The control that works backwards

The shake that is not resonance is about the dynamic aeroelastic failure: two structural modes coalescing in frequency and an oscillation that grows. Beside it in the same algebra sit two static failures, with no frequency and no oscillation anywhere in them, and only one of the two is famous.

Divergence is the famous one. The aerodynamic moment about the elastic axis grows as $U^2$ while the torsional spring does not, so above a speed the wing twists off. It needs the elastic axis behind the aerodynamic centre, and a wing with the two coincident cannot diverge at any speed whatever.

Reversal is the one that decides the design. Deflecting an aileron down adds lift and a nose-down pitching moment; the moment twists the wing to a smaller incidence; and above a certain dynamic pressure the lost twist beats the added lift and the aeroplane rolls the wrong way.

Two derivatives, from a solve #

The aerodynamics is a pair of numbers and they come from thin-aerofoil theory rather than from a tabulation.

The lift derivative $\mathrm dC_L/\mathrm d\delta$ is Glauert’s $2(\pi - \theta_h + \sin\theta_h)$ with $\cos\theta_h = 2E - 1$, and the panel solve reproduces it to 0.4 per cent across hinge positions from 0.6 to 0.9 of the chord. That is a check rather than a result — the closed form is exact — and it licenses the second derivative, which has no closed form anybody quotes.

$\mathrm dC_M/\mathrm d\delta$ about the quarter chord is negative at every hinge position, and that sign is the whole essay. It is the same nose-down moment a deflected flap produces, met here as a structural input rather than as a high-lift device. A flap deflected down carries its extra lift near the trailing edge, so the extra load acts behind the quarter chord, so the moment is nose-down — always, whatever the flap’s size.

One equation, and the two speeds #

The structure is a torsional spring: the wing twists until the elastic restoring moment balances the aerodynamic one.

$$K_\alpha\theta = qSc\left(C_{m\delta}\,\delta + C_{M\alpha}\,\theta\right) \quad\Longrightarrow\quad \theta = \frac{qSc\,C_{m\delta}\,\delta}{K_\alpha - qSc\,C_{M\alpha}},$$

with $C_{M\alpha} = a_w(x_{ea} - x_{ac})/c$ the moment of the incidence-generated lift about the elastic axis.

Two things go wrong with that expression at two different dynamic pressures.

Divergence is the denominator vanishing: at $q_{\rm div} = K_\alpha/ScC_{M\alpha}$ the twist is unbounded for any deflection, and indeed for none. It requires $C_{M\alpha} > 0$, which requires the elastic axis behind the aerodynamic centre.

Reversal is the rolling effectiveness vanishing:

$$\eta(q) = \frac{a_\delta + a_\alpha\,\partial\theta/\partial\delta}{a_\delta},$$

which is 1 at rest and passes through zero when the twist’s lift cancels the aileron’s own.

Effectiveness at zero is exactly 1, at the computed reversal pressure it is $5\times10^{-16}$, and thirty per cent beyond it is −0.86: the aileron is producing 86 per cent of its rigid rolling moment in the wrong direction.

What the two failures have in common, and how they differ #

The two static problems come out of one equation and it is worth writing the difference precisely, because “divergence” and “reversal” are usually taught as separate topics.

Divergence is the denominator vanishing — a property of the homogeneous problem, so it happens with no control input at all and would happen to a wing with no ailerons. It is a genuine instability: the twist grows without bound, the structure fails, and the failure mode is the wing departing the aircraft.

Reversal is the numerator vanishing — a property of the forced problem, so it needs an input and produces no instability whatever. Nothing grows. The wing is perfectly happy; it is simply doing the opposite of what the pilot asked.

That distinction matters for how each is found. A divergence speed is an eigenvalue and shows up in any stability analysis of the structure. A reversal speed is a zero of a transfer function and shows up only if somebody computes the transfer function — which is why reversal has historically been discovered in flight rather than on paper.

And it matters for what a margin means. A margin against divergence is a margin against a catastrophe. A margin against reversal is a margin against a control that has become useless, and useless is reached well before backwards: the effectiveness is down to a half at about sixty per cent of the reversal pressure, and a roll rate halved is already a handling problem.

Reversal always comes first #

For the wing computed here the reversal pressure is 0.4998 of the divergence pressure, and the ratio has a property worth checking rather than assuming.

Both speeds are proportional to the torsional stiffness, so the ratio is not: making the wing seven times stiffer multiplies both by seven and leaves the ratio unchanged to a part in 10¹². That is checked in the solve, and it means a stiffness increase cannot buy reversal margin relative to divergence. It buys both, in proportion, and the ordering between them is a property of the aerodynamics and the elastic axis position.

Everywhere behind the aerodynamic centre, reversal is first. So a wing designed to a divergence margin is a wing whose ailerons have already reversed.

The wing that cannot diverge #

Now the case that makes the point sharply.

Put the elastic axis at the quarter chord. Then $C_{M\alpha} = 0$, the denominator never vanishes, and the divergence speed is infinite: the aerodynamic moment about the elastic axis does not depend on incidence at all, so there is nothing to run away.

And it still reverses, at twice the ordinary wing’s pressure. The driving term is $C_{m\delta}$ — the aileron’s own nose-down moment — which is there whatever the elastic axis is doing, and removing the incidence’s contribution to the twist only halves the effect.

Removing the wing’s ability to diverge removes half the problem and none of the failure. That is the essay’s result and it is not a marginal case: forward-swept and highly loaded wings are routinely tailored to move the elastic axis forward for exactly the divergence reason, and reversal has to be checked separately every time.

The twist that does it #

It is negative — nose-down — and grows in magnitude, and at the reversal pressure the lift it removes is exactly the lift the aileron adds.

Nothing is failing. The control surface is working, the structure is working, and between them they arrive at nothing. That is a different kind of engineering failure from a divergence or a flutter: it is not an instability at all, it is a cancellation, and no amount of structural damping or margin will change it because there is nothing dynamic in it.

What a designer does about it #

Three responses, and each of them is a way of changing one of the two derivatives.

Move the aileron inboard. The twist is largest at the tip, so an inboard aileron on a stiffer part of the wing reverses at a higher speed. Almost every transport aircraft has a high-speed inboard aileron and a low-speed outboard one for exactly this reason, and the outboard pair is locked out above a certain speed.

Use spoilers instead. A spoiler works by making the flow separate rather than by adding circulation, so it has no reversing moment of its own. A spoiler produces a rolling moment by destroying lift on one side rather than by adding a nose-down moment, so it has no $C_{m\delta}$ of the reversing sign and does not reverse. Every large transport uses spoilers for roll at high speed.

And tailor the structure. A composite wing can be laid up so that bending produces a favourable twist — bend-twist coupling — which changes the effective $C_{M\alpha}$ without moving the physical elastic axis. That is what made the forward-swept X-29 possible, and it is the same tool applied to a different one of the two static problems.

The fourth response: stop using a flap #

Each of the three fixes above changes a derivative. There is a fourth that removes one.

An all-moving surface rotates in its entirety about a pivot, so the load it generates acts near its own aerodynamic centre rather than near a trailing edge. The nose-down couple that drives reversal is a property of a flap — extra load carried well aft of the quarter chord — and a surface with no flap in it does not have one. The numerator cannot vanish, because the term that would cancel the lift is not there.

That is a large part of why fast aircraft use all-moving tailplanes and use them differentially for roll. Above Mach one a trailing-edge flap is losing on a second count as well: with no upstream influence, the pressure change it makes is confined to the flap itself instead of being felt over the whole chord, so its effectiveness collapses at the same time as the wing’s flexibility is turning against it.

The price is that the problem moves rather than disappearing. The whole surface is now a wing on a torsional spring — its actuator and linkage — with a divergence speed of its own about that pivot. That is a stiffness requirement on a jack and a backlash requirement on a joint, which is a more tractable piece of engineering than the torsion box of a wing, and it is a requirement rather than a reprieve.

Reading the effectiveness curve as a design tool #

The curve is more useful than its zero, and it is worth reading across rather than at the crossing.

The effectiveness falls from 1 roughly linearly at first, so a wing at half its reversal pressure has lost something like a third of its rolling power. That is the number a handling-qualities requirement is written against — a minimum roll rate at a given speed — and it is reached long before the reversal speed appears in any calculation.

Two consequences follow. The design constraint is a fraction of the reversal pressure rather than the pressure itself, typically requiring the effectiveness to stay above some value at the design dive speed; and the constraint is on the wing’s torsional stiffness, because that is the only term available once the aerodynamics and the aileron geometry are fixed.

That makes reversal a structural requirement expressed as an aerodynamic one, which is why it shows up in a wing’s spar sizing beside the load cases. On a high-aspect-ratio wing it is often the sizing case for the torsion box, and it competes directly with the weight saving that the aspect ratio was chosen for in the first place — the same trade the span is the whole story sets up from the drag side.

And it is why a flexible wing has its ailerons where it does. The rolling moment an aileron produces goes as its distance from the centreline, so a designer wants it outboard; the twist it produces goes as the inverse of the local torsional stiffness, which is smallest outboard. Those pull in opposite directions, and the answer on a large aircraft is two sets of ailerons and a speed at which the outboard pair is disabled.

Where the model stops #

Rigid chord, one torsional degree of freedom. A real wing bends as well as twists, the two are coupled on a swept wing, and the bending’s contribution to the tip incidence is comparable with the twist’s.

Steady aerodynamics. Both static failures are computed with steady derivatives, which is right for them by definition and is not right for the dynamic problem next door. A wing near reversal is also near a region where the aileron’s unsteady effectiveness differs from its steady one, and the reduced frequency is the number that says by how much.

Incompressible derivatives. $C_{m\delta}$ and $a_\delta$ both change with Mach number, and the reversal speed of a transonic wing is a transonic calculation.

And a two-dimensional section standing in for a wing. The real problem is a spanwise one: the torsional stiffness varies along the span, the aileron occupies part of it, and the reversal condition is an eigenvalue of a distributed system rather than a root of one algebraic equation.

Where else the same cancellation appears #

Control reversal is the clean case and the mechanism — an input that produces a beneficial effect and a detrimental one, with the second growing faster — is not confined to ailerons.

Elevator effectiveness on a flexible fuselage. Deflecting an elevator produces a tail load, and the tail load bends the rear fuselage in a direction that reduces the tail’s own incidence. The same arithmetic gives an elevator reversal speed, usually well above the aileron’s, and the same fix — a stiffer structure or an all-moving surface.

Rudder effectiveness on a swept fin, which twists under its own load in the same way and for the same reason.

And a rotor’s blade pitch, which is the extreme case: a blade is long, thin and torsionally soft, its pitch input is at the root, and the aerodynamic moment distributed along it twists the tip away from what the swashplate asked for. The effect is large enough that rotor blades are designed with their elastic axis, mass axis and quarter chord as nearly coincident as manufacturing allows.

In every one of those the collapse being made is “a control input produces a proportional response”, and the residual is the structure’s own reply to the load the input created. That is a feedback the rigid calculation has no term for, and its gain grows as $q$ while the input’s does not.

Three static aeroelastic problems, one algebra #

It is worth listing the family, because the essay has now touched all of it.

Divergence — the twist unbounded, denominator zero, requires an aft elastic axis.

Reversal — the effectiveness zero, numerator zero, requires only the aileron’s own moment.

And load redistribution, which is not a failure at all: a swept wing twists nose-down under load — which shifts its own spanwise loading inboard, and reduces its root bending moment. That is a benefit of the same mechanism, it is designed for deliberately, and it is why a large swept wing’s structural load case is not the one a rigid calculation gives.

All three are the same equation with a different quantity asked about, and the moral is the one this whole phase has been circling: a single number — here the divergence speed — is a collapse, and what it discards is the other two questions the same algebra answers.

The one place the ratio is fixed #

The result the algebra gives that is worth carrying away is not either speed. It is that their ratio is a property of the section and not of the structure: with the elastic axis where it was put here, $q_{\mathrm{rev}}/q_{\mathrm{div}} = 0.4998$, and the stiffness has cancelled out of it entirely.

That cancellation is what makes reversal a design constraint rather than a design variable. Doubling the torsion box’s stiffness moves both speeds up by the same factor and leaves the ratio where it was; moving the elastic axis forward raises the divergence speed without limit — at the quarter chord the denominator never vanishes and divergence disappears — while barely touching the reversal speed, which is why the ratio is the honest statement of the trade and either speed alone is not.

And it says which failure to design against. Reversal always arrives first at a realistic elastic axis, so a wing sized to be free of divergence with a margin may still reverse inside its envelope. A structure that treats the two as one aeroelastic requirement, checked by raising stiffness until divergence clears, has checked the wrong root of the same quadratic — and the check will pass, because the divergence speed it computes is real and the aeroplane will never reach it.

Where the problem came from #

Control reversal was found the hard way, on aircraft. The most consequential case is the Supermarine Spitfire, whose fabric-covered ailerons on early marks reversed at high speed and whose roll rate therefore collapsed above about 300 mph — the metal-skinned ailerons introduced in 1941 raised the reversal speed and roughly doubled the roll rate, which mattered a great deal in 1941.

The theory is Cox and Pugsley’s, from 1932, and it predates most of the accidents. Roxbee Cox and Pugsley wrote the two static problems down together and were explicit that reversal comes first; the divergence half of their work is the half that entered the textbooks.

That asymmetry is worth noticing. Divergence is dramatic and reversal is not, so divergence is what gets taught — and reversal is the one that constrains almost every high-speed wing ever built.

What this leaves #

This phase set out to look for what a collapse discards, and this is the last of them. The divergence speed is a perfectly good number about a wing’s torsional stiffness, and what it discards is that the same stiffness, the same aerodynamics and the same equation have a second root that comes first — and that a wing engineered to have no first root at all still has the second.