The lever that moves the damper and not the band

The damper sized for the earthquake put a damper whose force goes as the 0.3 power of its speed on a brace as stiff as the storey, sized it to the brace’s best at the design earthquake’s amplitude, and followed it down to the amplitudes a building actually spends its life at. At the design amplitude it gave the storey 22.7% of critical damping. At a tenth of it, 1.9%; at a hundredth, 1.4%, less than the 2% the storey had with no damper at all. A damper of low exponent has a size that grows as its stroke shrinks, and in the wind, whose resonant part is what damping is for, it grows until it locks and its brace becomes a strut. Sizing it for a different amplitude only moved its band — a decade wide — along the amplitude axis.

The essay ended on a remedy used in practice. The damper locks because its stroke is tiny; a toggle or a scissor linkage between the brace and the damper multiplies the stroke by two or three. If the damper moves three times further it behaves as a smaller damper, and a smaller damper locks later. The question was whether such a lever can move the band down far enough to reach the wind without its own compliance closing the ceiling, and whether a lever whose ratio changes with the motion could widen the band instead of moving it.

The answer to the first half is no, and the reason is a line of virtual work. The answer to the second half is that the right kind of ratio would make the damper linear, and a toggle has a ratio of roughly that kind — on one side of its stroke, up to the drift at which it locks.

A lever makes the damper larger, not smaller #

A linkage of ratio $f$ turns a stroke $u$ across the storey’s brace into a stroke $fu$ across the damper. The work done on the damper is the work done on the storey, so a damper force $F$ appears at the storey as $fF$. For a damper whose force is $C|v|^\alpha$, the storey therefore feels

$$F_s = f\,C\,|f v|^{\alpha} = f^{1+\alpha}\,C\,|v|^{\alpha},$$

a damper of the same exponent and a constant $f^{1+\alpha}$ times larger. At a ratio of 3 and $\alpha = 0.3$ that is 4.17 times larger. The damper’s own stroke is three times longer, and its own equivalent size — the size measured at its own end of the linkage — does fall by $3^{0.7} = 2.2$, as the earlier essay said; but the lever multiplies its force by three on the way back to the storey and its stroke by three on the way in, and the storey sees the product.

A larger damper locks sooner, so the band moves up, away from the wind. The damper that gave 22.7% at the design amplitude with no lever gives 6.4% behind a ratio of 2 and 2.3% behind a ratio of 3, because at the design amplitude it is now well up the locking side of its brace’s curve. Its best has moved to 3.8 and 7.5 times the design amplitude. At a hundredth of the design amplitude all three give the same 1.4%: locked is locked, whatever the lever.

The size of the shift comes from the same scaling. A damper’s equivalent size at the storey goes as $C_s X^{\alpha-1}$, so multiplying $C_s$ by $f^{1+\alpha}$ is the same as moving to an amplitude $f^{(1+\alpha)/(1-\alpha)}$ times larger: $f^{1.86}$ for $\alpha = 0.3$, which is 3.6 at a ratio of 2 and 7.7 at 3. The figure’s peaks sit close to there; the difference is the stiffened frequency, which shifts with the size.

Re-sized, the lever changes nothing the storey can feel #

So the damper behind the lever has to be re-sized: a smaller one, chosen so that it sits at its brace’s best at the design amplitude, as the unamplified damper did.

The re-sized damper’s constant is $f^{-(1+\alpha)}$ of the original’s — 0.24 at a ratio of 3 — so that $f^{1+\alpha}$ times it is the original again. The storey then sees exactly the damper it had before, and every curve the earlier essay drew is unchanged: 22.7% at the design amplitude, 1.9% at a tenth of it, 1.4% at a hundredth. The band is a property of the damper as the storey sees it, and the lever has been chosen precisely so that the storey sees the same damper.

What changes is the hardware. The damper carries a third of the force and strokes three times as far. In the storey the earlier essay used — 50 kN/mm, a period of a second, a 40 mm design amplitude — the unamplified damper carried 990 kN over a 33 mm stroke with a constant of $1{,}510\ \mathrm{kN\,(s/m)^{0.3}}$. Behind a ratio of 3 it carries 330 kN over 98 mm, with a constant of about 360. That is a real saving: a damper’s cost goes mostly with its force, and a smaller, longer device is cheaper and easier to fit than a large short one. It is the reason amplifiers are used. But it is a saving in the damper, not a change in the building’s damping.

And in the wind, the re-sized damper behind the lever strokes three times as far as the unamplified one: three times the four hundredths of a micrometre that the earlier essay found, which is still far less than the thickness of the damper’s own oil film. Multiplying a stroke that has locked by three leaves it locked, and an ambient survey of the finished building still measures the braces as struts, stiffer than the model said.

The lever’s own looseness, multiplied by the square #

The linkage is not rigid. Its links stretch and bend, its pins have clearance, and the frame it is fixed to moves under it. All of that is a spring in series with the damper, at the damper’s end of the lever.

A spring of stiffness $k_l$ at the damper’s end stretches by $F/k_l$ under the damper’s force $F$. Seen from the storey, the force is $fF$ and the stretch, carried back through the lever, is $F/(f k_l)$, so the storey sees a stiffness of $k_l/f^2$ — the linkage’s compliance arrives multiplied by the square of the ratio. It sits in series with the brace, and a damper is only as good as its brace: the brace’s stiffness sets the most damping any damper on it can deliver.

A linkage ten times as stiff as the brace sounds stiff. Behind a ratio of 3 it reaches the storey as $10/9$ of the brace’s stiffness, in series with the brace, and the combined spring is $1/(1 + 0.9) = 0.53$ of the brace alone. The ceiling falls from 22.7% to 13.8%. To keep it above 21% at a ratio of 3 the linkage has to be about a hundred times as stiff as the brace it serves, which for links and pins working at a third of the brace’s force means links of comparable section to the brace itself and pins with almost no clearance.

The same rule is familiar elsewhere. An electrical transformer reflects an impedance through the square of its turns ratio; a gearbox reflects a load’s inertia through the square of its gear ratio. A mechanical lever does the same to whatever stiffness sits on its far side, and a stroke amplifier is a lever built to have a large ratio.

What would widen the band #

The band has the width it has because the damper’s size at the storey goes as $C_s X^{\alpha-1}$: a fixed constant and a stroke falling with amplitude. Fix the ratio and nothing changes that power. Let the ratio change with the amplitude, $f = f_0 X^\beta$, and the storey’s damper becomes

$$C_s X^{\alpha - 1} \propto f^{1+\alpha} X^{\alpha - 1} \propto X^{\beta(1+\alpha) + \alpha - 1},$$

which no longer depends on the amplitude at all when $\beta = (1-\alpha)/(1+\alpha)$. For $\alpha = 0.3$ that is 0.54.

With $\beta = 0.54$ the storey has 22.7% everywhere, from a thousandth of the design amplitude to a hundred times it. The damper is acting exactly as a linear one would, because the lever is undoing, amplitude by amplitude, the dependence its exponent built in. Half that exponent, 0.27, widens the band without flattening it: the damping at ten times the design amplitude almost doubles, from 7.8% to 14.6%, but at a hundredth it barely moves.

A ratio that grows with the motion is what the low exponent needs. It would let a damper cap its force in the large earthquake — the force at the storey still grows with the lever, but the damper’s own force is set by its own law — and still damp the wind, so that the one number that stops a resonance would be the same number at every amplitude. The price is a ratio twelve times as large at the design amplitude as at a hundredth of it: a lever that is nearly rigid in the wind and multiplies the earthquake’s drift by twelve.

A toggle’s ratio does change #

A toggle brace is two links pinned together at a knee, one pinned to a bottom corner of the bay and the other to the opposite top corner, with the damper running from the knee to the other bottom corner. As the storey drifts the top corner moves sideways, the links rotate, and the knee moves across the bay much further than the corner did — that is the amplification.

The ratio is not fixed. Drifted one way, the links straighten towards the diagonal and the knee’s motion per millimetre of drift grows; drifted the other, they fold and it falls. With the knee 400 mm off the diagonal the ratio is 2.71 at rest, 4.42 with the storey drifted 30 mm in the straightening direction and 2.04 in the folding one. And the growth has a limit of the worst kind. At 53 mm of drift the links lie straight, the knee’s motion per millimetre of drift becomes infinite, and the toggle locks: no further drift is possible without the links stretching.

This is the flat pair of bars that the essays on determinacy found rigid to every count and still free to move. Two bars in a straight line pinned at their ends offer no first-order resistance to a sideways push at their joint, and the force needed to hold the joint against it is unbounded; no count of members sees it. A toggle is that configuration approached on purpose, because the kinematics that make the force infinite at straight make the stroke ratio large near it. A toggle clamp and a stone crusher use the same geometry for the force; the damper brace uses it for the stroke.

So a toggle’s ratio does grow with the motion, which is the property the band needs. But it grows only on one side of the cycle, it falls on the other, and on the side where it grows it is heading for a lock. A storey moving ±30 mm drives the 400 mm toggle between a ratio of about 2 and about 4.4 on every cycle — a damper that is alternately larger and smaller as the building sways one way and the other, which no damping ratio describes and no symmetric band can be read from.

The larger the ratio, the sooner it locks #

The ratio at rest and the drift to the lock are tied together by the same geometry.

A larger ratio needs the knee nearer the diagonal, and a knee nearer the diagonal has less drift to go before the links are straight. A ratio of 2 leaves 86 mm; 3 leaves 45; 4 leaves 27; 8 leaves 8. A storey designed to drift 40 mm in its earthquake can carry a toggle of ratio about 3.2 at most — and at that ratio it reaches the lock at the design drift, with no margin for the larger earthquake a collapse check asks about. The usable ratios are the ones that keep the lock well beyond the largest drift, which in practice means two to three.

A toggle whose ratio would grow by twelve times over two decades of amplitude, as the linear damper needs, would have to sit close to straight at rest — so close that the wind’s fraction of a millimetre moves it measurably along its curve — and the same toggle would lock within a millimetre or two of drift. The geometry that makes the ratio grow is the geometry that ends the stroke. The band cannot be widened that way, because the drift range over which the ratio grows usefully is exactly the drift range the toggle has before it jams.

By hand, for the 400 mm toggle #

The bay is 6,000 mm wide and 3,500 mm high, so its diagonal is 6,946 mm long. The knee sits at mid-diagonal, 400 mm off it towards the damper’s corner, so the two links are each $\sqrt{3{,}473^2 + 400^2} = 3{,}496$ mm long, together 6,992 mm — 46 mm more than the diagonal.

The links lie straight when the distance between their outer pins grows to 6,992 mm. The top corner moves horizontally with the drift $s$, so that distance is $\sqrt{(6{,}000 + s)^2 + 3{,}500^2}$, which reaches 6,992 at $s = 53$ mm. The lock is at 53 mm of drift, as the figure has it.

Near rest, a drift $s$ does three things to the knee. It lengthens the diagonal by $s \cos 30.3° = 0.864s$, the angle being the diagonal’s slope, and each link takes up half of that by swinging towards the diagonal: with a half-diagonal of 3,473 mm and an offset of 400, the knee moves towards the diagonal by $3{,}473/400 \times 0.432s = 3.75s$. The damper runs from the knee to its corner at 33° to that motion and lengthens by 0.84 of it, $3.15s$. The lengthening also slides the knee along the diagonal by $0.432s$, which shortens the damper by $0.24s$; and the top corner’s sideways motion turns the diagonal, carrying the knee $0.25s$ the other way across it, which shortens the damper by $0.21s$. Together, $3.15 - 0.24 - 0.21 = 2.70$, against the computed 2.71.

What the lever is for #

A stroke amplifier is a good device, used for a good reason. It lets a small damper, with a third of the force and a quarter of the constant, do the work of a large one, and it lets the damper sit in a bay where a diagonal brace would block a corridor. What it does not do is change the damper the storey feels: the storey sees whatever the lever and the damper make between them, and a damper sized to the same design amplitude makes the same thing. The band, the locking in the wind and the frequency that rises as the damper locks are all properties of that combination, and are unchanged.

What it adds is the linkage’s compliance, multiplied by the square of the ratio, and a ratio that varies over the stroke in a way the linear calculation cannot see. Both argue for a modest ratio, two or three, in a linkage as stiff as the brace it serves — the regime in which amplifiers are built.

What the single storey leaves out #

The motion is harmonic and the linkage is counted with its ratio at rest. Every damping figure but the toggle’s uses a ratio fixed over the stroke. A real toggle’s ratio varies through each cycle, between about 2 and 4.4 for the 400 mm toggle at 30 mm, and the energy it extracts per cycle is the integral over that variation, which a single equivalent damper only approximates. Near the lock the approximation fails entirely: the damper’s force, multiplied by a ratio growing without bound, reaches the links as a force growing without bound, and the links yield or buckle before the drawing’s geometry is reached.

The linkage’s stiffness is a single spring. Real links are pinned, and pins have clearance. A clearance is a spring of zero stiffness over a small stroke — the stroke the wind produces — and multiplied by the square of the ratio it can swallow the wind’s motion before the damper sees any of it. The clearance in the earlier essay’s oil film and the clearance in the linkage’s pins are the same kind of problem, at the same scale, and the lever multiplies the second.

The damper’s law holds at every stroke. The ideal power law is what the essay on sizing used; real dampers have seal friction and a breakaway force, which behind a lever reach the storey multiplied by the ratio. A damper that does nothing below its breakaway force is a friction device at small amplitudes, and the lever multiplies the force at which it lets go.

Still open: the device that is linear on purpose #

The damper locks in the wind because its exponent is low, and nothing between it and the storey can undo that without a ratio that grows twelve times over the motion. The device that damps the wind without a brace at all is a tuned mass, and the earlier essay set it beside the dampers as the alternative that does not compete for brace stiffness. But a tuned mass sized for the wind is mistuned by the earthquake, whose storey drifts stiffen and soften the frame it is tuned to, and its stroke in the earthquake is limited by the room it sits in. Whether a building can carry a tuned mass for its comfort and low-exponent dampers for its safety, with the mass’s stroke stopped by bumpers before the earthquake reaches its design drift — and whether the bumpers then turn the mass into one more impact the frame must carry — is the question a building that needs both kinds of damping puts next.