A bent wing twists itself: aft sweep out of trouble, forward sweep into it

The wind a swept wing feels gave the reason a wing is swept at all: its boundary layer and its pressure respond only to the component of the stream normal to the span. The sweep a root does not have followed that into the parts of a swept wing where it fails. In a passage on what’s missing from its lattice, it recorded a sentence about a stiffness: a forward-swept wing’s bending under load raises the incidence at its tip, that raises the load and the bending, and aft sweep’s version of the same thing is self-correcting, “one of the quiet reasons sweepback is structurally the easy direction”.

That sentence can be turned into a calculation, and doing so shows how strong the effect is. A straight wing has a speed at which it diverges: the elastic restoring moment on the twist can’t keep up with the aerodynamic moment that twist generates. The shake that is not resonance set that out. Adding sweep doesn’t change a word of the straight wing’s physics, in the sense that a section still has a lift and a moment. It adds one line of geometry, and that one line changes the whole of the problem’s character. The question is how much of a change is available for a few degrees of sweep, and the answer is surprisingly much.

The line of geometry #

Take a wing that’s a uniform beam from a clamped root to a free tip, along an axis swept at $\Lambda$ from the normal to the stream. It carries lift along its length. Consider a section cut parallel to the stream, because that’s the section whose angle the stream sees. The beam can do two things to it. It can twist through $\theta$ about its axis, and it can bend, with a slope $h'$ along the span.

For an unswept wing, bending just raises a section without tilting it relative to the stream. With sweep, it does not. Take two points on the section, one at the leading edge and the other at the trailing edge. For an aft-swept wing, the trailing-edge point is further outboard along the beam than the leading-edge point. So when the beam bends upward, the trailing edge rises more than the leading edge. The stream section tilts nose-down. The incidence it sees is

$$\alpha=\alpha_0+\theta\cos\Lambda-h'\sin\Lambda ,$$

and the last term is the whole subject. With $\Lambda>0$ it is a washout, proportional to the bending slope. With $\Lambda<0$, it’s a wash-in. Lift then depends on that incidence, and the bending and twist in turn depend on the lift:

$$EI\,h''''=\ell,\qquad -GJ\,\theta''=e\,\ell,\qquad \ell=q\,c\,a\cos^2\!\Lambda\;\alpha ,$$

where $e$ is the distance by which the aerodynamic centre stands ahead of the elastic axis. Diederich and Budiansky solved essentially this system in 1948, and it’s the textbook statement.

Without sweep, $\theta$ alone couples the lift back to itself, through a closed form: in this essay’s scaled units the divergence pressure is $(\pi/2)^2/(\varepsilon r)$, with $\varepsilon=e/L$ and $r=EI/GJ$. For $\varepsilon=0.05$ and $r=2$ it’s 24.674. The calculation by finite differences gives 24.671. With sweep, the bending term comes in and the closed form goes. An aft-swept wing’s divergence runs off quickly to high pressures; a forward-swept wing’s falls, and it doesn’t go back up.

The steepness is what’s most striking. A straight wing is nowhere near a stable region of this map. At ten degrees aft sweep the divergence pressure has gone up by 28 times, and at nineteen degrees it’s vanished. The reason is that bending is a larger effect than torsion on a slender wing, since a beam with a long span has a large slope for a small tip deflection. Even a small share of it, through $\sin\Lambda$, overwhelms the twist.

What it does to the lift below divergence #

A wing that never reaches divergence still feels all this. The flexibility acts throughout the speed range, as a change to the wing’s total lift at a stated incidence.

So at a fixed incidence, a straight wing and a forward-swept one lift more as speed rises, and their curves run away. An aft-swept wing lifts less. That’s what the phrase “lift effectiveness” means, and it’s also a form of gust relief. An aft-swept wing meeting an upward gust bends upward, washes itself out and lifts less than a rigid one would. It’s done with no moving parts, and it’s working at any speed, in proportion to the load. Large jets count on it: a swept airliner’s load in a gust is meaningfully smaller than a rigid calculation says, and a passenger who sees the wingtip flex in turbulence is watching it work.

There’s a price for that as well. The control that works backwards has the mechanism that makes an aileron less effective at speed, and on an aft-swept wing the bending adds to this. A trailing-edge aileron deflected downward lifts the outer wing, which bends it upward and washes out the section: so an aileron’s rolling moment is less than a rigid wing’s by a fraction that grows with speed, and on the first swept jets, it led to inboard ailerons and spoilers for high-speed flight. The same coupling also moves the centre of lift. A washed-out tip loses load relative to the root, and on an aft-swept wing the root lies forward. So the lift’s centre moves forward as speed rises, and with it the wing’s contribution to stability falls.

A rule of thumb in one line #

The size of the critical sweep can be estimated without any calculation, by comparing the two deformations’ contributions directly. Under a lift per unit span $\ell$, spread over a span $L$, a clamped beam has a tip slope of $\ell L^3/6EI$, and the same lift, acting at the offset $e$, gives a tip twist of $e\ell L^2/2GJ$. Set the bending slope, multiplied by $\tan\Lambda$, equal to the twist, since the first is what sweep brings in and the second is what it must cancel:

$$\tan\Lambda_c\approx\frac{3e}{L}\,\frac{EI}{GJ}.$$

For $e/L=0.05$ and $EI/GJ=2$, that’s a sweep of 16.7°, against 18.7° from the full calculation. It’s remarkable that something so crude does that well. It says why slender wings are so easily rid of divergence: the bending slope goes up with the span cubed, against the twist’s square, so the longer the wing the less sweep it takes, for the same section. The offset $e$ being small compared with the span is what makes the numbers small: a wing’s chord is a small part of its length, and the offset a small part of the chord. Across the range of the later figure, the estimate and the full calculation stay together: 4.3° against 3.2° at a stiffness ratio of a half, 8.5° against 7° at one, 31° against 34° at twice the offset. And it gives the sensitivities of that figure directly. Twice the stiffness ratio means about twice the tangent, and the same for twice the offset.

The same line also explains the speed-independence of that critical sweep. Both of these effects are proportional to the load, so the speed cancels: at the critical sweep there’s no feedback at any dynamic pressure, and the system behaves like a rigid wing to first order. It’s a very clean point on the design map, the sweep at which a wing has neither wash-in nor washout at the tip. The tip’s own figure above shows it: at 15° aft it’s 1.04, within a few per cent of rigid.

Where the load goes #

Along the span, the difference is unambiguous.

At the same dynamic pressure, a tip that’s 15° forward is at 2.69 times its rigid incidence, and a tip that’s 45° aft is at 0.76. That’s more than a threefold range at one speed, from the same section and the same beam. For the aft case, this relieves the root’s bending moment at a given total lift, since the load moves inboard. A designer would count it as a direct saving of structure. A spar that must lift itself stated what a span costs in spar weight, and an aft-swept wing gets back some of that cost from its own flexibility. In the forward case the tips carry it the wrong way, so the spar pays twice: more load at the tip, with a larger moment arm, and it has to be stiff enough to stop that from going further.

It’s helpful to see the two parts of that coupling side by side.

Twist doesn’t change sign with sweep: with the aerodynamic centre ahead of the elastic axis, lift always twists a section nose-up, so a swept wing’s torsional coupling has the same sign as a straight wing’s. It’s the bending slope that differs, and it’s negative for aft sweep. At about seventeen degrees these two are equal in size and the coupling’s net sign changes. At larger sweep, bending dominates. That’s the entire mechanism in one pair of lines.

The size of that relief is easily read off the earlier figure. A wing 30° aft has 0.81 of a rigid wing’s lift at a dynamic pressure of 10, so a gust that would have added a hundred units of lift to a rigid wing adds eighty-one, and the root’s bending moment falls by more than that, since the lift has also moved inboard. Nobody has had to design anything to obtain it, and it costs nothing in drag. It’s a structural allowance that falls out of the sweep.

Why stiffness can’t simply be added #

The obvious fix for a forward-swept wing would be to make it stiff enough. The problem is that it takes too much. Bending stiffness has to be raised enough to reduce $h'$, and it’s expensive, because a beam’s stiffness goes with the cube of its depth and a wing of a low drag is thin. Aluminium wings in forward-swept experiments of that era had to be heavy. Junkers flew a forward-swept jet bomber, the Ju 287, in Germany in 1944. The Hansa Jet of 1964 had about 15° of forward sweep with a heavy structure, and it remained a curiosity.

Krone saw the way out in 1975. Composite skin has stiffness that depends on direction. A wing’s skin could have its fibres laid at an angle to the beam’s axis, so that bending itself twists the section. Rotated appropriately, that twist cancels the wash-in. The Grumman X-29 flew in 1984 with its forward sweep of about 30° kept from diverging entirely by that bending-twist coupling. It had no need of a heavier structure. Weisshaar worked out the theory of that tailoring. No design choice elsewhere in the aircraft could have bought it. Bending is now wanted to produce a twist in the right sense, and $h'$ goes into the incidence by way of the material as well as the geometry.

Forward sweep also has things to offer aerodynamically. Its isobars lie differently at the root and tip, and its boundary layer is drawn inboard rather than outboard, so stalling begins near the root. Its tips stay flying, unlike those of an aft-swept wing, which tend to stall first and pitch up. Nobody expected the structure to allow those advantages until the 1970s.

A number for how much sweep it takes #

The critical sweep depends on two ratios: how stiff the wing is in bending relative to torsion, and how large the offset between aerodynamic centre and elastic axis is. A wing that’s relatively more flexible in bending than torsion has a smaller critical sweep, because that’s the term doing the work, and a larger offset, which strengthens torsion’s destabilising feedback, needs more. Real wing box sections are near the middle of this figure. So all conventional swept transports have long since passed divergence as a limit, and flutter or control reversal are what sets their speeds instead: as the speed where the damping is zero sets out, a dynamic instability doesn’t care about this static argument.

A stiffness that’s already enough to prevent flutter at 30° aft would mean there’s plenty of margin. It’s the same story as with a pivoted tip, a device which does on purpose, with a hinge and at a small part of the span, what an aft-swept wing does with its whole structure.

What the picture cannot show #

A real wing’s load. Strip theory has each section’s lift depending only on its own incidence. A real wing’s lift has a three-dimensional loading, in which a section’s incidence is influenced by the rest of the span. Lift falls towards the tip regardless, and the total is smaller than strip theory’s. So the divergence pressures here are lower than a lifting-surface calculation’s. Diederich and Budiansky’s factors from lifting-line corrections are of order one, and they don’t change the sign or the rapid rise with sweep.

A wing’s taper. Everything’s uniform. A real wing tapers towards the tip, which moves the load inboard and lowers bending at the tip. That reduces the effect at a given sweep, and it changes the numbers in the figure by amounts that a designer would care about. The sign and the near-vertical rise in the critical region are robust.

Fuselage, engines and dynamics. There’s no inertia here, so nothing here is flutter. Under-wing engines add a forward mass and a moment. Nor does the beam have shear deformation, a root’s actual fixity, or any chord’s bending. The real X-29 had a canard, an instability by design, and a flight-control system that kept it flying, so the wing’s structure was only one of the things that made it possible.

Compressibility. The whole calculation is incompressible. Lift slope rises towards Mach one, and so does the aeroelastic effect with it. At high subsonic speeds, the shift of the aerodynamic centre also alters $e$, in a direction that’s stabilising.

The convention #

Sweep is positive aft. The dynamic pressure is scaled by $EI/(c\,a\,L^3)$, with $L$ the length of the beam along its axis, $c$ its chord normal to that axis and $a$ the section’s lift slope. Lift effectiveness is the total lift over a rigid wing’s at the same root incidence. Effective incidence is in units of that rigid incidence, and it’s taken on the stream section. The stiffness ratio $EI/GJ$ of 2 and the offset $e/L$ of 0.05 are representative of a slender, thin wing box. The critical sweep is the smallest with no divergence below 3000, in the same units.

How it was checked #

The straight wing’s closed form is a check of the whole assembly, and it needs both the torsion equations and the free-tip boundary conditions to be right. The convergence at 20° forward confirms the coupling term’s discretisation, since a straight wing doesn’t exercise it. The model refuses a wing with no stiffness, no offset or a grid too coarse to carry the bending’s fourth derivative.

Who found it, and when #

The Ju 287 flew in 1944. Diederich and Budiansky’s analysis is from 1948, in a NACA note. Krone’s dissertation on aeroelastic tailoring is from 1975, Weisshaar’s theory from around 1980, and the X-29’s first flight from 1984. The gust relief of aft-swept wings has been part of airworthiness loads calculations since the 1950s.

Still open: what a real wing’s lift does to the crossing #

Strip theory is the weakest part of the calculation. The next step would replace it with the lattice used in the earlier essay on the root, so that a section’s lift is computed from the whole wing’s vorticity, while keeping the same elastic beam. It would ask how far a real three-dimensional load moves the critical sweep, and whether the effect is the same for a low-aspect-ratio wing, for which strip theory would be quite wrong and whose lift already falls towards the tip, and so how slender a wing has to be before the near-vertical rise in the first figure still holds.