• 대한수학회논문집
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• 제12권3호
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• pp.645-653
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• 1997

We study centrally symmetric orthogonal polynomials satisfying an admissible partial differential equation of the form $$ Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y = \lambda_n y, $$ where $A, B, \cdots, E$ are polynomials independent of n and $\lambda_n$ is the eignevalue parameter depending on n. We show that they are either the product of Hermite polymials or the circle polynomials up to a complex linear change of variables. Also we give some properties of them.

• 대한기계학회논문집
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• 제16권8호
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• pp.1485-1492
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• 1992

본 연구에서는 Rayleigh-Ritz method를 사용하여 타원형(원형 포함) 평판의 자유 진동 문제를 해석한다. admissible함수로서는 단순 다항식의 곱(products of simple polynomials)을 사용한다. 이 함수는 다항식의 첫 항의 차수를 0,1 또는 2로 함으로써 각각 자유, 단순 지지 또는 고정인 경계 조건을 간편히 처리할 수 있게 하며, 에너지식에 포함된 적분값을 계산함에 있어서 점화식을 유도하여 사용함으로써 계산을 매우 간편히 수행할 수 있게 한다. 본문의 이론 해석은 직교 이방성 타원형 평판에 대해서 제시된다. 그러나 수치 결과는, 이방성의 다양한 조합에 대한 많은 결과를 제시하는 것은 그다지 의미가 없는 관계로, 등방성 평판에 대해서 주어진다. 등방성 평판의 여러 가지 세장비에 대한 결과는 설계 데이터로써 활용될 수 있을 것이다. 덧붙여 대표적으로 세장비가 0.5인 평판에 대해서 nodal pattern이 그림으로 제시된다.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 2003년도 봄 학술발표회 논문집
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• pp.35-42
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• 2003

This paper provides the first known flexural vibration data for thick (Mindlin) rectangular plates having V-notches. The V-notch has bending moment and shear force singularities at its sharp corner due to the transverse vibratory bending motion. Based upon Mindlin plate theory, in which transverse shear deformation and rotary inertia effects are considered, the Ritz procedure is employed with a hybrid set of admissible functions assumed for the rotational and transverse vibratory displacements. This set includes: (1) a mathematically complete set of admissible algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained; and (2) an admissible set of Mindlin corner functions which account for the bending moment and shear force singularities at the sharp corner of the V-notch. Extensive convergence studies demonstrate the necessity of adding the Mindlin corner functions to achieve accurate frequencies for rectangular plates having sharp V-notches.

• 한국콘크리트학회:학술대회논문집
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• 한국콘크리트학회 1998년도 가을 학술발표논문집(II)
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• pp.412-417
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• 1998

This paper provides accurate flexural vibration solutions for thick (Mindlin) sectorial plates. A Ritz method is employed which incorporates a complete set of admissible algebraic-trigonometric polynomials in conjunction with an admissible set of Mindlin “corner functions". These corner functions model the singular vibratory moments and shear forces, which simultaneously exist at the vertex of corner angle exceeding 180$^{\circ}$. The first set guarantees convergence to the exact frequencies as sufficient terms are taken. The second set represents the corner singularities, and accelerates convergence substantially. Numerical results are obtained for completely free sectorial plates. Accurate frequencies are presented for a wide spectrum of vertex angles (90$^{\circ}$, 180$^{\circ}$, 270$^{\circ}$, 300$^{\circ}$, 330$^{\circ}$, 350$^{\circ}$, 35 5$^{\circ}$,and 359$^{\circ}$)and thickness ratios.tios.

• 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.

• 대한수학회보
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• 제40권3호
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• pp.509-520
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• 2003

Let p be a prime number, $Q_{p}$ the field of p-adic numbers, K a finite extension of $Q_{p}$, $\bar{K}}$ a fixed algebraic closure of K and $C_{p}$ the completion of K with respect to the p-adic valuation. Let E be a closed subfield of $C_{p}$, containing K. Given elements $t_1$...,$t_{r}$ $\in$ E for which the field K($t_1$...,$t_{r}$) is dense in E, we construct integral bases of E over K.

• 대한기계학회논문집
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• 제18권8호
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• pp.1967-1973
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• 1994

The free vibration of a thin plate with three different boundary conditions is discussed in this paper. A semi-analytical approach to the plate problems has been exploited using computer algebra system(CAS). The approximate solutions are assumed as algebraic polynomials that satisfy the appropriate boundary conditions. In order to solve problems, Galerkin method is used, which is known as an ineffective tool for practical engineering problems, being involved with a large number of multiple integration and differentiation. All the admissible functions used in this paper are generated automatically by CAS otherwise a tedious algebraic manipulations should be done by hand. One, six and fifteen-term solutions in terms of frequency parameters are presented and compared with exact solutions. Even using one-term solution, the comparison with existing data shows good agreement and accuracy of the present method.

• 한국소음진동공학회논문집
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• 제14권4호
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• pp.336-343
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• 2004

This paper reports the first known free vibration data for thin rectangular plates with V-notches. The classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets include (1) mathematically complete algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained, and (2) corner functions which account for the bending moment singularities at the sharp reentrant corner of the Y-notch. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of nondirectional frequencies for rectangular plates having the V-notch. In this paper, accurate frequencies and normalized contours of vibratory transverse displacement are presented for various notched plates, so that the effect of corner stress singularities may be understood.

• Nuclear Engineering and Technology
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• 제31권2호
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• pp.214-225
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• 1999

An accurate method is presented for flexural vibrations of sectorial plates having simply supported-free and free-free radial edges, when the circular edge is either clamped or simply supported. The classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets consist of : (1) mathematically complete algebraic-trigonometric polynomials which gurantee convergence to exact frequencies as sufficient terms are retained, and (2) comer functions which account for the bending moment singularities at re-entrant comer of the radial edges having arbitrary edge conditions. Accurate (at least four significant figures) frequencies and normalized contours of the transverse vibratory displacement are presented for the spectra of corner angles [90$^{\circ}$, 180$^{\circ}$(semi-circular), 270$^{\circ}$, 300$^{\circ}$, 330$^{\circ}$, 350$^{\circ}$, 355$^{\circ}$, 360$^{\circ}$ (complete circular)] causing a re-entrant comer of the radial edges. Future solutions drawn from alternative numerical procedures and finite element techniques may be compared with these accurate results.

• 한국소음진동공학회논문집
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• 제25권3호
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• pp.207-215
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• 2015

This paper deals with a theoretical method to calculate natural frequencies of a fixed-free rectangular tank partially in contact with an outer water gap. Orthogonal polynomials satisfying the boundary conditions of the tank are used as admissible functions in the Rayleigh-Ritz method. A quarter model of the liquid-coupled system is constructed and it is simplified to a line supported flat plate in contact with the liquid. The liquid displacement potential functions satisfying the Laplace equation and water boundary conditions are derived, and the finite Fourier transform is accomplished in conjunction with the compatibility requirement along the contacting interfaces between the tank and water. An eigenvalue problem is derived so that the natural frequencies of the wet rectangular tank can be extracted. The predictions from the proposed analytical method show good agreement with the finite element analysis results.