The shake that is not resonance

Flutter destroys structures in seconds. A wing that has flown for a thousand hours reaches a speed a few knots above the one it flew at yesterday and comes apart before the pilot has finished noticing.

The explanation usually given is that the airflow excites the wing at its natural frequency, and the explanation is not merely incomplete: it names a mechanism that is not present. A steady airstream contains no oscillation. There is nothing periodic in it to resonate with, no forcing at any frequency, and no driver. Whatever is happening, it is not a wine glass and a singer.

The smallest system that can do it #

The model is the classical typical section: a rigid aerofoil on two springs, free to move up and down and to pitch about an elastic axis. Two degrees of freedom, and the aerodynamic force on it is the one thin-aerofoil theory computes with a lag added.

That is not a simplification of the real problem. It is the minimum, and the calculation here demonstrates it by refusing to answer for one. With plunge held fixed, the pitching equation’s damping coefficient is $-2a\pi\rho U b^3(\tfrac12 - a)$, and for any elastic axis ahead of mid-chord — which is every wing ever built — that is positive. The section is damped in pitch at every speed and cannot flutter in pitch alone, and the solver says so rather than returning a number.

Ask the same of plunge alone and the answer is the same: the plunge damping is $2\pi\rho U b$, which is positive and grows with speed. Each motion separately is stabilised by the air. Together they are not. That is the whole subject in two sentences, and it is why flutter has to be understood as a coupling rather than as anything happening to a single mode.

The aerodynamics used is quasi-steady — Theodorsen’s unsteady forces with the lift deficiency set to one, which is the assumption an essay of its own puts a number on — and the section’s equations then form a quadratic eigenvalue problem in the Laplace variable, whose characteristic polynomial is a quartic with real coefficients. Its four roots come in two conjugate pairs, and the whole story is what those pairs do as the speed rises.

Coalescence #

The signature is frequency coalescence, and it is the direct evidence against the resonance account.

At zero speed the two frequencies are the structure’s own — set by the bending and torsional stiffnesses and the masses, and quite different from each other. As the speed rises they approach. Not because anything is tuning them: because the aerodynamic forces have added off-diagonal terms to the stiffness and mass matrices, and the eigenvalues of the coupled system are not the eigenvalues of the uncoupled one.

The flutter frequency is neither of the two the structure started with. It is 95.6 radians per second here, between the 48.2 and 111.0 of the two rest modes. There is no frequency in the problem for anything to be resonant with, and the frequency that appears is manufactured by the coupling.

The gate requires the spread to have closed by at least fifteen per cent between rest and flutter, which is a weak requirement that a resonance account cannot satisfy at all, because a resonance does not move a structure’s natural frequencies.

The energy, and the phase that decides its sign #

The eigenvalue calculation says when. It does not say why, and the why is an energy argument that can be done independently.

Prescribe a harmonic motion: $h = h_0\cos\omega t$ and $\alpha = \alpha_0\cos(\omega t + \phi)$. Compute the work the aerodynamic forces do on the section over one cycle, by quadrature. If it is positive, the airstream is putting energy into the wing.

The term that is the coupling — the lift the incidence produces, acting on the plunge velocity — does work proportional to sin φ, and the solver requires that identity to a part in a million. In phase or in antiphase the work is exactly zero, however violent the motion is.

It is not how much force there is; it is when it arrives. A wing that pitches nose-up while it is rising is at a larger incidence on the way up than on the way down, so the upward force is larger while it is moving up — and force times velocity, integrated, is positive. A wing that pitches nose-up while falling gives the energy back.

The eigenvalue route and the energy route are then required to agree. Taking the flutter mode’s own amplitude ratio and phase from the eigenvector, and finding the speed at which the net work over a cycle — aerodynamic input minus structural dissipation — passes through zero, gives 80.843 metres per second against the eigenvalue crossing’s 80.843. The gate requires them within five per cent; they agree to the printed digits.

What the reduced frequency says about the model #

There is a number that decides whether any of the aerodynamics used here is legitimate, and it is worth reading off before going further.

The reduced frequency $k = \omega b/U$ measures how far the section travels in one cycle compared with its own size. At the flutter condition here it is 0.59, which is emphatically not small: the section is moving through less than two semi-chords per radian of its own oscillation, so the wake it has just shed is still nearby and still influencing it.

That is the regime the reduced frequency essay identifies as strongly unsteady, and it is exactly the regime in which the quasi-steady assumption — that the lift follows the incidence with no lag — is worst. Theodorsen’s function reduces the circulatory lift by about forty per cent and retards it by about fifteen degrees at k = 0.59, and both of those move the answer. The lag itself is the lift that arrives late, and the two indicial functions that carry it are not one function.

So the flutter speed computed here is a number about a model rather than a number about a wing, and the essay’s claims are correspondingly about mechanisms. What survives the substitution of the proper unsteady aerodynamics: the coalescence, the sign of the energy transfer, the phase identity, the necessity of two degrees of freedom, and the direction and rough magnitude of the mass-balance effect. What does not: the speed.

Saying which conclusions are robust to the model and which are not is the difference between a demonstration and a design calculation, and this is the former.

Why mass balance works #

Every control surface on every aeroplane has a lump of lead in its nose, and every flutter certification is partly an exercise in where the masses are. A rotor blade carries the same lead for the same reason, on a section whose incidence changes once a revolution.

The reason is on that figure and it is the phase. The static unbalance — the distance of the centre of mass behind the elastic axis — is what couples the plunge and pitch equations through the inertia: accelerate the section upwards and, if its mass is behind the hinge, the inertia pitches it nose-down. That inertial coupling is what sets the phase between the two motions, and moving the mass forward changes it.

The sweep is unambiguous. At two tenths of a semi-chord aft the section flutters at 81 metres per second; at a twentieth ahead, at 171; at a tenth ahead, not at all. The gate requires the flutter speed to rise monotonically as the mass moves forward and requires the flutter to be gone at the last station.

And then it requires the thing that makes the result honest: the balanced section still fails. At 204 metres per second it diverges — a real root crossing zero rather than a complex pair, an aeroelastic failure with no oscillation in it at all, in which the aerodynamic moment overcomes the torsional spring and the wing twists off. The solver separates the two by the imaginary part of the crossing root and reports which it found, and the divergence speed matches its closed form, $U_D^2 = K_\alpha / 2\pi\rho b^2(\tfrac12+a)$, to a part in ten thousand.

A fix that removes one failure and reveals the one underneath it is the ordinary situation in this subject — a control surface on a wing that twists reverses before it diverges, for the same reason and with the two speeds in the same order — and a solver that reported the divergence as a flutter speed would say the mass balance had not worked.

Three things the model gets right that a resonance account cannot #

Flutter has a speed, not a frequency. There is a well-defined airspeed above which the wing is unstable and below which it is not, and the frequency is an output. A resonance account predicts a frequency at which something bad happens and has no natural way to produce a threshold speed.

Damping matters, and it decides the answer. With no structural damping at all this quasi-steady model gives the torsion mode a positive growth rate at every speed — the lift deficiency has been set to one, which throws away the lag that stabilises the mode at low reduced frequency, and the artefact is a wing that flutters at walking pace. A real structure dissipates one or two per cent of critical, and that is what the airstream has to beat. So the flutter speed depends on the damping ratio, which is the single least-well-known number in the whole calculation.

That is not a defect of the model, it is why flutter clearance is a flight-test programme rather than a calculation: the aerodynamics is known far better than the number that decides the answer.

The threshold is a crossing, not a peak. What the eigenvalue calculation returns is the speed where the damping is exactly zero, which is a root condition rather than a maximum, and that is why it can be located to the printed digits by two routes that share nothing but the model.

And separating the frequencies is a real cure, for the right reason. Making the bending and torsion frequencies further apart raises the flutter speed — the coupling has more work to do to bring them together. That is the piece of engineering practice the resonance account gets accidentally right, and it gets it right for the wrong reason: not because the excitation is being avoided, but because coalescence is being made harder.

The two failures, side by side #

Aeroelasticity has two static-and-dynamic pairs and it is worth keeping them apart, because the remedies differ.

Divergence is static. The aerodynamic moment about the elastic axis grows with $U^2$ while the torsional spring does not, and above a speed the moment wins and the wing twists off. No oscillation is involved, and the closed form is a ratio of a stiffness to a dynamic pressure. The cure is torsional stiffness, and the reason a monoplane could not be built before stressed-skin construction is precisely this: a wire-braced wing has very little torsional stiffness, and the Fokker monoplanes that failed in 1917 failed this way.

Flutter is dynamic and needs the coupling and the phase, as above. The cures are mass balance, frequency separation and damping.

Control reversal is the static form of a control-surface problem: deflecting an aileron twists the wing in the direction that opposes the roll it was asking for, and above a speed the twist wins.

And control-surface flutter is the dynamic form, which is the one the lead weights are for.

All four scale with dynamic pressure against a stiffness, so all four are speed thresholds and all four are made worse by making a structure lighter — which is the permanent tension in aircraft design, since every kilogram removed from a wing lowers all four speeds.

How the boundary is actually found, and why the obvious method is unsafe #

The observation that the damping crosses zero shallowly is not a curiosity about a figure; it is the central difficulty of flutter flight testing, and the response to it comes out of the same quartic this essay is built on.

Clearance is established by envelope expansion. Fly at a speed inside the cleared range, excite the structure — a rap on the stick, a control-surface frequency sweep, a small inertial shaker — and measure how fast the response decays. That gives the modal damping at that speed. Step the speed up a little, repeat, and plot damping against speed or dynamic pressure. The flutter boundary is where the curve reaches zero, and the aeroplane is flown towards it in increments while somebody watches the trend.

The obvious extrapolation is the dangerous one. Damping against speed is frequently flat over most of the envelope and then turns down steeply in the last few per cent, for exactly the reason the root locus shows: the two roots wander slowly towards each other and then one of them turns sharply right. A straight line fitted through the well-behaved region predicts a boundary comfortably above the true one, and the test point that reveals the error is the one past it.

The repair is to extrapolate a better-behaved quantity, and one is available from the characteristic polynomial itself. Stability of a quartic with real coefficients is decided by a Routh–Hurwitz combination of its coefficients, and that combination — the flutter margin — vanishes at flutter like the damping does, while varying far more nearly linearly with dynamic pressure over the whole approach. Zimmerman and Weissenburger set it out in 1964, and it is still the standard basis for deciding whether the next test point is safe.

It is a small and satisfying inversion. The quartic’s roots are what the flutter is; its coefficients are what can be safely extrapolated towards. The physics is in the roots and the engineering is in a symmetric function of them.

What the model does not contain #

Quasi-steady aerodynamics. The lift deficiency function is set to one, which is exact only in the limit of zero reduced frequency and is a poor approximation at the reduced frequency of 0.59 the flutter mode here actually has. Theodorsen’s function would supply the lag and would change the number; what it would not change is any of the mechanism.

Two degrees of freedom. A real wing has an infinite number, and its flutter mode is a combination of several bending and torsional modes with control surfaces participating. The typical section is the smallest system that has the phenomenon and it is not a model of any particular wing.

A rigid section on springs. No spanwise structure, no mode shapes, no aspect ratio and no three-dimensional aerodynamics. A real flutter calculation couples a finite-element model to an unsteady aerodynamic influence matrix and is one of the larger computations in aircraft design.

Incompressible. Transonic flutter, where a shock moves on the section during the cycle, is a different and much harder problem, and the transonic dip in the flutter boundary is one of the reasons flight testing goes right to the edge of the envelope.

Linear throughout. The calculation says whether small motions grow. What happens once they are large — limit-cycle oscillation, in which a nonlinearity arrests the growth at some amplitude — is a distinct phenomenon and is often what is actually observed.

And no structural failure. The model says the amplitude grows. It has nothing to say about what breaks or when.

Who found it, and when #

Flutter was found by aircraft coming apart. The Handley Page O/400 bomber’s tail fluttered in 1916; Lanchester and Bairstow investigated, identified the coupling between the fuselage torsion and the elevators, and the cure was to connect the two elevators with a stiff torque tube — a stiffness fix, found by understanding the mechanism, within months.

Theodorsen’s 1935 NACA report gave the unsteady aerodynamics in closed form and made the calculation possible in principle. Frazer and Duncan’s The Flutter of Aeroplane Wings (1928) had already set out the eigenvalue formulation and introduced matrix methods into aeronautics to do it — one of the earliest engineering uses of matrix algebra as a computational tool.

The resonance explanation seems to date from popularisation rather than from the technical literature, which never held it. It is easy to see why it spread: it is short, it uses a word everybody knows, and it gets the practical advice roughly right. An explanation can be wrong about the mechanism and useful about the remedy, and that combination is the hardest kind to dislodge — which is the reason this site tests the famous explanations rather than ignoring them.

Where the ladder goes next #

A wing on its own is very nearly indifferent to its own incidence — the moment about its aerodynamic centre does not change with angle, which is exactly the property that makes it useless as a stabiliser. Stability comes from a second surface far enough behind, and the number that decides how far and how big turns out to contain a downwash gradient everybody quotes as a limit at infinity.