The law that throws the tuning away

A tuned mass helps by being late: hang a few per cent of a structure’s mass on a spring tuned near the structure’s frequency, damp it properly, and the structure’s resonant peak falls by a factor of several. An actuator can do the same job by pushing against the ground in proportion to the structure’s velocity, until a delay in its loop of about a quarter of a period turns its damping into driving. And an actuator that pushes against the tuned mass instead of the ground, fed by the structure’s velocity, is worse than either: it pumps the tuned mass’s own mode, goes unstable at a gain of a per cent and a half with no delay at all, and is worse than the passive mass at every gain below that.

That essay ended on a sentence — a feedback law designed for a structure without the device that applies it is designed for the wrong structure — and on a question: what a law that does know about the mass can achieve, and how much of that survives delay.

The mode the simple law pumped #

The trouble with the simple law has a precise location. The structure with its tuned mass has two modes, close together on either side of the structure’s own frequency, and in one of them the mass swings hard against the structure. An actuator that measures the structure’s velocity and pushes the structure against the mass puts its reaction on a body moving differently from the one it measured. In that mode the reaction does positive work on the mass, and once the gain is large enough to beat the mass’s own damping, that mode grows — a motion that feeds itself, built on purpose.

So the remedy, if the actuator is to stay between structure and mass, is to make sure the mass has no mode of its own to be pumped. The simplest way to do that is drastic: cancel its spring. Give the actuator a term in the mass’s displacement relative to the structure, with a gain exactly equal and opposite to the tuning spring’s stiffness, and the mass is no longer tuned to anything. It is a free mass, riding on the structure on its dashpot alone, and the actuator pushes off it.

The law is then

$$f_a = -\left[\, g_s\,\dot x_1 + g_r\,(\dot x_1 - \dot x_2) + k_r\,(x_1 - x_2) \,\right], \qquad k_r = -k_2,$$

with $x_1$ the structure, $x_2$ the mass and $k_2$ the tuning spring: a push against the structure’s velocity, a relative damping between the two bodies, and the cancellation. Everything below is a structure of one mode with 1 per cent damping carrying a 2 per cent mass, tuned and damped by Den Hartog’s rules, as in the two essays before this one.

Every such law is stable #

Sweep the two remaining gains over a grid of 1,681 laws — structure-velocity gains up to 0.6, relative damping from a twentieth of a per cent to a tenth — and test each one for stability by the argument principle, which counts the closed loop’s unstable roots without finding them. All 1,681 are stable, at every gain, including gains twenty times the one that destroyed the simple hybrid.

The reason is worth having in words, because it is the reason the cancellation works. A free mass, pushed by an oscillating force, accelerates in phase with the force and therefore moves in quadrature with it: its velocity is a quarter of a cycle away from the force pushing it. Power is force times velocity, and the average over a cycle of a product in quadrature is zero. So the actuator’s reaction on the free mass does no net work on it, cycle after cycle, however hard it pushes. There is nothing for the reaction to pump. The structure meanwhile is pushed against its own velocity, which is damping, and the relative-damping term is collocated — it measures exactly the motion it acts on — which is the arrangement that cannot be unstable without a delay.

The simple hybrid failed because its reaction landed on a tuned mass, whose velocity is not in quadrature with the force near its own frequency. Cancel the tuning and that frequency is gone.

What is thrown away #

The price of stability is the tuning. The passive mass’s whole effect comes from its being tuned: near the structure’s frequency it swings a quarter of a cycle behind, and its spring pushes back on the structure exactly when that helps. A cancelled spring does none of that. Whatever the free mass contributes, the actuator now has to supply, by pushing.

Hold the law to the two limits that bind a real device — the actuator’s force, as a fraction of the force shaking the structure, and the mass’s stroke, the distance it can travel relative to the structure — and the best law on the grid can be read off. At the passive mass’s own stroke and half the applied force, it is 7.38 against the passive mass’s 8.72: better, by about a sixth. Put another way, it is as good as a passive mass of 3.0 per cent, half as heavy again as the one it uses.

The response curve shows how it does it. The passive mass splits the structure’s resonance into two peaks of nearly equal height either side of the structure’s frequency; the law has no tuned mass and so no split, just one broad peak at the structure’s own frequency, held down by damping that the actuator supplies.

A third of the load before it breaks even #

The force needed is the real cost, and it is large. With the actuator allowed a fifth of the force applied to the structure, the best law’s peak is 9.34 — worse than the passive mass whose tuning it discarded. At a quarter, 8.91, still worse. It takes three tenths of the applied force, 8.54, just to break even; half gives 7.38; seven tenths gives 6.61, and past that the stroke, not the force, is the limit and the curve stops falling.

That is a large actuator. The force shaking a tall building in wind is a sizeable fraction of its weight in the governing mode, and an actuator rated for a third of it, reacting against a mass of 2 per cent, accelerates that mass at many times the structure’s own acceleration. The passive mass needed no actuator at all to reach 8.72. The active law spends its first third of the applied force getting back to where the passive mass started.

Most of it is bought with room #

Relax the other limit and the picture changes. Hold the force to half the applied force and let the mass travel further than the passive mass does: at a tenth more stroke the peak drops from 7.38 to 5.58; at a quarter more, 4.84; at half as much again, 4.49 — as good as a passive mass of 9.3 per cent, more than four times the mass actually carried. Beyond about one and three quarters of the passive stroke the force becomes the limit and further room buys nothing; the frontier flattens at 4.42.

So the frontier has two limbs and they are not equally steep. Force buys about a sixth of the peak at fixed stroke, and stroke, once force is available, buys most of the rest. This is the honest reading of what active mass drivers are for. The passive mass is already close to the best that 2 per cent of mass can do in 45 static deflections of travel; what an actuator allows is using more travel than a passive mass would ever ask for, because a passive mass tuned to swing further would have to be less damped and would then make the structure worse. A law that knows the mass can drive it further without that penalty.

The map in the first figure shows the same thing as a cloud. Every law with the spring cancelled sits at some stroke and some peak, and the cloud’s lower edge falls steeply as the stroke grows — while the passive mass sits on that edge, not above it. At its own stroke the passive mass is nearly as good as anything; it is the room beyond it that the actuator turns into performance.

Why the passive mass cannot use the room #

The comparison is only fair if the passive mass is also given the extra room, and it cannot use it. A passive mass travels further only if it is damped less, and damping less is exactly what the only thing that stops it argues against. Take the 2 per cent mass and cut its dashpot to seven tenths of Den Hartog’s value: its stroke rises to 56.5 static deflections — the quarter more that took the active law to 4.84 — and the structure’s peak rises with it, to 9.47. Halve the dashpot and the stroke is 72.5 and the peak 11.17. For a passive mass, room and performance pull in opposite directions; the optimum damping is already the point where giving the mass more travel stops helping and starts hurting.

The passive way to reach a peak of 4.49 is a heavier mass: 9.3 per cent of the structure’s mass, Den Hartog-tuned, does it — and travels only 11.9 static deflections, a quarter of what the 2 per cent mass needs at its optimum. So the active law and the heavy passive mass reach the same peak by opposite routes. One carries a fifth of the weight and moves it more than five times as far, with an actuator rated for half the force on the structure; the other carries the weight and needs no power at all. On a tall building, where the top floors are the most valuable space and a mass of several hundred tonnes has to be held up somewhere, the choice is a choice between floor area and stroke, and it is made on those terms rather than on the response curve.

What centres a free mass #

A free mass has a practical problem the tuned one did not. Its spring held it in the middle of its travel; without the spring, nothing does — except damping. The mass’s stroke under the law makes the point. Near resonance it is about the passive mass’s, as the cap requires, but it is spread over a much wider band, and at low frequency it does not fall away: it tends to a constant, 25.3 times the structure’s static deflection, which is the structure-velocity gain divided by the total damping between the two bodies — the passive dashpot’s and the law’s relative-velocity term together.

So the relative-damping gain is not there to damp anything in particular. It is there to keep the free mass from wandering toward the end of its travel under slow load, and a law that set it to zero would have a mass that drifted without limit under a static push. Real active mass drivers carry a weak centring term for exactly this reason, and it is what a cancelled spring makes necessary.

With the spring cancelled, the delay hardly matters #

The last question is the one that ruined the grounded actuator: delay. Every real loop has some — sensors, filters, computation, the actuator’s own response — and the grounded actuator lost its damping at a quarter of a period.

The modest law, held to the passive stroke, does not care. With a quarter of a period of delay its peak is 7.36, a shade better than undelayed, and a time-domain run with that delay dies away as an impulse should. Its structure-velocity gain is small, so the delay’s phase lag rotates a small force, and the free mass’s indifference to the force’s timing does the rest.

The stronger law, given half as much stroke again, behaves differently. Its structure-velocity gain is nearly three times as large, and at that gain the actuator pushing against the free mass is effectively an actuator pushing against the ground — the mass is heavy enough, at the structure’s frequency, to be a ground — and it inherits the ground actuator’s limit: unstable from 0.22 of a period, a little before the quarter period of the essay on delay. The performance that room buys is paid for twice, once in stroke and once in tolerance to delay.

The power, by hand #

The quadrature argument is short enough to check. Let the actuator push the free mass with $f\cos\omega t$. The mass’s acceleration is $(f/\mu)\cos\omega t$, its velocity $(f/\mu\omega)\sin\omega t$, and the power the force delivers to it is $f\cos\omega t \times (f/\mu\omega)\sin\omega t = (f^2/2\mu\omega)\sin 2\omega t$, which averages to nothing over a cycle. A tuned mass at its own frequency has a velocity in phase with a force at that frequency — that is what resonance is — and the same force then delivers $f^2/(2c)$ on average, positive, whatever its sign convention, which is the pumping the simple hybrid did.

The low-frequency stroke is equally short. At a frequency so low that the mass’s inertia is negligible, the forces on it must balance: the damping between the bodies, $(c_2 + g_r)(\dot x_1 - \dot x_2)$, against the actuator’s push, $g_s\dot x_1$. So the stroke is $g_s/(c_2 + g_r)$ times the structure’s motion: with $g_s = 0.120$, $c_2 = 0.0034$ and $g_r = 0.0014$, about 25.

One mode, a perfect cancellation, and an actuator with no limit #

One mode. The structure is a single mode. A real building has more than one period, and a spring-cancelling law that pushes hard enough on the structure’s velocity will push on them too; whether it damps or drives them depends on where the actuator sits in their mode shapes, which is the coupling a force between two parts of a structure always makes.

An exact cancellation. The law cancels the tuning spring exactly. A cancellation that is a few per cent off leaves the mass a weak spring of either sign: slightly positive, a very low tuned frequency that the law can pump; slightly negative, a mass that is statically unstable until the centring damping catches it. The stability of every law on the grid is a property of the exact cancellation, and how much error it tolerates is a separate calculation.

An ideal actuator. The force is whatever the law asks, up to the cap, with no saturation, no stroke stop and no dynamics of its own beyond the delay. Saturation is the usual way an active damper fails in a storm: it reaches its force limit, stops following the law, and for that stretch the mass is a free mass with no tuning and no control.

The stops, the storm and the second mode #

They cannot show the end of the stroke. Every stroke here is a peak of a steady response. A storm is not steady — the wind is a spectrum, not a sine wave — and a mass allowed one and a half times the passive travel will reach its stops in the largest gusts; what the structure does when the mass hits them is not linear and is not in any of these curves.

They cannot show the power supply. An active damper needs power at the moment it is most needed, which is the moment the grid is most likely to fail. A law that has cancelled the tuning leaves nothing behind when the power goes; a hybrid that keeps a passive mass underneath and only adds to it degrades to that mass. That is the practical argument for keeping the tuning and accepting a law of lower performance, and it cuts directly against the cancellation.

And they cannot show a second mode. A free mass driven against the structure in the first mode pushes on the second mode as well, and the second mode has no tuned mass and less damping.

Throwing the tuning away #

Cancel the mass’s spring and every law is stable. A free mass moves in quadrature with the force on it, so the actuator’s reaction does no net work on it, and there is no mode left to pump.

But the tuning was the mass’s whole contribution. Held to the passive stroke, the law needs three tenths of the applied force to break even, and half to reach 7.38 against 8.72 — as good as a 3.0 per cent mass from a 2 per cent one.

Room is where the advantage is. With half the applied force and half as much stroke again, the peak falls to 4.49, as good as a 9.3 per cent passive mass; then the force runs out.

And room costs tolerance. The modest law shrugs off a quarter period of delay; the one that uses the extra room is unstable from 0.22 of a period, like an actuator against the ground.

Still open: a law that keeps the tuning #

The cancellation is the bluntest way to make the mass safe to push. It is not the only one. A law could keep most of the tuning spring — so that, with the power off, the mass is still a respectable passive damper — and add only enough relative feedback to keep its own mode from being pumped by the structure-velocity term. Somewhere between the passive mass and the free one is a family of laws that fail gracefully, and the question is how much of the free mass’s advantage survives in it: whether a law that keeps nine tenths of the tuning can still turn extra stroke into performance, or whether the room is only usable once the tuning is gone.