The top floor the bearings shake

The essay on damping that belongs to no mode reduced an isolated building to two masses: a superstructure with one mode, and a slab on bearings with most of the damping in them. The usual analysis gives each mode its own damping ratio and throws the rest of the damping matrix away. It got the building’s periods and damping right to a fifth of a per cent, and its storey drift at the superstructure’s frequency six times too small. Past about a third of critical damping at the bearings, extra damping stopped reducing the drift at all.

It ended on the question a taller building asks. A superstructure of several storeys has several structural modes, each tied to the isolation mode through the bearings’ dashpot by its own off-diagonal term. Are those terms largest in the first structural mode, or do they grow in the higher ones? The answer is that they fall. The error they cause rises all the same, it gathers at the top of the building, and it turns the flattening the two-mass model found into a reversal.

Six storeys on a slab #

The building for the rest of this essay has six storeys of 500 tonnes each on a 500-tonne slab. The superstructure’s first period on a fixed base is 0.6 s, and the bearings give the whole 3,500 tonnes a period of 2.5 s. The superstructure has 2 per cent damping, arranged in proportion to its storey stiffnesses, so that on a fixed base it would decouple exactly. Every departure from classical behaviour here is therefore the bearings’.

The isolation mode is what isolation is for: the building moving as a block on soft bearings, at a period long enough to sit below most of an earthquake’s energy. The first essay on isolation is about that mode. The structural modes are the building flexing above its bearings, and in every one of them the slab is not still. It swings against the floors above by about as much as the roof does. The slab weighs no more than one floor and sits on springs about forty times softer than a storey, so it is the other free end of the chain. That is what puts the bearings’ dashpot into every structural mode.

The damping ratios under each mode are the diagonal terms the classical analysis keeps. The first structural mode’s 8.5 per cent is four times what the superstructure was given, because the slab drags the bearings’ dashpot through that mode. The next ones rise again, to 9.4 and 11.5 per cent. That is not the bearings. It is the superstructure’s own stiffness-proportional damping, which grows with frequency.

The coupling falls #

The number the previous essay watched is the off-diagonal term of the modal damping matrix, measured against the diagonal terms either side of it. It says how strongly one mode’s damping force pushes on the other mode.

The question as the previous essay posed it has a clear answer. The coupling falls with mode number, from 0.74 for the first structural mode, which is the two-mass model’s value almost exactly, to 0.20 by the fourth and nearly nothing by the sixth. The bearings’ share of each mode’s own damping falls the same way, from more than half in the first to nothing in the last.

The reason is in the way the number is built. The coupling term itself is the bearing dashpot times the slab’s movement in the isolation mode times its movement in the structural mode. The slab moves about as much in every structural mode, so the raw term falls only slowly, by about a third from the first structural mode to the fourth. What falls fast is the ratio, because its denominator contains each mode’s own damping. The superstructure’s stiffness-proportional damping rises with every mode’s frequency, and the ratio falls as the modes’ own damping outgrows the coupling. If the classical analysis’s error were set by this ratio, the higher modes would be the safe ones.

And the error rises #

It is not set by the coupling term, and the frequency response shows it plainly.

The two curves agree where isolation lives, around the isolation mode at 2.5 rad/s. Above it they part, and the gap between them grows at every structural mode in turn. At the first it is a factor of 3.3, close to the two-mass model’s. By the fourth it is 10.9. The mode tied most loosely to the bearings is the one the classical analysis gets most wrong.

Why a looser tie does more harm #

The resolution is in what each structural mode is being compared with. In the classical analysis a structural mode responds to the ground directly, through its participation factor — how much of the building’s mass it moves in the direction of the ground. For a building on bearings those factors are tiny, because the isolation mode has taken almost all the mass. The first structural mode carries 0.05 per cent of it, the second 0.003 and the third 0.0006. Each higher mode is reached by the ground about ten times less than the one below.

The part the classical analysis throws away arrives by another route. The isolation mode, carrying nearly all the mass, moves the slab. The slab’s velocity drives the bearings’ dashpot, and the off-diagonal term hands a share of that dashpot force to each structural mode. That raw term, before any normalising, is about the same size for all of them. So each structural mode receives roughly the same push through the dashpot, while its own push from the ground falls tenfold a mode. What the analysis discards is a growing fraction of what it keeps, and the ratio of the two is the error.

The estimate can be written in one line, and each of its factors is one step of that route. The raw coupling term is the share of the dashpot that lands on the mode. The isolation mode’s participation is how hard the ground drives the slab. Dividing by the mode’s frequency turns the slab’s displacement at that frequency into the velocity the dashpot sees, because far above its own frequency the isolation mode’s displacement falls as the frequency squared and its velocity as the frequency. Dividing by the mode’s own participation compares the result with what the ground delivers to that mode directly. Put together, it is the raw coupling term times the isolation mode’s participation, divided by the mode’s frequency and its own participation. It gives 2.9, 5.6, 8.1 and 10.1 for the four modes at 20 per cent damping, against 3.3, 6.2, 8.6 and 10.9 from the full solution. The normalised coupling of the previous figure falls because the modes’ own damping grows faster than the coupling shrinks. It measures one damping against another. The quantity that matters is a damping force against a ground force, and that one grows.

The figure also shows the error scaling with the bearings’ damping. Doubling it from 10 to 20 per cent roughly doubles every ratio, and doubling again to 40 roughly doubles them again. The coupling force is proportional to the dashpot, and nothing else in the estimate depends on it.

Where it lands in the building #

A factor of ten at the fourth mode’s frequency sounds alarming, but the fourth mode carries little of the building’s response. What matters is how the errors add up in each storey.

Integrated over a broadband input, the errors are smaller than the peak ratios, because a broadband input spreads its energy across frequencies rather than concentrating it where one mode resonates, and because most of any storey’s drift comes from the isolation mode and the first structural mode, where the two analyses are closest. They grow steadily up the building. The first storey above the slab is understated by 7.7 per cent at 20 per cent bearing damping. The top storey is understated by 16.2, twice as much. At 40 per cent damping the understatement is 28 per cent at the bottom and 63 at the top.

The shape of that profile follows from the mode shapes. The first structural mode bends the whole building. The higher ones concentrate their drift toward the top, where the chain is free, and they are the modes with the largest errors. So the storeys the classical analysis gets most wrong are the upper ones, which are the storeys with the least structure and often the most equipment. The spectrum a floor hands on to whatever is fixed to it is built from exactly these motions.

The consequence for that equipment is direct. A floor spectrum is the response spectrum of a floor’s motion — the peak response of a small oscillator bolted to the floor, against its period. It is peaked at the building’s structural frequencies, because those are the frequencies at which the floor itself moves most. Computed from a classical analysis of this building, it would understate the peaks at the second, third and fourth structural modes by roughly the factors in the peaks figure. Those are the peaks at which a stiff piece of equipment — a chiller on vibration mounts, a rack of instruments, a tall cabinet — is most likely to resonate. The building’s frame would be designed correctly. The contents on its roof would be designed for a floor that moves several times less than the real one at exactly their own frequencies.

More damping, more shaking #

The previous essay’s most practical finding was that past about a third of critical, extra bearing damping stopped reducing the drift. On the taller building the same sweep does something stronger.

The two lines agree up to about 10 per cent damping and then part. The classical analysis promises that every increment of damping at the bearings buys a further fall in the roof’s acceleration, all the way to 58 per cent, where it predicts a third of the lightly damped value. The exact analysis finds a minimum at about 28 per cent, 0.44 of the lightly damped value, and then the curve turns up. At 58 per cent the roof is shaking 14 per cent harder than at the minimum, and half again harder than the classical analysis says.

The surprising connection is that this was already known, for a different structure, in its simplest form. A machine on springs transmits less force to the floor as its mount’s damping rises, but only below $\sqrt{2}$ times the mount’s frequency. Above it, the same damping transmits more, and a heavily damped mount running fast is four times worse than a lightly damped one. A dashpot’s force grows with the speed across it, so at high frequency it stops isolating and starts coupling.

An isolation bearing is that mount turned upside down, with the ground as the machine. The isolation mode sits at 2.5 s, and every structural mode is far above $\sqrt{2}$ times its frequency, on the side of the curve where damping transmits. The classical analysis keeps half of this. It keeps the isolation mode’s own transmissibility — the roof riding the bearings as a block — which is why even the classical line eventually stops falling, near 57 per cent. What it discards is the part the dashpot transmits into the structural modes. The off-diagonal term is the transmissibility curve’s right-hand side, reaching the modes that flex, and it is the part that grows with every storey added.

The best damping depends on the building #

If there is a minimum, there is a best damping, and it is not a property of the bearings alone.

For a building of one storey the best damping is 48 per cent, where the curve is so flat that any value from a third to a half is nearly as good. For three storeys it is 32 per cent, and for ten, 28. The curve falls quickly over the first few storeys and then settles. The taller building has more structural modes for the bearings’ dashpot to drive, and a larger share of its roof’s motion comes from them, so the point at which extra damping starts to cost falls with height.

The one-storey building’s flat optimum and the ten-storey building’s sharper one differ for a reason the transmissibility curve also explains. With one structural mode, the damping’s cost is paid at one frequency, and it grows slowly as the damping rises. With many, the cost is paid at every structural frequency at once, each of them further up the right-hand side of the curve where damping hurts most. The sum turns up sooner and more steeply.

For a designer the lesson is uncomfortable. Isolation systems are specified with an equivalent damping ratio, and a supplier’s bearing that offers more is easy to read as a better bearing. For a single-storey building the reading is harmless. For a ten-storey building, damping above about a quarter of critical is being paid for twice — once in the bearings, and once in the roof’s contents. And the classical analysis, asked the same question, puts the best damping at about 57 per cent whatever the building’s height — twice the exact answer for ten storeys — so an isolation system tuned by it is tuned for a building that does not exist.

What an analysis does about it #

The previous essay listed three ways of treating non-classical damping, and the taller building changes their ranking. A response-history analysis with the bearings modelled as their own elements has no modal decomposition to get wrong. For an isolated building of more than a couple of storeys whose floor accelerations matter, it is the analysis whose answer is not in doubt. A modal analysis with complex modes also gets it right, at the cost of shapes that are sequences rather than shapes, now for every structural mode, and of a combination rule that has to carry the phase between every pair of them. The third option was to keep the classical analysis and know which quantity it is wrong about. That was reasonable for two masses, where the error was confined to one structural mode. Here it grows up the modes and up the building, and the quantity it is wrong about — the upper floors’ motion at the structural frequencies — is exactly what a floor spectrum is built from.

What the figures leave out #

The bearings are dashpots. Real isolators dissipate energy by the yielding of lead or the hysteresis of rubber, and their force does not grow with velocity the way a dashpot’s does. That removes the pure high-frequency transmission modelled here. It replaces it with the corners of a yielding loop, which carry high-frequency content of their own, and with a stiffness that changes during the earthquake. The mechanism is real for viscous dampers added to an isolation system, which is a common way of adding damping. For hysteretic bearings it is a qualitative warning rather than a number.

The input is white noise, cut off at three times the third structural mode’s frequency. Raising the cut-off to three and then to nine times the highest mode’s frequency changes none of the root-mean-square values in their third figure, so the cut-off is not doing the work. A real earthquake has little energy at the fourth mode’s 68 rad/s — about 11 Hz — which its spectrum shows and which reduces that mode’s contribution in both analyses. Whether a particular record shows the reversal depends on its content between the first and third structural modes. The figures establish the mechanism and its direction, not its size at any site.

The superstructure is a shear building of identical storeys. A real frame has storeys of different stiffness and mass, floors that rotate, and modes in two directions and in torsion. Each of those adds structural modes for the dashpot to drive, which is the direction this essay’s result already points.

The assumption the reversal rests on #

Everything here rests on the superstructure’s own damping being classical, in proportion to its stiffness, so that every departure from classical behaviour is the bearings’. That assumption makes the result clean. It also makes it conservative in one way, because a superstructure with damping concentrated in a few storeys would add its own off-diagonal terms. It is not conservative in another: stiffness-proportional damping gives the higher modes a great deal of damping, 11.5 per cent in the third structural mode here. Real buildings do not damp their high modes that strongly, and with less damping of their own those modes would respond even more to what the bearings feed them.

Still open: a building with no preferred direction #

Isolated buildings are very often square in plan, with the same lateral system in both directions, because the bearings do not care which way the ground moves and a symmetric superstructure avoids twisting on them. A building like that has two translational modes for every storey mode, one each way, with exactly the same period.

Two modes with one period are not two mode shapes. Any pair of directions at right angles is as good as any other, and a computer asked for the building’s modes returns whichever pair the rounding in its arithmetic prefers. Everything above assumed that a building’s modes are its own. A doubly symmetric building does not have modes of its own, and what an analysis does with the pair it is handed is the next question.