• Title/Summary/Keyword: nonlinear ordinary differential equation

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Nonlinear free vibration analysis of a composite beam reinforced by carbon nanotubes

  • M., Alimoradzadeh;S.D., Akbas
    • Steel and Composite Structures
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    • v.46 no.3
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    • pp.335-344
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    • 2023
  • This investigation presents nonlinear free vibration of a carbon nanotube reinforced composite beam based on the Von Kármán nonlinearity and the Euler-Bernoulli beam theory The material properties of the structure is considered as made of a polymeric matrix by reinforced carbon nanotubes according to different material distributions. The governing equations of the nonlinear vibration problem is delivered by using Hamilton's principle and the Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. The nonlinear natural frequency and the nonlinear free response of the system is obtained with the effect of different patterns of reinforcement.

THE METHOD OF QUASILINEARIZATION AND A THREE-POINT BOUNDARY VALUE PROBLEM

  • Eloe, Paul W.;Gao, Yang
    • Journal of the Korean Mathematical Society
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    • v.39 no.2
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    • pp.319-330
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    • 2002
  • The method of quasilinearization generates a monotone iteration scheme whose iterates converge quadratically to a unique solution of the problem at hand. In this paper, we apply the method to two families of three-point boundary value problems for second order ordinary differential equations: Linear boundary conditions and nonlinear boundary conditions are addressed independently. For linear boundary conditions, an appropriate Green\`s function is constructed. Fer nonlinear boundary conditions, we show that these nonlinearities can be addressed similarly to the nonlinearities in the differential equation.

Superharmonic and subharmonic resonances of a carbon nanotube-reinforced composite beam

  • Alimoradzadeh, M.;Akbas, S.D.
    • Advances in nano research
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    • v.12 no.4
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    • pp.353-363
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    • 2022
  • This paper presents an investigation about superharmonic and subharmonic resonances of a carbon nanotube reinforced composite beam subjected to lateral harmonic load with damping effect based on the modified couple stress theory. As reinforcing phase, three different types of single walled carbon nanotubes (CNTs) distribution are considered through the thickness in polymeric matrix. The governing nonlinear dynamic equation is derived based on the von Kármán nonlinearity with using of Hamilton's principle. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. Effects of different patterns of reinforcement, volume fraction, excitation force and the length scale parameter on the frequency-response curves of the carbon nanotube reinforced composite beam are investigated. The results show that volume fraction and the distribution of CNTs play an important role on superharmonic and subharmonic resonances of the carbon nanotube reinforced composite beams.

Stochastic Response of a Hinged-Clamped Beam (Hinged-clamped 보의 확률적 응답특성)

  • Cho, Duk-Sang
    • 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.

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Chaotic Vibration of a Curved Pipe Conveying Oscillatory Flow (조화진동유동을 포함한 곡선 파이프 계의 혼돈 운동 연구)

  • 박철희;홍성철;김태정
    • Journal of KSNVE
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    • v.7 no.3
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    • pp.489-498
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    • 1997
  • In this paper, chaotic motions of a curved pipe conveying oscillatory flow are theoretically investigated. The nonliear partial differential equation of motion is derived by Newton's method. The transformed nonlinear ordinary differential equation is a type of Hill's equation, which has the external and parametric excitation with a same frequency. Bifurcation curves of chaotic motion of the piping systems are obtained by applying Melnikov's method. Numerical simulations are performed to demonstrate theoretical results and show the strange attractor of the chaotic motion.

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Boundary Control of an Axially Moving Nonlinear Tensioned Elastic String (인장력하에서 길이방향으로 이동하는 비선형 탄성현의 경계제어)

  • 박선규;이숙재;홍금식
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.28 no.1
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    • pp.11-21
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    • 2004
  • In this paper, an active vibration control of a tensioned elastic axially moving string is investigated. The dynamics of the translating string ale described by a non-linear partial differential equation coupled with an ordinary differential equation. The time varying control in the form of the right boundary transverse motions is suggested to stabilize the transverse vibration of the translating continuum. A control law based on Lyapunov's second method is derived. Exponential stability of the translating string under boundary control is verified. The effectiveness of the proposed controller is shown through the simulations.

THE VARIATIONAL HOMOTOPY PERTURBATION METHOD FOR ANALYTIC TREATMENT FOR LINEAR AND NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

  • Matinfar, Mashallah;Mahdavi, M.;Raeisi, Z.
    • Journal of applied mathematics & informatics
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    • v.28 no.3_4
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    • pp.845-862
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    • 2010
  • In a recent paper, M.A. Noor et al. (Hindawi publishing corporation, Mathematical Problems in Engineering, Volume 2008, Article ID 696734, 11 pages, doi:10.1155/2008/696734) proposed the variational homotopy perturbation method (VHPM) for solving higher dimentional initial boundary value problems. In this paper, we consider the proposed method for analytic treatment of the linear and nonlinear ordinary differential equations, homogeneous or inhomogeneous. The results reveal that the proposed method is very effective and simple and can be applied for other linear and nonlinear problems in mathematical.

Inverse Compensation of Hysteresis in Ferromagnetic Materials (강자성체의 히스테리시스 역 보상 모델)

  • 박영우;한광섭
    • Proceedings of the Korean Society of Precision Engineering Conference
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    • 2004.10a
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    • pp.1470-1474
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    • 2004
  • This paper addresses the development of inverse compensation techniques for a class of ferromagnetic transducers including magnetostrictive actuators. In this work, hysteresis is modeled through the domain wall theory originally proposed by Jiles and Atherton[1]. This model is based on the quantification of the energy required to translate domain walls pinned at inclusions in the material with the magnetization at a given field level specified through the solution of an ordinary differential equation. A complementary differential equation is then employed to compute the inverse which can be used to compensate for hysteresis and nonlinear dynamics in control design.

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On the large amplitude free vibrations of axially loaded Euler-Bernoulli beams

  • Bayat, Mahmoud;Pakar, Iman;Bayat, Mahdi
    • Steel and Composite Structures
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    • v.14 no.1
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    • pp.73-83
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    • 2013
  • In this paper Hamiltonian Approach (HA) have been used to analysis the nonlinear free vibration of Simply-Supported (S-S) and for the Clamped-Clamped (C-C) Euler-Bernoulli beams fixed at one end subjected to the axial loads. First we used Galerkin's method to obtain an ordinary differential equation from the governing nonlinear partial differential equation. The effect of different parameter such as variation of amplitude to the obtained on the non-linear frequency is considered. Comparison of HA with Runge-Kutta 4th leads to highly accurate solutions. It is predicted that Hamiltonian Approach can be applied easily for nonlinear problems in engineering.

MONOTONE ITERATION SCHEME FOR IMPULSIVE THREE-POINT NONLINEAR BOUNDARY VALUE PROBLEMS WITH QUADRATIC CONVERGENCE

  • Ahmad, Bashir;Alsaedi, Ahmed;Garout, Doa'a
    • Journal of the Korean Mathematical Society
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    • v.45 no.5
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    • pp.1275-1295
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    • 2008
  • In this paper, we consider an impulsive nonlinear second order ordinary differential equation with nonlinear three-point boundary conditions and develop a monotone iteration scheme by relaxing the convexity assumption on the function involved in the differential equation and the concavity assumption on nonlinearities in the boundary conditions. In fact, we obtain monotone sequences of iterates (approximate solutions) converging quadratically to the unique solution of the impulsive three-point boundary value problem.