Technical Note · Measurement and Analysis
Measuring Transmissibility
How a vibration controller turns two accelerometer signals into a trustworthy transmissibility curve, how coherence tells you whether to believe it, and how to use it to qualify fixtures and detect damage.
Key takeaways
- Transmissibility is the ratio of motion at a response point to motion at a reference point, usually the control accelerometer, as a function of frequency. It has a magnitude and a phase.
- With sine it is measured one frequency at a time through tracking filters. With random it is estimated from averaged cross-spectra, which gives every frequency at once plus a coherence check.
- The H₁ estimator is unbiased by noise on the response channel; H₂ is unbiased by noise on the reference channel. H₁ is the usual default because the control channel is normally the clean one.
- Coherence near 1 means the measurement is trustworthy; low coherence flags noise, nonlinearity or unrelated vibration. It also sets how many averages are needed.
- Transmissibility is the tool for fixture surveys, isolator checks, resonance identification and pre/post-test damage detection.
Our Resonance note explains what a transmissibility curve means physically: the rigid, resonant and isolation regions, Q and the 90° phase signature. This note covers the practical side: how the curve is measured, what can go wrong, and how to check it.
Transmissibility versus frequency response function
Transmissibility and the frequency response function (FRF) are closely related and often confused. Both are ratios of an output to an input across frequency; they differ in what the input is.
| Transmissibility | Frequency response function (FRF) | |
|---|---|---|
| Ratio | Motion ÷ motion (e.g. response acceleration ÷ control acceleration) | Motion ÷ force (accelerance, mobility or compliance) |
| Units | Dimensionless (g/g) | g/lbf, (m/s²)/N, etc. |
| Typical setup | Test article mounted on a shaker; reference accelerometer on the table or fixture | Impact hammer or shaker with a force gauge at the drive point |
| Depends on | The structure and how it is excited and supported | The structure and its boundary conditions |
| Main uses | Fixture qualification, isolator evaluation, resonance search, damage detection | Modal analysis: natural frequencies, damping and mode shapes |
On a shaker the force at the interface is rarely measured, but the controller always knows the control acceleration. That makes transmissibility the natural measurement for vibration test labs.
What a measured transmissibility looks like
Figures 1 and 2 show a measured transmissibility for a test article with two modes, estimated from a random test with 64 averages and compared with the true system.
Magnitude is usually plotted on log-log axes so that a 0.03 dip and an 8× peak are both readable. Phase should always be inspected with magnitude: two closely spaced modes, or a mode and a fixture resonance, are often easier to separate by their phase behavior than by amplitude alone.
Sine or random: two ways to measure it
A sine sweep measures transmissibility directly: at each frequency the controller compares the tracking-filtered amplitude and phase of the response channel with those of the control channel. Random testing excites all frequencies at once, and transmissibility is estimated from averaged spectra.
| Swept or stepped sine | Random | |
|---|---|---|
| How it is computed | Ratio of tracking-filtered amplitudes and phases at the drive frequency | Ratio of averaged cross- and auto-spectra (H₁ or H₂) |
| Signal-to-noise | Excellent: all drive energy at one frequency | Lower: energy spread across the band |
| Speed | Slow; sweep rate limited by resonance build-up and filter settling | All frequencies at once; time set by resolution and averages |
| Quality check | Repeatability between sweeps | Coherence at every frequency |
| Nonlinear articles | Shows amplitude-dependent behavior at each level tested | Gives a best linear average over the amplitude distribution |
| Typical use | Low-level resonance searches before and after qualification tests | Fixture surveys, response monitoring during random tests |
Our Tracking Filters note explains how filter bandwidth and sweep rate affect sine-based measurements. The rest of this note focuses on the random case, where most of the subtleties lie.
Estimating transmissibility from random data
Simply dividing the response spectrum by the reference spectrum does not work with random data: it discards phase, and any noise on either channel adds directly to the result. Instead, the controller averages three spectra over many data blocks:
- Gxx, the auto-spectrum (PSD) of the reference channel x;
- Gyy, the auto-spectrum of the response channel y;
- Gxy, the cross-spectrum between them, which keeps only the part of y that is consistently related to x, including its phase.
Uncorrelated noise averages out of the cross-spectrum but not out of the auto-spectra. That leads to two standard estimators, each immune to noise on one channel:
Noise on the response channel: use H₁
Response accelerometers often sit where the motion is small, such as at anti-resonances, on isolated components or in the isolation region above resonance. There, measurement noise can be comparable to the signal.
Noise on the reference channel: H₂ is unbiased
When the reference channel is the noisy one, the roles reverse. This happens when transmissibility is taken between two response points (for example, across an isolator where the "input" side is itself at a resonance), or when the reference accelerometer is poorly mounted or low in sensitivity.
| Situation | Preferred estimator | Reason |
|---|---|---|
| Reference is the control accelerometer on the table or fixture (most shaker tests) | H₁ | The controlled reference channel has high signal-to-noise; response channels are where noise lives. |
| Accurate resonance peak amplitudes needed, reference may be noisy | H₂ | Unbiased by reference noise, so peaks are not underestimated. |
| Transmissibility between two response points (e.g. across an isolator) | Compare H₁ and H₂ | Both channels may be noisy; the true value lies between the two estimates. |
| Coherence near 1 at all frequencies of interest | Either | H₁ and H₂ agree when coherence is high, since H₁/H₂ = γ². |
Coherence: should you believe the curve?
The same three spectra give the coherence function, a number between 0 and 1 at every frequency that answers: how much of the response is linearly explained by the reference?
| Cause | Where it shows | Remedy |
|---|---|---|
| Channel noise | Anti-resonances, isolation regions, low test levels | Higher-sensitivity accelerometer, better range setting, more averages |
| Leakage (resolution too coarse) | At sharp, lightly damped resonances | Finer frequency resolution; Hanning window |
| Nonlinearity (rattle, impacts, friction, loose joints) | Broad areas, often level-dependent | Inspect hardware; compare transmissibility at several levels |
| Unrelated vibration sources | Specific frequencies (cooling fans, facility machinery) | Identify and isolate the source |
| Time delay between channels | Increasingly at high frequency | Use synchronously sampled, phase-matched channels |
Our existing notes on the importance of coherence and phase and coherence in vibration control cover coherence in the broader context of control and multi-exciter testing.
How many averages are enough?
Coherence also determines how much averaging is required. The random error in the transmissibility magnitude shrinks with the number of averages and grows as coherence falls:
| Coherence γ² | 4 averages | 16 averages | 64 averages | 256 averages |
|---|---|---|---|---|
| 0.99 | 3.6% | 1.8% | 0.9% | 0.4% |
| 0.95 | 8.1% | 4.1% | 2.0% | 1.0% |
| 0.90 | 11.8% | 5.9% | 2.9% | 1.5% |
| 0.80 | 17.7% | 8.8% | 4.4% | 2.2% |
| 0.60 | 28.9% | 14.4% | 7.2% | 3.6% |
Quadrupling the averages halves the error. Where coherence is high, a few dozen averages give a precise curve; where it is low, no reasonable amount of averaging will fix the problem, and the cause in Table 4 should be addressed instead.
Application: fixture surveys
A fixture is supposed to deliver the same motion to every mounting point. A fixture survey checks this by measuring transmissibility from the control accelerometer to accelerometers at the mounting points, typically with a low-level random or sine sweep before the test article is installed (or with a mass simulator).
The usable range is where all points stay within an agreed band, commonly ±3 dB, of the reference. If the test extends beyond it, the options are to stiffen or damp the fixture, to use multiple control accelerometers with averaging so the controlled level represents the whole interface, or to agree response limits with the specification owner. Our upcoming fixture design note will cover this in more detail.
Application: pre- and post-test resonance comparison
Many qualification procedures bracket a durability test with low-level resonance searches. If the structure has been damaged, by a cracked solder joint, a loosened fastener or a fatigued bracket, its stiffness drops and its damping usually rises, and the transmissibility shows it.
| Change | Likely meaning |
|---|---|
| Natural frequency down | Loss of stiffness: cracking, loosening, yielded joints. Frequency scales with √stiffness, so a 7% drop means roughly 13% less stiffness. |
| Peak transmissibility down, resonance broader | Increased damping, often from friction in a newly loose joint. |
| New peak appears | A component has come loose or a new rattle path has formed. |
| Phase or coherence changes without frequency change | Developing nonlinearity, such as intermittent contact. |
| Change only in one mode | Damage localized in the region that mode exercises. |
Acceptance criteria for frequency shift vary by program, so the allowable change should come from the test specification. Because the comparison relies on small differences, both searches must use the same accelerometer locations, mounting, level, sweep rate and analysis settings.
Common misconceptions
| Misconception | Reality |
|---|---|
| "Transmissibility is a property of the test article alone." | It also depends on how the article is supported and where the reference is measured. Change the fixture or reference point and the curve changes. |
| "Response PSD divided by control PSD gives transmissibility." | That gives |H|² only when both channels are noise-free, and it loses phase. Use H₁ or H₂ from cross-spectra. |
| "Low coherence means the controller is not working." | Coherence describes the relationship between two channels. Low values at anti-resonances are expected and simply mean the curve is uncertain there. |
| "More averages always fix a noisy curve." | Averaging reduces random error, not bias. Noise on the wrong channel biases H₁ or H₂, and nonlinearity cannot be averaged away. |
| "A transmissibility peak is always a test article resonance." | It may be a fixture, slip table, head expander or accelerometer-mount resonance. Fixture surveys and phase help separate them. |
Quick reference
| Quantity | Formula | Rule of thumb |
|---|---|---|
| H₁ estimator | Gxy / Gxx | Default when the reference is the control channel |
| H₂ estimator | Gyy / Gyx | Better resonance peaks when the reference is noisy |
| Coherence | |Gxy|² / (GxxGyy) = H₁/H₂ | Above about 0.9 for reliable data |
| Random error of |H| | √(1−γ²) / (|γ|√(2nd)) | ×4 averages → ½ the error |
| Stiffness change from frequency shift | k2/k1 = (f2/f1)² | 5% frequency drop ≈ 10% stiffness loss |
| Fixture usable range | All points within ±3 dB (typical) | Check with a survey before the test |
How Panther helps
- Transmissibility during any test: Panther computes transmissibility, phase and coherence between response channels and the control channel during sine and random tests, so fixture and article behavior is visible while the test runs.
- Phase-accurate channels: a single master clock gives channel-to-channel phase match on the order of 0.01°, keeping transmissibility phase and coherence valid up to high frequencies, including across multiple chassis.
- Low-noise measurement: 24-bit acquisition with more than 110 dB of dynamic range keeps anti-resonances and isolation regions measurable instead of lost in channel noise.
- Comparison and analysis: Multi-Source Data Definition overlays live and stored results for pre/post-test comparisons, and Panther Analyzer provides transfer-function and modal tools for deeper investigation.
References and further reading
- J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures, 4th ed., Wiley.
- J. S. Bendat and A. G. Piersol, Engineering Applications of Correlation and Spectral Analysis, 2nd ed., Wiley.
- D. J. Ewins, Modal Testing: Theory, Practice and Application, 2nd ed., Research Studies Press.
- C. M. Harris and A. G. Piersol (eds.), Harris' Shock and Vibration Handbook, 5th ed., McGraw-Hill.
- Spectral Dynamics Technical Library: Resonance; Importance of Coherence in Vibration Control; Coherence and Phase Matching in Vibration Testing; FFT Fundamentals.
