The deck that forgets its own rhythm

A deck moving at a stay’s anchorage swings the stay’s tension, and at twice the stay’s frequency that swing pumps the stay with no force across it at all. The condition for growth came out as one comparison: a quarter of the swing in tension against the stay’s damping ratio. For the 120 m stay at 3,500 kN, a deck moving ±10 mm vertically swings the tension by 1.39 per cent, and against the stay’s own 0.1 per cent damping it grows tenfold every 147 s.

That answer rested on one assumption, named and left open where it was made: the deck moves as a sine, at one frequency, for as long as the stay takes to answer it. Every growth it drew took minutes, and for all of those minutes the deck’s frequency had to sit within a third of a per cent of twice the stay’s.

A deck moved by gusting wind does not do that. Its motion is the response of one of its own modes to a push that arrives at every frequency at once, and a lightly damped mode answers such a push by ringing at its own frequency — but with an amplitude that swells and fades and a phase that drifts. The question left open was whether that motion holds the stay inside the band long enough. It has an answer, and the answer turns out not to depend on how far the deck moves so much as on how long it remembers how it was moving.

A deck that rings without keeping time #

The model of the deck here is the simplest that has the right character. One vertical mode of the deck, with a natural frequency exactly twice the stay’s — 2.01 Hz — and a damping ratio of its own, 1 per cent, is pushed by white noise and scaled to an rms displacement at the anchorage of 7.1 mm. That rms is chosen to match the sine it replaces: a sine of ±10 mm has an rms of $10/\sqrt{2}$.

The trace has the frequency the sine has; at a glance the cycles are the same width. What differs is everything slower than a cycle. The amplitude rises above twice the rms and falls to a fraction of it within a quarter of a minute, and across each fade the phase slips, so a stretch of motion before a quiet moment is not in step with the stretch after it.

How long the deck holds its phase is set by its damping. A mode damped at $\zeta_d$ forgets its motion in a time of about one over $\zeta_d$ times its circular frequency — the time over which its free vibration decays by a factor of e, and over which the random push replaces what was there with something new. For this deck that is 7.9 s, sixteen of its cycles.

Set that beside the stay’s clock. Under the ±10 mm sine the stay grew at 0.0157 a second, so one factor of e took 64 s, and growing from a millimetre to a few hundred took several minutes. The deck keeps its rhythm for eight seconds; the stay needs a minute to answer one. Whatever the stay does over a minute, it does to an average of eight or so unrelated stretches of deck motion, and averaging is exactly the operation under which a sine and a random signal of the same rms stop looking alike.

The same rms, two outcomes #

The first test is direct. Integrate the stay and the deck together, the stay starting at a millimetre with its own 0.1 per cent damping, and compare it against the sine of the same rms.

The sine does what it did before, a straight line on these axes, tenfold every 147 s — growth with no ceiling, unlike a resonance driven across a structure, whose limit the damping sets. The random deck with the same rms does something else: the stay’s amplitude wanders upward for a few minutes, reaches 5 mm, and then decays for the rest of the run, ending at a tenth of a millimetre after twenty minutes. A random deck of twice the rms, 14 mm, does grow the stay — but at 0.009 a second rather than 0.0157, and raggedly, with steps up when the deck happens to hold its phase and steps back when it does not.

The dashed lines come from averaging, and they are worth deriving because they carry the whole result. Write the stay’s motion as an amplitude and a phase, and ask how the amplitude changes under a tension that swings by a random fraction ε(t). Its logarithm changes at a rate proportional to ε(t) times the sine of twice the stay’s phase. Over a cycle that averages to nothing, unless ε has a component that keeps step with twice the stay’s frequency, so the part that survives the averaging is proportional to how much of ε’s variance lies at exactly that frequency. That quantity is the spectral density of the tension swing at 2ω, Φ(2ω), and the growth rate of a typical history comes out as

$$\lambda = -\zeta\omega + \frac{\pi\omega^2}{4}\,\Phi(2\omega).$$

The first term is the stay’s damping, as before. The second replaces the quarter of the swing, and it contains no amplitude at all — only a density, the variance per unit of frequency at one precise frequency.

For a deck that is a damped mode at twice the stay’s frequency, the density at its own frequency is the variance divided by 2π $\zeta_d$ times that frequency: the same variance, spread over a band whose width is set by the deck’s damping. Putting that in gives a requirement that can be written in one line. A typical history of the stay holds still when ζ exceeds $\sigma_\varepsilon^2/16\zeta_d$: the square of the rms swing over sixteen times the deck’s own damping ratio.

For the 7.1 mm deck, $\sigma_\varepsilon$ is 0.99 per cent and the requirement is 0.061 per cent. The sine of the same rms asks for a quarter of its amplitude, 0.348 per cent — nearly six times as much — and the stay’s 0.1 per cent sits between the two. That is why the same rms grows the stay as a sine and lets it settle as a random deck. The integration knows nothing about any of this, and its rates are what the averaging predicts: −0.0018 a second against −0.0025 at 7.1 mm, +0.0090 against +0.0087 at 14 mm.

A square, not a size #

The requirement for the sine grows with the deck’s movement; the requirement for the random deck grows with its square. That difference is the first thing a designer would want to see drawn.

Against the stay’s own damping, the sine starts a growth at an rms of 2.0 mm — the ±2.9 mm of amplitude found before. The random deck starts one at 9.1 mm, four and a half times as much movement. At small movements the parabola lies far below the line, which is the regime that matters for a stay in service: a deck moving a few millimetres in a moderate wind asks almost nothing of the stay’s damping, because the square of a small swing is very small.

The dots test the parabola without trusting it. Each is the growth rate of the stay with no damping at all, averaged over five long histories, divided by ω. That division is exact rather than an approximation, because damping in this equation only ever subtracts: writing the stay’s displacement as e^(−ζωt) times another variable leaves that variable undamped, at a frequency lower by a factor $\sqrt{1-\zeta^2}$. So the undamped growth over ω is the damping that just holds the stay still. At 7.5 mm the integration asks for 0.06 per cent against the parabola’s 0.07, and at 15 mm for 0.25 against 0.27.

Beyond about 20 mm the dots fall below the parabola: 0.49 per cent at 22.5 mm against 0.61, and 0.71 at 30 mm against 1.1. The averaging that produced the parabola assumes that the stay changes slowly compared with how quickly the deck forgets, and at a large swing the predicted growth becomes fast enough to break that. Past that point the stay can grow during a single coherent stretch of deck motion and lose ground in the next, and the average overstates it. The parabola is the answer where it holds and a safe bound just beyond.

The dotted parabola is the mean square, and it answers the form in which the question was first asked. Averaging shows more than a rate. The logarithm of the stay’s amplitude does not climb or fall steadily; it takes a random walk with that rate as its drift, and its variance grows at a rate κ equal to the second term of λ — the same density at 2ω. The spread is there in the integrations: for a 14 mm deck damped at 2 per cent, 24 histories of 3,000 s each scatter in their growth rates by 0.00175 a second, and a walk of that variance predicts 0.00158. A walk of logarithms is skewed when undone. The mean of the amplitude’s square is carried by the rare histories that happened to take a long run upward, and it grows at twice the drift plus twice the variance rate. The mean square needs exactly twice the damping a typical history does: 0.12 per cent for the 7.1 mm deck, still a third of the sine’s.

Which of the two is the requirement is a choice about what is being protected. A typical history says what a stay will usually do on a windy day; the mean square is what a fatigue count that weights each day by its squared amplitude sees, because it is dominated by the few days in which the deck happened to keep its rhythm. Either way, the sine overstates it by a factor of three or more at the rms of service.

The deck’s own damping is the hidden variable #

The requirement $\sigma_\varepsilon^2/16\zeta_d$ has the deck’s damping in the denominator, and nothing in the sine’s requirement mentions the deck at all. That makes the deck’s damping the variable the stay now depends on most.

Read along the curve from the right. A deck damped at 2.5 per cent forgets its rhythm in about three seconds and the stay needs 0.03 per cent. At 1 per cent, 0.06. As the deck’s damping falls the deck remembers longer and the same variance is packed into a narrower band around its own frequency, so more of it sits at exactly twice the stay’s, and the stay needs more. Halve the deck’s damping and the stay’s requirement doubles.

The curve cannot go on rising for ever, and the dots show where it stops. A deck with almost no damping rings at one frequency with an amplitude that barely changes from minute to minute; for all the stay can tell, that is a sine. The averaged curve would cross the sine’s requirement at a deck damping of 0.17 per cent, which is exactly where its own assumption fails: the deck now remembers its rhythm for longer than the stay takes to answer it. The integrations bend away from the curve before that and climb towards the sine from below — 0.18 per cent at a deck damping of 0.2, 0.12 at 0.4 — without ever reaching it. A random deck never asks more of the stay than a tuned sine of the same rms, and a lightly damped one asks almost as much.

That makes the open question from the sine’s case concrete. Whether a wandering deck holds a stay inside the band long enough depends on one ratio: the deck’s memory, one over $\zeta_d$ times its frequency, against the stay’s growth time. With the deck at 1 per cent the ratio is 8 s to 64 s and the requirement collapses to a sixth. With the deck at 0.2 per cent the deck remembers for 40 s, the two clocks are comparable, and the stay sees something close to a sine.

The histories show what that looks like. Under the lightly damped deck the stay at 7.1 mm grows in steps: a rise of a factor of ten in a couple of minutes while the deck keeps its phase, a fall when the phase slips, another rise. Its overall rate is 0.005 a second, a third of the sine’s, and the averaged formula, which assumes forgetting is fast, would have said more than twice that. This is the regime the open question was worried about, and it is where the deck is coherent rather than random.

Where in that range a real deck sits depends on which mode of motion is being driven and by what. A deck mode measured on a finished bridge typically has structural damping of a fraction of a per cent to a per cent or two, and in wind it gains aerodynamic damping that rises with the wind speed. Buffeting — the resonant part of a deck’s response to the turbulence in the wind — is the broad-band, forgetful case that the parabola describes. Vortex-induced lock-in is the opposite: the shedding synchronises with the deck, its frequency is held by the deck’s own, and the motion is about as close to a sine as a structure in wind ever gets. The sine’s requirement is right for that case and generous for the other.

The band spreads while it lowers #

The sine’s danger was narrow. Its region of growth was a third of a per cent wide, so a deck a little off twice the stay’s frequency did nothing however hard it moved — the reason a temperature change of a degree or two could carry a stay out of it. A random deck spreads its variance over a band, and that spreading reaches across the tuning as well.

The sine’s curve is a tall spike, 0.348 per cent at the exact ratio and zero a third of a per cent either side. The random deck’s curves are low and broad. With the deck damped at 1 per cent the requirement is 0.061 per cent at tuning, 0.057 per cent a quarter of a per cent off, 0.049 at half a per cent off and 0.031 at 1 per cent off. A deck 1 per cent from twice the stay’s frequency, which the sine would have ignored entirely, still asks for half of what the random deck asks at its peak.

The trade is not symmetric in its consequences. Lower everywhere near tuning means a stay with any reasonable damping is safe near tuning under buffeting; broader means the safety no longer depends on knowing the frequencies to a third of a per cent. The sine’s analysis ended on the observation that the band was narrower than any frequency is known to in advance, so a list of stays against deck frequencies could say which stays to watch but not which were inside. Against a random deck that observation loses its force: the requirement varies so slowly with tuning that a stay within a per cent or so of the ratio can be checked at the peak value and nothing is lost.

The damper, checked again #

A viscous damper near the anchorage gives this stay at most about 1 per cent of critical, wherever it is sized, because of where it sits. Against the sine that ceiling was enough to raise the deck movement the stay can take from ±2.9 mm to ±29 mm. Against a random deck the same damping goes further, because the requirement rises as a square.

With the damper in place the stay grows under a sine of 20.7 mm rms, ±29 mm in amplitude, and under the random deck only past 29 mm rms — an hourly peak of more than 120 mm at the anchorage, a movement that would be noticed by everyone on the bridge and is not a condition of service. The mean square reaches the damper’s 1.02 per cent at 20.5 mm rms, close to the sine’s figure. On this reading the damper that was sized against rain and wind already covers a buffeted deck with a wide margin, and the case it does not cover — the lightly damped, coherent deck — is the one the sine already described.

The margin over the parabola is also safe in the direction that matters. Past 20 mm the integrated requirement falls below the parabola rather than above it, so a check made with $\sigma_\varepsilon^2/16\zeta_d$ overstates the damping a strongly moving random deck needs; it does not understate it.

What the peak was hiding #

The reason all this matters in practice is the number a deck measurement usually reports. A record of a deck’s movement in wind is reported by its largest excursion, and the largest excursion of a random deck is several times its rms. For a deck moving with a 7.1 mm rms at 2 Hz, the largest of the 7,200 cycles in an hour is about 30 mm.

Read as the amplitude of a sine, 30 mm swings the tension by 4.2 per cent, and the stay would need a quarter of that, 1.04 per cent, to hold still — more than its damper can give. The random deck that produced the 30 mm needs 0.061 per cent for a typical history and 0.12 per cent for its mean square. The peak overstates the requirement seventeen times, because the peak is the size of a moment the deck does not hold, and growth needs the deck to hold its rhythm for longer than any one moment.

That is the answer to the question as it was left. What decides whether the stay grows is how much of the deck’s motion falls at exactly twice the stay’s frequency, measured as a density, and how long the deck keeps that share in step — not how large the motion becomes. The quantity to extract from a deck record is therefore not its peak but its spectrum near twice each stay’s frequency, or equivalently the rms and the damping of the deck mode that sits there. Both are routinely identified from ambient vibration of a finished bridge, by the same kind of measurement that reads a stay’s tension off its frequency.

What the random deck leaves out #

One deck mode, one stay mode. The deck’s motion at the anchorage is drawn as one mode; a real deck has several, each with its own period, and those near twice a stay’s frequency each add their own density at 2ω. Because the requirement adds up from densities rather than amplitudes, adding modes is a sum of terms of this shape.

White-noise forcing. The wind’s push is broad but not flat. For the narrow band that matters, set by the deck’s damping, the push is close to flat; the approximation fails only if the turbulence has a sharp feature near the deck’s frequency.

Linear stay. The stretching that stopped the sine’s growth at a few hundred millimetres is left out here, because the question is whether growth starts. A stay that grows in bursts under a coherent deck would be capped by the same mechanism within each burst.

Constant tension. The stay’s own frequency is taken as fixed. Temperature and traffic change it slowly; against the sine that mattered because the band was narrow, and against the random deck it matters little because the requirement varies slowly with tuning.

Viscous damping in the stay. As before, a strand bundle’s own damping is small and varies with amplitude, and the wind can add or remove damping. The requirement is a damping ratio; whatever supplies it, the comparison is the same.

Still open: what the deck’s traffic does to the rhythm #

Wind supplies a push with no rhythm of its own, so the deck’s memory is set entirely by the deck’s damping. Traffic is different. A vehicle crossing a span applies a moving load whose passage excites the deck’s modes in a transient that decays with the deck’s damping, and a stream of vehicles produces a sequence of such transients at intervals set by the traffic’s spacing and speed. Whether a dense, regular stream — a convoy, or a queue moving at constant speed — produces deck motion whose phase is held across vehicles, and so behaves for the stay like the coherent case rather than the forgetful one, is a question about the spacing of vehicles relative to the deck’s period, and none of the models here contains it.