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See detailNonlinear MDOF System Characterization and Identification using the Hilbert- Huang Transform: Experimental Demonstration
Kerschen, Gaëtan ULg; Vakakis, Alexander F.; Lee, Young.S et al

in 25th International Modal Analysis Conference, Orlando, 2007 (2007)

The Hilbert transform is one of the most successful approaches to tracking the varying nature of vibration of a large class of nonlinear systems, thanks to the extraction of backbone curves from ... [more ▼]

The Hilbert transform is one of the most successful approaches to tracking the varying nature of vibration of a large class of nonlinear systems, thanks to the extraction of backbone curves from experimental data. Because signals with multiple frequency components do not admit a well-behaved Hilbert transform, this transform is inherently limited to the analysis of single-degree-of-freedom systems; this shortcoming is potentially overcome by the Hilbert-Huang transform (HHT). In this study, the joint application of the complexification-averaging method and the HHT enables us to develop a new technique, the slow-flow model identification method. Through an experimental application, we demonstrate that the proposed method is adequate for characterizing and identifying multi-degreeof-freedom nonlinear systems. [less ▲]

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See detailNonlinear modal analysis and energy localization in a bladed disk assembly
Georgiades, Fotios; Peeters, Maxime ULg; Kerschen, Gaëtan ULg et al

in ASME Turbo Expo 2008: Power for Land, Sea and Air, Berlin, 2008 (2008, June)

The objective of this study is to carry out modal analysis of nonlinear periodic structures using nonlinear normal modes (NNMs). The NNMs are computed numerically with a method developed in [18] that is ... [more ▼]

The objective of this study is to carry out modal analysis of nonlinear periodic structures using nonlinear normal modes (NNMs). The NNMs are computed numerically with a method developed in [18] that is using a combination of two techniques: a shooting procedureand a method for the continuation of periodic motion. The proposed methodology is applied to a simplified model of a perfectly cyclic bladed disk assembly with 30 sectors. The analysis shows that the considered model structure features NNMs characterized by strong energy localization in a few sectors. This feature has no linear counterpart, and its occurrence is associated with the frequencyenergy dependence of nonlinear oscillations. [less ▲]

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See detailNonlinear modal analysis of a full-scale aircraft
Kerschen, Gaëtan ULg; Peeters; Golinval, Jean-Claude ULg et al

in Journal of Aircraft (2013), 50

Nonlinear normal modes (NNMs), which are defined as a nonlinearextension of the concept of linear normal modes, are a rigorous tool for nonlinear modal analysis. The objective of this paper is to ... [more ▼]

Nonlinear normal modes (NNMs), which are defined as a nonlinearextension of the concept of linear normal modes, are a rigorous tool for nonlinear modal analysis. The objective of this paper is to demonstrate that the computation of NNMs and of their oscillation frequencies can now be achieved for complex, real-world aerospace structures. The application considered in this study is the airframe of the Morane-Saulnier Paris aircraft. Ground vibration tests of this aircraft exhibited softening nonlinearities in the connection between the external fuel tanks and the wing tips. The NNMs of this aircraft are computed from a reduced-order nonlinear finite element model using a numerical algorithm combining shooting and pseudo-arclength continuation. Several NNMs, involving, e.g., wing bending, wing torsion and tail bending, are presented, which highlights that the aircraft can exhibit very interesting nonlinear phenomena. Specifically, it is shown that modes with distinct linear frequencies can interact and generate additional nonlinear modes with no linear counterpart. [less ▲]

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See detailNonlinear modal analysis of aerospace structures
Peeters, Maxime ULg; Kerschen, Gaëtan ULg; Golinval, Jean-Claude ULg et al

in Proceedings of the 3rd International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (2011)

The dynamic systems theory is well-established for linear systems and can rely on mature tools such as the theories of linear operators and linear integral transforms. This is why theoretical and ... [more ▼]

The dynamic systems theory is well-established for linear systems and can rely on mature tools such as the theories of linear operators and linear integral transforms. This is why theoretical and experimental modal analysis, i.e., the computation of vibration modes from a mathematical model and from experimental data, respectively, is really quite sophisticated and advanced. Even though linear modal analysis served, and is still serving, the structural dynamics community for applications ranging from bridges to satellites, it is commonly accepted that nonlinearity is a frequent occurrence in engineering structures. Because linear modal analysis fails in the presence of nonlinear dynamical phenomena, the development of a practical nonlinear analog of modal analysis would be an extremely timely contribution; it is clearly missing in the structural dynamics literature. A new framework for nonlinear modal analysis of real-world structures, which includes the computation of nonlinear modes from finite element models, is introduced in this paper. This framework will permit a rigorous, yet understandable by the practicing engineer, analysis of nonlinear dynamical phenomena. It will also provide solid theoretical foundations for extending finite element model validation to nonlinear aerospace structures. [less ▲]

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See detailNonlinear Modal Analysis of Conservative and Nonconservative Aerospace Structures
Renson, Ludovic ULg

Doctoral thesis (2014)

The concept of nonlinear normal modes (NNMs) provides a solid and rigorous theoretical framework for the analysis of the nonlinear oscillations of mechanical systems. If NNMs have been studied since more ... [more ▼]

The concept of nonlinear normal modes (NNMs) provides a solid and rigorous theoretical framework for the analysis of the nonlinear oscillations of mechanical systems. If NNMs have been studied since more than fifty years, it is only very recently that contributions dealing with their numerical calculation have been reported in the literature. Although these methods pave the way for the application of NNMs to more complex systems, they have not yet reached the necessary maturity. In this context, the purpose of this research is (i) to further investigate the performance of an existing method for computing the NNMs of conservative systems and (ii) to propose two new methods for the computation of NNMs of nonconservative systems. The first contribution of this thesis is to calculate the NNMs of a real-life aerospace structure, the SmallSat spacecraft developed by EADS Astrium. An algorithm that combines an advanced shooting method with the pseudo-arclength continuation technique is utilized. We show that the NNMs provide a very useful interpretation of the strongly nonlinear dynamics of the spacecraft. One specific contribution is to numerically reproduce with great fidelity several interactions between modes with noncommensurate linear frequencies that were observed experimentally. The second original contribution of this thesis is to develop two new methods for computing the NNMs of damped systems. The first method solves the partial differential equations (PDEs) governing the geometry of the NNM. The PDEs are recognized as hyperbolic, and it is shown that they require appropriate numerical treatments including specific boundary conditions. The proposed method combines a streamline upwind Petrov-Galerkin finite-element formulation with a resolution strategy based on annular domains to grow sequentially the manifold. The algorithm is demonstrated using a wide variety of systems ranging from two-degree-of-freedom to multi-degree-of-freedom nonlinear systems with linear and nonlinear damping. The applicability of the algorithm to complex real-life structures is demonstrated using a full-scale aircraft. The second method presented in this work computes a NNM as a collection of trajectories defined with boundary value problems (BVPs). The method has the distinctive advantage that it does not rely on a parameterization of the NNM. It is demonstrated on two-degree-of-freedom examples featuring linear and nonlinear damping. [less ▲]

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See detailNonlinear modal analysis of the SmallSat spacecraft
Renson, Ludovic ULg; Kerschen, Gaëtan ULg; Newerla, Alfred

in Proceedings of the SEM IMAC XXX Conference (2012)

Non-linear elements are present in practically all spacecraft structures. The assumption of a (quasi-)linear structure is nevertheless adequate for structural analyses and design verification purposes in ... [more ▼]

Non-linear elements are present in practically all spacecraft structures. The assumption of a (quasi-)linear structure is nevertheless adequate for structural analyses and design verification purposes in those cases where these structural non-linearities are relatively weak or not substantially activated by the mechanical environments encountered during the launch or during ground testing. However, when significant non-linear effects in spacecraft structures are no longer negligible then linear modal analysis will not be able to handle non-linear dynamical phenomena in an adequate manner: the development of a non-linear analogue of linear modal analysis becomes an urgent and important issue. The objective of this paper is to show that nonlinear normal modes (NNMs) represent a useful and practical tool in this context. A full-scale spacecraft structure is considered and is modeled using the finite element method. Its NNMs are computed using advanced numerical algorithms, and the resulting dynamics is then carefully analyzed. Nonlinear phenomena with no linear counterpart including nonlinear modal interactions are also highlighted. [less ▲]

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See detailNonlinear Modal Analysis of the SmallSat Spacecraft
Renson, Ludovic ULg; Noël, Jean-Philippe ULg; Kerschen, Gaëtan ULg et al

in Proceedings of the European Conference on Spacecraft Structures, Materials & Environmental Testing (2012, March)

Non-linear elements are present in practically all spacecraft structures. When such non-linear effects are important linear modal analysis can no longer be applied. The development of a non-linear ... [more ▼]

Non-linear elements are present in practically all spacecraft structures. When such non-linear effects are important linear modal analysis can no longer be applied. The development of a non-linear analogue of linear modal analysis is therefore an important endeavor. The objective of this paper is to show that nonlinear normal modes (NNMs) represent a useful and practical tool in this context. A full-scale spacecraft structure is considered and is modeled using the finite element method. Its NNMs are computed using advanced numerical algorithms, and the resulting dynamics is then carefully analyzed. Nonlinear phenomena with no linear counterpart including nonlinear modal interactions are also highlighted. [less ▲]

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See detailNonlinear normal modes and band zones in granular chains with no precompression
Jayaprakash, K.R.; Starosvetsky, Y.; Vakakis, A.F. et al

in Nonlinear Dynamics (2011), 63

We study standing waves (nonlinear normal modes—NNMs) and band zones in finite granular chains composed of spherical granular beads in Hertzian contact, with fixed boundary conditions. Although these are ... [more ▼]

We study standing waves (nonlinear normal modes—NNMs) and band zones in finite granular chains composed of spherical granular beads in Hertzian contact, with fixed boundary conditions. Although these are homogeneous dynamical systems in the notation of Rosenberg, we show that the discontinuous nature of the dynamics leads to interesting effects such as separation between beads, NNMs that appear as traveling waves (these are characterized as pseudo-waves), and localization phenomena. In the limit of infinite extent, we study band zones, i.e., pass and stop bands in the frequency–energy plane of these dynamical systems, and classify the essentially nonlinear responses that occur in these bands. Moreover, we show how the topologies of these bands significantly affect the forced dynamics of these granular media subject to narrowband excitations. This work provides a classification of the coherent (regular) intrinsic dynamics of one-dimensional homogeneous granular chains with no pre-compression, and provides a rigorous theoretical foundation for further systematic study of the dynamics of granular systems, e.g., the effects of disorders or clearances, discrete breathers, nonlinear localized modes, and high-frequency scattering by local disorders. Moreover, it contributes toward the design of granular media as shock protectors, and in the passive mitigation of transmission of unwanted disturbances. [less ▲]

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See detailNonlinear Normal Modes of a Full-Scale Aircraft
Peeters, Maxime ULg; Kerschen, Gaëtan ULg; Golinval, Jean-Claude ULg et al

in 29th International Modal Analysis Conference, Jacksonville, 2011 (2011)

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See detailNonlinear Normal Modes of Nonconservative Systems
Renson, Ludovic ULg; Kerschen, Gaëtan ULg

in Proceedings of the SEM IMAC XXXI Conference (2013)

Linear modal analysis is a mature tool enjoying various applications ranging from bridges to satellites. Nevertheless, modal analysis fails in the presence of nonlinear dynamical phenomena and the ... [more ▼]

Linear modal analysis is a mature tool enjoying various applications ranging from bridges to satellites. Nevertheless, modal analysis fails in the presence of nonlinear dynamical phenomena and the development of a practical nonlinear analog of modal analysis is a current research topic. Recently, numerical techniques (e.g., harmonic balance, continuation of periodic solutions) were developed for the computation of nonlinear normal modes (NNMs). Because these methods are limited to conservative systems, the present study targets the computation of NNMs for nonconservative systems. Their definition as invariant manifolds in phase space is considered. Specifically, a new finite element technique is proposed to solve the set of partial differential equations governing the manifold geometry. [less ▲]

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See detailNonlinear normal modes of real-world structures: application to a full-scale aircraft
Peeters, Maxime ULg; Kerschen, Gaëtan ULg; Golinval, Jean-Claude ULg et al

in Proceedings of the ASME International Design Engineering Technical Conferences (2011)

The objective of this paper is to demonstrate that the numerical computation of the nonlinear normal modes (NNMs) of complex real-world structures is now within reach. The application considered in this ... [more ▼]

The objective of this paper is to demonstrate that the numerical computation of the nonlinear normal modes (NNMs) of complex real-world structures is now within reach. The application considered in this study is the airframe of the Morane-Saulnier Paris aircraft, whose ground vibration tests have exhibited some nonlinear structural behaviors. The finite element model of this aircraft, elaborated from drawings, has more than 80000 degrees of freedom, and softening nonlinearities exist in the connection between the external fuel tanks and the wing tips. From this model, a reduced-order model, which is accurate in the [0-100Hz] range, is constructed using the Craig- Bampton technique. The NNMs of the reduced model are then computed using a numerical algorithm combining shooting and pseudo-arclength continuation. The results show that the NNMs of this full-scale structure can be computed accurately even in strongly nonlinear regimes and with a reasonable computational burden. Nonlinear modal interactions are also highlighted by the algorithm and are discussed. [less ▲]

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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 normal modes, Part 2: Toward a pratical computation using numerical continuation techniques
Peeters, Maxime ULg; Viguié, Régis ULg; Sérandour, Guillaume ULg et al

in Mechanical Systems & Signal Processing (2009), 23

The conceptofnonlinearnormalmodes(NNMs)isdiscussedinthepresentpaperandits companion,PartI.OnereasonofthestilllimiteduseofNNMsinstructuraldynamicsis that theircomputationisoftenregardedasimpractical ... [more ▼]

The conceptofnonlinearnormalmodes(NNMs)isdiscussedinthepresentpaperandits companion,PartI.OnereasonofthestilllimiteduseofNNMsinstructuraldynamicsis that theircomputationisoftenregardedasimpractical.However,whenresortingto numericalalgorithms,weshowthattheNNMcomputationispossiblewithlimited implementationeffort,whichpavesthewaytoapracticalmethodfordeterminingthe NNMsofnonlinearmechanicalsystems.Theproposedalgorithmreliesontwomain techniques,namelyashootingprocedureandamethodforthecontinuationofNNM motions. Thealgorithmisdemonstratedusingfourdifferentmechanicalsystems,a weakly andastronglynonlineartwo-degree-of-freedomsystem,asimplifieddiscrete modelofanonlinearbladeddiskandanonlinearcantileverbeamdiscretizedbythefinite elementmethod. [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 detailNonlinear Normal Modes, Part II: Practical Computation using Numerical Continuation Techniques
Peeters, Maxime ULg; Viguié, Régis ULg; Sérandour, Guillaume 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 companion, Part I. One reason of the still limited use of NNMs in structural dynamics is that their computation is ... [more ▼]

The concept of nonlinear normal modes (NNMs) is discussed in the present paper and its companion, Part I. One reason of the still limited use of NNMs in structural dynamics is that their computation is often regarded as impractical. However, when resorting to numerical algorithms, we show that the NNM computation is possible with limited implementation effort, which paves the way to a practical method for determining the NNMs of nonlinear mechanical systems. The proposed algorithm relies on two main techniques, namely a shooting procedure and a method for the continuation of NNM motions. The algorithm is demonstrated using two different mechanical systems, a nonlinear two-degree-of-freedom system and a nonlinear cantilever beam discretized by the finite element method. [less ▲]

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See detailNonlinear Normal Modes: A Finite Element Approach
Renson, Ludovic ULg; Kerschen, Gaëtan ULg

Conference (2011)

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See detailNonlinear numerical analysis of ship impact on lock gates
Gernay, Thomas ULg

in Van Schel, L (Ed.) PIANC Yearbook 2010 (2011)

This paper focuses on ship impact analysis on lock gate. First, the lock gate structure is determined by an elastic design and optimization process. Then, a nonlinear numerical analysis by finite elements ... [more ▼]

This paper focuses on ship impact analysis on lock gate. First, the lock gate structure is determined by an elastic design and optimization process. Then, a nonlinear numerical analysis by finite elements is conducted to study the response of the lock gate in case of ship impact. It is shown that the elastically optimized structure is not able to resist significant impacts, because of the buckling of its reinforcement elements; the gate is thus reinforced. Finally, several ship impact analyses are conducted on the reinforced gate and they highlight the influence of the stiffener dimensions and the impact zone on the structural behavior of the gate. The results of the numerical analyses underline the importance of the development of a global plastic mechanism with the purpose of dissipating a large amount of energy. [less ▲]

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See detailNonlinear optical susceptibilities, Raman efficiencies, and electro-optic tensors from first-principles density functional perturbation theory
Veithen, M.; Gonze, X.; Ghosez, Philippe ULg

in Physical Review. B, Condensed Matter and Materials Physics (2005), 71

The nonlinear response of infinite periodic solids to homogenous electric fields and collective atomic displacements is discussed in the framework of density functional perturbation theory. The approach ... [more ▼]

The nonlinear response of infinite periodic solids to homogenous electric fields and collective atomic displacements is discussed in the framework of density functional perturbation theory. The approach is based on the 2n+1 theorem applied to an electric-field-dependent energy functional. We report the expressions for the calculation of the nonlinear optical susceptibilities, Raman scattering efficiencies, and electro-optic (EO) coefficients. Different formulations of third-order energy derivatives are examined and their convergence with respect to the k-point sampling is discussed. We apply our method to a few simple cases and compare our results to those obtained with distinct techniques. Finally, we discuss the effect of a scissors correction on the EO coefficients and nonlinear optical susceptibilities. [less ▲]

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