The tank that follows the building

The tank that must be shallow to be slow put a tuned liquid damper in the core of a 5 s tower: a rectangular tank 10 m long in the direction of sway, whose deep-water period of 3.58 s is too fast for the building, so that its depth has to tune it. With 1.73 m of water weighing 2 per cent of the building’s modal mass, and screens giving the sloshing 7 per cent damping, it cut the building’s peak response to a harmonic push from 50 times its static deflection to 10.

That essay ended on a property it had found almost incidentally. A shallow tank’s period depends on its depth so steeply that a tank can be retuned by pumping water in or out. And buildings do not keep the period they were designed for. They arrive stiffer than their models, because partitions, cladding and stairs work as structure whether or not anyone drew them as structure — stiffer than the model said found the measured stiffness of buildings consistently above the computed one — and they soften as their concrete cracks under the first storms, as the stiffness that a cracked section keeps is spent. Whether a tank tuned by its water level can follow a building through all of that, and what it costs in water and in wave, is the question.

What a damper does when the building is somewhere else #

A tuned damper works by being out of step with its building at exactly the right amount, so that its spring pulls back on the structure when the structure moves and its damping removes the energy the exchange collects. The mass that helps by being late found that the right amount is narrow: Den Hartog’s tuning puts the damper’s frequency at $1/(1+\mu)$ of the structure’s, μ being the ratio of the moving mass to the structure’s, and with μ of a few per cent there is not much room either side.

For the core tank, the sloshing mass is not all its water. A share of the water moves with the walls, and only the first mode’s convective mass, $m_c = m \cdot 8\tanh(\pi h/L)/(\pi^3 h/L)$, hangs on the sloshing spring. At the design depth that is 74 per cent of 2 per cent, so the effective μ is 1.5 per cent and the tank is tuned to 0.985 of the building’s frequency. A building whose period turns out 10 per cent shorter or longer puts the tank at 0.89 or 1.08 of its frequency instead — far outside the tuning band of a 1.5 per cent mass.

Pumping it back #

The tank’s first sloshing frequency is $\omega^2 = (\pi g/L)\tanh(\pi h/L)$, so its period falls as the water deepens, steeply when the water is shallow and hardly at all once it is deep. Retuning the tank to a building of period $T_b$ means finding the depth at which $\omega = (2\pi/T_b)/(1 + \mu_c)$, with $\mu_c$ itself depending on the depth, because the tank’s plan is fixed and its water’s mass goes up and down with its level.

The curve is steep where the building drifts slower and shallow where it drifts faster. A building that turns out to have a period of 6 s — 20 per cent softer than designed — needs the water let down by a third, from 1.73 m to 1.16. One that turns out to be 4.2 s needs it raised by more than half, to 2.74 m, and the curve flattens towards the 3.58 s floor that no depth can cross: deep water’s period is set by the tank’s length alone. Following a stiffer building costs water, and eventually costs more water than any depth can give. Following a softer one only needs water let out.

The mass ratio moves with the depth, and the figure gives it: 3.2 per cent at 4.2 s, 1.3 per cent at 6 s. A drained tank is a lighter damper, and the obvious worry is that the lighter damper loses whatever the retuning gains.

Retuned, the damper is still a damper #

The dashed curve is the tank left alone. At its design period it holds the building’s peak to 10 times its static deflection, and ten per cent either side it lets it rise to 22: a building that turns out to be 4.5 s or 5.5 s has lost more than half of what its damper was for. At 6 s the peak is 34, and at 6.5 s it is 41 — most of the way back to the bare building’s 50.

The solid curve is the same tank retuned by its level, and it hardly moves: 9.6 at 4.5 s, 10.8 at 5.5 s, 11.7 at 6 s. The light curve is the comparison that isolates the water: retuned to the same depths but with its mass held at the design 2 per cent, as if the tank could change its period without changing its water. The two differ by 1.7 at 6 s. The water let out costs a tenth of what the retuning repairs.

The reason is in the convective share. Shallow water sloshes more of itself — the share rises from 74 per cent at 1.73 m to 78 per cent at 1.16 — so the sloshing mass falls more slowly than the water does: keeping two-thirds of the water keeps 70 per cent of the sloshing mass, and a tuned damper’s best peak rises only as the inverse square root of its mass. A damper’s tuning band, by contrast, is a few per cent wide, and outside it the peak climbs towards the bare building’s. One of those two penalties is gentle and the other is not.

Filling works the other way. A stiffer building gets a tank with more water in it, and below the design period the retuned tank is slightly better than the design one — 9.2 at 4 s — because it is heavier.

What the retuning repairs, and what the water costs #

The two bars at each period are the whole of the trade, and they are of different orders. At 5.5 s retuning repairs 11.7 and the water costs 0.8; at 6.5 s, 28.1 and 2.7. On the stiff side the water’s bar turns negative — filling adds mass, and the heavier tank is a better damper than the design one — so there the only question is whether the tank can hold it.

The building’s own damping is 1 per cent throughout, which is low and typical for a tall building at serviceability amplitudes. A building with 2 per cent would need its damper less and would lose less by detuning, but the ratio between the two bars would be much the same: it is set by the tuning band of a 1.5 per cent sloshing mass, and that is a property of the tank, not of the building.

The building that has moved away from its tank #

The light curve shows what a detuned damper is. Left at its design depth, the tank’s sloshing mode sits at 1.18 of the 6 s building’s frequency, well above it, and the response has one large peak just below the building’s own frequency and a small one beside the tank’s: the damper is doing a little at a frequency the building does not care about. Retuned, the two peaks of a tuned damper return, both low, with the sloshing’s 7 per cent damping between them.

A building’s period can be measured in service far more easily than predicted beforehand — an accelerometer on the top floor and a few hours of ambient wind give it to a per cent or two, and the period nobody chose found how directly it follows from the stiffness and mass that are actually there. A tank tuned by its level turns that measurement into a setting. That is the practical difference from a tuned mass, whose spring is steel and is changed only by replacing it, and from a tank tuned by its width, whose walls are concrete.

How far the building may drift before the valve is opened #

The dashed curve also says how often a tank needs attention. Its peak is 10 at the design period, 12.6 at 4.85 s and 12.1 at 5.1 s, 15.1 at 4.75 s and 15.8 at 5.25 s. If a peak half as large again as the design one — 15 against 10, still a third of the bare building’s — is accepted as the tolerance, the tank may be left alone while the building’s period stays within about 5 per cent of its setting, and must be retuned once it moves further. Measured periods are good to a per cent or two, so the tolerance is three or four times the measurement’s own uncertainty: a retuning rule written as “reset the level when the measured period has moved 5 per cent from the last setting” is one a building owner can actually follow.

That puts the retuning on the calendar of the building’s changes rather than on a fixed schedule. A tower’s period moves most in its first few years — the fit-out, the first storms that crack its cores, the first winters — and very little afterwards, so the visits come early and then stop. It is the same shape as the arm that is reset by hand, whose outrigger shims were reset most often in the years when its creep was fastest, but with a cheaper visit: a valve and a level gauge rather than jacks and shims.

The tolerance is a property of the sloshing mass, not of the building. A heavier tank has a wider tuning band and would tolerate more drift; a building with more damping of its own would be less sensitive to its damper’s being out of tune at all. The damper is only as good as its brace found the same narrowness in a viscous damper’s useful range, for a different reason: every device that works by its timing has a band, and the building has to stay inside it.

Walls that stop the water #

A tank is built with the freeboard its design depth needs and a little more. Its walls decide how far it can follow a stiffer building.

With walls for 2.2 m of water, the tank follows a stiffer building down to about 4.5 s and then stops, full, and detunes from there: at 3.9 s it lets the peak reach 25, better than the 34.8 of the design depth only because the extra water is better than none. A building whose period might turn out a fifth shorter than designed, 4 s, needs a tank with walls for 3.2 m — nearly twice its design depth of water, and a heavier load on the floor that carries it.

That asymmetry is the opposite of the one designers usually guard against. The familiar fear is a building softer than its model, and for a level-tuned tank a softer building is easy. The building that defeats the tank is the stiff one — and the evidence of measured buildings is that the stiff one is the common one, at least for the small amplitudes a comfort damper is there for.

The wave in shallower water #

The earlier essay found its limit in the wave. Computed linearly, the wave at the wall of the 5 s tank at 20 thousandths of g was a quarter of its depth, and on an 8 s building seven-tenths, by which point the linear tuning that designed the tank no longer describes it. Retuning by the level moves along the same curve.

At 15 thousandths of g, around the acceleration that guidance on occupants’ comfort allows a residential tower in a storm it expects once a decade, the design tank’s wave is 0.19 of its depth. Retuned for a 6 s building it is 0.33, and for 6.5 s, 0.41: shallower water, the same building acceleration at a lower frequency and therefore a larger displacement, and a wave that is a larger share of a smaller depth. The stiff side goes the other way, to 0.11 at 4.2 s in deeper water.

The dashed curve is the detuned tank, and it is misleadingly reassuring: a tank that has fallen out of tune barely sloshes, because it is no longer being driven at its own frequency. A tank that is doing its job makes waves; one that makes none has stopped working. Retuning restores the damping and the wave together.

What happens past a third of the depth is beyond this calculation. Shallow-water sloshing becomes nonlinear as its wave grows — its period shifts with amplitude, the crest steepens into a travelling bore, and breaking adds damping that the screens’ 7 per cent did not count on. Some of that is benign, since a breaking wave is a damper too; some of it is not, since a tank whose period depends on how hard it is shaken cannot be tuned by one number. A retuned shallow tank is a damper whose design check moves from the period to the wave.

The retuned depth by hand #

For a building of 6 s, the retuned sloshing mass ratio is about 1.0 per cent, so the tank’s period should be $6 \times 1.01 = 6.06$ s and its frequency $\omega = 1.037$ rad/s. Then $\omega^2 = 1.074 = (\pi g/L)\tanh(\pi h/L) = 3.082 \tanh(\pi h/10)$, so $\tanh(\pi h/10) = 0.349$, $\pi h/10 = 0.364$ and $h = 1.16$ m.

At that depth the water weighs $0.02 \times 1.16/1.73 = 1.34$ per cent of the modal mass. Its convective share is $8 \times 0.349/(\pi^3 \times 0.116) = 0.78$, so the sloshing mass is $1.04$ per cent — the figure assumed at the start — against 1.48 per cent at the design depth. The tank has lost a third of its water and 30 per cent of its sloshing mass.

The design tank left at 1.73 m on the same building has its frequency at $\sqrt{3.082 \tanh(0.544)} = 1.237$ rad/s, which is 1.18 of the building’s 1.047. Den Hartog’s tuning for a 1.5 per cent mass wants 0.985. The error in tuning is 20 per cent, for a damper whose useful band is a few per cent wide.

What a linear slosh rests on #

One building mode, one sloshing mode, linear water. The building is a single mode of 1 per cent damping driven harmonically; the tank’s higher sloshing modes, the impulsive water’s effect on the building’s own period, and any torsion are left out. Retuning the water level also changes the impulsive mass riding with the building, which shifts the building’s period by a fraction of a per cent at these mass ratios — small, but in the direction that a measurement taken after the change would catch.

The damping of the sloshing is held at 7 per cent at every depth. Screens’ damping depends on the water’s velocity through them, which changes with depth and amplitude, and the best damping for a lighter damper is a little lower. Both are second-order beside the tuning.

The period is assumed known. It is measured with the building in service, at small amplitudes; a building’s period lengthens a little at larger amplitudes, as its cladding joints slip and its concrete cracks further, so the period a storm sees is not quite the one an ambient survey records.

Still open: the water that is moved during the storm #

Every retuning here is made in calm weather, once the building’s period has been measured, and left. But a building’s period changes within a storm as well as over years: it lengthens as the amplitude grows, and shortens again as the storm passes. A tank with a pump, a level sensor and an accelerometer could follow that too, letting water out as the building softens under the gusts — which is the step from a passive damper to a semi-active one, of the kind the water tower that could damp itself and the tank that is its own tower never needed, because their water was set by the town’s demand. Whether a tank that retunes during a storm, with a pump that moves a few cubic metres a minute, can follow a period that drifts over the minutes of a storm’s peak, or whether the water’s own inertia and the wave it is already making put any useful change out of reach, is the question a valve that works between storms puts to a pump that works during one.