• Structural Engineering and Mechanics
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• v.82 no.6
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• pp.747-756
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• 2022

This study presents the nonlinear free and forced vibrations of a cracked atomic force microscopy (AFM) cantilever by using the modified couple stress. The cracked section of the AFM cantilever is considered and modeled as rotational spring. In the frame work of Euler-Bernoulli beam theory, Von-Karman type of geometric nonlinear equation and the modified couple stress theory, the nonlinear equation of motion for the cracked AFM is derived by Hamilton's principle and then discretized by using the Galerkin's method. The semi-inverse method is utilized for analysis nonlinear free oscillation of the system. Then the method of multiple scale is employed to investigate primary resonance of the system. Some numerical examples are presented to illustrate the effects of some parameters such as depth of the crack, length scale parameter, Tip-Mass, the magnitude and the location of the external excitation force on the nonlinear free and forced vibration behavior of the system.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2002.10a
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• pp.185-192
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• 2002

In this study, we choose the sinusoidal shaped arch with pin-ends subjected to sinusoidal distributed excitation to investigate the fundamental mechanism of the dynamic instability. We derive the nonlinear equations of motion to investigate the instability phenomenon of arch structures and Identify the buckling characteristics of sinusoidal shaped arch structures through the nonlinear eigenvalue analysis with discreted equations of motion by Galerkin's method. We examine that phenomenons which direct snapping and indirect snapping with backbone curves to understand occurrence paths of the dynamic buckling.

• Advances in nano research
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• v.6 no.4
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• pp.323-337
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• 2018

The nonlinear free vibration of a nanorod subjected to finite strain is investigated. The governing equation of motion in material configuration in terms of displacement is determined. By means of Galerkin method, the Fourier series solutions satisfying some typical boundary conditions are determined. The amplitude-frequency relationship and interaction between the modes are studied. The effects of nonlocal elasticity are shown for different length of nanotubes and nonlocal parameter. The results show that nonlocal effects lead to additional internal modal interaction for nanorod vibrations.

• Structural Engineering and Mechanics
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• v.61 no.2
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• pp.193-207
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• 2017

In this study, the vibration of an electrostatically actuated micro cantilever beam is analyzed in which a viscoelastic layer covers a portion of the micro beam length. This proposed model is considered as the main element of mass and pollutant micro sensors. The nonlinear motion equation is extracted by means of Hamilton principle, considering nonlinear shortening effect for Euler-Bernoulli beam. The non-linear effects of electrostatic excitation, geometry and inertia have been taken into account. The viscoelastic model is assumed as Kelvin-Voigt model. The motion equation is discretized by Galerkin approach. The linear free vibration mode shapes of non-uniform micro beam i.e. the linear mode shape of the system by considering the geometric and inertia effects of viscoelastic layer, have been employed as comparison function in the process of the motion equation discretization. The discretized equation of motion is solved by the use of multiple scale method of perturbation theory and the results are compared with the results of numerical Runge-Kutta approach. The frequency response variations for different lengths and thicknesses of the viscoelastic layer have been founded. The results indicate that if a constant volume of viscoelastic layer is to be deposited on the micro beam for mass or gas sensor applications, then a modified configuration may be found by using the analysis of this paper.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.4 no.3
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• pp.53-61
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• 1984

A nonlinear formulation of a beam finite element(NB6) on the total Lagrangian mode for the geometrically nonlinear analysis of two-dimensional elastic framed structures is presented. The NB6 beam element has been degenerated from the three-dimensional continuum by introducing the deep beam assumptions and consists of three reference nodes and three relative nodes. The element characteristics are derived by discretizing the beam equations of motion using the Galerkin weighted residual method and are reduced-integrated repeatedly for each loading step by the Newton-Raphson iteration techpique. Several numerical examples are given to demonstrate the accuracy and versatility of the proposed nonlinear NB6 beam element.

• Geomechanics and Engineering
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• v.32 no.2
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• pp.125-135
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• 2023

Nonlinear vibration analysis of composite beam reinforced by carbon nanotubes resting on the nonlinear viscoelastic foundation is investigated in this study. The material properties of the composite beam is considered as a polymeric matrix by reinforced carbon nanotubes according to different distributions. With using Hamilton's principle, the governing nonlinear partial differential equations are derived based on the Euler-Bernoulli beam theory. In the nonlinear kinematic assumption, the Von Kármán nonlinearity is used. 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. In addition, the effects of different patterns of reinforcement, linear and nonlinear damping coefficients of the viscoelastic foundation on the nonlinear vibration responses and phase trajectory of the carbon nanotube reinforced composite beam are investigated.

• Structural Engineering and Mechanics
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• v.75 no.1
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• pp.87-100
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• 2020

The present paper investigates the combination resonance behavior of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells with internal and external functionally graded stiffeners under two-term large amplitude excitations. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model with a nonlinear cubic stiffness, to account for the vibration hardening/softening phenomena and damping considerations. With regard to classical plate theory of shells, von-Kármán equation and Hook law, the relations of stress-strain are derived for shell and stiffeners. The spiral stiffeners of the cylindrical shell are modeled according to the smeared stiffener technique. According to the Galerkin method, the discretized motion equation is obtained. The combination resonance is obtained by using the multiple scales method. Finally, the influences of the stiffeners angles, foundation type, the nonlinear elastic foundation coefficients, material distribution, and excitation amplitude on the system resonances are investigated comprehensively.

• Structural Engineering and Mechanics
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• v.64 no.2
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• pp.259-270
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• 2017

The stochastic vibration response of the sandwich beam with the nonlinear adjustable visco-elastomer core and supported mass under stochastic support motion excitations is studied. The nonlinear dynamic properties of the visco-elastomer core are considered. The nonlinear partial differential equations for the horizontal and vertical coupling motions of the sandwich beam are derived. An analytical solution method for the stochastic vibration response of the nonlinear sandwich beam is developed. The nonlinear partial differential equations are converted into the nonlinear ordinary differential equations representing the nonlinear stochastic multi-degree-of-freedom system by using the Galerkin method. The nonlinear stochastic system is converted further into the equivalent quasi-linear system by using the statistic linearization method. The frequency-response function, response spectral density and mean square response expressions of the nonlinear sandwich beam are obtained. Numerical results are given to illustrate new stochastic vibration response characteristics and response reduction capability of the sandwich beam with the nonlinear visco-elastomer core and supported mass under stochastic support motion excitations. The influences of geometric and physical parameters on the stochastic response of the nonlinear sandwich beam are discussed, and the numerical results of the nonlinear sandwich beam are compared with those of the sandwich beam with linear visco-elastomer core.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.423-438
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• 1994

We employ the Galerkin method to solve the nonlinear Urysohn integral equation (1.1) x(t) = f(t) ＋ $∫_{D}$ k(t, s, x(s))ds (t $\in$ D), where D is a bounded domain in $R^{d}$ , the function f and k are known and x is the solution to be determined. We assume that D has a locally Lipschitz boundary ([1, p. 67]). We can rewrite (1.1) in operator notation as x = f ＋ Kx. We consider (1.1) as an operator equation on $L_{\infty$}$(D) and assume that K is defined on the closure $\Omega$ of a bounded open set $\Omega$ ⊂ $L_{\infty}$(D). Throughout our analysis we put the following assumptions on (1.1).(omitted)(1.1).(omitted)

• Structural Engineering and Mechanics
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• v.37 no.4
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• pp.367-383
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• 2011

In this paper, nonlinear partial differential equations of motion for a hybrid composite plate subjected to initial stresses on elastic foundations are established to investigate its nonlinear vibration behavior. Pasternak foundation and Winkler foundations are used to represent the plate-foundation interaction. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the example problems. The governing equations of motion are reduced to the time-dependent ordinary differential equations by the Galerkin's method. Then, the Runge-Kutta method is used to evaluate the nonlinear vibration frequency and frequency ratio of hybrid composite plates. The nonlinear vibration behavior is affected by foundation stiffness, initial stress, vibration amplitude and the thickness ratio of layer. The effects of various parameters on the nonlinear vibration of hybrid laminated plate are investigated and discussed.