A general solution procedure for the forced vibrations of a system with cubic nonlinearities: Three-to-one internal resonances with external excitation

Özhan, B. Burak; Pakdemi̇rli̇, Mehmet

doi:10.1016/j.jsv.2010.01.010

Cited by 30 publications

(2 citation statements)

References 20 publications

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“…Pakdemirli and Boyaci (2003) researched the nonlinear transverse quivering of the beam formation with a supporting nonlinearity, while Tsiatas (2010) looked at how the presence of a nonlinear elastic base affected the beam structure’s ability to vibrate in a transverse direction. Vibration features and change responses of beam structures with cubic nonlinearities were widely investigated in Refs (Ghayesh et al, 2011, 2012; Özhan and Pakdemirli, 2010), which found that cubic nonlinearities significantly influenced the dynamic responses of beam frameworks. Ghayesh (2012) researched the nonlinear dynamic output of a simply-supported beam structure, in which the nonlinearity was introduced through a nonlinear supporting spring.…”

Section: Introductionmentioning

confidence: 99%

Transverse forced nonlinear vibration analysis of a double-beam system with a supporting nonlinearity

Zhao

et al. 2022

Journal of Vibration and Control

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To exploit the potential application of supporting nonlinearity in marine engineering, an attempt is made to establish the transverse forced vibration analysis model of a double-beam system supported by a spring-mass system that is nonlinear. This kind of vibration system consists of two beam sections, boundary supports, a coupling component, and a nonlinear spring-mass arrangement. The variational approach and the generalized Hamiltonian concept are used to develop the governing equations of such a double-beam system. The Galerkin truncation method (GTM) is a technique for obtaining the governing equations’ residual equations. By solving the associated residual equations numerically, the nonlinear responses of the double-beam system can be figured out. The GTM has good solidity and correctness in the prediction of the vibration system’s forced transverse vibration. The dynamic responses of the double-beam structure supported by a spring-mass system that is nonlinear are subtle to their initial calculation values. Appropriate parameters of the nonlinear support will subdue the level of vibration at the boundary of the double-beam system. In contrast, unsuitable parameters of the nonlinear support motivate complex dynamic responses of the double-beam system and harmfully influence the vibration repression at the boundary of the vibration system.

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Section: Introductionmentioning

confidence: 99%

Transverse forced nonlinear vibration analysis of a double-beam system with a supporting nonlinearity

Zhao

et al. 2022

Journal of Vibration and Control

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“…These early studies were pursued and extended by: Chen and co-workers (Tang et al 2009, Chen and Ding 2010, Huang et al 2011, Yang et al 2012, who considered string and different beam models of the system and employed different analytical and numerical methods; Marynowski and co-workers Kapitaniak 2002, Marynowski andKapitaniak 2007), who considered several energy dissipation mechanisms in the model; Pellicano and Vestroni (Pellicano and Vestroni 2002), who investigated the dynamics of high-speed axially moving systems; Suweken and Van Horssen (Suweken and Van Horssen 2003), who investigated the vibrations of the system with weak nonlinearity and found several internal resonances in the system dynamics; Huang et al (2011), who employed the method of harmonic balance to investigate the system dynamics; Pakdemirli and co-workers (Pakdemirli et al 1994, Pakdemirli and Ulsoy 1997, Pakdemirli and Özkaya 1998, Öz et al 2001, Pakdemirli and BoyacI 2003, Burak Özhan and Pakdemirli 2010, who conducted systematic research in this area by employing some perturbation techniques; Stylianou and Tabarrok (Stylianou and Tabarrok 1994), who used the finite element method to examine the system dynamics; and Nguyen and Hong (2011), who developed a novel control algorithm to suppress the transverse vibrations of an axially moving web.…”

Section: Introductionmentioning

confidence: 99%

Internal resonance and nonlinear response of an axially moving beam: two numerical techniques

Ghayesh¹,

Amabili²

2012

Coupled Systems Mechanics

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The nonlinear resonant response of an axially moving beam is investigated in this paper via two different numerical techniques: the pseudo-arclength continuation technique and direct time integration. In particular, the response is examined for the system in the neighborhood of a three-to-one internal resonance between the first two modes as well as for the case where it is not. The equation of motion is reduced into a set of nonlinear ordinary differential equation via the Galerkin technique. This set is solved using the pseudo-arclength continuation technique and the results are confirmed through use of direct time integration. Vibration characteristics of the system are presented in the form of frequency-response curves, time histories, phase-plane diagrams, and fast Fourier transforms (FFTs).

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Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams

Ding

2011

Acta Mech Sin

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Steady-state periodical response is investigated for an axially moving viscoelastic beam with hybrid supports via approximate analysis with numerical confirmation. It is assumed that the excitation is spatially uniform and temporally harmonic. The transverse motion of axially moving beams is governed by a nonlinear partial-differential equation and a nonlinear integro-partial-differential equation. The material time derivative is used in the viscoelastic constitutive relation. The method of multiple scales is applied to the governing equations to investigate primary resonances under general boundary conditions. It is demonstrated that the mode uninvolved in the resonance has no effect on the steady-state response. Numerical examples are presented to demonstrate the effects of the boundary constraint stiffness on the amplitude and the stability of the steady-state response. The results derived for two governing equations are qualitatively the same, but quantitatively different. The differential quadrature schemes are developed to verify those results via the method of multiple scales.

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A general solution procedure for the forced vibrations of a system with cubic nonlinearities: Three-to-one internal resonances with external excitation

Cited by 30 publications

References 20 publications

Transverse forced nonlinear vibration analysis of a double-beam system with a supporting nonlinearity

Transverse forced nonlinear vibration analysis of a double-beam system with a supporting nonlinearity

Internal resonance and nonlinear response of an axially moving beam: two numerical techniques

Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams

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