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See detailClassification of periodic orbits of two-dimensional homogeneous granular crystals with no pre-compression
Detroux, Thibaut ULg; Starosvetsky, Yuli; Kerschen, Gaëtan ULg et al

in Nonlinear Dynamics (2014), 76(April 2014), 673-696

In the present study we classify the periodic orbits of a squarely packed, uncompressed and undamped, homogeneous granular crystal, assuming that all elastic granules oscillate with the same frequency (i ... [more ▼]

In the present study we classify the periodic orbits of a squarely packed, uncompressed and undamped, homogeneous granular crystal, assuming that all elastic granules oscillate with the same frequency (i.e., under condition of 1:1 resonance); this type of Hamiltonian periodic orbits have been labeled as nonlinear normal modes. To this end we formulate an auxiliary system which consists of a two-dimensional, vibro-impact lattice composed of non-uniform “effective particles” oscillating in an anti-phase fashion. The analysis is based on the idea of balancing linear momentum in both horizontal and vertical directions for separate, groups of particles, whereby each such a group is represented by the single effective particle of the auxiliary system. It is important to emphasize that the auxiliary model can be defined for general finite, squarely packed granular crystals composed of n rows and m columns. The auxiliary model is successful in predicting the total number of such periodic orbits, as well as the amplitude ratios for different periodic regimes including strongly localized ones. In fact this methodology enables one to systematically study the generation of mode localization in these strongly nonlinear, highly degenerate dynamical systems. Good correspondence between the results of the theoretical model and direct numerical simulations is observed. The results presented herein can be further extended to study the intrinsic dynamics of the more complex granular materials, such as heterogeneous two-dimensional and three-dimensional granular crystals and multi-layered structures. [less ▲]

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See detailEfficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures:Part II, analytical study
Sapsis, T. P.; Vakakis, Alexander F.; Gendelman, O. V. et al

in Journal of Sound & Vibration (2009)

We study targeted energy transfer in a two degree-of-freedom damped system under the conditionof1:1transient resonance capture. The system consists of a linear oscillator strongly coupled to an ... [more ▼]

We study targeted energy transfer in a two degree-of-freedom damped system under the conditionof1:1transient resonance capture. The system consists of a linear oscillator strongly coupled to an essentially nonlinear attachment or nonlinear energy sink. In a companion paper[ Quinnetal., Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with1:1resonance captures: part I, Journal of Sound and Vibration 311 (2008)1228–1248]we studied the under lying structure of the Hamiltonian dynamics of this system ,and showed that for sufficiently small values of viscous damping , nonlinear damped transitions are strongly influenced by the under lying topological structure of periodic and quasi periodic or bits of the Hamiltonian system. In this work direct analytical treatment of the governing strongly nonlinear damped equations of motion is performed through slow/fast partitions of the transient responses, in order to investigate analytically the parameter region of optimal targeted energy transfer .To this end, we determine the characteristic time scales of the dynamics that influence the capacity of the nonlinear attachment to passively absorb and locally dissipate broad band energy from the linear oscillator. Then, we prove that optimal targeted energy transfer is realized for initial energies close to the neighbourhood of a homo clinic or bit of the under lying Hamiltonian system. We study analytically transient orbits resulting as perturbations of the homo clinic or bit in the weak lydamped system, and show that this yields an additional slow-time scale in the averaged dynamics, and leads to optimal targeted energy transfer from the linear oscillator to the nonlinear energy sink in a single ‘‘super-slow’’ half-cycle. We show that at higher energies, this ‘‘super-slow’’ half-cycle is replaced by strong nonlinear beats, which lead to significant but suboptimal targeted energy transfer efficiency. Finally, we investigate numerically targeted energy transfer efficiency in this system over a wide range of system parameter sand verify the analytical predictions. 2009 Elsevier Ltd .All rights reserved. [less ▲]

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See detailNonlinear Targeted Energy Transfer in Mechanical and Structural Systems II
Vakakis, Alexander F.; Gendelman, O. V.; Bergman, L. A. et al

Book published by Springer (2009)

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See detailNonlinear normal modes, Part 1: Useful framework for the structural dynamicist
Kerschen, Gaëtan ULg; Peeters, Maxime ULg; Golinval, Jean-Claude ULg et al

in Mechanical Systems & Signal Processing (2009), 23

The conceptofnonlinearnormalmodes(NNMs)isdiscussedinthepresentpaperandits companion,PartII.BecausethereisvirtuallynoapplicationoftheNNMstolarge-scale engineeringstructures ... [more ▼]

The conceptofnonlinearnormalmodes(NNMs)isdiscussedinthepresentpaperandits companion,PartII.BecausethereisvirtuallynoapplicationoftheNNMstolarge-scale engineeringstructures,thesepapersareanattempttohighlightseveralaspectsthat mightdrivetheirdevelopmentinthefuture.Specifically,wesupportthat(i)numerical methodsforthecontinuationofperiodicsolutionspavethewayforaneffectiveand practical computationofNNMs,and(ii)time–frequencyanalysisisparticularlysuitable fortheanalysisoftheresultingdynamics.Anotherobjectiveofthepresentpaperisto describe,insimpleterms,andtoillustratethefundamentalpropertiesofNNMs.Thisis achievedtoconvincethestructuraldynamicistnotnecessarilyacquaintedwiththemthat theyareausefulframeworkfortheanalysisofnonlinearvibratingstructures. & 2008ElsevierLtd.Allrightsreserved [less ▲]

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See detailNonlinear Targeted Energy Transfer in Mechanical and Structural Systems I
Vakakis, Alexander F.; Gendelman, O. V.; Bergman, L. A. et al

Book published by Springer (2009)

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See detailTargeted energy transfer for suppressing aeroelastic instability due to unsteady lift
Lee, Y. S.; Vakakis, Alexander F.; Bergman, L. A. et al

in Targeted energy transfer for suppressing aeroelastic instability due to unsteady lift, Palm Springs, 2009 (2009)

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See detailUsing passive nonlinear targeted energy transfer to stabilize drill-stringsystems
Viguié, Régis ULg; Kerschen, Gaëtan ULg; Golinval, Jean-Claude ULg et al

in International Colloquium on Non-Linear Dynamics of Deep Drill-String Systems, Liège, 2009 (2009)

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See detailAPPLYING L. MANEVITCH’S COMPLEXIFICATION – AVERAGING METHOD TO ANALYZE CONDITIONS FOR OPTIMAL TARGETED ENERGY TRANSFER IN COUPLED OSCILLATORS WITH ESSENTIAL STIFFNESS NONLINEARITY
Sapsis, Themistoklis; Quinn, D. Dane; Gendelman, Oleg et al

in International Conference on Nonlinear Phenomena in Polymer Solids and Low-Dimensional Systems. Moscou, 2008 (2008, July)

We study targeted energy transfer (TET) [1] from a linear damped oscillator (LO) to a light attachment with essential stiffness nonlinearity, caused by 1:1 transient resonance capture (TRC). First, we ... [more ▼]

We study targeted energy transfer (TET) [1] from a linear damped oscillator (LO) to a light attachment with essential stiffness nonlinearity, caused by 1:1 transient resonance capture (TRC). First, we study the underlying Hamiltonian dynamics and show that for sufficiently weak damping, the nonlinear damped transitions of the system are strongly influenced by the underlying topological structure of periodic and quasiperiodic orbits of the hamiltonian system. Then, we formulate conditions that lead to effective or even optimal TET from the linear system to the nonlinear attachment. Direct analytical treatment of the governing strongly nonlinear damped equations of motion is performed by applying L. Manevitch’s complexification – averaging (CX-A) method [2] to perform slow-fast partition of the transient responses, and analytically model the dynamics in the region of optimal TET. This analysis determines the characteristic time scales of the dynamics that influence the capacity of the nonlinear attachment to passively absorb and locally dissipate broadband energy from the linear oscillator in an optimal fashion. [less ▲]

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See detailSUPPRESSION OF AEROELASTIC INSTABILITIES BY BROADBAND PASSIVE TARGETED ENERGY TRANSFERS
Lee, Young S.; McFarland, D. Michael; Vakakis, Alexander F. et al

in Sixth EUROMECH Nonlinear Dynamics Conference, Saint Petersbourg, 2008 (2008, July)

We study passive and nonlinear targeted energy transfers induced by transient resonant interactions between an essentially nonlinear attachment and an in-flow rigid wing model. We show that it is feasible ... [more ▼]

We study passive and nonlinear targeted energy transfers induced by transient resonant interactions between an essentially nonlinear attachment and an in-flow rigid wing model. We show that it is feasible to partially or even completely suppress aeroelastic instabilities in the wing (limit cycle oscillations-LCOs) by passively transferring broadband vibration energy from the wing to the attachment in a one-way irreversible fashion. We study the nonlinear dynamical mechanisms that govern TET and show that they are series of transient or sustained resonance captures in different resonance manifolds of the dynamics. Aeroelastic instability suppression is performed by partially or completely eliminating the triggering mechanism for aeroelastic instability. Through numerical parametric studies we identify three main mechanisms for suppressing aeroelastic instability, and investigate them in detail, both numerically by Empirical Mode decomposition (EMD), and analytically by slow/fast partitions of the transient dynamics. [less ▲]

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See detailPassive non linear TET and its application to vibration absorption: a review
Lee, Y. S.; Vakakis, Alexander F.; Bergman, L. A. et al

in Proceedings of the Institution of Mechanical Engineers - Part K - Journal of Multi-body Dynamics (2008), 222

This review paper discusses recent efforts to passively move unwanted energy from a primary structure to a local essentially non-linear attachment (termed a non-linear energy sink) by utilizing targeted ... [more ▼]

This review paper discusses recent efforts to passively move unwanted energy from a primary structure to a local essentially non-linear attachment (termed a non-linear energy sink) by utilizing targeted energy transfer (TET) (or non-linear energy pumping). First, fundamental theoretical aspects of TET will be discussed, including the essentially non-linear governing dynamical mechanisms for TET. Then, results of experimental studies that validate the TET phenomenon will be presented. Finally, some current engineering applications of TET will be discussed. The concept of TET may be regarded as contrary to current common engineering practise, which generally views non-linearities in engineering systems as either unwanted or, at most, as small perturbations of linear behaviour. Essentially non-linear stiffness elements are intentionally introduced in the design that give rise to new dynamical phenomena that are very beneficial to the design objectives and have no counterparts in linear theory. Care, of course, is taken to avoid some of the unwanted dynamic effects that such elements may introduce, such as chaotic responses or other responses that are contrary to the design objectives. [less ▲]

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See detailToward a fundamental understanding of the Hilbert-Huang Transform in nonlinear dynamics
Kerschen, Gaëtan ULg; Vakakis, Alexander F.; Lee, Y. S. et al

in Journal of Vibration & Control (2008)

The Hilbert–Huang transform(HHT) has been shown to be effective for characterizing a wide range of nonstationary signals in terms of elemental components through what has been called the empirical mode ... [more ▼]

The Hilbert–Huang transform(HHT) has been shown to be effective for characterizing a wide range of nonstationary signals in terms of elemental components through what has been called the empirical mode decomposition (EMD). The HHT has been utilized extensively despite the absence of a serious analytical foundation, as it provides a concise basis for the analysis of strongly nonlinear systems. In this paper, an attempt is made to provide the missing theoretical link, showing the relationship between the EMD and the slow-flow equations of a system. The slow-flow reduced-order model is established by performing a partition between slow and fast dynamics using the complexification-averaging technique in order to derive a dynamical system described by slowly-varying amplitudes and phases. These slow-flow variables can also be extracted directly from the experimental measurements using the Hilbert transform coupled with the EMD. The comparison between the experimental and analytical results forms the basis of a novel nonlinear system identification method, termed the slow-flow model identification (SFMI) method. Through numerical and experimental application examples, we demonstrate that the proposed method is effective for characterization and parameter estimation of multi-degree-of-freedom nonlinear systems. [less ▲]

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See detailImpulsive periodic and quasi-period orbits in coupled oscillators with essential nonlinearity
Kerschen, Gaëtan ULg; Gendelman, Oleg; Vakakis, Alexander F. et al

in Communications in Nonlinear Science & Numerical Simulation (2008), 13

We study the impulsive responses of a grounded linear oscillator coupled to a light nonlinear attachment through an essentially nonlinear (nonlinearizable) stiffness. We analyze the periodic and quasi ... [more ▼]

We study the impulsive responses of a grounded linear oscillator coupled to a light nonlinear attachment through an essentially nonlinear (nonlinearizable) stiffness. We analyze the periodic and quasi-periodic dynamics of the undamped system forced by a single impulse on the linear oscillator and being initially at rest, by considering separately low-, moderateand high-energy impulsive motions. The motivation for studying the impulsive dynamics of this system centers on passive targeted energy transfer properties of the corresponding weakly damped one, that is, of the possibility of one-way, irreversible transfer of energy from the linear oscillator to the nonlinear attachment. A rather surprising aspect of this work is the complexity of the analysis required to study the impulsive dynamics of this system, due to its high degeneracy, as it undergoes a co-dimension three bifurcation. 2006 Elsevier B.V. All rights reserved. [less ▲]

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See detailEfficiency of TET in coupled oscillators associated with 1: resonance: Part 1
Quinn, D. Dane; Gendelman, Oleg; Kerschen, Gaëtan ULg et al

in Journal of Sound & Vibration (2008), 311

We study targeted energy transfers and nonlinear transitions in the damped dynamics of a two degree-of-freedom system of coupled oscillators (a linear oscillator with a lightweight, essentially nonlinear ... [more ▼]

We study targeted energy transfers and nonlinear transitions in the damped dynamics of a two degree-of-freedom system of coupled oscillators (a linear oscillator with a lightweight, essentially nonlinear, ungrounded attachment), caused by 1:1 resonance captures of the dynamics. Part I of this work deals with the underlying structure of the Hamiltonian dynamics of the system, and demonstrates that, for sufficiently small values of viscous damping, the damped transitions are strongly influenced by the underlying topological structure of periodic and quasiperiodic orbits of the corresponding Hamiltonian system. Focusing exclusively on 1:1 resonance captures in the system, it is shown that the topology of these damped transitions affect drastically the efficiency of passive energy transfer from the linear system to the nonlinear attachment. Then, a detailed computational study of the different types of nonlinear transitions that occur in the weakly damped system is presented, together with an analytical treatment of the nonlinear stability of certain families of periodic solutions of the underlying Hamiltonian system that strongly influence the said transitions. As a result of these studies, conditions on the system and forcing parameters that lead to effective or even optimal energy transfer from the linear system to the nonlinear attachment are determined. In Part II of this work, direct analytical treatment of the governing strongly nonlinear damped equations of motion is performed, in order to analytically model the dynamics in the region of optimal energy transfer, and to determine the characteristic time scales of the dynamics that influence the capacity of the nonlinear attachment to passively absorb and locally dissipate broadband energy from the linear oscillator. r 2007 Elsevier Ltd. All rights reserved [less ▲]

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See detailNonlinear Normal Modes, Part I: An Attempt To Demystify Them
Kerschen, Gaëtan ULg; Peeters, Maxime ULg; Golinval, Jean-Claude ULg et al

in 26th International Modal Analysis Conference, Orlando, 2008 (2008)

The concept of nonlinear normal modes (NNMs) is discussed in the present paper and its companions, Parts II and III. Because there is virtually no application of the NNMs to large-scale engineering ... [more ▼]

The concept of nonlinear normal modes (NNMs) is discussed in the present paper and its companions, Parts II and III. Because there is virtually no application of the NNMs to large-scale engineering structures, these papers are an attempt to highlight several aspects that might drive their development in the future. Specifically, we support that (i) numerical methods for the continuation of periodic solutions pave the way for an effective and practical computation of NNMs, and (ii) time-frequency analysis is particularly suitable for the analysis of the resulting dynamics. Another objective of the present paper is to describe, in simple terms, and to illustrate the fundamental properties of NNMs. This is achieved to convince the structural dynamicist not necessarily acquainted with them that they are a useful framework for the analysis of nonlinear vibrating structures. [less ▲]

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See detailEfficient targeted energy transfers in coupled nonlinear oscillators through 1:1 transcient resonance captures:
Sapsis, Themistoklis; Quinn, D. Dane; Gendelman, Oleg et al

in Sixth EUROMECH Nonlinear Dynamics Conference, Saint Petersbourg, 2008 (2008)

We study targeted energy transfer (TET) in a two degree-of-freedom damped system caused by 1:1 transient resonance capture (TRC). The system consists of a linear oscillator strongly coupled to an ... [more ▼]

We study targeted energy transfer (TET) in a two degree-of-freedom damped system caused by 1:1 transient resonance capture (TRC). The system consists of a linear oscillator strongly coupled to an essentially nonlinear attachment. First, we study the underlying structure of the Hamiltonian dynamics of the system, and then show that, for sufficiently small values of viscous damping, the nonlinear damped transitions are strongly influenced by the underlying topological structure of periodic and quasiperiodic orbits of the hamiltonian system. Then, a detailed computational study of the different types of nonlinear transitions that occur in the weakly damped system is presented. As a result of these studies, conditions that lead to effective or even optimal TET from the linear system to the nonlinear attachment are determined. Finally, direct analytical treatment of the governing strongly nonlinear damped equations of motion is performed through slow/fast partition of the transient responses, in order to analytically model the dynamics the region of optimal TET, and to determine the characteristic time scales of the dynamics that influence the capacity of the nonlinear attachment to passively absorb and locally dissipate broadband energy from the linear oscillator. [less ▲]

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See detailTheoretical Understanding of Targeted Energy Transfers for Suppressing Aeroelastic Instabilities in a Nonlinear Aeroelastic Test Apparatus
Lee, Young S.; McFarland, D. Michael; Kerschen, Gaëtan ULg et al

in 45th Technical Meeting of the Society of Engineering Science, Urbana Champaign, 2008 (2008)

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See detailEnhancing robustness of aerolastic instability suppression using MDOF energy sinks
Lee, Young S.; Vakakis, Alexander F.; Bergman, Lawrence A. et al

in AIAA Journal (2008), 46(6), 1371-1394

In this last of a three paper sequence, we use simultaneous multimodal broadband targeted energy transfers to multi-degree-of-freedom nonlinear energy sinks to improve the robustness of aeroelastic ... [more ▼]

In this last of a three paper sequence, we use simultaneous multimodal broadband targeted energy transfers to multi-degree-of-freedom nonlinear energy sinks to improve the robustness of aeroelastic instability suppression of a rigid wing with structural nonlinearities. A numerical bifurcation analysis of limit cycle oscillations of the wing with the multi-degree-of-freedom nonlinear energy sinks attached shows that controlling the lower parameter value for limit point cycle bifurcation to occur above Hopf bifurcation is crucial to enhancing the robustness of limit cycle oscillation suppression. We demonstrate that multi-degree-of-freedom nonlinear energy sinks can greatly enhance the robustness of limit cycle oscillation suppression, compared with single-degree-of-freedom nonlinear energy sinks (which were studied in our previous papers), with a much smaller total mass.We also investigate the nonlinear modal interactions that occur between the aeroelastic modes and the multi-degree-of-freedom nonlinear energy sinks, in an effort to gain a physical understanding of the mechanisms governing instability suppression. We demonstrate that a properly designed multi-degree-of-freedom nonlinear energy sink provides robustness of aeroelastic instability suppression by efficiently, passively, and rapidly transferring a significant portion of unwanted vibration energy to the furthest mass of the nonlinear energy sink. Consideration of other types of multi-degree-of-freedom nonlinear energy sinks suggests that the robustness enhancement is achieved by the concentrated mass effect of the attached nonlinear energy sinks. [less ▲]

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See detailIdentification of the Slow-Flow Dynamics of Transonic Aeroelastic Response from Time-Series Data
Lee, Y. S.; Vakakis, Alexander F.; McFarland, D. M. et al

in 45th Technical Meeting of the Society of Engineering Science, Urbana Champaign, 2008 (2008)

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See detailEmpirical Mode Decomposition in the Reduced-Order Modeling of Aeroelastic Systems
Lee, Young S; McFarland, D. Michael; Vakakis, Alexander F. et al

in 49th AIAA Structural Dynamics and Materials Conference, Shaumberg, 2008 (2008)

A relationship between IntrinsicMode Functions (IMFs), derived from the Empirical Mode Decomposition (EMD), and the slow-flow model of a nonlinear dynamical system has been exploited in the development of ... [more ▼]

A relationship between IntrinsicMode Functions (IMFs), derived from the Empirical Mode Decomposition (EMD), and the slow-flow model of a nonlinear dynamical system has been exploited in the development of the Slow Flow Model Identification (SFMI) method for strongly nonlinear systems, in which the physical parameters of such systems are identified from experimental data. Both the slow flows and IMFs provide the means to expand a general multicomponent signal in terms of a series of simpler, dominant, monocomponent signals. The slow flows are obtained analytically, for example through application of the method of complexification and averaging (CxA), which transforms the equations of motion into a set of approximate equations in amplitude and phase for each modeled frequency component. In contrast, the EMD characterizes a signal through the envelope and phase of its elemental components, the IMFs. Thus, between nonlinear transitions, the equations derived using the CxA method govern the amplitude and phase of the modeled IMFs. Application of SFMI has, until now, been limited to low-dimensional systems subjected to impulsive excitation. Herein, the method is extended to identification of a planar rigid airfoil [less ▲]

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See detailTargeted Energy Transfers for Suppressing Aeroelastic Instability due to Unsteady Lift
Vakakis, Alexander F.; Bergman, Lawrence A.; McFarland, D. M. et al

in AVT-152 Symposium on Limit Cycle Oscillations, Norway, 2008 (2008)

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