• Transactions of the Korean Society of Mechanical Engineers A
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• v.31 no.1 s.256
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• pp.97-104
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• 2007

Vibration analysis of a rigid body supported by in-parallel linear springs can be greatly simplified by utilizing the conditions for a plane of symmetry. The vibration modes of an oscillatory system having plane of symmetry are classified into the in-plane and out-of-plane modes. From the viewpoint of screw theory, they represent respectively the vibration axes perpendicular to the plane of symmetry and lying in the plane of symmetry. In this paper, the sets of orthogonal and mutually intersecting three springs are used as resilient support of a rigid body. The geometrical conditions for the system to have a plane of symmetry and diagonalized stiffness matrix are presented. From the orthogonality of the vibration modes with respect to the inertia matrix, the geometrical relation between the reaction wrenches and the vibration modes are derived. This geometrical relation is then used to get the cubic design equation for the design of out-of-plane modes. The numerical design example of engine mounts is presented in order to explain the suggested design technique.

• The Journal of the Acoustical Society of Korea
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• v.17 no.2E
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• pp.47-52
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• 1998

An analysis of dynamic responses is carried out on monoclinic anisotropic system due to a buried harmonic line source. The load is in the form of a normal stress acting along an arbitrary axis on the plane of symmetry within the orthotropic materials: In case that the line load is acting along the symmetry axis normal to the plane of symmetry, plane wave equation is coupled with verital shear wave and longitudinal wave. However, if the line load is acting along an arbitrary axis normal to the plane of symmetry, plane wave equation is coupled with vertical shear wave, longitudinal wave and horizontal shear wave. We first considered the equation of motion in a reference coordinate system, where the line load is coincident with a symmetry axis of the orthotropic material. Then the equation of motion is transformed into one with respect to general coordinate system with azimuthal angle by using transformation tensor. Plane wave solutions of monoclinic systems are derived for infinite media. Finally complete solutions for the plane harmonic wave are obtained by calculating the inverse of the integral transforms, in which bulk wave poles are avoided by deforming the contour of the integration to the complex plane. Numerical results for examples of orthotropic material belonging to monoclinic symmetry are demonstrated.

• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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• pp.1111-1117
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• 2011

Using the moving plane method, we establish the radial symmetry of topological one-vortex solutions of a semilinear elliptic system arising from the self-dual Maxwell-Chern-Simons O(3) model.

• Journal of the korean Society of Automotive Engineers
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• v.10 no.2
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• pp.61-69
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• 1988

The finite-difference three-dimensional boundary layer procedure of Chang and Patel is modified and applied to solve the boundary layer development on the automobile surface. The inviscid pressure distribution needed to solve the boundary layer equations is obtained by using a low order panel method. The plane of symmetry boundary layer exhibits the strong streamline divergence up to the midbody and convergence thereafter. The streamline divergence in front of the windshield helps the boundary layer to overcome the sever adverse pressure gradient and avoid the separation. The relaxation of the pressure right after the top of the wind-shield, on the other hand, makes the overly thinned boundary layer to readjust and prompts the streamlines to converge into the symmetry plane before the external streamlines do. The three-dimensional characteristics are less apparent after the midbody and the boundary layer is similar to that of the two-dimensional flow. The results of the off-plane-of-symmetry boundary layer are also presented.

• Journal of computational fluids engineering
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• v.2 no.2
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• pp.1-8
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• 1997

Two versions of anisotropic k-ε turbulence model are incorporated in the modified k-ε model of Sohn et al. to avoid the need for the experimental normal stress value in the model and applied to convergent and divergent flows with strong and adverse pressure gradients in the plane of symmetry of a body of revolution. The models are the nonlinear k-ε model of Speziale and the anisotropic model of Nisizima & Yoshizawa. All of the models yield satisfactory results for relatively complex flow on a plane-of-symmetry boundary layer. The results of the models are compared with those results of experimental normal stress value.

• Journal of the Korean Mathematical Society
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• v.37 no.5
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• pp.751-761
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• 2000

Symmetry properties of positive solutions for semilinear elliptic problems in n are considered. We give a symmetry result for the problem in the feneral case, and then derive various results for certain classes of demilinear elliptic equations. We employ the moving plane method based on the maximum principle on unbounded domains to obtain the result on symmetry.

• East Asian mathematical journal
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• v.37 no.4
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• pp.355-379
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• 2021

In this article, we investigated plane figures introduced in elementary school mathematics in the perspective of traditional classification, and also analyzed plane figures focused on the invariance of plane figures out of traditional classification. In the view of traditional classification, how to treat trapezoids was a key argument. In the current mathematics curriculum of the elementary school mathematics, the concept of sliding, flipping, and turning are introduced as part of development activities of spatial sense, but it is rare to apply them directly to figures. For example, how are squares and rectangles different in terms of symmetry? One of the main purposes of geometry learning is the classification of figures. Thus, the activity of classifying plane figures from a symmetrical point of view has sufficiently educational significance from Klein's point of view.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.24 no.1 s.173
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• pp.110-117
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• 2000

Vibration modes obtained from a modal analysis can be better explained from a screw theoretical standpoint. A vibration mode can be geometrically interpreted as a pure rotation about the vibration center in a plane and as the twisting motion on a screw in a three dimensional space. This paper, presents the method to diagonalize a spatial stiffness matrix by use of a parallel axis congruence transformation. It also describes that the stiffness matrix diagonalized by a congruence transformation, can have the planes of symmetry depending on the location of the center of elasticity. For a plane of symmetry, any vibration mode can be expressed by the axis of vibration. Analytical solutions for the axis of vibration has been derived.

• Journal of the Society of Naval Architects of Korea
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• v.47 no.6
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• pp.843-849
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• 2010

This paper presents definition of symmetry of a ship section where three symmetries are proposed: material, geometric, and load symmetries. Precise terminologies of centroid, moment plane, and neutral axis plane are also defined. It is suggested that force vector equilibrium as well as force equilibrium are necessary condition to determine new position of neutral axis due to translational and rotational mobility. It is also stated that new reference datum of ENMP(elastic neutral moment plane), PNMP(fully plastic moment plane), ENAP(elastic neutral axis plane), and INAP(inelastic neutral moment plane) are required to define asymmetric section properties such as second moment of area, elastic section modulus, yield moment, fully plastic moment, and ultimate moment. Since collision-induced damage and flooding-induced biaxial bending moment produce typical asymmetry of section, the section properties are calculated for a typical VLCC. Geometry asymmetry is determined from ABS and DNV rules and two moment planes of 0/30 degs are assumed for load asymmetry. It is proved that the property reduction ratios directly calculated from second moment of area are usually larger than area reduction ratio. Reduction ratio of ultimate moment capacity shows almost linearly proportional to area reduction ratio. Mobility of elastic and inelastic neutral axis planes is visually provided.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.15 no.10 s.103
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• pp.1186-1194
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• 2005

In this paper, we have systematically formulated the equations concerned to the in-plane and out-of-plane motions and deformations of a thick circular beam by using the kinetic and strain energies in order to analyse natural frequencies of a thick ring. The effects of variation of radius of curvature across the cross-section and also the effects of bending shear, extension and twist are considered. The equations of motion for natural vibration analysis of a ring are obtained utilizing the cyclic symmetry of vibration modes of the ring. The frequencies calculated using thick ring model and thin ring model are compared and discussed with the ones obtained from finite element analysis using the method of cyclic symmetry with 20-node hexahedral solid elements for rings with the different ratio of radial thickness to mean radius.