• Steel and Composite Structures
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• 제29권5호
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• pp.647-657
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• 2018

Geometric imperfections may be created during the production process or setting borders of single-layer graphene sheets (SLGSs). Vacancy defects are an instance of geometric imperfection, so investigating the effect of these vacancies on the mechanical properties of single-layer graphene is extremely important. Since very few studies have been conducted on the structure of imperfect graphene (with the vacancy defect) as an anisotropic structure, further study of this defective structure seems imperative. Due to the vacancy defects and for the proper assessment of mechanical properties, the graphene structure should be considered anisotropic in certain states. The present study investigates the effects of site and size of vacancy defects on the mechanical properties of graphene as an anisotropic structure using the lekhnitskii interaction coefficients and Molecular Dynamic approach. The effect of temperature on the severity of the SLGS becoming anisotropic is also investigated in this study. The results reveal that the amount of temperature has a big effect on the severity of the structure getting anisotropic even for a graphene without any defects. The effect of aspect ratio, temperature and also size and site of vacancy defects on the material properties of the graphene are studied in this research work. According to the present study, using material properties of flawless graphene for imperfect structure can lead to inaccurate results.

• 대한토목학회논문집
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• 제32권6A호
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• pp.327-336
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• 2012

본 논문은 완성계 강사장교의 기하학적 특성이 극한 거동에 미치는 영향을 분석한다. 사장교는 구조형식의 특성에 따라 매우 효율적인 구조체로 알려져 있지만, 이러한 구조 특성에 따라 구조물의 극한 거동에 영향을 미치는 다양한 비선형성과 함께 복잡한 구조거동을 보인다. 본 연구에서는 거더 및 주탑의 단면 크기, 케이블 배치 형식 및 케이블 단면적 변화에 따른 완성계 강사장교의 극한 거동에 대해 다룬다. 선행연구를 통해 도출된 극한거동에 지배적인 활하중에 대해 각 인자의 변화에 대한 매개변수연구를 수행하였다. 활하중에 대한 완성계 사장교의 합리적인 해석을 위해 초기형상해석-활하중해석을 거치는 2단계 해석법을 통해 극한 해석을 수행하였다. 해석에 고려된 사장교 모델은 총 920.0 m의 지간장을 갖는 강사장교이고 케이블 배치각도에 따른 거동분석을 위해 방사형 사장교와 팬 형 사장교 모델을 이용하였다. 본 해석 연구를 통해 각 기하학적 특성 인자 변화에 따른 완성계 강사장교의 극한거동 변화 특성을 도출하였다.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 1994년도 Proceedings of the Korea Automatic Control Conference, 9th (KACC) ; Taejeon, Korea; 17-20 Oct. 1994
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• pp.628-633
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• 1994

A new Riemannian geometric model for the controlled plant is proposed by imbedding the control vector space in the state space, so as to reduce the dimension of the model. This geometric model is derived by replacing the orthogonal straight coordinate axes on the state space of a linear system with the curvilinear coordinate axes. Therefore the integral manifold of the geometric model becomes homeomorphic to that of fictitious linear system. For the lower dimensional Riemannian geometric model, a nonlinear optimal regulator with a quadratic form performance index which contains the Riemannian metric tensor is designed. Since the integral manifold of the nonlinear regulator is determined to be homeomorphic to that of the linear regulator, it is expected that the basic properties of the linear regulator such as feedback structure, stability and robustness are to be reflected in those of the nonlinear regulator. To apply the above regulator theory to a real nonlinear plant, it is discussed how to distort the curvilinear coordinate axes on which a nonlinear plant behaves as a linear system. Consequently, a partial differential equation with respect to the homeomorphism is derived. Finally, the computational algorithm for the nonlinear optimal regulator is discussed and a numerical example is shown.

• Steel and Composite Structures
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• 제45권5호
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• pp.621-640
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• 2022

In the present article, the vibration response of a geometrically imperfect bi-directional functionally graded plate (2D-FGP) with geometric discontinuities and micro-structural defects (porosities) has been investigated. A porosity model has been developed to incorporate the effective material properties of the bi-directional FGP which varies in two directions i.e. along the axial and transverse direction. The geometric discontinuity is also introduced in the plate in the form of a circular cut-out at the center of the plate. The structural kinematic formulation is based on the non-polynomial trigonometric higher-order shear deformation theory (HSDT). Finite element formulation is done using C° continuous Lagrangian quadrilateral four-noded element with seven degrees of freedom per node. The equations of motion have been derived using a variational approach. Convergence and validation studies have been documented to confirm the accuracy and efficiency of the present formulation. A detailed investigation study has been done to evaluate the influence of the circular cut-out, geometric imperfection, porosity inclusions, partial supports, volume fraction indexes (along with the thickness and length), and geometrical configurations on the vibration response of 2D-FGP. It is concluded that after a particular cut-out dimension, the vibration response of the 2D FGP exhibits non-monotonic behavior.

• 대한수학회논문집
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• 제36권4호
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• pp.801-815
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• 2021

Ellipses have a lot of interesting geometric properties. It is quite natural to ask whether such properties of ellipses and some related ones characterize ellipses. In this paper, we study some chord properties and area properties of ellipses. As a result, using the curvature and the support function of a strictly convex curve, we establish four characterization theorems of ellipses and hyperbolas centered at the origin.

• 대한수학교육학회지:수학교육학연구
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• 제8권1호
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• pp.291-298
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• 1998

This study deals with the problem of proof education in the middle school geometry bby examining van Hiele#s geometric thought level theory and Freudenthal#s mathematization teaching theory. The implications that have been revealed by examining the theory of van Hie이 and Freudenthal are as follows. First of all, the proof education at present that follows the order of #definition-theorem-proof#should be reconsidered. This order of proof-teaching may have the danger that fix the proof education poorly and formally by imposing the ready-made mathematics as the mere record of proof on students rather than suggesting the proof as the real thought activity. Hence we should encourage students in reinventing #proving#as the means of organization and mathematization. Second, proof-learning can not start by introducing the term of proof only. We should recognize proof-learning as a gradual process which forms with understanding the meaning of proof on the basic of the various activities, such as observation of geometric figures, analysis of the properties of geometric figures and construction of the relationship among those properties. Moreover students should be given this natural ground of proof.

• 대한수학교육학회지:학교수학
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• 제10권1호
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• pp.43-61
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• 2008

현 교육과정에서 이차함수 그래프에 관한 교수-학습은 대수적 조작에 의한 완전제곱형식으로의 변환을 강조하고 이미 선행한 일차함수의 그래프와는 무관하게 다루어지고 있다. 본 논문은 이차함수 그래프의 구조적 특성이라 할 수 있는 대칭성, 꼭지점 및 합동에 대한 기하적인 근거를 일차함수의 그래프에 기초하여 분석하고 이것의 결과를 구체적 이차함수에 대해 그 꼭지점의 좌표 및 절편을 찾는 데 적용한다. 본 연구는 이차함수 그래프의 구조적 특성을 일차함수의 그래프와 기하적으로 연결시키는 데 의의가 있으며 그 기하적인 과정은 완전제곱형식의 대수적 절차로 연결된다. 이러한 연결은 일차함수의 그래프에 대한 이해가 이차함수의 그래프에 대한 이해로 전이되어 이차함수에 대한 기하적인 이해를 넓히는 교수-학습방법이 될 수 있을 것이다.

• 한국수학사학회지
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• 제21권4호
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• pp.169-182
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• 2008

칠교판을 관찰하고 사용하는 경험을 통해 칠교판 조각들 사이의 길이, 각도, 모양, 넓이 등과 같은 기학학적인 특성을 파악하고, 이를 이용하여 칠교판을 활용한 활동에서 조각들을 유의미하게 분류하고, 조각의 사용에 대한 조합 등을 구하여 체계적으로 문제를 해결할 수 있다 이러한 과정을 학생들이 직접 경험할 수 있도록 구체적인 발문 형태의 문제로 제공함으로써 칠교판을 학교수학에서 효율적으로 활용하기 위한 기초 자료로 제시할 수 있다. 이는 수학자가 새로운 정리를 발견하듯이, 소박하고 직관적인 상태에서 도형들의 특징을 파악하고, 학생들 수준에 맞는 활동을 통해 도형과 도형 사이의 관계를 유추하여 주어진 문제의 해답을 시행착오에 의존하는 것이 아니라 논리적으로 추론하여 체계적으로 해답을 찾는 경험을 제공하는 과학적인 지도 방법이다.

• Steel and Composite Structures
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• 제44권2호
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• pp.155-169
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• 2022

In the present paper an analytical model was developed to study the non-linear vibrations of Functionally Graded Carbon Nanotube (FG-CNT) reinforced doubly-curved shallow shells using the Multiple Scales Method (MSM). The nonlinear partial differential equations of motion are based on the FGM shallow shell hypothesis, the non-linear geometric Von-Karman relationships, and the Galerkin method to reduce the partial differential equations associated with simply supported boundary conditions. The novelty of the present model is the simultaneous prediction of the natural frequencies and their mode shapes versus different curvatures (cylindrical, spherical, conical, and plate) and the different types of FG-CNTs. In addition to combining the vibration analysis with optimization algorithms based on the genetic algorithm, a design optimization methode was developed to maximize the natural frequencies. By considering the expression of the non-dimensional frequency as an objective optimization function, a genetic algorithm program was developed by valuing the mechanical properties, the geometric properties and the FG-CNT configuration of shallow double curvature shells. The results obtained show that the curvature, the volume fraction and the types of NTC distribution have considerable effects on the variation of the Dimensionless Fundamental Linear Frequency (DFLF). The frequency response of the shallow shells of the FG-CNTRC showed two types of nonlinear hardening and softening which are strongly influenced by the change in the fundamental vibration mode. In GA optimization, the mechanical properties and geometric properties in the transverse direction, the volume fraction, and types of distribution of CNTs have a considerable effect on the fundamental frequencies of shallow double-curvature shells. Where the difference between optimized and not optimized DFLF can reach 13.26%.

• 한국수학사학회지
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• 제17권1호
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• pp.87-96
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• 2004

This article dealt with two approaches, algebraic and geometric approaches in terms of Pythagoreans theorem. As mathematics evolves, many theorems had been developed beginning with geometric approaches. However, the algebraic techniques that survive these days are so powerful and generalized in school curriculum. So, if students have more chances to see mathematical properties in geometrical ways, they can experience how beautiful and meaningful they are through the process of the advent of them. Also, it was to try to develop an insight into their applications to other problems.