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Structural Health Monitoring · SHM-06  |  Confidence: A
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MAC, COMAC and Frequency: Three Indicators, Three Different Answers

The same building, before and after an earthquake. MAC says "the shapes did not change". Frequency says "something certainly changed". COMAC says where. Reading any one of them alone leads to a different — and wrong — conclusion.

Confidence: A Engine: engine/shm.js Tests: tests/shm.test.js — 50/50 Reading: ~14 min Last verified: 2026-09-10 Türkçe: bu yazının Türkçesi
Correction — see SHM-03 Below, the first-mode frequency drops are discussed as evidence of damage. A later analysis showed that they are not. In SHM-03 the natural scatter of the same building was measured across fourteen events, and the first-mode shifts turned out to stay inside that ordinary variation. The COMAC argument in this article is unaffected: two independent directions pointing at the same floor cannot be produced by scatter. Read this article together with SHM-03.

1. Framing the problem: what are we measuring?

Structural health monitoring asks a simple question: has anything changed in this structure? The measurable quantities are modal parameters — frequencies and mode shapes. But each of them answers a different question:

2. MAC — the Modal Assurance Criterion

MAC measures the similarity of two mode shape vectors:

MAC(φA, φB) = |φATφB|² / [(φATφA)(φBTφB)] It is the square of the cosine of the angle between the two vectors. 1 means the shapes are identical; 0 means orthogonal.

MAC's main use is mode pairing. The modes identified from two measurements may not come out in the same order; without pairing them correctly, comparing mode 1 with mode 1 is meaningless.

3. MAC's blind spot: scale independence

Blind spot MAC is independent of scale. A shape multiplied by two gives MAC = 1 against the original. That is not a fault — in operational modal analysis the absolute amplitude cannot be determined anyway, so scale independence is exactly what is needed. But it means MAC by definition cannot see a uniform change in magnitude.

The consequence for damage detection: damage that changes the whole shape proportionally does not appear in MAC. What MAC sees is a change in the proportions of the shape.

4. COMAC — where did the change occur?

MAC produces a single number per mode. COMAC inverts the question: rather than per mode, it computes per coordinate — that is, per measurement point.

COMAC(j) = (Σ|φA,j·φB,j|)² / [Σφ²A,j · Σφ²B,j] Summed over all modes at coordinate j. A value near 1 means that point behaves consistently across the two states; a lower value means it does not.
COMAC's own blind spot The formula contains an absolute value. A coordinate whose sign flips but whose magnitude stays the same therefore appears "consistent". This matters particularly in torsional modes and in mode pairs of symmetric structures.

5. Frequency shift and the stiffness inference

Frequency is related to stiffness and mass:

f ∝ √(k/m)   ⇒   Δk/k ≈ 2·Δf/f (for small changes) A 1% drop in frequency corresponds to roughly a 2% drop in stiffness — provided the mass has not changed.

That proviso is not decorative. If after an earthquake debris is removed, partitions demolished, or the building emptied, the mass has changed and the stiffness inference is invalid. The engine carries this assumption with every result rather than leaving it to the reader.

6. Application: an instrumented building

Records from an instrumented building before and after an earthquake. Sensors are located on the —, and the two horizontal directions are processed separately.

6.1 Frequencies

DirectionModeBefore [Hz]After [Hz]ShiftImplied stiffness
computing…
 The stiffness column assumes the mass is unchanged. See the correction note at the top of the article for whether these shifts constitute evidence.

6.2 MAC

DirectionMode 1Mode 2Mode 3Pairing
computing…
 Diagonal MAC values. All are very close to 1.

6.3 COMAC

FloorNSEW
computing…
 Damage indicator per coordinate. Arrows mark the most divergent floor in each direction.

7. Three indicators, three different answers

Summary of the casecomputing…

The lesson is the article's title. Had you read MAC alone you would have concluded "nothing changed". Had you read frequency alone you would have said "something changed" but not where. Only COMAC gives a location — and its strength comes from two independent directions pointing at the same floor.

That last point deserves emphasis, because it is what survives the correction in SHM-03. The frequency shifts turned out to stay inside the natural scatter. But the NS and EW measurements are physically independent, and scatter does not make two independent measurements agree on the same floor.

8. Common mistakes

  1. Reading a single indicator. The whole subject of this article. Each answers a different question.
  2. Expecting MAC to see everything. It is scale-independent by design and cannot see a proportional change.
  3. Forgetting the mass assumption in the stiffness inference. If the mass changed after the event, Δk/k ≈ 2Δf/f is invalid.
  4. Reading a frequency shift without comparing it to the scatter. Exactly the error this article itself made; see SHM-03.
  5. Ignoring COMAC's absolute value. A sign flip does not show up.
  6. Ignoring temperature. Temperature alone can shift frequencies by a few percent; without recording it, damage and season cannot be separated.

9. Test yourself

  1. Why is MAC scale-independent, and why is that a requirement rather than a defect in operational modal analysis?
  2. What question does COMAC answer that MAC cannot?
  3. Under what condition is the inference Δk/k ≈ 2Δf/f valid?
  4. Diagonal MAC values are all above 0.99, yet the frequencies have dropped. How can both be true at once?
  5. Why is the agreement of two independent directions stronger evidence than a single frequency shift?
  6. Which change would COMAC fail to see?

References

  1. MAC and COMAC definitions — standard modal analysis results, independent of any national code.
  2. The relation f ∝ √(k/m) and its linearisation for small changes.
  3. Data: modal parameters of an instrumented building before and after an earthquake, derived from records of a public national strong-motion network.

All figures in this article are produced by engine/shm.js and separately pinned in tests/shm.test.js (50/50). The same comparison can be repeated with your own mode shapes in the calculation tool. The correction note at the top of this article is a later addition; the original claim and its correction are both kept visible rather than the text being quietly rewritten.

Open the modal comparison tool → SHM-03: when is a shift evidence? → Türkçe okuyun →
archi-civil.com — MAC, COMAC and Frequency · SHM-06 · Confidence A · Printed:
Each indicator answers a different question; reading one alone gives a different and wrong conclusion.