Structural Engineering & Statics
where the load goes
Concept

Parametric excitation — where it appears

Driving a vibration by changing a property of the structure in time, such as a cable's tension, rather than by pushing on it. It is strongest when the property swings at twice the natural frequency, and growth then depends on how large the swing is against the damping.

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

The frequencies at which a moving deck makes a stay grow. Stability of the first mode of a 120 m stay at 3,500 kN inclined at 25°, whose own frequency is 1.01 Hz, when the deck at its anchorage moves at a frequency Ω against the stay's ω, plotted against the swing in tension the movement produces. Inside each shaded region a small disturbance grows. Each region's tip is at a swing of four times the damping ratio — 0.40%, 2.0% and 4.0% for damping of 0.10%, 0.50% and 1.0% — and it widens as the swing grows, to a deck between 1.970 and 2.030 times the stay's frequency at a 6.0% swing with 0.10% damping. With no damping, dashed, the region reaches down to no swing at all. A deck moving ±5, ±10 and ±20 mm vertically swings the tension by 0.70%, 1.4% and 2.8%. At ±10 mm and exactly twice the stay's frequency, marked, the stay grows with damping of 0.10% and settles with 0.50% and 1.0%.

The stay shaken along its own length

A deck that moves at a stay's anchorage pushes nothing across the stay. It stretches the stay along its own line and lets it go again, so the tension swings, and at twice the stay's frequency that swing drives the stay with no sideways force at all. Whether the swing grows is one comparison — a quarter of the tension swing against the damping ratio — and it is a comparison the capped damper wins.

dynamics · Cable dynamics 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.

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.

dynamics · Cable dynamics

Named alongside it

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

Cable dynamicsDampingResonanceStay cableViscous damperFatigueGeometric stiffnessNatural frequencySpectrumTaut string

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