• International Journal of Precision Engineering and Manufacturing
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• 제3권3호
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• pp.44-49
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• 2002

A study of free vibration of axisymmetric circular plates based on Mindlin theory using a pseudospectral method is presented. The analysis is based on Chebyshev polynomials that are widely used in the fluid mechanics research community. Clamped, simply supported and flee boundary conditions are considered, and numerical results are presented for various thickness-to-radius ratios.

• International Journal of Naval Architecture and Ocean Engineering
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• 제4권3호
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• pp.267-280
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• 2012

An approximate method based on an assumed mode method has been presented for the free vibration analysis of a rectangular plate with arbitrary edge constraints. In the presented method, natural frequencies and their mode shapes of the plate are calculated by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange's equations of motion. Characteristic orthogonal polynomials having the property of Timoshenko beam functions which satisfies edge constraints corresponding to those of the objective plate are used. In order to examine the accuracy of the proposed method, numerical examples of the rectangular plates with various thicknesses and edge constraints have been presented. The results have shown good agreement with those of other methods such as an analytic solution, an approximate solution, and a finite element analysis.

• Coupled systems mechanics
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• 제5권1호
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• pp.59-86
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• 2016

This paper deals with the free vibration analysis of a dynamical coupled system: flexible gravity dam- compressible rectangular reservoir. The finite element method is used to compute the natural frequencies and modal shapes of the system. Firstly, the reservoir and subsequently the dam is modeled by classical 8-node elements and the natural frequencies plus modal shapes are calculated. Afterwards, a new 21-node element is introduced and the same procedure is conducted in which an efficient method is employed to carry out the integration operations. Finally, the coupled dam-reservoir system is modeled by solely one 21-node element and the free vibration of dam-reservoir interaction system is investigated. As an important result, it is clearly concluded that the one high-order element treats more precisely than the eight-node elements, since the first one utilizes fifth-degree polynomials to construct the shape functions and the second implements polynomials of degree two.

• Structural Engineering and Mechanics
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• 제55권4호
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• pp.785-799
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• 2015

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of hyperboloidal shells free at the top edge and clamped at the bottom edge like a hyperboloidal cooling tower by the Ritz method based upon the circular cylindrical coordinate system instead of related 3-D shell coordinates which are normal and tangent to the shell midsurface. The Legendre polynomials are used as admissible displacements. Convergence to four-digit exactitude is demonstrated. Natural frequencies from the present 3-D analysis are also compared with those of straight beams with circular cross section, complete (not truncated) conical shells, and circular cylindrical shells as special cases of hyperboloidal shells from the classical beam theory, 2-D thin shell theory, and other 3-D methods.

• Kyungpook Mathematical Journal
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• 제55권4호
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• pp.807-816
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• 2015

Let $P(z)={\limits\sum_{j=0}^{n}}a_jz^j$ be a polynomial of degree n and Re $a_j={\alpha}_j$, Im $a_j=B_j$. In this paper, we have obtained a zero-free region for polynomials in terms of ${\alpha}_j$ and ${\beta}_j$ and also obtain the bound for number of zeros that can lie in a prescribed region.

• Journal of Mechanical Science and Technology
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• 제19권9호
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• pp.1753-1760
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• 2005

The pseudo spectral method is applied to the free vibration analysis of double-span Timoshenko beams. The analysis is based on the Chebyshev polynomials. Each section of the double-span beam has its own basis functions, and the continuity conditions at the intermediate support as well as the boundary conditions are treated separately as the constraints of the basis functions. Natural frequencies are provided for different thickness-to-length ratios and for different span ratios, which agree with those of Euler-Bernoulli beams when the thickness-to-length ratio is small but deviate considerably as the thickness-to-length ratio grows larger.

• Structural Engineering and Mechanics
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• 제7권1호
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• pp.53-67
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• 1999

The natural frequencies and modes of free vibration of specially orthotropic elliptic and circular plates are analysed using the Rayleigh-Ritz method. The assumed functions used are two-dimensional boundary characteristic orthogonal polynomials which are generated using the Gram-Schmidt orthogonalization procedure. The first five natural frequencies are reported here for various values of aspect ratio of the ellipse. Results are given for various boundary conditions at the edges i.e., the boundary may be any of clamped, simply-supported or fret. Numerical results are presented here for several orthotropic material properties. For rectilinear orthotropic circular plates, a few results are available in the existing literature, which are compared with the present results and are found to be in good agreement.

• 대한수학회보
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• 제57권5호
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• pp.1165-1176
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• 2020

Let M be a module over a commutative ring R. In this paper, we study Int(R, M), the module of integer-valued polynomials on M over R, and Int_M(R), the ring of integer-valued polynomials on R over M. We establish some properties of Krull dimensions of Int(R, M) and Int_M(R). We also determine when Int(R, M) and Int_M(R) are nontrivial. Among the other results, it is shown that Int(ℤ, M) is not Noetherian module over Int_M(ℤ) ∩ Int(ℤ), where M is a finitely generated ℤ-module.

• 한국소음진동공학회논문집
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• 제24권12호
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• pp.948-953
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• 2014

Many studies using the assumed mode method have been found for the free vibration analysis of stiffened plate with known elastic boundary conditions. However many local structures such as tank edges and equipment foundations consist of connected structures and it is very difficult to find suitable elastic boundary conditions. In this study combined polynomials which satisfy simply and fixedly supported boundary conditions are proposed. The proposed method has been applied to tanks which bounded by bulkhead and a deck. The results of this study shows good agreements with these obtain by the FEA S/W(Patran/Nastran).

• 충청수학회지
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• 제30권1호
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• pp.15-19
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• 2017

We introduce the $M{\ddot{o}}bius$ functions ${\mu}k$ of order k and give the asymptotic formula for the summatory function associated to theses functions in function field case.