Structural Engineering & Statics
where the load goes
Concept

Viscous damper — where it appears

A device whose resisting force is proportional to the velocity across it, so that it dissipates energy on every cycle of motion. Fitted near a stay cable's anchorage it has an optimum size, and the damping it can supply is capped by its position however large it is made.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

What a damper at the anchorage can do, and the ceiling it cannot pass. Modal damping against damper size for a 200 m stay at 4500 kN, with the damper 4 m from the anchorage — 2.0 per cent of the length. Each curve is a mode, found as a complex root of the taut string with a viscous damper in it rather than from a formula. Every one of them peaks at 1.00 per cent of critical, which is x/2L exactly, and the peaks are at different damper sizes — a higher mode wants a smaller damper, because it moves faster at the same amplitude. The curves are flat near their peaks: half the optimum coefficient gives 80 per cent of the ceiling, and so does twice it.

The damper that is too near the end

A stay cable has almost no damping of its own, so it is given a damper — and the damper cannot go where the motion is, because the middle of a two-hundred-metre stay is a hundred metres above the road. What it can supply is then decided by one length, and no amount of damper changes it.

dynamics · Cable dynamics 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 link that keeps two buildings apart has made them one. The largest closing movement between a 500 t building with a 0.8 s period and a 300 t building with a 1.2 s period, 50 mm apart, under one 1.0 s sine pulse of 0.50 g, and each building's largest displacement, against the size of a viscous damper joining them across the gap, from 0.01 MN·s/m to 541.3 MN·s/m. With no link they close by 515 mm, and the stiffer building moves 220 mm and the softer 419 mm. The link that takes the most energy out, 0.64 MN·s/m, still lets them close by 214 mm. The least that keeps the 50 mm gap is 4.8 MN·s/m, where the stiffer building moves 269 mm and the softer 280 mm. At the largest link the two move together, 281 mm and 281 mm.

The damper that ends up as a joint

A damper across the gap between two buildings acts on exactly the motion the gap is sized for, and it can be sized for two different things. The size that takes the most energy out of the pair still lets the buildings collide. The size that keeps them apart has nearly stopped moving: it has joined them into one building, and the stiffer of the two pays for it in drift.

dynamics · Pounding 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 A filler that knows the speed meets the collision hardest at its start. The contact force against the depth crushed in the hardest collision, 500 t and 300 t closing at 2.85 m/s with 761 kJ, on a 40 mm filler between 500 t and 300 t, crushable over 24 mm: uniform (dashed), with 50 per cent of its strength at that speed depending on the speed, and with all of it. Each is as strong as it must be to stop the collision in 24 mm. Uniform: 30.5 MN at first contact; 50 per cent on speed: 38.4 MN at first contact; all on speed: 63.4 MN at first contact. The crushing speed is highest at the first instant and falls to nothing, so a filler whose strength follows it spends its force at the start; the same area under a falling line needs a higher peak than under a level one, and a purely viscous filler's is twice the uniform's.

The filler that knows how fast it is hit

A filler graded through its depth meets the small collisions softly and the hard one with a strong layer that has to be very strong, because every kilojoule the soft layer misses must be absorbed in less depth. A filler whose strength rises with the speed it is crushed at ranks collisions by speed instead, with no layers at all, and treats the middling collisions better than grading does. But it meets every collision hardest at its first instant, when the crushing speed is greatest, and a force that falls as the collision proceeds needs a higher peak to absorb the same energy in the same depth. A purely viscous filler that stops the hardest collision in 24 mm hits it with exactly twice the force of a uniform one.

dynamics · Pounding A damper too large stops damping. The damping of a structure with 2.0% of its own, against the size of a viscous damper held by a brace — the damper's force over the brace's at the structure's own frequency, on a logarithmic scale — for braces a quarter as stiff as the structure, as stiff, and four times as stiff. With the brace at 0.25 the damping peaks at 7.9% for a damper of 0.86; with the brace at 1.00 the damping peaks at 22.7% for a damper of 0.58; with the brace at 4.00 the damping peaks at 63.9% for a damper of 0.29. Either side of the peak it falls: a small damper barely moves, and a large one barely strokes — it locks, and the brace carries the motion as a spring.

The damper is only as good as its brace

A viscous damper added to a frame is held by a brace, and the brace is a spring in series with it. That pair has a ceiling: the most damping it can add is (√(1+κ) − 1)/2, κ being the brace's stiffness over the frame's, reached at one damper size and lost on either side of it. A brace as stiff as the frame allows 20.7%; a quarter as stiff, 5.9%. A damper larger than the best one is not better — it strokes less, locks, and turns the brace into a strut that moves the resonance instead of flattening it.

dynamics · Damping Sized for one amplitude, a nonlinear damper damps only a band around it. The damping of a structure with 2.0% of its own and a brace as stiff as the structure, against the storey's amplitude over the amplitude the damper was sized for, on a logarithmic scale, for dampers whose force goes as the velocity to the power α, each sized to the brace's best at the design amplitude. All four give 22.7% there. The linear damper gives it at every amplitude. A tenth of the design amplitude, α = 0.5 gives 6.4%, α = 0.3 1.9% and α = 0.15 1.4%; a hundredth of it, 1.9%, 1.4% and 1.4% — below the structure's own, because the locked brace has stiffened it.

The damper sized for the earthquake

A fluid damper whose force goes as its velocity to the power 0.3 is chosen because it caps the force a large earthquake can push through its brace. The same exponent makes its effective size depend on how far it strokes. Sized to its brace's best at the earthquake's amplitude, it adds 20.7% of critical damping there, 5.4% at three-tenths of it, and nothing at all at a tenth — where the brace has locked, the building is stiffer, and the motion its occupants feel meets a resonance that is as sharp as an undamped frame's. Giving half the brace to a linear damper keeps 9.3% under the wind for 1.2 points at the earthquake.

dynamics · Damping The same damper behind a lever is best at a larger motion. The damping of one storey with 2% of its own damping and a brace as stiff as the storey, with an α = 0.3 damper sized to its brace's best at the design amplitude with no amplifier, against the storey's amplitude, with the same damper behind stroke amplifiers of ratio 1, 2 and 3. A ratio f makes the damper act on the storey as one f to the power 1 + α larger — 2.46 and 4.17 times — so it locks more, and its band moves up by about f to the power (1 + α)/(1 − α): its best is near 3.8 and 7.5 times the design amplitude. At a hundredth of the design amplitude all three give 1.4%; at the design amplitude 22.7%, 6.4% and 2.3%.

The lever that moves the damper and not the band

A damper whose force goes as a low power of its speed is sized for the earthquake and locks in the wind. A toggle or scissor linkage multiplies its stroke, and it seemed it might move the band of amplitudes the damper serves down towards the wind. It cannot. Behind a lever of ratio f the same damper acts on the storey as one f to the power 1 + α larger, so its band moves up, not down; re-sized to the design amplitude, it gives the storey exactly the curve it had without the lever, and what has changed is the damper — a quarter of the size, a third of the force, three times the stroke. What would widen the band is a ratio that grows with the motion, and a toggle has one — until its links lie straight and it locks.

dynamics · Damping

Named alongside it

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

DampingResonanceBraceCable dynamicsEnergy dissipationNatural periodStay cableEigenvalueFatigueLoss factorModal dampingMode shape

All concepts