Natural Frequencies of Orthotropic Rectangular Plates Obtained by Iterative Reduction of the Partial Differential Equation

Sakata, Toshiyuki; Takahashi, Kazuhiro; Bhat, Rama

doi:10.1006/jsvi.1996.9999

Cited by 37 publications

(16 citation statements)

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“…In Table IV, comparison studies are carried out for rectangular plates with four combinations of simply supported, clamped and free edges. It is shown that DQFEM solutions agree very closely with the Kantorovich solutions [88] for CSCS and FSFS plates, with the double trigonometric series solution [89] for CCCC plate, even to all available significant digits, and with FEM results for FFFF plate to four significant digits, this implies that the DQFEM precision is independent of boundary conditions. Table V presents comparison studies of five triangular plates with five combinations of simply supported, clamped and free edges, and one corner-supported triangular plate.…”

Section: Numerical Comparisonsmentioning

confidence: 49%

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

Xing

Liu

2009

Numerical Meth Engineering

134

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SUMMARYBased on the differential quadrature (DQ) rule, the Gauss Lobatto quadrature rule and the variational principle, a DQ finite element method (DQFEM) is proposed for the free vibration analysis of thin plates. The DQFEM is a highly accurate and rapidly converging approach, and is distinct from the differential quadrature element method (DQEM) and the quadrature element method (QEM) by employing the function values themselves in the trial function for the title problem.The DQFEM, without using shape functions, essentially combines the high accuracy of the differential quadrature method (DQM) with the generality of the standard finite element formulation, and has superior accuracy to the standard FEM and FDM, and superior efficiency to the p-version FEM and QEM in calculating the stiffness and mass matrices.By incorporating the reformulated DQ rules for general curvilinear quadrilaterals domains into the DQFEM, a curvilinear quadrilateral DQ finite plate element is also proposed. The inter-element compatibility conditions as well as multiple boundary conditions can be implemented, simply and conveniently as in FEM, through modifying the nodal parameters when required at boundary grid points using the DQ rules. Thus, the DQFEM is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter-element compatibilities. A series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the DQFEM.

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Section: Numerical Comparisonsmentioning

confidence: 49%

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

Xing

Liu

2009

Numerical Meth Engineering

134

View full text Add to dashboard Cite

show abstract

“…There is an analogy between the current S-DSM and the boundary element and the spectral methods, in that the size of Table 3 Dimensionless natural frequency parameters 4xa 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qh=D11 p for CCCC orthotropic plates with different aspect ratios and orthotropy ratios. The results computed by the current S-DSM are compared with published results by other methods [13][14][15]. the DS matrix is significantly reduced.…”

Section: Analysis Within Medium To High Frequency Ranges and Some Obsmentioning

confidence: 90%

An exact spectral-dynamic stiffness method for free flexural vibration analysis of orthotropic composite plate assemblies – Part II: Applications

Xiang

Banerjee

2015

Composite Structures

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a b s t r a c tAn exact spectral-dynamic stiffness method (S-DSM) for free vibration analysis of composite plates and plate assemblies has been proposed in Part I of this two-part paper. The main purpose of this Part II paper is twofold: (i) To validate and demonstrate the superiority of the proposed S-DSM and (ii) To establish exact benchmark solutions for free vibration of composite plate-like structures. The S-DSM is applied to a number of problems covering orthotropic composite plates and plate assemblies. It is demonstrated that the S-DSM gives exact solutions with high computational efficiency within low as well as high frequency ranges. The applications are completely general and the new development can handle complex plate shapes with any boundary conditions.

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“…These include the Rayleigh-Ritz method [22][23][24][25], Fourier series-based analytical method [26][27][28][29] and superposition method [30][31][32] amongst others [33][34][35][36]. The Rayleigh-Ritz method has been frequently used [22][23][24][25] because of its flexibility and conceptual simplicity [37].…”

Section: Mathematical Basis Variation Based Differentiation Based (Stmentioning

confidence: 99%

An exact spectral-dynamic stiffness method for free flexural vibration analysis of orthotropic composite plate assemblies – Part I: Theory

Xiang

Banerjee

2015

Composite Structures

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a b s t r a c tA spectral-dynamic stiffness method (S-DSM) for exact free vibration analysis of general orthotropic composite plate-like structures is presented in the paper. The method combines the advantages of the classical dynamic stiffness method (DSM) with those of the spectral method, and it resembles the finite element and the boundary element methods. The formulation is based on the exact general solution of the governing differential equation, which provides complete flexibility to describe any arbitrary boundary conditions. The dynamic stiffness formulation is essentially accomplished through a mixed-variable approach in a symbolic form with explicit expressions rendering physical meanings. Then a systematic procedure for plate assemblies under arbitrary boundary constraints is described. Finally, a set of novel techniques are proposed to enhance the Wittrick-Williams algorithm by resolving the fully-clamped plate problem. The validation of the theory and its applications to a wide range of engineering structures are demonstrated in Part II of this two-part paper.

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Natural Frequencies of Orthotropic Rectangular Plates Obtained by Iterative Reduction of the Partial Differential Equation

Cited by 37 publications

References 0 publications

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

An exact spectral-dynamic stiffness method for free flexural vibration analysis of orthotropic composite plate assemblies – Part II: Applications

An exact spectral-dynamic stiffness method for free flexural vibration analysis of orthotropic composite plate assemblies – Part I: Theory

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