• Wind and Structures
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• v.28 no.6
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• pp.347-354
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• 2019

This study provides an approximate analytical solution to the fractional vibration problem of thin plate governing anomalous motion of plate subjected to in-plane loads. The method of variable separable is employed to transform the fractional partial differential equations under consideration into a fractional ordinary differential equation in temporal variable and a bi-harmonic plate equation in spatial variable. The technique of conformable fractional derivative is utilized to solve the resulting fractional differential equation and the approach of finite sine integral transform method is used to solve the accompanying bi-harmonic plate equation. The deflection field which measures the transverse displacement of the plate is expressed in terms of product of Bessel and trigonometric functions via the temporal and spatial variables respectively. The obtained solution reduces to the solution of the free vibration problem of thin plate in literature. This work shows that conformable fractional derivative is an efficient mathematical tool for tracking analytical solution of fractional partial differential equation governing anomalous vibration of thin plates.

• Proceedings of the KSME Conference
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• 2000.04a
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• pp.625-630
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• 2000

In this paper, the governing equation and the boundary conditions of deploying rods are derived by using Hamilton's principle. The Galerkin method using the comparison function of the instantaneous natural modes is adopted by which the governing equation is discretized. Based on the discretized equations, the time integration analysis is performed and the longitudinal vibrations for the deploying and the retrieving velocity are analyzed.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.383-405
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• 2004

We study the incomprssible Navier Stokes equations for the flow inside contraction geometry. The governing equations are expressed in the vorticity-stream function formulations. A rectangular computational domain is arised by elliptic grid generation technique. The numerical solution is based on a technique of automatic numerical generation of acurvilinear coordinate system by transforming the governing equation into computational plane. The transformed equations are approximated using central differences and solved simultaneously by successive over relaxation iteration. The time dependent of the vorticity equation solved by using explicit marching procedure. We will apply the technique on several irregular-shapes.

• Magazine of the Korean Society of Agricultural Engineers
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• v.34 no.4
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• pp.97-105
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• 1992

The main purpose of this paper is to present the buckling loads of tapered columns due to dynamic concept. The ordinary differential equation governing the bucking loads for tapered columns is derived on the basis of dynamic concept. Three kinds of cross sectional shape are considered in the governing equation. The Improved Euler method and Determinant Search method are used to perform the integration of the differential equation and to determine the buckling loads, respectively. The hinged-hinged, hinged-clamped, clamped-clamped and free-clamped end constraints are applied in numerical examples. The buckling loads are reported as the function of section ratio, and the effects of cross-sectional shapes are investigated. The buckling load equation, which are fitted by numerical data, are proposed as a function of section ratio. It is expected that these equations can be utilized in structural engineering field.

• Coupled systems mechanics
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• v.11 no.6
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• pp.485-504
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• 2022

This paper presents nonlinear oscillations of a carbon nanotube reinforced composite beam subjected to lateral harmonic load with damping effect based on the modified couple stress theory. As reinforcing phase, three different types of single walled carbon nanotubes distribution are considered through the thickness in polymeric matrix. The non-linear strain-displacement relationship is considered in the von Kármán nonlinearity. The governing nonlinear dynamic equation is derived with using of Hamilton's principle.The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. The frequency response equation and the forced vibration response of the system are obtained. Effects of patterns of reinforcement, volume fraction, excitation force and the length scale parameter on the nonlinear responses of the carbon nanotube reinforced composite beam are investigated.

• Interaction and multiscale mechanics
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• v.1 no.3
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• pp.329-337
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• 2008

This paper presents the derivation of the generalized governing equations in theory of elasticity, their weak forms and the some applications in the numerical analysis of structural mechanics. Unlike the differential equations in classical elasticity theory, the generalized equations of the equilibrium and compatibility equations presented here take the form of integral equations, and the generalized equilibrium equations contain the classical differential equations and the boundary conditions in a single equation. By using appropriate test functions, the weak forms of these generalized governing equations can be established. It can be shown that various variational principles in structural analysis are merely the special cases of these weak forms of generalized governing equations in elasticity. The present weak forms of elasticity equations extend greatly the choices of the trial functions for approximate solutions in the numerical analysis of various engineering problems. Therefore, the weak forms of generalized governing equations in elasticity provide a powerful modeling tool in the computational structural mechanics.

• Journal of Korean Society of Steel Construction
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• v.22 no.3
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• pp.241-251
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• 2010

This paper presents the development of a new analytical solution to the governing differential equation for isotropic annular sector plates subjected to uniform loading in a three-dimensional polar coordinate system. The 4th order governing partial differential equation (PDE) was converted to an ordinary differential equation (ODE) by assuming the Levy-type series solution form and the subsequent mathematical operations. Finally, a series-type solution was assembled with homogeneous and nonhomogeneous solution parts after operating real values and complex conjugates derived from the characteristic equation. To demonstrate the convergence rate and the accuracy of the featured method, several examples with various sector angles were selected and solved. The deflections and internal moments in the example annular sector plates that were obtained from the proposed solution were compared with those obtained from other analytical studies and numerical analyses using the finite element analysis package program, ABAQUS. Very good agreement with the results of other analytical and numerical methodologies was shown.

• Journal of the Korean Society for Aeronautical & Space Sciences
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• v.49 no.6
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• pp.449-456
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• 2021

In this paper, we present a strategy to extend solution capability of an existing low Mach number preconditioned compressible solver to incompressible flows with a little modification. To this end, the energy equation that is of the same form of the total energy equation of compressible flows is used. The energy equation is obtained by a linear combination of the thermal energy equation, the continuity equation and the mechanical energy equation. Subsequently, a modified artificial compressibility method in conjunction with a time marching technique is applied to these incompressible governing equations for steady flow solutions. It is found that the Roe average of the common governing equations is equally valid for both the compressible and incompressible flow conditions. The extension of an existing compressible solver to incompressible flows does not affect the original compressible flow analysis. Validity for incompressible flow analysis of the extended solver is examined for various inviscid, laminar and turbulent flows.

• Proceedings of the Korean Nuclear Society Conference
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• 1997.10a
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• pp.599-604
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• 1997

One of important inconsistencies in the six-equation model predictions has been found to be the force experienced by a single bubble placed in a convergent stream of liquid. Various sets of governing equations yield different amount of forces to hold the bubble stationary in a convergent nozzle. By using the first order potential flow theory, it is found that the six-equation model can not be used to estimate the force experienced by a deformed bubble. The theoretical value of the particle stress of a bubble in a convergent nozzle flow has been found to be a function of the Weber number when bubble distortion is allowed. This force has been calculated by using different sets of governing equations and compared with the theoretical value. It is suggested in this study that the bubble size distribution function can be used to remove the presented inconsistency by relating the interfacial variables with different moments of the bubble size distribution function. This study also shows that the inconsistencies in the thermal-hydraulic governing equation can be removed by mechanistic modeling of the phasic interface.

• Structural Engineering and Mechanics
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• v.43 no.5
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• pp.561-582
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• 2012

Despite popularity of FEM in analysis of static and dynamic structural problems and the routine applicability of FE softwares, analytical methods based on simple mathematical relations is still largely sought by many researchers and practicing engineers around the world. Development of such analytical methods for analysis of free vibration of non-prismatic beams is also of primary concern. In this paper a new and simple method is proposed for determination of vibration frequencies of non-prismatic beams under variable axial forces. The governing differential equation is first obtained and, according to a harmonic vibration, is converted into a single variable equation in terms of location. Through repetitive integrations, integral equation for the weak form of governing equation is derived. The integration constants are determined using the boundary conditions applied to the problem. The mode shape functions are approximated by a power series. Substitution of the power series into the integral equation transforms it into a system of linear algebraic equations. Natural frequencies are determined using a non-trivial solution for system of equations. Presented method is formulated for beams having various end conditions and is extended for determination of the buckling load of non-prismatic beams. The efficiency and convergence rate of the current approach are investigated through comparison of the numerical results obtained to those obtained using available finite element software.