Lab · Live model · 03

A wing tip on a flared hinge.

The flow and the gravity start together, as in the paper. The wing bends up, the tip lags behind it, folds past its final angle and settles: 45 degrees at 10 degrees of incidence, 25 degrees at 5. The hinge is flared, so the fold reduces the incidence of the tip, and the tip settles at the angle where its lift carries it.

Hinge flare, 0° to 45°; the paper uses 10°. 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 view is from the front, along the flow. The inner wing of 12 metres is a beam in its first bending mode, clamped on the left. The tip of 4 metres is a rigid body on the hinge. The accent arrows are the lift along the wing and on the tip, normal to each surface; the grey arrow is the weight of the tip. The faint tips are the tip a half second and a quarter second before.

The plan view at the top right is the paper's Fig. 17: the inner wing, the tip, the hinge line flared from the flow, and the flow from above. The trace below it is the fold angle against time, which is the paper's Fig. 18. In that figure the two codes of the paper, ANCF and SHARPy, lie on top of each other; the scene runs one model and draws it in both styles, as a line and as marks.

The model

The tip turns on the hinge under its lift, its weight, the inertial load from the rising hinge, and the damping of the joint:

\ddot{Θ} = A α_{tip} - W (1 + \frac{\ddot{z_{h}}}{g}) \cos Θ - c (\dot{Θ} - \dot{ψ_{h}})

the tip on its hinge: its lift, its weight with the inertial load, and the damping of the joint

α_{tip} = α - \arctan (\tan θ \cdot \sin β)

the incidence left on the tip after the fold

θ = Θ - ψ_{h}

the fold, measured from the wing at the hinge

\ddot{z_{h}} + 2 ζ ω \dot{z_{h}} + ω^{2} (z_{h} - R α) = 0

the inner wing, in its first bending mode

A = 12 s⁻² per rad, W = 0.0752 s⁻², β = 10 °, R = 14 m per rad

calibrated to the two cases of the paper

A fold of $θ$ about a hinge flared by $β$ turns the chord of the tip by $\arctan (\tan θ \cdot \sin β)$ , which is the relation of the flared folding wing tip. The tip folds until the incidence left on it makes just the lift that carries its weight. With no flare nothing stops it, and it goes to its stop. The inertial term holds the tip back while the hinge accelerates upward, which is the dip at the start of the paper's figure. $A$ and $W$ are calibrated so that the coast angles are those of the paper, and the time runs at the speed of the paper's figure.

The control

The slider holds the flare of the hinge line. At 10 degrees the model gives the two cases of the paper. With more flare the fold is smaller and the joint is stiffer. With less flare the tip folds further, and below a few degrees it reaches its stop. Auto alternates the two cases of the paper, 10 and 5 degrees of incidence, every 7 seconds.

Where it comes from

The scene draws the background of the paper Absolute Nodal Coordinate Formulation for Nonlinear Multibody Modeling of Flared Hinged Wings.