• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 1996년도 추계학술대회논문집; 한국과학기술회관, 8 Nov. 1996
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• pp.275-281
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• 1996

In order to examine the validity of an asymptotic solution obtained from the method of multiple scales, we investigate a third-order subharmonic resonance response of a bar constrained by a nonlinear spring to a harmonic excitation. The motion of the bar is governed by a linear partial differential equation with a nonlinear boundary condition. The nonlinear boundary value problem is solved by using the finite difference method. The numerical solution is compared with the asymptotic solution.

• Smart Structures and Systems
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• 제22권1호
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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.

• Kyungpook Mathematical Journal
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• 제48권4호
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• pp.593-611
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• 2008

Global attractivity and oscillatory behavior of the following nonlinear impulsive parabolic differential equation which is a general form of many population models $$\array{\{{{\frac {{\partial}u(t,x)}{{\partial}t}=\Delta}u(t,x)-{\delta}u(t,x)+f(u(t-\tau,x)),\;t{\neq}t_k,\\u(t^+_k,x)-u(t_k,x)=g_k(u(t_k,x)),\;k{\in}I_\infty,}\;\;\;\;\;\;\;\;(*)$$ are considered. Some new sufficient conditions for global attractivity and oscillation of the solutions of (*) with Neumann boundary condition are established. These results no only are true but also improve and complement existing results for (*) without diffusion or impulses. Moreover, when these results are applied to the Nicholson's blowflies model and the model of Hematopoiesis, some new results are obtained.

• Smart Structures and Systems
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• 제26권3호
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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.

• 한국항해항만학회지
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• 제33권9호
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• pp.629-634
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• 2009

Vessels stability problems need to resolve the nonlinear mathematical models of rolling motion. For nonlinear systems subjected to random excitations, there are very few special cases can obtain the exact solutions. In this paper, the specific differential equations of rolling motion for intact ship considering the restoring and damping moment have researched firstly. Then the partial stochastic linearization method is applied to study the response statistics of nonlinear ship rolling motion in beam seas. The ship rolling nonlinear stochastic differential equation is then solved approximately by keeping the equivalent damping coefficient as a parameter and nonlinear response of the ship is determined in the frequency domain by a linear analysis method finally.

• 한국항해항만학회:학술대회논문집
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• 한국항해항만학회 2009년도 추계학술대회
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• pp.239-240
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• 2009

Vessels stability problems need to resolve the nonlinear mathematical models of rolling motion. For nonlinear systems subjected to random excitations, there are very few special cases can obtain the exact solutions. In this paper, the specific differential equations of rolling motion for intact ship considering the restoring and damping moment have researched firstly. Then the partial stochastic linearization method is applied to study the response statistics of nonlinear ship rolling motion in beam seas. The ship rolling nonlinear stochastic differential equation is then solved approximately by keeping the equivalent damping coefficient as a parameter and nonlinear response of the ship is determined in the frequency domain by a linear analysis method finally.

• 충청수학회지
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• 제24권2호
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• pp.359-370
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• 2011

In this work, we obtain new solitary wave solutions for some nonlinear partial differential equations. The Jacobi elliptic function rational expansion method is used to establish new solitary wave solutions for the combined KdV-mKdV and Klein-Gordon equations. The results reveal that Jacobi elliptic function rational expansion method is very effective and powerful tool for solving nonlinear evolution equations arising in mathematical physics.

• 대한기계학회논문집A
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• 제21권2호
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• pp.337-345
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• 1997

An analysis is presented for the response of a beam constrained by a nonlinear spring to a harmonic excitation. The system is governed by a linear partial differential equation with a nonlinear boundary condition. The method of multiple scales is used to reduce the nonlinear boundary value problem to a system of autonomous ordinary differential equations of the amplitudes and phases. The case of the third-order subharmonic resonance is considered in this study. The autonomous system is used to determine the steady-state responses and their stability.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제4권1호
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• pp.11-22
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• 2000

Nonhysteretic infiltration in nonswelling soil is modelled by the Burgers equation under appropriate physical conditions. For this nonlinear partial differential equation the modal approximation scheme is used for estimating parameters such as soil water diffusivity and hydraulic conductivity. The parameter estimation convergence is proved, and numerical experiments are performed.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제3권1호
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• pp.9-18
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• 1996

In this paper, we consider the existence of a solution to the elliptic nonlinear partial differential equation ${\Delta}u＋H({\chi})e^{u}$ = 0 (H $\neq$ 0) (1) on a compact manifold without boundary. This equation is related to the problem of a pointwise conformal deformation of metrics on two dimensional compact connected manifolds.(omitted)