The tank that must be shallow to be slow

The tank that is its own tower found that a standpipe’s water cannot be tuned to its shaft. Only a lens near the surface sloshes, that lens stops growing once the water is deeper than a radius, and so its period is set by the width alone: 3.3 s, against a shaft that sways at 0.38. The standpipe was far too stiff for its own water. It ended on the opposite case: a tall building, slender and slow, with a tank in its core whose width is chosen rather than inherited. Whether a deep, narrow tank can be tuned to such a building by its width alone — since its depth cannot help — was the question.

It cannot, for a reason that sits in one equation, and the way it fails decides what a tuned liquid damper in a building has to be.

A tank is a pendulum whose length is its width #

A tuned liquid damper is a mass that helps by being late: a mass on a spring, hung from the structure, and the split between the water that does that and the water that only rides along is the period the liquid has of its own; tuned to sway at nearly the structure’s frequency so that it moves against the structure and draws energy out of it. In a tank the mass is the water that sloshes and the spring is gravity acting on its tilted surface. For a rectangular tank of length $L$ in the direction of sway, holding water $h$ deep, the first sloshing mode has

$$\omega_1^2 = \frac{\pi g}{L}\tanh\frac{\pi h}{L}.$$

When the water is deep the hyperbolic tangent is one and the period is $2\pi\sqrt{L/\pi g}$: the depth has dropped out, and the width sets everything. The water swings like a pendulum whose length is proportional to the tank’s width. A wider tank is slower; nothing else is available.

A tall building’s first mode has a period of roughly a tenth of a second per storey, so a sixty-storey tower sways at five or six seconds. Its core, which has to hold the lifts, stairs and risers as well as any tank, might give a tank 10 m in the direction of sway. Deep, such a tank sloshes at 3.58 s. To reach a 5 s building a deep tank would have to be 19.5 m long; to reach an 8 s one, 50 m — longer than many towers are wide. A deep tank narrow enough to fit in a core is too fast for the building, and a deep tank slow enough is too wide to fit. Width alone cannot tune it.

The depth has to do it #

The equation’s other term is the escape. Make the water shallow and the hyperbolic tangent falls below one, the spring softens, and the period lengthens. A 10 m tank can be made as slow as any building needs by giving it less water.

Read the other way, the figure says something less obvious. Depth can only slow a tank down; deep water is the fastest a tank of that width can ever be. So a tank’s width sets a floor under the periods it can reach. A 10 m tank can serve any building slower than 3.58 s and none faster; a 6 m tank any slower than 2.77 s. For a stiff, fast building the only way to raise a tank’s frequency is to make it narrower — to divide the water into cells, which is the cure the standpipe essay found absurd for a reservoir and which is entirely ordinary for a damper. A 3 s building, say, cannot be served by the 10 m tank at all; split it by a wall across the middle into two tanks 5 m long, or into cells 6 m long, and each cell, deep at 2.8 s or so, can be slowed to 3 s by its depth — 2.43 m of water in a 6 m cell, with 54 per cent of it sloshing. The wall costs nothing but the wall, and it is how a tank too wide for a stiff building is made narrow enough.

For a slow building, the width is fixed by the core and the depth does the tuning: 1.80 m of water to slosh at 5 s, 1.18 m at 6, 0.65 m at 8.

That is why the tuned liquid dampers built into tall buildings are shallow tanks. It is not a choice of style. In a tank narrow enough to fit, only shallow water is slow enough to be tuned.

Shallow water does more of the work #

Making the tank shallow has a second consequence, and it is a gift. In deep water only a lens near the surface sloshes; the rest moves with the walls and adds mass to the structure without doing anything. In shallow water almost all of it sloshes.

The share that sloshes is $8\tanh(\pi h/L)/(\pi^3 h/L)$, which tends to $8/\pi^2$, 81 per cent, as the water gets shallow. Tuned to a 5 s building, 73 per cent of the water in the 10 m tank sloshes; to an 8 s one, 80 per cent. A deep tank 10 m long and 10 m deep would slosh only a quarter of its water, and the standpipe’s lens was 8 per cent of its column. The slower the building, the shallower the tank and the more of its water is a damper rather than a load.

For the damper that is the number that matters. The mass ratio that sets a tuned damper’s effectiveness is the moving mass over the structure’s modal mass, and every tonne of water that moves with the walls is a tonne the floor carries for nothing. A tank holding water weighing 2 per cent of the building’s modal mass, tuned to 5 s, gives a sloshing mass of 1.5 per cent.

What the tuned tank does #

With the sloshing mass known and its frequency set, the tank is a tuned mass damper and can be judged as one. Den Hartog’s tuning sets the damper slightly below the building, at $1/(1 + \mu)$ of its frequency, which here is 0.98; that asks for 1.73 m of water rather than the 1.80 m that would match the building exactly.

The bare building, with one per cent of damping, amplifies a resonant force fifty times — the arithmetic of the only thing that stops it, which is that a resonance is limited by its damping and nothing else. The tank, tuned, splits that resonance into two. With plain water, whose own sloshing damping is about half a per cent, the two peaks are 34 each: the tank has moved the resonance rather than removed it, which is exactly what an undamped tuned mass does. With the sloshing damped at 7 per cent the two peaks merge into a flat-topped hump of 10 — a fifth of the bare building’s response.

The damping is not an accessory. Below a per cent the tank does little; the best response needs about 7 per cent of critical damping on the sloshing, which is what the water tower that could damp itself found a screen across a tank can give. Too much damping is as bad as too little: a sloshing mass damped heavily is locked to the walls, and at 30 per cent the tank has become dead weight again. A tuned liquid damper is three decisions — the width the core allows, the depth that tunes it, and screens or baffles that damp it — and the first two cannot be made separately.

The plan a shallow tank needs #

The depth that tunes the tank has a price in plan. The damper’s effectiveness is set by its mass ratio, so the water it needs is fixed by the building rather than by the tank; a shallower tank must spread the same water over more floor. Take a tall building whose first mode has a modal mass of 40,000 t and give it water weighing 2 per cent of that, 800 t. Tuned to 4 s, in 3.29 m of water, the tank covers 243 m² — two and a half tanks 10 m square. Tuned to 5 s, in 1.73 m, 462 m², nearly five. Tuned to 6 s, 699 m²; to 8 s, in 0.63 m of water, 1,280 m², thirteen tanks 10 m square, on floors at the top of the building that are among its most valuable.

The sloshing mass barely changes across that range — 486 t at 4 s, 640 t at 8 — because the shallower water uses more of itself. What changes is the plan. A slower building needs a shallower tank, and a shallower tank needs a larger plan for the same mass, faster than the square of the period over the range drawn.

The two formulas combine into one that says exactly where the cost comes from. The sloshing mass is $\rho L B h$ times its share, $8\tanh(\pi h/L)/(\pi^3 h/L)$, and the tuning fixes $\tanh(\pi h/L) = \omega^2 L/\pi g$. Put together, the sloshing mass per square metre of tank is

$$\frac{m_1}{LB} = \frac{8\rho\,\omega^2 L^2}{\pi^4 g},$$

which contains no depth at all. For a building of a given frequency, a tank’s useful mass per square metre is set by its length in the direction of sway, and grows as its square: 1.32 t/m² for a 10 m tank on a 5 s building, 0.52 t/m² on an 8 s one, four times as much for a tank twice as long. The core’s width does not stop the tank being tuned; it limits how much damper each square metre of tank can hold, and the slower the building, the less. That is why the dampers of very tall, slow towers are so often solid masses on pendulums instead: a pendulum’s length tunes it at any mass, and its mass sits in a block rather than in a lake.

The wave the damper is made of #

Everything so far is linear: the sloshing frequency, the share of water that moves and the damping are all derived for a wave small against the water’s depth. A tuned damper works by moving a lot, and the question is how much.

The convective mass’s motion relative to the tank makes a wave at the wall of $(4/\pi)\tanh(\pi h/L)$ times that motion. In the tuned tank its relative motion is several times the building’s, which is the point of tuning. So the wave can be estimated from the building’s own motion, and a building’s motion is specified as an acceleration: comfort criteria for tall buildings are written in thousandths of g, and a residential tower might be asked to keep its peak below 10 to 20 thousandths of g in a storm that comes once a year or once a decade.

On a 5 s building at 20 thousandths of g, the linear wave is a quarter of the water’s depth. On an 8 s building it is seven tenths, and it passes the depth itself at about 28 thousandths. Two things drive it. A slower building moves further for the same acceleration — its displacement is the acceleration over the square of its frequency — and a slower building needs shallower water. The wave grows as the depth shrinks.

The linear theory describes a wave small against its depth, and the damper asks for one comparable to it. Long before the ratio reaches one, shallow water stops behaving like the mass on a spring the tuning assumed: the wave steepens into a travelling bore that runs from wall to wall, the period moves with the wave’s height, and the water starts to break. A breaking wave dissipates energy, which is why some shallow tanks need no screens at all; but it does so at amplitudes the linear model cannot see and with a frequency the linear model did not choose. A tuned liquid damper in a slow building is designed, in practice, with nonlinear sloshing models calibrated on shaking-table tests, because the linear one is wrong precisely where the damper does its work.

The tank by hand #

The 10 m tank’s deep period is $2\pi\sqrt{10/(\pi \times 9.81)} = 3.58$ s. To slosh at 5 s it needs $\tanh(\pi h/10) = (2\pi/5)^2 \times 10/(\pi \times 9.81) = 0.512$, so $\pi h/10 = 0.566$ and $h = 1.80$ m. At that depth the sloshing share is $8 \times 0.512/(\pi^3 \times 0.180) = 0.73$.

A deep tank slow enough has $L = gT^2/4\pi$: $9.81 \times 25/12.57 = 19.5$ m for 5 s, and $9.81 \times 64/12.57 = 50$ m for 8 s.

At 20 thousandths of g a 5 s building moves $0.196/(2\pi/5)^2 = 0.124$ m. At the response peak the tuned tank’s sloshing mass moves 5.8 times that relative to its walls, 0.72 m, and with $\tanh(\pi \times 1.73/10) = 0.50$ the wave at the wall is $(4/\pi) \times 0.50 \times 0.72 = 0.45$ m: on 1.73 m of water, the figure’s 0.26.

One mode, one tank, linear water #

The building is one mode. A tall building’s second mode is several times faster than its first, and a tank tuned to the first does nothing for it. A building whose comfort is governed by torsion, or by a second mode in the across-wind direction, needs a tank aimed at that.

The tank is two-dimensional. A rectangular tank in a building sloshes in both plan directions, at frequencies set by its two lengths; a square tank serves both directions of a square tower with one depth, and an oblong tank tunes one direction and not the other. A tank that has to serve both directions of an oblong tower needs its depth chosen as a compromise or its plan divided.

The excitation is harmonic. Wind on a tall building is a spectrum rather than a single frequency, a broadband load whose resonant part the damper attacks, and the response is a random process whose peaks are a few times its standard deviation. The comparison between bare and damped holds for that too, as it did for the water tower; the wave estimate is for the peaks.

The walls are rigid and the water is a potential flow. The tank’s walls move with the building and do not bend, and the water has no viscosity of its own beyond the half per cent that plain water’s damping stands for; the boundary layers at the walls and floor of a shallow tank are a larger share of its depth than of a deep one’s, which adds a little damping the model does not count.

Everything is linear, and the last figure is there to say how far that holds. The model’s frequencies, masses and damping are right for small waves and wrong for the waves the damper makes in the storms it was built for.

Breaking water and slatted screens #

They cannot show the water breaking, which is where a shallow tank’s damping really comes from at large amplitude, and they cannot show the screens: slatted panels across the tank whose damping depends on the flow through them and therefore, again, on the amplitude. They cannot show the tank’s own weight in the core — 2 per cent of a tall building’s modal mass is several hundred tonnes of water at the top of the structure, carried by the core in every load case including the ones in which the tank is doing nothing. And they cannot show the alternative a designer is always weighing against it: the switch that hands the mass back and the other active and passive solid-mass dampers, which tune with a spring and a pendulum length rather than a depth, and which do not break.

Still open: the tank that is tuned by being emptied #

A shallow tank’s period depends on its depth so steeply — from 5 s to 8 s as 1.73 m of water falls to 0.63 m — that a tank can be retuned by pumping water in or out. A building whose period drifts — stiffening as its cladding goes on, softening as its concrete cracks, lengthening as its occupants’ furniture arrives — could keep its damper tuned by adjusting a valve. Whether a tank tuned by its water level can follow a building through its first years, whether the mass ratio lost by draining it costs more than the detuning it repairs, and how the wave, already near the depth on a slow building, behaves in the shallower water that slower tuning needs, is the question a damper whose spring is its own depth leaves for a building that does not keep still.