The lag that makes flutter possible

The speed where the damping is exactly zero computes this collection’s flutter boundary, and the module behind it carries a note in its own docstring that this essay exists to act on:

With none at all, this quasi-steady model gives the torsion mode a real part of +0.005 at one metre per second and every speed above it: 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.

The repair was two per cent of structural damping, which is physically reasonable and covers the artefact rather than removing it. This essay removes it, by putting back the thing that was thrown away — and the thing is a memory.

What the deficiency is #

A wing changing its incidence sheds vorticity, and the shed vorticity sits in the wake behind it inducing a downwash on the wing that made it. So the circulation a wing has is less than the circulation its instantaneous incidence would give it, by an amount that depends on how much wake is still nearby — which is to say, on how recently the incidence changed.

For sinusoidal motion at reduced frequency $k$ that reduction is a single complex number, Theodorsen’s $C(k)$: a magnitude, which is how much of the quasi-steady lift survives, and a phase, which is how late it arrives.

Its two limits say what it is. At zero frequency it is exactly one: the motion is so slow that the wake has convected to infinity and there is nothing left to interfere. At high frequency it is exactly a half: the shed vorticity is still right behind the wing and cancels half of what the incidence asked for. In between the magnitude falls monotonically and the phase reaches about fifteen degrees of lag.

It is the Wagner function, transformed #

The same memory has a time-domain form which this collection already uses: the lift that arrives late computes the growth of circulation after an impulsive start, and compares it with Wagner’s function.

The two are a Fourier pair. Computing $C(k)$ from its closed form and computing it by numerically transforming the Wagner function this site already exports gives answers agreeing to 5·10⁻⁸ across the range of frequencies a flutter calculation visits.

That is worth doing rather than assuming, and one detail of it is worth recording. The transform of the Wagner function does not converge — the function tends to one rather than to zero — so what has to be integrated is the function minus one, with the constant’s transform added back in closed form. Integrating the function itself over a long window and correcting the tail is wrong by order one, and the error does not fall as the window is lengthened. A truncated non-convergent integral does not converge slowly; it does not converge.

What happens when it is set to one #

The quasi-steady model is $C = 1$, which is the zero-frequency limit applied at every frequency. Since the magnitude of $C$ at the reduced frequencies flutter actually happens at is around 0.7, that is not a small error, and its effect on the stability is not small either.

With no structural damping at all, the quasi-steady section’s worst root has a positive real part at every speed — 5·10⁻³ at one metre a second, rising steadily. The same section with the lag in place has a negative real part until 129.5 metres a second.

So the quasi-steady model does not merely get the flutter speed wrong. It gets the existence of a stable régime wrong, and the two per cent of structural damping that repairs it is doing work no structure should have to do.

What the lag was worth #

Four numbers for one section:

• quasi-steady, no structural damping: unstable at rest;
• quasi-steady, two per cent damping: 80.8 m/s — the figure this collection has been quoting;
• with the lag, no structural damping: 129.5 m/s;
• with the lag, two per cent damping: 131.0 m/s.

The last two are close together, which is the point. With the lag in place the structural damping barely matters, because the aerodynamics is supplying the stability. Without it the structural damping is the only thing standing between the model and nonsense, so the answer depends on a number nobody knows well — and the previous essay says so, correctly, in a paragraph about why flutter clearance is a flight-test programme.

That paragraph is still true and its reason has changed. The sensitivity to structural damping is genuine near the flutter boundary; the dominance of it in the quasi-steady model was an artefact.

The wake as a feedback path, drawn properly #

There is a way of reading the deficiency that makes the whole thing obvious, and it is worth setting out because it transfers to every problem of this kind.

A wing’s circulation responds to its incidence. The change in circulation is shed into the wake. The wake induces a downwash on the wing, which changes its effective incidence. That is a feedback loop, and $C(k)$ is its closed-loop gain.

Read that way the two limits are immediate. At low frequency the shed vorticity has convected far away before it can do anything, the loop gain is zero, and $C = 1$ — the open-loop answer. At high frequency the shed vorticity is still at the trailing edge, the feedback is at its strongest, and it removes exactly half.

The phase is the transport delay: the wake is being convected past at a finite speed, so what it induces now was shed a while ago. A transport delay in a feedback path is the classic route to instability in a control system, and here it is the route to stability, because the sign of the feedback is negative — the shed vorticity opposes the change that produced it, which is Kelvin’s theorem stated as a control loop.

That is also why the effect saturates rather than growing without limit: the loop gain cannot exceed one half, so no amount of frequency removes more than half the circulatory lift.

Why the lag stabilises rather than destabilising #

It is worth asking why a lag helps, since a lag is usually what makes a feedback loop unstable.

Flutter is an energy argument. Over a cycle, the airstream does net work on the wing if the lift is in phase with the wing’s velocity rather than with its displacement. In the binary flutter of a typical section that phasing is produced by the coupling between plunge and pitch, and the aerodynamic force is what carries energy between the two modes.

$C(k)$ does two things to that. It reduces the circulatory lift, which reduces the coupling; and it retards it, which moves the lift out of phase with the wing’s velocity and into phase with its displacement. Both act to reduce the work done per cycle at the frequencies where flutter would otherwise occur, and the second is the larger.

At the flutter point the reduced frequency is 0.254, where the deficiency has magnitude 0.726 and a phase of −15.5 degrees. Twenty-seven per cent of the circulatory lift has been removed and fifteen degrees of lag added — which is why setting $C$ to one moves the answer by sixty per cent rather than by a few.

What the solver computed, and how it was checked #

The aeroelastic matrices are this collection’s own, from the module behind the speed where the damping is exactly zero. The circulatory entries — the ones that come through the wake — are separated out and multiplied by $C(k)$; the rest, which is the added mass and the terms that do not pass through the wake, is left alone. Setting $C = 1$ returns the original matrices exactly, which is the first check.

The eigenvalue problem is then nonlinear, because the matrices depend on the frequency being solved for. It is closed by the standard p-k iteration: guess a reduced frequency, form the complex matrices, find the roots of the resulting quartic, take the frequency back out, repeat.

That needs a complex polynomial root finder. The one this collection already has takes real coefficients, so a Durand-Kerner iteration for complex ones is written here rather than generalising the other — which is called from a check that asserts on its own behaviour and should not be disturbed.

Three checks. That the two representations of the memory agree, to 2·10⁻³, and that its two limits are one and a half. That the artefact is present without the lag — the check fails if the quasi-steady section is stable at two metres a second, because then there would be nothing to repair. And that the same section with the lag is stable there, and flutters somewhere above.

One mistake is recorded because it is easy to make and silent. The quasi-steady comparison is $C \to 1$, which is the low-frequency limit. The first version passed a very large reduced frequency to get “no lag”, which gives $C \to 1/2$ — halving every circulatory term rather than removing the lag, and producing a stable section for entirely the wrong reason.

What this changes about the earlier essay #

Three things, and it is worth being plain about them.

The flutter speed for that section is 131 m/s rather than 80.8. The mechanism described there — a crossing of the damping through zero, at a frequency between the two structural ones — is unchanged and correct. The speed is not.

The dependence on structural damping is much weaker than reported. That essay says the flutter speed depends on the damping ratio and that this is why flutter clearance is flown rather than calculated. The dependence is real; in the quasi-steady model it was overwhelming.

And the sub-critical extrapolation there is affected. That essay extrapolates a damping trend to predict a flutter speed and finds the prediction optimistic. The trend it extrapolates is a quasi-steady one; whether the real trend is better behaved is a question this essay does not answer, and it is recorded as owed.

What kind of memory this is #

It is worth placing the deficiency among the other memories this collection has measured, because it is a different species from most of them.

It is linear, so the whole of it is one kernel and the response to any motion is a convolution. That is not true of the dynamic-stall memory in two lifts at one incidence, where the state variable is a separation point moving on a nonlinear static curve, and where no single kernel describes the response.

It is bounded, in the sense that it removes at most half. Compare the viscous memory in the theory with no memory in it, whose kernel falls as an inverse square root and has no time constant at all.

And it is entirely made of vorticity, which is the first of the three doors that essay inventories. A wing has a memory because it sheds; a cylinder started from rest does not shed and has none, which is everything about the start, except one vector.

So the deficiency is the cleanest available example of a fluid memory: one mechanism, one kernel, bounded, linear, and with both of its limits exactly known.

Where the same repair is owed elsewhere #

The general form of the mistake is worth naming, because this collection can be searched for it. Wherever a steady aerodynamic derivative is used in a problem with a time scale in it, the wake’s memory has been set to one, and the size of the error is $1 - |C(k)|$ at the reduced frequency of the motion.

At $k = 0.05$ that is ten per cent. At $k = 0.2$ it is twenty-six. At $k = 0.5$ it is forty-one. So the question to ask of any unsteady calculation is what its reduced frequency is, and what of order one is worth is this collection’s account of why that question is the one that decides whether a term may be dropped.

What a designer takes from this #

Three practical statements, and the first is the one that decides how much of this matters.

Compute the reduced frequency before choosing a model. It is $\omega b / U$ — the wing’s own half-chord divided by the distance the air travels in a radian of the motion. A control-surface oscillation on a transport wing is around 0.05 to 0.15; a flutter mode is 0.1 to 0.4; a rotor blade meeting a gust is far higher. Below about 0.02 the deficiency is within two per cent of one and the quasi-steady model is fine.

Do not compensate an aerodynamic deficiency with structural damping. The two have different frequency dependencies, so a damping chosen to fix one speed is wrong at every other speed, and the match at the chosen point conceals the fact that the mechanism is wrong.

And treat a quasi-steady flutter speed as a lower bound rather than an estimate. In this case it is sixty-two per cent low, and low in the safe direction — but the safety is accidental, since the deficiency’s phase can move a coupled system either way depending on which modes are involved.

What the picture cannot show #

The deficiency is drawn as a magnitude and a phase, which is two real numbers standing for one complex one, and the drawing therefore cannot show the thing that matters most about it: that the two are not independent. They are the real and imaginary parts of one analytic function, tied together by the requirement that the time-domain response be causal, and no wing can have one without the other.

Nothing here draws the wake either. The vorticity that produces the deficiency is a sheet stretching downstream, and its geometry — flat, convecting at the free-stream speed, not rolling up — is an assumption of Theodorsen’s theory that the figures do not display.

Who found it, and when #

Theodorsen’s function is from 1935, in the report that gave aeroelasticity its standard method. Wagner’s indicial function is from 1925 and the two were connected explicitly only afterwards. R. T. Jones’s rational approximation — the two-lag form used here — is from 1938, and it exists because the exact function is a ratio of Hankel functions and a designer in 1938 could not evaluate those quickly.

The p-k method is later still and is a practical device rather than a theory: it makes a frequency-domain aerodynamic model usable in a problem that is not, in fact, oscillating at a fixed amplitude.

Limits recorded rather than smoothed over #

Jones’s approximation, not the exact function. The two differ by about one per cent in magnitude around $k = 0.3$. That is well inside the difference this essay is about and it is not zero, and the same approximation is used here as in the time domain so that the two representations are consistent with each other by construction.

Incompressible. $C(k)$ is the incompressible lift deficiency. A compressible one exists, is different, and matters above about Mach 0.5 — where transonic effects arrive and neither is right.

A flat plate, and a flat wake. Theodorsen’s theory has the wake leaving the trailing edge along the chord line and convecting at the free-stream speed without rolling up. That is a good approximation at small amplitude and it is not one at the amplitudes flutter reaches.

The typical section is two degrees of freedom. A real wing has many, and its flutter mode is a combination of several — which is why a measured flutter speed is not the typical section’s and why the section is a teaching device with the right mechanism in it rather than a model of an aeroplane. The shake that is not resonance is the account of what the mechanism actually is.

And the p-k method is a device. It is not a solution of the aeroelastic system; it is a consistent frequency assumption iterated to a fixed point, and its damping values away from the neutral point are approximate. The neutral point itself — where the real part crosses zero — is where the assumption is exactly satisfied, so the flutter speed is the number it computes best.