• 응용통계연구
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• 제24권5호
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• pp.963-977
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• 2011

다변량 자료를 분석함에 있어 자료의 차원을 축소하는데 활용되는 중요한 툴 중 하나인 PCA 분석(주성분 분석, Principal Component Analysis)을 실시간으로 처리해야 하는 적용 분야가 최근 늘고 있다. PCA 분석에서는 표본 공분산 행렬의 고유값과 고유벡터를 도출하는 것이 관건인데, 자료의 양이 방대하며 고차원인 경우 이를 실시간으로 수행하기에는 어려움이 따른다. 이러한 문제점을 해결하기 위해서 Erdogmus 등 (2004)는 일차 섭동 이론(first order perturbation theory)을 활용하여 공분산 행렬의 고유값과 고유벡터를 추정하는 Recursive PCA 방법을 제안했다. 이 방법은 추가된 자료의 양이 많지 않은 경우는 상당히 정확하지만, 추가된 자료의 양이 많아짐에 따라 오차도 커진다는 한계를 가지고 있다. 본 논문은 공분산 행렬의 고유값과 고유벡터가 가지고 있는 수학적 관계를 이용하여 Erdogmus 등 (2004)가 제안한 Recursive PCA 방법을 수정한 Modi ed Recursive PCA 방법을 제안하다. 또한, 모의 실험을 통해 Recursive PCA 방법과 Modi ed Recursive PCA 방법에서의 고유값과 고유벡터 추정값의 정확도를 비교해 보았으며 그 결과 기존 Recursive PCA 방법 보다 정확한 추정이 가능함을 확인할 수 있었다.

• 대한기계학회논문집A
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• 제26권11호
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• pp.2270-2276
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• 2002

Dynamic stability of a rotary oscillating cantilever beam is presented in this study. Using the stretch deformation instead of the conventional axial deformation, three linear partial differential equations are derived from Hamilton's principle and transformed into dimensionless forms. Stability diagrams of the first order approximate solutions are obtained by using the multiple scale perturbation method. The stability diagrams show that relatively large unstable regions exist near the combination of the first chordwise bending natural frequency and the first stretch natural frequency. This result is verified by using the generalized -$\alpha$ method.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2001년도 추계학술대회논문집 I
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• pp.335-340
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• 2001

Dynamic stability of a rotary oscillating cantilever beam is presented in this study. Using the stretch deformation instead of the conventional axial deformation, three linear partial differential equations are derived from Hamilton's principle and transformed into dimensionless forms. Stability diagrams of the first order approximate solutions are obtained by using the multiple scale perturbation method. The stability diagrams show that relatively large unstable regions exist near the combination of the first chordwise bending natural frequency and the first stretch natural frequency. This result is verified by using the generalized-${\alpha}$ method.

• Journal of applied mathematics & informatics
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• 제27권1_2호
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• pp.441-452
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• 2009

In this paper, a fifth order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been using for delay. Similar boundary value problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, first we use Taylor approximation to tackle terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system and is solved using the boundary conditions. Several numerical examples are solved and compared with exact solution. It is observed that present method approximates the exact solution very well.

• Structural Engineering and Mechanics
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• 제20권3호
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• pp.293-312
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• 2005

In this paper, free vibration of rotating beam with random properties is studied. The cross-sectional area, elasticity modulus, moment of inertia, shear modulus and density are modeled as random fields and the rotational speed as a random variable. To study uncertainty, stochastic finite element method based on second order perturbation method is applied. To discretize random fields, the three methods of midpoint, interpolation and local average are applied and compared. The effects of rotational speed, setting angle, random property variances, discretization scheme, number of elements, correlation of random fields, correlation function form and correlation length on "Coefficient of Variation" (C.O.V.) of first mode eigenvalue are investigated completely. To determine the significant random properties on the variation of first mode eigenvalue the sensitivity analysis is performed. The results are studied for both Timoshenko and Bernoulli-Euler rotating beam. It is shown that the C.O.V. of first mode eigenvalue of Timoshenko and Bernoulli-Euler rotating beams are approximately identical. Also, compared to uncorrelated random fields, the correlated case has larger C.O.V. value. Another important result is, where correlation length is small, the convergence rate is lower and more number of elements are necessary for convergence of final response.

• Advances in aircraft and spacecraft science
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• 제3권4호
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• pp.471-502
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• 2016

This paper presents the dynamic instability analysis of un-damped elastically supported piezoelectric functionally graded (FG) beams subjected to in-plane static and dynamic periodic thermomechanical loadings with uncertain system properties. The elastic foundation model is assumed as one parameter Pasternak foundation with Winkler cubic nonlinearity. The piezoelectric FG beam is subjected to non-uniform temperature distribution with temperature dependent material properties. The Young's modulus and Poison's ratio of ceramic, metal and piezoelectric, density of respective ceramic and metal, volume fraction exponent and foundation parameters are taken as uncertain system properties. The basic nonlinear formulation of the beam is based on higher order shear deformation theory (HSDT) with von-Karman strain kinematics. The governing deterministic static and dynamic random instability equation and regions is solved by Bolotin's approach with Newmark's time integration method combined with first order perturbation technique (FOPT). Typical numerical results in terms of the mean and standard deviation of dynamic instability analysis are presented to examine the effect of slenderness ratios, volume fraction exponents, foundation parameters, amplitude ratios, temperature increments and position of piezoelectric layers by changing the random system properties. The correctness of the present stochastic model is examined by comparing the results with direct Monte Caro simulation (MCS).

• Structural Engineering and Mechanics
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• 제16권6호
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• pp.731-748
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• 2003

In this paper a thick cylindrical shell with varying thickness which is subjected to static non-uniform internal pressure is analyzed. At first, equilibrium equations of the shell have been derived by the energy principle and by considering the first order theory of Mirsky-Herrmann which includes transverse shear deformation. Then the governing equations which are, a system of differential equations with varying coefficients have been solved analytically with the boundary layer technique of the perturbation theory. In spite of complexity of modeling the conditions near the boundaries, the method of this paper is very capable of providing a closed form solution even near the boundaries. Displacement predictions are in a good agreement with the calculated finite elements and other analytical results. The convergence of solution is very fast and the amount of calculations is less than the Frobenius method.

• Journal of applied mathematics & informatics
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• 제28권1_2호
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• pp.411-423
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• 2010

In this paper, a numerical method for singular perturbation problems arising in chemical reactor theory for general singularly perturbed two point boundary value problems with boundary layer at one end(left or right) of the underlying interval is presented. The original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument. By using the trapezoidal formula we obtain a three term recurrence relation, which is solved using Thomas Algorithm. To demonstrate the applicability of the method, we have solved four linear (two left and two right end boundary layer) and one nonlinear problems. From the results, it is observed that the present method approximates the exact or the asymptotic expansion solution very well.

• 전산구조공학
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• 제7권3호
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• pp.115-122
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• 1994

불확정 구조계획의 선형 고유치 문제는 재료정수나 경게조건 및 외부하중 등에 결정론적으로 사용할 수 없는 확률량을 포함하고 있다. 변동량을 내포한 고유치 문제의 해석은 기대치에 대한 지배 방정식과 변동량 결정방정식을 고려해야 한다. 비선형성이 심한 구조계를 선형화할 대 일차 및 이차 변동값을 반영함으로 고유치의 정도를 향상시킬 수 있다. 매개변수에 불확정성을 포함한 고유치 문제는 최적설계 정식화에서 변동된 값을 고료해 줌으로 신뢰성 있는 설계가 된다. 최적설계 알고리즘 중에는 목적함수와 제한 조건식의 설계 민감도를 요한다. 이차 기울기에 근거를 둔 최적설계 수행시에 변동량에 고려하여 제한식으로 설정하고, 설계 민감도를 구할 수 있는 방법을 제시하였다.

• 대한조선학회논문집
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• 제58권6호
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• pp.348-357
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• 2021

In this paper, the generalized hydrodynamic force and response of a flexible body are calculated at different reference coordinate systems. We generalize the equation of motion for a flexible body by using the conservation of momentum (Mei et al., 2005). To obtain the equations in the generalized mode, two different reference coordinates are adopted. The first is the body-fixed coordinate system by a rigid body motion. The other is the inertial coordinate system which has been adopted for the analysis. Using the perturbation scheme in the weakly-nonlinear assumption, the equations of motion are expanded up to second-order quantities and several second-order forces are obtained. Numerical tests are conducted for the flexible barge model in head waves and the vertical bending is only considered in the hydroelastic responses. The results show that the linear response does not have the difference between the two formulations. On the other hand, second-order quantities have different values for which the rigid body motion is relatively large. However, the total summation of second-order quantities has not shown a large difference at each reference coordinate system.