• Magazine of the Korean Society of Agricultural Engineers
• /
• v.38 no.3
• /
• pp.50-57
• /
• 1996

A numerical method is presented to obtain the natural frequencies and mode shapes of the arbitrary tapered beams with static deflection due to arbitrary distributed dead loads. The differential equation governing free vibration of such beams is derived and solved numerically. The double integration method using the trapezoidal rule is used to solve the static behaviour of beams loaded arbitrary distributed dead load. Also, the Improved Euler method and the determinant search method are used to integrate the differential equation subjected to the boundary conditions and to determine the natural frequencies of the beams, respectively. In the numerical examples, the various geometries of the beams are considered : (1) linearly tapered beams as the arbitrary variable cross-section, (2) the triangular, sinusoidal and uniform loads as the arbitrary distributed dead loads and (3) the hinged-hinged, clamped-clamped and hinged-clamped ends as the end constraints. All numerical results are shown as the non-dimensional forms of the system parameters. The lowest three natural frequencies versus load parameter, slenderness ratio and section ratio are reported in figures. And for the comparison purpose, the typical mode shapes with and without the effects of static deflection are presented in the figure. According to the numerical results obtained in this analysis, the following conclusions may be drawn : (1) the natural frequencies increase when the effects of static deflections are included, (2) the effects are larger at the lower modes than the higher ones and (3) it should be betteF to include the effect of static deflection for calculating the frequencies when the beams are supported by both hinged ends or one hinged end.

• Structural Engineering and Mechanics
• /
• v.53 no.3
• /
• pp.537-573
• /
• 2015

Multiple-step beams carrying intermediate lumped masses with/without rotary inertias are widely used in engineering applications, but in the literature for free vibration analysis of such structural systems; Bernoulli-Euler Beam Theory (BEBT) without axial force effect is used. The literature regarding the free vibration analysis of Bernoulli-Euler single-span beams carrying a number of spring-mass systems, Bernoulli-Euler multiple-step and multi-span beams carrying multiple spring-mass systems and multiple point masses are plenty, but that of Timoshenko multiple-step beams carrying intermediate lumped masses and/or rotary inertias with axial force effect is fewer. The purpose of this paper is to utilize Numerical Assembly Technique (NAT) and Differential Transform Method (DTM) to determine the exact natural frequencies and mode shapes of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and/or rotary inertias. The model allows analyzing the influence of the shear and axial force effects, intermediate lumped masses and rotary inertias on the free vibration analysis of the multiple-step beams by using Timoshenko Beam Theory (TBT). At first, the coefficient matrices for the intermediate lumped mass with rotary inertia, the step change in cross-section, left-end support and right-end support of the multiple-step Timoshenko beam are derived from the analytical solution. After the derivation of the coefficient matrices, NAT is used to establish the overall coefficient matrix for the whole vibrating system. Finally, equating the overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the differential equations of the motion. The calculated natural frequencies of Timoshenko multiple-step beam carrying intermediate lumped masses and/or rotary inertias for the different values of axial force are given in tables. The first five mode shapes are presented in graphs. The effects of axial force, intermediate lumped masses and rotary inertias on the free vibration analysis of Timoshenko multiple-step beam are investigated.

• Journal of Korean Society of Steel Construction
• /
• v.11 no.2 s.39
• /
• pp.131-142
• /
• 1999

The general behavior of the curved girder including the warping effects can be presented as the series of differential equations developed by Vlasov. Generally, bending moment is the most important factor for engineer to decide the section of the girder. In order to accommodate easiness of the structural analysis for the curved girder bridge, this paper suggest the ratios of bending moment of curved gilder to that of straight girder. These ratios are presented by an approximate formula setting central angle ${\theta}(L/R)$ as a variable. The approximate formula of the maximum bending moment ratios and influence lines of all stress resultants can be used to design the three-span curved girder bridges.

• KSCE Journal of Civil and Environmental Engineering Research
• /
• v.26 no.3A
• /
• pp.533-538
• /
• 2006

This study deals with the static and dynamic stability analyses of simple tapered columns with constant volume. The crosssections of column taper are the regular polygons whose depths are varied with the parabolic functional fashion. The hingedhinged end constraint is chosen as the boundary condition of the column. The non-dimensional ordinary differential equation governing free vibrations of such column subjected to an axial load is derived and solved numerically. From numerical results, the relationships between natural frequencies and section ratios are obtained, from which the configurations of dynamic optimal shapes of columns and the strongest columns are extracted.

• Journal of Korean Society of Steel Construction
• /
• v.13 no.6
• /
• pp.683-691
• /
• 2001

Numerical methods are developed for solving the elastica and buckling load of tapered columns with shear deformation, subjected to a compressive end load. The linear, parabolic and sinusoidal tapers with the regular polygon cross-sections are considered, whose material volume and span length are always held constant. The differential equations governing the elastica of buckled column are derived. The Runge-Kutta method is used to integrate the differential equations, and the Regula-Falsi method is used to determine the rotation at left end and the buckling load, respectively. The numerical methods developed herein for computing the elastica and the buckling loads of the columns are found to be efficient and reliable.

• Bulletin of the Korean Chemical Society
• /
• v.31 no.10
• /
• pp.2841-2848
• /
• 2010

Quasiclassical trajectory (QCT) method has been used to investigate stereodynamics information of the reaction $O(^1D)+H_2{\rightarrow}\;OH$+H on the DK (Dobbyn and Knowles) potential energy surface (PES) at a collision energy of 23.06 kcal/mol, with the initial quantum state of reactant $H_2$ being set for v = 0 (vibration quantum number) and j = 0-5 (rotation quantum number). The PDDCSs (polarization dependent differential cross sections) and the distributions of P($\theta_r$), P($\phi_r$), P($\theta_r$, $\phi_r$) have been presented in this work. The results demonstrate that the products are both forward and backward scattered. As j increases, the backward scattering becomes weaker while the forward scattering becomes slightly stronger. The distribution of P($\theta_r$) indicates that the product rotational angular momentum j' tends to align along the direction perpendicular to the reagent relative velocity vector k, but this kind of product alignment is found to be rather insensitive to j. Furthermore, the distribution of P($\phi_r$) indicates that the rotational angular momentum vector of the OH product is preferentially oriented along the positive direction of y-axis, and such product orientation becomes stronger with increasing j.

• Proceedings of the Korean Institute of Electrical and Electronic Material Engineers Conference
• /
• 2001.05c
• /
• pp.201-203
• /
• 2001

This paper describes the information for quantitative simulation of weakly ionized plasma. We must grasp the meaning of the plasma state condition to utilize engineering application and to understand materials of plasma state. In this paper, the drift velocity of electron in Xenon gas calculated for range of E/N values from 0.01~500[Td] at the temperature is $300[^{\circ}K]$ and pressure is 1[Torr], using a set of electron collision cross sections determined by the authors and the values of drift velocity of electrons are obtained for TOF, PT, SST sampling method of Backward Prolongation by two term approximation Boltzmann equation method. it has also been used to predict swarm parameter using the values of cross section as input. The result of Boltzmann equation, the drift velocity of electrons, has been compared with experimental data by L. S. Frost and A. V. Phelps for a range of E/N. The swarm parameter from the study are expected to server as a critical test of current theories of low energy scattering by atoms and molecules.

• Advances in nano research
• /
• v.8 no.1
• /
• pp.13-24
• /
• 2020

In this paper, for the first time based on the nonlocal strain gradient theory the effect of size dependency in torsional vibration of bi-direction functionally graded (FG) nonlinear nano-cone is study. The material properties were assumed to vary according to the arbitrary function in radial and axial directions. The Navier equation and boundary conditions of the size-dependent bidirectional FG nonlinear nano-cone were derived by Hamilton's principle. These equations were solved by employing the generalized differential quadrature method (GDQM). The presented model can turn into the classical model if the material length scale parameters are taken to be zero. The effects of some parameters, such as inhomogeneity constant, cross-sectional area parameter and small-scale parameters, were studied. As an essential result of this study can be stated that an FG nano-cone model based on the nonlocal elasticity theory behaves softer and based on the strain gradient theory behaves harder.

• Proceedings of the KSR Conference
• /
• 2011.05a
• /
• pp.979-987
• /
• 2011

The Ballast Track has been widely applied in traditionally due to low initial cost and abundant elastic property. But the disadvantages of Ballast track are; labor-intensive and costly maintenance, weak in high-speed and heavy axial load, in additionally need wide cross section of tunnel and massive substructure in viaduct. Therefore, recent applications tend to more and more towards slab track such as Gyeungbu high speed rail and existing line. The slab track increased the stability, resistance and durability of track, and save maintenance cost compare to the Ballast Track. But the slab track have weakness of track elongation by sub-ballast differential settlement and that threat safety of train operation. Therefor the slab track need to prevent cracks in concrete ballast for insure the durability of slab track. In this paper, review main items and its expected effects of the slab track maintenance standards that control sub-ballast settlement and concrete ballast cracks.

• Advances in aircraft and spacecraft science
• /
• v.3 no.4
• /
• pp.379-397
• /
• 2016

In this manuscript, free vibrations of a unidirectional composite orthotropic Timoshenko beam based on finite strain have been studied. Using Green-Lagrange strain tensor and comprising all of the nonlinear terms of the tensor and also applying Hamilton's principle, equations of motion and boundary conditions of the beam are obtained. Using separation method in single-harmonic state, time and locative variables are separated from each other and finally, the equations of motion and boundary conditions are gained according to locative variable. To solve the equations, generalized differential quadrature method (GDQM) is applied and then, deflection and cross-section rotation of the beam in linear and nonlinear states are drawn and compared with each other. Also, frequencies of carbon/epoxy and glass/epoxy composite beams for different boundary conditions on the basis of the finite strain are calculated. The calculated frequencies of the nonlinear free vibration of the beam utilizing finite strain assumption for various geometries have been compared to von Karman one.