Lab · Live model · 02

Flutter across the angle of attack.

The wing sits at the trim of its angle of attack and receives the perturbation of one degree that the paper applies. Between 3 and 4.6 degrees the perturbation grows, through the coupling of the second bending mode with the first torsion mode; below and above that band it decays. The parametric model of the paper reproduces this stability boundary across the angle of attack.

Angle of attack, 0.5° to 8°. In Auto the model sets the parameter and runs the cases of the paper; in Hold the slider sets it. The line below the control says what the model does at this moment. With reduced motion the model shows a fixed frame, and the controls still work.

What the drawing shows

The wing sits at the trim shape of its angle. The faint fan behind it is the trims from 1 to 8 degrees, as in the paper's Fig. 2. The thin lines that trail the wing are its shape over the last 1.2 seconds: a ripple of the second bending mode that grows or decays. The tip twists with that ripple, a quarter cycle behind it, as the coupled flutter mode does.

The upper chart is the velocity of the tip against time, as in the paper's Fig. 7 and Fig. 10: the oscillation grows inside the band and dies outside it. The lower chart is the largest real part of the eigenvalues against the angle of attack, read from the paper's Fig. 6. Where it is above zero the wing flutters, and that band is shaded. The marker is the angle of the moment.

The model

The first bending mode carries the trim and is stable. The second bending mode grows or decays at the rate the paper gives for the angle:

\ddot{q_{1}} + 2 ζ ω_{1} \dot{q_{1}} + ω_{1}^{2} (q_{1} - q_{trim} (α)) = 0

the first bending mode, stable

\ddot{q_{2}} - 2 [μ (α) - g {(\frac{q_{2}}{Q})}^{2}] \dot{q_{2}} + ω_{2}^{2} q_{2} = 0

the second bending mode, with a soft limit on its amplitude

μ (α) > 0 for 3.0 ° \leq α \leq 4.6 °

the growth rate of the paper, read from its Fig. 6

q_{trim} (α) = 0.062 α - 0.0008 α^{2}

the rise of the tip at the trim, as a fraction of the span

θ_{tip} = \frac{G \dot{q_{2}}}{ω_{2}}

the twist, a quarter cycle behind the bending

On the wing the second bending mode is at 29 Hz; on the screen it is at 2.4 Hz, and the growth rates are scaled by the same ratio, so the growth in one cycle is the one the paper finds. The term in g is a soft limit that holds the unstable motion at a small amplitude; the paper sees the peaks of the unstable response saturate in the same manner. The trim comes from the flutter chart the paper reproduces, in which the tip rises to almost half the span at 8 degrees.

The control

The slider sets the angle, and each new angle is a new trim with the perturbation of one degree. Auto runs the four cases of the paper's Fig. 10, 11 seconds each: 1.75 degrees before the band, 4 degrees inside it, 5 degrees at its end, and 7.5 degrees after it.

Where it comes from

The scene draws the background of the paper Data-Driven Parametric Aeroelastic Modeling of the Pazy Wing.