Structural Engineering & Statics
where the load goes
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Tuned mass damper — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 3% of the mass, hung on a spring, against the peak it removes. The magnification of a structure with 1.0% damping, with and without a tuned mass damper of 3.0% of its mass, tuned to 0.9709 of its frequency with 10.5% damping of its own. The bare peak is 50; with the absorber the single peak becomes two of 7.34, a reduction to 15% — a factor of 6.8. The marked points at frequency ratios 0.923 and 1.044 are the fixed points: the response there is the same whatever damping the absorber is given, which is what makes the optimum a question with an answer. The absorber's own stroke at the worse of the two peaks is 29.57 times the structure's static deflection, and that stroke is what decides whether it fits.

    The mass that helps by being late

    Hang three per cent of a building's mass from a spring in its roof, tune the spring so the mass arrives a quarter-cycle behind the motion, and the peak response falls by a factor of seven. Nothing was strengthened and nothing was stiffened.

    part 1 · dynamics
  2. One actuator, a tower and a floor. The peak amplification of a mode with 1 per cent damping under velocity feedback of gain 0.10, against the mode's frequency on a logarithmic scale, for control loops whose delay is 10, 25, 50 milliseconds. The dotted line is the mode without control, 50. With 10 ms: 4.6 at 0.2 Hz, 4.5 at 2 Hz and 5.0 at 8 Hz; with 25 ms: 4.6 at 0.2 Hz, 4.6 at 2 Hz and 19.0 at 8 Hz, unstable from 9.8 Hz; with 50 ms: 4.5 at 0.2 Hz, 5.3 at 2 Hz and unstable at 8 Hz, unstable from 4.9 Hz. A delay is a fixed time and a period is not, so the same loop that damps a tall building's sway is too late for a floor.

    The actuator that arrives late

    A tuned mass damps a structure because its force arrives a quarter of a cycle behind the motion. Replace the mass with an actuator told to push against the structure's velocity, and the same quarter-cycle is fatal: the push that was damping becomes stiffness, the stiffness becomes a source of energy, and a gain that would have cut the response tenfold makes the structure vibrate by itself. The delay is a few hundredths of a second, which is nothing to a tower and everything to a floor.

    part 2 · dynamics
  3. 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.

    part 3 · dynamics

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