Analysis of the 1:1 resonant energy exchanges between coupled oscillators with rheologies

Savadkoohi, Alireza Ture; Lamarque, Claude‐Henri; Weiss, Mathieu; Vaurigaud, Bastien; Charlemagne, S.

doi:10.1007/s11071-016-2792-3

Cited by 19 publications

(18 citation statements)

References 34 publications

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“…Eq. 28 corresponds to the slow invariant manifold (SIM) of the system that is a geometrical surface for all system behaviors, including periodic [17] and strongly modulated responses [18]. The SIM is plotted in Fig.…”

Section: Slow Invariant Manifold Of the Systemmentioning

confidence: 99%

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Nonlinear Vibratory Energy Exchanges between a Two-Degree-of-Freedom Pendulum and a Nonlinear Absorber

2019

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The multi-time scale responses of a two degrees of freedom pendulum coupled with a nonlinear absorber is studied. The absorber is positioned in an arbitrary direction with respect to those of the pendulum oscillations. Phase-dependent slow invariant manifold of the system and its stable zones are traced at fast time scale while studying system responses at slow time scale around the slow invariant manifold leads to detection of equilibrium and singular points. Moreover, the amplitude-frequency curves of the system are detected showing the possibility of existence of isolated branches, which could correspond to high energy levels for pendulum oscillations. All analytic developments are confronted with numerical results collected from direct numerical integration of system equations. Depending on characteristics of external excitations, the system can face periodic or modulated regimes.

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Section: Slow Invariant Manifold Of the Systemmentioning

confidence: 99%

“…Equilibrium points of the system are obtained via considering the Eqs. 40 and 41 and evolution of the SIM at τ 1 time scale [17]:…”

Section: Equilibrium Pointsmentioning

confidence: 99%

Nonlinear Vibratory Energy Exchanges between a Two-Degree-of-Freedom Pendulum and a Nonlinear Absorber

2019

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“…To find the equilibrium points of the system we have to solve the system of equations formed by the Eqs. 41 and 42 and the evolution of the SIM at τ 1 time scale [8]:…”

Section: Stability Of the Simmentioning

confidence: 99%

“…One of them called nonlinear energy sink (NES) [5,6], has a purely nonlinear restoring force function (cubic nonlinearity in its original form) with no linear part. Its interactions with the main system lead to strongly modulated response (SMR) oscillations [7] or periodic regime [8]. It is possible to replace the polynomial nonlinearity by a non-smooth piecewise linear function [9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Passive control of a two degrees-of-freedom pendulum by a non-smooth absorber

2019

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A pendulum, which can oscillate in two directions, is subjected to a generalized external force. A non-smooth absorber is coupled to the pendulum with an arbitrary location and orientation. The equations of the system are derived and are treated with a multiple scale method. At fast time scale, the topology of the slow invariant manifold is described with its stable and unstable zones. The equilibrium and singular points of the system are detected at the first slow time scale. The responses of the main system, given as a function of the frequency of the external force, show reductions of the vibration levels. The analytic predictions are compared by direct numerical time integration of the equations of the system. They illustrate the operationality of the non-smooth absorber in several cases.

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“…To treat this system of N nonlinear equations, the method described in [33] is used, keeping the equations in a discrete form. Three analytical tools are implemented:…”

Section: Description Of the System And Explanation Of The Analytical mentioning

confidence: 99%

Vibratory control of a linear system by addition of a chain of nonlinear oscillators

2017

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An N -degree-of-freedom model consisting of a single-degree-of-freedom linear system coupled to a chain of (N − 1) light nonlinear oscillators is studied. The connection between the chain and the singledegree-of-freedom system is supposed to be linear. Time multi-scale system behaviors at fast and slow time scales are investigated and lead to the detection of the slow invariant manifold and equilibrium and singular points. These points correspond to periodic regimes and strongly modulated responses, respectively. These analytical developments are used to provide evidence of transfer of vibratory energy of the main system to the chain in the form of localized modes during periodic regimes and extreme energy exchanges between modes when the overall structure faces singularities. Furthermore, analytical predictions at slow time scale and nonlinear normal modes of the system are compared with numerical results obtained from direct time integration of the system equations, showing a good agreement between them. Finally, we present a procedure showing how these analytical developments can be used to study a system where the main structure is replaced by a multi-degree-of-freedom linear system, by projecting its dynamics on one of its modes.

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Analysis of the 1:1 resonant energy exchanges between coupled oscillators with rheologies

Cited by 19 publications

References 34 publications

Nonlinear Vibratory Energy Exchanges between a Two-Degree-of-Freedom Pendulum and a Nonlinear Absorber

Nonlinear Vibratory Energy Exchanges between a Two-Degree-of-Freedom Pendulum and a Nonlinear Absorber

Passive control of a two degrees-of-freedom pendulum by a non-smooth absorber

Vibratory control of a linear system by addition of a chain of nonlinear oscillators

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