Structural Engineering & Statics
where the load goes
Concept

Mass ratio — where it appears

A tuned-mass damper's mass as a share of the mass of the structure it damps. It sets both the best tuning and the best damping, and a larger ratio makes the damper less sensitive to being slightly mistuned.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

3% of the mass, hung on a spring, against the peak it removes. The magnification of a structure with 1.0% damping, with and without a tuned mass damper of 3.0% of its mass, tuned to 0.9709 of its frequency with 10.5% damping of its own. The bare peak is 50; with the absorber the single peak becomes two of 7.34, a reduction to 15% — a factor of 6.8. The marked points at frequency ratios 0.923 and 1.044 are the fixed points: the response there is the same whatever damping the absorber is given, which is what makes the optimum a question with an answer. The absorber's own stroke at the worse of the two peaks is 29.57 times the structure's static deflection, and that stroke is what decides whether it fits.

The mass that helps by being late

Hang three per cent of a building's mass from a spring in its roof, tune the spring so the mass arrives a quarter-cycle behind the motion, and the peak response falls by a factor of seven. Nothing was strengthened and nothing was stiffened.

dynamics · Tuned mass damper The water splits the tower's resonance, and a baffle flattens it. For a 12 m tank 8 m deep on a tower of period 2.5 s with 2% damping, 150 t of container, full: the base shear per unit of harmonic ground acceleration against the shaking's angular frequency, as a share of the frozen-water tower's peak. Frozen to its container, the water makes one resonance; free, it is a second mass on a spring tuned 0.69 times the tower's frequency, against Den Hartog's optimum of 0.71, and it splits the resonance in two — with half a per cent of damping both peaks are higher than the frozen one, 1.83 times it. A baffle giving 10% brings the peak to 0.21; the best damping for the rms, 27%, to 0.11.

The water tower that could damp itself

A full water tower's sloshing water is a second mass on a spring, hung from the container at very nearly the tuning a designed tuned-mass damper would be given, and it weighs two-fifths as much as what it hangs from. Plain water has half a per cent of damping, so it splits the tower's resonance into two sharp peaks and helps nothing. A screen across the tank that gives it a tenth makes the water a damper: the tower's rms base shear falls to four-tenths of what it would be with the water frozen, and the same screen keeps working as the town empties the tank.

dynamics · Sloshing Pumping water in or out keeps the damper, and the lost water barely matters. The peak amplification of a building with 1 per cent damping and a 10 m tank sized for a 5.0 s building, 1.73 m of water weighing 2.0 per cent of the modal mass, its sloshing damped at 7 per cent, against the period the building actually has: the tank left at its design depth (dashed), retuned by its water level with the water's mass following the depth (solid), and retuned with the mass held at the design value (light); dotted, the design period; bare, the building would reach 50. At 4.5 s, 21.7 left alone and 9.6 retuned; at 5.5 s, 22.6 and 10.8; at 6.0 s, 34.0 and 11.7, of which the water let out costs 1.7.

The tank that follows the building

A sloshing damper is tuned to a period computed before the building exists, and the building that is built has another one — stiffer at handover, with its partitions and cladding working, softer once its concrete has cracked. A tank whose plan is fixed and whose period is set by its depth can be brought back by opening a valve. Ten per cent off tune, the tank left alone lets the building's peak response more than double; pumped back to the right depth, it holds within a tenth of the design value, and the water it gains or loses barely matters. What does matter is that a slower building needs shallower water, and in shallow water the wave the damper is made of grows towards the depth it is sloshing in.

dynamics · Sloshing

Named alongside it

The objects these essays reach for when they reach for this one.

DampingResonanceTuned mass damperConvective massSloshingDetuningFixed pointsImpulsive massMode shapeNatural periodServiceabilitySpectrum

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