Theme

The load that will not hold still — page 5

Essays 97 to 98 of 98 on this thread, in the same order.

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.

A sine's requirement grows with the deck, and a random deck's with its square. The damping a 120 m stay at 3,500 kN, 1.01 Hz needs to hold still, against the rms of the deck's movement at the anchorage, with the deck tuned to twice the stay. A sine of that rms: √2·σ·(dε/dx)/4, a straight line. Against it, a random deck with 1.0% damping of its own: the swing's variance over sixteen times the deck's damping ratio, a parabola for a typical history and, dotted, twice that for the mean square, with dots from integrating the stay and the deck together over five 2,000-second histories each. With the stay's own 0.10% the sine grows above an rms of 2 mm and the random deck above 9.1 mm — 6.4 mm for its mean square. At 7.5 mm the integration asks for 0.06% against an average of 0.07%, at 15 mm the integration asks for 0.25% against an average of 0.27%, at 23 mm the integration asks for 0.49% against an average of 0.61%, past the range the average claims and at 30 mm the integration asks for 0.71% against an average of 1.1%, past the range the average claims.

Dynamics

The deck that forgets its own rhythm

A deck at twice a stay's frequency grows the stay whenever a quarter of the tension swing beats the damping. But a deck pushed by gusts is not a sine: its amplitude comes and goes and its phase drifts, and it holds a rhythm for seconds where the stay needs minutes to answer one. Against that deck the requirement is not a quarter of the swing but its square over sixteen times the deck's own damping — a sixth of the sine's at service, and a number set by how much the deck remembers rather than by how far it moves.

All themes