394 E. Verstraelen et al. Time [s] 0 5 10 15 20 25 30 Angle [rad] -0.1 -0.05 0 0.05 0.1 U = 0 m/s U = 5.5 m/s U = 8.7 m/s Time [s] 20 10 0 Pitch angle [rad] -0.15 -0.05 0 0.05 0.15 U = 14.8 m/s U = 13.3 m/s U = 12.4 m/s a b Fig. 35.5 Time series of the pitch response of the system at sub-critical and super-critical airspeeds. (a) Sub critical pitch response. (b) Super critical pitch response Airspeed [m/s] 0 2 4 6 8 10 12 Imag(λ) 0 5 10 15 20 25 Airspeed [m/s] 0 2 4 6 8 10 12 -Real(λ) 0 0.5 1 1.5 Exp (Pitch) Exp (Flap) Model (Pitch) Model (flap) a b Fig. 35.6 Variation of the pitch and flap modal parameters with the airspeed. (a) Pulsation. (b) Decay rate of an aeroelastic system undergoing flutter: the frequency gap between the two modes decreases as the airspeed is increased until the modes are close enough to interact and cause flutter. The real part—identified using an exponential fitting of the Hilbert transform of the response—is also typical of aeroelastic systems: the flap damping strongly increases while the pitch damping increases at first then decreases until it drops to zero and flutter occurs. In this case, the model seems to over-estimate the damping, however it must be noted that damping is very difficult to identify and even harder to model; considering the simplicity of the model, the estimate is satisfactory. Once the flutter speed is reached, the average damping of the system drops to zero and limit cycle oscillations are observed. Figure 35.7a displays the bifurcation diagrams in pitch and flap angle of the system obtained from wind tunnel observations (triangles and dots) and from the numerical continuation analysis of Eq. (35.8). The airspeed was increased then decreased in the wind-tunnel to look for hysteresis effects but none were found. Both experimental and numerical results exhibit a supercritical Hopf bifurcation at 11.5 m/s, followed by increases in LCO amplitude and frequency, consistent with cubic hardening stiffness. At 13.5 m/s a discontinuity occurs in the experimental pitch measurements; both the amplitude and frequency increase sharply. This phenomenon is due to dynamic stall and is overlooked by the reduced order model, which uses linear aerodynamics. The flap displacement undergoes a gentle increase in amplitude that is predicted with reasonably accuracy by the model. Because of the coupling between the pitch and the flap modes, at 13.5 m/s a jump is observed also in the flap response. However, the increase in amplitude is not significant.
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