The damper that is too near the end

The force read off a frequency uses a stay cable as an instrument: its frequency is a tension divided by a mass, so measuring one gives the other. This is about the other property of the same object, which is the one that causes trouble.

A stay cable has almost no damping. Its logarithmic decrement is a fraction of a per cent — perhaps 0.1 per cent of critical for a long stay with no ancillaries on it — because there is nothing in a steel cable under tension to dissipate energy: no joints, no friction, no cracking, and very little material damping. Damping is measured rather than designed everywhere in this field, and a stay is the one member whose measured value is small enough to be a design problem on its own. So a stay responds enormously to anything that excites it at one of its frequencies, and there are several such things: vortices shed at its own frequency, the rivulet of water that forms on it in rain and wind, and the deck moving under it at twice a stay frequency.

The remedy is a damper, and the damper cannot go where a damper should go.

Two per cent of the length buys one per cent of critical damping. That is the whole result, it is exact enough for design, and it is the reason stay-cable damping is a detailing problem rather than a mechanical-engineering one.

Which free body produced the number #

The free body is the cable, cut at both anchorages, treated as a taut string: a tension that does not change, a mass per unit length, and no bending stiffness.

Crossing the cut at the damper’s position is a shear discontinuity. The string’s slope changes there by whatever the damper’s force requires, and the damper’s force is its coefficient times the velocity at that point — which is what makes the problem complex rather than real: the force is out of phase with the displacement, so no real mode shape can satisfy it.

Writing the string as a sine on each side of the damper, matching the displacement and balancing the force, collapses to one equation:

$$\sin(kL) + i\,\eta\,\sin(kx_1)\sin(k(L-x_1)) = 0, \qquad \eta = \frac{c}{\sqrt{T\mu}}$$

with $k$ complex. Its real part is the wavenumber and its imaginary part is the decay, and the damping ratio is the ratio of the two. Every curve on this page is a root of that equation found numerically, and nothing about it is a fitted rule.

The undamped case is $\eta = 0$, which returns $\sin(kL)=0$ and the familiar $n\pi/L$. The infinitely stiff case is $\eta \to \infty$, which returns $\sin(kx_1)\sin(k(L-x_1)) = 0$ — the modes of the two segments separately, all of them real, all undamped. Both extremes have zero damping, which is why there is a maximum in between.

Why a support is not a damper #

That pair of limits is worth sitting with, because it is where the ceiling comes from and it is not obvious.

A very large damper does not move. It is a support, and a string with a support at $x_1$ has real modes and no dissipation whatever — the damper is not dissipating because nothing is happening at its location.

A very small damper moves with the string but exerts almost no force, so it dissipates almost nothing either.

Between the two there is a best size, at which the damper both moves and pushes. What is available at that size is bounded by how much motion there is at $x_1$ in the first place, and for a mode whose shape is $\sin(n\pi x/L)$ that is $\sin(n\pi x_1/L) \approx n\pi x_1/L$ — small, because $x_1/L$ is small.

Work it through and the modal damping comes out at $x_1/2L$, with the mode number cancelling: a higher mode has more motion at the damper and more cycles to dissipate over, and the two effects cancel exactly. The ceiling is the same for every mode.

Why the higher modes want a smaller damper #

The one feature of the first figure that is not explained by the ceiling is the spread of the peaks along the horizontal axis, and it is worth a paragraph because it decides how a single damper is sized.

A damper’s force is its coefficient times a velocity. At a given amplitude, a mode at four times the frequency moves four times as fast at the damper, so it produces four times the force from the same device — and a device that is optimal for mode 1 is over-damping mode 4 by that factor. The optimum coefficients therefore fall roughly as $1/n$: on the stay drawn they are 15.9, 8.6, 5.4 and 4.0 in the dimensionless variable, which is $L/(n\pi x_1)$ to two figures for every one of them.

That is a closed form worth having, because it turns the design into arithmetic. The optimum coefficient is $c^\ast = \sqrt{T\mu}\,L/(n\pi x_1)$, so a damper sized for mode 1 and a damper sized for mode 4 differ by four times — and the flatness of the peaks is what makes a single choice tolerable across the range.

Sizing for the middle of the range in use is the usual practice: a damper set at the optimum for mode 2 or 3 sits within 20 per cent of the ceiling for modes 1 to 5, which spans the frequencies rain-wind and vortex excitation actually occupy on a long stay.

What that costs on a real bridge #

The numbers make the constraint concrete, and it is tighter than it looks.

A damper is mounted between the cable and a guide pipe or a deck-level anchorage, so its distance from the end is set by architecture rather than by mechanics: two to four metres is ordinary, six is a large intervention, and ten is a visible piece of equipment on the deck.

On a 200 m stay, four metres is 2 per cent of the length and the ceiling is 1 per cent of critical. On a 100 m stay the same four metres is 4 per cent and gives 2 per cent. A long stay is the one that needs damping most and the one a damper helps least, because the constraint is a distance and the requirement is a fraction.

Is 1 per cent enough? For rain-wind vibration, mostly yes — the accepted criterion is a Scruton number rather than a damping ratio, and 1 per cent of critical on a heavy stay clears it. For fatigue at the anchorage under long-term low-amplitude vibration it is marginal, and it is the reason the other interventions exist.

The two things done when a damper is not enough #

Which is the practical consequence of a ceiling: when the ceiling is too low, the answer is not a better damper.

Cross-ties. Ropes connecting several stays into a network change the modes themselves — a stay tied at two points is a different string with a shorter free length, and its frequencies rise out of the excitable range. It is the one intervention that changes the eigenvalue problem rather than damping it, and it is visible on any recent cable-stayed bridge as a set of secondary cables in the plane of the stays.

And surface treatment. Helical fillets or a dimpled sheath disrupt the water rivulet that causes rain-wind vibration, which removes the excitation instead of the response. That is a different kind of fix entirely and often a cheaper one, because it is a property of the extruded sheath rather than a piece of equipment.

The hierarchy those two make with the damper is worth stating: remove the excitation if it can be identified, change the modes if the geometry allows, and damp what is left — with the last of the three subject to a ceiling nobody can raise.

The measurement that settles it #

None of this would be worth much if the ceiling could not be checked on a real cable, and it can — with the same instrument the rung below uses.

A stay is plucked or left to the wind, an accelerometer records it, and the decay of each mode is read off the record. That gives a damping ratio per mode directly, and it gives it before and after a damper is fitted. The measurements published for instrumented bridges land where the figures here say: a bare stay at 0.05 to 0.2 per cent, a damped one at 0.5 to 1.5, and the ratio between them very nearly the $x/2L$ of the installation.

What the measurement also settles is the part of the model most open to doubt. A real damper is not viscous, its mounting is not rigid, and the cable is not a taut string — so the honest expectation would be that the exact solution of an idealised problem overstates what is delivered. It does, by 10 to 30 per cent, and the shape of the answer survives: every mode within a few per cent of the same ceiling, and a ceiling proportional to the damper’s distance from the end.

That is the sense in which the idealisation earns its place. It is not accurate; it is structurally right, and it identifies the one variable worth arguing about at a design meeting.

What a designer does with a ceiling #

There is a particular discipline that follows from a limit of this kind, and it is worth stating because it is the opposite of the usual one.

Most structural quantities are improved by making a component bigger. Damping supplied at the anchorage is not, and the design conversation therefore has to be about geometry: how far out along the cable the damper can be taken, whether it can be internal to the guide pipe or has to be external, and whether the architecture will accept the bracket.

A metre of position is worth more than any conceivable improvement in the device. On the 200 m stay, moving the damper from 4 m to 6 m raises the achievable damping from 1.02 to 1.55 per cent — half as much again, for a bracket. No damper made can do that from where it was.

That is the same shape of finding as a brace that need not be strong — a component whose requirement is not its own capacity — and it belongs to the same family of arguments as the ideal stiffness of a brace: what the device can do is bounded by where the structure lets it act, and the bound is a property of the mode rather than of the device.

Where the model stops #

A taut string has no bending stiffness. A real stay has some, which raises the higher frequencies slightly and — much more importantly here — puts a bending moment into the cable near the anchorage. That is the region where the damper is, so the model is least accurate exactly where the intervention is.

And the damper is not viscous. Real stay dampers are viscous, friction, or elastomeric, and only the first behaves as this model assumes. A friction damper has a threshold below which it does not slip and above which it does, so its effective damping depends on the amplitude — which turns a linear eigenvalue problem into one where the answer depends on how hard the wind is blowing.

The cable is straight. A stay sags, its static profile is a catenary, and its dynamics couple to that sag through the same mechanism that gives a cable its geometric stiffness. The effect is small for a taut stay and large for a slack one, and the boundary is the Irvine parameter rather than anything on this page.

And there is no deck. A stay is anchored at both ends to structures that move, and a stay whose anchorage moves is being driven rather than left alone — the parametric excitation case, which is a Mathieu equation and behaves quite unlike anything here.

What the pictures cannot show #

They cannot show the amplitude, because an eigenvalue problem has none. What the damping ratio decides is how fast a motion decays and therefore how large a steady-state response is for a given excitation, and the excitation is a wind statistic rather than a load case.

They also cannot show what actually matters at the anchorage, which is fatigue. A stay vibrating at 1 Hz for a hundred nights a year accumulates ten million cycles in a decade, at a stress range set by the curvature of the mode near the anchorage — and the damper, sitting where it does, both reduces the amplitude and imposes a local curvature of its own. The detail category at the anchorage is what the whole exercise is protecting, and it appears nowhere in the calculation.

The assumption the figure rests on #

That the damper is a point.

It is not: it is a device with a length, a bracket, a bearing and a connection to the structure, and every one of those is a flexibility in series with the dashpot it contains. A damper with a soft mounting is a spring and a dashpot in series, which is a Maxwell element rather than a viscous one, and its effective coefficient falls with frequency.

The consequence is the usual one when a component is drawn as an ideal element: the delivered coefficient is smaller than the specified one, and by an amount nobody measures. It matters less here than it would elsewhere, and for a reason the figures give directly — the peak is flat. Being wrong by a factor of two about the coefficient costs 20 per cent of the ceiling, which on a design already limited by a length is a second-order concern.

That flatness is the most reassuring property of the whole arrangement, and it is worth carrying to other tuned devices: a tuned mass damper is not flat at all, and mistuning it by a few per cent costs a great deal. The difference is that a tuned mass has a frequency to be wrong about and a dashpot does not.

The history, which is a bridge that would not stop moving #

Stay-cable damping became a design subject in the 1980s and 1990s for a simple reason: cable-stayed bridges got long enough for their stays to be excited by ordinary weather, and several of them moved alarmingly in the first months of service.

The mechanism took a while to identify. Rain-and-wind-induced vibration — amplitudes of a metre, at wind speeds of 8 to 15 m/s, only when it is raining — was first documented in Japan in the early 1980s and looked like nothing in the aeroelastic literature: too slow for flutter, at the wrong Scruton number for galloping, and dependent on a variable nobody had put in a wind tunnel. The rivulet of water running down the upper surface of the sheath turned out to be a movable aerodynamic feature, and the cable and the rivulet together are an oscillator with a negative damping term.

The $x/2L$ result arrived alongside it, from Pacheco and his colleagues in 1993, as a universal curve: plot modal damping normalised by $x/L$ against damper size normalised by the cable’s properties, and every mode of every cable falls on one line with a peak of one half. That is the figure at the top of this page, drawn from the exact roots rather than from the fitted curve, and its value in 1993 was that it turned a problem needing a complex eigenvalue solution into a chart.

What is worth carrying from the sequence is the shape of it. A universal curve is what a problem looks like once the right variables have been found, and finding them is the work — the tension, the mass, the mode number and the coefficient all had to be shown not to matter separately before the one length that does could be seen.

The ladder from here #

Later rungs on this anchor: cross-ties in full, where several stays become one system with a sparser spectrum and the connections carry forces of their own. Rain-wind vibration as a mechanism, with the rivulet as a moving boundary and the aerodynamic derivative it changes. The Maxwell damper — a dashpot in series with a spring — which has its own optimum and its own reduced ceiling. Parametric excitation from a moving anchorage, where the tension varies in time and the stability boundary is a Mathieu chart. And the same ceiling argument applied elsewhere: any damper placed where a mode is small is limited by the mode rather than by itself, which is why a tuned mass goes at the top of a building and a viscous wall goes where the drift is.