A pivoted wingtip follows the lift only with no spring stiffness and no camber

A wing that twists with its speed washes out the wrong way flew a trainer’s sixty-minute sortie — climb, cruise, slow flight and stalls, circuits — and asked what twist a flexible wing should have. Each segment flies at its own lift coefficient, and washout is right at one lift coefficient: the best twist for each is in proportion to the lift coefficient it flies at, so slow flight wants the most. A fixed twist cannot follow, and on that sortie its compromise costs 1.41 points of induced energy on top of the planform’s own floor. A wing’s natural twist does follow something — the dynamic pressure, through its sections’ camber moment — but in the wrong direction, adding washout at cruise where the least is wanted.

It ended on the one passive device that answers to angle rather than to pressure: a tip panel pivoted on a spanwise hinge near its own aerodynamic centre and held by a spring. The moment about the hinge then depends on the panel’s lift, the lift on its incidence, and the panel settles at an angle the flight decides. The question was how much of the spread cost such a device recovers, and what it gives away in a gust.

The answer has a sharp edge. A panel whose section carries no camber moment, held by a constant torque rather than a stiff spring, recovers all of it: its own lift coefficient stays in exact proportion to the wing’s, and every segment flies at its own best twist. Every departure from those two conditions costs, and one of them costs more than the panel can ever earn back.

A panel balanced on a hinge #

The wing is the earlier essay’s trainer: rectangular, aspect ratio 8, sixteen square metres, a tonne, flown over the same four segments at lift coefficients of 0.33 at cruise, 0.82 in the climb, 0.92 in the circuits and 1.28 in slow flight. Its outer quarter of each semi-span is a separate panel that can rotate nose-down — wash out — relative to the inner wing, about a hinge that runs along the span.

The lifting line is Prandtl’s — the model the price of having ends built — solved by Glauert’s method with the panel’s step of incidence put in as a spanwise right-hand side; it needs many terms to converge on a step, and 160 odd harmonics leave the slope of the best law changed by two parts in a thousand when the count is raised to 240. Everything the line gives is linear in the wing’s incidence and the panel’s washout: the wing’s lift coefficient, the panel’s mean section lift coefficient, and the induced drag as a quadratic form in the lift coefficient and the washout.

The panel’s equilibrium is a moment balance about its hinge. The hinge sits a distance $\varepsilon c$ ahead of the panel’s aerodynamic centre, so the panel’s lift pulls its trailing edge down with a moment $\varepsilon c\, q S_p c_p$, where $c_p$ is its section lift coefficient and $S_p$ its area. Its section’s camber adds a moment $c_{m0} q S_p c$, nose-down for an ordinary cambered section. A spring resists the washout with a stiffness $K$, and a preload holds the panel nose-up with a constant torque. In level flight the dynamic pressure is the weight over the wing area and the lift coefficient, so each segment gives one linear equation for the panel’s washout, and the sortie’s induced energy follows from the same quadratic form the earlier essays used, weighted by distance flown.

A hinge ahead of the aerodynamic centre makes the panel statically stable — a nose-up disturbance raises its lift, which pulls the nose back down — and inside the chord means $\varepsilon$ is at most a quarter. Both conditions are kept throughout. The stiffness and the preload are measured in units of the cruise’s dynamic pressure times the panel’s chord and area, so a stiffness of 1 is a spring whose torque per radian equals that product: for this trainer about five kilonewton-metres per radian.

A constant torque makes the panel’s lift proportional #

The first figure draws the panel’s washout against the lift coefficient. The faint line is what each lift coefficient wants for a panel of this size, 5.31 degrees per unit of lift coefficient — less than the earlier essay’s 7.87 for a twist spread linearly along the span, because a step at the tip does its work in a narrower region. The four dots are the sortie’s segments, from 1.76 degrees at cruise to 6.77 in slow flight. The dashed line is the best fixed panel washout, 2.31 degrees, and it costs 2.65 per cent of an elliptic wing’s induced energy.

At the dial’s first setting, no stiffness, the symmetric panel’s curve lies exactly on the faint line, and the sortie’s induced energy is 1.340 per cent — the floor, every segment at its own optimum. The reason is a line of algebra. With no camber moment and no spring stiffness, the hinge balance reads: dynamic pressure times hinge offset times panel lift coefficient equals a constant torque. So the panel’s lift coefficient is that torque divided by the dynamic pressure, and in level flight the dynamic pressure is inversely proportional to the wing’s lift coefficient. The panel holds a lift coefficient in fixed proportion to the wing’s. A loading whose every part scales with the wing’s lift coefficient has the same shape at every speed, and the twist that produces a fixed shape is exactly the twist in proportion to the lift coefficient that each segment wants. The preload sets the proportion, and so the slope; the hinge offset only trades against it.

Without the preload the panel would hold a fixed lift coefficient of its own, set by the hinge’s balance against nothing, and wash out at 24.6 degrees per unit of the wing’s lift coefficient — the right direction at nearly five times the right rate. That is the free-floating wingtip in its simplest form, and it is a gust alleviator, not an induced-drag device; the preload is what turns one into the other.

The free panel’s loading, drawn #

The second figure shows what the panel does to the loading. It plots the section lift coefficient along the semi-span, divided by the wing’s lift coefficient, for slow flight at 1.28 and cruise at 0.33. With the fixed washout the two curves differ: the same 2.31 degrees is too little washout for slow flight, so the outer wing is overloaded there, and too much for cruise, so it is underloaded. With the free panel the curves for the two segments are the same shape — the panel washes out 6.77 degrees in slow flight and 1.76 at cruise, and the loading divided by the lift coefficient does not change.

Neither is elliptic. A step of incidence puts a kink in the loading at the hinge line, which an elliptic loading does not have, and that is why the floor here, 1.34 per cent, is above the earlier essay’s 0.90 for a smoothly twisted wing. The pivot follows the lift coefficient perfectly and has the wrong shape to do it with. Against the earlier essay’s numbers that is still a gain: the free panel’s 1.34 per cent beats the smooth wing’s best fixed twist, 2.31, by nearly a point.

Stiffness gives the spread back #

A real hinge needs a spring — to hold the panel against flutter, to give a pilot something predictable, to stop it swinging on the ground. The third figure prices it. For the symmetric panel, a stiffness of 0.03 cruise dynamic pressures costs a fortieth of a point, 1.36 per cent; 0.1 costs 1.49, 0.3 costs 1.83, 1 costs 2.27, and by 3 the panel is at 2.50, most of the way back to the fixed twist. The dial on the first figure shows why: a stiffness adds a term to the panel’s equilibrium that does not scale with the dynamic pressure, and as it grows the panel’s washout curve flattens towards a constant. A spring is a fixed twist laid on top of whatever the aerodynamics would choose.

The scale is set by the panel’s own aerodynamic stiffness, the rate at which its hinge moment grows with its washout, which is proportional to the dynamic pressure and the hinge offset. A spring much softer than that leaves the aerodynamics in charge; one much stiffer locks the panel. For this trainer the dividing line is a few kilonewton-metres per radian at cruise, and lower in slow flight, where the dynamic pressure is a quarter of cruise’s and the same spring dominates four times as much — which is the segment where the following matters most.

A cambered section is the wrong law #

The fourth figure changes the panel’s section. A symmetric section has no pitching moment about its aerodynamic centre; an ordinary cambered section, such as a trainer’s, has a nose-down moment, a coefficient of about −0.05. About the hinge that moment adds a term proportional to the dynamic pressure and independent of the lift, and its sign is the one the earlier essay found ruinous: more washout as the aeroplane speeds up.

The free panel with a camber moment of −0.02 spends 2.62 per cent, no better than the fixed twist; at −0.05 it spends 9.31 per cent — three and a half times the fixed twist’s cost, and more than an untwisted panel’s 6.76. The hinge offset cannot fix it, because the camber term is divided by the offset and the offset is bounded by the chord: within the quarter-chord a hinge can move, the camber’s dynamic-pressure washout is several degrees at cruise. A spring helps only by locking the panel, and the best stiffness for a −0.05 section leaves it above the fixed twist at every stiffness the search tried.

A reflexed section does the opposite. A positive camber moment of +0.02 adds washout in slow flight, where it is wanted, and a modest spring can trim the rest: at a stiffness of 0.3 the reflexed panel reaches 1.41 per cent. So the claim that a pivoting tip adapts by itself is right for one class of section and badly wrong for the class most wings use. What a pivoted panel follows is its hinge balance, and only a section with no moment of its own makes that balance a statement about lift alone. The design rule is the same one that makes a flying wing’s sections reflexed, for a different reason: there the moment has to be removed for trim, as no elevon reaches past a quarter chord found when it priced trimming a section by its own trailing edge; here it has to be removed for the law.

How much of the wing should pivot #

The panel so far was the outer quarter of the semi-span. The fifth figure moves its inboard edge. A small panel, from 0.92 of the semi-span outward, does too little: its best fixed washout and its floor are both above 3 per cent, since there is too little wing at the tip to reshape the loading. A large one, from 0.4 outward, puts its step too far inboard, where the kink in the loading costs more than the following saves: a floor of 3.16 per cent. Between them the floor is lowest, 1.27 per cent, for a panel from 0.70 of the semi-span — the outer thirty per cent.

The fixed washout’s cost follows the same shape a little higher, 2.59 per cent at the best edge, so the gap between fixed and free — what the pivot is worth — stays near 1.3 points across the range that matters. A pivot is worth having on any reasonable fraction of the span; the fraction mainly decides how close to the smooth wing’s floor the result can get.

A gust, quasi-steadily #

The sixth figure turns to the earlier essay’s other question. A vertical gust raises the incidence of the whole wing at once. A locked panel adds its share of the extra lift; a free one rotates to keep its hinge balance, which with no stiffness means keeping its lift coefficient where the dynamic pressure puts it, so it adds almost none. Quasi-steadily the free wing passes 0.722 of a locked wing’s gust lift. The outer panel carries only 19 per cent of a locked wing’s gust lift, so more than the panel’s own share is shed: the panel’s washout also takes lift off the wing inboard of the hinge, where the step in the loading changes the downwash, and about a third of the relief comes from there. A designer reading the root bending moment would see more than that, since the shed lift is outboard.

Stiffness takes the relief back: 0.762 of the locked wing’s lift at a stiffness of 0.1, 0.895 at 1, 0.984 at 10. The figure is the same trade as the third one, seen from the structure’s side. The softness that lets the panel follow the sortie’s slow changes is the softness that lets it shed a gust’s fast one; a spring stiff enough to be harmless in a gust has given up most of the induced-drag gain. And the gust lift shed at the tip is shed where its moment about the wing root is largest — which is the load one wing root feels most directly.

The calculation is quasi-steady, and that is the gap it leaves. A real gust arrives over a time comparable to the panel’s own response, and a wing averages a gust through its own loading only once the loading has had time to form. A panel whose only restoring moment is aerodynamic has a pitch frequency that rises with speed, and whether it answers a gust or resonates with one depends on its inertia and where its mass sits relative to its hinge.

The convention: washout, stiffness and preload #

Washout is the outer panel’s nose-down rotation relative to the inner wing, in degrees. The sortie’s induced energy is per cent above an elliptic wing’s over the same sortie, summed by distance flown with each segment weighted as the earlier essays weighted it, at a fixed weight. A stiffness of 1 is a spring torque per radian equal to the cruise’s dynamic pressure times the panel’s chord and area; a preload is a constant torque in the same units. The hinge offset is the hinge’s distance ahead of the panel’s aerodynamic centre in chords, at most a quarter. The camber moment coefficient is the section’s pitching moment about its aerodynamic centre, negative nose-down. “Best” means the hinge offset and preload that minimise the sortie’s induced energy for the stated stiffness.

What the calculation does not price #

Flutter. The free panel is restrained by nothing but its own aerodynamics, and a surface free to pitch on a hinge can flutter if its centre of mass lies behind the hinge. Mass balance — a weight ahead of the hinge — is the classical cure, and it is not modelled. This is the largest omission, and the shake that is not resonance is the reason it cannot be waved away.

Profile drag, friction and the gap. The hinge line has a gap or a seal; the deflected panel changes the profile drag of its own sections; the hinge has friction, which would make the panel stick until its moment exceeds a threshold. None of those is counted.

Turns and accelerations. In a turn the dynamic pressure and the lift coefficient both change, and the panel’s balance answers to the load factor through the lift; the sortie here is flown level, as the earlier essay’s was.

Roll. A pivoting tip changes the wing’s roll damping and its aileron’s effectiveness if the aileron is on the panel. A free panel at each tip also answers differently on the two sides in a sideslip.

Who tried it #

Wingtips free to pivot have been tried for roll control and for gust relief for most of a century: floating wing-tip ailerons were tested in American wind tunnels in the 1930s, the Freewing aircraft of the 1980s and 1990s pivoted the whole wing ahead of its aerodynamic centre so that it held its own lift coefficient through gusts, and a subscale demonstrator flown by Airbus around 2020 freed folding wingtips on a flared hinge for gust relief. The finding here — that a constant preload turns a free tip from a gust device into an induced-drag one, and that it does so exactly only for a section with no camber moment — is this calculation’s, from the lifting-line model above, and it inherits the quasi-steady, rigid-panel assumptions stated.

How the panel was checked #

The untwisted rectangular wing’s span efficiency, 0.937, is the shared lifting line’s value for aspect ratio 8. A panel that spans the whole wing is no twist at all, only a change of incidence, and couples to the induced drag not at all — to rounding error. A free panel with no preload holds its section lift coefficient at the camber moment over the hinge offset in every segment, to the sixteenth decimal. And the free symmetric panel’s sortie energy equals the floor computed directly by giving each segment its own optimum.

Still open: the panel that flutters #

The free panel’s gain rests on a softness that is also a freedom to oscillate. Its pitch about the hinge is restrained by a moment proportional to the dynamic pressure, so its natural frequency rises with speed, and coupled to the wing’s bending — the tip moves up and down as the wing flexes, which changes the panel’s incidence — it can extract energy from the stream the way the lag that makes flutter possible describes for a whole wing. The next calculation would give the panel a mass, an inertia and a centre of mass relative to its hinge, couple it to the wing’s first bending mode with unsteady aerodynamics, and ask how much mass balance the free panel needs to stay clear of flutter up to the trainer’s never-exceed speed, and what that mass costs in the weight the sortie’s induced energy is measured at.