On a model reduction method for computing forced response using non-linear normal modes

This paper presents a method for constructing reduced order models using non-linear normal modes (NNM) in the context of non-linear vibrations. Starting from a discretised version of the non-linear problem, the non-linear normal modes (NNM) of the structure are computed using the Harmonic Balance Method (HBM). A two parameters (amplitude and phase) parametrization of the NNM is introduces and they are then used for the forced response construction, assuming that the solution involves only a single (non-linear) resonant mode. The solution is then eventually corrected by linear terms which helps decreasing the error outside resonance , in particular around anti-resonances. The procedure results in two algebraic equations containing only two variables, one controlling the amplitude of vibration and the other controlling the phase, thus leading to a drastic reduction in the number of degrees of freedom. The procedure is illustrated on a simple, but representative, example. It is shown that a single mode approximation is sufficient for computing a good approximation of the forced response around a particular mode

On a new harmonic selection technique for harmonic balance method

International audienceThis paper is intended to present a new harmonic selection technique when solving nonlinear dynamic systems with the harmonic balance method. This technique belongs to the class of method called the adaptive harmonic balance method (AHBM). The harmonic selection is based on the use of a tangent predictor and relies on a stepwise regression procedure that allows for a dynamic management of the number of selected harmonics via an addition or removal procedure. The efficiency of this method relative to the classical harmonic balance method (HBM) is then evaluated through examples; this later step will indicate that AHBM can significantly reduce the number of variables, thus leading to computational time savings without deteriorating solution quality

Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases

International audienceThis paper is devoted to the study of vibration of mechanical systems with geometric nonlinearities. The harmonic balance method is used to derive systems of polynomial equations whose solutions give the frequency component of the possible steady states. Groebner basis methods are used for computing all solutions of polynomial systems. This approach allows to reduce the complete system to an unique polynomial equation in one variable driving all solutions of the problem. In addition, in order to decrease the number of variables, we propose to first work on the undamped system, and recover solution of the damped system using a continuation on the damping parameter. The search for multiple solutions is illustrated on a simple system, where the influence of the retained number of harmonic is studied. Finally, the procedure is applied on a simple cyclic system and we give a representation of the multiple states versus frequency

Energy Localization in Periodic Structures: Application to Centrifugal Pendulum Vibration Absorber

International audienceIn this paper we study the non-linear dynamic of centrifugal pendulum vibration absorbers (CPVA), and we pay a special a ention to localized state solutions. e prediction of such states of vibration, and their stability, is of particular importance because they can lead to inefficient behavior for the CPVA and/or unforeseen stress levels. Using an approximated equation for the pendulums dynamics, we derive initial conditions to put the system in localized states. Following an analytic study of the non-linear normal mode of the system, the resolution of the approximated equation is carried out in the frequency domain using the Harmonic Balance Method and the Asymptotic Numeric Method. For sufficiently low inertia ratio, we show that the system can possess stable localized states

Multiple spatially localized dynamical states in friction-excited oscillator chains

International audienceFriction-induced vibrations are known to affect many engineering applications. Here, we study a chain of friction-excited oscillators with nearest neighbor elastic coupling. The excitation is provided by a moving belt which moves at a certain velocity v d while friction is modelled with an exponentially decaying friction law. It is shown that in a certain range of driving velocities, multiple stable spatially localized solutions exist whose dynamical behavior (i.e. regular or irregular) depends on the number of oscillators involved in the vibration. The classical non-repeatability of friction-induced vibration problems can be interpreted in light of those multiple stable dynamical states. These states are found within a "snaking-like" bifurcation pattern. Contrary to the classical Anderson localization phenomenon, here the underlying linear system is perfectly homogeneous and localization is solely triggered by the friction nonlinearity

Dynamique non linéaire des structures mécaniques: application aux systèmes à symétrie cyclique

In an industrial context, the design of new mechanical systems requires long design processes in order to define and to anticipate the behavior of all the constitutive parts. In the particular case of aeronautical structures such as plane engines, design is especially critical since they have to meet various and strict needs (life duration, performances...). Then, anticipating vibratory behavior is very important as this provides information about cyclic solicitations and fatigue. Most often, numerical models are used to mimic the structure and mechanical behavior is simulated by solving a set of differential equations. In the case of industrial structures, such models can be quite large and their resolution very time-consuming. Moreover, in order to model experimental behavior realistically, it is often necessary to take nonlinear phenomena into account and thus increase the required computational effort. The work presented in this PhD deals with the study of mechanical nonlinear systems. It focuses on two principal directions : model reduction and multiple solutions computation. The goal of the first direction is to contribute to the building of numerical reduced order models usable in industrial context and to propose tools to exploit an interpret them. Particularly, Galerkin projection methods are investigated in the context of nonlinear systems reduction, showing that those methods are, under certain conditions, able to give a reliable picture of full system behavior. In the case of the harmonic balance method, complementary methods are also proposed to reduce the size of the algebraic equations system by using harmonic selection techniques. The presented methods are firstly illustrated and compared on a simple nonlinear beam example ; they are then applied to an industrial model of open rotor blade. The second direction of this work deals with the computation of multiple solutions arising in nonlinear dynamical systems. Indeed, it has been shown that such systems can present different stable configurations for a given solicitation. The objective here is to provide tools for computing such multiple solutions. We only consider the case of periodic solutions for systems with polynomial nonlinearities, treated with harmonic balance method. These hypotheses enable one to search for multiple states as solutions of polynomial algebraic systems of equations, for which some methods exist to compute the entire set of solutions. In particular, we propose to use methods relying on Groebner basis computation, in order to compute the whole set of solutions. The proposed methods are illustrated and compared on simple examples, showing that even such simple systems can present very complex dynamical behavior.D'un point de vue industriel, la mise en place de nouvelles architectures de systèmes mécaniques nécessite un long processus de conception permettant de définir et d'anticiper le comportement. Dans le cas particulier des systèmes aéronautiques tels que les moteurs d'avions, un certain nombre de pièces sont particulièrement sensibles car elles doivent répondre à des impératifs stricts en termes d'encombrement, de performance et de tenue mécanique. Dans ce contexte, la prévision du comportement vibratoire revêt une importance particulière puisqu'elle permet d'évaluer le niveau des sollicitations cycliques appliquées sur le système et guide ainsi la détection en amont d'éventuels problèmes de fatigue des matériaux. La plupart du temps, des modèles numériques sont utilisés pour représenter les structures, et le comportement est simulé en résolvant un ensemble d'équations. Pour atteindre un niveau de détail répondant au besoin industriel, ces modèles peuvent être particulièrement gros, et la résolution des équations associées demande des ressources et des temps de calcul considérables. De plus, pour rendre compte au mieux des comportements observés expérimentalement, il est souvent nécessaire de prendre en compte des phénomènes non-linéaires, ce qui augmente encore la difficulté. Les travaux présentés dans ce manuscrit concernent cette problématique du comportement vibratoire des structures non-linéaires et s'orientent autour de deux axes : la réduction de modèle et le calcul des solutions multiples. L'objectif du premier axe est de contribuer à la construction de modèles numériques non-linéaires réduits utilisables en conception de systèmes industriels et de proposer des outils d'exploitation et d'interprétation de ces modèles. En particulier, on considère le cas des méthodes de projection de Galerkin et on montre qu'elles sont à même de construire des modèles réduits réalistes. Des méthodes complémentaires de réduction de modèles sont également présentées dans le cas particulier de la recherche de solutions par la méthode de la balance harmonique (HBM) : on s'intéressera en particulier à des méthodes de sélection d'harmoniques. Après avoir comparé les différentes méthodes proposées sur un exemple simple de poutre non-linéaire, elles sont appliquées à un modèle de structure industrielle représentant une aube d'hélice d'open rotor. Le second axe de ces travaux concerne le calcul de solutions multiples pour les systèmes dynamiques non-linéaires. Une particularité de ces systèmes est en effet de présenter plusieurs configurations stables pour un état de sollicitation donné. Il s'agira ici de proposer des méthodes de calcul permettant de dresser la liste exhaustive des solutions possibles. Le travail présenté se concentre sur la recherche de solutions périodiques par la méthode de la balance harmonique pour des systèmes possédant des non-linéarités polynomiales. Ces restrictions conduisent à la résolution de systèmes polynomiaux pour lesquels il existe des méthodes permettant de calculer l'ensemble des solutions. En particulier, on propose l'utilisation originale de méthodes basées sur le calcul de bases de Groebner pour la résolution de systèmes polynomiaux issus de la mécanique. Les différentes méthodes présentées sont illustrées et comparées sur des exemples simples. Les résultats montrent que même pour des systèmes simples, le comportement dynamique peut être très complexe

On a model reduction method for computing forced response using non-linear normal modes

This paper presents a method for constructing reduced order models using non-linear normal modes (NNM) in the context of non-linear vibrations. Starting from a discretised version of the non-linear problem, the non-linear normal modes (NNM) of the structure are computed using the Harmonic Balance Method (HBM). A two parameters (amplitude and phase) parametrization of the NNM is introduces and they are then used for the forced response construction, assuming that the solution involves only a single (non-linear) resonant mode. The solution is then eventually corrected by linear terms which helps decreasing the error outside resonance , in particular around anti-resonances. The procedure results in two algebraic equations containing only two variables, one controlling the amplitude of vibration and the other controlling the phase, thus leading to a drastic reduction in the number of degrees of freedom. The procedure is illustrated on a simple, but representative, example. It is shown that a single mode approximation is sufficient for computing a good approximation of the forced response around a particular mode

Model reduction with nonlinear normal modes

International audienc

Vibration Analysis of a Nonlinear System With Cyclic Symmetry

International audienceThis work is devoted to the study of non linear dynamics of structures with cyclic symmetry under geometrical nonlinearity using the harmonic balance method (HBM). In order to study the influence of the non-linearity due to large deflection of blades a simplified model has been developed. It leads to nonlinear differential equations of the second order, linearly coupled, in which the nonlinearity appears by cubic terms. Periodic solutions in both free and forced cases are sought by the HBM coupled with an arc length continuation and stability analysis. In this study, a specific attention has been paid to the evaluation of nonlinear modes and to the influence of excitation on dynamic responses. Indeed, several cases of excitation have been analyzed: punctual one and tuned or detuned low engine order. The paper shows that for a localized, or sufficiently detuned, excitation, several solutions can coexist, some of them being represented by closed curves in the Frequency-Amplitude domain. Those different kinds of solution meet up when increasing the force amplitude, leading to forced nonlinear localization. As the closed curves are not tied with the basic nonlinear solution they are easily missed. They were calculated using a sequential continuation with the force amplitude as a parameter