Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure

Haddow, Anne; Barr, A.D.S.; Mook, Dean T.

doi:10.1016/0022-460x(84)90272-4

Cited by 200 publications

(94 citation statements)

References 6 publications

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“…Another novel method of vibration suppression, which is closely tied to the "autoparametric" vibration absorber, makes use of the so-called saturation effect in systems that possess a 2:1 internal resonance. This phenomenon, which was first uncovered in the context of ship motions in the 1970s by Nayfeh, Mook, and Marshall (7) and later studied in the context of structural systems by Haddow, Barr, and Mook (121) , relies on the presence of quadratic nonlinearities in the system. In these systems, as the excitation amplitude increases, the response of the directly excited vibration mode saturates at an essentially fixed amplitude, while that of the other vibration mode (the indirectly excited mode whose natural frequency is one half of that the directly excited mode) grows and "soaks up" the vibration energy.…”

Section: Journal Of System Design and Dynamicsmentioning

confidence: 98%

A Review of Nonlinear Dynamics of Mechanical Systems in Year 2008

Shaw

Balachandran

2008

JSDD

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In this article, we provide an overview of a selection of topics of current interest in nonlinear dynamics and vibrations of mechanical systems. Specifically, we cover the traditional topics of structural dynamics, rotating systems, vibration control, vehicle dynamics, and machining, as well as some topics that have emerged more recently, namely micro-and nano-electromechanical systems, piecewise smooth systems, and structural health monitoring and system identification. In the introductory and closing sections, we offer our perspective on the state of affairs in nonlinear dynamics and its utility in the study of mechanical systems.

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Section: Journal Of System Design and Dynamicsmentioning

confidence: 98%

A Review of Nonlinear Dynamics of Mechanical Systems in Year 2008

Shaw

Balachandran

2008

JSDD

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“…This second-order internal resonance involves quadratic nonlinearity, and is now classical, since the first report of its effect on the response of a ship system by Froude [10,22]. References [11,21,23,24,35] provide a complete picture of analytical solutions and experimental observations. Note that we use here the terminology "1:2" resonance to name that case whereas it is often denoted 2:1 resonance in other studies.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances

et al. 2013

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. thus displaying a 1:2:4 internal resonance. The second system exhibits a 1:2:2 internal resonance, so that the frequency relationship reads ω 3 ω 2 2ω 1 . Multiple scales method is used to solve analytically the forced oscillations for the two models excited on each degree of freedom at primary resonance. A thorough analytical study is proposed, with a particular emphasis on the stability of the solutions. Parametric investigations allow to get a complete picture of the dynamics of the two systems. Results are systematically compared to the classical 1:2 resonance, in order to understand how the presence of a third oscillator modifies the nonlinear dynamics and favors the presence of unstable periodic orbits.

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“…Several different analytical and numerical approaches have been used to approach this type of problem, such as perturbation methods [18], nonlinear normal modes [13,22,23,25] or normal form analysis [2,11,21]. The majority of work in the literature on modal interaction is based on 2-DoF nonlinear systems where two modes interact, see for example [3,5,[8][9][10]13,29], although for continuous systems higher numbers of modes can typically be retained in the approximation, see for example [24]. Some work has been carried out on 3-DoF lumped mass systems in the context of nonlinear vibration suppression [12].…”

Section: Introductionmentioning

confidence: 99%

$$N-1$$ N - 1 modal interactions of a three-degree-of-freedom system with cubic elastic nonlinearities

et al. 2015

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In this paper the N − 1 nonlinear modal interactions that occur in a nonlinear three-degree-offreedom lumped mass system, where N = 3, are considered. The nonlinearity comes from springs with weakly nonlinear cubic terms. Here, the case where all the natural frequencies of the underlying linear system are close (i.e. ω n1 : ω n2 : ω n3 ≈ 1 : 1 : 1) is considered. However, due to the symmetries of the system under consideration, only N − 1 modes interact. Depending on the sign and magnitude of the nonlinear stiffness parameters, the subsequent responses can be classified using backbone curves that represent the resonances of the underlying undamped, unforced system. These backbone curves, which we estimate analytically, are then related to the forced response of the system around resonance in the frequency domain. The forced responses are computed using the continuation software AUTO-07p. A comparison of the results gives insights into the multi-modal interactions and shows how the frequency response of the system is related to those branches of the backbone curves that represent such interactions.

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Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure

Cited by 200 publications

References 6 publications

A Review of Nonlinear Dynamics of Mechanical Systems in Year 2008

A Review of Nonlinear Dynamics of Mechanical Systems in Year 2008

Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances

$$N-1$$ N - 1 modal interactions of a three-degree-of-freedom system with cubic elastic nonlinearities

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