The Limit Cycle Oscillations (LCO) of a two-dimensional airfoil, including a control surface with freeplay restoring force at the trailing edge, are investigated theoretically. The model has three degrees of freedom: airfoil plunge h, airfoil pitch alpha, and control surface flap angle beta. The pitch and plunge motions are governed by the stiffness of wing in deflection and the torsional stiffness around its hinge line. The flap motion is governed by its torsional (linear) stiffness around its hinge line and by a certain angular size ranging from -delta to +delta where the stiffness is zero (freeplay region). The aeroelastic equations of motion can be formulated as a set of first-order ordinary differential equations. The dynamic response is investigated by a time integration method. When delta becomes negligibly small, the aeroelastic system becomes strictly linear. In this case, the system stability limit is indicated by the flutter velocity. At small flap stiffnesses, two distinct flutter velocities and frequencies are observed, corresponding to pitch dominated and plunge dominated motion. At large flap stiffnesses, there is only one flutter velocity that corresponds to the plunge dominated motion. When a freeplay is present, there are various types of nonzero motions for a velocity U below the flutter speed, including LCOs with or without higher harmonics and chaos. There are two abrupt changes: one at a certain value U = U1, when the zero motion becomes unstable and the system settles down at a nonperiodic LCO, and the other one at U = U2 > U1, where the system motion changes from nonperiodic to periodic motion. The first jump at U = U1 is a subcritical bifurcation, where a hysteresis is observed, whereas no hysteresis is observed in the second jump. In the hysteresis region (U1 < U < U2), the onset of LCO depends on the initial conditions in flap rotation. For large initial flap angle, the steady states are LCOs for all initial flap angle velocities considered in this study (including zero velocity). For small initial flap angles, however, the smaller the initial flap angle is, the larger is the initial flap angle velocity needed to excite LCOs. For very small flap angles, no LCO is found. The effect of the initial pitch angle in the hysteresis region appears much more complex. The existence of 'zero-motion islands' in the pitch-pitch velocity diagram far away from the origin indicates that for large initial pitch angle displacement and angle velocity the motion may not be excited to an LCO. However, for the same flow condition, a small initial pitch angle displacement and velocity may be able to excite an LCO. As the flow velocity increases, the far away islands of zero motions disappear. A general derivation of two harmonic balance techniques, including the first (HB1) and the third (HB3) harmonics, respectively, has been developed specifically for the freeplay model. In the HB1 approach, a single polynomial equation for the fundamental frequency is obtained, and the solution of the Fourier coefficients are explicitly expressed in terms of the frequency. In the HB3 approach, a system of three nonlinear equations for the frequency and two augmented variables describing the freeplay range within one period of flap rotation is obtained. The solutions of the Fourier coefficients are explicit functions of the frequency and two augmented variables. Generally, the HB3 improves on the HB1 results and are in excellent agreement with the time integration results.


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    Titel :

    Harmonic balance approach for an airfoil with a freeplay control surface


    Weitere Titelangaben:

    Beschreibung der Flatterschwingungen einer Tragfläche mit einer frei beweglichen Landeklappe mit der Methode der harmonischen Balance


    Beteiligte:
    Liu, Liping (Autor:in) / Dowell, Earl H. (Autor:in)

    Erschienen in:

    AIAA Journal ; 43 , 4 ; 802-815


    Erscheinungsdatum :

    2005


    Format / Umfang :

    14 Seiten, 17 Bilder, 2 Tabellen, 18 Quellen




    Medientyp :

    Aufsatz (Zeitschrift)


    Format :

    Print


    Sprache :

    Englisch






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