Where dynamics actually matters in practice

Three kinds of loading

Periodic loading repeats. Rotating machinery, a reciprocating pump, a person walking at a steady pace. It is the dangerous class, because if its frequency happens to coincide with a natural frequency the response builds up cycle after cycle. Resonance is only possible with loading that repeats.

Transient loading happens once. A dropped load, a vehicle impact, a blast, a sudden change of support. The structure responds and then rings down. The peak response is what matters, and it usually occurs within the first cycle.

Random loading has no repeating pattern. Wind turbulence, earthquake ground motion, a crowd. It contains energy over a broad band of frequencies, so it will excite whatever natural frequencies lie in that band — but it does so intermittently rather than building up steadily.

The sources, and what they do

Pedestrian loading. People walk at 1.6 to 2.4 Hz, and run at 2 to 3.5 Hz. A footbridge or a long-span floor with a vertical natural frequency in that range will be excited directly. Worse, the second harmonic of walking lands around 4 Hz, which catches many composite floors. Lateral excitation is a separate problem at about half the pacing rate — around 0.9 Hz — and it is the mechanism behind the well-known lateral instability of some slender footbridges under crowds.

Machinery. A machine running at 1 500 rpm forces at 25 Hz. Rotating unbalance is the usual mechanism, and its force amplitude grows with the square of the speed, so a machine that is quiet at half speed can be violent at full speed. The critical period is start-up and shut-down, when the machine sweeps through every frequency below its running speed — including, possibly, the structure's.

Traffic and rail. Vehicles excite bridges both by their own weight moving across (which is a moving-load problem, not necessarily a vibration one) and by bouncing on their suspensions at 1 to 4 Hz. Rail is more demanding: sleeper passing frequencies and wheel flats produce narrow-band excitation at frequencies that can be predicted and must be checked.

Wind. Two distinct mechanisms, often confused. Buffeting by turbulence is broad-band random loading, and it excites whatever is there. Vortex shedding is periodic: a bluff body sheds vortices at a frequency proportional to wind speed, and at one particular speed that frequency coincides with a natural frequency and the response locks in. Chimneys, masts and circular sections are the classic sufferers.

Construction vibration. Piling, compaction, demolition and blasting produce ground-borne vibration that is transmitted into neighbouring structures. The concern is rarely the structure itself; it is the occupants, the sensitive equipment and — increasingly — the legal threshold agreed before work started.

Impact and blast. Short-duration, high-magnitude transient loading. Blast is included in this course as context only: it needs nonlinear analysis at strain rates where material properties themselves change, and it is a specialist subject.

Earthquake. Ground acceleration applied at the base, broad-band, lasting tens of seconds. The demanding case, and the reason for Stage C of this course.

Fatigue: the long-term consequence

A vibrating structure is a structure being cycled. At 3 Hz, a footbridge experiences about 10 million cycles a year of whatever stress range the vibration produces. Steel details have finite fatigue lives at stress ranges far below yield, and a detail that is perfectly adequate for a static check can accumulate damage indefinitely under a stress range of a few tens of newtons per square millimetre.

Fatigue is not a separate subject that happens to involve vibration. It is what vibration does over time, and the link between them is the stress RANGE and the number of cycles — both of which come out of a dynamic analysis and neither of which appears in a static one.

Monitoring: what it can and cannot tell you

Accelerometers on a structure will give you, reliably: natural frequencies, mode shapes if enough sensors are used, and damping ratios at the amplitude of the measurement.

They will not give you, without a great deal more work: the stress in any member, the loading that caused the response, or the damping at any other amplitude. Damping in particular is amplitude-dependent — a small-amplitude ambient test on a building typically reports 1 to 2%, and the same building in a storm may show 3 to 5%, because friction in cladding, partitions and joints only engages once things move enough.

That gap between measured and design damping is a recurring theme in this course, and it is the reason damping is the least reliable number in any dynamic model.