The water tower that could damp itself

The tank that sloshes the tower put a tank of water on a tower and found that the two cannot be calculated apart. On the ground a tank’s water divides into an impulsive part that moves with the wall and a convective part that sloshes at a long period of its own, and the two responses are added. On a tower whose own period is seconds, the convective water hangs off a container that is itself swaying, and the tank and the tower are one system with two modes. Adding the ground tank’s numbers overstated the base shear by up to 48 per cent.

Its last section saw the shape of a better question. The convective water on a tower is a mass on a spring hung from the structure — which is exactly what a tuned-mass damper is. Some tall buildings carry a tank of water in their roof for that purpose, with screens across it to damp the sloshing. A water tower is the same device with a very large mass and no damping, and the essay asked whether a tower’s own water, with a baffle chosen for one fill level, could cut its base shear — and what that would be worth when the level is whatever the town drew yesterday.

A damper already tuned and never damped #

The tower drawn is the earlier essay’s: a tank 12 m across holding water 8 m deep, a 150 t container, on a tower whose period with the tank full is 2.5 s, with 2 per cent of damping in its own structure. Full, the water’s convective part is 304 t, sloshing at 3.65 s; the container and the impulsive water riding with it are 751 t.

The mass that helps by being late worked through the design of a tuned-mass damper: a small mass on a spring whose frequency is set a little below the structure’s, with damping chosen so that the structure’s resonance is replaced by two lower, broader peaks. Den Hartog’s rule gives both numbers from the mass ratio $\mu$: tuning $\omega_d/\omega_s = 1/(1 + \mu)$ and damping $\zeta = \sqrt{3\mu/8(1 + \mu)^3}$.

For this tower the mass ratio is 0.40. Den Hartog’s tuning is 0.71. The water’s own tuning, which nobody chose, is 0.69. A designer setting out to build a tuned-mass damper for this tower could hardly do better: a mass two-fifths of the structure’s, tuned within 3 per cent of the optimum. Den Hartog’s damping for it is 23 per cent. Water in a plain cylindrical tank has about half of one per cent.

Why the tuning is not luck #

It would be easy to read the 0.69 as a coincidence of this one tower. It is closer to a property of water towers in general. A tank’s sloshing period is set almost entirely by its radius, growing as its square root once the water is deeper than about half the radius: a tank 8 m across sloshes at about 3.0 s, 12 m at 3.7 s, 18 m at 4.5 s, 24 m at 5.3 s. A tower built to hold such a tank up, tens of metres in the air, with a stiffness chosen for the wind and the weight, usually has a period of a second or two to three. So the water is almost always slower than the tower, by a factor somewhere around two-thirds — and Den Hartog’s optimum for the mass ratios that tanks produce, 0.3 to 0.6, is a tuning between about 0.6 and 0.8.

The mass ratio is no accident either. The share of a tank’s water that sloshes is set by the tank’s proportions, from about 30 per cent in a deep narrow tank to more than 40 in a wide one, and the share that rides with the wall is the rest. The damper’s mass and the structure’s mass are both water, divided by the same shape. A water tower is, by its proportions, a tuned-mass damper with a mass ratio a building designer would never be allowed — and no damping at all.

That last part is the whole problem. A tuned-mass damper in a tall building is designed around its damping: the mass is the easy part, and the dashpots, or the screens, or the oil, are what make it a damper. The only thing that stops it made the general case that damping is the one property of a structure that removes energy rather than storing or moving it. The water stores and moves energy beautifully. It removes almost none.

Two sharp peaks where there was one #

The tower’s response to harmonic ground shaking at each frequency shows what the water does. With the water frozen to its container — all 1,055 t moving as one — the tower has one resonance, at 2.1 rad/s, damped by its own 2 per cent. With the water free to slosh, the system has two modes, one with the water swinging against the tower at 1.5 rad/s and one with them together at 2.9 rad/s. The single peak becomes two.

With plain water’s half per cent of damping, both peaks are sharp, and the lower one is 1.83 times as high as the frozen tower’s peak. The water has not damped anything. It has moved the tower’s resonance to two other frequencies and made one of them worse: the lower mode is mostly the water swinging, and the water brings almost no damping of its own to it.

Give the sloshing 10 per cent of damping and both peaks collapse: the worst harmonic shaking now produces 0.21 of the frozen tower’s peak. At 27 per cent, the damping that minimises the response to broadband shaking, it is 0.11, and the curve has no peak worth the name. The water was always a tuned mass. What it lacked was a way to lose energy.

How much damping, and from what #

Earthquake ground motion is not a sine, and the measure that matters for a broad-band shaking is the root-mean-square response: the area under the square of the transfer function, which a sharp peak contributes to in proportion to its height times its width. Under that measure plain water is no better than frozen water — 1.01 times its rms — because the two sharp peaks together have about the area of the single one.

As the sloshing’s damping rises, the rms falls steadily: 0.71 at 2 per cent, 0.51 at 5, 0.40 at 10, and least, 0.34, at 27 per cent, close to Den Hartog’s 23. Above that it rises again, because a sloshing mass damped too heavily is dragged along by the container and stops acting as a separate mass at all; in the limit it is frozen water again. The harmonic peak, which only cares about the worst frequency, wants more damping still, 40 per cent.

Where the damping comes from is the practical question, and it is the one this calculation takes as given. A vertical screen of slotted plate or mesh across the tank makes the sloshing water push through it, and the pressure drop across the screen dissipates energy in proportion to the square of the flow through it. The damping it gives depends on the screen’s solidity and on how large the sloshing is, and the figures here hold it at one value. Ten per cent would take a substantial screen; 27 per cent would want more than one. So the useful reading is the shape of the curve: most of the benefit is in the first ten per cent of damping, and the tower does not need the optimum to get most of it.

What a screen is worth in the tower’s own terms #

A tower designer thinks of damping as a property of the structure, and it helps to translate the screen into that language. For a structure with one mode under broadband shaking, the rms base shear falls as the square root of the damping ratio: quadruple the damping and the rms halves. The screened tank’s 0.40 of the frozen tower’s rms is therefore what the frozen tower would achieve with its own damping raised from 2 per cent to about 12 per cent; the best damping’s 0.34 is worth about 18 per cent.

That is a great deal. Twelve per cent of critical damping in a steel or concrete tower would take a substantial set of viscous dampers, braced into the structure and maintained for its life. Here it is bought with a slotted plate across a tank that is already there, full of a mass that is already moving. It is the same comparison as made weaker on purpose drew for base isolation, which buys its reduction by moving the structure’s period away from the shaking: the screened tank leaves the periods roughly where they were and removes energy instead.

The translation has a limit, and it is the one the transfer function shows. A frozen tower with 12 per cent damping has one broad peak; the screened tower has two broad humps, one for each mode, and most of the mass moves together in neither of them. A response spectrum read at one period, as the earlier essay read it, will give a different answer from the rms, because it asks about one period at a time and the screened tower’s response is spread over two.

Emptying the tank does two things at once #

The difficulty the earlier essay named is that a water tower’s level is not fixed. Every damper designed for a building is tuned to one structure and stays tuned. A water tower draws down by day and refills by night, and its water’s period and mass both change with the depth.

The sloshing slows as the tank empties, because shallow water sloshes slowly: at a tenth full its frequency is 0.16 of the tower’s, far below any useful tuning. But the mass ratio rises at the same time. A deep tank’s water is mostly impulsive, moving with the wall; a shallow one’s is mostly convective, sloshing. As the depth falls, the impulsive part that adds to the tower’s own mass falls faster than the convective part that is the damper, and the ratio between them climbs from 0.40 full to 0.85 at middling fills. A heavier damper is less sensitive to its tuning, because the two peaks it produces are further apart and broader.

The two effects nearly cancel. One screen, giving the same 10 per cent of damping at every depth, keeps the tower’s rms base shear between 0.37 and 0.64 of the frozen-water tower’s from a tenth full to full. Above a third full it is within 41 per cent of the best that fill could have had with damping chosen for it; nearly empty, where the water is so detuned that only very heavy damping would help, it is further from the best but still far below frozen. Plain water, by contrast, sits within a fifth of frozen at every fill: it never helps much at any depth.

So the answer to the earlier essay’s worry is that the changing level matters much less than it seems to. A tank emptied far enough to detune its water has also made its water a larger share of the tower, and a larger damper forgives a worse tuning.

The tower the water suits #

The tuning is set by the tank and the tower together, so the same tank on a different tower is a different damper. The screen does most on a tower near 2.3 s, where the full tank’s water is tuned 0.64 times the tower’s frequency. On a stiff 0.8 s tower the water is far too slow to be tuned, but the screen still takes the rms to 0.62 of frozen, because separating a heavily damped convective mass from the tower takes it partly out of the tower’s mode. On a flexible 4.4 s tower, slower than the sloshing, the water is tuned above the tower and helps least.

Plain water’s line is the earlier essay’s result seen from a different side. On a stiff tower the free water lowers the base shear a little; on a flexible one it raises it, by 12 per cent at 4.4 s. That essay found that adding the ground tank’s numbers overstates the shear; this one finds that the undamped coupling itself can make things worse, and that the coupling is worth having only once the water can lose energy.

A calmer wave, not a taller one #

A tuned-mass damper works by moving: the damper mass swings further than the structure, and its damping dissipates energy in that relative motion. In a tank the relative motion is the slosh, and a sloshing wave that rises too far hits the roof, as the essays on the liquid that has a period of its own worked out for freeboard. It would be a poor damper that needed a taller wave to work.

It does not. With 10 per cent of damping the rms of the wave is 0.34 of plain water’s; at 27 per cent, 0.21. Plain water’s undamped slosh was riding on its own resonance, swinging hugely at the one frequency where it was tuned, and damping it removes that far more than the damper’s extra work adds. The freeboard that a plain tank needs for its sloshing is an upper bound for a screened one.

What the calculation takes for granted #

The screen’s damping is linear and constant. A real screen’s force goes as the square of the flow, so its damping rises with the size of the sloshing, and a screen tuned for a design earthquake gives less in a small one. The damper is only as good as its brace ended on the same problem for a viscous damper sized for one excitation and living through another.

The ground shaking is white noise. A real earthquake’s spectrum is shaped, with most of its energy at periods shorter than this tower’s, and the spectrum is not a load is about why a response computed against a broad-band input is a measure of the structure rather than a design value. The ratios to the frozen tower are the result here, not the base shears.

The sloshing is linear. A wave a few per cent of the tank’s radius high behaves as the linear theory says; a larger one steepens, its period lengthens a little, and near the wall it breaks, which dissipates energy by itself. Plain water in a large earthquake is therefore somewhat better damped than half a per cent, and a screened tank’s damping depends on the wave’s size as well as the screen’s. Both effects move the plain-water line towards the screened one in the largest shaking and leave the small-shaking result — where the plain tank is no damper at all — as it is.

Only the first sloshing mode is a damper. The higher modes are much shorter and carry little mass, and they are left out as the earlier essays left them out.

The tower is one mass on a spring. A tall tower has its own higher modes, and a damper tuned to the first does nothing for them.

Still open: the tower that is also a tank #

The towers here hold a tank at the top. A different structure holds the water in its own shaft — a standpipe, a reservoir built as a tall cylinder, a concrete tower whose upper storeys are the tank — so that the water fills a height comparable with the structure’s own and its sloshing mass is spread along the part of the structure that sways most. Whether a tall column of water inside a swaying shaft is still one tuned mass, whether its sloshing period is set by the shaft’s diameter rather than by any depth the town controls, and whether a screen inside it could make the shaft damp itself as the screened tank damps the tower, is the question the tank at the top of a tower puts to the tank that is the tower.