35 Experimental Passive Flutter Mitigation Using a Linear Tuned Vibrations Absorber 393 spanwise strips of width dy. These infinitesimal forces and moments are computed assuming 2D aerodynamics. The strip theory assumption leads to the following formulation for the flap and pitch aerodynamic moments M D Z s2 s1 y dL.t/ (35.5) M D Z s2 s1 dM .t/ (35.6) s1 being the distance between the flap axis and the root of the wing ands2 the distance between the flap axis and the tip of the wing. The lift force dLand pitch moment dM of a strip can be computed using any 2D unsteady aerodynamic formulation. Unsteady aerodynamic modelling based on Wagner’s function [12] was chosen here because the reduced frequency of oscillation k D !b U 1:5 >> 0:02 is too large to use quasi-steady aerodynamics while Theodorsen theory is defined in frequency domain, which makes it difficult to use on a nonlinear system. The growth of circulation around a flat uncambered airfoil after a step change of incidence is approximated by Wagner’s function ˆ.t/ D1 1e 1Ut b 2e 2Ut b (35.7) Integrating Eqs. (35.5) and (35.6) with Fung’s lift and moment expressions [12] and applying a transformation to replace the wake integrals by aerodynamic state variables [13] yields the complete equations of motion I S S I C b2 ƒ3=3 abƒ2=2 abƒ2=2 b2ƒ1.a2 C1=8/ R R C c c c c C Ub 2 3 ˆ.0/ƒ3 bƒ2Œ1=2 ˆ.0/.1 a=2/ bƒ2ˆ.0/.aC1=2/ ƒ1b2.a 1=2/Œ1C2ˆ.0/.aC1=2/ P P C " k k k k C U2 2 3 ƒ3Œ bPˆ.0/ U bƒ2Œˆ.0/ C bPˆ.0/ U .1=2 a/ bƒ2Œ bPˆ.0/ U .aC1=2/ 2ƒ3.aC1=2/Œˆ.0/ bPˆ.0/ U .a 1=2/ !# C 2 6 6 6 4 U3 0 B B @ 2 2 1 1ƒ3=3b 2 1 1ƒ2.2aC1/=2 2 2 2 2ƒ3=3b 2 2 2ƒ2.2aC1/=2 ƒ2. 1 1 2 1 1=2Ca 2 1 1/ 2b 1 1ƒ1.aC1=2/. 1.a 1=2/ C1/ ƒ2. 2 2 2 2 2=2Ca 2 2 2/ 2b 2 2ƒ1.aC1=2/. 2.a 1=2/ C1/ 1 C C A T3 7 7 7 5 0 B B @ w1 w2 w3 w4 1 C C A D M ;ext.t/ M ;ext.t/ C2 UbPˆ.t/ ƒ3 .0/=3C. 3b 2 xf /ƒ2 .0/=2 .aC1=2/bŒƒ2 .0/=2C. 3b 2 xf /ƒ1 .0/ 0 knl;3 3 (35.8) with ƒj D s j 2 s j 1. The left hand side of Eq. (35.8) includes inertia, damping, stiffness terms and the aerodynamic state proportionality matrix. The right hand side comprises three terms, the external loads (set to zero since self-excited oscillations are under investigation), a transient term, which is also set to zero since it quickly decays in time, and a term related to the structural nonlinearities. The nonlinear equations of motion are solved using a numerical continuation algorithm based on a finite difference formulation [14]. 35.3 Aeroelastic Analysis The system’s aeroelastic behaviour is first studied at sub-critical airspeeds, where the average damping is positive and leads to decaying motions, as demonstrated by the pitch response time histories plotted in Fig. 35.5a for airspeeds of 0, 5.5 and 8.7 m/s. At higher airspeeds, the response is a self-excited limit cycle oscillation (LCO). This behaviour is demonstrated in Fig. 35.5b which plots pitch response time histories at airspeeds of 12.4, 13.3 and 14.8 m/s. Figure 35.6 displays the variation of the modal parameters of the pitch and flap modes of the NLPFW with airspeed according to experimental measurements (black dots) and the analytical model (plain lines). The imaginary part of the response—identified using Fast Fourier Transform—is accurately reproduced by the model and shows the typical features
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