• Structural Engineering and Mechanics
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• v.68 no.2
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• pp.203-213
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• 2018

This paper presents a study of geometric nonlinear forced vibration of carbon nano-tubes (CNTs) reinforcement composite plates on nonlinear elastic foundations. The plate is bonded with piezoelectric layers. The von Karman geometric nonlinearity assumptions with classical plate theory are employed to obtain the governing equations. The Galerkin and homotopy perturbation method (HPM) are utilized to investigate the effect of carbon nano-tubes volume fractions, large amplitude vibrations, elastic foundation parameters, piezoelectric applied voltage on frequency ratio and primary resonance. The results indicate that the carbon nano-tube volume fraction, applied voltage and elastic foundation parameters have significant effect on the hardening response of carbon nanotubes reinforced composite (CNTRC) plates.

• Bulletin of the Korean Mathematical Society
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• v.32 no.1
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• pp.45-56
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• 1995

D. H. Gottlieb [1, 2] studied the subgroups $G_n(X)$ of homotopy groups $\pi_n(X)$. In [5, 7, 10], the authors introduced subgroups $G_n(X, A)$ and $G_n^{Rel}(X, A) of \pi_n(X)$ and $\pi_n(X, A)$ respectively and showed that they fit together into a sequence $$ \cdots \to G_n(A) \longrightarrow^{i_*} G_n(X, A) \longrightarrow^{j_*} G_n^{Rel}(X, A) \longrightarrow^\partial $$ $$ \cdots \to G_1^{Rel}(X, A) \to G_0(A) \ to G_0(X, A) $$ where $i_*, j_*$ and \partial$ are restrictions of the usual homomorphisms of the homotopy sequence $$ \cdot \to^\partial \pi_n(A) \longrightarrow^{i_*} \pi_n(X) \longrightarrow^{j_*} \pi_n(X, A) \to \cdot \to \pi_0(A) \to \pi_0(X) $$.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1311-1327
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• 2010

We introduce the concept of cyclic morphisms with respect to a morphism in the category of pairs as a generalization of the concept of cyclic maps and we use the concept to obtain certain sets of homotopy classes in the category of pairs. For these sets, we get complete or partial answers to the following questions: (1) Is the concept the most general concept in the class of all concepts of generalized Gottlieb subsets introduced by many authors until now? (2) Are they homotopy invariants in the category of pairs? (3) When do they have a group structure?.

• Steel and Composite Structures
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• v.35 no.6
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• pp.717-727
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• 2020

The homotopy perturbation method (HPM) to predict the pre- and post-buckling behaviour of simply supported steel beams with rectangular hollow section (RHS) is presented in this paper. The non-linear differential equations solved by HPM derive from a kinematics where large twist and cross-sections distortions are considered. The results (linear and non-linear paths) given by the present HPM are compared to those provided by the Newton-Raphson algorithm with arc length and by the commercial FEM code Abaqus. To investigate the effect of cross-sectional distortion of beams, some numerical examples are presented.

• Structural Engineering and Mechanics
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• v.5 no.5
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• pp.577-586
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• 1997

In nonlinear finite element stability analysis of structures, the foremost necessary procedure is the computation to precisely locate a singular equilibrium point, at which the instability occurs. The present study describes global and local procedures for the computation of stability points including bifurcation points and limit points. The starting point, at which the procedure will be initiated, may be close to or arbitrarily far away from the target point. It may also be an equilibrium point or non-equilibrium point. Apart from the usual equilibrium path, bypass and homotopy path are proposed as the global path to the stability point. A local iterative method is necessary, when it is inspected that the computed path point is sufficiently close to the stability point.

• 제어로봇시스템학회:학술대회논문집
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• 1990.10b
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• pp.1145-1149
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• 1990

This paper proposes a numerical method for redundant manipulators using predetermined optimal resolution. In order to obtain optimal joint trajectories, it is desirable to formulate redundancy resolution as an optimization problem having an integral cost criterion. We predetermine the trajectories of redundant joints in terms of the Nth partial sum of the Fourier series, which lead to the solution in the desirable homotopy class. Then optimal coefficients of the Fourier series, which yield the optimal solution within the predetermined class, are searched by the Powell's method. The proposed method is applied to a 3-link planar manipulator for cyclic tasks in Cartesian space. As the results, we can obtain the optimal solution in the desirable homotopy class without topological liftings of the solution. To show the validity of the proposed method, we analyze both optimal and extremal solutions by the Fast Fourier Transform (FFT) and discuss joint trajectories on the phase plane.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.2
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• pp.109-121
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• 2009

In this paper, we use He's polynomials for solving higher order integro differential equations (IDES) by converting them to an equivalent system of integral equations. The He's polynomials which are easier to calculate and are compatible to Adomian's polynomials are found by using homotopy perturbation method. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Several examples are given to verify the reliability and efficiency of the proposed method.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.2
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• pp.125-136
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• 2018

In this article, a homotopy perturbation transform method (HPTM) and the Laplace transform combined with Taylor expansion method are presented for solving Volterra integral equations with a convolution kernel. The (HPTM) is innovative in Laplace transform algorithm and makes the calculation much simpler while in the Laplace transform and Taylor expansion method we first convert the integral equation to an algebraic equation using Laplace transform then we find its numerical inversion by power series. The numerical solution obtained by the proposed methods indicate that the approaches are easy computationally and its implementation very attractive. The methods are described and numerical examples are given to illustrate its accuracy and stability.

• Journal of the Korean Mathematical Society
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• v.33 no.1
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• pp.15-23
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• 1996

Let C be a category of (pointed) spaces. For $X, Y \in C$ we denote the wedge (or one point union) by $X \vee Y$ and the cartesian product by $X \times Y$. Let $Z \in C$; we say that Z cancels with respect to wedge (resp. cartesian product) and C, if for all $X, Y \in C$ the existence of a homotopy equivalence $X \vee Z \to Y \vee Z$ implies the existence of a homotopy equivalence $X \to Y$ (resp. for cartesian product). If this does not hold, we say that there is a non-cancellation phenomenon involving Z (and C).

• Structural Engineering and Mechanics
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• v.44 no.3
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• pp.405-416
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• 2012

In this study the investigation of large deflections subject in compliant mechanisms is presented using homotopy perturbation method (HPM). The main purpose is to propose a convenient method of solution for the large deflection problem in compliant mechanisms in order to overcome the difficulty and complexity of conventional methods, as well as for the purpose of mathematical modeling and optimization. For simplicity, a cantilever beam of linear elastic material under horizontal, vertical and bending moment end point load is considered. The results show that the applied method is very accurate and capable for cantilever beams and can be used for a large category of practical problems for the aim of optimization. Also the consequence of effective parameters on the large deflection is analyzed and presented.