Structural Engineering & Statics
where the load goes
Concept

Feedback — where it appears

Making a force depend on a measurement of the motion it is acting on, so that the response feeds back into its own cause. In structural control it is what turns an actuator into a damper, and the delay in measuring and acting decides whether the loop calms the structure or drives it.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

What the actuator pushes against decides its limit. The largest feedback gain, as the damping it would add to the bare mode, at which the loop stays stable, against the delay as a fraction of the period, on a logarithmic scale, for a mode of 1.00 per cent damping carrying a tuned mass of 2.00 per cent of its modal mass at Den Hartog's tuning. Pushing against the ground, fed by the structure's velocity: 2.41 at a delay of 0.05, 0.25 at 0.2. Pushing against the tuned mass, fed by the same velocity: 0.0156 at almost no delay, rising to 0.091 at a quarter of a period. Pushing against the tuned mass, fed by the velocity across it: unlimited with no delay, 0.048 at 0.05, 0.0073 at 0.2.

The actuator that pushes against the mass

An actuator that damps a structure by feeding back its velocity has to push against something. Against the ground, it is stable up to large gains until a delay of a quarter period removes its damping. Against a tuned mass — the arrangement meant to keep a passive damper underneath — the same feedback goes unstable at a gain of a per cent and a half with no delay at all, makes the structure worse than the passive mass at every gain below that, and drives the mass three or four times as far. What limits it is not when it acts but what it pushes against.

dynamics · Tuned mass damper Every law with the spring cancelled, stroke against peak. Each point is one law for a structure of one mode with 1.0 per cent damping and a tuned mass of 2.0 per cent of its mass, tuned and damped by Den Hartog's rules with the mass's spring cancelled, its structure-velocity and relative-velocity gains on a grid: its peak response against the mass's peak stroke, both as multiples of the static deflection. Filled, the 731 laws that need no more than 0.50 of the applied force; faint, the 730 that need more. The passive mass is the ring, at 45.3 and 8.72. The lower edge of the filled cloud is the best a law of this kind can do for each stroke; at the passive stroke it is 7.38, and it falls steeply as the mass is given more room.

The law that throws the tuning away

An actuator that pushes against a tuned mass and listens only to the structure pumps the mass's own mode. A law that also listens to the mass can be made safe in one move — cancel the mass's spring, so that it has no mode to pump — and every such law is stable. But cancelling the spring throws away the tuning that made the mass useful, and the actuator has to earn it back: below a third of the force on the structure, the law is worse than the passive mass it replaced. What it buys above that, it buys mostly by letting the mass travel further.

dynamics · Tuned mass damper The moment the law lets go. The response of a structure of one mode with 1 per cent damping and a tuned mass of 2 per cent of its mass, its spring cancelled by an actuator held to 0.50 of the applied force and a stroke of 1.50 times the passive mass's, per unit static deflection, under a steady harmonic force at 0.90 of the structure's frequency, through a health check failing after sixty periods, when the actuator lets go and the mass is left on its own spring. With the law running the structure moves 4.63; left to the passive mass in a steady state it would move 6.97. In the cycles just after the switch it moves 8.78, more than either: the free mass's large stroke is handed to a spring that holds it at a frequency the forcing is not at, and the two beat before the passive state settles.

The switch that hands the mass back

An actuated tuned mass that cancels its own spring does far better than the passive mass, and the passive mass is always underneath it, so the obvious safeguard is a switch: when the stroke nears its limit or the controller fails, let go and leave the mass on its spring. The switch does not land on the passive mass. For several periods after it, the structure moves more than it would with the law or with the passive mass alone — up to the passive mass's own worst peak, at frequencies where the passive mass would have been comfortable. Letting go gradually is worse still, and a switch that is slow lets a growing storm through.

dynamics · Tuned mass damper Taking hold of a swinging mass throws it. The largest stroke of a tuned mass of 2 per cent, as a share of its limit, after the free-mass law takes hold of it while it swings on its own spring and damper in a steady storm at the structure's frequency, against the instant in the structure's period at which the law takes hold; dashed, the law waiting for the stroke to cross zero, and for the mass to come to rest relative to the structure. Depending on the instant the stroke reaches 1.39 to 3.12 times the limit; waiting for zero stroke, 2.81; waiting for rest, 2.27.

Taking hold is harder than letting go

A tuned mass driven by a law that cancels its spring is safe to let go of: the transient is the same whenever the switch falls, and the passive mass takes over. Switching the law back on is the reverse operation and behaves nothing like it. The mass the spring was holding is thrown up to three times past its stroke limit, by an amount that depends on the instant, and a switch that is allowed to come back in a storm near its threshold comes back hundreds of times.

dynamics · Tuned mass damper

Named alongside it

The objects these essays reach for when they reach for this one.

ResonanceTuned mass damperActive controlActuatorCollocationDampingStabilityStrokeTime delayTransientBeating

All concepts