Case study
Impact Hammer FRF Low Coherence Troubleshooting Case Study
Impact-hammer FRF case study on low coherence, H1/H2 estimator disagreement, double-hit diagnosis, damping error, retest and release evidence.
This case study follows an impact-hammer modal test that initially produced a plausible frequency response function but failed the data-quality review. The problem was not that the bracket had no mode near the target frequency. The problem was that low coherence and estimator disagreement made the first damping estimate unfit for release.
The scenario is a generic machine-support bracket tested after a stiffness modification. The engineering team needs a defensible FRF around the first bending mode before approving the change for commissioning.
Case Context
| Item | Engineering relevance |
|---|---|
| Structure | Welded machine-support bracket on a test fixture. |
| Test method | Instrumented impact hammer with one accelerometer response channel. |
| Target band | 80\ \text{Hz} to 160\ \text{Hz}. |
| Decision | Confirm the first bending mode frequency and damping for a vibration release note. |
| Acceptance rule | Coherence near the target mode must be at least 0.90, and H1/H2 FRF magnitudes must agree within 10 percent. |
| Original symptom | FRF peak near 120\ \text{Hz} looked real, but coherence was low. |
| Risk | A poor test could overstate damping and hide a resonance problem. |
The test looked convincing at first because the FRF peak was narrow, repeatable on the plot and near the frequency predicted by the finite-element model. The time record told a different story: several hammer strikes had a small second contact after the main hit.
Test Boundary
The case is about modal-test evidence quality, not about redesigning the bracket. The boundary is the measured acceleration-over-force FRF between one hammer input point and one response point on a defined fixture. The result may support a commissioning note, model-correlation update or troubleshooting decision only for that boundary condition.
The boundary excludes several adjacent questions:
- whether every bracket mode has been identified;
- whether the finite-element model has been fully updated;
- whether the fixture represents all field mounting conditions;
- whether nonlinear contact, preload or weld cracking exists outside the tested level;
- whether operational vibration under rotating machinery forcing is acceptable.
Those questions may require shaker testing, multiple response points, operating deflection shapes, run-up data or strain evidence. This case asks a narrower but critical question: can the single impact-hammer FRF be used to release one modal frequency and damping estimate?
Original Spectral Data
At the frequency line near 120\ \text{Hz}, the processed spectra from the original test were:
| Quantity | Symbol | Value |
|---|---|---|
| force autospectrum | G_{ff} | 18.0\ \text{N}^2 |
| acceleration autospectrum | G_{aa} | 0.720\ (\text{m}/\text{s}^2)^2 |
| cross-spectrum magnitude | $ | G_{af} |
| H1 peak frequency | f_n | 120\ \text{Hz} |
| initial half-power frequencies | f_1,\ f_2 | 116\ \text{Hz},\ 127\ \text{Hz} |
The H1 estimator magnitude is:
The H2 estimator magnitude is:
The estimator spread is:
That fails the 10 percent agreement rule. The two estimators are not telling the same story about the response/input ratio.
The disagreement matters because the estimators respond differently to measurement errors. In a simplified single-input test, (H_1) is usually more robust to output noise, while (H_2) is usually more robust to input noise. A large spread does not identify the cause by itself, but it tells the reviewer that the spectra are not a clean single-input single-output representation of the structure.
Step 1: Check Coherence
The magnitude-squared coherence is:
Substitute the original spectra:
This fails the local acceptance rule:
If the low coherence were caused mainly by uncorrelated output noise in a simple single-input case, the approximate SNR screen would be:
In decibels:
Engineering comment: this is not strong evidence for damping extraction. The FRF peak may still locate a real mode, but the response/input relation at the decision frequency is too weak to support a release value.
Raw Time-Record Gate
The spectra should not be reviewed without the time records. A low-coherence impact test can be caused by many mechanisms that look similar after averaging:
| Time-record symptom | Likely effect on FRF |
|---|---|
| double hit | distorted force spectrum and phase |
| clipped force pulse | underestimated input energy |
| weak hit | poor signal-to-noise ratio near the target band |
| cable slap | output not caused only by structural response |
| drifting fixture contact | boundary condition changes between averages |
| late ringing in force channel | input not close to an impulse |
For this case, the raw force records show occasional secondary contact after the main hammer impact. Those records should be rejected before averaging. Keeping them because the final FRF peak looks plausible is a process failure: bad averages can create a clean-looking but biased modal estimate.
The gate should be explicit. Accept a record only if the force pulse is single-contact, unclipped, contains enough energy through the target band, and the response channel is not saturated or contaminated by cable motion. Rejecting bad records at acquisition is better than explaining a weak averaged result after the test article has been removed.
Step 2: Estimate the Damping Error
Using the first data set, the half-power damping estimate would be:
That damping value looks comfortable for release, but it is built on low-coherence data and large H1/H2 disagreement. The team therefore treats it as a rejected estimate, not as a conservative structural property.
The likely causes were visible in the raw records:
- a secondary hammer contact after the main hit;
- weak force energy around the target band on several averages;
- inconsistent response phase between records;
- a response cable moving against the fixture during impacts.
The damping error is the most dangerous part of the first result. Natural frequency is often less sensitive to a moderate FRF-quality problem than damping, because damping depends on peak width and local curve shape. A double hit can broaden or distort the peak enough to make the structure look safer than it is. For release, an optimistic damping estimate is worse than a missing estimate because it can hide a resonance exposure.
Step 3: Retest with Corrected Setup
The corrective actions were deliberately small:
- Reject double-hit records at acquisition instead of averaging them.
- Use a hammer tip that gives better force energy through 160\ \text{Hz}.
- Add cable strain relief near the accelerometer.
- Keep the boundary condition unchanged.
- Repeat the same input and response coordinates.
The retest spectra near the same mode were:
| Quantity | Symbol | Value |
|---|---|---|
| force autospectrum | G_{ff} | 32.0\ \text{N}^2 |
| acceleration autospectrum | G_{aa} | 1.40\ (\text{m}/\text{s}^2)^2 |
| cross-spectrum magnitude | $ | G_{af} |
| retest peak frequency | f_n | 121.1\ \text{Hz} |
| retest half-power frequencies | f_1,\ f_2 | 118.7\ \text{Hz},\ 123.5\ \text{Hz} |
The retest coherence is:
The estimator magnitudes are:
The estimator spread is now:
The damping estimate becomes:
Retest Acceptance Matrix
The retest should not be accepted only because the numbers improved. It should pass the same predeclared gates that rejected the first data set.
| Gate | Original data | Retest data | Decision |
|---|---|---|---|
| coherence near mode | 0.640 | 0.958 | retest passes |
| H1/H2 estimator spread | 56.3\% | 4.4\% | retest passes |
| damping estimate | 4.6\% | 2.0\% | use retest only |
| hit quality | double hits present | single-contact records accepted | retest passes |
| boundary condition | nominally unchanged | unchanged and documented | comparable |
This matrix is useful because it separates two decisions: rejecting the first data and accepting the second. A retest can fix one issue and introduce another. If the boundary condition changes, a better coherence value may not be comparable to the first result. If the input point changes, the same mode may be excited differently. If the accelerometer is moved, the apparent modal amplitude and anti-resonance shape can change.
For this case, the boundary was controlled and the same input/response coordinates were used. That makes the retest a valid replacement rather than a separate experiment.
Release Decision
The original data set is rejected for damping release because it fails both quality gates:
and:
The retest data set passes:
and:
The accepted modal evidence is therefore the retest result: first bending mode at approximately 121.1\ \text{Hz} with damping near 2.0\% for the stated fixture condition. The release note must also state that the first test overestimated damping because the FRF quality was inadequate.
Release Evidence Package
The release package should include more than the final FRF plot:
- raw accepted and rejected force time records;
- rejection rule for double hits, clipping and weak force input;
- impact tip, hammer range and force-window settings;
- accelerometer model, mounting method, cable routing and range;
- input/output coordinates and fixture condition photographs or notes;
- averaging, windowing, frequency resolution and bandwidth settings;
- H1 and H2 FRF overlays around the target band;
- coherence plot with the acceptance threshold shown;
- damping extraction method and frequency picks;
- statement that the finite-element model uses retest damping only.
This evidence lets another engineer reconstruct why the first estimate was rejected and why the second estimate can support the commissioning note. Without the rejected-record evidence, a later reviewer may see two damping values and choose the more convenient one.
Retest Triggers
Repeat or escalate the modal test if any of these conditions appear:
| Trigger | Engineering concern |
|---|---|
| coherence remains below threshold | the input/output relation is not release-quality |
| H1/H2 spread remains high | input noise, output noise or nonlinearity is unresolved |
| damping changes strongly with impact level | the bracket or fixture may be nonlinear |
| peak frequency shifts between averages | boundary condition or contact state is changing |
| cable or sensor motion is visible | response is not only structural acceleration |
| mode shape evidence conflicts with model | the measured peak may not be the expected mode |
If escalation is needed, move from a single-point hammer test toward multiple response points, roving hammer measurements, shaker excitation, improved fixturing or operational vibration data. The right escalation depends on the release decision: commissioning a speed range needs stronger evidence than identifying a troubleshooting direction.
Lessons for Engineering Review
A clean FRF peak is not enough. The release evidence must include input quality, response quality, coherence, estimator consistency, raw time records and setup repeatability.
Low coherence is not a number to hide in an appendix. It is a diagnostic signal. In this case it changed the engineering conclusion from “bracket damping is comfortably high” to “the original damping estimate is invalid and the bracket must be judged from retest data.”
The result is also a useful boundary on model correlation. A finite-element update should not be tuned to match the rejected 4.6\% damping estimate. It should use the accepted frequency and damping evidence from the retest, with uncertainty tied to the measurement chain and fixture condition.