• Journal of applied mathematics & informatics
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• v.28 no.1_2
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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.

• Proceedings of the Earthquake Engineering Society of Korea Conference
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• 2001.09a
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• pp.243-250
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• 2001

Industrial machines are sometimes exposed to the danger of earthquake. In the design of a mechanical system, this factor should be accounted for from the viewpoint of reliability. A method to analyze a complex nonlinear structure system under random excitation is proposed. First, the actual random excitation, such as earthquake, is approximated to the corresponding Gaussian process far the statistical analysis. The modal equations of overall system are expanded sequentially. Then, the perturbed equations are synthesized into the overall system and solved in probabilistic way. Several statistical properties of a random process that are of interest in random vibration applications are reviewed in accordance with nonlinear stochastic problem. The obtained statistical properties of the nonlinear random vibration are evaluated in each substructure. Comparing with the results of the numerical simulation proved the efficiency of the proposed method.

• Structural Engineering and Mechanics
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• v.54 no.5
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• pp.987-998
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• 2015

In this paper, free vibrations of a clamped-clamped double-walled carbon nanotube (DWNT) under axial force is studied. By utilizing Euler-Bernoulli beam theory, each layer of DWNT is modeled as a beam. In this analysis, nonlinear form of interlayer van der Waals (vdW) forces and nonlinearities aroused from mid-plane stretching are also considered in the equations of motion. Further, direct application of multiple scales perturbation method is utilized to solve the obtained equations and to analyze free vibrations of the DWNT. Therefore, analytical expressions are found for vibrations of each layer. Linear and nonlinear natural frequencies of the system and vibration amplitude ratios of inner to outer layers are also obtained. Finally, the results are compared with the results obtained by Galerkin method.

• Steel and Composite Structures
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• v.44 no.3
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• pp.371-388
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• 2022

The present study tackles the problem of forced vibration of imperfect axially functionally graded shell structure with truncated conical geometry. The linear and nonlinear large-deflection of the structure are considered in the mathematical formulation using von-Kármán models. Modified coupled stress method and principle of minimum virtual work are employed in the modeling to obtain the final governing equations. In addition, formulations of classical elasticity theory are also presented. Different functions, including the linear, convex, and exponential cross-section shapes, are considered in the grading material modeling along the thickness direction. The grading properties of the material are a direct result of the porosity change in the thickness direction. Vibration responses of the structure are calculated using the semi-analytical method of a couple of homotopy perturbation methods (HPM) and the generalized differential quadrature method (GDQM). Contradicting effects of small-scale, porosity, and volume fraction parameters on the nonlinear amplitude, frequency ratio, dynamic deflection, resonance frequency, and natural frequency are observed for shell structure under various boundary conditions.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.26 no.11
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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.

• Journal of the Korean Society of Fisheries and Ocean Technology
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• v.32 no.4
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• pp.432-446
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• 1996

A stochastic Hamilton variational principle(SHVP) is formulated for dynamic problems of linear continuum. The SHVP allows incorporation of probabilistic distributions into the finite element analysis. The formulation is simplified by transformation of correlated random variables to a set of uncorrelated random variables through a standard eigenproblem. A procedure based on the Fourier analysis and synthesis is presented for eliminating secularities from the perturbation approach. In addition to, a method to analyse stochastic design sensitivity for structural dynamics is present. A combination of the adjoint variable approach and the second order perturbation method is used in the finite element codes. An alternative form of the constraint functional that holds for all times is introduced to consider the time response of dynamic sensitivity. The algorithms developed can readily be adapted to existing deterministic finite element codes. The numerical results for stochastic analysis by proceeding approach of cantilever, 2D-frame and 3D-frame illustrates in this paper.

• Structural Engineering and Mechanics
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• v.9 no.2
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• pp.181-193
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• 2000

The objective of this research is to propose a new global damage detection parameter, termed as the static defect energy (SDE). This candidate parameter possesses the ability to detect, locate and quantify structural damage. To have a full understanding about this parameter and its applications, the scope of work can be divided into several tasks: theory and formulation, numerical simulation studies, experimental verification and feasibility studies. This paper only deals with the first part of the task. Brief introduction will be given to the dynamic defect energy (DDE) after systematically reviewing the previous works. Process of applying the perturbation method to the oscillatory system to obtain a static expression will be followed. Two implementation methods can be used to obtain SDE equations and the diagrams. Both results are equally good for damage detection.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2001.11a
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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.

• Coupled systems mechanics
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• v.7 no.5
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• pp.635-647
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• 2018

Dynamic problems arising from the Euler-Bernoulli beam model with inhomogeneous elastic properties are considered. The method of Green's function and perturbation theory are employed to find the deflection in the beam correct to the first-order. Eigenvalue problems appearing from transverse vibrations of inhomogeneous beams in linear and nonlinear cases are also discussed.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.24 no.4 s.175
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• pp.1065-1074
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• 2000

A robust nonlinear $H_2$/$H_{\infty}$ control method for a parallel inverted pendulum with structured perturbation and dry friction is proposed. By the random input describing function techniques, the nonlinear dry friction is approximated into the quasi-linear system. Introducing the quadratic robustness theorem, the robust $H_2$/$H_{\infty}$ control system is constructed for the quasi-linear perturbed system. But it is difficult to design a controller due to the nonlinear correction term in Riccati equation. With some transformations on the Riccati equation containing nonlinear correction term, the design of the robust nonlinear controller can be done easily. Hence when the stiffness and mass of the parallel inverted pendulum vary in certain ranges, the proposed control scheme has the robustness for both the structured perturbation and dry friction. The results of computer simulation show the effectiveness of our proposed control method.