The damping that belongs to no mode

Everything modal analysis does rests on one piece of luck. The mass matrix and the stiffness matrix of a structure are both diagonalised by its undamped mode shapes, so in modal coordinates the equations of motion come apart into independent oscillators, one per mode. The damping matrix has no reason to be diagonalised by the same shapes, and in general it is not.

The standard response is to assume that it is. Each mode is given a damping ratio directly — two per cent, five per cent, whatever a decay test suggests — as though the damping matrix were whatever combination of mass and stiffness makes that true, and the terms that would couple one mode to another are never computed. The first essay on mode shapes named this as a fix rather than a derivation, and suggested it was not the largest approximation in the calculation, given how little is known about where damping comes from.

That is right for a structure whose damping is spread through it in roughly the same way as its stiffness. It is wrong for a structure that has been deliberately built with most of its damping in one place, and the most common such structure is one sitting on isolation bearings.

A building on bearings, as two masses #

Reduce an isolated building to its two essential parts. A superstructure of 1,000 tonnes, which on a fixed base would have a period of 0.4 s and two per cent damping. Under it a base slab of 200 tonnes, standing on bearings whose stiffness gives the whole 1,200 tonnes a period of 2.5 s, and whose lead cores or high-damping rubber give that motion twenty per cent damping.

Those numbers make a storey stiffness of 246,740 kN/m and a bearing stiffness of 7,580 kN/m — the bearings are thirty-three times softer, which is the point of them — and dashpots of 628 kN·s/m across the storey and 1,206 kN·s/m across the bearings. The bearing dashpot is twice the storey’s, on a spring thirty-three times softer. In proportion to its own stiffness it is sixty-three times as strong.

That ratio is the problem in one number. Damping that could be diagonalised by the mode shapes would have to be the same multiple of stiffness everywhere, or of mass, or a fixed combination of the two. This building’s damping is in a wildly different proportion to stiffness at the bearings than in the storey above, and no combination of its mass and stiffness matrices can reproduce it.

The term that is thrown away #

Transform the damping matrix by the undamped mode shapes and look at what comes out.

The diagonal entries become the modes’ damping ratios: 19.4 per cent for the isolation mode and 11.4 per cent for the structural one. The second is already a surprise. The superstructure was built with two per cent, and its mode has been handed more than five times that — because in the structural mode, as will be seen, the base slab moves five times as far as the roof, and the slab is where the dashpot is.

The off-diagonal entry is the one the classical analysis discards, and it is not small. Measured against the geometric mean of the two it sits between, it is 0.75. The coupling term is three-quarters the size of the terms that are kept. A structure with damping in proportion to its stiffness would have a zero there. This one has a number that, if it were on the diagonal, nobody would dream of dropping.

Even the case that looks uniform is not. Give the bearings two per cent as well, so that both parts of the building are damped at the same ratio, and the off-diagonal term is still 0.29 of the diagonal mean, and the structural mode’s damping comes out at 5.5 per cent. The same ratio on two springs of very different stiffness under two very different masses is not proportional damping. Proportional damping is a condition on the matrices, not on the ratios people write on drawings.

Which dashpot put each number there #

The matrix in the figure is small enough to build by hand, and building it says where the coupling comes from.

Normalise the undamped modes so that each has unit generalised mass. The isolation mode moves the slab by 0.0283 and the roof by 0.0290 — the storey barely deforms, 0.0007 between them. The structural mode moves the slab by −0.0648 and the roof by 0.0126, so the storey deforms by 0.0775 and the slab goes the other way five times as far as the roof.

Each dashpot contributes to an entry of the modal damping matrix its coefficient times the product of the relative movements it sees in the two modes. The bearing dashpot sees the slab’s movement; the storey dashpot sees the difference between roof and slab.

The isolation mode’s own term. Bearings: $1{,}206 \times 0.0283^2 = 0.963$. Storey: $628 \times 0.0007^2$, which is nothing. So the isolation mode’s damping is the bearings’, as it should be.

The structural mode’s own term. Bearings: $1{,}206 \times 0.0648^2 = 5.07$. Storey: $628 \times 0.0775^2 = 3.77$. Fifty-seven per cent of the superstructure’s modal damping comes from the bearings, because in that mode the slab moves further than the roof and drags the bearing dashpot with it. That is where the 11.4 per cent came from, and it is why the two per cent the superstructure was built with is not the damping its mode has.

The coupling term. Bearings: $1{,}206 \times 0.0283 \times (-0.0648) = -2.210$. Storey: $628 \times 0.0007 \times 0.0775 = 0.036$. The total is −2.174, and the bearing dashpot supplies all of it — the storey’s contribution is 1.6 per cent, and in the other direction. The storey dashpot, which is proportional to the storey stiffness exactly as the storey’s own damping would be in a fixed-base building, contributes almost nothing to the coupling, because in the isolation mode the storey hardly deforms.

That is the whole mechanism in three multiplications. A dashpot couples two modes in proportion to the product of what it sees in each. The bearing dashpot sees large movement in both modes — the isolation mode because the whole building moves on it, the structural mode because the light slab is flung about under the heavy superstructure — and so it couples them strongly. A dashpot that saw large movement in only one mode would couple nothing, whatever its size.

It also says what would reduce the coupling. A heavier base slab moves less in the structural mode, which cuts both the bearings’ share of that mode’s damping and the coupling term. A stiffer superstructure pushes the structural mode further from the isolation frequency, which does not reduce the coupling term but reduces what a given coupling force can do. A tuned mass damper puts its dashpot on a soft spring too, and the same three multiplications apply to it.

What an exact mode looks like #

Solving the coupled equations exactly needs no approximation, only a different kind of eigenvalue problem. Instead of $\det(K - \omega^2 M) = 0$, the characteristic equation is $\det(\lambda^2 M + \lambda C + K) = 0$ — a quartic in $\lambda$ for two masses — and its roots come in complex-conjugate pairs. Each pair gives a frequency, from its magnitude, and a damping ratio, from how far its real part is from zero.

The roots can be checked without trusting whatever found them. Their sum has to be minus the trace of $M^{-1}C$, which for this building is $-9.80$, and their product has to be $\det K/\det M$, which is 9,351. Both hold to the precision of the arithmetic.

The frequencies and damping ratios that come out are almost exactly the classical ones: 2.523 s and 19.41 per cent against 2.527 s and 19.38; 0.1620 s and 11.38 per cent against 0.1620 s and 11.36. If modal properties were the test of an analysis, the classical one would pass with honours.

The mode shapes are what differ. Substituting a complex root back into the equations gives a complex ratio between the slab’s motion and the roof’s, and a complex ratio means the two do not reach their peaks at the same instant.

In the isolation mode the difference is 0.6 degrees and the mode looks like any other: the building rides the bearings almost as a block.

In the structural mode, the slab and the roof move in opposite directions — and not quite. They are 8.8 degrees away from exact opposition. A real mode shape has nodes, points that do not move at all, and they stay put while the mode vibrates. In this mode the place where the line between slab and roof crosses the undisplaced axis moves through the cycle, from 79 to 87 per cent of the way up the storey and back. The mode does not have a shape. It has a sequence of shapes, and the drawing of it at twelve instants is a fan that does not share a pivot.

This is what “complex mode” means physically, and it is also why these modes are hard to use. A mode shape with a fixed node can be drawn, compared with a measurement, and read as a deflected shape. A complex one is a travelling pattern, and the energy in it moves along the structure during each cycle instead of sloshing back and forth in place.

The response that is six times what the modes predict #

The modal properties agree and the shapes disagree by a few degrees, which sounds harmless. The frequency response of the storey drift shows it is not.

The two curves lie on each other through the isolation mode. That is the region isolation is designed for — ground motion near 2.5 s — and it is where the classical analysis is nearly right: the peak drift per unit acceleration there is 11.44 mm in the exact response against 10.70 in the classical one, six and a half per cent short.

After the isolation peak the curves part. By the structural mode, at 39 rad/s, the exact drift is 6.1 times the classical. And above it the exact response decays in proportion to a lower power of frequency than the classical one, so the gap keeps widening for as far as the ground has energy.

The mechanism is visible in the equations once it has been pointed out. The bearing dashpot applies a force proportional to the slab’s velocity. In the isolation mode the slab moves a great deal, so that force is large, and in modal coordinates it is split between the two modes by the off-diagonal term. The classical analysis gives the isolation mode its share, as damping, and discards the share that lands on the structural mode. But that discarded share is not damping from the structural mode’s point of view. It is a force, applied at the frequency of whatever the slab is doing, and it drives the superstructure.

The bearings, in other words, are not only removing energy from the building’s motion. They are also a path by which the ground’s high-frequency content reaches the superstructure, through the dashpot, and the path grows with the dashpot.

The drift is the quantity this essay uses because it is the storey’s force divided by its stiffness. The same mechanism shows more strongly in the floor accelerations that govern the contents of an isolated building — hospitals, laboratories, museums, the buildings most often isolated in the first place — because acceleration weights the high frequencies further.

The damping that stops helping #

The consequence that matters in design is drawn below. It takes the storey drift’s root-mean-square under a broadband input and plots it as the bearings’ damping is raised from two to fifty per cent, both analyses divided by the exact value at two per cent.

At ten per cent the two agree: 0.457 exact against 0.448 classical. By twenty per cent they have separated, 0.343 against 0.316, and the classical analysis is eight per cent short. By thirty it is fifteen per cent short and by fifty thirty-one per cent. The classical curve keeps falling — every increment of bearing damping looks like it buys a further reduction — and at fifty per cent it predicts 0.20. The exact curve flattens: 0.305 at thirty per cent, 0.293 at forty, 0.291 at fifty.

Split the drift into the part below 10 rad/s, which is the isolation mode’s, and the part above, which is the structural mode’s, and the reason is plain. The low-frequency part falls with bearing damping in both analyses, as it should. The high-frequency part in the classical analysis falls a little too, because the structural mode’s diagonal damping rises. In the exact analysis the high-frequency part rises sixfold, from $7.3 \times 10^{-4}$ at two per cent damping to $4.2 \times 10^{-3}$ at fifty. Past about thirty per cent the gain in the first part and the loss in the second nearly cancel, and extra damping at the bearings buys nothing.

It is a result the literature on isolation reached in the 1990s — Kelly argued it directly — usually stated in floor accelerations: heavily damped isolators raise the higher-mode response of the structure they carry. What the two analyses side by side show is why it was not obvious. Every quantity a classical modal analysis reports about the modes is correct — periods, damping ratios, participation factors, effective masses — and the one thing it gets wrong is invisible in all of them: the coupling term, which it never computed.

When the classical analysis is safe #

It would be a mistake to take from this that modal analysis with modal damping is untrustworthy in general. The size of the error is predictable, and it is usually small.

It is governed by two things. The first is how far the damping departs from proportional, which is what the off-diagonal term measures: 0.29 of the diagonal mean for this building with both parts damped at two per cent, 0.75 at twenty per cent on the bearings. The second is how much the modes are separated in frequency, because a coupling force at the isolation mode’s frequency acting on a mode fifteen times stiffer produces little response unless something else supplies energy near that mode’s own frequency.

With both parts at two per cent, the white-noise drift error is a tenth of a per cent. With the bearings at ten per cent it is two per cent. The classical analysis is a good approximation for any structure whose damping comes from the same material and details throughout, and a poor one only where a deliberate, concentrated source of damping sits on a much softer spring than the rest of the structure. That describes base isolation, supplemental viscous dampers in one storey or one bay, soil radiating energy away from a foundation, and a stiff superstructure on a flexible tower.

The picture at forty per cent is what the growing error looks like in the modes themselves. The structural mode’s departure from opposition has doubled, the node’s travel has more than doubled, and — on the response curve — the exact drift at the structural mode is 11.8 times the classical one.

What an analysis does about it #

Three responses are available and they trade accuracy against familiarity.

Solve the coupled equations in time. A response-history analysis with the bearings modelled as their own elements, with their own damping, has no modal decomposition to approximate. This is what the design of isolated buildings in practice uses, and the reason is largely the one drawn here, though it is more often stated as the need to model the bearings’ nonlinearity.

Use the complex modes. The state-space formulation doubles the size of the eigenvalue problem and returns complex eigenvectors that do diagonalise the damped system. Modal superposition then works exactly, at the cost of shapes that are not shapes, participation factors that are complex, and a combination rule that has to account for the phase between the parts of each mode. Combining modal peaks correctly was difficult enough with real modes.

Keep the classical analysis and know which quantity it is wrong about. The isolation mode and everything about it — bearing displacement, base shear at the isolation period — is accurate. The structural mode’s response is not, and it is the drift and floor acceleration of the superstructure that carry the error. A design governed by the first can use the classical result; one governed by the second should not.

What the figures cannot show #

The building is two masses. A real superstructure has several storeys and several structural modes, and the coupling through the bearing dashpot reaches all of them. Higher modes are further from the isolation frequency and carry less mass, so the effect on each is smaller, but there are more of them.

The dashpots are linear. Real isolation bearings dissipate energy by yielding lead or by hysteresis in rubber, not by viscosity. An equivalent viscous damping ratio of twenty per cent describes the energy removed per cycle at one amplitude, and a yielding bearing transmits force to the slab through a force–displacement loop whose corners contain high-frequency content of their own. The linear model understates that path, if anything.

The input is white noise. The root-mean-square drift is computed for a broadband input of equal intensity at all frequencies. A real ground motion, as its spectrum shows, has less energy at 39 rad/s than at 2.5 s, which reduces the structural mode’s share of the response in both analyses. Whether the exact flattening survives a given record depends on that record’s content near the structural frequency, and the figures establish the direction and the mechanism, not the size of the error for any particular site.

The frequency response is drawn to a finite frequency. The curves stop at three times the structural mode’s frequency. The root-mean-square values were checked against an integral taken forty times further and do not change in their fourth significant figure.

Where the model stops #

Linear, and time-invariant. Complex modes are a property of a linear system with constant matrices. A lead-rubber bearing’s stiffness and damping change with its displacement, so the modes change during the earthquake, and neither the classical nor the complex modal description is more than a snapshot.

Horizontal motion in one direction. Bearings respond in both horizontal directions at once, and a yielding bearing’s force in one direction depends on its displacement in the other. The two-mass model has neither.

A rigid slab. The base slab is a mass. Real isolation slabs are flexible between bearings and can have vertical and rocking modes of their own, which a lumped model cannot represent.

No vertical isolation. The bearings are soft horizontally and very stiff vertically, and nothing here says anything about vertical ground motion, which isolation does not reduce.

Still open: where the node goes in a taller building #

A two-mass building has one structural mode and one node to watch. A building of several storeys on bearings has several structural modes, each with nodes, and each coupled to the isolation mode by its own off-diagonal term. Whether those terms are largest for the first structural mode or grow with mode number — whether the bearings feed the higher modes more because they are further from resonance or less because they carry less mass — decides whether the flattening seen here is a property of the fundamental structural mode or something that gets worse in the higher modes, and it is the question a taller isolated building actually asks.