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Measuring Torsion: Testing the Difference Signal
Two sensors on the same floor: their mean gives translation, their difference gives rotation. But the same operation that extracts rotation also amplifies noise. A peak in the difference signal is not torsion until it has been tested.
1. Why two sensors?
In SHM-04 we saw that the sensors sit as pairs on adjacent floors, and that this choice costs high-mode resolution. The reason for the pair is here: a single sensor cannot see rotation.
A floor moves as a rigid diaphragm: it translates and it rotates. A single accelerometer records the sum of both and cannot separate them. Two sensors at a known separation can.
2. Decomposition and its hidden cost
That asymmetry is why the difference signal has to be treated with suspicion. A peak seen there may well be noise that the operation itself created.
3. A signal that looked bad at first
When the difference signal was first computed the picture was discouraging. The ratio of its amplitude to that of the mean signal was around 1.5, and the broadband correlation between the two channels sat between 0.2 and 0.6 — values that in most contexts would be read as "this is noise".
The temptation at that point is to give up, or worse, to declare a peak in the difference signal to be the torsional mode without further checking. Neither is right.
4. Coherence and the control group
The correct move is to look at coherence not over the whole band but within the modal band — and to compare it against a control.
| Channel pair | at 0.55 Hz | at 1.83 Hz | Reading |
|---|---|---|---|
| computing… | |||
The control group is what makes this comparison work. Two sensors on different floors in the same direction must, in a global translational mode, be almost perfectly coherent. And they are. Two sensors on the same floor are highly coherent but noticeably less so — and that residual difference is exactly the rotation component.
5. The decisive test: is the difference consistent between floors?
Coherence within a floor is suggestive but not conclusive. The decisive question is different: if the difference signal is real torsion, the difference on one floor must agree with the difference on another floor — because torsion is a global mode.
If the difference signal were noise, differences on separate floors would be uncorrelated. Noise does not agree between floors.
| Signal | at 0.55 Hz | at 0.60 Hz | at 1.83 Hz |
|---|---|---|---|
| computing… | |||
6. The torsional frequencies
| Mode | Translational | Torsional | Ratio Ω | State |
|---|---|---|---|---|
| computing… | ||||
An unexpected by-product: the torsional frequency was determined more cleanly than the translational one. Its coefficient of variation across the events is —, with no outliers at all. The reason is that the difference operation removes the common component, so what remains is a narrow, well-defined peak.
7. What does the torsion/translation ratio tell us?
8. Coupling in the second mode
The second row of the table is the more interesting one. There the ratio is very close to 1 — the torsional and translational frequencies nearly coincide.
This also connects back to SHM-02: two modes closer together than 1/T cannot be resolved at all. Here they are resolvable, but only just — which is another reason the coupling has to be reported rather than glossed over.
9. Method: what to do before taking a difference
- Know that the difference amplifies noise. A peak in the difference signal is a candidate, not a result.
- Look at coherence in the modal band, not broadband. Broadband coherence is dominated by regions where there is no signal.
- Use a control group. A pair known to be coherent — different floors, same direction — calibrates what "high coherence" means for this data.
- Test consistency between floors. This is the decisive criterion: torsion is global, noise is not.
- Check the ratio and report coupling. Ω ≈ 1 means the two modes cannot be interpreted separately.
- Verify the sensor separation. The rotation amplitude is divided by d; an error in d scales the whole result.
10. Test yourself
- Why can a single accelerometer not separate translation and rotation?
- Why does the mean suppress noise while the difference amplifies it? By how much?
- You found a peak in the difference signal. What two tests would you run before calling it torsion?
- Why is a control group needed? What would be missing without one?
- What does Ω < 1 mean structurally, and why is it unfavourable?
- Why was the torsional frequency determined more cleanly than the translational one?
References
- Rigid diaphragm assumption and the decomposition of translation and rotation from a sensor pair — standard structural dynamics.
- Coherence function and its use in modal identification.
- Data: records of an instrumented building. The published data set contains the torsional analysis outputs, not the raw acceleration records.
All figures in this article are produced by engine/oma.js and separately pinned in tests/oma.test.js (42/42). The torsion/translation ratio can be computed for your own frequencies in the calculation tool.
The difference amplifies noise by √2; a peak in it is a candidate, not a result.