The actuator that pushes against the mass

The actuator that arrives late replaced a tuned mass with an actuator: measure the structure’s velocity, push back on it in proportion, and the structure is damped as heavily as the actuator is strong. Delay was its undoing. Velocity feedback that arrives late adds damping in proportion to the cosine of the delay’s phase, so a quarter of a period of delay removes all of it, and more than that turns damping into its opposite.

It ended on the arrangement practice settled on: keep a passive tuned mass, and put the actuator on it. The passive mass does most of the work and is stable on its own; the actuator only nudges it. It asked “whether a delay that would destabilise pure velocity feedback is harmless in a hybrid — because the mass’s own quarter-cycle of lag and the actuator’s add up to something the mass can absorb — or whether the hybrid inherits the same limit a little later.”

The answer turns out to be neither. Put the same actuator on the mass and it has a far lower limit than delay ever gave it, and it has that limit with no delay at all.

Three places to push #

The structure is a single mode of unit mass and unit frequency with 1 per cent damping, carrying a tuned mass of 2 per cent of its modal mass, tuned and damped by Den Hartog’s rules. An actuator applies a force proportional to a velocity measured at an earlier time. The gain is written as the damping it would add to the bare mode if it had no delay — a gain of 0.05 is five per cent of critical — so that the three arrangements are compared on one scale.

Against the ground. The actuator pushes on the structure and reacts against something that does not move, and it is fed by the structure’s own velocity. This is the delay essay’s actuator; the tuned mass is left out.

Against the tuned mass, fed by the structure. The actuator sits between the structure and the tuned mass, pushing on the structure and reacting on the mass — the active mass driver — and it is still fed by the structure’s velocity.

Against the tuned mass, fed by the velocity across it. The same actuator between structure and mass, fed by the relative velocity of the two — an active damper in parallel with the passive one.

What each can stand #

The grounded actuator is the delay essay’s curve: stable to very large gains when the delay is small — 2.41, more than twice critical damping, at a twentieth of a period — and falling steeply as the delay approaches a quarter of a period, where its damping has gone.

The actuator on the tuned mass, fed by the structure, is a different kind of curve. At almost no delay it is stable only below a gain of 0.0156 — a hundred and fifty times less than the grounded actuator with a twentieth of a period of delay. And its limit rises as the delay grows, to 0.091 at a quarter of a period, before falling again. The hybrid is not limited by its delay. It is limited by what it pushes against, and a delay, far from being its undoing, moves it away from its limit.

Why pushing against the mass is different #

The explanation is in who does work on whom. Against the ground, the actuator pushes on the structure with a force opposite to its velocity, and the only work it does is on the structure, negative, always: it can only take energy out. That is what makes undelayed velocity feedback unconditionally stable, and it is why the delay essay’s instability needed a delay at all.

Against the tuned mass, the actuator pushes the structure one way and the mass the other with the same force. On the structure it does negative work, as before. On the mass it does positive work whenever the mass is moving in the direction of the structure’s velocity — and in one of the two modes the tuned mass creates, the mass and the structure move together, the mass with a much larger amplitude than the structure. In that mode the actuator’s total work over a cycle is the structure’s velocity times the difference between the structure’s velocity and the mass’s, and the mass’s is larger: the actuator pumps energy into the mode it is meant to be damping, through the very mass that was meant to make it safe.

The pumping is opposed by the tuned mass’s own passive damper, and the instability arrives when the actuator puts in more than that damper takes out. That is why the limit is so low. It is set not by the structure but by the tuned mass’s own damping, which for a mass of 2 per cent is 8.4 per cent of the mass’s own critical, on a mass a fiftieth of the structure’s.

The same accounting explains the third arrangement. Fed by the velocity across the mass, the actuator pushes on each body with a force opposite to their relative velocity, and its work is the force times that relative velocity, negative always. It measures what it pushes on — the actuator and the sensor are collocated — and like the grounded actuator it is stable at any gain until it is delayed. Delay then does to it exactly what it did to the grounded one, and sooner, because the relative motion it measures is fast.

Worse than the passive mass, at every gain #

Stability is only a boundary. The more telling result is what the hybrid does inside it.

The passive tuned mass leaves the structure’s response with two equal peaks, at 8.72 times the static deflection, which is what Den Hartog’s tuning is: the best that mass can do. Add the actuator at half a per cent of gain and one peak rises to 13.42; at one per cent, to 27.01. The peak that rises is the one belonging to the mode the actuator pumps. The actuator never helps. Every gain it is given, from nothing up to its limit, leaves the structure responding more than the passive mass alone would have let it.

The curves say the same thing at every delay, and they say it most mildly at the longest. With a fifth of a period of delay, one per cent of gain costs only 9 per cent on the peak response, because the delay has turned the actuator’s force partly into a stiffness rather than a negative damping of the pumped mode. A delay that would ruin a grounded actuator protects this one — not because the delay is good, but because the feedback it delays was wrong for this arrangement from the start.

The mass the actuator drives #

The tuned mass’s own travel shows where the pumped energy goes. With the passive mass alone its stroke peaks at 45 times the structure’s static deflection — a tuned mass moves a long way, which is why it needs room. The actuator at half a per cent raises that to 74, at one per cent to 183: four times the travel for a structure that is responding three times as much. In a real damper the stroke is limited by end stops, and a mass driven into them is a mass hammering the structure it was put there to calm.

A limit set by the mass #

The limit scales with the mass pushed against. Across tuned masses from half a per cent to ten per cent of the modal mass, the largest stable gain is between nine tenths and two thirds of the mass ratio: an actuator can add, in damping, not much more than two thirds of the mass it reacts against. For a tuned mass of half a per cent of the modal mass — a size tall buildings do carry — that is less than half a per cent of damping, and the feedback that seemed the obvious thing to try cannot even deliver that without first making the structure worse.

A man-made flutter #

The mechanism has been met before in this collection, delivered by nature rather than by a controller. A structure goes unstable in wind when the air’s force on one of its motions does positive work on another — the motion that feeds itself is a force proportional to a velocity, applied where it does not oppose that velocity, and the energy it pumps into a mode grows until something else limits it. A footbridge goes unstable under a crowd when the walkers’ sideways force falls into step with the deck’s sway and pushes in the direction the deck is already moving.

Even the wind that brings its own frequency, which is a forced vibration rather than a self-excited one, becomes self-excited once the structure’s motion starts to organise the vortices it sheds. The actuator on the tuned mass is the same thing built on purpose. It applies a force proportional to one body’s velocity to another body, and in the mode where the other body moves with the first and further, that force does positive work. What makes the grounded actuator safe and this one not is the same thing that separates a damper from a flutter mechanism: whether the force is applied to the motion it measures. The structure’s several periods — here two, the structure’s and the mass’s, split by the tuning — are what give it a mode to pump.

A check that needs no model #

The accounting that explains the instability also gives a test that can be applied to any proposed arrangement before it is analysed. For each actuator, write down the power it delivers: its force times the velocity of every point it pushes on, summed with signs. If the force is proportional to the velocity it measures, and it pushes only on the point it measures, the power is minus the gain times that velocity squared — negative, whatever the motion — and without delay the loop cannot be unstable. That is the grounded actuator and the actuator fed by the velocity across the mass it pushes on.

If the actuator measures one velocity and pushes on a point whose velocity is different, the power is the gain times a product of two different velocities, and there is some motion of the structure in which it is positive. Whether that motion is one of the structure’s modes, lightly enough damped for the actuator to win, is then the question an analysis has to answer — and for an actuator reacting against a small tuned mass the answer is yes, at a gain of about two thirds of the mass ratio.

Measure what the actuator pushes on, or include in the law everything it pushes on. Either restores the negative power the grounded actuator had for free; the simple hybrid does neither.

Why hybrids are built anyway, and how #

None of this condemns the active mass driver. It condemns feeding it the structure’s velocity alone. The hybrid dampers that have been built on tall buildings and bridges are controlled by laws that include the tuned mass’s own motion — its displacement and velocity relative to the structure — alongside the structure’s, and the gains are designed with the tuned mass inside the model rather than outside it. Such a law can make the mass behave as though it were heavier than it is, or its damping better tuned than a passive dashpot allows, and it is stable because the controller knows about the mode it could otherwise pump.

The simple law fails for a reason that is general. A feedback law designed for a structure without the device that applies it is designed for the wrong structure. The grounded actuator and the structure form one system whose modes are the structure’s, so feeding back the structure’s velocity damps them. The actuator on a tuned mass and the structure form a system with a new mode, the mass’s, and feeding back the structure’s velocity alone does not see it. The same trap is familiar from damping that belongs to no mode: a force applied between two parts of a structure couples its modes, and a law that assumes they stay separate is assuming away the coupling it creates.

The one-line version, by hand #

The reason the limit scales with the mass can be seen from the work done in the pumped mode. Let the structure move with velocity amplitude $v_1$ and the mass with $v_2$, nearly in phase, $v_2 \gg v_1$. The actuator, with force $g v_1$, does work at a rate of about $g v_1 (v_2 - v_1)/2$ on the pair, averaged over a cycle, and the tuned mass’s dashpot $c_d = 2\zeta_d \mu\omega$ removes $c_d (v_2 - v_1)^2 / 2$. The actuator wins when $g v_1 > c_d (v_2 - v_1)$, so the critical gain is $c_d$ times the ratio of the relative velocity to the structure’s in the pumped mode. Under Den Hartog’s tuning the damper $c_d = 2\zeta_d\mu\omega$ grows as $\mu^{3/2}$, since $\zeta_d$ grows as $\sqrt\mu$, and the mass’s travel relative to the structure’s falls roughly as $1/\sqrt\mu$: their product, the critical gain, goes as $\mu$. For the 2 per cent mass the damper is 0.0033 and the critical gain 0.031, so in the pumped mode the relative velocity is about nine times the structure’s. The scaling and the smallness of the limit both follow; the factor of two thirds to nine tenths is what the figure measured.

A single mode, a linear actuator and a rigid reaction #

The structure is one mode. A real building has several, most of its mass moving together in the first and the rest spread over modes of shorter period, and a tuned mass is tuned to one; the actuator’s force reaches the others through the structure, and with a structure’s velocity fed back from one sensor it reaches them with whatever phase their own motion at that sensor has. The mode-by-mode limits here are the ones for the tuned mode.

The actuator is ideal. It produces the force it is told to, at the delay it is given and no other lag; real actuators have their own dynamics — a hydraulic ram’s oil column is a spring, an electric motor’s current has a time constant — and each adds phase in the way delay does.

The tuned mass is a rigid body on a linear spring and dashpot. Real ones are pendulums, sliding masses on rails with friction, and tanks of water, each with nonlinearities that change the pumped mode’s damping with its amplitude.

What the pictures cannot show #

That the critical gains are for a perfectly tuned mass. A tuned mass that has drifted off its tuning — through a change in the structure’s frequency as it is fitted out, or its occupants change — has a pumped mode of a different shape, and the limit moves. The figures show the arrangement at its best, and it is already unstable at a per cent and a half.

Nor can they show the end stops. A tuned mass’s travel is limited by the building it sits in, and the strokes in the fourth figure — 183 times the structure’s static deflection at one per cent of gain — are strokes a real damper could not make. Long before the loop went unstable, the mass would be striking its stops, which is a nonlinearity that none of these linear curves follows.

Still open: the law that knows about the mass #

The actuator failed here because its law did not include the tuned mass’s motion. A law that does — feeding back the relative displacement and velocity across the mass as well as the structure’s velocity, with gains chosen together — can be stable, and can outperform the passive mass. How much it can outperform it, and how much of that advantage survives the delays that ruined the grounded actuator, is the question that decides whether an active mass driver is worth its power supply and its failure modes, or whether the passive mass, which cannot pump anything, was the better answer all along.