Two modes that are really a plane

The twist the combination rule invents was about two modes whose periods are close. A floor whose stiffness sits a little off its centre of mass has a translational mode and a torsional mode a few per cent apart. The square root of the sum of squares, which treats every pair of modes as unrelated, then invents a twist that the complete quadratic combination correctly removes. The two modes there were distinct: close in period, different in shape, each a genuine property of the floor.

This essay is about the limit where the gap closes completely. The previous essay ended on the case that makes it common rather than exotic: a building equally stiff in both directions, which isolated buildings, towers and many ordinary frames are by design. Such a building has two translational modes with exactly one period. At that point something happens that no amount of closeness prepares for. The two modes stop being two shapes and become a plane of shapes. The pair an analysis prints is a choice, and results that depend on the pair depend on the choice.

Every direction is a mode #

Reduce the building to one floor, translating in two directions, with its torsion far enough away to ignore — the single-storey floor the twist essay used, made symmetric. The floor is square, 24 m on a side, and its lateral system is equally stiff in x and in y, with a period of 0.8 s either way. Its stiffness matrix is the same number on the diagonal and zero off it, a multiple of the identity.

A mode is a direction in which the floor can vibrate on its own, with the restoring force pointing back along the displacement. For a stiffness that is a multiple of the identity, every displacement produces a restoring force exactly opposite to it. Pushed along x, the floor swings along x. Pushed along 30°, it swings along 30°. Every direction is a mode, all with the same period, and there is no first one.

The eigenvalue problem says the same thing in its own language. The two eigenvalues are equal, and the eigenvectors belonging to a repeated eigenvalue are not two vectors but a whole plane — any vector in it, and any pair at right angles as a basis. A solver asked for two mode shapes must return two, so it returns a pair, and which pair depends on the order in which it did its arithmetic.

Decided by a millionth #

A real floor is never exactly symmetric, and it might seem that the slightest asymmetry would settle the matter. It does settle it, and that is the difficulty.

Stiffen the floor by one part in a million along some direction and the modes snap to that direction and the one at right angles to it. The softer direction has the longer period and is the first mode. The periods separate by one part in a million. The directions separate by up to a right angle — whichever way the millionth pointed.

That is the defining feature of a repeated eigenvalue, and it is the reverse of how modes normally behave. A distinct mode changes in proportion to a change in the structure, divided by how far its period is from its neighbours’. That is why modes a few per cent apart are sensitive and modes far apart are robust. Here the distance is zero. A vanishing cause produces a finite effect. The size of the asymmetry decides how far apart the periods are. Only its direction decides the modes, and it decides them completely.

The mechanism is short enough to state. When two modes share a period, a small change in the structure acts first within the plane of those two modes, and there it is a small two-by-two stiffness of its own. That small matrix has principal directions — the directions in which it pushes back along the displacement — and those become the modes. Doubling the change doubles the small matrix and leaves its principal directions where they were. So the modes follow the pattern of the change and ignore its size. For a distinct mode the same change is divided by the distance to the next period, and a change of a millionth moves a mode by about a millionth over that distance. With the distance at zero, the division is by nothing.

The same thing happens in a structure that never moves. A square section has the same second moment about every axis through its centroid, so every axis is principal, and a beam of square section bends in the plane of its load whatever that plane is. That is the static form of this degeneracy. A square column is its buckling form. Its critical load is the same about every axis, so the axis it buckles about is chosen by whatever imperfection it has, and a perfect one would have no preferred axis. The building’s two translational modes, the square beam’s principal axes and the square column’s buckling direction are one mathematical object: a symmetric matrix with a repeated eigenvalue, whose eigenvectors are a plane until something outside the problem picks a pair.

The practical form of this is unnerving the first time it is met. Two engineers model the same symmetric building in two programs and get the same periods to every digit printed, and first mode shapes pointing in different directions. Both are right. Each program’s eigenvalue solver has returned an orthogonal pair from the plane, chosen by the order of its rotations and the rounding in its last digits. A third run of either program on a model renumbered in a different order may return a third pair. The model is not unstable, and neither is the solver. The question it was asked has no single answer, and each gave one of the correct ones.

Measured buildings behave the same way. An ambient-vibration survey of a square tower usually reports its first two translational modes with periods close together, along directions rotated some way from the tower’s axes. A rotation like that is not a defect in the survey or in the tower. It says that some small asymmetry — a core a little off centre, one face with more glazing, a stair on one side — happens to outweigh the others. The rotation shows which way that asymmetry points, and says nothing about its size.

A combination rule that depends on the pair #

The pair matters as soon as an analysis combines modal responses, because a design spectrum gives a peak for each mode and not the times at which the peaks occur.

Under ground motion along 30°, the exact response of a floor that is a mode in every direction is a swing along 30° itself. Per unit spectral displacement, that is 0.87 along x and 0.50 along y. Any analysis that adds the two modes’ contributions with their signs gets that, in any basis, because the two contributions are two components of one vector.

The square root of the sum of squares does not add with signs. It adds magnitudes in quadrature, which is correct for two modes whose peaks are unrelated in time. Two modes of one period are the opposite of unrelated. They reach their peaks together, always. Combined that way, the floor’s x-response depends on the pair the solver returned. With a pair along the axes, or along the direction of the ground motion, it happens to come out right. At 150° it comes out 0.61 along x — 30 per cent too small — and 0.79 along y, 58 per cent too large. Over all possible pairs the x-response runs from 0.35 to 0.94.

Two properties survive. The square-root combination gets the size of the displacement right in every basis, because a vector’s length does not depend on the axes it is written in. And the complete quadratic combination gets everything right, because for two modes of equal period its correlation coefficient is exactly one, and it adds them with their signs. What goes wrong is the direction. A corner column or a bracing line in one direction is designed from a component, and the component is what depends on the choice.

With the ground motion along the diagonal, the exact response is 0.71 along each axis, and the square-root combination swings from 0.50 to 0.87 along x. The two figures together show what the error depends on. It is not the direction of the ground motion or the direction of the pair, but the angle between them. It vanishes when the pair lies along the ground motion or across it, and it is worst when the pair sits at 45° to it.

That includes the case a design code actually asks for, which is ground motion along each of the building’s axes in turn. With the ground moving along x and a solver that happened to return its pair along the diagonals — as it may, for a square building — the square-root combination gives the response along x as 0.71 of its true value, 29 per cent short. It puts a spurious 0.71 across it, along y, where the true response is nothing at all. The ordinary analysis, run exactly as intended on a correct model, is right or wrong by nearly a third depending on a choice nobody made.

A push, and the path the roof takes #

The dynamics of the floor show the degeneracy directly, without any combination rule.

Exactly symmetric, the floor swings along the line it was pushed on and nowhere else, because that line is a mode. Make it 1 per cent stiffer along x and 1 per cent softer along y, and the push along 30° is no longer a mode. It is two modes, one along x and one along y, with periods about 1 per cent apart. They start in step and slowly drift out of it. The swing opens into an ellipse, turns into a swing along the mirror direction, and comes back. The beat period is the inverse of the frequency difference: 100 cycles for a 1 per cent split and 25 for a 4 per cent one.

That precession is what a measurement of a nominally symmetric tower actually records. It is not a sign of anything wrong. It is the signature of two modes whose periods are almost equal, beating. The direction of the ellipse’s long axis at any moment is not a mode shape. The mode shapes are the axes about which the precession is symmetric, and finding them needs a record long enough to see at least one full beat, which at 1 per cent is a hundred cycles.

When two frequencies are one #

Damping sets a floor under all of this. Two modes whose periods differ by less than the damping blurs a resonance cannot be told apart by any measurement of the structure’s response, whatever the eigenvalue problem says.

With 2 per cent damping, a floor stiffened by 1 per cent one way and softened by as much the other — its two frequencies about 1 per cent apart — makes one peak, indistinguishable from a single mode. At 4 per cent the peak has split into two. The rule is that two peaks can be seen when their frequencies differ by more than about twice the damping ratio, because that is the width of each one.

The factor of two comes from the width of a resonance. A mode with damping ratio ζ responds at more than half its peak power over a band of frequencies about 2ζ wide, centred on its own frequency. Two such bands closer than their width overlap into one, and the dip between them that would show two peaks is filled in.

With 5 per cent damping the same 4 per cent asymmetry that separated the peaks before is invisible, and it takes 10 per cent. That makes the degeneracy physical rather than mathematical. Below a split of about twice the damping, the two modes’ directions are not observable. No test on the building could find them, because the building’s response to any excitation is indistinguishable from the response of a building with no preferred direction at all. The eigenvalue problem still returns a definite pair. But the pair is a statement about the model’s stiffness matrix, not about anything the structure does.

The model that fails a test it passed #

That has a consequence for the most common way a model is compared with a structure.

The modal assurance criterion compares two mode shapes by the square of the cosine of the angle between them, and it is the standard way a model is checked against a building. A value above about 0.9 is conventionally taken as a match. For a doubly symmetric floor, a model that is right to one part in a million scores 1.00 if the built floor’s asymmetry happens to lie along one of its axes, 0.50 if it lies along a diagonal, and zero if it lies at right angles. The model’s periods match the measurement to a millionth throughout.

The low score is not evidence that the model is wrong. It is the degeneracy being measured by an instrument that assumes there is none. The practical danger is in model updating. An automated procedure that adjusts a model’s stiffnesses until its mode shapes match measured ones will find the direction of the model’s first mode far too sensitive to be useful. It will happily introduce a small, fictitious asymmetry to rotate the model’s modes onto the measured pair, and report a better model. What it has done is fit the direction of a millionth. For nearly symmetric structures the comparison has to be made between the planes the mode pairs span, which do agree, rather than between individual shapes. The angle between two planes is well defined and is a genuine property of the two structures. For the floor here, whatever the direction of its asymmetry, the measured pair and the model’s pair span the same plane, and the angle between them is zero. A comparison built that way scores the correct model as correct, at every direction, and would score a genuinely wrong one — a model whose translational pair was contaminated by torsion, say — as wrong.

What the figures assume #

The floor is two translations and nothing else. A real symmetric building also has a torsional mode, usually at a shorter period. If that period comes close to the translational pair, the degeneracy becomes a threefold near-degeneracy, which is the torsional coupling of the close-mode case and the degeneracy of this one together.

The asymmetry is a perturbation of the stiffness alone. An asymmetry of mass does the same thing. So does an asymmetry in the damping, which need not belong to either mode and which can rotate the directions in which the response decays even when the periods are equal.

The ground motion has one direction. Real ground motion has components in both horizontal directions at once, usually combined in design by rules for orthogonal effects — full effect one way and a fraction of it the other. Those rules exist precisely because the direction of an earthquake is unknown. A doubly symmetric building makes the same point from the structural side: the direction of its response is not known either, and not only because of the ground.

The assumption that makes a mode a property #

Every use of mode shapes so far has rested on an assumption that is usually too obvious to state: that a structure’s modes are properties of the structure, so that a model and a measurement can be compared shape by shape and a combination rule applied mode by mode. That assumption holds when the periods are distinct by more than the damping blurs them. For a doubly symmetric structure it fails in the most complete way. The periods are properties of the structure, and so is the plane the pair spans, but the pair itself is not, and neither is anything calculated from the pair alone.

The safe practice follows. Combine repeated or nearly repeated modes with the complete quadratic combination, whose correlation coefficient carries the information the pair lacks. Compare models with measurements by the frequencies and by the planes that pairs of shapes span, not shape by shape. And read a precessing record as two modes beating rather than as a mode whose direction is unstable.

Still open: a mode that notices what a frequency does not #

The degeneracy here is a case where the frequencies are robust and the shapes are fragile. The next question is the reverse: a mode shape that registers a change the frequency hardly feels. Take a tenth of the stiffness out of one storey of a ten-storey building and the first frequency moves by about a per cent. The mode’s curvature moves by ten times as much, in the damaged storey and nowhere else. Which of the two a structure’s owner can afford to measure decides whether the damage is ever found.