When Is a Frequency Shift Evidence?

1. An open question

SHM-06 compared the modal parameters of a building before and after an earthquake. Frequencies had fallen in every mode, and that fall was presented as evidence of damage. But the article left one question unanswered:

This article measures it. Fourteen separate earthquake records of the same building are processed, and the building's own natural scatter is quantified.

2. What is operational modal analysis?

In classical modal testing the structure is excited by a known force — a shaker or an instrumented hammer — and the transfer function between input and output is computed. For a building that is impractical: you cannot shake a twenty-storey structure with a known force.

Operational modal analysis (OMA) works only from the output. The excitation is ambient: wind, traffic, an earthquake. The assumption is that the excitation is broadband and roughly white, so that the peaks in the response belong to the structure rather than to the input.

3. FDD: decomposition in the frequency domain

Frequency Domain Decomposition builds the cross-spectral density matrix of all channels at each frequency and decomposes it. At a modal frequency the matrix becomes nearly rank-one, and its dominant eigenvector is the mode shape.

4. The data: fourteen events

The records come from an instrumented building: — channels, sampled at — Hz. Fourteen separate earthquake events spread over a period of years.

What makes this data valuable is that it is not a single before/after pair. Fourteen records make the building's own variability measurable — which is the missing ingredient of the whole argument.

5. What can be done without a channel map?

At the time of this analysis the sensor layout drawing was not in hand: which channel sits on which floor and in which direction was unknown. That does not block frequency identification, because a modal frequency is a global property — it appears in every channel.

What it does block is mode shape: without knowing the position of a coordinate, the shape cannot be drawn. So this article deliberately confines itself to frequency. The channel map was obtained later and the shapes were processed in SHM-04.

6. The identified frequencies

7. Automatic peak picking went wrong four times

Look at the marked values. Automatic peak picking made four erroneous identifications out of fifty-six — a rate of about 7%. The reason is usually the same: in a weak record the modal peak is not sufficiently above the noise, and the algorithm locks onto a neighbouring peak.

8. Natural scatter

After outlier removal, here is what the same building's own variation looks like:

The coefficient of variation is around 1–2%. That is, the frequency of the same building in the same state varies at this level from event to event. The physical sources are known: excitation amplitude (a building softens slightly under larger motion), temperature, and identification error. With this data set they cannot be separated — and that is precisely why the result is used as an elimination criterion, not a physical explanation.

9. Consequence: correcting SHM-06

Now the shifts reported in SHM-06 can be placed against this scatter.

An important qualification: this does not mean the building is undamaged. It means that the frequency shifts alone are not sufficient evidence of damage. SHM-06 has a second, independent line of argument — the COMAC results — and that argument is unaffected, because two independent directions pointing at the same floor cannot be produced by scatter.

10. A larger shift is not always more detectable

There is a counter-intuitive result in the table. The EW first mode shift (−2.12%) is larger than the NS first mode shift (−1.60%). Yet it is less detectable.

The practical translation: in a monitoring system, the mode with the largest shift is not necessarily the most informative. The most informative is the one whose baseline is quietest.

11. What to do in practice

1. A single undamaged record is not a baseline. Enough repetitions are needed to measure the scatter. One reading gives no threshold.
2. Reject general thresholds. There is no such rule as "a 5% drop means damage". The threshold is determined from that structure's own measurements.
3. Use a robust criterion for outliers. Mean and standard deviation are corrupted by the very value you are trying to detect.
4. Report detectability, not the shift. "−2.1%" says nothing on its own; "1.15 times the scatter" does.
5. Look for a second, independent line of evidence. A spatial indicator such as COMAC is stronger than a single scalar shift.
6. Record the environmental variables. Temperature in particular is a known source of frequency variation; without it, scatter and damage cannot be separated.

12. Test yourself

1. Why can the absolute amplitude of a mode shape not be determined in operational modal analysis?
2. Why are outliers removed with the median absolute deviation rather than the standard deviation?
3. A −3% shift was reported and the scatter is 2%. Is this evidence? What would you need to say it is?
4. A mode with a larger shift is less detectable. How is that possible?
5. Why does a frequency measurement not require a channel map, while a mode shape does?
6. The frequency shifts turned out not to be evidence. Does it follow that the building is undamaged?