The motion that feeds itself

An iced power line in a steady winter wind swings through metres, slowly, for hours. A rectangular signal gantry in a steady breeze oscillates until its welds crack. A suspension bridge deck in a steady 19 m/s wind twists itself apart.

None of those is resonance, and the reason is the word steady. A steady wind contains no frequency for a structure to resonate with. The oscillation has to be manufactured, and the structure manufactures it.

Den Hartog’s criterion #

Consider a section moving across a steady wind at velocity $\dot y$. Relative to the section, the wind is no longer horizontal: it arrives at an angle $\alpha \approx \dot y / U$, because the section’s own motion has added a component.

If the section’s lift changes with angle of attack, that induced angle produces a change in lift. And if the lift falls as the angle rises — which is what happens on a bluff section past a certain angle, and never on a well-behaved aerofoil — the change in force is in the same direction as the motion.

Writing the aerodynamic force as a damping term gives the equation of motion

$$m\ddot y + \left(c + \tfrac{1}{2}\rho U D \left[\frac{dC_L}{d\alpha} + C_D\right]\right)\dot y + ky = 0$$

and the bracket is Den Hartog’s criterion. When

$$\frac{dC_L}{d\alpha} + C_D < 0$$

the aerodynamic damping is negative, and since it grows with $U$ there is a speed at which it overcomes the structure’s own:

$$U_{\text{crit}} = \frac{4 m \zeta \omega}{\rho D \left|dC_L/d\alpha + C_D\right|}$$

Three things about that expression deserve attention, and each contradicts an intuition.

The criterion is about the SHAPE. Whether a section can gallop at all is decided by two aerodynamic coefficients and nothing else. A circular cylinder cannot — it has no angle of attack, so $dC_L/d\alpha$ is zero and the criterion is satisfied by the drag alone. A square section can, a D-section can, and an iced conductor can, because the ice makes an aerofoil of an object that was a circle.

The frequency does not appear except through the damping. There is no coincidence to avoid, no band to stay out of, and nothing to be tuned away from. This is what most distinguishes it from every other problem in this field.

Mass and damping appear as a product, exactly as in the Scruton number, and for the same reason: what matters is how much the structure can dissipate against how much the air can supply.

What growth without a driver looks like #

That figure is the whole distinction, and it is worth stating as a pair of tests a reader can apply to any reported failure.

A resonant response requires an excitation at a matching frequency; is bounded by damping at $1/2\zeta$; grows to a steady amplitude and stays; and disappears when the excitation moves off the frequency.

A self-excited response requires only that a parameter exceed a threshold; is not bounded by anything linear; grows exponentially from any disturbance; and cannot be removed by detuning.

Tacoma Narrows, and why the textbooks are wrong #

The bridge that collapsed on 7 November 1940 is the most famous illustration of resonance in engineering, and it was not resonance.

The wind that day was around 19 m/s and steady. The motion that destroyed the deck was torsional — the deck twisting about its own axis, one edge up and the other down — and it developed after the bridge had been oscillating vertically for some hours, following a change in the deck’s behaviour when a mid-span tie slipped. The torsional motion grew over about an hour to angles beyond 30 degrees.

The mechanism was aeroelastic: the deck’s own rotation changed the flow around it in a way that added energy to the rotation. It is usually classified as torsional flutter, which is the same family as galloping with an angular coordinate rather than a linear one, and complicated by the deck’s H-shaped section shedding vortices at the same time. The deck was stiffened by shallow plate girders rather than by a truss, on a main span of 853 m, giving a depth-to-span ratio of about one in three hundred and fifty and a torsional stiffness lower than any comparable bridge then standing — which is the property every subsequent long-span deck has been designed to have more of.

What it certainly was not is a vortex-shedding resonance. The shedding frequency of the deck at that wind speed was around 1 Hz, and the torsional mode was at 0.2 Hz. The numbers do not match, and they were checked at the time.

The reason the wrong explanation survived is instructive, and it is not carelessness. The film shows a structure oscillating at a steady frequency with growing amplitude, and that is exactly what resonance looks like. The two mechanisms produce visually identical evidence, and the only way to distinguish them is by asking what the excitation was — which requires knowing that a steady wind has no frequency in it, and that a growing oscillation therefore has to be self-excited.

This site’s rule about naming the free body has a version for this field: name the excitation, and say what its frequency is. If the answer is “the structure’s own”, the problem is not resonance.

What it took to establish that #

The correction to the Tacoma story was not a matter of anyone rereading the film. It came from the wind tunnel work that followed the collapse and from the aeroelastic theory that the aircraft industry had been building since the 1920s, when flutter began destroying aeroplanes for exactly the same reason.

That transfer is worth noticing. The mechanism that destroyed a bridge had been understood, analysed and designed against by another profession for fifteen years, in a field that shared no journals, no conferences and no vocabulary with structural engineering. The bridge engineers’ apparatus for wind was a static pressure; the aeronautical engineers’ was a set of motion-dependent forces measured on a model.

It is the strongest argument this collection can offer for the figure-first habit it is built on: the two professions were drawing different free bodies. One drew a deck with a pressure on it; the other drew a section with forces that depended on how the section was moving. Everything else followed from that choice, and the choice was invisible to both of them until a bridge fell down.

What flutter needs that galloping does not #

Galloping is the one-degree-of-freedom member of the family and the only one with a criterion this simple. Classical flutter — of an aircraft wing, and of a bridge deck — needs two degrees of freedom that couple.

The mechanism is that vertical motion and rotation feed each other: rotating changes the lift, which drives the vertical motion, which changes the angle of attack, which drives the rotation. Below the critical speed the two motions are out of phase in a way that dissipates energy and above it they are in phase in a way that absorbs it, and the transition is a speed at which two of the system’s modes coalesce.

Two consequences worth carrying:

Flutter needs the two frequencies to be close, which makes it the one member of this family for which a mode list is the first thing to look at. A deck whose torsional frequency is well above its vertical one is much less susceptible, which is why modern long-span decks are torsionally stiff box girders rather than open trusses or plate girders, and why the ratio of the two frequencies is a headline number in a long-span design.

It cannot be found by any calculation that treats the wind as a load. The critical speed comes out of an eigenvalue problem in which the aerodynamic forces are functions of the motion — the flutter derivatives, measured in a wind tunnel on a section model. That measurement is a routine part of long-span bridge design and exists because of 1940.

The comparison is worth making explicit because it is diagnostic. If a structure’s problem can be drawn as a peak on a magnification curve, damping bounds it and a tuned mass can reduce it. If it can only be drawn as a crossing, then damping does not bound the response — it sets the speed at which the response begins, which is a different thing to buy and a different thing to check.

The family, and why it belongs with buckling #

Three problems, in three industries, with three names: a crowd on a footbridge, a wind on a bluff section, and a flow over a coupled pair of modes. In each of them the force is a function of the motion; in each there is a threshold in a parameter; and in each, below the threshold nothing happens at all.

That is not the structure of a load problem. It is the structure of a stability problem, and the resemblance to buckling is exact rather than poetic:

A column that is strong enough and falls over anyway and a deck that is strong enough and tears itself apart anyway are the same failure of the same kind of thinking: a check on the response to a given load says nothing whatever about whether the equilibrium the response is measured from is stable.

What the picture cannot show #

The coefficients are measurements. $dC_L/d\alpha$ and $C_D$ come from wind tunnel tests on the section, they vary with Reynolds number and with turbulence, and for an iced conductor they vary with the shape of the ice. The criterion is exact and its inputs are not.

The growth stops somewhere. The linear model grows without bound. In practice galloping settles into a large-amplitude limit cycle, because the aerodynamic coefficients change once the amplitude is big enough that the induced angle is no longer small. The structure oscillates hugely and steadily rather than accelerating to destruction — which is why iced conductors gallop for hours and are usually still there afterwards.

One mode is assumed. The criterion above is written for a section translating in one direction. A real member gallops in whichever mode is least stable, and for a long cable or conductor that is a mode with several half-waves along the span — which changes the effective mass and the damping, and therefore the critical speed, without changing the criterion.

Turbulence matters and is not in the model. Atmospheric turbulence disrupts the coherent flow that galloping needs, so a section in rough flow is more stable than in smooth. The dangerous conditions are again steady ones — which is the third time in this field that the worst case has turned out to be a mild, steady, unremarkable day rather than a storm.

Where the ladder goes #

This is the last rung of the field’s foundation and it points at the whole of it. The essays before it built up a way of asking what a structure does about a load; this one is where that question stops being the right one.

The immediate direction is the flutter analysis proper — the two-degree-of-freedom eigenvalue problem with measured aerodynamic derivatives, which is how every long-span bridge deck is now checked and how the aircraft industry has worked since the 1930s.

The wider direction is the classification. This field opened with a load that arrives suddenly and a factor of two, and it ends with a load that does not arrive at all until the structure invites it. Between the two lie every intermediate case: a load with a frequency, a load with a spectrum, a load that is a displacement of the ground, and a load supplied by the people using the structure. What organises them is not what causes the load but how the load is related to the response — independent of it, statistically related to it, or manufactured by it — and that is the ordering worth carrying out of this field into any other.

Read that way, the field has a shape. The first four essays are about a structure meeting a load it does not affect. The middle five are about loads described statistically, where the structure and the excitation are still independent but the answer is a distribution. And the last three are about loads that would not exist if the structure held still — which is the point at which the free body has to include the thing applying the force, and the point at which dynamics stops being statics with an extra term.