Vibrations non linéaires géométriques de structures minces. Modèles d'ordre réduit et transition vers le chaos

Les travaux présentés dans ce mémoire concernent les vibrations non linéaires géométriques (grande amplitude)des milieux minces, et plus particulièrement les plaques et les coques. Le premier chapitre présente les modèlesutilisés dans le document, en rappelant les hypothèses qui président à leur établissement. Le chapitre 2 est entièrement consacréà la théorie des modes non linéaire et à son application pour établirdes modèles d'ordre réduit pour les vibrations de coques en non linéaire géométrique.La définition d'un mode non linéaire (MNL) comme variété invariante de l'espace des phasesest rappelée, puis une méthode, fondée sur la théorie des formes normales, et permettantde calculer aisément les MNLs, est présentée. Son application au cas des vibrations libresmontre qu'elle permet à moindre coût une prédiction juste de la tendance de non-linéarité(comportement raidissant/assouplissant). L'utilisation des MNLs comme base réduite montreson excellent comportement pour diminuer drastiquement le nombre de degrés de libertés (ddls)pour le cas des vibrations forcées, harmoniques et basse fréquence. Le chapitre 3 traite de la transitionvers le chaos observée lorsqu'on augmente l'amplitude d'un forçage harmonique pour les structures minces.Le cas générique observé à partir de nombreuses expériences est d'abord rappelé, montrant deux bifurcationsnettes menant d'un régime périodique à une régime quasi-périodique, puis chaotique.La première bifurcation est analysée théoriquement et expérimentalement pour les cas particuliersde deux résonances internes. Enfin le régime chaotique est étudié à l'aide du formalismede la turbulence d'ondes

Identification of mode couplings in nonlinear vibrations of the steelpan

The authors are grateful to Bertrand David (Telecom-ParisTech) for computing the code allowing the STFT filtering procedure used in Section 5.1. The filter has been designed in the framework of the PAFI project (Plateforme d’Aide la facture Instrumentale, www.pafi.fr) which is also thanked.The vibrations and sounds produced by two notes of a double second steelpan are investigated, the main objective being to quantify the nonlinear energy exchanges occurring between vibration modes that are responsible of the peculiar sound of the instrument. A modal analysis first reveals the particular tuning of the modes and the systematic occurence of degenerate modes, from the second one, this feature being a consequence of the tuning and the mode localization. Forced vibrations experiments are then performed to follow precisely the energy exchange between harmonics of the vibration and thus quantify properly the mode couplings. In particular, it is found that energy exchanges are numerous, resulting in complicated frequency response curves even for very small levels of vibration amplitude. Simple models displaying 1:2:2 and 1:2:4 internal resonance are then fitted to the measurements, allowing to identify the values of the nonlinear quadratic coupling coefficients resulting from the geometric nonlinearity. The identified 1:2:4 model is finally used to recover the time domain variations of an impacted note in normal playing condition, resulting in an excellent agreement for the temporal behaviour of the first four harmonics

Idiophones à plaques et coques. Partie II. Instruments fortement non linéaires : cymbales, tam-tams et plaques tonnerre

International audienceDans cet article nous nous intéressons aux idio-phones à plaques et coques dont le comportement est fortement non linéaire. Nous traiterons du cas des cymbales, des gongs – et plus particulièrement des tam-tams – ainsi que de la plaque tonnerre. Ces instruments sont distingués des précédents, dits « à faible non-linéarité », car la physique vibratoire mise en jeu est radicalement différente, beaucoup plus complexe et très fortement non linéaire. Au niveau perceptif, cette distinction est immédiatement notable à l'oreille car les instruments que nous considérons dans cet article n'ont pas de hauteur définie. Leur son caractéristique est décrit comme très brillant : le contenu fréquentiel est tellement riche qu'on ne peut leur attribuer une hauteur précise, on observe plutôt un continuum de fréquences

Nonlinear vibrations of steelpans: analysis of mode coupling in view of modal sound synthesis.

Steelpans are musical percussions made from steel barrels. During the manufacturing, the metal is stretched and bended, to produce a set of thin shells that are the differents notes of the instrument. In normal playing, each note is struck, and the sound reveals some nonlinear characteristics which give its peculiar tone to the instrument. In this paper, an experimental approach is first presented in order to show the complex dynamics existing in steelpan’s vibrations. Then two models, based on typical modal interactions, are proposed to quantify these nonlinearities. Finally, one of them is observed in free oscillations simulations, in order to compare the internal resonance model to the steelpan vibrations behaviour in normal playing. The aim is to identify the important modes participating in the vibrations in view of building reduced-order models for modal sound synthesis

Statistics of power injection in a plate set into chaotic vibration

A vibrating plate is set into a chaotic state of wave turbulence by either a periodic or a random local forcing. Correlations between the forcing and the local velocity response of the plate at the forcing point are studied. Statistical models with fairly good agreement with the experiments are proposed for each forcing. Both distributions of injected power have a logarithmic cusp for zero power, while the tails are Gaussian for the periodic driving and exponential for the random one. The distributions of injected work over long time intervals are investigated in the framework of the fluctuation theorem, also known as the Gallavotti-Cohen theorem. It appears that the conclusions of the theorem are verified only for the periodic, deterministic forcing. Using independent estimates of the phase space contraction, this result is discussed in the light of available theoretical framework

A complete vibroacoustic model for the nonlinear response of imperfect circular plates : application to sound synthesis

International audienceA complete vibroacoustic model is presented in order to compute numerically the sound pressure generated by a thin circular plate vibrating with large amplitude motions. The vibratory part relies on a modal approach for the von Kármán thin plate equations. A special emphasis is put in this paper on the inclusion of a geometrical imperfection describing the shape of the circular plate, hence extending previous results for perfect plates to the generic case of imperfect plates and shallow shells. A conservative scheme is used in order to integrate in time the modal equations of motion for the imperfect plate. The acoustic radiation is taken into account by using a finite difference approach for the sound field. The vibroacoustic coupling gives rise to a complete model which is applied for the purpose of sound synthesis of cymbals and gong-like instruments. Simulation results are shown in order to investigate the influence of the geometric imperfection on the sound produced. Plates with different profiles are compared and focus is set on the ability of the imperfection to favour the appearance of the turbulent regime

A phenomenological model for predicting the effect of damping on wave turbulence spectra in vibrating plates

International audienceThin plates vibrating at large amplitudes may exhibit a strongly nonlinear regime that has to be studied within the framework of wave turbulence. Experimental studies have revealed the importance of the damping on the spectra of wave turbulence , which precludes for a direct comparison with the theoretical results, that assumes a Hamiltonian dynamics. A phenomenological model is here introduced so as to predict the effect of the damping on the turbulence spectra. Self-similar solutions are found and the cutoff frequency is expressed as function of the damping rate and the injected power

Design of a Magnetic Vibration Absorber with Tunable Stiffnesses

International audienceThe design and characterisation of a magnetic vibration absorber (MVA), completely relying on magnetic forces, is addressed. A distinctive feature of the absorber is the ability of tuning the linear stiffness together with the nonlinear cubic and quintic stiffnesses by means of repulsive magnets located in the axis of the main vibrating magnetic mass, together with a set of corrective magnets located off the main axis. The tuning methodology is passive and relies only on three geometrical parameters. Consequently the MVA can be adjusted to design either a nonlinear tuned vibration absorber (NLTVA), a nonlinear energy sink (NES), or a bi-stable absorber with negative linear stiffness. The expressions of the stiffnesses are given from a multipole expansion of the magnetic fields of repulsive and corrective magnets. A complete static and dynamic characterisation is performed, showing the robustness of the modelling together with the ability of the MVA to work properly in different vibratory regimes, thus making it a suitable candidate for passive vibration mitigation in a wide variety of contexts

Effects of geometrical nonlinearities on the acoustic black hole effect

International audienceThe Acoustic Black Hole effect (ABH) is a passive vibration damping technique without added mass based on flexural waves properties in thin structures with variable thickness. The usual implementation on plates is a region where the thickness is reduced with a power law profile, covered with a visco-elastic layer. The inhomogoneity induces a decrease of the wave speed and an increase of the amplitude in the small thickness region, which makes the energy dissipation more efficient due to the absorbing layer. The wave amplitude in the ABH easily reaches the plate thickness and is the origin of geometrical nonlinearities. These nonlin-earities can generate coupling between linear beam eigenmodes of the structure and induce energy transfer between low and high frequency regime. This phenomenon may be used to increase the efficiency of the ABH treatment in the low frequency regime where it is usually inefficient. An experimental investigation shows that the ABH termination displays a nonlinear behaviour and allows for modal coupling. A strongly nonlinear regime can also be observed, which is associated with Wave Turbulence. A model of nonlinear ABH beam as von Kármán plate of variable thickness and a modal resolution of the problem confirm the observed effects and gives more insights on these results