Inﬂuence d’un NLTVA sur l’instabilité de ﬂottement

Inﬂuence d’un NLTVA sur l’instabilité de ﬂottement

L’inﬂuence d’un amortisseur non-linéaire de vibration accordée (NLTVA) sur l’instabilité de ﬂot- tement de proﬁl d’aile est étudié. En particulier, son eﬀet sur le seuil de l’instabilité, l’amplitude des cycles limites se développant après le seuil d’instabilité et la potentielle sous-criticalité de la bifurcation, sont analysés. Dans cette optique, le mouvement du proﬁl d’aile est étudié à l’aide du modèle classique à deux degrés de liberté en utilisant une approche linéaire pour l’estimation des eﬀorts aérodynamiques. Aﬁn d’assurer l’apparition de cycles limites dans le régime post-ﬂottement, des non linéarités cubiques sont incluses dans le modèle structurel. L’inﬂuence de chaque paramètre du NLTVA est étudié et une règle d’optimisation de ces paramètres est proposée. L’étude révèle la capacité du NLTVA à retarder de façon conséquente le seuil de l’instabilité, ainsi qu’à assurer la super-criticalité de la bifurcation et réduire l’amplitude des cycles limites.Ce chapitre est composé d’un article écrit en anglais et soumis à la revue Journal of Computa- tional Nonlinear Dynamics. La première partie de l’article présente les équations du mouvement du système couplant le proﬁl d’aile et le NLTVA, le deuxième chapitre s’intéresse à l’analyse de stabilité linéaire du proﬁl d’aile à deux degrés de liberté couplé au NLTVA et la troisième partie détaille l’étude de l’inﬂuence du NLTVA dans le régime post-ﬂottement.

Flutter control of a two-degrees-of-freedom airfoil using a nonlinear tuned vibration absorber

When slender structures, such as bridge deck or airfoil, undergo wind excitation, a ﬂutter insta- bility, triggered by a Hopf bifurcation, may arise. The ﬂow velocity for which the instability starts is called ﬂutter velocity. This phenomenon is detrimental and may even lead to fatal vibrations of the structure.Numerous strategies have been investigated in the past in order to control the ﬂutter instability. Most of them concern active control techniques, such as governing a trailing edge ﬂap on the wing [1, 2, 3] or using piezoelectric actuators [4]. The reader may refer to the Dowell’s book on aeroe- lasticity [5] for more details on ﬂutter active control. The alternative strategy is the use of passive control techniques, which is also the aim of the present study. This research topic is relatively recent. Most eﬀorts have been devoted to bridge deck passive control [6, 7, 8].The most classical passive device for controlling ﬂutter instabilities is the tuned vibration absorber initially developed by Den Hartog [9], and generally known as TMD (Tuned Mass Damper) or TVA (Tuned Vibration absorber). This device consists of a small lumped mass attached to the primary structure through a linear spring and a damper. If its eigenfrequency and damping ratio are correctly tuned, it can signiﬁcantly shift the ﬂutter speed [10]. Nevertheless, the TMD has no eﬀect on the nonlinearities arising in the post-critical regime.Lee et al. [11, 12] proposed to use a nonlinear energy sink (NES) to control the ﬂutter instability.

The NES has been initially investigated in the ﬁeld of vibration control [13, 14, 15], it consists of an oscillator linked to the primary structure through an essential nonlinear stiﬀness (i.e. with no linear part), leading to targeted energy transfer [16]. Applied to the ﬂutter control, the method shows an important reduction in the amplitude of the LCOs [11, 12]. Nevertheless, it has no eﬀect on the ﬂutter velocity, and needs a certain amount of energy to be activated.In the present study, the use of a nonlinear tuned vibration absorber (NLTVA), as proposed by Viguié and Kerschen [17] and Habib et al [18] is investigated. The distinctive feature of the NLTVA is that it possesses both a linear and a nonlinear stiﬀness. The linear part of the absorber enables to recover the optimal control given by a TMD, whereas its nonlinear part can be tuned in order to enhance the control of the primary system nonlinearities, using a so-called principle of similarity [19]. Its eﬀectiveness has been proven on a Van der Pol-Duﬃng oscillator [20]. Moreover, several experimental realizations of the NLTVA have been recently explored [21, 22] and a tuning rule for the NLTVA parameters coupled with a primary system having polynomial nonlinearities has been proposed using the Den Hartog’s equal peak method [19].In the present context of the ﬂutter instability, two main design rules are followed for optimizing the NLTVA characteristics. First, the linear stiﬀness of the absorber is tuned in order to increase as much as possible the critical ﬂutter velocity, thus repelling the appearance of LCO to larger ﬂow velocities. This step consists in ﬁnding the optimal values of a TMD in order to control the ﬂutter instability, for which preliminary results can be found in [23] for a speciﬁc set of airfoil parameters.

Secondly, the nonlinear stiﬀness of the NLTVA may be tuned in order to enforce supercriticality, as well as to reduce the LCO amplitudes. The aim of the present study is thus to extend the results presented in [20], obtained for a Van der Pol-Duﬃng oscillator as primary structure. To get closer to a real airfoil situation, the primary structure is here selected as the classical two degrees-of-freedom aeroelastic system, with pitch and plunge motions as generalized coordinates [5]. The quasi-steady theory is used to estimate the aerodynamic loads. In order to take into account the nonlinear structural behavior for large displacements, the model includes cubic stiﬀness terms in the pitch and plunge motion. This allows the system to develop LCO in the post-critical regime.The ﬁrst section is devoted to the linear stability analysis. The optimization of the linear char- acteristics of the vibration absorber is investigated, showing in particular how the ﬂutter velocity can be repelled to larger values using a well-tuned absorber. The airfoil parameters are then varied and a design rule is proposed in order to ﬁnd easily the optimized linear values. Then the inﬂuence of the nonlinear part of the NLTVA is investigated. Two diﬀerent cases are speciﬁcally studied for the airfoil nonlinear restoring force: a hardening and a softening scenario. These results give a number of fruitful strategies in order to control passively the ﬂutter instability by means of a nonlinear vibration absorber.