The damper is only as good as its brace

The only thing that stops it put the damping ratio at the centre of structural dynamics: the response at resonance is one over twice it, every forced vibration is limited by it, and it is the one parameter of the whole subject that nobody designs, specifies or measures before the structure is built. It ended on the exception — a structure whose own damping is too low can be given more, deliberately, in a device sized for the purpose.

The commonest such device is a viscous damper: a piston in a cylinder of silicone oil, resisting with a force proportional to the speed it is pushed at, fixed across a storey on a diagonal brace or a chevron. Its coefficient is chosen, specified, tested and certified. And what it adds to the structure is not set by its coefficient alone, because the damper does not act on the structure directly. It acts through the brace that holds it, and the brace is a spring.

A dashpot in series with a spring #

The structure is reduced to its essentials: a mass on a spring, with a little damping of its own — 2% — and a natural frequency $\omega_0$. Across it is a brace of stiffness $\kappa$ times the structure’s, and in the brace a damper of coefficient $c$. The damper and the brace are in series: the force through them is the same, and the storey’s movement is shared between the brace’s stretch and the damper’s stroke.

The damper’s size is best measured against the brace: $x = c\,\omega_0/k_{\text{brace}}$, the force the damper would make at the structure’s own frequency over the force the brace would make for the same movement. A small $x$ is a damper that strokes freely while the brace barely stretches; a large $x$ is a damper that resists so hard that the brace does the stretching instead.

Every curve has a peak. With a brace as stiff as the structure, the damping rises from 2% with no damper to 22.7% at a damper of size 0.58, and then falls. At ten times the best size it is back to 4.4%; with the damper’s force a hundred times the brace’s, 1.6% — less than the structure had on its own. A damper too large stops damping.

Why too large is too small #

The damper dissipates energy in proportion to its force times its stroke. A small damper strokes freely but resists weakly, so its force is small. A large damper resists so strongly that it hardly strokes at all: the brace beside it stretches instead, and a brace stores energy and gives it back. Between the two is the size at which force and stroke together are largest, and that is the peak.

At the large end the damper has locked. The brace, with an immovable damper in it, is just a strut: it adds stiffness, $\kappa$, to the structure and nothing else. The structure is now stiffer — its frequency rises to $\sqrt{1 + \kappa}$ times its own — and its own damping, a fixed coefficient on a stiffer spring, is a smaller ratio of critical than it was. That is why the locked damper leaves the structure with less damping than no damper at all.

The frequency is the clearest symptom of which side of the peak a damper is on. A damper doing its job raises the frequency part of the way, to $(1 + \kappa)^{1/4}$ — 1.189 here. A damper that has locked raises it all the way to $\sqrt{1+\kappa}$. A measured frequency close to the second, on a structure fitted with dampers, is a measurement that the dampers have stopped working — seized, overfilled, or simply specified too large.

Where the energy goes in one cycle #

The peak can be seen in the energy. In each cycle of motion at amplitude $X$ the damper dissipates $\pi c\,\omega\,u^2$, where $u$ is its own stroke, and the stroke is what is left of $X$ after the brace has stretched. The brace’s stretch and the damper’s stroke carry the same force, so they are in the ratio of the damper’s resistance to the brace’s, which is $x$: of the storey’s movement, a share $1/\sqrt{1+x^2}$ is stroke, in magnitude, and the rest is brace.

So a damper twice the best size strokes noticeably less, and its larger coefficient does not make up for it; one ten times the best barely strokes at all, and the energy it dissipates — its coefficient times the square of a stroke that has almost vanished — falls as $1/x$. At the best size, about 0.6 for a brace as stiff as the structure, the damper takes most of the movement and resists it as hard as it can while still taking it. The ratio $x$ is the whole of the design: it is the damper’s coefficient made dimensionless by the brace that has to hold it.

The ceiling the brace sets #

The peak is a ceiling, and the brace’s stiffness alone decides its height.

The exact ceiling, solved from the cubic that governs the structure, the brace and the damper together, is

$$\zeta_{\max} = \frac{\sqrt{1 + \kappa} - 1}{2},$$

added to the structure’s own, reached with a damper of size $x = (1+\kappa)^{-3/4}$ at a frequency $(1+\kappa)^{1/4}$. A brace a tenth as stiff as the structure allows 2.4%, which is hardly worth fitting a damper for. A quarter as stiff, 5.9%. As stiff as the structure, 20.7%. No damper, however large or well made, adds more than its brace allows.

The usual estimate takes the loss factor of the damper and brace at the structure’s original frequency and halves it, $\kappa/4\sqrt{1+\kappa}$. For a soft brace it is the exact answer’s first term and agrees with it. For a stiff one it is low — 17.7% against 20.7% at $\kappa = 1$, and 44.7% against 61.8% at 4 — because the damper raises the frequency, and at the raised frequency it works harder than the estimate read at the original one. The estimate errs on the safe side, which is the side estimates should err on, but it can cost a designer a stiffer brace than the damper needed.

Turned round, the ceiling is a requirement on the brace. Ten per cent of added damping needs $\kappa = (1 + 2\times 0.1)^2 - 1 = 0.44$; twenty per cent needs 0.96. A diagonal brace in a storey of a steel frame is easily as stiff as the frame’s columns in sway; a brace on a long, slender bracket off a floor slab, or a damper hung from a flexible outrigger, is not, and its damper is limited before it is chosen.

What the response sees #

The damping ratio is a summary. What it summarises is the response to a vibration at the structure’s frequency, and the three dampers produce three very different responses.

The small damper leaves a resonance peak of 7.2 at the structure’s own frequency. The best one flattens it to 2.0, spread across a band of frequencies around 1.19. The large one has a peak of 7.9, as sharp as the small damper’s, at $\sqrt 2$ times the original frequency: it has not removed the resonance, only moved it. A structure excited by footfall, by wind or by a rotating machine at its original frequency would be relieved by the locked damper, by detuning; one excited across a band, by an earthquake or turbulent wind, would find the new resonance as easily as the old.

The free decay says the same thing in time. The best damper stops the structure in a cycle and a half; the small one takes five; the locked one takes eleven and a half, oscillating faster all the while.

A soft brace #

With a brace a quarter as stiff as the structure the ceiling falls to 5.9%, and the picture is the same one at a smaller scale.

The best damper brings the peak from 15.4 to 6.0 — useful, but a factor of two and a half where the stiff brace gave a factor of three and a half — and the large one again leaves a peak nearly as high as the small one’s, slightly moved. The brace is a quarter as stiff and the damper is still a good damper; what has gone is the brace’s ability to make the damper stroke.

Where in the storey the damper goes #

A damper on a diagonal brace acts along the diagonal, and the storey moves horizontally. Both the damper and the brace see the storey’s drift multiplied by the cosine of the diagonal’s angle, so both their contributions to the storey are multiplied by its square: the brace’s stiffness in sway is $k_{\text{axial}}\cos^2\theta$, and the damper’s coefficient in sway is $c\cos^2\theta$. The ratio $x$ is unchanged by the angle; $\kappa$ is not. A diagonal at 45° delivers half its axial stiffness to the storey, and the ceiling is set by that half.

A chevron arrangement mounts the damper horizontally between the apex of an inverted V and the floor beam above, so the damper sees the full drift — but the chevron’s two braces are now the connection, and their stiffness is in series with the beam they meet at, which bends under the force the braces leave behind. Toggle and scissor arrangements use geometry to amplify the damper’s stroke beyond the storey’s drift, which raises its effective coefficient without a larger damper — and puts more slender members, and more pins with play in them, in series with it.

Each arrangement is a different $\kappa$ for the same steel. The choice between them is a choice of where the series spring is and how stiff it can be made, which is the ceiling.

Tested on a rigid rig, installed on a soft one #

The damper’s coefficient is a certified number because it is measured: each damper, or a sample of them, is driven back and forth on a test rig at specified amplitudes and frequencies, and its force is recorded against its velocity. The rig is built to be far stiffer than the damper, so that all the movement is the damper’s stroke — on the rig, $\kappa$ is effectively infinite, and the coefficient measured is the damper’s alone.

In the building the same damper sits on a brace whose $\kappa$ is of order one. The certified coefficient is correct and the damping it delivers is a fraction of what the coefficient alone would suggest, by an amount the test cannot show. A specification that names the coefficient and leaves the brace to someone else’s drawing has specified the half of the pair that was easiest to measure.

Fluid dampers of this kind came to buildings from the shock absorbers of military and aerospace equipment, where they were mounted on airframes and gun carriages stiff enough that the question did not arise. In a building the mounting is part of the structure, designed by a different calculation from the damper’s, and the series spring is the price of putting a machine part into a frame.

The same ceiling elsewhere #

The structure of this result recurs wherever a damper is connected to what it damps through something elastic.

A damper near the end of a stay cable can add at most half its distance from the anchorage over the cable’s length, however large it is, because the cable near the anchorage barely moves and a damper there barely strokes; the cable’s own stiffness plays the brace. A tuned mass damper has its own ceiling set by its mass ratio, and its damper too has an optimum beyond which it locks the mass to the structure and the damper becomes useless. A link between two buildings has the same optimum and the same locking, with the second building as the brace.

The common element is a series connection: the damper’s force has to pass through something that deflects under it, and the deflection is stroke the damper does not get. In every case the size that maximises damping is the one at which the damper’s resistance and the connection’s are of the same order, and in every case a larger damper is a stiffer connection rather than a better damper.

The ceiling, by hand #

At the best size for a brace as stiff as the structure, $x = 2^{-3/4} = 0.595$, so the damper’s coefficient is $c = 0.595\,k_{\text{brace}}/\omega_0$. For a storey of stiffness 50 kN/mm with a brace of the same stiffness and a period of 1 s, $\omega_0 = 6.28$ rad/s and $c = 0.595 \times 50{,}000/6.28 = 4{,}700$ kN·s/m — a damper of ordinary size for a building, and one that adds $(\sqrt 2 - 1)/2 = 20.7\%$ of critical damping.

The same damper on a brace a quarter as stiff is $x = 0.595 \times 4 = 2.4$, well past that brace’s best size of $1.25^{-3/4} = 0.85$: it adds about 3%, of a possible 5.9%. Choosing the damper first and the brace afterwards is how that happens.

Linear, one mode, one storey #

The model is the simplest that contains the series connection, and real dampers depart from it.

A damper is not a friction device. A friction damper caps its force and dissipates energy only once its slip load is reached, so its optimum is a slip force rather than a coefficient, and below that force it is a strut. The viscous damper’s virtue is that it acts at any amplitude; its vice is the series spring it needs.

Fluid dampers are often not linear. Their force goes as the velocity to a power below one — 0.3 to 0.5 is common — which caps the force at high speed and changes the optimum; the locking at large size remains.

The coefficient moves with the oil. A damper’s fluid is chosen for a viscosity that changes little with temperature, but little is not nothing, and a damper on an exposed bridge sees a range of tens of degrees in a year. Near the best size the damping curve is flat at its top, so a coefficient that drifts by a fifth either way costs little; well past it, on the locking side, the same drift moves the damping much more — which is one more reason to size a damper for its peak and not for its force.

A building has many storeys and modes. Each storey’s damper adds damping to each mode in proportion to how much that storey drifts in it; a damper in a storey that a mode barely moves does nothing for that mode, and the dampers’ sizes are distributed up the height accordingly.

The connection is more than the brace. The gusset plates, the bolts’ slip, the floor diaphragm and the column’s own shortening are all in series with the damper, and every one of them is a softer brace; a measured brace stiffness well below the drawn one is usual, and it lowers the ceiling.

Earthquakes are not harmonic. A structure on the ground’s spectrum benefits from added damping through a spectrum that falls as damping rises — and, for a structure whose design accepts yielding, from a smaller displacement demand — and loses from the stiffening that raises its frequency into a stronger part of the spectrum. The best damper does both, a little; the locked damper does only the second.

Still open: the damper that is sized for the earthquake and fails the wind #

A building’s dampers are sized for one excitation and live through another. Sized for a design earthquake, a nonlinear damper is near its best at large velocities; under everyday wind, at velocities a hundred times smaller, a damper whose force goes as a low power of velocity is relatively much stiffer — nearly locked — and adds little damping to the motion that occupants feel. Whether one set of dampers can be at its best for both, or whether a building that needs damping for comfort and for safety needs two kinds of device sized against two different braces, is the question of what a damper’s size should be measured against when the motion it damps ranges over two orders of magnitude.