The aeroplane rises out of its own gust

The lift at the mean angle held the aeroplane still and asked what its wing averages over the angles a gust puts it through. That was the right question for the wing’s lift curve and the wrong one for the aeroplane, because an aeroplane in a gust does not stay still. The gust pushes the wing up, the lift rises, and the whole machine accelerates upward. In rising it meets the air at a smaller angle than the gust gave it, and part of the load the gust was delivering is taken away by the aeroplane’s own response.

That is the gust alleviation factor: the peak load an aeroplane actually reaches in a gust, as a fraction of the load a fixed wing would feel at the gust’s full angle. It is one of the oldest numbers in structural design, it has been in the airworthiness codes in one form or another since the 1950s, and it turns out to rest on a calculation short enough to redo from scratch.

The one number that decides #

The aeroplane is taken as rigid, free to move up and down but not to pitch, flying level into a vertical gust whose speed rises and falls as one minus a cosine over a distance $2H$. Its equation of motion is Newton’s law with the lift on the right-hand side, and when everything is measured in the natural units — distance in semichords, vertical speed as an angle — one group is left over:

$$\mu = \frac{2\,(W/S)}{\rho\,\bar c\,a\,g},$$

the mass parameter: the wing loading over the product of air density, chord, lift-curve slope and gravity. It is the ratio of the aeroplane’s mass per unit wing area to the mass of a slab of air one chord thick over that area, weighted by how readily the wing turns angle into lift. A large $\mu$ is a heavy aeroplane for the air it flies in, which responds sluggishly and keeps most of the gust’s load; a small one is a light aeroplane that rises quickly and escapes much of it.

For a light single at sea level — a wing loading of 670 pascals, a chord of a metre and a half, a lift slope of 4.6 per radian — the mass parameter is 16. For an airliner at eleven kilometres — six kilopascals, four and a half metres, five per radian — it is 149. The two differ by a factor of nine, and a factor of three and a third of that is the thinner air at altitude.

The lift does not arrive at once #

Newton’s law alone would give the answer if the lift followed the angle of attack instantly. It does not, and this is where the two indicial functions come in. The lift on a wing that has flown into a sharp-edged gust grows from nothing to its steady value over several semichords — Küssner’s function — because the gust reaches the leading edge first and the vorticity the wing sheds has to be carried away. The lift due to the wing’s own change of motion builds differently, starting from half its final value — Wagner’s function — because a change of motion is felt over the whole chord at once. The lift arrives late in both cases, and the two lateness laws are different.

Whether the lag matters at all is the question slow enough to be steady answers with the reduced frequency, the gust’s rate of change measured against the time the air takes to cross half a chord. A gust twenty-five chords long has a reduced frequency of $\pi/25$, about an eighth, which is well inside the range where the lift’s lag is a first-order effect and outside the range where quasi-steady lift would do. Gusts a few chords long are further in still. The design gusts of civil aviation sit exactly where neither the quasi-steady nor the fully impulsive picture holds, which is why the indicial functions were developed for them in the first place.

So the lift at any moment is a convolution of the gust’s history with Küssner’s function, less a convolution of the aeroplane’s own vertical speed with Wagner’s. Both functions are taken here as R. T. Jones’s sums of two exponentials, the same fits used wherever this subject needs them, and each exponential turns its convolution into one extra ordinary differential equation. The aeroplane’s vertical speed, four lag states and the gust are marched together with a midpoint rule at a fiftieth of a semichord.

The marcher was checked against the one case with a closed form. With quasi-steady lift and a step gust, an aeroplane free to rise has a load that decays as $e^{-s/2\mu}$ exactly, and the marcher reproduces it to $10^{-9}$. Held still with quasi-steady lift it reproduces the full load, one, to rounding.

The factor, computed #

At a mass parameter of 5 the peak load is 0.43 of the quasi-steady figure; at 20, 0.71; at 100, 0.86; and as the aeroplane becomes infinitely heavy it tends to 0.903, which is not one because even a wing held perfectly still does not see the full load of a gust 25 chords long — the lift lags the gust and the peak has passed before the lift has caught up.

The two effects are separable in the figure. The upper curve is the aeroplane’s rise with instantaneous lift: pure inertia and Newton’s law. The curve below it adds the lag of the lift, and Pratt’s fit lies just under that. At small mass parameters the rise dominates, because a light aeroplane is lifted out of the gust before the lag matters; at large ones the lag dominates, because a heavy aeroplane barely moves and all that is left is the aerodynamics.

Pratt did exactly this calculation in 1953, for a gust of 12.5 chords’ gradient distance, and summarised it with the fit

$$K_g = \frac{0.88\,\mu}{5.3 + \mu},$$

which went into the airworthiness codes and stayed there for forty years. Against the calculation redone here it sits within 2.6 per cent at every mass parameter from 5 to 200, and below it throughout — the conservative side, for a load. The residual is almost all at the heavy end: the fit’s 0.88 as $\mu$ grows without limit is the lag of the lift alone, and with Jones’s approximation to Küssner’s function that lag is 0.903. The two calculations differ in the indicial function, not in the dynamics.

Which gust does the most harm #

The fit has one gust length built into it, and the aeroplane does not.

A short gust is over before the lift has built, so it delivers little whatever the aeroplane does; a wing held still sees only 0.47 of the load of a gust one chord long. A long gust gives the lift all the time it needs — but it also gives the aeroplane time to rise, and a light aeroplane rises right out of it: the light single keeps only 0.40 of the load of a gust fifty chords long. Between the two there is a worst length. For the light single it is about eight chords, twelve metres, where it keeps 0.70; for the airliner it is about twenty-three chords, a hundred metres, where it keeps 0.90. At Pratt’s fixed 12.5 chords each is about three per cent below its own worst.

That is why the codes stopped using one gust. Modern certification replaced the single gust of fixed length with a family of discrete gusts over a range of gradient distances, each run through the aeroplane’s own dynamic response, and the design load is the worst of them — a search over exactly the axis of this figure. The fixed-length formula was a good summary of the aeroplanes of its day and a statement about none of them in particular.

Why a light aeroplane is a rough ride #

The alleviation factor favours the light aeroplane, and the light aeroplane is still the one that throws its passengers about. The two statements are about different things, and putting numbers on both shows why.

The load a gust delivers, before any alleviation, is the lift-curve slope times the angle the gust adds times the dynamic pressure, divided by the weight on each square metre of wing. The lift curve’s slope is much the same for both aeroplanes. The angle is the gust’s speed over the aeroplane’s, so the slower aeroplane meets the same gust at a steeper angle. The dynamic pressure favours the faster one. And the wing loading — 670 pascals against six thousand — is nine times larger for the airliner. Put together, a five-metre-a-second gust would give the light single 1.16 g of extra load at the quasi-steady value and the airliner 0.19 g.

Alleviation then takes a third of the light single’s load and an eighth of the airliner’s, and the light single still ends at 0.79 g against 0.17. The light aeroplane escapes more of each gust and meets far more of it to begin with, and the second effect wins by a factor of four. It also rises 1.4 metres through the gust where the airliner rises eighteen centimetres, and it comes out of the gust still rising faster than the surrounding air, which is the negative load at the end of its trace — the lurch after the bump that passengers in small aircraft know well.

That is the same group read two ways. The mass parameter measures how heavy the aeroplane is for the air it flies in; a large one means a sluggish response, which keeps more of the load in proportion but is heavily loaded to start with. Wing loading is, in this sense, a ride-quality parameter as much as a performance one, and it is why aeroplanes designed to fly low and slow in turbulent air carry as much weight on each square metre of wing as their field performance will allow.

The same aeroplane, higher #

The mass parameter has the air’s density in its denominator, so the same aeroplane is effectively heavier at altitude and escapes less of each gust. For the stated airliner the factor rises from 0.81 at sea level to 0.87 at eleven kilometres. That works against the other thing altitude does, which is to reduce the dynamic pressure at a given true airspeed, and it is one reason gust loads are assessed across the whole flight envelope rather than at one condition — the envelope’s corners are different aeroplanes as far as a gust is concerned.

The formula is in the definition of the gusts #

There is a subtlety about how the factor was used that makes it more robust than its assumptions deserve. The gust speeds that design is carried out against were not measured directly. They were inferred from accelerometer records carried by airliners in service, by running the gust-load formula backwards: a recorded acceleration, divided by the formula’s load per unit gust speed — alleviation factor included — gave a derived gust velocity. Those derived velocities were then used, with the same formula, to predict the loads on new aeroplanes.

So an error in the alleviation factor partly cancels. An aeroplane similar to the ones that gathered the data gets back very nearly the loads those aeroplanes measured, whatever the factor is. What does not cancel is the difference between aeroplanes: a new design whose mass parameter, gust response or flexibility differs from the fleet that measured the gusts inherits the error in full. That is the honest reading of a 2.6 per cent discrepancy between a fit and its calculation. Within the population it was fitted on, it hardly matters; outside it, the fit is as good as its physics and no better.

What the picture cannot show #

The gust in every figure is a single, clean one-minus-cosine pulse, uniform across the span. Real turbulence is a continuous random field, and an aeroplane flying through it meets a spectrum of gusts at once; the alleviation of that is a filter rather than a factor, and the chord’s own averaging of short gusts is part of it. A gust also varies across the span, which averages it again on a wing much wider than the gust — a subject of its own.

Nor do the figures show the atmosphere’s side of the bargain. A gust five metres a second strong is a moderate one; the gusts design is carried out against are several times stronger at low altitude and fall off with height, and how often an aeroplane meets each strength is a statistical statement about the air that no calculation here touches. The factor says what an aeroplane does with a gust once it has met one.

The negative load at the end of the light single’s history is real within the model — the aeroplane is still rising when the gust has passed — and it is the reason gust cases produce down loads after up loads. It is also exactly where a real aeroplane’s pitching motion, which is left out here, would change the answer most.

Where the model stops #

No pitching. A real aeroplane pitches in a gust: the tailplane meets the gust later than the wing, and the aeroplane’s static stability turns it into or out of the gust. Pratt left pitch out for the same reason it is left out here, and the modern tuned-gust analyses put it back.

No flexibility. A flexible wing bends in a gust, and the bending changes its twist and its local angle of attack. For large aeroplanes that is a first-order effect on the loads, and it is why gust analysis became a structural-dynamics calculation. The rigid factor is the aerodynamic half.

Two-dimensional lift functions. Küssner’s and Wagner’s functions are for an aerofoil of infinite span. A finite wing’s lift builds faster, because the wake’s influence is weaker, and its lift-curve slope is lower; the mass parameter uses the finite wing’s slope and the indicial functions do not.

The convention the numbers depend on #

The alleviation factor is defined against the quasi-steady load at the gust’s full angle, with the aeroplane held still. The mass parameter uses the mean chord and the aeroplane’s own lift-curve slope, and distances are in chords for gust lengths and in semichords inside the lift functions, because that is what the lift functions are written in. The gradient distance $H$ is half the gust’s length; Pratt’s 12.5 chords is a gust 25 chords long.

Who found it, and when #

Küssner published his function in 1936 and Wagner his in 1925. Early gust-load formulas, from the 1930s, used a sharp-edged gust and an empirical alleviation factor; Pratt’s report of 1953 replaced the sharp edge with the one-minus-cosine gust and derived the factor from exactly the calculation above, and Pratt and Walker re-derived the gust velocities from the accelerometer records of the preceding two decades with the new factor in 1954. Discrete tuned gusts with dynamic response replaced the fixed-length formula in the civil codes in the late twentieth century.

Still open: what pitching adds #

The calculation that would follow gives the aeroplane its second degree of freedom. With pitch, the gust reaches the tailplane some chords after the wing — a time the tail’s distance behind the wing and the airspeed decide — and the aeroplane’s static margin sets whether it noses into the gust or away from it. A stable aeroplane noses into a vertical gust, which raises the wing’s angle and works against the alleviation computed here; how much depends on the tail volume and on how the tail’s delay compares with the gust’s length.

The question with a number in it is the ratio of the pitching aeroplane’s peak load to the plunging one’s, across the mass parameter and the static margin, for the gust length that is worst in each case. Beside it is the flutter essay’s mass ratio, which is the same group seen from the structure’s side: a heavy wing for its air has little aerodynamic damping, and the same heaviness that keeps a gust’s load is what lets a flutter mode grow.