The mass that helps by being late

The response at resonance is one over twice the damping ratio, and a structure’s damping is a property nobody designed. That leaves two ways out: make the structure so stiff that nothing can excite it, or give it damping it did not have.

The second is far cheaper, and the standard way of doing it is strange enough to be worth the whole essay: hang another mass on another spring, and tune it.

Why an added mass does anything #

The absorber does not absorb. The name is misleading and it is worth dismantling before the arithmetic.

A tuned mass damper is a second mass on a spring, hung from the structure and free to move relative to it. When the structure moves, the absorber moves too — but not in step. Tuned so that its own natural frequency is near the structure’s, at the frequency that matters it lags by about a quarter of a cycle, and its inertia force acts against the structure’s motion.

So the structure feels a force opposing its movement, supplied by something that is merely hanging there. That force is what reduces the response. The energy then has to go somewhere, and it goes into the absorber’s own dashpot, which is stretching and compressing at the difference between the two motions.

Two consequences follow and both are counter-intuitive.

The absorber has to move a great deal. On the 3% absorber above, the relative stroke at the worse of the two peaks is 29.6 times the structure’s static deflection. The absorber travels much further than the structure does, because it is the relative motion that does the work.

The absorber’s own damping must be moderate. With none at all, the absorber is a perfect vibration neutraliser at exactly one frequency and useless at every other — the two-degree-of-freedom system has two undamped resonances of its own, and the response peaks at 55. With too much, the absorber is effectively clamped to the structure and simply adds to its mass. Between them there is an optimum.

The fixed points, which make the problem solvable #

Here is the elegant part, and it is why the optimum can be written down rather than searched for.

Sweep the absorber’s damping from zero to enormous and watch the response curve move. Almost everywhere it changes — but there are two frequencies at which it does not. At those two points every possible absorber damping gives exactly the same response, and their positions depend only on the mass ratio and the tuning.

That is Den Hartog’s observation, and it converts the design into a two-step argument.

First, choose the tuning so that the two fixed points are at equal height — which happens at

$$f_{\text{opt}} = \frac{1}{1+\mu}$$

so a 3% absorber is tuned 2.9% below the structure’s own frequency rather than to it.

Second, choose the damping so that the curve peaks at those points rather than between or beyond them:

$$\zeta_{\text{opt}} = \sqrt{\frac{3\mu}{8(1+\mu)}}$$

which for 3% is 10.5%. The resulting peak is $\sqrt{1+2/\mu}$ — 8.23 for a 3% absorber, against the measured 7.34 for a structure that also has 1% damping of its own.

The reason this is worth admiring is that no calculation of the optimum could have been formulated without the fixed points. They are a property of the system that nothing in the design brief mentions, and finding them made a two-parameter optimisation into two one-line formulae.

What the mass ratio buys #

The returns diminish as $1/\sqrt{\mu}$, which is why real installations cluster between 0.5% and 2% of the modal mass — and why the stroke column matters as much as the response column. A 0.5% absorber gives a factor of 3.3 and needs to travel 143 times the structure’s static deflection, which on a tall building is metres. The mass ratio is chosen by the space available for the absorber to move in, at least as often as by the response required.

Mistuning is the failure mode #

That sensitivity is the reason a tuned mass damper is commissioned rather than designed. The frequency it has to match is the structure’s, and that frequency is not known in advance to better than ten or twenty per cent — the mass depends on occupancy, the stiffness on cracking and on non-structural components, and the end conditions on details nobody drew.

So the practice is: build the structure, measure its frequency, and then tune the absorber to what was measured. Most installations have adjustable stiffness for exactly this, and several have been retuned years later after refurbishment changed the mass.

There is a second consequence, less often stated. A tuned mass damper protects one mode. A structure with two modes close in frequency, or with modes in two directions, needs either a device per mode or an absorber deliberately detuned to cover a band at lower efficiency — which is why a structure’s mode list is the first thing consulted when one is specified.

What it is used for, and what it is not #

The applications are where damping is scarce and the consequence of resonance is discomfort rather than collapse:

Tall buildings under wind, where the resonant part of the response is the part that makes people at the top feel unwell, and where it is inversely proportional to the damping. This is the largest market for the device and the one that produced the enormous pendulums now installed in several very tall towers, hung through four or five storeys and visible to visitors.

Footbridges, both for the vertical bouncing and for the lateral instability — where raising the total damping raises the critical crowd in proportion.

Chimneys and masts, against vortex shedding, where the Scruton number contains the damping linearly and a device weighing a fraction of a per cent can move a structure across the threshold below which the response is negligible.

Long-span floors, where a device of a few hundred kilograms in a ceiling void can fix a response factor that would otherwise need the whole floor redesigned — and where the modal mass is small enough that a useful mass ratio weighs less than the furniture.

What they are not used for is earthquake resistance, and the reason is instructive. An absorber needs several cycles at a steady frequency to develop its relative motion, and a strong-motion record does not offer that: the largest demands arrive in the first few cycles, before the absorber has begun to move. Devices for seismic use are therefore ones that work instantly — dissipating through yielding, friction, or viscous fluid — rather than by tuning.

Where the idea came from #

The device is Hermann Frahm’s, patented in 1909 for reducing the rolling of ships — a tank of water arranged so that its own sloshing opposed the roll — and it arrived in structures much later. Den Hartog’s analysis, in Mechanical Vibrations in the 1930s and 40s, is what turned it from a trick into a design procedure by finding the fixed points and the optimum that follows from them.

The route it took is worth a sentence, because it is a common one in this subject. The problem was solved first in a field where the excitation is regular and known — ships, engines, machine tools — and imported into structures once structures became light enough to need it. The same is true of the response spectrum, which came from seismology, and of the gust factor, which came from communications theory.

The variants, and one that has no spring #

The physical realisations vary more than the theory does.

A pendulum rather than a spring — a mass hung on cables, whose frequency is set by the length. Simple, adjustable by changing the length, and the standard for very tall buildings, where a pendulum of the required period is naturally several storeys tall.

A liquid column in a U-shaped tank, where the mass is water and the tuning is set by the tank’s geometry. It needs no bearings and cannot seize.

A sloshing tank, where the water’s own surface wave is the tuned mode. The tuning is fixed by the tank’s length and the water depth, and the damping by baffles. A building that needed a water tank anyway can have one that is also a damper, which is the cheapest version of this idea there is.

The last two are worth noticing because they have no moving mechanical parts at all: the absorber is water in a box, tuned by the shape of the box.

What the picture cannot show #

The curves are steady state. They assume a harmonic force applied indefinitely, which is right for a machine and approximately right for wind. Under a transient — a gust, a bang, an earthquake — the absorber’s benefit is much smaller, because it has not started moving.

The absorber is drawn as free. It has end stops, and a device that hits them stops being tuned and becomes an impact. Sizing the stroke for the design event, and deciding what happens beyond it, is most of the engineering.

Nothing here draws the structure. The absorber’s mass is applied at one point — the roof, usually — and its effectiveness depends on the mode shape at that point. A device at a node of the mode it was meant to damp does nothing at all, which has happened.

The mass ratio is modal, not total. The 3% is three per cent of the effective modal mass of the mode being damped, which for a first mode is around 85% of the building’s mass and for a second mode is a tenth of it. An absorber sized as a percentage of the whole building is oversized for the first mode and hopelessly undersized for the second, and the participation table is where the right denominator is found.

Measured as added damping #

The last figure suggests the way absorbers are actually specified, and it is worth making explicit because it connects this rung to every other one in the field.

An absorber’s effect on the peak response can be quoted as the equivalent damping ratio the structure would need to achieve the same peak without one. A 3% absorber that takes the peak from 50 to 7.3 is worth about 5.8 points of damping — a structure at 1% behaving like a structure at 6.8%.

That is a convenient number for two reasons. It goes straight into every other calculation in this field: the resonant part of a wind response, the critical crowd on a footbridge, the Scruton number for a chimney, the response factor for a floor. And it is measurable on the finished structure by exactly the test that measures anything else — displace it, release it, and watch the decay.

The catch is that the equivalence holds at one frequency. A structure with an absorber has a shape of response quite unlike a structure with more damping — two peaks rather than one, and a much narrower band of benefit — so quoting an equivalent damping is a summary rather than a substitution, and it is a summary that is only accurate near the tuning.

Where the ladder goes #

Two directions, and they run in opposite senses.

Upward in sophistication: an active system replaces the tuned spring and dashpot with an actuator that pushes the mass wherever a controller says. It is not limited by tuning, can protect several modes, and introduces a failure mode no passive device has — a control system that adds energy at the wrong moment.

Downward in ambition, and closer to home: a tuned mass damper is a second degree of freedom added on purpose, and everything the field says about two-degree-of-freedom systems applies to it. The absorber and the structure are two coupled oscillators, and the two peaks in every figure above are the two modes of the combined system. Nothing new is happening — the system has been given a second mode, arranged so that both of them are worse at resonating than the one it replaced.

That reading is worth keeping, because it is the honest one. A tuned mass damper is not a mysterious device that removes vibration; it is a deliberate second degree of freedom, whose coupling to the first is arranged so that neither can respond strongly on its own. Everything else in this field has been about accepting the modes a structure happens to have. This is the one rung where they are chosen.