Applications To Plane Stress Problemsmentioning

confidence: 92%

Section: Applications To Plane Stress Problemsmentioning

confidence: 99%

Section: Applications To Plane Stress Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Xiang

Banerjee

2017

Mechanical Systems and Signal Processing

View full text Add to dashboard Cite

show abstract

3D Free Vibration Analysis of Functionally Graded Plates with Arbitrary Boundary Conditions in Thermal Environment

et al. 2021

View full text Add to dashboard Cite

An analytical method for free vibration of functionally graded (FG) plates with arbitrary boundary conditions in thermal environments is developed based on 3D elasticity theory. Material properties of the FG plate are assumed to be temperature dependent but graded along with its thickness. At the edges of the plate, three different sets of linear springs are introduced, so that its boundary condition can be altered by changing the spring stiffness. Each displacement component of the plate is expanded as a standard Fourier cosine series, supplemented with closed‐form auxiliary functions. With thermal environment effects duly accounted for, the energy method is used to derive eigenvalue equations, whereas the Rayleigh−Ritz procedure is utilized to calculate the natural frequencies of the FG plate. Numerical examples show that the present method converges quickly and gives satisfactory results. The method is then utilized to calculate the natural frequencies of FG plates having different aspect ratios, volume fractions, and elastic boundaries.

show abstract

On Fourier series for higher order (partial) derivatives of functions

Sun

Zhang

2016

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term‐by‐term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto‐dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2rth order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges

Cited by 129 publications

References 10 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

3D Free Vibration Analysis of Functionally Graded Plates with Arbitrary Boundary Conditions in Thermal Environment

On Fourier series for higher order (partial) derivatives of functions

Contact Info

Product

Resources

About