Half a spring is neither tuning nor freedom

A tuned mass helps by being late: a small mass on a spring, tuned a little below the structure’s frequency and damped just enough, moves out of phase with the structure and pushes back on it at every cycle. A 2 per cent mass, tuned and damped by Den Hartog’s rules, takes a lightly damped mode’s resonant peak of 50 down to 8.72. An actuator between the mass and the structure can do better, and the way to make it do better safely is to cancel the mass’s spring in the law: the actuator applies a force equal and opposite to the spring’s, the mass has no mode of its own to be pumped, and every feedback law of that kind is stable. The price is that the tuning is thrown away, and the actuator has to earn it back: with half the applied force and the passive mass’s stroke the best law peaks at 7.38, and with half as much stroke again it reaches 4.65.

That essay ended on the obvious compromise. Cancelling the whole spring is the bluntest way to make the mass safe to push. A law could cancel only part of it — keep most of the tuning, so that the mass is still a respectable damper of its own, and add just enough feedback to improve on it. Somewhere between the passive mass and the free one there should be a family of laws that get most of both.

This essay searches that family.

The family, searched #

The law is the cancelling law with one change: the relative-displacement term cancels only a share of the tuning spring, $k_r = -(1 - \kappa)k_2$, so that $\kappa$ is the share left in place — 0 for the free mass, 1 for a mass whose spring is untouched. For each $\kappa$ the same grid of 441 laws is searched as before: a structure-velocity gain from nothing to 0.6, a relative-damping gain over two and a half decades. Of the stable laws whose actuator force stays within half the applied force and whose stroke stays within one and a half times the passive mass’s, the one with the lowest peak is kept.

There is no sweet spot. The best law gets steadily worse as more spring is kept — not smoothly, because the grid is coarse, but without a single share that beats the free mass. Half the spring kept costs a quarter of the free mass’s advantage over the passive mass; four fifths kept costs nearly three quarters of it. And from 85 per cent upward the best stable law is worse than having no law at all: 9.88 at 85 per cent, 12.24 at nine tenths. With the whole spring kept, the best the grid can do is the passive mass with a touch of extra relative damping, 8.81 — the law has nothing to add.

The compromise is not halfway between two good designs. It is a valley between them — or, read as a peak response, a ridge: the two ends of the family are the two designs worth building, and the middle is worse than either.

What the partial cancellation does to the mass #

The reason is in what a partial cancellation is. The spring left in place is $\kappa k_2$, so while the actuator runs the mass is tuned to $\sqrt\kappa$ of the frequency Den Hartog chose for it.

With nine tenths of the spring kept, the mass is tuned to 0.95 of its proper frequency, which is enough to move one of the passive mass’s two equal peaks up and the other down. The best law at that share has almost no structure-velocity gain — any more and it is unstable — so nothing pulls the high peak back down, and the structure is left with a resonance at 1.02 of its natural frequency standing at 12.24. A tuned mass is notoriously sensitive to its tuning, and mistuning by a few per cent undoes much of what it does — the sensitivity a vibration absorber shares with every tuned device; a partial cancellation is a deliberate mistuning applied by the controller.

With the spring fully cancelled the mass has no tuning to lose. It is a free mass pushed by a law, and the law can put its response wherever the force and the stroke allow, which is what the free mass’s curve shows: a single, lower peak below the natural frequency. With half the spring kept, the mass has half its tuning and half its freedom, and the curve sits between.

The laws that run away #

The partial cancellation also hands back the thing the full cancellation removed: a mode of the mass’s own.

With the spring cancelled, no law on the grid is unstable, which was the cancelling law’s whole point. Keep a share of the spring and the mass has a natural frequency again, and a law that feeds the structure’s velocity into the actuator is now pushing a mass that can resonate. The structure-velocity term that does all the useful work in the free-mass law becomes, past a certain gain, a pump on the mass’s mode, exactly as it was in the actuator that pushed against the mass with no law for the mass at all. The more spring kept, the lower that gain, and the larger the share of the grid that runs away: a quarter of it with half the spring kept, three fifths with the whole spring.

The map shows how little room is left. The unstable region takes the bottom right of the grid — high structure-velocity gain with little relative damping to keep the mass’s mode quiet — and the stable laws that also respect the force and stroke limits occupy a narrow strip near the left edge, where the structure-velocity gain is small. The best law sits on that strip’s boundary, which is the shape every constrained optimum has, and the strip is narrow enough that a gain error of a factor of two moves a law off it in one direction or the other.

The safe way from one end to the other #

The map also says how to get from the passive mass to the free one without passing through trouble, which matters when an actuated mass is commissioned on a real building and its gains are raised in steps.

Along the left edge of the map, where the structure-velocity gain is zero, every law is stable at every share of spring. There the actuator only changes the spring and adds relative damping, and a mass on a positive spring with positive damping is a passive system that cannot run away. Along the bottom of the family, with the spring fully cancelled, every law is stable at every gain — the cancelling law’s own result. So the path that goes down the left edge first, cancelling the spring with no structure-velocity feedback at all, and then along the bottom, raising that feedback once the cancellation is complete, never meets an unstable law.

The path that does both together — a little less spring and a little more gain at each step, the natural way to ramp up a controller — crosses the unstable region in the middle of the family, where a quarter to a half of the laws run away. The order of the two moves decides whether the commissioning is safe, and the intermediate laws a cautious engineer would pause at to check the response are the very ones that are not.

What the room is worth #

The cancelling law’s essay found that most of what an actuator buys, it buys with stroke: giving the mass more room to travel is what turns force into performance.

The free mass turns room into performance up to about one and a half times the passive stroke, and then the force limit stops it. The half-kept law stops sooner: past 1.5 times the stroke it has nothing more to spend the room on, because the gains that would use it are unstable. The four-fifths law is worse than the passive mass at the passive stroke and only catches up with it when it is given half as much room again. Extra stroke is worth something only once the spring is gone, because stroke is what a free mass is for and what a tuned one was trying not to need.

The delay, and why it flatters the wrong laws #

One thing does improve as the spring is kept, and it is the thing that makes the compromise look attractive in a robustness study.

The free mass’s best law loses its stability when the loop is a fifth of a period late — the delay that defeats every actuator eventually — and the part-cancelled laws survive longer: three tenths of a period with half the spring kept, nearly half with four fifths. A study that ranked laws by their tolerance of delay would rank the compromises above the free mass.

It would be reading the wrong thing. The part-cancelled laws tolerate delay because their gains are small, and their gains are small because larger ones are unstable. A law that is robust because it is doing almost nothing is robust in the way a disconnected actuator is robust. The comparison that means something holds the delay fixed and asks for the best performance. At a loop delay of a tenth of a period, well inside every law’s margin, the best free-mass law peaks at 2.87, the best half-kept law at 3.98 and the best four-fifths law at 6.91: the order is unchanged. (A modest delay happens to help every law in this family, by shifting the phase of the feedback towards the one a damper wants; that is a property of these gains, and the margin to the delay that destabilises them is what an engineer would actually budget.)

The power off, which every law survives #

The compromise was proposed for a failure, and the failure is the one thing it does not need to guard against.

In every law of this family the cancellation is done by the actuator. The spring itself is a physical spring between the mass and the structure, tuned by Den Hartog’s rule, and it is whole all the time. When the power goes off, the actuator’s force goes to zero, the cancellation goes with it, and the mass is the passive tuned mass again — 8.72 — whatever share of the spring the law was cancelling. A law that cancels the whole spring falls back to exactly the same passive mass as a law that cancels a tenth of it.

So the compromise buys nothing on power loss and costs a great deal with the power on. Its one real advantage is the delay margin just discussed, and that is better bought by limiting the free mass’s gains than by keeping some of the spring. The failures a partly cancelled law does change are the ones in which the actuator stays powered and does the wrong thing — a saturated amplifier, a sensor that drifts — and there a law with a quarter of its grid unstable is the more dangerous one to have.

The compromise done the other way round #

There is a second way to build the compromise, and it is the version a designer worried about the mistuning would reach for: make the physical spring stiffer, $k_2/\kappa$, so that after the law cancels its share the powered mass is tuned exactly as Den Hartog wanted. That puts the mistuning where the first version did not have it — in the unpowered state.

The arithmetic answers it without a new search. With the power on, the spring the mass feels is $k_2$, so the law’s family is the one with the whole spring kept, whose best member is the passive mass with a touch of extra damping: 8.81. With the power off, the mass is on a spring too stiff by $1/\kappa$ and tuned to $1/\sqrt\kappa$ of where it should be. For a spring stiffened to cover a cancellation of a tenth, the unpowered peak is 13.4; for one covering half, 41.6, nearly the 50 of the structure with no mass at all.

So the version that protects the powered state ruins the unpowered one and buys nothing in either. A tuned mass works because of the spring it has; a free mass works because it has none. Every law in between pays for some of each and gets the benefit of neither, and moving the spring’s stiffness around only chooses which state pays.

The arithmetic of a partial cancellation #

The retuning is a single line. The mass’s natural frequency on the spring left in place is $\sqrt{\kappa k_2/m_2}$, and Den Hartog’s tuning put $\sqrt{k_2/m_2}$ at $1/(1 + \mu) = 0.980$ of the structure’s frequency for a 2 per cent mass. With nine tenths of the spring the mass is at $0.980\sqrt{0.9} = 0.930$; with half, at 0.693. The passive mass’s two equal peaks depend on that tuning to within a few per cent, so a nine-tenths law has detuned the mass by about as much as a tuned mass can stand, and a half law has detuned it so far that the mass is no longer tuned at all — it is a heavily damped, softly sprung mass that the law has to drive.

That is the picture to carry. Near $\kappa = 1$ the mass is a slightly mistuned passive damper with a weak law on it, and mistuning costs more than a weak law can win back. Near $\kappa = 0$ it is a free mass with a strong law, and the law decides everything. In between, the mass is neither tuned enough to work by itself nor free enough for the law to place its response where it likes.

One mode, a perfect actuator and a grid that is coarse #

The search rests on choices that bound what it can claim.

The grid is coarse. Twenty-one values of each gain, over the same ranges the cancelling law used. A finer grid would find slightly better laws at every share, and smooth the unevenness in the family’s curve; it would not move the passive mass’s line, and the part-cancelled laws near the top of the family are worse than it by margins of 13 to 40 per cent, far more than a finer grid would recover.

The law has three terms. Structure velocity, relative velocity and relative displacement. A law that also fed back the structure’s displacement, or that changed its gains with frequency, is a larger family, and a part-cancelled law in that family might do better. The finding here is about the family the compromise was proposed in.

The structure has one mode and the actuator is perfect. A real structure’s second mode, and an actuator with its own dynamics, both narrow the stable region further — and narrow it most for the laws with a mode of the mass’s own to excite.

The end stops, and who pays for the power #

The figures cannot show the mass’s travel limit being reached. Every law here is held to a stroke, but the stroke is computed under a steady harmonic force; a gust or an earthquake that drives the structure harder than the design force drives the mass into its end stops, and a tuned mass on its stops is a mass bolted to the structure, doing nothing. The free mass uses more stroke to do its work, so it reaches its stops sooner in an overload, and that is the one argument for keeping some of the spring that this calculation does not answer.

They also cannot show cost. A free mass’s actuator does all the work all the time; a passive mass’s spring does it for nothing — and the damping the passive mass relies on is the one quantity in the subject nobody designs, while the actuator’s is a gain somebody sets. The cancelling law needs half the applied force to beat the passive mass, which on a tower under wind is a powered actuator of several hundred kilonewtons running continuously.

Still open: the mass that keeps its spring and switches its law #

Every law here is fixed. A practical alternative to a compromise is a switch: run the free-mass law when the motion is small and the stroke has room to spare, and drop the cancellation — handing the structure back to the passive mass — when the stroke approaches its limit or the controller’s health check fails. The passive mass is always there underneath, so the switch only has to choose between two designs, each good on its own terms. Whether the transition itself — a mass moving freely one instant and spring-mounted the next — injects a transient worse than either state, and how quickly the switch has to act for the overload it is protecting against, is a question about the moment of change rather than about either law.