What a mode shape notices that a frequency does not

The previous essay was about a building whose frequencies are robust and whose mode shapes are fragile: a millionth of asymmetry swings its modes through a right angle and leaves its periods where they were. This essay is about the reverse sensitivity, and the reverse is the one an owner of a structure cares about. Something inside the building has lost stiffness — a cracked wall, a yielded brace, a connection that has worked loose. The question is which of the building’s dynamic properties will say so.

The obvious candidate is the frequency. It is the property every question in dynamics turns on, and the easiest thing about a structure to measure. An accelerometer on the roof, a few minutes of ambient vibration and a spectrum give a first frequency to a fraction of a per cent without closing the building, and stiffness is what the frequency measures. So monitor the frequency, and when it falls, look for damage. The trouble is how little it falls.

A tenth of one storey, and the frequencies #

Take a ten-storey shear building with equal floors and equal storeys, and remove a tenth of the stiffness from one storey, which is a great deal of damage.

In the first storey, where it does most, the damage moves the first frequency by 1.04 per cent. In the tenth, it moves it by 0.02 per cent, which is below anything that could be measured on a real building. The second and third modes do a little better in some storeys and worse in others, and none of the thirty combinations moves any frequency by more than that 1.04 per cent.

A one per cent change is measurable in a laboratory. On a building it is not reliable, because a building’s frequencies wander on their own. The stiffness of concrete and of the non-structural fabric that clads it changes with temperature and moisture. Measured frequencies of real structures move over a year by amounts of this size and larger. The documented case is the Z24 bridge in Switzerland, monitored for a year before it was deliberately damaged, whose frequencies moved more with frost than with anything the damage then did to them.

Why the frequency is so forgiving #

The pattern in the first figure is not random, and the reason for it is one of the oldest results in structural dynamics.

A natural frequency squared is a ratio: the strain energy stored in the mode’s shape, divided by the kinetic energy of the same shape moving at unit rate. Rayleigh’s quotient is that ratio. Its property that matters here is that, to first order, a small change in stiffness changes the frequency without changing the shape. The strain energy is a sum over storeys of each storey’s stiffness times its drift squared. Cut one storey’s stiffness by a fraction, and the squared frequency falls by that fraction times the storey’s share of the total strain energy.

The shares are very unequal. The first mode of a shear building works its lower storeys hardest: the first storey stores 18.9 per cent of the mode’s strain energy, and the top storey 0.4. So a tenth of the first storey’s stiffness lowers the squared frequency by about 1.9 per cent and the frequency by about 0.95 — the exact figure is 1.04, the first-order estimate being slightly low. The same tenth in the top storey lowers the frequency by 0.02 per cent. A frequency is a sum over the whole building, weighted by where the mode works hardest, so it is least sensitive exactly where the mode is quietest.

The first-order estimate’s small shortfall is informative too. It assumes the shape does not change, so the damaged storey keeps its old share of the drift. In fact a softened storey draws a little more of the mode’s deformation into itself, which is the change in shape the rest of this essay is about. Concentrating deformation where the building is softest lowers the stored energy for a given motion a little further than the frozen shape allows. That is why the exact fall is 1.04 per cent rather than 0.95. Rayleigh’s quotient is stationary at the true shape, so the correction is second order and small, and it always goes the same way: the true frequency is lower than the first-order estimate.

That also says which damage a frequency can never see. A storey’s share of a mode’s strain energy is at most the mode’s whole, and in a tall building no storey holds much of any mode’s energy. Loss of stiffness in one member of a hundred barely registers, however complete. Only damage spread through much of the structure — a general loss of stiffness from repeated cracking, or from fire — moves the frequency by a fraction comparable to itself.

Three frequencies instead of one #

There is a way to get more out of frequencies, and it follows directly from the energy shares. Each mode keeps its strain energy in different storeys, which is why the higher modes carry what the first leaves out. The first works the bottom of the building hardest. The second has a node partway up and works the storeys near its peaks of curvature, and the third has two nodes and works three places. So a loss of stiffness in one storey moves the three frequencies by three different amounts, in proportions set by that storey’s three energy shares — and the proportions do not depend on how large the loss is.

A tenth of the third storey lowers the first three frequencies by 0.91, 0.20 and 0.09 per cent. Three tenths of it lowers them by 3.38, 0.73 and 0.37. The drops are four times larger and the ratios between them are the same: the second mode’s drop is 0.22 of the first’s both times, the third’s 0.10 and 0.11. A loss in the eighth storey gives ratios near 5 and 3 instead, whatever its size, because the eighth storey is near where the second mode bends hardest. The pattern of ratios is a fingerprint of the storey, and the size of the drops is a measure of the damage. Separating the two was proposed for locating damage from frequencies alone in the late 1970s. It works in a model.

It does not escape the environment, which moves every frequency together and so enters every ratio. And it needs the higher frequencies measured as precisely as the first, which ambient vibration makes harder. But it shows that the frequency’s blindness is not total. A single frequency is blind. Several frequencies, read against the energy shares of their modes, can see where, even when none of them can see much.

The shape barely moves either #

If the frequency is dull, perhaps the shape is sharper. A damaged storey is softer, so the mode should lean more there.

It does lean more, and the lean is invisible. Scaled so the roof moves the same, the two shapes differ by at most 0.92 per cent of the roof’s movement at any floor. The standard single-number comparison of two shapes, the modal assurance criterion, is 0.99997 — a match that would pass any comparison a test engineer runs. A model of the undamaged building and a measurement of the damaged one would be declared to agree.

The small difference is not spread evenly, though, and that is where the useful signal is. Below the damaged storey the two shapes are nearly identical. Above it, every floor is displaced a little further in the damaged building, by the extra lean the soft storey allows, and the difference then stays roughly constant up to the roof. A shape is an integral of the storey drifts from the ground up. The damage has put a step into the integrand, and integration smooths a step into a kink too gentle to see.

The curvature moves where the damage is #

Undo the integration and the step comes back.

The storey drift is the difference between the movement of the floor above and the floor below, which is the shear building’s equivalent of a beam’s curvature. In the damaged building the third storey’s drift rises by 9.6 per cent. The other storeys change by less than 2 per cent. The drift finds the damage, and says where it is, which the frequency’s fall of 0.91 per cent for this storey could not do even if it were measurable, because a frequency is one number for the whole building.

This is the basis of a family of damage-detection methods that compare mode-shape curvatures. They were proposed in the early 1990s for beams, where curvature is the second derivative of a mode shape, and they have been refined ever since. The figure shows why they work. The size of the change is not set by where the storey is. It is set by what the damage did to that storey.

Even at the top #

The frequency could not see a storey near the roof. The drift can.

With the damage moved up to the eighth storey, the first frequency falls by a fifth of a per cent. The eighth storey’s drift rises by 10.7 per cent, and every other storey’s changes by less than one. The drift’s sensitivity has hardly moved, while the frequency’s has fallen fivefold.

The reason is the surprising connection this essay turns on, and it makes the drift less a clever signal than a direct measurement. In its own mode, a building is loaded by its own inertia: each floor’s mass times its acceleration, which is the floor’s mode-shape ordinate times the frequency squared. Those forces set the shear in each storey. A 10 per cent loss in one storey barely changes the shape, so it barely changes the forces, and so it barely changes the shear in any storey. The damaged storey carries almost the same shear through 10 per cent less stiffness, and its drift rises by the ratio of the two stiffnesses — one over 0.9, eleven per cent — less a little because the frequency has fallen too.

The numbers bear the reasoning out. The stiffness ratio alone predicts 11.1 per cent. In the third storey, the first frequency has fallen by 0.9 per cent, so the squared frequency and with it every inertia force have fallen by nearly 2, and the drift rises by 9.6. In the eighth, the frequency has fallen by 0.2 per cent, the inertia forces by 0.4, and the drift rises by 10.7. What little the drift loses is exactly what the frequency gained, which is why it loses least where the frequency sees least.

The mode shape’s drift is a static stiffness test of every storey at once, loaded by the building’s own inertia. It is the measurement a load test makes with jacks and dial gauges, made instead by the building vibrating, with the load supplied free by its mass. A frequency averages that test over the whole building and reports the average. The drift reports the result storey by storey.

The measurement nobody makes #

A drift is the difference between two floors’ movements, and so measuring it means measuring two floors. The price of the drift’s sensitivity is an instrument on every floor, and precision in each one.

The frequency clears a one per cent yardstick only for damage in the first storey. The drift, with every floor measured to one per cent of the roof’s movement, does no better, and for a reason that is easy to miss. In the first mode the drift of an upper storey is a small fraction of the roof’s movement — the top storey’s is 2.2 per cent of it, against 15 per cent for the first storey’s. Two measurements each uncertain by one per cent of the roof, subtracted to give a quantity of 2.2 per cent of the roof, give that quantity to within more than half of itself. A tenth of it is lost in that. The signal is strong in relative terms and small in absolute ones, and the absolute size is what the instruments see.

Measure every floor five times more precisely, to 0.2 per cent of the roof’s movement, and the drift finds a 10 per cent loss in nine storeys of ten. The frequency, whose limit is set by the environment rather than by the instrument, finds it in one. So the method works, on two conditions. There must be an instrument on every floor, or at least on the floors either side of every storey that matters. And the instruments must be synchronised and calibrated well enough that the difference between two of them means something at the level of a fifth of a per cent. Almost no building has either, and a monitoring system built around a single roof accelerometer has chosen, without saying so, the one instrument this figure shows to be nearly blind.

What an owner who wanted to find it would install #

The figures make the design of a monitoring system a matter of arithmetic rather than of preference. The frequency is cheap and blind below the lowest storeys. The drift is sensitive everywhere and needs two synchronised instruments per storey at a precision of a fraction of a per cent of the roof’s movement. Between them lie choices a designer can actually make.

Instrument the storeys where damage is expected rather than all of them — a soft ground storey, a transfer level, the storey above a change of section — with a pair of sensors either side of each. Measure the drift directly rather than as a difference, with a displacement or tilt sensor across the storey, which removes the subtraction that made the noise so costly. And record long enough to identify the first mode’s shape at those storeys, not only its frequency. None of this needs anything exotic. What it needs is a decision, made when the building is designed, about which storeys matter — the decision that a roof accelerometer allows a building’s owner to avoid making.

What the figures assume #

The building is a shear building. Its floors translate and do not rotate, so a storey’s drift is a direct measure of that storey’s stiffness. In a real frame, rotation of the joints spreads a local loss of stiffness over the columns above and below, and the curvature signal is broader and weaker. In a slender tower whose deformation is mostly flexural, the equivalent of drift is the curvature of the tower, which is the second difference of the mode shape. That makes it more sensitive to noise, since three measured floors enter every value rather than two.

Damage is a uniform loss of stiffness in one storey. Real damage is local and often non-linear. A crack that opens under tension and closes under compression — a breathing crack — makes the stiffness depend on the direction of motion. A loosened connection adds friction and changes the damping more than the stiffness. The linear model captures the first-order effect of lost stiffness and nothing else.

The noise is in the ordinates only. Measured mode shapes have errors from the modal identification itself, from normalisation, from sensors that are not quite where the model thinks, and from environmental effects on the shape as well as on the frequency. The yardsticks in the last two figures are therefore optimistic floors, not realistic ceilings.

The assumption that makes damage visible at all #

Everything here rests on damage being a loss of stiffness, because stiffness is the only thing a linear dynamic measurement can see. Much of the damage that matters is not that, or not yet. Reinforcement corroding inside concrete that has not yet cracked, a fatigue crack in a steel detail before it has grown to a significant fraction of the section, a bearing that has seized — these are changes in strength, durability or boundary condition that leave the stiffness nearly untouched. Neither instrument in this essay sees them. The frequency sees stiffness averaged over the building, and the drift sees it storey by storey. A fatigue crack in a steel detail is designed to be found by inspection for exactly this reason: no measurement of the structure’s stiffness will find it first.

Still open: the modes at zero frequency #

Every structure in these essays has been held to the ground, so every mode has had a frequency above zero and every shape has had somewhere to bend from. A structure that is not held — a span being launched out over a river, a segment hanging from a crane, a pontoon — has modes at exactly zero frequency, in which it moves as a rigid body and stores no strain energy at all. Its stiffness matrix is singular, and its strain-energy shares, the quantity this essay turned on, are undefined for those modes. They are real modes nonetheless: a mass count counts them, and they turn out to carry every kilogram of the structure the moment the ground moves uniformly beneath it.