Taking hold is harder than letting go

The switch that hands the mass back followed an actively driven tuned mass through the moment its controller lets go. The law it runs cancels the mass’s own spring and turns it into a free reaction mass, which does far better than the passive mass for as long as the stroke lasts; when the stroke nears its limit, or a health check fails, the actuator releases and the mass is left on its spring and damper, Den Hartog’s passive absorber. That essay found a transient in the cycles after the release that briefly took the structure past what either state alone would allow, and found that it hardly depended on the instant of release, and that easing the release made it worse.

Every switch in that essay was one-way. The law let go and never came back, so the structure finished every storm on its passive mass. A practical controller does not work like that. Once the motion has fallen, it takes hold again, because the law is the better state and the storm that tripped it may have passed. This essay is about the reverse transient, and about what happens to a switch that is allowed to act in both directions in a storm that will not make up its mind.

The reverse of letting go #

The model is the one the previous essays used: one structural mode of unit mass and frequency with 1 per cent damping, a tuned mass of 2 per cent on a spring and damper tuned by Den Hartog’s rules, and an actuator between them. With the law on, the actuator pushes with

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

the last term cancelling the physical spring, and the gains are the best ones the partial-spring search found with the actuator’s force held to half the applied force and the stroke to 1.5 times the passive mass’s. The stroke $s = x_1 - x_2$ is reported as a share of that limit. A harmonic force at the structure’s own frequency builds up over sixty periods with the actuator off, so that the mass swings in the passive steady state, and at a chosen instant the law takes hold.

Letting go and taking hold look symmetric on paper, and they are not. When the law lets go, it removes a force the spring was not supplying and leaves the spring to catch the mass: whatever stroke the mass has, the spring is there to bring it back. When the law takes hold, it removes the spring. A mass that was swinging on a spring, at the instant the law comes in, is a mass with no restoring force, a large relative velocity and nothing but damping between it and the end stop.

The stroke after taking hold reaches between 1.39 and 3.12 times the limit, and every instant in the cycle throws the mass past the end of its travel. Taking hold of a swinging mass throws it, and the throw is not a small overshoot to be absorbed by a margin. A mass that needs three times its designed room is a mass that hits its end stops, and every impact on an end stop is a force the building was not designed to receive at the floor where the damper sits.

Unlike the release, the throw depends strongly on when it happens. The curve repeats twice a period, because the stroke does, and its lowest values come a little after the stroke passes through zero, its highest a quarter-period later. That is the opposite of the release, which the earlier essay found to lie between 8.53 and 8.77 wherever in the cycle it fell. The release is insensitive to the instant because the free mass carries its departure from the spring’s equilibrium continuously. The take-over is sensitive because it inherits the passive mass’s state, which oscillates, and the law then has to absorb whatever that state was.

Waiting for a good moment does not find one #

If the instant matters, a controller can wait for a good one. Two natural rules suggest themselves. Wait for the stroke to cross zero, so that the spring being cancelled carries no force at the instant it is cancelled. Or wait for the mass to come to rest relative to the structure, at the end of a swing, so that it is released with no velocity.

Neither finds the best instant. Waiting for zero stroke gives 2.81 times the limit: the spring force is nothing, but the relative velocity is at its largest, and a free mass handed its largest velocity coasts furthest. Waiting for rest gives 2.27: the velocity is nothing but the spring was at its fullest stretch, and once it is cancelled nothing but damping and the structure’s own push can bring the mass back from there. The best instant on the curve, 1.39, lies between the two, a little after the stroke passes zero, where neither the spring’s force nor the relative velocity is at its worst — and it lies there only for this law, this storm and this mass. A rule that has to be tuned to the storm is not a rule a controller can apply in one it has not seen.

Easing the law in #

The release had an obvious remedy that turned out to be worse than the disease: fading the actuator’s force out over several periods let the structure drift towards the passive mass’s worst peak through a long intermediate state. Easing the law in is the corresponding remedy here, and it does the opposite.

Over five periods the worst throw falls from 3.12 to 1.41 times the limit; over ten, to 1.26; over twenty, to 1.08. The asymmetry with the release is the same asymmetry as before, seen from the other side. Fading the law out leaves a partly cancelled spring for a while, and a partly cancelled spring is neither tuning nor freedom — the worst state the earlier search found. Fading the law in passes through the same partial states, but it starts from a mass that is already tuned and swinging moderately, and it removes the spring slowly enough for the law’s velocity terms to take up the mass’s momentum as the spring lets go of it. What hurts the release helps the take-over.

Twenty periods of fade still leave the stroke 8 per cent past its limit in this storm, and a twenty-period fade is a long time: at a structural period of five seconds it is nearly two minutes, during which the building has neither the law’s performance nor a clean passive mass. The fade buys back most of the throw and none of the speed.

A storm that hovers #

The question that matters in practice is not one take-over but many. A switch that lets go at 0.8 of the stroke limit and takes hold again as soon as the stroke is below 0.8 is a switch that can act on every cycle, if the storm’s level sits where the law’s stroke straddles the threshold. Real wind and real earthquakes do that: their amplitude wanders, as the damper sized for the earthquake had to allow for, and a building near its design storm spends a good part of it close to whatever level a designer chose.

The storm here hovers: a harmonic force at the structure’s frequency, whose amplitude swings ±30 per cent about 1.1 times the design’s over a slow cycle of about thirty periods. The law on its own — never switched — keeps the structure to 5.07 times its static deflection through it, with its stroke peaking at 0.95 of the limit. It never needed to let go. The passive mass alone allows 10.84.

The switch acts 282 times. Each time the swell of the storm carries the law’s stroke over 0.8, the actuator lets go; within a fraction of a cycle the passive mass has pulled the stroke back under 0.8, and the actuator takes hold again, throwing the mass, which carries the stroke over 0.8, and so on through every crest of the storm. The marks along the foot of the figure come in bursts at each swell. The structure’s own response hardly suffers — it peaks at 5.48 against the law’s 5.07 — because the trip catches each throw within a cycle. The actuator suffers instead: 282 reversals of its force in one storm, each a step of a substantial fraction of the applied load, at the floor where the mass is mounted.

This is chatter — the same instability at a threshold that makes a friction damper’s slip load a design decision rather than a detail — and every control engineer’s first remedy for it is a band: re-arm the law only once the stroke has fallen well below the trip, and only after it has stayed there for a while.

The band that stops the chatter #

Without a hold, the band hardly matters: the structure’s peak stays near the law’s at every re-arming level, and the switch still acts more than two hundred times, because a mass on its passive spring at this storm’s level passes through small strokes twice a cycle and finds a moment under any band almost at once. With a hold of one period — the stroke must stay below the re-arming level for a whole period before the law may return — the chatter collapses: from 282 switches to 123 with no band, 48 at a re-arming level of 0.6, 13 at 0.5 and a single switch at 0.4 and below. And the structure’s peak rises to meet the passive mass. At a re-arming level of 0.4 the switch acts once, lets go at the first swell, and never takes hold again, because the passive mass in this storm never brings its stroke under 0.4 of the limit for a whole period.

The band that stops the chatter keeps the law off. The two outcomes are not a trade with a sweet spot between them. Every band either leaves the switch chattering, or leaves the mass passive for whatever remains of the storm, or — between the two — does some of each. A band is a statement about how sure the controller must be that the storm has passed, and in a storm that is not passing, any certainty worth having is never reached.

A storm the law cannot carry #

In the storm above the law did not need to let go; its own stroke never exceeded 0.95 of the limit, and the switch was acting on a threshold set inside the law’s normal operation. In a larger storm the switch is genuinely needed. At 1.5 times the design’s, swinging by the same ±30 per cent, the law never switched would keep the structure to 6.91 times its static deflection but would carry the mass to 1.30 times its limit, onto its stops.

Now there is no setting that works. Re-armed at once, the switch chatters more than four hundred times and the structure peaks at 8.3 to 9.3 — better than the passive mass, at the price of a mass that is thrown on every return. With a one-period hold and a narrow band, the structure peaks at 16.05, worse than the passive mass’s 14.78, because each of the 51 returns throws the mass and the throw arrives at the crest of a swell. Only a band wide enough to keep the law off for good brings the structure back to the passive mass’s 14.67. In a storm the law cannot carry, the best two-way switch is a one-way switch.

The setting that ends the chatter #

The first storm had a different lesson in it, and the last figure reads it directly. The law, never switched, peaked at 0.95 of its stroke limit. A trip set at 0.8 is a trip that the law reaches in normal operation, at every swell of a storm only a tenth above design.

Tripped at 0.6 of the limit, with no band, the switch acts 520 times; at 0.8, 282; at 0.9, 122; at 1.0 and above, never. The band changes the count below the law’s own stroke and changes nothing above it, because above it there is nothing to chatter about. The chatter is not a property of the switch’s re-arming; it is the trip sitting inside the law’s operating range. A trip has to sit above the stroke the law reaches in the storms it is meant to carry, and the margin a designer wants against the end stops then has to come from the stops themselves — a buffer, a longer stroke, a mass that can run a little past its working range without damage — rather than from a threshold that throws the law away whenever the storm swells.

That changes what the switch is for. It is not a stroke limiter that cycles the law on and off at the edge of its travel. It is a fault response: it lets go when something is wrong — a storm beyond the law’s capacity, a sensor that has failed, a stroke that has truly reached the stops — and it takes hold again only when there is reason to believe the cause has gone, slowly, over several periods.

The throw, by hand #

The size of the throw can be estimated in two parts without the simulation, and the two parts are the two things the law inherits when it takes hold.

The first is the mass’s own motion. The law cancels the spring completely, so after it takes hold nothing pulls the mass back towards the middle of its travel except damping: the passive damper’s $c_2 = 0.0033$ and the law’s relative term $g_r = 0.0092$, together $c_r = 0.0125$ in the model’s units. A mass of $\mu = 0.02$ moving at relative velocity $v$ and resisted only by $c_r$ coasts a further $\mu v / c_r$ before it stops — a coasting length of $\mu/c_r = 1.6$ time units. In the passive steady state the stroke amplitude is 0.66 of the limit and the relative velocity amplitude $0.98$ times that, so a mass taken hold of at zero stroke, when its velocity is largest, coasts about $1.6 \times 0.98 \times 0.66 = 1.03$ of the limit on its own.

The second is the structure. At the instant the law takes hold, the structure is moving at the passive mass’s amplitude, 7.85 times its static deflection, and the law was designed for a structure moving at its own, 3.58. The free-mass law pushes the mass in proportion to the structure’s velocity, so in the cycles it spends bringing the structure down it asks for roughly $7.85/3.58 = 2.2$ times the stroke it was sized for, which was 0.67 of the limit: about 1.47 of the limit.

Added, the two give about 2.5 times the limit at the worst instant, against the simulation’s 3.12, and the coasting part alone explains why waiting for zero stroke is a bad rule. Taking hold at rest removes the coasting part but leaves the mass at the full passive stroke when the structure’s demand arrives, which is the simulation’s 2.27. Easing the law in works on the second part: it lets the structure come down from the passive amplitude while the spring still holds the mass, so the law never asks for the stroke that bringing the structure down at once would need.

Which free body the actuator sees #

Cut the actuator out of the model. On its two ends act equal and opposite forces, one on the structure’s floor and one on the mass. With the law on, the force is the cancelled spring’s plus the law’s velocity terms; with it off, nothing. Every switch is a step in that force, applied at once to the floor. In the chattering storm the step happens 282 times, and each is the actuator’s whole force at that instant — a force the law was designed to keep near half the amplitude of the applied load, appearing or vanishing within one time step at the floor that carries the mass.

That is the half of chatter that the structure’s peak cannot show. The figures report the structure’s displacement, which is what the damper exists to reduce, and by that measure chatter was nearly harmless in the first storm. The actuator, its mounts and the floor under it see a fatigue load the design never considered, and a building whose comfort system operates near its threshold on most windy days would apply it for the life of the building.

What the single mode cannot show #

The structure is one mode. A real building’s other modes are excited by every step in the actuator’s force, and a chattering switch is a broadband source; whether that matters depends on how close the next mode is and how much of the mass’s force it sees.

The end stops are not modelled. A stroke past the limit is reported as a number. In a real damper it is an impact on a buffer, which changes the transient entirely and puts a short, large force into the floor. The throws in the first figure are therefore statements about where the mass would go if nothing stopped it, which is a measure of how hard the stops would be hit.

The storm is a modulated sine. Real wind is broadband and real earthquakes are short. A broadband storm wanders across the threshold at random rather than on a slow cycle, which spreads the switching out but does not remove it; a short earthquake gives a switch with a long hold no chance to return at all, which is what a cautious designer would want anyway.

The law is fixed. A controller that schedules its gains with the stroke — softening its spring cancellation as the stroke grows, rather than switching it off — avoids the step altogether, and is the subject the next question raises.

Still open: a law that softens instead of switching #

Everything here is a switch: the spring is either cancelled or it is not, and every transient comes from the instant between. A law that cancels less of the spring as the stroke grows — all of it at small strokes, none at the limit — never steps, and it would look at the edge of its travel exactly like the partly cancelled springs that the partial-spring search found worst of all. Whether a stroke-dependent cancellation can stay near the free-mass law’s performance in a moderate storm and degrade towards the passive mass in a large one without passing through the unstable partial laws on the way — and whether a law that is nonlinear in its own stroke keeps any of the stability margins against delay that the linear laws were chosen for — is the question that would replace the switch with a slope.