Dynamics of coupled linear and essentially nonlinear oscillators with substantially different masses

Gendelman, Oleg; Gorlov, D. V.; Manevitch, Leonid I.; Musienko, Andrey I.

doi:10.1016/j.jsv.2004.09.021

Cited by 68 publications

(33 citation statements)

References 13 publications

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“…If we take into account viscous friction, we can formulate the problem of almost irreversible energy transfer from an initially excited mass to the second, smaller one, known as the energy pumping problem [13][14][15][16][17].…”

Section: Lpts In Damped Systemmentioning

confidence: 99%

“…A real application of the energy pumping phenomenon for the defence of various structural elements is found if a protected system, which can be described, for instance, by the linear model, is connected to a strongly nonlinear element that has a relatively small mass [16]. In such a case elimination of the damping leads to the existence of beats corresponding to LPTs or close trajectories.…”

Section: Lpts In Damped Systemmentioning

confidence: 99%

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New approach to beating phenomenon in coupled nonlinear oscillatory chains

Manevitch¹

2006

Arch Appl Mech

128

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We propose an adequate analytical approach to the description of nonlinear vibration with strong energy exchange between weakly coupled oscillators. The fundamental notion of the limiting phase trajectory (LPT) corresponding to full energy transfer is introduced. In a certain sense this is alternative to the nonlinear normal mode (NNM) characterized by conservation of energy. Many well-known approximations based on NNMs turn out to be valid for the case of weak energy exchange, and the proposed approach can be used for the description of nonlinear processes with strong energy exchange between weakly coupled oscillators or oscillatory chains. Such a description is formally similar to that of a vibro-impact process and can be considered as generating such a process when dealing with a description of other processes with intensive energy transfer. We consider subsequently two weakly coupled conservative, damped and forced nonlinear oscillators as well as a system of two weakly coupled oscillatory chains. It is shown that two dynamic transitions occur in the system under consideration when the nonlinearity parameter increases. The first, well-known, transition corresponds to the bifurcation of one of the cooperative nonlinear normal modes (NNMs) and leads to its instability and the appearance of two stable normal modes with predominant excitation of one of oscillators. The second transition is characterized by the instability of the the limiting phase trajectory (LPT), corresponding to complete energy transfer between the oscillators. As a result of such a transition the complete energy transfer is broken, and energy localization on one of oscillators (whereas it was initially concentrated on it) becomes possible instead of beating. This motion with the most intensive energy transfer turns out to be as important as NNMs for the explanation of dynamic transformations and the description of closely related regimes.

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Section: Lpts In Damped Systemmentioning

confidence: 99%

Section: Lpts In Damped Systemmentioning

confidence: 99%

New approach to beating phenomenon in coupled nonlinear oscillatory chains

Manevitch¹

2006

Arch Appl Mech

128

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“…Targeted energy transfer has been observed in these strongly nonlinear systems, with the attachments being commonly referred to as nonlinear energy sinks (NESs) [5,6]. Targeted energy transfer describes the nearly irreversible passive transfer of a significant amount of energy initially stored in a linear structure to an appropriately designed strongly nonlinear lightweight attachment which acts, in essence, as passive broadband adaptive boundary controller [7][8][9][10][11][12]. The complex dynamics of these systems results from the capacity of the essentially nonlinear attachment to engage in resonance captures with modes of the linear structure over an extensive range of frequencies and energies.…”

Section: Introductionmentioning

confidence: 99%

Sustained High-Frequency Dynamic Instability of a Nonlinear System of Coupled Oscillators Forced by Single or Repeated Impulses: Theoretical and Experimental Results

Remick¹,

Vakakis

Bergman³

et al. 2013

Journal of Vibration and Acoustics

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This report describes the impulsive dynamics of a system of two coupled oscillators with essential (nonlinearizable) stiffness nonlinearity. The system considered consists of a grounded weakly damped linear oscillator coupled to a light-weight weakly damped oscillating attachment with essential cubic stiffness nonlinearity arising purely from geometry and kinematics. It has been found that under specific impulse excitations the transient damped dynamics of this system tracks a high-frequency impulsive orbit manifold -IOM in the frequency-energy plane. The IOM extends over finite frequency and energy ranges, consisting of a countable infinity of periodic orbits and an uncountable infinity of quasi-periodic orbits of the underlying Hamiltonian system and being initially at rest and subjected to an impulsive force on the linear oscillator. Damped, non-resonant dynamics tracking the IOM then resembles continuous resonance scattering, in effect quickly transitioning between multiple resonance captures over finite frequency and energy ranges. Dynamic instability arises at bifurcation points along this damped transition, causing bursts in the response of the nonlinear light oscillator which resemble selfexcited resonances. It is shown that for appropriate parameter design the system remains in a state of sustained high-frequency dynamic instability under the action of repeated impulses. In turn, this sustained instability results in strong energy transfers from the directly excited oscillator to the lightweight nonlinear attachment, a feature that can be employed in energy harvesting applications. The theoretical predictions are confirmed by experimental results.

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“…To obtain the steady state response of Equations (25) and (26), the following change of variable is assumed: (27) By substituting the above equation into Equations (25) and (26), the equations are rewritten in the form of:…”

Section: Problem Formulationmentioning

confidence: 99%

“…Several authors, including Vakakis et al [27] or Mikhlin and Reshetnikova [29] used the concept of Nonlinear Normal Modes (NNM) and their stability to explain this result. The concept of nonlinear normal modes was first introduced by Rosenberg [33] and has been the subject of many investigations in the past years.…”

Section: Introductionmentioning

confidence: 99%

Effects of mistuning on dynamic behavior of nonlinear cyclic systems with lump masses and cubic nonlinearity

Estabragh

Rad

Rahimi

2016

Mechanics & Industry

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-Mistuning in cyclic symmetric systems increases severely the forced response of system and splits the modes. This paper concerns with nonlinear behavior of mistuned cyclic systems. A nonlinear, mistuned model based on the method of multiple scales is proposed and formulated in which nonlinearity and mistuning parameter is assumed to be in of low order. Next, two mistuned systems were considered and solved by the multiple scale technique. Numerical results demonstrate that mistuning can lead to repeating and scattering of jump phenomena during the excitation frequency whereas in tuned cyclic system it occurs simultaneously (synchronously).

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Dynamics of coupled linear and essentially nonlinear oscillators with substantially different masses

Cited by 68 publications

References 13 publications

New approach to beating phenomenon in coupled nonlinear oscillatory chains

New approach to beating phenomenon in coupled nonlinear oscillatory chains

Sustained High-Frequency Dynamic Instability of a Nonlinear System of Coupled Oscillators Forced by Single or Repeated Impulses: Theoretical and Experimental Results

Effects of mistuning on dynamic behavior of nonlinear cyclic systems with lump masses and cubic nonlinearity

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