FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / Wormholes sÃo estruturas exÃticas que podem ligar regiÃes longÃnquas do espaÃo e por isso sÃo bastante visados como objeto de pesquisa. Pode-se mostrar que buracos negros possuem modos de vibraÃÃo amortecidos, os chamados de modos quase-normais. Esses modos sÃo responsÃveis por irradiar as ondas gravitacionais para espaÃo. Isso se deve ao fato de pequenas perturbaÃÃes na mÃtrica poderem ser descritas pelas equaÃÃes de Einstein, em forma de equaÃÃes de onda. Esse comportamento à semelhante para wormholes, onde devemos encontrar um potencial de Regge-Wheeler associado a equaÃÃo de onda. Nosso objetivo nesse trabalho serà tentar ver se à possÃvel encontrarmos um potencial do tipo Regge-Wheeler para um wormhole, que se supÃem ser sustentado pela matÃria escura na regiÃo dos halos galÃcticos. E por consequÃncia, verificarmos se à possÃvel determinar modos quase-normais usando o mÃtodo WKB, para esse wormhole em particular. / Wormholes are exotic structures that can connect distant regions of space and therefore are highly targeted as a research object. It can be shown that black holes have damped vibration modes, The so-called quasi-normal modes. These modes are responsible for radiating gravitational waves to space. This is due to the fact that small disturbances in the metric can be described by Einsteinâs equations, in the form of wave equations. This behavior is similar for wormholes, where we must find a Regge-Wheeler potential associated with the wave equation. Our goal in this work will be to see if it is possible to find Regge-Wheeler potential for a wormhole, Which are supposed to be sustained by dark matter in the region of galactic halos. And consequently, Verify if it is possible to determine quasi-normal modes using the WKB method, for that particular wormhole.

Στη γραμμική εκδοχή της γενικής θεωρίας της σχετικότητας, θεωρεί κανείς τις διαταραχές κάποιας μετρικής γύρω από κάποιο χωροχρονικό υπόβαθρο. Κρατώντας όρους διαταραχών μέχρι και πρώτης τάξεως, οδηγείται κανείς στις γραμμικές εξισώσεις Einstein. Σε αυτό το πλαίσιο αποδεικνύεται μια σχέση δυισμού ανάμεσα στα διάφορα στοιχεία του τανυστή Weyl, αντίστοιχη με το δυισμό ανάμεσα στην ηλεκτρική και τη μαγνητική ροή της ηλεκτρομαγνητικής θεωρίας του Maxwell. Στην εργασία αυτή κάνουμε μία ανασκόπηση της έρευνας που έχει γίνει μέχρι τώρα αναφορικά με αυτές τις σχέσεις δυισμού. Πιο συγκεκριμένα, εξετάζουμε την ισχύ των σχέσεων στον Anti-de Sitter χωρόχρονο και επισημαίνουμε το τρόπο με τον οποίο κατασκευάζει κανείς δυικές δομές από τις εκφράσεις για τις διαταραχές. Επίσης, χρησιμοποιώντας τη τεχνική της ολογραφικής επανακανονικοποίησης, εξετάζουμε το σύμμορφο σύνορο του χωροχρόνου. Βρίσκουμε εκεί μια σχέση δυισμού ανάμεσα στα στοιχεία του τανυστή ενέργειας-ορμής και του τανυστή Cotton της αντίστοιχης Chern - Simons θεωρίας, η οποία αποδεικνύεται ότι είναι άμεση συνέπεια του δυισμού στο AdS υπόβαθρο. Τέλος, εφαρμόζουμε την ίδια συλλογιστική στο Schwarzschild - Anti-de Sitter υπόβαθρο, όπου η παρουσία της μελανής οπής διαφοροποιεί τις συνοριακές συνθήκες του προβλήματος. Λόγω αυτού του γεγονός δεν μπορεί να πει κανείς με σιγουριά εάν μπορούν να διατυπωθούν σχέσεις δυισμού σε αυτή τη περίπτωση. Παρόλα αυτά βρίσκουμε ότι ισχύουν σχέσεις δυισμού στο σύμμορφο σύνορο παρόμοιες με αυτές του AdS υποβάθρου, πράγμα που σημαίνει ότι στο σύστημα παραμένει κάποια συμμετρία από τη γραμμική θεωρία. Η εργασία καταλήγει σε σχόλια και μία εκτενή συζήτηση για τις πιθανές μελλοντικές κατευθύνσεις. / In the linear version of the general theory of relativity, one considers metric perturbations around a fixed background. Keeping terms up to first order of perturbation leads to the linearized Einstein equations. In this framework it has been proved that a duality between the various elements of the Weyl tensor holds. This duality is similar to the one between the electric and magnetic fluxes of Maxwell's electromagnetism. In the present work we review the status of these dualities for non trivial backgrounds. We examine the Anti-de Sitter background, where we point out the way to explicitly construct dual configurations using the metric perturbation expressions. Using the holographic renormalization technique, we examine the conformal boundary where a duality between the components of the energy-momentum tensor and the Cotton tensor of the corresponding Chern - Simons theory holds. It is then proved that this duality is a direct consequence of the electric/magnetic duality in the bulk, in the case of the AdS background. Finally, we apply same procedure to the Schwarzschild - Anti-de Sitter background, where the presence of the black hole changes the boundary conditions of the problem. This simple fact makes it impossible say whether such a duality exists in this case. Nevertheless, we find that a duality similar to that of the AdS background still holds for the conformal boundary, which means that there is a remnant of symmetry from the linear theory. We conclude with comments and a extensive discussion on possible future directions.

Pour les solutions transitoires répétées des structures géométriquement nonlinéaires l’effort numérique présente souvent une contrainte importante. Ainsi, l’introduction d’un modèle d’ordre réduit, qui prend en compte les effets nonlinéaires et qui accélère considérablement les calculs, s’avère souvent nécessaire.Ce travail aboutit à une méthode qui permet des solutions transitoires accélérées, fidèles et paramétrables, à travers d’un modèle réduit de la structure initiale. La structure est discrétisée et son équilibre dynamique décrit par une équation matricielle. La projection sur une base réduite est introduite afin d’obtenir un modèle réduit. Une étude numérique complète sur plusieurs bases communes démontre que la simple introduction d’une base constante ne suffit pas pour prendre en compte le comportement nonlinéaire. Trois exigences sont déduites pour une solution transitoire accélérée, fidèle et paramétrable. L’algorithme de solution doit permettre un suivi de l’évolution nonlinéaire de la solution transitoire, la solution doit être autonome des termes nonlinéaires en éléments finis et la base doit être adaptée à des paramètres externes.Trois approches sont mises en place, chacune répondant à une exigence. Ces approches sont assemblées dans la méthode intégrée. Les approches sont la mise-à-jour et augmentation de la base , la formulation polynomiale des termes nonlinéaires et l’interpolation de la base. Un algorithme de type Newmark forme le cadre de la méthode intégrée. L’application de la méthode intégrée sur des cas test en élément finis géométriquement nonlinéaires confirme qu’elle répond au but initial d’obtenir des solutions transitoires accélérées, fidèles et paramétrables. / For repeated transient solutions of geometrically nonlinear structures the numerical effort often poses a major obstacle. Thus, the introduction of a reduced order model, which takes the nonlinear effects into account and accelerates the calculations considerably, is often necessary.This work yields a method that allows for rapid, accurate and parameterisable solutions by means of a reduced model of the original structure. The structure is discretised and its dynamic equilibrium described by a matrix equation. The projection on a reduced basis is introduced to obtain the reduced model. A comprehensive numerical study on several common reduced bases shows that the simple introduction of a constant basis is not sufficient to account for the nonlinear behaviour. Three requirements for an rapid, accurate and parameterisable solution are derived. The solution algorithm has to take into account the nonlinear evolution of the solution, the solution has to be independent of the nonlinear finite element terms and the basis has to be adapted to external parameters.Three approaches are provided, each responding to one requirement. These approaches are assembled to the integrated method. The approaches are the update and augmentation of the basis, the polynomial formulation of the nonlinear terms and the interpolation of the basis. A Newmark-type time-marching algorithm provides the frame of the integrated method. The application of the integrated method on test-cases with geometrically nonlinear finite elements confirms that this method leads to the initial aim of a rapid, accurate and parameterisable transient solution.

OLIVEIRA, R. dos. Wormhole do halo central da galáxia e modos quase-normais. 2017. 79 f. Dissertação (Mestrado em Física) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Giordana Silva (giordana.nascimento@gmail.com) on 2017-04-12T22:32:15Z No. of bitstreams: 1 2017_dis_roliveira.pdf: 2016508 bytes, checksum: b13e51c1946445379ebd0aa6da75f18b (MD5) / Approved for entry into archive by Giordana Silva (giordana.nascimento@gmail.com) on 2017-04-12T22:32:44Z (GMT) No. of bitstreams: 1 2017_dis_roliveira.pdf: 2016508 bytes, checksum: b13e51c1946445379ebd0aa6da75f18b (MD5) / Made available in DSpace on 2017-04-12T22:32:44Z (GMT). No. of bitstreams: 1 2017_dis_roliveira.pdf: 2016508 bytes, checksum: b13e51c1946445379ebd0aa6da75f18b (MD5) Previous issue date: 2017 / Wormholes are exotic structures that can connect distant regions of space and therefore are highly targeted as a research object. It can be shown that black holes have damped vibration modes, The so-called quasi-normal modes. These modes are responsible for radiating gravitational waves to space. This is due to the fact that small disturbances in the metric can be described by Einstein’s equations, in the form of wave equations. This behavior is similar for wormholes, where we must find a Regge-Wheeler potential associated with the wave equation. Our goal in this work will be to see if it is possible to find Regge-Wheeler potential for a wormhole, Which are supposed to be sustained by dark matter in the region of galactic halos. And consequently, Verify if it is possible to determine quasi-normal modes using the WKB method, for that particular wormhole. / Wormholes são estruturas exóticas que podem ligar regiões longínquas do espaço e por isso são bastante visados como objeto de pesquisa. Pode-se mostrar que buracos negros possuem modos de vibração amortecidos, os chamados de modos quase-normais. Esses modos são responsáveis por irradiar as ondas gravitacionais para espaço. Isso se deve ao fato de pequenas perturbações na métrica poderem ser descritas pelas equações de Einstein, em forma de equações de onda. Esse comportamento é semelhante para wormholes, onde devemos encontrar um potencial de Regge-Wheeler associado a equação de onda. Nosso objetivo nesse trabalho será tentar ver se é possível encontrarmos um potencial do tipo Regge-Wheeler para um wormhole, que se supõem ser sustentado pela matéria escura na região dos halos galácticos. E por consequência, verificarmos se é possível determinar modos quase-normais usando o método WKB, para esse wormhole em particular.

L'utilisation de systèmes légers non linéaires permet de réaliser le contrôle vibratoire de structures subissant des oscillations non acceptables en termes de confort pour l'usager ou de sécurité de l'ouvrage. L'étude des puits d'énergie non linéaires, ou « Nonlinear Energy Sinks » (NES), a notamment fait l'objet de nombreuses recherches depuis le début des années 2000. Sa non-linéarité lui confère des capacités de pompage énergétique large bande, c'est-à-dire pour un large intervalle de fréquences de sollicitation, ce qui représente un avantage significatif en comparaison des absorbeurs comme l'amortisseur à masse accordée. Le but de ce manuscrit est d'étudier le couplage de chaîne d'oscillateurs non linéaires à des systèmes dynamiques linéaires soumis à des sollicitations harmoniques et d'analyser d'une part le comportement global du système, et d'autre part les potentialités de contrôle passif de telles chaînes. Une méthodologie analytique générale est présentée, puis appliquée à des exemples où des absorbeurs à non-linéarités cubiques à un, puis à N degrés de liberté sont attachés à un oscillateur linéaire. Une variation de cette méthodologie adoptant une vision continue de la chaîne est ensuite proposée. Enfin, un dispositif expérimental étudie le comportement d'un modèle réduit de bâtiment à un étage couplé à une chaîne de huit oscillateurs non linéaires. / Nonlinear light oscillators can be used for performing vibratory passive control of structures undergoing unacceptable oscillations in terms of comfort and safety. The study of Nonlinear Energy Sinks (NES) has been especially subject to an important research effort since the beginning of the 2000s. Its essential nonlinearity enables it to achieve large-band energy pumping, which is a significant advantage in comparison with classical Tuned Mass Dampers. In this manuscript, nonlinear chains of oscillators coupled to linear systems under harmonic excitation are studied. The main goal is to understand the behavior of the whole system and find evidence of passive control abilities of such chains. First of all, a general analytical methodology is presented and applied to examples where single and multi-degree-of-freedom absorbers with cubic nonlinearities are linked to a linear oscillator. A modification of this approach by considering the chain in the form of a continuous approximation is then proposed. Finally, an experimental device composed of a single storey reduced-scale building coupled to a chain of eight nonlinear oscillators is investigated.

[pt] Vigas flexíveis são encontradas com cada vez mais frequência em diferentes indústrias, uma vez que novos projetos têm buscado por estruturas mais longas e leves. Isso pode ser uma consequência direta das novas demandas estruturais nos projetos, ou uma simples consequência do engajamento das indústrias em programas de redução de custo (utilização de menos materiais). Em geral, vigas flexíveis são modeladas sob hipóteses de grandes deslocamentos, grandes rotações, mas com pequenas deformações. Essas hipóteses permitem que o equacionamento da dinâmica de vigas flexíveis seja feito através de elementos finitos co-rotacionais. A formulação co-rotacional decompõe o movimento das estruturas flexíveis em duas partes: uma contendo o movimento de corpo rígido e outra com uma (pequena) deformação elástica. Dessa forma, a não-linearidade geométrica causada pelos grandes deslocamentos e rotações das seções transversais das vigas podem ser computadas de forma eficiente. Uma das inovações dessa tese é o uso direto das equações de movimentos geradas pelos elementos finitos co-rotacionais no cálculo dos modos normais não-lineares (MNNs). Até agora, a maioria das análises dinâmicas com elementos finitos co-rotacionais foram restritas à integração das equações de movimento. O conhecimento de MNNs é útil na análise de sistemas não-lineares pois permitem um detalhado entendimento das vibrações nos regimes não-lineares. Com eles, pode-se, por exemplo, prever comportamentos de enrijecimento/relaxamento, localização de respostas, interação entre modos, existência de isolas, etc. A definição de Rosenberg sobre MNNs como sendo soluções periódicas (não necessariamente síncronas) do sistema é adotado na tese. Os métodos do Balanço Harmônico e do Tiro são apresentados e utilizados no cálculo de soluções periódicas de sistemas não-lineares. Um procedimento de continuação numérica é implementado para computar os MNN eficientemente para diferentes níveis de energia. Exemplos numéricos mostram a capacidade do método proposto quando aplicado aos elementos finitos co-rotacionais. / [en] Flexible beams are becoming ubiquitous in several industrial applications, as new projects often aim for lighter and longer structures. This fact is directly related to the new challenging demands on structural performances, or it is a simple consequence of the engagement of industries in cost reduction programs (usage of less material). Flexible beams are usually modeled under the assumption of large displacements, finite rotations, but with small strains. Such hypotheses allow the equation of motion to be built using co-rotational finite elements. The co-rotational formulation decomposes the total motion of a flexible structure into two parts: a rigid body displacement and an elastic (small) deformation. This way, the geometric nonlinearity caused by the large displacements and rotations of the beam s cross sections can be efficiently computed. One of the novelties of this thesis is the direct usage of the equation of motion generated by a co-rotational finite element formulation in the computation of nonlinear normal modes (NNM). So far, most of the dynamic analyses with co-rotation finite element models were restricted to numerical integrations of the equation of motion. The knowledge of NNMs can be beneficial in the analysis of any nonlinear structure since it allows a thoroughly understanding of the vibratory response in the nonlinear regime. They can be used, for example, to predict a hardening/softening behavior, a localization of the responses, the interactions between modes, the existence of isolas, etc. The Rosenberg s definition of NNM as periodic solutions (non-necessarily synchronous motion) is adopted here. The Harmonic Balance method and the Shooting methods are presented and used to compute periodic solutions of nonlinear systems. A numerical path continuation scheme is implemented to efficiently compute NNMs at different energy levels. Numerical examples show the capability of the proposed method when applied to co-rotational beam elements.

No description available.

[pt] As torres estaiadas estão entre as estruturas mais altas construídas pelo homem. Estas estruturas usualmente são muito esbeltas e a interação cabos/mastro leva a comportamentos altamente não lineares. Devido a sua complexidade, modelos simplificados são desenvolvidos para as simulações dessas estruturas. Um modelo discreto de dois graus de liberdade investigado por diversos autores apresenta fenômenos característicos de estruturas não lineares, como a superabundância de modos normais não lineares similares e modos normais não similares (NNMs), bifurcações de NNMs, ressonância interna e interação modal. O presente trabalho visa investigar o comportamento de um modelo estrutural contínuo de uma torre estaiada com um a três níveis de estais. O método dos elementos finitos (MEF) com uma formulação não linear é usado para realizar análises paramétricas da influência na resposta estática e dinâmica, linear e não linear, das características geométricas e físicas dos cabos, do peso próprio dos cabos e do mastro e de imperfeições iniciais nas frequências naturais e carga crítica da torre. As simetrias geradas pela distribuição uniforme dos cabos têm grande influência na resposta, dando origem a cargas críticas e frequências naturais coincidentes. Isso gera interação modal na flambagem e ressonância interna 1:1, aumentando o efeito da não linearidade geométrica na resposta. Uma análise qualitativa é desenvolvida, comparando as respostas da análise de vibração não linear do modelo contínuo com as do modelo de dois graus de liberdade. Essa análise comparativa indica a existência de múltiplos NNMs e multimodos. A influência desses modos e simetrias inerentes à torre é investigada através de uma análise paramétrica da torre sob excitação harmônica lateral. Os resultados mostram que a torre exibe uma resposta altamente não linear, mesmo sob baixos níveis de carga, o que deve ser considerado com cuidado na fase de projeto e indica a necessidade de investigações adicionais da resposta dinâmica não-linear dessas estruturas, considerando as diferentes distribuições dos cabos utilizadas na prática. / [en] The guyed towers are among the tallest man-made structures. These structures are usually very slender and their guy/mast interaction leads to highly nonlinear behaviors. Due this, simplified models are developed for simulating these structures. The discrete model of tow-degree of freedom investigated by several authors exhibits characteristic phenomena of nonlinear structures such as a superabundance of similar nonlinear normal modes and non-similar normal modes (NMNs), bifurcations of NMNs, internal resonance, and modal interaction. The present work aims to investigate the behavior of a continuous structural model of a tower with one to three guyed levels. The nonlinear finite element method (FEM) is used to parametric analyzes of the influence on static and dynamic responses, linear and nonlinear, of the geometric and materials characteristics of the guys, of the mast and guys self-weight and initial imperfections of the tower s natural frequencies and critical loads. The symmetries generated by the uniform distribution of guys have a great influence on the response, given rise to coincident critical loads and natural frequencies. This generates modal interaction in the buckling and 1:1 internal resonance, increasing the effect of the geometric nonlinearity on the response. A qualitative analysis is developed, comparing as the response of the nonlinear vibration of the continuous model as those of the two degrees of freedom model. This comparative analysis indicates the existence of the multiple NNMs and multimodes. The influence of theses modes and tower inherent symmetries are investigated through a parametric analysis of the tower under lateral harmonic excitation. tower modes. The results show that the tower exhibits a highly nonlinear response, even at low load levels, which must be considered with care in the design stage and indicates the necessary of further investigations of the nonlinear dynamic response of these structures considering the different guys distribution used in practice.

L'enjeu des travaux effectués au cours de cette thèse est l'extraction in silico de molécules potentiellement intéressantes dans le processus d'inhibition du récepteur tyrosine kinase c-Met. La faculté de cette protéine à interagir dans les phénomènes d'embryogenèse et de réparation tissulaires rendent son inhibition cruciale dans les traitements contre les développements tumoraux où c-Met se trouve impliquée. Pour cela, la stratégie employée implique l'utilisation de méthodes in silico de conception rationnelle de médicaments. Nous avons utilisé comme support les multiples structures cristallographiques publiées sur la ProteinData Base. Un travail de modélisation par homologie fut tout d'abord nécessaire pour combler les lacunes des structures cristallographiques collectées. Afin d'échantillonner au mieux l'espace conformationnel de la kinase c-Met et de caractériser sa flexibilité, une longue campagne de simulation de Dynamique Moléculaire fut menée. Pour compléter ces simulations, nous avons également utilisé la méthode des modes normaux de vibration. De ces 2 approches, nous avons extrait un ensemble de 10 conformères considérés comme les plus représentatifs de l'espace conformationnel simulé pour la kinase c-Met et avons proposé un mode de fonctionnement de ce récepteur. Utilisant les conformations représentatives, nous avons ensuite mené une importante campagne de criblage virtuel sur plusieurs chimiothèques constituant environ 70.000 composés. L'analyse des résultats de l'arrimage moléculaire nous a conduits à la sélection de composés intéressants possédant théoriquement une bonne affinité pour la kinase c-Met. Ces molécules ont été soumises aux tests expérimentaux. / The challenge of this PhD work is the in silico identification of potentially interesting molecules concerning the inhibitory process of tyrosine kinase receptor c-Met. The faculty of this protein to interact in embryogenesis and tissue repair phenomena makes its inhibition crucial for treatments against tumor development in which c-Met is involved. For that purpose, the employed strategy involves the use of several in silico methods for rational drug design. As the basement of this work, we used the multiple crystal structures published in the ProteinData Base (PDB). A preliminary homology modeling work was needed to fill gaps in the crystal structures. To sample at best the c-Met kinase conformational space and to characterize its flexibility, a long Molecular Dynamics (MD) simulation campaign was carried out both on apo and holo forms of available crystal structures. To complete these simulations, part of this work consisted to use normal modes of vibration (NM) method. From these two approaches (DM and NM), we extracted a set of 10 conformers considered as the most representative of the kinase simulated conformational space and we suggested a mode of operation of this kinase. Using extracted conformations from the conformational sampling has enabled us to conduct an extensive campaign on several virtual screening libraries constituting a total of approximately 70,000 compounds. Analysis of the molecular docking results has led us to the selection of several theoretically interesting molecules with good potential affinity for c-Met kinase. These molecules were submitted to experimental tests performed by the biologist team associated to our work.

Le travail de cette thèse a été réalisé dans le cadre d'une collaboration entre EDF R&D et le LMA de Marseille (CNRS). Le but était de développer des outils théoriques et numériques pour le calcul de modes non-linéaires de structures industrielles possédant des non-linéarités localisées à jeu. La méthode de calcul utilisée est une combinaison de la méthode d'équilibrage harmonique (EH) et de la méthode asymptotique numérique (MAN), appelée EHMAN. Elle est réputée pour sa robustesse sur les problèmes réguliers. L'enjeu de ce travail de thèse est de l'appliquer sur des problèmes non-réguliers régularisés de type butée à jeu pour lequel un grand nombre d'harmonique est nécessaire. Des améliorations ont été apportées à la méthode de base pour rendre effectif le traitement de modèles à "grand" nombre de degrés de liberté (DDL). Les développements réalisés pendant la thèse ont été capitalisés par la création de nouveaux opérateurs dans Code_Aster.Une étude approfondie d'un système à 2 degrés de liberté a permis de faire émerger quelques caractéristiques des systèmes non-linéaires à jeu. Celles-ci ont servi entre autre à établir une méthodologie pour l'étude de systèmes à grand nombre de DDL. Pour finir, la potentialité des modes non-linéaires comme outil de diagnostic vibratoire est démontrée avec l'étude d'un tube cintré de générateur de vapeur. Le calcul des modes non-linéaires a monté l'existence d'une interaction entre un mode hors-plan (basse fréquence) et un mode plan (haute fréquence) expliquant des régimes vibratoires non-standards. Ce résultat, impossible à obtenir avec les outils de l'analyse modale linéaire, est confirmé expérimentalement. / This work is a collaboration between EDF R&D and the Laboratory of Mechanics and Acoustics. The objective is to develop theoretical and numerical tools to compute nonlinear normal modes (NNMs) of structures with localized nonlinearities.We use an approach combining the harmonic balance and the asymptotic numerical methods, known for its robustness principally for smooth systems. Regularization techniques are used to apply this approach for the study of nonsmooth problems. Moreover, several aspects of the method are improved to allow the computation of NNMs for systems with a high number of degrees of freedom (DOF). Finally, the method is implemented in Code_Aster, an open-source finite element solver developed by EDF R&D.The nonlinear normal modes of a two degrees-of-freedom system are studied and some original characteristics are observed. These observations are then used to develop a methodology for the study of systems with a high number of DOFs. The developed method is finally used to compute the NNMs for a model U-tube of a nuclear plant steam generator. The analysis of the NNMs reveals the presence of an interaction between an out-of-plane (low frequency) and an in-plane (high frequency) modes, a result also confirmed by the experiment. This modal interaction is not possible using linear modal analysis and confirms the interest of NNMs as a diagnostic tool in structural dynamics.