The line of ties that stops short

A stay cable responds violently to anything that arrives at one of its frequencies, and its frequencies are an arithmetic progression with no gaps in it, so something always does. The damper that is too near the end showed why the obvious remedy is capped: a damper at the anchorage can supply at most its distance along the stay over twice the length, which is about one per cent of critical. It closed on the remedy that is not capped that way. Ropes strung across several stays join them into a network, and a network has different modes from the stays it is made of. The usual sentence that follows is that the ties lift the frequencies out of the excitable range.

That sentence is half true, and which half depends on something the sentence does not mention: whether the line of ties is anchored to anything.

A stiffer tie stops helping at 0.82 Hz #

The three stays drawn sit side by side on one plane of a fan. The shortest is the 120 m stay at 3,500 kN that had its tension read off its own frequency; the other two are longer, more heavily tensioned and heavier. On their own their fundamentals are 0.73, 0.85 and 1.01 Hz. A line of ties crosses each a third of the way up from its deck anchorage, and each tie is a spring between two neighbouring stays.

The longest stay is the problem, because its fundamental is the lowest and it is the one most likely to meet a low-frequency excitation. Tying it to its neighbours does lift its first mode. With ties of a kilonewton a metre the network’s first frequency is 0.74 Hz; at a meganewton a metre it is 0.82. After that the curve is flat. At a thousand meganewtons a metre, which is a tie far stiffer than any rope, it is still 0.82 Hz.

The stiffness axis spans six decades, and over the last three the ties are effectively rigid. Rigid ties do not make a rigid network. The first frequency has stopped at a value that is not any stay’s frequency and not any obvious combination of them, well short of the shortest stay’s 1.01 Hz.

Why no free line can pass the shortest stay #

The limit is not a feature of these three stays. It follows from Rayleigh’s method, which bounds a structure’s first frequency from above by any shape the structure could deflect into.

Choose a trial shape for the whole network in which each stay takes the shape of its own fundamental, a half sine along its length. Scale each stay’s sine so that all the tie points move by the same amount. In that shape no tie stretches, because every tie has both its ends displaced equally, so the ties store no energy and contribute nothing to the quotient. What is left is each stay’s own strain energy over each stay’s own kinetic energy:

$$\omega_1^2 \;\le\; \frac{\sum_i m_i\,\omega_i^2}{\sum_i m_i},$$

where $\omega_i$ is stay $i$’s own fundamental and $m_i$ the mass it moves in the trial shape. The right-hand side is a weighted mean of the stays’ own squared fundamentals, and a weighted mean cannot exceed the largest of them. For these stays the bound is 0.84 Hz, and the network’s true first frequency, with the ties as stiff as anything, is 0.82.

The other side is simpler. A tie is a spring, and adding a spring to a structure can only raise its frequencies, so the first frequency is never below the longest stay’s own. A free line of ties leaves the network’s first frequency somewhere between the lowest and the highest of the stays’ own fundamentals, whatever the ties are made of and wherever they cross. No tie stiffness changes that, and no position does either.

The contrast with a machine block that sways and rocks at once is exact, and it is worth having both in mind. There the two motions were joined through the block’s mass, because the mass sat above the base, and the joined system had one frequency below both separate ones and one above both. Here the stays are joined through springs between them, and a spring between two parts can only average the parts. Coupling through inertia spreads frequencies outward; coupling through a stiffness that nothing outside holds keeps the lowest one inside.

The stay as a spring whose stiffness changes sign #

What each stay presents to a tie is a dynamic stiffness: the force a tie would have to apply at the crossing point, at a given frequency, to move that point by a unit amount. For a taut string crossed at a distance $a$ from one end, with $b = L - a$ the rest,

$$K(\omega) = \frac{T\,k\,\sin kL}{\sin ka\,\sin kb}, \qquad k = \omega\sqrt{\mu/T}.$$

It is zero at the stay’s own frequencies, where $\sin kL$ vanishes: the stay moves freely there and needs no force at the crossing to do so. It is infinite at the frequencies of its two segments, where $\sin ka$ or $\sin kb$ vanishes: at those, the crossing point would have to be a node, and no finite force can move it. Between the two it changes sign, which is what makes a network’s frequencies interleave with the stays’ own rather than simply rise. Every frequency drawn here is a root of the matrix of these stiffnesses plus the ties, counted rather than searched for, so that two stays with the same frequency cannot hide a root from the count.

The stays move together, and moving together stretches nothing #

The mode itself shows why the bound is nearly reached. The three stays move in phase, each close to its own half sine, and the tie points move within 1.4 mm of one another. The ties are stiff and barely stretched. Stiffening them further has almost nothing to act on.

That is the whole mechanism of a free line in its first mode. The network finds a shape in which the stays move as nearly together as their different lengths allow, and it pays for the remaining mismatch with a small stretch of the ties. The frequency it settles at is a compromise between the stays’ own: the long stay is lifted from 0.73 Hz and the short stay is pulled down from 1.01. A stay that was clear of a 0.9 Hz excitation on its own is now part of a mode at 0.82 Hz, carrying 14 per cent of that mode’s motion.

The tie forces in this mode are small, 4.5 and 3.8 kN for every 100 mm of motion in the longest stay, for the same reason the frequency gain is small. A tie stores energy only by being stretched, and the energy it stores is exactly what the frequency gains over the stays moving freely. A tie that carries almost nothing has changed almost nothing.

Identical stays are the case where ties do nothing #

The mismatch between the stays’ own frequencies is what the ties have to work with, and the limiting case makes that plain.

Two stays of the same length, tension and mass, tied at the same point, have a first mode in which both move identically. The tie is never stretched in it, so the mode does not know the tie is there. The network’s first frequency is exactly each stay’s own 0.85 Hz, at any stiffness. The bound from the trial shape is reached with equality, because the trial shape is the true one.

Real stays in one plane are never identical, since each runs from a different point on the pylon to a different point on the deck. But neighbouring stays in a harp or a fan are often close, and the closer their frequencies are, the narrower the band a free line can move the first mode within. The benefit of a free line comes from the stays being different, and a group of similar stays gets little of it.

The modes that stretch a tie are the ones it moves #

The first mode is one of several, and the others behave differently, because in them the stays do not move together.

In the second mode the two longer stays move against each other where the tie joins them. That tie is stretched hard, and the mode has been pushed up to 1.17 Hz — above both of those stays’ own fundamentals — because the tie resists exactly the relative motion the mode consists of. This is the mode the usual sentence about cross-ties describes, and it is not the lowest.

It is also the mode in which a tie earns its force. At 14.4 kN for every 100 mm of stay motion, an amplitude of the size recorded in rain-wind vibration, several hundred millimetres, puts tens of kilonewtons into the tie on every cycle, alternately in tension and in compression. A rope cannot carry compression, so a tie is installed with a pretension, and the pretension has to exceed the largest dynamic force the tie will see. A tie that goes slack stops being a spring for part of every cycle, the network loses the stiffness the calculation assumed, and the tie snaps taut again on the return with an impact. That is the same failure as a guy that goes soft under the load it was meant to hold, arriving here on every cycle.

The pretension is not free either. At 500 mm of motion in this mode the tie’s force swings by 72 kN, so its pretension has to be at least that, and a tie pretensioned to 72 kN pulls sideways on the 180 m stay at the crossing whether anything is moving or not. A sideways force on a member in tension kinks it by the ratio of the two forces — the deviation force a tendon exerts where it changes direction, running the other way — so the stay is bent at the tie by 72 over 5,500 kN, about 0.013 radians. Small, but permanent, and in a stay whose anchorage detail was designed for a straight cable.

The spectrum keeps its lack of gaps #

The trouble with a stay was never only its first mode. A taut string’s modes are evenly spaced, so a stay 180 m long has a frequency every 0.73 Hz for ever, and an excitation that arrives anywhere in its range finds a mode within half a spacing. Vortex shedding, which brings its own frequency with the wind speed, sweeps across that range as the wind changes and meets one mode after another.

Joining three stays does not open gaps in that spectrum. The three stays between them have 11 frequencies below 3.5 Hz, and the network has 10. The frequencies have moved, interleaved and in places paired up, but the density is essentially what it was.

The reason is again a bound, and it is the same one that made the tie’s force its price. Each tie adds a single spring between two points, which is the smallest possible change to a structure’s stiffness, and a change of that size can push at most one frequency from below any given frequency to above it. Two ties can remove at most two frequencies from below 3.5 Hz, or below any other frequency, out of eleven. A stay’s spectrum is dense because a string has infinitely many modes. A handful of springs added to an infinite number of modes cannot make it sparse.

Where the line crosses changes how close each spring comes to its one. With the line a fifth of the way up each stay instead of a third, the free line removes both of the frequencies it is allowed from below 3.5 Hz, leaving 9, and the anchored line removes three of the four it is allowed, leaving 8. That is close to the most a line of two ties and two deck springs can do, and it still leaves eight or nine frequencies in a band where the excitation can arrive anywhere.

So the ties do not make a stay detunable in the way a beam with its large gap above the first mode is detunable. What they change is which modes exist in the low range, and how much each stay moves in them. Each of those modes still has only the stays’ own damping, which is almost none, and a device that acts on one mode at a time — a damper sized for it, or a mass tuned to it — meets the same density from the other side, because there is always another mode within reach.

Anchoring the line is what lifts the first mode #

The bound had one condition: the line of ties is free at its ends. Every spring in it runs between two stays. Carry the line on to the deck or the pylon, and the springs at its ends run between a stay and something that does not move.

The trial shape that proved the bound is no longer available. Moving all the tie points together now stretches the springs to the deck, so the shape stores energy in them and the quotient no longer reduces to a mean of the stays’ own frequencies. The first frequency goes to 1.09 Hz, above every stay’s own fundamental and half as much again as the longest stay’s. With the same anchoring, the two identical stays that a free line could not help go from 0.85 to 1.27 Hz.

The count has not changed, though. The anchored network still has 10 frequencies below 3.5 Hz. The deck springs removed the lowest frequencies and left the density above them as it was.

What anchoring really does is shorten the longest stay #

The mode that is now the lowest is not a network mode in any useful sense. It belongs to the longest stay alone, and in it the stay’s tie point has almost stopped moving. The stay is vibrating as the 120 m of its length above the tie, with the tie point acting as a near-support, and a 120 m length of that stay has a fundamental of 1.09 Hz.

That is the same limit the stay damper reaches when it is made too large: a device stiff enough stops moving and becomes a support, and a support divides a stay into two shorter stays. For a damper that was the failure, because a support dissipates nothing. For a line of ties it is the success, because a shorter stay has a higher frequency. An anchored line of ties lifts the first mode by turning the tie points into supports and the longest stay into its longest segment, and the frequency it reaches is that segment’s.

That also says where the next limit is. The anchored line can lift the longest stay’s first mode no higher than the fundamental of the longer of its two segments, which is set by where the line crosses it. A line at a third of the way up gives the segment two thirds of the length and the stay half as much frequency again.

Moved to mid-length on every stay, the same anchored line lifts the first frequency to 1.43 Hz, close to the 1.46 Hz of the longest stay’s 90 m half. It pays in two ways. The spring carrying the line to the deck beside that stay now takes 29.5 kN for every 100 mm of motion rather than 10.9, so the tie line needs nearly three times the pretension. And a line at mid-length sits at a node of every even mode of every stay, where it cannot act on them at all: the free network’s second frequency with the line there is 1.46 Hz, which is simply the longest stay’s own second mode, untouched. The position that lifts the first mode furthest is the one that leaves half of the others alone.

What the taut-string network leaves out #

The stays have no sag and no bending stiffness. Sag stiffens the low symmetric modes of a long stay through the cable’s elastic stretch, and it matters most for exactly the longest stays whose first mode the ties are meant to lift. Bending stiffness matters for the short ones.

The ties have no mass and no modes of their own. A long tie rope is itself a string with its own frequencies, and when one of them is near a network frequency the tie stops behaving as a spring.

The ties are always taut and always linear. A tie that slackens is a different structure for part of each cycle, and the network then has no single set of frequencies at all.

The motion is in the plane of the ties. A tie resists motion along its own line and very little across it, so a line of ties in the plane of the stays does almost nothing to motion out of that plane, where the stays keep their own frequencies unchanged.

Nothing moves at the anchorages. A deck and a pylon that move change the stays’ tension in time, which is a separate excitation with its own stability boundaries.

Still open: how much damping a line of ties adds, and whether it survives going slack #

Every network drawn here is undamped, and a line of ties is known in practice to add damping as well as stiffness — through friction at its clamps, through the rope’s own internal friction, and through the impacts of a tie that slackens and snaps taut. None of that is a viscous dashpot, and none of it is bounded by the ceiling that caps a damper at the anchorage. Whether the damping a tie supplies is a property of its detail that can be designed, or a by-product of the very slackening its pretension is meant to prevent, is a question about the connection rather than the network, and it decides whether cross-ties are one intervention or two.