Dynamic interaction of a semi-infinite linear chain of coupled oscillators with a strongly nonlinear end attachment

Manevitch, Leonid I.; Gendelman, Oleg; Musienko, Andrey I.; Vakakis, Alexander F.; Bergman, Lawrence A.

doi:10.1016/s0167-2789(02)00806-0

Cited by 42 publications

(24 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Again one can present the solution of (19) as (15) where the smooth functions X i (τ ) and Y i (τ ) satisfy the equations ∂ ∂τ…”

Section: Lpts In Damped Systemmentioning

confidence: 99%

See 1 more Smart Citation

New approach to beating phenomenon in coupled nonlinear oscillatory chains

Manevitch¹

2006

Arch Appl Mech

128

View full text Add to dashboard Cite

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.

show abstract

“…Again one can present the solution of (19) as (15) where the smooth functions X i (τ ) and Y i (τ ) satisfy the equations ∂ ∂τ…”

Section: Lpts In Damped Systemmentioning

confidence: 99%

“…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%

New approach to beating phenomenon in coupled nonlinear oscillatory chains

Manevitch¹

2006

Arch Appl Mech

128

View full text Add to dashboard Cite

show abstract

“…They can engage either in sequential resonance captures of several modes of the primary system, known as resonance capture cascades [19,20], or in relaxation oscillations, known as strongly modulated response (SMR) cycles when modes of the main system are closely spaced [21]. One can find other examples in the work of Gendelman and Manevitch [22], Vakakis et al [23], Manevitch et al [24], and Rothos and Vakakis [25] who study energy pumping achieved by a nonlinear attachment connected to semi-infinite continuous string and rod and semi-infinite and infinite linear chains of oscillators, respectively. Nonetheless, it can be convenient for design reasons to couple multiple NES to the structure.…”

Section: Introductionmentioning

confidence: 99%

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

2017

View full text Add to dashboard Cite

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.

show abstract

“…In papers by Vakakis, Manevitch, Gendelman, Bergman, and others [21][22][23][24][25], theoretical investigation and some experimental verification on the use of nonlinear localization for reducing the transmitted vibrations in structures subjected to transient base motions has been presented. Conditions for the effective energy transfer and localization are discussed.…”

Section: Introductionmentioning

confidence: 99%

“…The experimental assembly, containing the main linear subsystem and the nonlinear absorber is described in [24]. Authors of the works [21][22][23][24][25] define the controlling process of the space energy transfer from the place of their initial appearance in the system under consideration to the other specific place where the energy, as the final result, is localized, as the "energy pumping." It means a controled, unidirectional process of the energy transfer to passive nonlinear receiver where the energy is localized and diminishes in time due to damping dissipation.…”

Section: Introductionmentioning

confidence: 99%

Transient in a two-DOF nonlinear system

2007

View full text Add to dashboard Cite

Dynamics of a system containing a linear oscillator, linearly coupled to an essentially nonlinear attachment, is considered. A damping is taken into account. It is assumed that some initial excitation implies vibrations of the linear oscillator. Envelops of the subsystem's kinetic energies are selected to use the numerical investigation of transient in the system. The parametrical optimization approach is used to obtain regions of effective energy transfer in the system parameter space. It is demonstrated that this efficient energy transfer may be obtained for a rather small value of the attachment mass.

show abstract

Dynamic interaction of a semi-infinite linear chain of coupled oscillators with a strongly nonlinear end attachment

Cited by 42 publications

References 20 publications

New approach to beating phenomenon in coupled nonlinear oscillatory chains

New approach to beating phenomenon in coupled nonlinear oscillatory chains

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

Transient in a two-DOF nonlinear system

Contact Info

Product

Resources

About