Structural Engineering & Statics
where the load goes
Concept

Beating — where it appears

The slow rise and fall of a vibration's amplitude when two motions of nearly equal frequency are added together. At the peaks of the beat they are in phase and add; between them they cancel, and the beat's period is set by the difference of their frequencies.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The resonance that ran out of time. The response of a 0.100 s oscillator at 2.0% damping while the driving frequency sweeps up through its own, plotted against the driving frequency rather than against time. The steady-state amplification is 1/2ζ = 25; this sweep reaches 23.6, which is 95% of it, because the time spent inside the half-power band is a limited number of build-up time constants. The whole answer depends on one group, β/ζ²ω², and over the range this figure's sibling sweeps the fraction falls from 100% to 41%. Two features fall out of the integration and neither is guessable from the steady-state picture: the peak arrives after the frequency has passed resonance, by 1.2% of it here, and the response beats afterwards at the difference between the two frequencies. A machine's instrument therefore reads its largest amplitude while it is already above its critical speed.

The resonance that ran out of time

A resonance amplifies by one over twice the damping, which for a lightly damped structure is fifty or a hundred. That is a steady-state answer and it takes time to arrive — with a time constant containing the same small damping — so nothing that sweeps through a resonance ever collects all of it, and past a certain rate the peak reached stops depending on the damping at all.

dynamics · Transient resonance 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

Named alongside it

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

ResonanceActuatorBuild-upCritical speedDampingDynamic amplificationFatigueFeedbackHalf-power bandwidthImpact factorModal massMoving load resonance

All concepts