The only thing that stops it

Push a structure at any frequency but its own and the answer is bounded by the structure. Push it at its own, and the structure has nothing to say about the answer at all.

Everywhere on that plot except a narrow band, the damping is irrelevant to two decimal places. Inside the band it is the only thing in the answer. The stiffness has cancelled, the mass has cancelled, and the multiplier is $1/2\zeta$.

That is an awkward arrangement, because $\zeta$ is the one number about a structure that cannot be computed from a drawing.

Where the number comes from #

At resonance the force and the velocity are exactly in step. Every cycle the force does work over the whole distance travelled, in both directions, and nothing about the spring or the mass can take any of it away — the spring gives back at the end of each half cycle exactly what it stored at the beginning, and the mass gives back exactly what it took.

So the amplitude grows until the energy leaving per cycle equals the energy arriving per cycle, and the only thing that removes energy is the damper.

The build-up rate matters as much as the destination. A structure takes roughly $1/(2\pi\zeta)$ cycles to reach its steady amplitude — about eight cycles at 2%, and forty at 0.4% — which is why a machine passing through a resonance on its way up to speed does far less than a machine sitting on one. It is also why a gust that lasts three seconds cannot resonate a tower whose build-up takes a minute, a point the wind essay turns into a number.

The peak is not quite where everyone puts it #

Two exact statements are worth having, precisely because both are usually replaced by approximations that are correct to more decimal places than anyone needs.

The magnification peaks at a frequency ratio of $\sqrt{1-2\zeta^2}$, not at 1, and its height is $1/(2\zeta\sqrt{1-\zeta^2})$, not $1/2\zeta$. At 2% damping that puts the peak at 0.9996 of the natural frequency and 0.04% above the simple formula.

Which is to say: the approximations are excellent, and the reason to keep the exact forms is that the only way to know an approximation is good is to have the other one beside it. Past $\zeta = 1/\sqrt{2}$ there is no peak at all — the response falls monotonically from the static value — and a structure damped that heavily has no resonance to speak of. Nothing structural is remotely close; a car suspension is, deliberately.

Three real structures make the range concrete: a welded steel chimney at 0.4%, a bolted steel frame at 2%, a concrete frame at 5%. Their peaks are 125, 25 and 10 — a factor of twelve between the first and the last, produced entirely by which materials the joints are made of and what is hung on the frame, and not one of those three numbers can be computed from a drawing. The first of them is worth integrating rather than reading off a curve.

Where a structure’s damping actually comes from #

Almost none of it is the material.

Steel’s own internal damping, measured on a specimen in a laboratory, corresponds to a damping ratio of a few tenths of a per cent. Whole steel buildings measure at 1–2%, so most of what a structure has is not in the steel. The rest comes from:

Friction in the joints. Every bolted connection has faying surfaces that slip microscopically as the structure sways, and each slip converts a little energy to heat. A slip-critical joint is the extreme case of a mechanism that exists everywhere.

Cladding, partitions and finishes. A frame moves; the non-structural things attached to it are dragged along, deform, rub and crack. This is thought to be the largest single contribution in a typical building, and it is contributed by exactly the components that the structural model deletes.

Radiation into the ground. A foundation shaking the soil beneath it sends energy away as waves that never come back. For a stiff structure on soft ground this can exceed everything else, and it is the one mechanism that has nothing to do with the structure at all.

None of that is on a drawing, none of it is specified, and none of it is inspected. The consequence is that damping is quoted by building type — a bolted steel frame at 2%, a welded one at 1%, a concrete frame at 3–5%, a slender steel chimney at 0.4% — and the ranges quoted for nominally identical structures span a factor of five.

How long it rings #

The same ratio that fixes the peak fixes the memory.

Low damping and long memory are the same statement, and both follow from the same $\zeta$. The practical version is that a lightly damped structure integrates its loading where a heavily damped one follows it, and the loads that give trouble are therefore different in kind: a stiff, well-damped frame is troubled by the largest gust, and a slender, lightly damped mast by the steadiest wind.

Measured, by hitting it #

The only way to know is to make the structure move and watch it stop.

The arithmetic is exact and takes one line. If the amplitude falls from $u_i$ to $u_{i+n}$ over $n$ cycles, the logarithmic decrement is

$$\delta = \frac{1}{n}\ln\frac{u_i}{u_{i+n}} \qquad \zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}}$$

and for the small values structures have, $\zeta \approx \delta/2\pi$.

Two practical points hide in that formula and both are counter-intuitive.

The decay is measured on a structure nobody is loading, so the measurement is cheap: a few people bouncing in step on a footbridge, or a heel-drop on a floor, or the ambient traffic on a bridge, is enough. What is difficult is not the excitation but having an instrument on the structure at all.

Damping is amplitude-dependent, so the measurement is only valid at the amplitude it was made at. Friction mechanisms need slip to dissipate, so a structure that barely moves has almost no damping and a structure moving a great deal has more. Measured damping typically rises with amplitude, which means a value measured from ambient vibration underestimates what will be available in a storm — and overestimates nothing, which is the useful direction.

The loop, which is what damping is #

Draw force against displacement rather than either against time, and the mechanism becomes visible.

The area of that ellipse is $\pi c \omega X^2$: proportional to the frequency, and to the square of the amplitude. That last dependence is what makes the steady state stable. Double the amplitude and the energy in per cycle doubles, while the energy out quadruples, so there is exactly one amplitude at which they balance — and the structure finds it.

It also explains why viscous damping is a strange model for a structure. A viscous damper dissipates in proportion to velocity, so its loop area is proportional to the frequency; friction in a joint does not care how fast the joint slides, so its loop area is not. The whole subject nevertheless uses the viscous model, for a reason worth being honest about: it is the only damping law that keeps the equation of motion linear, and the error is absorbed into the fact that $\zeta$ was a guess in the first place.

Why nobody specifies it #

It would seem obvious to write damping into a specification, the way strength and stiffness are written in. Nothing does, and the reason is instructive.

A specified property has to be verifiable, and verifying damping means exciting the finished structure and measuring the decay — after it is built, after the cladding is on, after the partitions are in, which is to say after every opportunity to do anything about a shortfall has passed. A steel section that arrives undersized can be rejected at the gate. A frame that turns out to have 0.8% damping instead of 2% cannot be rejected at all.

So damping is handled the way every unverifiable quantity is handled: the value used in design is chosen low, and the consequences of the choice are made cheap. A low assumed damping gives a high computed response, which sizes the structure conservatively for a resonance problem and — importantly — makes a serviceability prediction pessimistic rather than optimistic. When the complaint is about comfort, being wrong in the pessimistic direction costs money and being wrong the other way costs a rebuild.

There is one significant exception, and it is the reason connection detailing turns up in a dynamics essay: the mechanisms that supply damping are mechanisms that also do damage. Joints that slip repeatedly wear; cladding fixings that absorb energy fatigue. A structure that relies on its accidental damping over millions of cycles is relying on components that were never designed to be dissipating anything, which is the same argument the fatigue essays make from the other side.

It is not one number, it is one per mode #

Everything above quotes $\zeta$ as a property of the structure. It is a property of a mode, and a structure has as many of them as it has freedoms.

The reason is in the list of mechanisms. Every one of them — joint friction, cladding drag, radiation into the ground — depends on where the structure is deforming, and each mode deforms it somewhere different. A building’s first mode racks every storey a little; a higher mode concentrates the drift into fewer storeys and works those joints much harder while leaving the rest almost still. Measured damping usually rises with mode number for that reason, though not reliably enough to predict.

Which creates a difficulty in the analysis rather than in the structure, and the way it is resolved is worth knowing because it is the same compromise the viscous law was.

A matrix analysis needs a damping matrix $[C]$, and the whole economy of modal analysis depends on the modes being uncoupled — each one responding independently, so a structure with fifty freedoms becomes fifty separate one-degree problems. An arbitrary $[C]$ destroys that: the modes talk to one another and the decomposition is gone.

So $[C]$ is chosen from the small family that preserves it. The usual choice is Rayleigh damping,

$$[C] = \alpha[M] + \beta[K]$$

which is uncoupled by construction because both $[M]$ and $[K]$ already are. What it gives, mode by mode, is

$$\zeta_n = \frac{\alpha}{2\omega_n} + \frac{\beta\,\omega_n}{2}$$

— a curve that is high at low frequencies, high at high frequencies, and has a minimum in between. The two constants are fixed by choosing two frequencies at which the damping is right, and everything else follows.

Nothing about a real structure has that shape. What has been chosen is the damping distribution that keeps the mathematics separable, and the physics has been fitted to it at two points.

There is a diagnostic in the shape that costs nothing. The minimum of that curve sits at $\omega = \sqrt{\alpha/\beta}$, and the two branches rise away from it — so any mode far from the fitting points is being given a damping the modeller did not choose and probably did not look at. Plotting the modal damping the model has actually assigned, against mode number, takes a minute and is almost never done; what it shows is a U-shaped curve with two points on it that anybody agreed to.

The consequence is specific and it runs one way. Fit at the first and third modes of a building and the fifth is over-damped, often heavily — its response is suppressed by the model rather than by the structure. Since higher modes are what carry a floor’s footfall response, a bridge’s local deck vibration and a building’s response to a short-period earthquake, that suppression is not always harmless, and it is invisible unless somebody plots $\zeta_n$ against $\omega_n$ and looks at where the fitting points were.

The honest description is the one the essay has already made about the viscous law, one level up. The damping law was chosen because it is linear, and the damping matrix because it is diagonalisable — and in both cases the error is absorbed into the observation that $\zeta$ was a measured guess to begin with. Which is defensible, and is worth knowing about, because it means a modal damping value taken from a table and a modal damping value produced by a Rayleigh fit are not the same kind of number, and only the first was ever anybody’s estimate of a structure.

What the picture cannot show #

The magnification curve is drawn as though $\zeta$ were a number. Three things it hides:

The curve is a steady state that may never be reached. At 0.5% damping the peak is 100, and getting there takes about thirty cycles of uninterrupted forcing at exactly the right frequency. Real excitations wander.

The peak is narrow, and narrower the lower the damping. The half-power bandwidth is $2\zeta$ of the natural frequency, so a 1% damped structure has a resonance a fiftieth of an octave wide. That is what makes resonance both dangerous and rare: it needs a coincidence, and the coincidence gets less likely as the consequence gets worse.

The band is drawn against a frequency the structure only approximately has. The period itself carries a ±20% uncertainty, and a resonance peak 2% wide inside a frequency band 20% wide is a peak that cannot be dodged by calculation. It can only be dodged by measurement, or by making the peak lower.

Which is the design response, and it is the honest one: stop trying to avoid the frequency and reduce the consequence of hitting it. That means adding damping, which is what a tuned mass damper is for, and it is why the retrofit of a bridge that swayed was dampers rather than stiffening.

The one place structural damping is designed #

Everything above is about damping that arrives by accident. There is one context in which it is specified, and it is worth naming because it inverts the whole essay.

Under an earthquake a ductile structure yields, and yielding traces a hysteresis loop that is not an ellipse but a parallelogram — with an area that does not depend on frequency at all and is very much larger than any viscous mechanism can offer. That is the argument the seismic field is built on, and it converts damping from a property one hopes for into a mechanism one detailed for.

The same substitution appears in what happens to steel after it has yielded once: the material’s willingness to trace a loop rather than snap back is what makes energy dissipation available, and it is the same property, read once as residual stress and once as damping.

Drawn on the axes of the loop figure above, the yielding mechanism is a parallelogram rather than an ellipse, and the difference shows in two ways at once. Its area does not depend on how fast it is traced, which is what makes it available at the long periods buildings have. And it does not return to where it started — every circuit leaves the structure displaced — where the ellipse closes on itself and leaves nothing behind.

The comparison is the reason the two are never alternatives. Viscous damping is available at every amplitude and costs nothing to the structure; hysteretic damping is available only after yield and is paid for in permanent damage. A structure designed on the second has, in effect, agreed to be repaired.

Where the ladder goes #

Damping decides the answer to every question in this field, and it is measured rather than designed. Three consequences propagate outward from that.

Every response computed here should be quoted with its damping assumption attached, because a response at 2% and a response at 5% differ by a factor of two and a half, and the difference is not a modelling refinement.

A design that depends on damping for its safety is depending on a number nobody checked. A design that depends on it for its comfort — which is most floor and footbridge design — is on firmer ground, because the failure mode is complaint rather than collapse.

And a structure whose damping is too low to solve the problem can be given more, deliberately, in a device sized for the purpose — a viscous damper across a brace, a mass on a spring in the roof plant, a liquid tank tuned to slosh at the right rate. That is one of the field’s few genuinely engineered answers, and it is engineered precisely because the accidental version could not be relied on.

It is worth ending on how strange the arrangement is. The subject’s most consequential parameter is one that structural engineering does not design, does not specify, does not inspect and cannot guarantee — and every calculation in this field is quoted to three significant figures on top of it.