• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.21 no.1
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• pp.66-73
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• 2011

This paper deals with free vibrations of the tapered beams with the constant surface area. The surface area of the objective beams are always held constant regardless shape functions of the cross-sectional depth. The shape functions are chosen as the linear and parabolic ones. Ordinary differential equations governing free vibrations of such beams are derived and solved numerically for determining the natural frequencies. In the numerical examples, hinged-hinged, hinged-clamped and clamped-clamped end constraints are considered. As the numerical results, the relationships between non-dimensional frequency parameters and various beam parameters such as section ratio, surface area ratio, end constraint and taper type are reported in tables and figures. Especially, section ratios of the strongest beam are calculated, under which the maximum frequencies are achieved.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.22 no.3
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• pp.223-233
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• 2012

This paper deals with free vibrations of the tapered Timoshenko beam with constant volume, in which both the rotatory inertia and shear deformation are included. The cross section of the tapered beam is chosen as the regular polygon cross section whose depth is varied with the parabolic function. The ordinary differential equations governing free vibrations of such beam are derived based on the Timoshenko beam theory by decomposing the displacements. Governing equations are solved for determining the natural frequencies corresponding with their mode shapes. In the numerical examples, three end constraints of the hinged-hinged, hinged-clamped and clamped-clamped ends are considered. The effects of various beam parameters on natural frequencies are extensively discussed. The mode shapes of both the deflections and stress resultants are presented, in which the composing rates due to bending rotation and shear deformation are determined.

• Steel and Composite Structures
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• v.21 no.5
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• pp.999-1016
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• 2016

This study presents free vibration of beam made of porous material. The mechanical properties of the beam is variable in the thickness direction and the beam is investigated in three situations: poro/nonlinear nonsymmetric distribution, poro/nonlinear symmetric distribution, and poro/monotonous distribution. First, the governing equations of porous beam are derived using principle of virtual work based on Euler-Bernoulli theory. Then, the effect of pores compressibility on natural frequencies of the beam is studied by considering clamped-clamped, clamped-free and hinged-hinged boundary conditions. Moreover, the results are compared with homogeneous beam with the same boundary conditions. Finally, the effects of poroelastic parameters such as pores compressibility, coefficients of porosity and mass on natural frequencies has been considered separately and simultaneously.

• Journal of the Korean Society of Industry Convergence
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• v.3 no.1
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• pp.43-51
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• 2000

The response statistics of a hinged-clamped beam under broad-band random excitation is investigated. The random excitation is applied at the nodal point of the second mode. By using Galerkin's method the governing equation is reduced to a system of nonautonomous nonlinear ordinary differential equations. A method based upon the Markov vector approach is used to generate a general first-order differential equation in the dynamic moment of response coordinates. By means of the Gaussian and non-Gaussian closure methods the dynamic moment equations for the random responses of the system are reduced to a system of autonomous ordinary differential equations. The case of two mode interaction is considered in order to compare it with the case of three mode interaction. The analytical results for two and three mode interactions are also compared with results obtained by Monte Carlo simulation.

• Journal of KSNVE
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• v.9 no.6
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• pp.1193-1199
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• 1999

This paper deals with the free vibrations of circular curved beams with thin-walled cross-section. The differential equation for the coupled flexural-torsional vibrations of such beams with warping is solved numerically to obtain natural frequencies and mode shapes. The Runge-Kutta and determinant search methods, respectively, are used to solve the governing differential equation and to compute the eigenvalues. The lowest three natural frequencies and corresponding mode shapes are calculated for the thin-walled horizontally curved beams with hinged-hinged, hinged-clamped, and clamped-clamped end constraints. A wide range of opening angle of beam, warping parameter, and two different values of slenderness ratios are considered. Numerical results are compared with existing exact and numerical solutions by other methods.

• Journal of KSNVE
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• v.6 no.5
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• pp.587-594
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• 1996

The differential equation governing both the free vibrations and buckling loads of tapered beam-columns of regular polygon cross-section with constant volume were derived and solved numerically. The parabolic and sinusoidl tapers were chosen as the variable depth of cross-section for the tapered beam-column. In numerical examples, the clamped-clamped, hinged-clamped and hinged-hinged end constraints were considered. The variations of frequency parameters and first buckling load parameters with the non-dimensional system parameters are reported in figures, and typical vibrating mode shapes are presented. Also, the configurations of strongest columns were determined.

• Computational Structural Engineering
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• v.9 no.3
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• pp.135-143
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• 1996

The differential equations governing both the free vibrations and buckling loads of tapered beam-columns of circular cross-section with constant volume are derived and solved numerically. The effects of axial load are included in the differential equations. The parabolic equation is chosen as the variable radius of circular cross-section for the tapered beam-column. In numerical examples, the clamped-clamped, clamped-hinged and hinged-hinged end constraints are considered. The variations of the frequency parameters and buckling load parameters with the non-dimensional system parameters are presented in figures and the configurations of strongest columns are obtained.

• Journal of KSNVE
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• v.9 no.5
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• pp.1012-1018
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• 1999

This paper deals with the coupled flexural-torsional vibrations of Timoshenko beams with monosymmetric cross-section. The governing differtial equations for free vibration of such beams are derived and solved numerically to obtain frequencies and mode shapes. Numerical results are calculated for three specific examples of beams with free-free, clamped-free, hinged-hinged, clamped-hinged and clamped-clamped end constraints. The effect of warping stiffess on the natural frequencies and mode shapes is discussed and it is concluded that substantial error can be incurred if the effect is ignored.

• Structural Engineering and Mechanics
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• v.14 no.5
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• pp.611-624
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• 2002

In this paper, a complete solution is presented for dynamic plastic response of a rigid, perfectly plastic hinged-free beam, of which one end is simply supported or hinged and the other end free, subjected to a transverse strike by a travelling mass at an arbitrary location along its span. The governing differential equations are expressed in non-dimensional forms and solved numerically to obtain the instantaneous deflection of the beam and the plastic dissipated energy in the beam. The dynamic behavior for a hinged-free beam is more complicated than that of a free-free beam. It transpires that the mass ratio and impact position have significant influence on the final deformation. In the aspect of energy dissipation, unlike simply supported or clamped beams for which the plastic deformation consumes almost the total input energy, a considerable portion of the input energy would be transferred as rigid-body motion of hinged-free beam, and the energy dissipated in its plastic deformation is greatly reduced.

• Magazine of the Korean Society of Agricultural Engineers
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• v.42 no.6
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• pp.122-129
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• 2000

The purpose of this study is to investigate both the fundamental and some higher natural frequencies and mode shapes of compressive members resting on the linear elastic foundation. The model of compressive member is based on the classical Bernoulli-Euler beam theory. The differential equation governing free vibrations of such members subjected to an axial load is derived and solved numerically for calculating the natural frequencies and mode shapes. The Improved Euler method is used to integrate the differential equation and the Determinant Search method combined with the Regula-Falsi method to determine the natural frequencies, respectively. In numerical examples, the hinged-hinged, hinged-clamped, clamped-hinged and clamped-clamped end constraints are considered. The convergence analysis is conducted for determining the available step size in the Improved Euler method. The validation of theories developed herein is also conducted by comparing the numerical results between this study and SAP 90. The non-dimensional frequency parameters are presented as the non-dimensional system parameters: section ratio, modulus parameter and load parameter. Also typical mode shapes are presented.