The switch that hands the mass back

Half a spring is neither tuning nor freedom searched for a law that keeps part of a tuned mass’s spring, so that the mass would still be a respectable passive damper if the actuator lost power, and found the compromise the worst choice available. Every share of spring kept cost performance and made more of the laws unstable; from about 85 per cent kept, no stable law did as well as the passive mass with no law at all. The best law cancels the spring entirely and turns the mass into a free reaction mass, and the law that throws the tuning away found that law brings the structure’s peak from 8.72 times its static deflection with the passive mass to 4.65, if the actuator may push with half the applied force and the mass may travel half again as far as the passive mass does.

That essay left the alternative to a compromise: a switch. The physical spring is always there, so a law that cancels it is a law laid on top of a passive damper. Run the free-mass law when the motion is small, and when the stroke approaches its limit — or when the controller’s own health check fails — let go, and leave the mass on its spring. The switch only has to choose between two designs that are each good on their own terms. This essay asks what happens at the moment it chooses.

Two good states and the instant between them #

The structure is the one the tuned-mass essays have used: a single mode of unit mass and frequency with 1 per cent damping, a tuned mass of 2 per cent of its mass on a spring and damper tuned by Den Hartog’s rules, and an actuator between the two driven by the free-mass law — the mass’s spring cancelled, a gain on the structure’s velocity and a little relative damping to keep the mass centred, the best law the partial-spring essay found with the actuator held to half the applied force and the stroke to 1.5 times the passive mass’s. Here the law is followed in time rather than in frequency. A harmonic force rises over a few periods and holds, and at a chosen moment the actuator’s force drops to nothing.

The hero figure is the whole story in one run. For sixty periods the law holds the structure to 4.63 times its static deflection — about half what the passive mass would allow at this frequency. Then the health check fails and the actuator lets go. The structure does not move to the passive mass’s 6.97 and stay there. It overshoots, to 8.78 within a few periods, beats down over the next twenty, and only then settles at 6.97.

The structure moves more in the cycles after the switch than it would with either the law or the passive mass. It moves more, in fact, than the passive mass’s worst resonant peak, 8.72, even though the forcing is at a frequency where the passive mass, left alone from the start, would never have allowed that.

Why a free mass makes a bad start for a spring #

The overshoot has a simple cause. A free reaction mass does its work by moving a long way relative to the structure — 63 times the static deflection at this frequency, against 27 for the passive mass — and at a phase the law chooses rather than one the spring would set. At the instant the actuator lets go, the spring the law was cancelling takes hold of that relative displacement at once. The passive mass starts its life not from rest, and not from its own steady motion, but from a stroke more than twice its steady amplitude, at the wrong phase. Its free motion — at its own frequency, near the structure’s — is added to the forced motion at the forcing frequency, and the two beat. The structure’s peaks in the first few beats are the forced response plus most of the free one.

The figure shows where it matters. Near the structure’s frequency, between about 0.93 and 1.05, the passive mass is already near its own two peaks, and the switch adds little to them. Away from that band the passive mass is comfortable on its own — at 0.84 of the frequency it holds the structure to 3.96 — but the law has carried the mass far from where its spring would hold it, and the switch hands the passive system an initial condition that rings at its own frequencies. The structure peaks at 7.60 there, nearly twice what either state allows in steady operation. Above the frequency the same happens on a smaller scale: 6.03 at 1.12 of the frequency, against 4.66 for the passive mass.

So the switch’s worst case is not the storm the passive mass handles worst. It is the storm the law handled best: a forcing far enough from resonance that the passive mass would be comfortable, and close enough that the law was working the mass hard.

The instant hardly matters #

A switch that acts on a health check fails at a random instant; a switch that acts on the stroke fails at the instant the stroke reaches its threshold. If the transient depended on the instant, a well-timed switch could avoid it.

It does not. Wherever in the forcing’s cycle the actuator lets go, the structure’s peak in the cycles after lies between 8.53 and 8.77. A free mass carries its stroke at every instant of the cycle — when its relative displacement is small its relative velocity is large, and the passive system receives the same energy either way, as strain in the spring or motion of the mass. The transient is set by how far the law had carried the mass from where its spring would hold it, which the law does continuously, and not by the moment of letting go.

Letting go gradually #

If a sudden release starts a transient, the obvious remedy is not to release suddenly: fade the actuator’s force down to nothing over several periods, so that the spring takes hold of the mass gradually. The next figure tests it.

It is worse, and the longer the fade the worse it gets. Cut at once, the structure peaks at 8.53 after the switch. Faded over ten periods, 10.23; over a hundred, 13.48, and the mass’s stroke reaches 1.53 times the limit the switch was meant to protect.

The reason is the partial-spring essay’s finding, arriving from an unexpected direction. Scaling the actuator’s force down scales every term of the law together, and the term that cancels the spring with it: halfway through a fade the law is cancelling half the spring and applying half the structure-velocity gain. That is one of the part-cancelling laws, and most of those pump the mass’s own mode, because the spring that remains gives the mass a frequency for the structure-velocity term to drive. A fade is a slow walk through the worst laws in the family. The switch has to be a switch.

A fade that is safe would have to scale the terms separately — drop the structure-velocity gain first, while the spring is still fully cancelled, then restore the spring with relative damping alone — which is a second law and a sequence, not a fade, and a controller that has just failed its health check is not the thing to trust with a sequence.

A switch that takes its time #

A switch that acts on the stroke has to act before the stroke reaches its limit, and in a storm that is growing, the stroke grows while the switch waits.

The threshold here is 80 per cent of the stroke limit, which in a design storm the law never quite reaches but a larger storm does. With no delay the stroke stops close to its threshold in every storm: the switch acts, the passive mass takes over, and its own smaller stroke holds. With a delay the law keeps the mass free while the storm keeps growing its motion. A storm twice the design force carries the stroke past its limit if the switch takes two periods to act; 1.5 times the design, three periods; 1.2 times, five.

A period here is the structure’s: several seconds for a tall building, so these delays are seconds, not milliseconds. A switch that waits for a filtered stroke signal to cross its threshold, or for a fault to be confirmed, or for a supervisory controller to agree, spends periods doing it.

And a storm that grows fast #

How fast the storm grows decides how much the stroke gains during the delay.

A storm that reaches twice the design force in two periods — a squall, rather than a storm building over an hour — takes the stroke past its limit with half a period of delay. One that builds over sixteen periods allows two. The switch’s speed has to be set against the fastest growth of motion the mass will see, not against the design storm’s steady state, which the switch never needs to act in.

A switch’s threshold buys time only in proportion to how slowly the storm grows. The 20 per cent margin below the stroke limit is a margin in stroke; how long it lasts depends on how fast the stroke is climbing when it is crossed.

What the switch has to be designed for #

The figures change what a designer of a switched mass has to provide. The passive state the switch falls back to is usually sized as the passive mass would be on its own: a stroke budget for the passive mass’s motion in the design storm, end stops beyond it, and a structure checked for the passive mass’s response. None of those is the state the switch actually lands in. For the first several periods after the switch, the mass carries the active law’s stroke, not the passive mass’s, and the structure moves as much as the passive mass’s worst resonant peak, whatever the forcing frequency. The passive state’s stroke budget has to be the active law’s, and the structure’s check after a switch has to be the passive mass’s peak, not its response at the storm’s frequency. A stroke is a displacement the hardware must accommodate, not a force it must resist, and the earthquake asks for a displacement is the general form of that: what a moving system demands of its stops is set by how far it travels, and the switch hands over travel.

That is a cost the active law did not seem to have. The force that is capped on purpose found that a limit put on a force moves the demand somewhere else rather than removing it, and the switch is the same: the stroke limit the law respects in operation is a limit the passive state inherits the moment the law stops respecting it for itself.

The switch also has to know what it is protecting against. A stroke threshold protects the end stops, and it does so only if the switch acts faster than the storm grows the stroke. A health check protects against a law gone wrong, and a law gone wrong — a sign error, a sensor that has stuck — may be pumping the mass when the check fires, so that the stroke handed to the passive state is larger than any the law would have produced. The damper is only as good as its brace found that a damper’s value is set by the weakest stiffness in series with it; a switched mass’s value in a fault is set by the stroke it has when the fault is caught.

None of this argues against switching. The law’s steady performance is twice the passive mass’s, the transient lasts a few periods, and a structure checked for the passive mass’s peak is checked for something it would meet anyway near resonance. It argues against counting on the switch to land on the passive mass’s response at the storm’s frequency, which is the assumption that makes the passive mass look like a free safeguard.

A bound on the transient #

The size of the transient can be bounded from the strokes alone. In the steady active state at 0.9 of the frequency the mass’s stroke amplitude is 63.5 static deflections; the passive steady stroke there is 26.6. At the instant of the switch the passive system is handed the difference as free motion, about 37 static deflections of stroke. In the passive mass’s own two modes the mass moves about one over the square root of the mass ratio — about seven times — as far as the structure, so a stroke of 37 is a structure motion of about 37/8 ≈ 4.6.

If that free motion and the forced 6.97 peaked together at the instant of the switch, the structure would reach 11.6. They do not: the free motion is at the passive system’s own frequencies and the forced one at the forcing’s, so they come into phase only as they beat, a few periods later, and by then the tuned mass’s damping, eight per cent of critical, has taken most of the free motion away. The run finds the structure 1.81 above its forced response at the widest beat, between nothing and the 4.6 bound, and the bound is what makes the size of the transient intelligible: it is set by how much larger the law made the stroke than the passive mass would have, which is exactly what the law was designed to do.

Where the single-mode model stops #

The structure has one mode. A tall building’s second mode is several times the first’s frequency, and the switch’s transient, at the passive system’s frequencies near the first, does not excite it much; a structure whose modes are close would share the transient.

The forcing is harmonic and steady. Wind on a tall building is a broad band of frequencies whose largest part sits near the structure’s frequency, and the law’s large strokes away from resonance are a fraction of its work; the transient’s severity in a real storm depends on how much of the motion was at frequencies where the passive mass was comfortable and the law was not.

The switch is perfect. An actuator that lets go is modelled as a force dropping to nothing in one step. A real actuator that loses power may lock, or drag, or drop its force over a fraction of a second; locking makes the mass part of the structure, a different and worse state that neither design considered.

The end stops, the hysteresis and the occupants #

They cannot show the end stops, the buffers that catch a mass that has run out of stroke, which turn a stroke limit into an impact and are the last thing a switch is protecting. They cannot show the controller’s decision, which in practice is not a single threshold but a set of them with hysteresis, so that the law is not switched back on at once by the falling stroke of the transient it caused. And they cannot show the occupants, who notice the motion in the cycles after the switch more than either steady state, because it is the motion that changed.

Still open: the law that comes back #

Every switch here is one-way: the law lets go and the passive mass keeps the structure for the rest of the storm. A practical controller switches back once the motion has fallen — and switching a free-mass law on, onto a passive mass in mid-swing, is the reverse transient: the actuator’s first act is to cancel a spring that is holding the mass at some displacement, releasing the mass to fly with whatever velocity it has. Whether switching on is as violent as switching off, whether a controller that switches both ways can chatter between the two laws in a storm that sits near its threshold, and what hysteresis keeps it from doing so, is the question the one-way switch leaves. The actuator that arrives late is where delay first cost an actuated mass its margin, the actuator that pushes against the mass is where the mass’s own mode first showed it could be pumped, and the mass that helps by being late is the passive mass that every one of these laws comes back to.