• Nonlinear Functional Analysis and Applications
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• v.27 no.3
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• pp.679-689
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• 2022

In this paper, we establish sufficient conditions for the existence and uniqueness of solution for a class of initial boundary value problems with Dirichlet condition in regard to a category of fractional-order partial differential equations. The results are established by a method based on the theorem of Lax Milligram.

• Kyungpook Mathematical Journal
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• v.60 no.4
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• pp.853-867
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• 2020

The aim of this study was to analyze the momentum and heat transfer of a rotating nanofluid with conducting spherical dust particles. The fluid flows over a stretching surface under the influence of an external magnetic field. By applying similarity transformations, the governing partial differential equations were trans-formed into nonlinear coupled ordinary differential equations. These equations were solved with the built-in function bvp4c in MATLAB. Moreover, the effects of the rotation parameter ω, magnetic field parameter M, mass concentration of the dust particles α, and volume fraction of the nano particles 𝜙, on the velocity and temperature profiles of the fluid and dust particles were considered. The results agree well with those in published papers. According to the result the hikes in the rotation parameter ω decrease the local Nusselt number, and the increasing volume fraction of the nano particles 𝜙 increases the local Nusselt number. Moreover the friction factor along the x and y axes increases with increasing volume fraction of the nano particles 𝜙.

• Journal of Mechanical Science and Technology
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• v.19 no.spc1
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• pp.461-472
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• 2005

A formulation for flexible multibody systems (MBS) is investigated, where rigid MBS substructures are coupled with flexible bodies described by a nonlinear finite element (FE) approach. Several aspects that turned out to be crucial for the presented approach are discussed. The system describing equations are given in differential algebraic form (DAE), where many sophisticated solvers exist. In this paper the performance of several solvers is investigated regarding their suitability for the application to the usually highly stiff DAE. The substructures are connected with each other by nonlinear algebraic constraint equations. Further, partial derivatives of the constraints are required, which often leads to extensive algebraic trans-formations. Handcoding of analytically determined derivatives is compared to an approach utilizing algorithmic differentiation.

• Smart Structures and Systems
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• v.22 no.1
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• pp.105-120
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• 2018

In this paper, the nonlinear free and forced vibration responses of sandwich nano-beams with three various functionally graded (FG) patterns of reinforced carbon nanotubes (CNTs) face-sheets are investigated. The sandwich nano-beam is resting on nonlinear Visco-elastic foundation and is subjected to thermal and electrical loads. The nonlinear governing equations of motion are derived for an Euler-Bernoulli beam based on Hamilton principle and von Karman nonlinear relation. To analyze nonlinear vibration, Galerkin's decomposition technique is employed to convert the governing partial differential equation (PDE) to a nonlinear ordinary differential equation (ODE). Furthermore, the Multiple Times Scale (MTS) method is employed to find approximate solution for the nonlinear time, frequency and forced responses of the sandwich nano-beam. Comparison between results of this paper and previous published paper shows that our numerical results are in good agreement with literature. In addition, the nonlinear frequency, force response and nonlinear damping time response is carefully studied. The influences of important parameters such as nonlocal parameter, volume fraction of the CNTs, different patterns of CNTs, length scale parameter, Visco-Pasternak foundation parameter, applied voltage, longitudinal magnetic field and temperature change are investigated on the various responses. One can conclude that frequency of FG-AV pattern is greater than other used patterns.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.18 no.2
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• pp.160-168
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• 2008

This paper presents the study on the stability of nonlinear oscillations of a thin cantilever beam subject to harmonic base excitation in vertical direction. Two partial differential governing equations under combined parametric and external excitations were derived and converted into two-degree-of-freedom ordinary differential Mathieu equations by using the Galerkin method. We used the method of multiple scales in order to analyze one-to-one combination resonance. From these, we could obtain the eigenvalue problem and analyze the stability of the system. From the thin cantilever experiment using foamax, we could observe the nonlinear modes of bending, twisting, sway, and snap-through buckling. In addition to qualitative information, the experiment using aluminum gave also the quantitative information for the stability of combination resonance of a thin cantilever beam under parametric excitation.

• Smart Structures and Systems
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• v.26 no.3
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• pp.361-371
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• 2020

Exact solution for nonlinear behavior of clamped-clamped functionally graded (FG) buckled beams is presented. The effective material properties are considered to vary along the thickness direction according to exponential-law form. The in-plane inertia and damping are neglected, and hence the governing equations are reduced to a single nonlinear fourth-order partial-integral-differential equation. The von Kármán geometric nonlinearity has been considered in the formulation. Galerkin procedure is used to obtain a second order nonlinear ordinary equation with quadratic and cubic nonlinear terms. Based on the mode of the corresponding linear problem, which readily satisfy the boundary conditions, the frequencies for the nonlinear problem are obtained using the Jacobi elliptic functions. The effects of various parameters such as the Young's modulus ratio, the beam slenderness ratio, the vibration amplitude and the magnitude of axial load on the nonlinear behavior are examined.

• Structural Engineering and Mechanics
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• v.30 no.5
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• pp.603-617
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• 2008

Three dimensional coupled bending-torsion dynamic vibrations of thin-walled open section beam subjected to moving vehicle are investigated by transfer matrix method. Through adopting the idea of Newmark-${\beta}$ method, the partial differential equations of structural vibration can be transformed to the differential equations. Then, those differential equations are solved by transfer matrix method. An iterative scheme is proposed to deal with the coupled bending-torsion terms in the governing vibration equations. The accuracy of the presented method is verified through two numerical examples. Finally, with different eccentricities of vehicle, the torsional vibration of thin-walled open section beam and vertical and rolling vibration of truck body are investigated. It can be concluded from the numerical results that the torsional vibration of beam and rolling vibration of vehicle increase with the eccentricity of vehicle. Moreover, it can be observed that the torsional vibration of thin-walled open section beam may have a significant nonlinear influence on vertical vibration of truck body.

• Proceedings of the KIEE Conference
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• 2000.11d
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• pp.765-768
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• 2000

Linearization of the biped dynamic equations and design of linear controller for the linearized equations are studied in this paper. The biped robot with inverted pendulum type trunk, used to stabilize the dynamic balancing of the biped robot during dynamic walking period, is modelled with 14 DOF and simulated. Despite of well defined linear control theories so far, the linear control methods was limited to the applications for a walking robot, because they have been inherently strong nonlinear properties, such as a modeling parameter uncertainties, external forces as noise, inertial and Coriolis terms by three dimensional modeling and so on. To linearize the nonlinear equations of motion of biped robot on MIMO and time varying linear equations of motion, 1st order Taylor series is used to formulate the linear equation. And a 2nd order numerical perturbation method Is used to approximate partial differential equations. Using the linearized equations of motion, a linear controller is designed by pole placement method with feed forward compensation. Using the obtained linearized equations and linear controller, the continuous walking simulation is performed.

• Steel and Composite Structures
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• v.45 no.5
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• pp.641-652
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• 2022

Fluid-conveying tubes are widely used to transport oil and natural gas in industries. As an advanced composite material, functionally graded carbon nanotube-reinforced composites (FG-CNTRC) have great potential to empower the industry. However, nonlinear free vibration of the FG-CNTRC fluid-conveying pipe has not been attempted in thermal environment. In this paper, the nonlinear free vibration characteristic of functionally graded nanocomposite fluid-conveying pipe reinforced by single-walled carbon nanotubes (SWNTs) in thermal environment is investigated. The SWCNTs gradient distributed in the thickness direction of the pipe forms different reinforcement patterns. The material properties of the FG-CNTRC are estimated by rule of mixture. A higher-order shear deformation theory and Hamilton's variational principle are employed to derive the motion equations incorporating the thermal and fluid effects. A two-step perturbation method is implemented to obtain the closed-form asymptotic solutions for these nonlinear partial differential equations. The nonlinear frequencies under several reinforcement patterns are presented and discussed. We conduct a series of studies aimed at revealing the effects of the flow velocity, the environment temperature, the inner-outer diameter ratio, and the carbon nanotube volume fraction on the nature frequency.

• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.719-728
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• 2012

In this paper, we develop a numerical method to solving an optimal control problem, which is governed by a parabolic partial differential equations(PDEs). Our approach is to approximate the PDE problem to initial value problem(IVP) in $\mathbb{R}$. Then, the homogeneous part of IVP is solved using semigroup theory. In the next step, the convergence of this approach is verified by properties of one-parameter semigroup theory. In the rest of paper, the original optimal control problem is solved by utilizing the solution of homogeneous part. Finally one numerical example is given.