Structural Engineering & Statics
where the load goes
Concept

Frequency response — where it appears

A structure's steady response to a harmonic load or ground motion of unit size, plotted against the frequency of that excitation. Its peaks sit at the natural frequencies and their widths are set by damping, so it is where two close modes, or an analysis that has discarded part of the damping, can be compared with the exact response.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The damping that stops helping. The storey drift's white-noise root-mean-square as the bearings' damping is raised from 2 to 50 per cent, each divided by the exact value at 2 per cent. The classical analysis promises that every increment helps, and at 50 per cent predicts 0.20 of the lightly damped drift. The exact analysis flattens: 0.34 at 20 per cent and 0.29 at 50, because the damping force at the bearings is transmitted into the superstructure's own mode, which the classical analysis has decoupled from it.

The damping that belongs to no mode

Give every mode its own damping ratio and throw the rest of the damping matrix away, and an isolated building's periods and damping come out right to within a fifth of a per cent. Its storey drift at the superstructure's frequency comes out six times too small. The bearings' dashpot pushes on both modes at once, and the more of it there is the less extra damping buys: the classical analysis promises the drift keeps falling, and it stops.

dynamics · Mode shapes Bearing damping that shakes the roof harder. The roof's white-noise root-mean-square acceleration as the bearings' damping rises from 3 per cent to 58 per cent, exact and classical, both divided by the exact value at 3 per cent, for the six-storey building. The classical analysis falls all the way, to 0.33. The exact one falls to a minimum of 0.44 at about 28 per cent and then rises, to 0.50 at 58 per cent: past the minimum, every extra per cent of damping at the bearings shakes the top floor harder, while the classical analysis says it is still helping.

The top floor the bearings shake

On a building of several storeys, the damping at the isolation bearings is tied less tightly to each higher structural mode than to the one below. It still does more harm in each, because the ground barely reaches those modes on its own. The error the usual analysis throws away gathers in the upper storeys, and past about a third of critical damping at the bearings the roof shakes harder while the analysis says it is still helping. How much damping is best depends on how tall the building is.

dynamics · Mode shapes A mode that points wherever the asymmetry points. A square floor 24 m on a side, equally stiff in both directions, with a period of 0.80 s. Four copies are drawn on top of each other, each made stiffer by one part in 1,000,000 along a different direction: 0°, 20°, 45°, 70°. The first mode each one returns lies along 90°, 110°, 135°, 160° — at right angles to its stiffening, whatever the size of it — and the two periods differ by one part in 1,000,000. Four structures no instrument could tell apart have first modes pointing four different ways. The floor with no asymmetry at all has no first mode: every direction is one.

Two modes that are really a plane

A building equally stiff in both directions has two translational modes with one period, and they are not a pair of shapes but a whole plane of them. The pair an analysis returns is chosen by asymmetries of a millionth, so any result that depends on the pair — a square-root combination, a comparison with measured modes — inherits a choice the building never made. Damping then decides whether the difference can be seen at all.

dynamics · Mode shapes

Named alongside it

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

Modal analysisMode shapeOrthogonalityBase isolationDampingEigenvalueNatural periodResonanceDegrees of freedomModal combinationModal massPrincipal axes

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