Nonlinear Forced Vibration of a Viscoelastic Buckled Beam with 2 : 1 Internal Resonance

Xiong, Liuyang; Zhang, Guoce; Ding, Hu; Chen, Liqun

doi:10.1155/2014/906324

Cited by 11 publications

(3 citation statements)

References 18 publications

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“…The regular and chaotic vibrations of an axially moving viscoelastic string were presented [12]. Nonlinear forced vibration of a viscoelastic buckled beam [13] and of a viscoelastic pipe conveying fluid around curved equilibrium due to the supercritical flow [14] subjected to primary resonance in the presence of Two-to-One internal resonance is investigated.…”

Section: Short Review Of the Application Of The Galerkin Methodsmentioning

confidence: 99%

Convergence of the Galerkin Method for Nonlinear Dynamics of the Continuous Structure

Ding

2015

J. Adv. App. Comput. Math.

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The Galerkin truncation method is a powerful method for nonlinear dynamics analysis, and has been widely used to discretize the spatial differential operator. Due to more and more new fields of application, the research interest on the Galerkin method is still high today. In this paper, research on the convergence of Galerkin method for nonlinear dynamics of the continuous structure is thoroughly reviewed. At the beginning, the Galerkin method is briefly introduced. Then, the paper reviews the application of the truncation method. This paper also sums up the comparative study on the Galerkin method with other methods, such as the finite difference method (FDM), the finite element method (FEM), and the multiple time scales method. In the investigations concerning the convergence of the Galerkin method, this paper summarizes recent studies on nonlinear dynamics of the axially moving systems, the continua on the nonlinear foundation, and the belt-pulley systems. Finally, the truncation terms of Galerkin method for the continuous structure's nonlinear dynamics analysis is suggested for the future research applications.

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Section: Short Review Of the Application Of The Galerkin Methodsmentioning

confidence: 99%

Convergence of the Galerkin Method for Nonlinear Dynamics of the Continuous Structure

Ding

2015

J. Adv. App. Comput. Math.

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“…Huang et al [16] investigated the nonlinear vibration of a simply-supported curved beam with 1:1 internal resonance between the first and third modes subjected to the base harmonic excitation, and discussed various bifurcation phenomena and responses of symmetric and anti-symmetric modes. Xiong et al [17] studied the nonlinear dynamics of a viscoelastic buckled beam with 2:1 internal resonance, and observed a double-jumping phenomenon. Mao et al [18] investigated the local and global resonances of a super-critically axially moving beam with 3:1 internal resonance, and discussed the excitation effect on the internal resonance.…”

Section: Introductionmentioning

confidence: 99%

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

Huang

2020

Appl. Math. Mech.-Engl. Ed.

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The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated. The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton's principle. A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method. A high-dimensional model of the buckled beam is derived, concerning nonlinear coupling. The incremental harmonic balance (IHB) method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve, and the Floquet theory is used to analyze the stability of the periodic solutions. Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited. Bifurcations including the saddle-node, Hopf, perioddoubling, and symmetry-breaking bifurcations are observed. Furthermore, quasi-periodic motion is observed by using the fourth-order Runge-Kutta method, which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.

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“…More recently, Xiong et al [16] studied the influence of viscoelasticity on the two-to-one internal resonance of a post buckled beam. In their work, a two-term MTS expansion was applied on the discretized equation to analyze the effect of the viscoelastic damping.…”

Section: Introductionmentioning

confidence: 99%

Two-to-one internal resonance of an inclined marine riser under harmonic excitations

Alfosail

Younis

2018

Nonlinear Dyn

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In this paper, we study the two-to-one internal resonance of an inclined marine riser under harmonic excitations. The riser is modeled as an Euler-Bernoulli beam accounting for mid-plane stretching, self-weight, and an applied axial top tension. Due to the inclination, the self-weight load causes a static deflection of the riser, which can tune the frequency ratio between the third and first natural frequencies near two. The multiple time scales method is applied to study the nonlinear equation accounting for the system nonlinearity. The solution is then compared to a Galerkin solution showing good agreement. A further investigation is carried out by plotting the frequency response curves, the force response curves, and the steady state response of the multiple scales solution, in addition to the dynamical solution obtained by Galerkin, as they vary with the detuning parameters. The results reveal that the riser vibrations can undergo multiple Hopf bifurcations and experience quasi-periodic motion that can lead to chaotic behavior. These phenomena lead to complex vibrations of the riser, which can accelerate its fatigue failure.

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Nonlinear Forced Vibration of a Viscoelastic Buckled Beam with 2 : 1 Internal Resonance

Cited by 11 publications

References 18 publications

Convergence of the Galerkin Method for Nonlinear Dynamics of the Continuous Structure

Convergence of the Galerkin Method for Nonlinear Dynamics of the Continuous Structure

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

Two-to-one internal resonance of an inclined marine riser under harmonic excitations

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