Generator

The frequency-response generator

Rendered here at the parameters it defaults to, with every essay that calls it — which is the same list as the blast radius of changing it.
How much a harmonic force is magnified, at four damping ratiosDisplacement amplitude divided by the deflection the same force would produce if it were applied slowly, against the ratio of the forcing frequency to the structure's own, at 2%, 5%, 10%, 20% of critical damping. At the natural frequency the magnification is 25, 10, 5, 2.5 respectively — one over twice the damping ratio, and nothing else in the problem enters it.00.511.522.530510152025forcing frequency ÷ natural frequencyamplitude ÷ static deflection2% damping — 25.01× at the peak5% damping — 10.01× at the peak10% damping — 5.03× at the peak20% damping — 2.55× at the peak

7 essays call frequency-response. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

00.511.522.530510152025forcing frequency ÷ natural frequencyamplitude ÷ static deflection1% damping — 50× at the peak2% damping — 25.01× at the peak5% damping — 10.01× at the peak10% damping — 5.03× at the peak Dynamics

The only thing that stops it

Drive a structure at its own frequency and the amplitude grows without limit unless something takes energy out. What takes it out is damping, and damping is the one structural property that is never designed, never drawn, and never known until the thing is built.

2345678910051015202530floor frequency (Hz)response factor1× pace2× pace3× pace4× pace4 Hz — R = 106 Hz — R = 7 Dynamics

The floor that is strong and unusable

A floor can satisfy every strength check, deflect less than the limit, and still be rejected by the people who work on it — because somebody walking across it at two steps a second happens to be exciting it at exactly the rate it likes to move.

10⁻³10⁻²10⁻¹10¹00.20.40.60.81frequency (Hz)n·S(n), scaled to its own peakthe structure, 0.2 Hza stiffer one, 2 Hzbackground — the structure following the gusts: 70% of the turesonant — the structure ringing: 90% of the responsegust factor on the mean response: 4.39 Dynamics

The wind is a spectrum

A wind load is quoted as a pressure, which suggests something steady. It is not — the energy is spread across four decades of frequency, almost all of it in gusts lasting minutes, and a tall building takes ninety per cent of its response from the sliver of that energy sitting at its own frequency.

024681012141618200123shedding frequency (Hz)1.2 m across · St 0.18 · 0.9 Hzlock-in6 m/s02468101214161820050100150200wind speed (m/s)cross-wind amplitude (mm)0.40% damping — Sc = 11.2, peak 180.94 mm Dynamics

The wind that brings its own frequency

Every other load in this subject arrives at whatever rate it happens to arrive at. Vortex shedding arrives at a rate set by the wind speed — so for any chimney, mast or cable there is always a wind speed at which the shedding matches the structure exactly, and it is a breeze rather than a storm.

0.60.70.80.911.11.21.31.41.501020304050forcing frequency ÷ the structure's ownamplitude ÷ static deflectionno absorber — peak 503.0% absorber — peak 7.34a factor of 6.8, for 3.0% of the mass Dynamics

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.

00.511.522.533.540510152025forcing frequency ÷ natural frequencyforce out ÷ force in√2 — nothing gained, at any damping2% damping5% damping20% damping Dynamics

The machine that shakes the building

Put a machine on springs to keep its vibration out of the floor, and below a frequency ratio of root two the springs make things worse. Every transmissibility curve ever drawn passes through exactly one at that ratio, whatever the damping — so a soft mount either works well or fails badly, with nothing in between.

024681012-0.01-0.0050.005wind speed (m/s)total damping ratio0.60% damping · 1.2 Hz4.14 m/s — nothing leftbelow: a disturbance dies awayabove: the structure drives itself Dynamics

The motion that feeds itself

A steady wind contains no frequency at all, and it can destroy a bridge. The force that does it is manufactured by the structure's own movement, so there is no excitation to resonate with — there is a wind speed above which the equilibrium is unstable, and below which nothing happens.

The library, page 2 of 4 — where frequency-response sits