An Asymptotic Theory for a Weakly Nonlinear Beam Equation with a Quadratic Perturbation

Boertjens, G. J.; Horssen, Wim T. van

doi:10.1137/s0036139998336766

Cited by 46 publications

(33 citation statements)

References 18 publications

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“…For this purpose, either numerical [10] or asymptotic approaches [11][12][13][14] have been applied. The latter allow getting more adequate imagination of the qualitative/quantitative system dynamical behavior and this gives also a motivation for our paper.…”

Section: Introductionmentioning

confidence: 99%

Influence of geometric and physical nonlinearities on the internal resonances of a finite continuous rod with a microstructure

Andrianov

Awrejcewicz

Danishevskyy

et al. 2017

Journal of Sound and Vibration

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Abstract. In this work, nonlinear longitudinal vibrations of a finite composite rod are studied including geometric and physical nonlinearities. An original boundary value problem for a heterogeneous rod yielded by the macroscopic approximation obtained earlier by the higher-order asymptotic homogenization method is used. The effects of internal resonances and modes coupling are predicted, validated and analyzed. The defined novel continuous problem governed by PDEs is solved using space-discretization and the method of multiple time scales. We are aimed at understanding and analyzing how the presence of the microstructure influences the processes of mode interaction. It is shown that, depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell, different scenarios of the modes coupling can be realized.Additionally to the asymptotic solution, numerical simulation of the modes coupling is performed by means of the Runge-Kutta fourth-order method. The obtained numerical and analytical results demonstrate good qualitative agreement.

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Section: Introductionmentioning

confidence: 99%

Influence of geometric and physical nonlinearities on the internal resonances of a finite continuous rod with a microstructure

Andrianov

Awrejcewicz

Danishevskyy

et al. 2017

Journal of Sound and Vibration

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“…In this paper, beam-like problems will be considered. Bridges [5] and tall buildings [6] can be modelled by elastic beams.…”

Section: Introductionmentioning

confidence: 99%

On aspects of damping for a vertical beam with a tuned mass damper at the top

Hijmissen

Horssen

2006

Nonlinear Dyn

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In this paper, the wind-induced, horizontal vibrations of a vertical Euler-Bernoulli beam will be considered. At the top of the beam, a tuned mass damper (TMD) has been installed. The horizontal vibrations can be described by an initial-boundary value problem. Perturbation methods will be applied to construct approximations of the solutions of the initialboundary value problem, and it will be shown that the TMD uniformly damps the oscillation modes of the beam. In the analysis, it will be assumed that damping, wind-force, and gravity effects are small but not negligible.

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“…The analysis becomes more complicated for beam equations. These equations are discussed for instance in [3], [4], [5], [6], [10], [11], [18] and [19]. Only a little is known about weakly nonlinear plate equations.…”

Section: Introductionmentioning

confidence: 99%

On the Vibrations of a Simply Supported Square Plate on a Weakly Nonlinear Elastic Foundation

Zarubinskaya

Horssen

2005

Equadiff 2003

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An Asymptotic Theory for a Weakly Nonlinear Beam Equation with a Quadratic Perturbation

Cited by 46 publications

References 18 publications

Influence of geometric and physical nonlinearities on the internal resonances of a finite continuous rod with a microstructure

Influence of geometric and physical nonlinearities on the internal resonances of a finite continuous rod with a microstructure

On aspects of damping for a vertical beam with a tuned mass damper at the top

On the Vibrations of a Simply Supported Square Plate on a Weakly Nonlinear Elastic Foundation

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