Transfert d'énergie passif d'un système linéaire forcé vers une chaîne d'oscillateurs non linéaires

Le contrôle passif a pour but de réduire les vibrations d'une structure sans apport d'énergie afin de la protéger d'éventuels dégâts, voire de la rupture, ou plus simplement dans le but d'assurer le confort des usagers. L'amortisseur harmonique linéaire, ou Tuned Mass Damper (TMD) [1], est un système faisant partie des plus utilisés et des plus connus. Il est néanmoins avantageux d'utiliser des amortisseurs non linéaires (NES), qui permettent de diminuer sensiblement la masse ajoutée et de permettre un fonctionnement pour une gamme de fréquences de sollicitation plus large [2]. Récemment, des études ont prouvé l'efficacité d'un NES constitué de deux oscillateurs montés en série [3,4]. Le but de ce travail est d'étendre cette idée à l'étude d'une chaîne de NES. Un système linéaire, soumis à une sollicitation externe, est couplé à une chaîne de N oscillateurs non linéaires en série vers laquelle l'énergie vibratoire du système primaire est transférée. Les oscillateurs de la chaîne sont supposés très légers en comparaison du système principal. Une méthodologie analytique de traitement des équations de l'équilibre dynamique, issue de la généralisation d'une méthode appliquée à des systèmes à deux degrés de liberté [5], est présentée. Après l'utilisation d'une méthode dite de complexification-moyenne (complexification et méthode de Galerkine), une méthode d'échelles multiples en temps est mise en place. Des échelles de temps reliées entre elles par le faible ratio de masse entre oscillateurs non linéaires et système primaire sont introduites et le comportement du système est étudié à chacune de ces échelles. À l'échelle de temps rapide, la variété invariante lente, ou Slow Invariant Manifold (SIM), est calculée. Ce SIM rassemble tous les comportements asymptotiques vers lesquels le système pourrait tendre. À la première échelle de temps lente, les points d'équilibre et les points singuliers sont mis en évidence. Ils permettent de discriminer deux types de comportements asymptotiques que le système peut subir autour du SIM. Les points d'équilibre correspondent à des régimes périodiques, qui sont ici des modes non linéaires du système, et les points singuliers à des régimes quasi-périodiques caractérisés par des sauts répétés du système autour de zones instables du SIM, donnant lieu ici à des bifurcations entre modes. Ces points donnent une évaluation complète des amplitudes finales du système et permettent de vérifier qu'une part importante de l'énergie de la structure primaire est transférée vers la chaîne. Ces prédictions analytiques sont ensuite comparées et validées par des simulations numériques obtenues par intégration temporelle directe des équations initiales. Cette méthode peut par ailleurs s'appliquer par le biais de deux approches. La première considère la chaîne comme un assemblage discret d'oscillateurs quand la seconde l'appréhende comme un continuum. Cette approche continue remplace les N variables des masses de la chaîne par une fonction continue d'une variable d'espace, conduisant ainsi à l'étude d'équations aux dérivées partielles et non plus à un système de N équations.   [1] H. Frahm, Device for damping vibrations of bodies, US Patent 989,958 (1911). [2] O. V. Gendelman, A. F. Vakakis, Transitions from localization to nonlocalization in strongly nonlinear damped oscillators, Chaos, Solitons & Fractals 11 (2000) 1535?1542. [3] Quinn, D. D., et al., Equivalent modal damping, stiffening and energy exchanges in multi-degree-of-freedom systems with strongly nonlinear attachments, Proceedings of the Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics 226(K2) (2012) 122?146. [4] N.E. Wierschem, et al., Experimental Testing and Numerical Simulation of a Six-Story Structure Incorporating Two-Degree-of-Freedom Nonlinear Energy Sink, Journal of Structural Engineering 140 (2014) 04014027. [5] S. Charlemagne, et al., Interactions Between Two Coupled Nonlinear Forced Systems: Fast/Slow Dynamics, International Journal of Bifurcation and Chaos 26 (2016) 1650155

Mixed-forest species establishment in a monodominant forest in Central Africa: Implications for tropical forest invasibility

Background: Traits of non-dominant mixed-forest tree species and their synergies for successful co-occurrence in monodominant Gilbertiodendron dewevrei forest have not yet been investigated. Here we compared the tree species diversity of the monodominant forest with its adjacent mixed forest and then determined which fitness proxies and life history traits of the mixed-forest tree species were most associated with successful co-existence in the monodominant forest. Methodology/Principal Findings: We sampled all trees (diameter in breast height [dbh]≥10 cm) within 6x1 ha topographically homogenous areas of intact central African forest in SE Cameroon, three independent patches of G. dewevrei-dominated forest and three adjacent areas (450-800 m apart). Monodominant G. dewevrei forest had lower sample-controlled species richness, species density and population density than its adjacent mixed forest in terms of stems with dbh≥10 cm. Analysis of a suite of population-level characteristics, such as relative abundance and geographical distribution, and traits such as wood density, height, diameter at breast height, fruit/seed dispersal mechanism and light requirement-revealed after controlling for phylogeny, species that co-occur with G. dewevrei tend to have higher abundance in adjacent mixed forest, higher wood density and a lower light requirement. Conclusions/Significance: Our results suggest that certain traits (wood density and light requirement) and population-level characteristics (relative abundance) may increase the invasibility of a tree species into a tropical closed-canopy system. Such knowledge may assist in the pre-emptive identification of invasive tree species. © 2014 Peh et al

Nonlinear dynamics of a set of oscillators : application to control

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

Dynamique non linéaire d’un assemblage d’oscillateurs : application au contrôle

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.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

Experimental results on the vibratory energy exchanges between a linear system and a chain of nonlinear oscillators

International audienceExperimental results on a nonlinear chain coupled to a main system are presented. The chain is composed of eight moving masses, each one possesses local nonlinear restoring forcing function and global linear springs for coupling to other masses. The main system is coupled to the first mass of the chain via a linear spring. The main system is under external sinusoidal excitation with sweeping frequency around its targeted mode. Experimental results show that according to the amplitude of the excitation, the system can reach periodic or modulated regimes. The goal of using the nonlinear chain is to examine experimentally the possibility of localization of the vibratory energy of the main system into the chain for the aim of passive control

Vibrations induced by tunnel boring machine in urban areas: In situ measurements and methodology of analysis

Excavation with tunnel boring machine (TBM) can generate vibrations, causing damages to neighbouring buildings and disturbing the residents or the equipment. This problem is particularly challenging in urban areas, where TBMs are increasingly large in diameter and shallow in depth. In response to this problem, four experimental campaigns were carried out in different geotechnical contexts in France. The vibration measurements were acquired on the surface and inside the TBMs. These measurements are also complemented by few data in the literature. An original methodology of signal processing is proposed to characterize the amplitude of the particle velocities, as well as the frequency content of the signals to highlight the most energetic bands. The levels of vibrations are also compared with the thresholds existing in various European regulations concerning the impact on neighbouring structures and the disturbance to local residents

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

International audienceA N-degree-of-freedom model consisting of a single-degree-of-freedom linear system coupledto a chain of (N − 1) light nonlinear oscillators is studied. The connection between the chain and thesingle-degree-of-freedom system is supposed to be linear. Time multi-scale system behaviors at fastand slow time scales are investigated and lead to detection of the Slow Invariant Manifold (SIM) andequilibrium and singular points. These points correspond to periodic regimes and strongly modulatedresponses, respectively. These analytical developments are used to provide evidence of transfer of vibratoryenergy of the main system to the chain in the form of localized modes during periodic regimes andextreme 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 withnumerical results obtained from direct time integration of system equations, showing a good agreementbetween them. Finally, we present a procedure showing how these analytical developments can be usedto study a system where the main structure is replaced by a multi-degree-of-freedom linear system, byprojecting its dynamics on one of its modes

Interactions Between Two Coupled Nonlinear Forced Systems: Fast/Slow Dynamics

International audienceDynamics of a system formed by a linear structure coupled to a light nonlinear oscillator, bothsubjected to external excitations, is studied. Effects of the external forcing of the nonlinearoscillator are especially investigated. Complex geometry of the slow invariant manifold andequilibrium and singular points of the system are detected thanks to a multiple time scale strategyaround 1:1:1 resonance. Equilibrium points lead to periodic regimes while singular pointsare hints of strongly modulated response of the system characterized by repeated bifurcationsaround its stable zones. A method for detection of changes in mechanical properties of mainstructural system is explained. Numerical simulations obtained by direct integration of the systemare used to validate analytical predictions

Dynamics and energy exchanges between a linearoscillator and a nonlinear absorber with local and globalpotentials

International audienceThe dynamical behavior of a two degree-of-freedom system made up of a linearoscillator and a coupled nonlinear energy sink with nonlinear global and localpotentials is studied. The nonlinear global potential of the energy sink performsdirect interactions with the linear oscillator, while its local potential dependsonly on its own behavior during vibratory energy exchanges between two oscillators.A time multiple scale method around 1:1:1 resonance is used to detect slowinvariant manifold of the system, its equilibrium and singular points. Detectedequilibrium points permit us to predict periodic regime(s) while singular pointscan lead the system to strongly modulated responses characterized by persistentbifurcations. Several possible scenarios occurring during these strongly modulatedregimes are highlighted. All analytical predictions are compared withthose which are obtained by direct numerical integration of system equations

Dynamics of a linear system coupled to a chain of light nonlinear oscillators analyzed through a continuous approximation

personalized Share Link provided by elsevier:https://authors.elsevier.com/c/1X7i4c2EeSSF6International audienceThe continuous approximation is used in this work to describe the dynamics of a nonlinear chain of light oscillators coupled to a linear main system. A general methodology is applied to an example where the chain has local nonlinear restoring forces. The slow invariant manifold is detected at fast time scale. At slow time scale, equilibrium and singular points are sought around this manifold in order to predict periodic regimes and strongly modulated responses of the system. Analytical predictions are in good accordance with numerical results and represent a potent tool for designing nonlinear chains for passive control purposes