Nonlinear normal modes of a two degree of freedom oscillator with a bilateral elastic stop

A study of the non linear modes of a two degree of freedom mechanical system with bilateral elastic stop is considered. The issue related to the non-smoothness of the impact force is handled through a regularization technique. In order to obtain the Nonlinear Normal Mode (NNM), the harmonic balance method with a large number of harmonics, combined with the asymptotic numerical method, is used to solve the regularized problem. These methods are present in the software "package" MANLAB. The results are validated from periodic orbits obtained analytically in the time domain by direct integration of the non regular problem. The two NNMs starting respectively from the two linear normal modes of the associated underlying linear system are discussed. The energy-frequency plot is used to present a global vision of the behavior of the modes. The dynamics of the modes are also analyzed comparing each periodic orbits and modal lines. The first NNM shows an elaborate dynamics with the occurrence of multiple impacts per period. On the other hand, the second NNM presents a more simple dynamics with a localization of the displacement on the first mass

A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions

Combinig the harmonic balance method (HBM) and a continuation method is a well-known technique to follow the periodic solutions of dynamical systems when a control parameter is varied. However, since deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, the classical HBM is often limited to polynomial (quadratic and cubic) nonlinearities and/or a few harmonics. Several variations on the classical HBM, such as the incremental HBM or the alternating frequency/time domain HBM, have been presented in the literature to overcome this shortcoming. Here, we present an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with smooth equations. The main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the HBM. Once the equations have been rendered quadratic, it becomes obvious to derive the algebraic system and solve it by the so-called ANM continuation technique. Several classical examples are presented to illustrate the use of this numerical approach.Comment: PACS numbers: 02.30.Mv, 02.30.Nw, 02.30.Px, 02.60.-x, 02.70.-

Méthode Asymptotique Numérique adaptative pour la dynamique transitoire non-linéaire

National audienceCet article décrit le couplage de la Méthode Asymptotique Numérique (MAN) avec un schéma d’intégration temporelle permettant la dissipation numérique des hautes fréquences tout en conservant les moments linéaires et angulaires. Différentes stratégies sont présentées afin d’optimiser l’efficacité de la méthode proposée, dont l’adaptation de l’ordre des séries de la MAN à chaque pas de temps. L’algorithme obtenu est appliqué à des structures minces fortement non linéaires discrétisées par éléments finis

Passive suppression of helicopter ground resonance instability by means of a strongly nonlinear absorber

International audienceIn this paper, we study a problem of passive suppression of helicopter Ground Resonance (GR) using a single degree freedom Nonlinear Energy Sink (NES). GR is a dynamic instability involving the coupling of the blades motion in the rotational plane (i.e. the lag motion) and the helicopter fuselage motion. A reduced linear system reproducing GR instability is used. It is obtained using successively Coleman transformation and binormal transformation. The analysis of the steadystate responses of this model is performed when a NES is attached on the helicopter fuselage. The NES involves an essential cubic restoring force and a linear damping force. The analysis is achieved applying complexification-averaging method. The resulting slow-flow model is finally analyzed using multiple scale approach. Four steady-state responses corresponding to complete suppression, partial suppression through strongly modulated response, partial suppression through periodic response and no suppression of the GR are highlighted. An algorithm based on simple criterions is developed to predict these steady-state response regimes. Numerical simulations of the complete system confirm this analysis of the slow-flow dynamics. A parametric analysis of the influence of the NES damping coefficient and the rotor speed on the response regime is finally proposed

A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems

International audienceThe high-order purely frequency-based harmonic balance method (HBM) presented by Cochelin and Vergez (2009) [1] and extended by Karkar et al. (2013) [2] now allows to follow the periodic solutions of regularized non-smooth systems (stiff systems). This paper compares its convergence property to a reference method in applied mathematics: orthogonal collocation with piecewise polynomials. A first test is conducted on a nonlinear smooth 2 degree-of-freedom spring mass system, showing better convergence of the HBM. The second test is conducted on a one degree-of-freedom vibro-impact system with a very stiff regularization of the impact law. The HBM continuation of the nonlinear mode was found to be very robust, even with a very large number of harmonics. Surprisingly, the HBM was found to have a better convergence than the collocation method for this vibro-impact system

Exploration des solutions périodiques d'un modèle simple de corde frottée

National audienceAbstract : The HBM-ANM continuation of periodic solutions [Cochelin, Vergez 2009] is applied to a bowed string toy model with a regularized friction law. Several stable solutions can be found for the same parameters, which underlines the usefulness of continuation even with few modes. An investigation with a large number of modes requires optimization, such as harmonic selection with respect to the variables.Résumé : La continuation des solutions périodiques par couplage MAN-HBM [Cochelin, Vergez 2009] est appliquée à un modèle réduit de corde frottée avec une loi de friction régularisée. L'intérêt de la continuation est souligné même avec un nombre restreint de modes, donnant accès à plusieurs solutions stables pour les mêmes paramètres de jeu. L'étude plus complète avec un nombre de modes élevé nécessite une optimisation des méthodes, par exemple une sélection du nombre d'harmoniques différenciée selon les variables

Enhancing the dynamic range of Targeted Energy Transfer in acoustics using several nonlinear membrane absorbers

International audienceIn order to enhance the robustness and the energy range of efficiency of targeted energy transfer (TET) phenomena in acoustics, we discuss in this paper about the use of multiple nonlinear membrane absorbers in parallel. We show this way, mainly thanks to an experimental set-up with two membranes, that the different absorbers have additional effects that extend the efficiency and the possibilities of observation of TET. More precisely, we present the different behavior of the system under sinusoidal forcing and free oscillations, characterizing the phenomena for all input energies. The frequency responses are also presented, showing successive clipping of the original resonance peak of the system. A model is finally used to generalize these results to more than two NES and to simulate the case of several very similar membranes in parallel which shows how to extend the existence zone of TET

Modélisation de structures multi-rubans pour applications spatiales

National audienceOn s'intéresse à la modélisation de structures multi-rubans déployables. Partant d'un modèle 1D spécifique, on établit un outil métier capable de modéliser un mètre ruban seul afin d'effectuer une analyse des réponses possibles pour un jeu de conditions limites imposé. Cet outil sera par la suite étendu à la modélisation de structures multi-rubans, pour conduire une étude comparative de diverses solutions de déploiement envisagées

Modes non-linéaires de structures élastiques dotées de non-linéarités de contact

National audienceCet article propose une méthode de calcul des modes non-linéaires de structures élastiques dotées de non-linéaritées de contact localisées. Les lois de contact sont régularisées au moyen de relations polynomiales comportant un paramètre de régularisation. La procédure de calcul est basée sur le couplage de la méthode de balance harmonique et de la méthode asymptotique numérique