The damper sized for the earthquake

The damper is only as good as its brace found a ceiling. A viscous damper fixed across a storey on a diagonal brace is a dashpot in series with a spring, and the pair can add at most $(\sqrt{1+\kappa}-1)/2$ of critical damping to the structure, $\kappa$ being the brace’s stiffness over the structure’s: 20.7% for a brace as stiff as the frame. That ceiling is reached at one damper size. A smaller damper strokes freely and resists too weakly; a larger one resists so hard that it barely strokes, the brace stretches instead, and the damper has become a strut.

That calculation took the damper to be linear: its force proportional to the speed it is pushed at. Most dampers installed in buildings for earthquakes are not. Their orifices are shaped so that the force grows as a low power of the velocity, $F = C\,|v|^{\alpha}$ with $\alpha$ between about 0.2 and 0.5, and the reason is a good one. A larger earthquake than the design one drives the damper faster, and a linear damper’s force grows in proportion, while a damper of $\alpha = 0.3$ gives only $3^{0.3} = 1.39$ times its design force at three times its design velocity. The brace, the gusset, the bolts, the column and the foundation beneath all carry that force. Capping it caps all of them.

The exponent that caps the force at the top of the range has a consequence at the bottom, and it is the one that occupants notice.

A size that depends on how far the damper moves #

A damper that is not linear has no single coefficient, but over one cycle of steady motion it dissipates some definite energy, and a linear damper of some coefficient would dissipate the same. Equating the two gives the damper’s equivalent coefficient at that stroke amplitude $U$ and frequency $\omega$:

$$c_{\text{eq}} = \lambda_\alpha\, C\,(\omega U)^{\alpha - 1}, \qquad \lambda_\alpha = \frac{2^{2+\alpha}\,\Gamma^2(1+\alpha/2)}{\pi\,\Gamma(2+\alpha)}.$$

The constant $\lambda_\alpha$ is 1 for a linear damper and about 1.17 for $\alpha = 0.3$; it is a matter of the shape of the force over a cycle and changes little. What matters is the power. With $\alpha$ below one, $\alpha - 1$ is negative, so the smaller the stroke, the larger the damper. A damper of $\alpha = 0.3$ moving a tenth as far behaves as one $10^{0.7} = 5$ times the size.

In the series-damper calculation, size was everything. The damper’s size measured against its brace, $x = c\,\omega_0/k_{\text{brace}}$, decided where on the curve of damping it sat — on the rising side, at the peak, or on the locking side past it. A nonlinear damper does not sit at one point on that curve. It moves along it with the amplitude of the motion, and the direction it moves in for small motion is towards locking.

There is one more coupling. The stroke is not the storey’s movement; it is the storey’s movement less what the brace stretches, and the larger the damper the more of the movement the brace takes. So a smaller storey motion makes the damper larger, which makes it stroke even less of that motion, which makes it larger still. The size at each amplitude has to be found as a fixed point — the size that produces the stroke that produces that size. It always exists and is unique, because the stroke never falls faster than the size grows; the solution here is found by bisection at every amplitude.

The structure is the one the brace calculation used: a single storey with 2% damping of its own, a brace as stiff as the storey, and a damper sized so that at the design amplitude it is exactly at the brace’s best, $x = 2^{-3/4} = 0.59$. The design amplitude is the earthquake’s, and every other motion is measured against it.

All four dampers give 22.7% at the earthquake’s amplitude: the structure’s own 2% and the brace’s ceiling of 20.7% on top. The linear damper gives 22.7% at every amplitude, from a thousandth of the earthquake’s to ten times it, because its size does not depend on its stroke. The nonlinear ones fall away on both sides, and they fall fastest on the side of small motion. At a tenth of the design amplitude the damper of $\alpha = 0.3$ leaves the structure with 1.9% of critical damping. That is less than the 2% it had with no damper at all.

Why small is the losing direction #

The curves are not symmetric about the design amplitude, and the reason is in the sizes. Above the design amplitude the damper strokes further, its equivalent size falls, and it slides down the rising side of the brace’s curve towards a small damper — one that strokes freely and adds too little. That side of the curve is gentle: a damper a tenth of its best size still adds about a third of the ceiling. Below the design amplitude it climbs the locking side, and the locking side has no floor. Every factor on the size past the best takes damping away, until the damper is rigid and the brace is a strut.

The marks are the same damper. At three times the design amplitude it is a damper of size 0.27 and gives 15.0%; at the design amplitude, size 0.59 and 22.7%; at three-tenths, size 2.95 and 7.4%; at a tenth, size 36; at three-hundredths, 594. The step from three-tenths to a tenth of the motion multiplies its size by twelve, not by $3^{0.7} = 2.2$, because of the fixed point: a damper that has begun to lock strokes a smaller share of a smaller motion, and the two shrinkages compound. Once the damper is on the locking side, a fall in amplitude accelerates its own effect.

The damping it is left with, 1.4%, is not the structure’s own 2% either. It is the structure’s own damping coefficient acting on a stiffer structure — the frame plus the brace as a strut, with stiffness $1 + \kappa = 2$ times the frame’s — and a fixed coefficient on a stiffer spring is a smaller fraction of critical: $2\%/\sqrt{2} = 1.4\%$.

What a small motion is #

The design amplitude is the storey drift the damper was sized for, in an earthquake that the building is expected to meet perhaps once in its life. Most of the building’s motion over that life is something else: wind, which a tall building feels every day, which governs whether its occupants notice it, and which moves the storeys by a few hundredths or thousandths of what the design earthquake would. Comfort criteria are written in accelerations of a few thousandths of $g$ at the top floors, and a building near its threshold of perception is moving by millimetres where its earthquake design asks for drifts of tens of centimetres over its height.

So the region of the figure where the nonlinear dampers have given up is not an extreme. It is where the building spends nearly all of its time. The wind is a spectrum, and its resonant part — the part a damper is for — is controlled by exactly the damping the structure has at the amplitude the wind produces. A tall building that needs damping for the wind needs it at a hundredth of the earthquake’s amplitude, and there a damper of $\alpha = 0.3$ sized for the earthquake has none to give.

The frequency gives it away #

A locking damper does one more thing that can be measured. It turns its brace into a strut, and a strut stiffens the storey.

At the design amplitude every damper raises the frequency to 1.195 times the frame’s own, the partly-stiffened state of a damper at its best. The linear damper holds it there. The nonlinear one sweeps it: down towards the bare frame’s frequency for large motions, where the damper strokes freely and the brace barely participates, and up to $\sqrt{1+\kappa} = 1.414$ for small ones, where the brace is a strut.

That second value is what an ambient vibration survey finds. A finished building’s natural periods are measured from its response to wind, traffic and microtremor, at amplitudes far below anything in its design, and the measured frequencies of finished buildings come out above the computed ones for a list of real reasons — partitions, cladding, stairs, the stiffness of connections called pinned. A building with nonlinear dampers adds one more to the list: its braces are measured as struts. The 18% between 1.195 and 1.414 is a frequency the design never used, and the same survey will usually report the damping it finds as well. That damping is the locked value, a little below the structure’s own, and it says nothing about the damping the earthquake will meet.

The resonance that returns #

The damping ratio is a summary of a linearised structure. The structure itself is nonlinear, and the honest test of the summary is to drive the real equations with a harmonic force, sweep the frequency to find the worst one, and wait for the motion to settle.

The damper’s equation is the one that makes this easy. Its force is set by its stroke rate, but the force in the damper is also the force in the brace, which is set by how far the brace has stretched. So the stroke rate follows from the brace’s stretch directly — $\dot u = \operatorname{sign}(F)\,(|F|/C)^{1/\alpha}$, with $F$ the brace’s force — and the storey, the brace and the damper together are three first-order equations with nothing implicit in them.

With a linear damper the answer does not depend on the force: the steady peak is 1.96 times the static deflection at every force, because a linear structure scales. With the damper of $\alpha = 0.3$ it depends on almost nothing else. Near the design amplitude the peak is 1.99, the same as the linear damper’s. With the force reduced until the storey moves an eighth of the design amplitude the peak is 10.4 static deflections, and the linearised curve is still climbing at the left of the figure towards the locked frame’s $1/(2 \times 0.014) = 35$. Under a small force the resonance comes back, at the stiffened frequency, about as sharp as the frame would have had with no damper and a strut in its place.

The integrated dots and the linearised line agree to within 17% everywhere, and agree best at the two ends — near the design amplitude, where the damper is near linear in its working range, and deep in the locked state, where the damper hardly matters. Between them the true force is far from sinusoidal, and the linearisation, which counts only energy, is a little optimistic about the peak. The direction of the effect is not in doubt, and neither is its size.

Choosing a different amplitude to be best at #

The obvious response is to size the damper for the wind. A damper that is at its brace’s best at a hundredth of the earthquake’s amplitude gives the full ceiling there.

It does, and it pays for it at the other end. Sized at a hundredth of the earthquake’s amplitude the damper gives 22.7% to the wind and 3.2% to the earthquake, because in the earthquake it strokes so far that it has become a small damper — the gentle side of the curve, but a long way down it. Sized at a tenth, it gives 7.8% to the earthquake and 1.9% to the everyday wind, and is good at neither. The band of amplitudes a damper of given exponent can serve has a fixed width, and sizing it chooses only where the band sits. For $\alpha = 0.3$ on this brace the band in which it gives at least half its best runs from 0.42 to 4.2 times its design amplitude: one decade exactly. For $\alpha = 0.5$ it runs from 0.24 to 7.5, a decade and a half; for $\alpha = 0.15$, from 0.55 to 3.3. For the linear damper it has no edges at all.

The width is set by how fast the damper’s size changes with its stroke — the power $\alpha - 1$ — and by how sharp the brace’s curve is, which a stiffer brace makes no blunter. The motion a building has to be damped over runs across two or three decades of amplitude, from the comfort wind to the design earthquake. A single damper of low exponent spans one.

Two devices on one brace budget #

So the building that needs damping both for comfort and for safety is asking one device to cover a range only a linear one can. The alternative the question proposed is two kinds of device. A storey has a certain stiffness of bracing it can afford — a number of diagonals, each of a certain section, a certain connection stiffness — and that brace stiffness can be divided between two dampers: some to a nonlinear damper sized for the earthquake, the rest to a linear one sized for its own brace’s best.

The result is better than the arithmetic of halves suggests. Half the brace to a linear damper costs the earthquake only 1.2 points: the structure has 21.5% at the design amplitude, against 22.7% with the whole brace on the nonlinear damper. The reason is that the ceiling is concave in the brace stiffness. Two Maxwell elements, each on half the brace and each at its own best, extract nearly what one element on the whole brace would, because each half-brace’s ceiling, $(\sqrt{1.5}-1)/2$, is more than half the whole one’s. And at a hundredth of the earthquake’s amplitude, where the nonlinear damper has locked and become a strut, the linear damper on its half brace still works — on a structure now stiffened by the other half — and the structure keeps 9.3%. A quarter of the brace given to the linear damper keeps 4.9% at the wind for 21.8% at the earthquake.

The linear share also widens the band upwards. At ten times the design amplitude the shared pair still gives 16.4%, where the nonlinear damper alone has slid down to 7.8%, because the linear damper on its half brace does not lose its size as the motion grows.

What the low exponent was for #

The division is not free, and its price is the thing the exponent was chosen for.

At the design amplitude the three options put similar forces into their braces: the linear damper 1.17 times the nonlinear one’s, the shared pair 1.28. They differ above it. At three times the design amplitude — a rarer and larger earthquake, the one a collapse-prevention check is made for — the nonlinear damper’s force has grown to 1.40 times its design value and the linear damper’s to 3.51. The shared pair’s has grown to 2.82, because half of it now grows in proportion to velocity. A quarter share gives 2.20. The force cap that made the nonlinear damper attractive is mostly given back.

That is the real trade, stated in the terms of what each choice costs. A building damped for its earthquake alone with a low-exponent damper has its brace and connections sized for a force that barely grows beyond the design event, and it has nothing for the wind. A building that also needs comfort damping has to pay for it either in brace force at the extreme event — a linear share, sized up for the force it will see — or in a second kind of device altogether, of the kind the brace calculation did not consider: a tuned mass in the roof, which works at any amplitude because it is linear, and which a building can carry in addition to its dampers because it does not compete with them for brace stiffness.

The worked storey #

The figures are in units of the structure. In a storey of stiffness 50 kN/mm, with a period of 1 s and a brace of the same stiffness, designed for an earthquake storey amplitude of 40 mm, the damper at its brace’s best works at the raised frequency $1.195 \times 2\pi = 7.51$ rad/s with a stroke of $40/\sqrt{1 + (0.595 \times 1.195)^2} = 32.6$ mm, a peak velocity of 0.245 m/s. Its equivalent coefficient is $c_{\text{eq}} = 0.595 \times 50\,000/6.28 = 4\,740$ kN·s/m and its equivalent force 1,160 kN, so its true peak force is $1\,160/1.17 = 990$ kN, and its constant is $C = 990/0.245^{0.3} = 1\,510\ \mathrm{kN\,(s/m)^{0.3}}$.

Under a wind that moves the storey 0.4 mm — a hundredth of the earthquake — the brace, now carrying almost all the motion, sees about $50 \times 0.4 = 20$ kN. The stroke rate that would give the damper a force of 20 kN is $(20/1\,510)^{1/0.3}$ m/s, about $5.5 \times 10^{-7}$ m/s: a stroke amplitude of six hundredths of a micrometre at the locked frequency. With the factor $\lambda$ included the solution gives four hundredths. A linear damper of the same design would stroke 0.33 mm. The nonlinear damper’s piston moves less than the thickness of its own oil film, and whatever it does at that scale — seal friction, the oil’s compressibility, a little play in a pin — is outside its certified curve and not in its favour.

A single storey, harmonic motion, and an ideal law #

The model is one storey and one mode, and its other simplifications are worth naming.

The motion is harmonic. Equivalent linearisation is exact in energy for a sinusoidal stroke and approximate for anything else. Wind and earthquake motion are random, and the stochastic version of the same idea weights the damper by the motion’s standard deviation rather than its amplitude. The dependence on the power $\alpha - 1$ is the same, and with it the locking.

The damper follows its law down to zero velocity. Real fluid dampers have seal friction, a small breakaway force, and pressurised accumulators whose compliance is a spring in series with the orifice. All of these make a real damper at microscopic strokes stiffer than the ideal law, not softer, which moves the locking to larger amplitudes than the figures show.

One storey. In a building the dampers are spread up the height and each storey sees its own drift, much as a damper on a stay cable sees only the motion of the point it is fixed to; the argument applies storey by storey, and because wind and earthquake load the modes differently, the storeys whose dampers matter for comfort may not be the ones that matter for safety.

The brace’s stiffness is the drawn one. The connections, gussets and floor are all in series with it. The series-damper calculation found a softer brace lowers the ceiling; here a softer brace also lowers the brace force at which the damper locks, and the band in amplitude shifts with it.

A friction device is the limit $\alpha \to 0$. A friction damper is exactly a damper of zero exponent: it does nothing at all until its slip force is reached and is a strut below it. The curves of this essay steepen towards that limit as $\alpha$ falls — the band for $\alpha = 0.15$ is already narrower than a decade — and the friction damper is their end-point: a device that is fully on above one amplitude and fully off below it.

What the band says about damping as a number #

The essay on damping’s role observed that every response in structural dynamics should be quoted with its damping assumption attached. A building with nonlinear dampers makes that observation stricter. Its damping is not one number but a function of the amplitude, and quoting it requires stating the amplitude as well — and the amplitude in turn depends on the damping, through the response it limits.

The same is true of damping a building did not have designed into it. Damping in real structures grows with amplitude as cracks open, joints slip and partitions rub, which is why a damping ratio measured in an ambient test is usually lower than the one used for design. The nonlinear damper reverses that tendency: it gives most at large amplitude and least at small. Both effects say the same thing about measured damping. It belongs to the amplitude it was measured at.

A spectrum falls as damping rises, so the earthquake design is entitled to its 22.7% at the design amplitude, provided the design amplitude is where the response sits. For the displacement the earthquake asks for that is the case by construction. For the wind it is not, and for a building near the edge of what its occupants will accept, the band that misses the wind is the band that matters.

Still open: whether the stroke can be amplified instead #

The damper locks because its stroke at small motions is tiny, and its stroke is tiny because the brace carries the motion. A toggle or scissor arrangement turns a small storey drift into a larger damper stroke through geometry: it multiplies the stroke by a factor of two or three, and for a nonlinear damper that factor enters inside the power. A damper moving three times further behaves as one $3^{0.7} = 2.2$ times smaller, which moves the whole band down in amplitude without changing its width, at the price of more pins and slender links in series with the damper, each one softening the brace. Whether a geometric amplifier can move the band far enough to reach the wind without its own compliance closing the ceiling — and whether an amplifier whose ratio itself changes with amplitude, as some linkages do, could widen the band rather than only move it — is the question of whether the brace can be designed to make the damper’s size independent of the motion, which is what the linear damper had for free.