Lab · Live model · 01

A step in the angle of attack.

The wing receives a step in the angle of attack, as in the paper, and answers with a transient: the tip rises, and the force overshoots and settles. At these deflections the force is a nonlinear function of the shape of the wing, and that relation is what the model of the paper learns across the range of angles.

Angle of attack, 0° to 8°; each new value is a step. 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 is the Pazy wing of the papers: a strip of 5.5 chords with its ribs, seen from the front and a little from above, clamped at the root on the left. The arrows along the span are the load, normal to the surface. As the wing curls they tilt inward, and the vertical part of the force falls. The faint outlines are the wing 0.6, 0.4 and 0.2 seconds before, in the manner of the paper's Fig. 1, which shows the shape at four time steps.

The inset is the vertical force at the tip against time, which is the paper's Fig. 5. The dashed accent line is the level the nonlinear model settles at. The dashed grey line is the level the linear model gives, where the force is proportional to the angle. The gap between the two lines grows with the angle.

The model

The scene integrates the first two bending modes of a clamped-free beam under a load that follows the deformed surface:

\ddot{q_{i}} + 2 ζ_{i} ω_{i} \dot{q_{i}} + ω_{i}^{2} q_{i} = \frac{Q_{i}}{m}

the two bending modes, i = 1 and 2

Q_{i} = \int_{0}^{1} l (s) φ_{i} (s) d s

the generalised force on a mode

l (s) = K [α \cos ψ (s) - \frac{\dot{w} (s)}{U}]

the load per unit span: the incidence the surface sees, less the plunge rate

F_{z} = l (1) \cos ψ (1)

the vertical force at the tip, where s = 1

ψ (s) = \sum_{i} q_{i} φ_{i}^{'} (s)

the slope of the surface; the axis keeps its length, so the tip curls inward

K is set so that the linear model lifts the tip by half the span at 8 degrees. The slope enters twice: the surface sees less incidence as it turns away from the flow, and its force has a smaller vertical part. Both take the force below the straight line as the angle grows, and that gap is the nonlinearity the model of the paper captures. The rate term is the aerodynamic damping, so the transient decays, which is the stability constraint of the paper. The wing settles where the load and the stiffness balance, which is the steady-state constraint. The first mode is at 0.42 Hz on the screen; on the wing it is at 4.5 Hz.

The control

The slider is the step input: each new value is a step, and the wing answers with a transient. Auto runs the angles of the paper in turn: 1, 2, 4, 7 and 8 degrees, 7 seconds each. The paper trains its model on the steps to 1, 2, 7 and 8 degrees and tests it on the step to 4 degrees.

Where it comes from

The scene draws the background of the paper Physics-Informed Data-Driven Modelling of Nonlinear Aerodynamic Forces of the Pazy Wing.