The speed where the damping is exactly zero

The shake that is not resonance is the collection’s account of what flutter is: two structural modes coupled through the air, exchanging energy, with no external forcing anywhere. This essay is about where the boundary is, which is an exactly defined place, and about why that exactness is of so little help in finding it.

An algebraic condition, and what “exact” means here #

Write a wing section as two degrees of freedom — plunge and pitch — with quasi-steady aerodynamics. The equations of motion are linear with constant coefficients, so their solutions are exponentials $e^{\lambda t}$ and $\lambda$ satisfies a quartic with real coefficients.

Real coefficients means the roots come in conjugate pairs, so each pair is an oscillation with a growth rate. Flutter is the speed at which a pair’s real part passes through zero. Below it every mode decays and above it one grows, and the boundary is a single point on the speed axis defined by an algebraic condition on four numbers.

Bisecting on that condition puts the boundary at 80.843 metres a second, and at that speed the crossing root’s real part is $-4\times10^{-18}$ — which is what an exact algebraic condition looks like when a root finder has been pointed at it. It is not a tolerance and not a convergence criterion. There is one speed at which the real part is zero, and this is it.

The frequency tells the story of the mechanism. It starts near the torsional frequency and moves towards the bending one as the aerodynamic coupling grows; at the boundary it is 95.59 radians a second, between the structure’s own 50 and 100. That coalescence of two frequencies is what flutter is, and it is why no external forcing is needed — the energy comes from the airstream through the phase lag between the motion and the force, which is the lift that arrives late doing something destructive rather than merely slow.

What the quartic is, and why it has four roots #

The four roots are worth accounting for, because “the least damping” is a selection from them and the selection is not always the same mode.

The section has two degrees of freedom, so its mass, damping and stiffness matrices are two by two and the characteristic polynomial is quartic. At zero airspeed the four roots are two conjugate pairs at the structural frequencies, each with the structural damping — two per cent of critical here. As the speed rises, the aerodynamic terms enter the stiffness and damping matrices proportional to $U^2$ and $U$ respectively, and the roots move.

They do not move independently. The aerodynamic stiffness is not symmetric — the lift acts at the quarter chord and the section’s centre of mass is elsewhere, so a pitch produces a plunging force and a plunge produces a pitching moment, and the two are not equal. That asymmetry is the whole of flutter. A symmetric system cannot extract net energy from a steady stream over a cycle; an asymmetric one can, and the quartic’s coefficients carry the asymmetry.

So “the least damping” is whichever pair is currently losing, and the mode that flutters is not the mode that was least damped at rest. Watching the pair swap over as the speed rises is the usual diagnostic in a flutter analysis, and it is the same information the frequency plot above carries.

And the approach to it is nearly flat #

Now the difficulty, which was not what this essay was written expecting.

Two per cent below the flutter speed the least damping is 0.19 per cent of critical. Getting from plus two per cent of critical to minus two takes 61 per cent of the flutter speed: the damping is at +2 per cent at 0.53 of the boundary and at −2 per cent at 1.15 of it.

So the boundary is a knife edge in outcome and a very gentle slope in the quantity that locates it. That combination is the worst one available. If the damping were steep near the crossing, a measurement of it would fix the speed sharply; if the outcome were gradual, a small error in the speed would not matter. Neither is the case.

At the crossing the damping ratio changes by 0.0975 per unit of speed ratio. Read backwards: an uncertainty of half a per cent of critical damping — which is about as well as damping is extracted from a decaying flight-test response — leaves the flutter speed uncertain by five per cent. A tenth of a per cent still leaves one per cent, and one per cent of flutter speed is not a margin anybody is comfortable with.

What a margin has to cover #

Certification practice is to demonstrate freedom from flutter to 1.15 times the dive speed, and the fifteen per cent is often described as covering “uncertainty”. The figures above say what it is covering.

A five per cent uncertainty in the located boundary from the damping measurement alone; a similar figure from the structural model, since the boundary moves with the frequency ratio and the mass ratio; and whatever the aerodynamic model is worth, which for a quasi-steady analysis of a transonic wing is not much. Those do not add linearly, and fifteen per cent is not generous against them.

It also says which improvements are worth buying. Halving the damping-measurement error halves that term, which is the cheapest of the three and the one most often worked on. Improving the aerodynamic model does not narrow the measurement uncertainty at all — it moves the prediction, and the prediction and the measurement are two different quantities whose disagreement is the thing being managed. That distinction is the same one the instrument in the answer is about, and it is easy to lose in a programme where both are called “the flutter speed”.

Which is why extrapolating overshoots #

The standard flight-test procedure is to fly at increasing speeds, excite the structure, measure the damping of each mode from the decay, and extrapolate the trend to zero.

Doing that here with three points at 0.6, 0.7 and 0.8 of the flutter speed predicts 101.8 metres a second. The true boundary is 80.8. The extrapolation is 26 per cent high, and it is high — on the unsafe side — from a set of points that look perfectly collinear over the range they cover.

The reason is visible in the figure. The damping curve is convex: it falls slowly at first and then turns down, so a chord drawn through its early part passes below it and reaches zero late. Fitting higher-order curves helps and does not fix it, because the curvature is where the aerodynamic coupling becomes strong and there is no information about that in the sub-critical data.

This is not an argument against the method, which is the only one available and which flies with large margins for exactly this reason. It is an argument for the margins, and for the fact that the exactness of the boundary buys nothing at all when it comes to locating it experimentally.

The flatness is the mechanism’s, not this section’s #

Sweeping the ratio of the bending to torsional frequency across the family, every member takes between 57 and 63 per cent of its own flutter speed to cross from +2 per cent damping to −2. So the difficulty is a property of two coupled modes exchanging energy through a phase lag, rather than of any particular design.

The family shows two other things. The flutter speed has a minimum near a frequency ratio of 0.8, which is why designers separate the bending and torsional frequencies rather than tuning them together — the two modes are hardest to couple when their natural frequencies are far apart, and the aerodynamic coupling that flutter needs has more work to do.

And the rule of thumb that the flutter frequency lies between the two structural ones is a rule of thumb. At a frequency ratio of 0.95 it comes out at 112.7 radians a second, above both the 95 and the 100 it is supposed to sit between. Nothing forbids that; the coalescence is of the aeroelastic modes, whose frequencies are not the structural ones.

What the energy is doing #

There is a way to read the flatness that makes it feel less like an accident of the algebra.

Over one cycle of an oscillation, the work the air does on the structure is a difference of two nearly equal quantities: the energy the aerodynamic forces feed into the motion through their component in phase with velocity, and the energy the structural damping takes out. At the boundary those two are exactly equal — that is what zero damping means — and both of them are large.

A difference of two large quantities is a small quantity that moves slowly when either of its parts moves. Increasing the speed by two per cent changes the aerodynamic work by about four per cent, and four per cent of a large number is a small change in a difference that was already near zero. So the net is small and its derivative is small, and the flatness follows.

That is the same structure as what a fluid takes out of a swing, where the damping of an oscillating body is also a net of two competing terms — and it is why a measurement of net damping is a much worse instrument than a measurement of either part would be, if either part were measurable.

Removing the coupling instead of measuring it #

If the asymmetry is the whole of flutter, the obvious question is whether it can be taken out — and for one of the two couplings it can, cheaply enough that every aircraft does it.

The aerodynamic coupling is not negotiable: the lift acts near the quarter chord and that is where the section’s aerodynamics puts it. The inertial coupling is a choice. It comes from the centre of mass lying aft of the elastic axis, so that a bending acceleration produces a pitching moment; move the mass forward until the two axes coincide and that term vanishes. In the typical-section model, classical bending–torsion flutter goes with it.

Which is why control surfaces carry lead. A mass balance weight ahead of the hinge line pulls the surface’s centre of mass forward onto its hinge, removing the coupling between the wing’s bending and the surface’s rotation; it is dead weight carried for the life of the airframe, it is required rather than optional, and it is one of the very few places in aeronautics where mass is added deliberately.

It also explains two things that look like other decisions. Engines hung on pylons ahead of the wing move its mass axis forward and raise the flutter speed. And free play in a control circuit is dangerous out of proportion to its size, because a surface that is loose is a surface whose balance weight is no longer attached to the thing it was balancing.

The other boundary, which has no frequency in it #

There is a second instability and it is exact in a different way. Divergence is where the quartic’s constant term vanishes — the aerodynamic moment overcomes the torsional stiffness and the section simply twists until something breaks. That is a static condition with no time in it at all: no frequency, no phase lag, no damping. A root goes to the origin rather than crossing the imaginary axis.

For this section divergence sits at 204.1 metres a second against a flutter speed of 80.8, a ratio of 2.52. That ordering is the usual one — flutter is the boundary that gets designed against, and the essay on the control that works backwards is about a third speed, control reversal, that is related to the divergence condition rather than to this one.

The contrast between the two boundaries is worth keeping. Divergence is a root in the elementary sense: the constant term crosses zero transversally, so measuring the constant term to a part in a thousand locates the speed to a part in two thousand. Flutter is a crossing of a quantity that is nearly stationary there, and it is not.

Why the boundary is a point and the danger is not #

One last distinction, because the two get conflated.

The boundary is a point: below it every disturbance decays and above it one grows, and there is no band in which the answer is “sometimes”. That is a property of a linear system, and it is exact.

The danger is not a point, because what matters in flight is not whether a disturbance decays but how fast. A mode at 0.19 per cent of critical damping is stable and is, for practical purposes, undamped: a gust sets it ringing and it rings for hundreds of cycles, long enough to be a fatigue problem, a handling problem and a pilot-induced-oscillation problem before it is a flutter problem.

So the useful boundary in service is not the one this essay locates. It is a damping level — three per cent of critical is a common requirement — and the figures above say that the three per cent line sits at about half the flutter speed, not just below it. The exactly located point is the one the analysis computes; the one the aircraft is flown to is a long way underneath it.

The pattern this belongs to #

The essay’s two halves — an exactly located boundary and a badly located measurement of it — are an instance of something general, and the general statement is short.

A quantity located by a root is known as well as the function is: if $g$ crosses zero with slope $s$ and is known to $\varepsilon$, the crossing is known to $\varepsilon/|s|$. A quantity located by a stationary point is known only to $\sqrt{2\varepsilon/c}$ with $c$ the curvature — the square root, and half the significant figures gone.

Flutter is a root, so it should be in the good class. What puts it in the bad one is that its slope is small: $|s| = 0.0975$ per unit speed ratio, so the divisor is a tenth rather than a one and the uncertainty is ten times the measurement error rather than equal to it. A root with a small slope behaves like a stationary point without being one.

That distinction — between a boundary that is exact and a boundary that is findable — turns up throughout this collection, and where a pure number comes from is where it is taken apart properly. Here the practical form of it is a flight-test margin.

What is not claimed #

Quasi-steady aerodynamics. The forces here are proportional to the instantaneous angle of attack including the plunge rate, with no wake memory. Real flutter analysis uses Theodorsen’s function or a full unsteady method, and the reduced frequency at the boundary here is high enough that the correction is not small. The numbers above are the model’s; the shapes — the crossing, the flatness, the convexity of the damping curve — are what the essay is about, and they survive the correction.

The section is a typical section, not a wing. Two degrees of freedom at one spanwise station with lumped mass and stiffness is the standard teaching model, and it captures the coupling mechanism and not the spanwise distribution of it. A real wing’s flutter mode has a shape along the span, and the aerodynamic forces on it are three-dimensional — which brings back the wing the equation is really solving and its change of aspect ratio, since compressibility and finite span both alter the lift-curve slope that the aerodynamic stiffness is built from.

Two degrees of freedom. A wing has many, and the mode that flutters is often not either of the two lowest. Adding modes moves the boundary and does not change the character of the crossing.

No structural nonlinearity. Free play in a control circuit, or a stiffness that changes with amplitude, produces limit-cycle oscillation rather than divergent flutter, and then the boundary is not a point on the speed axis at all — it is a region, and the amplitude matters.

And the extrapolation figure depends on where the points are taken. Three points at 0.85 and above extrapolate better than three at 0.6 to 0.8. The essay’s claim is that the curve is convex and that extrapolating from well below is optimistic, not that 26 per cent is a universal number.