Glossary term
Frequency Response Function
Complex response-to-excitation ratio measured or computed versus frequency, used in modal testing, vibration analysis and structural dynamic validation.
Definition
modelA frequency response function is a complex ratio between a measured response and an applied excitation as a function of frequency.
In vibration and modal testing, a frequency response function, often abbreviated FRF, relates a response such as displacement, velocity, acceleration or strain to an excitation such as force or base motion. FRFs preserve both magnitude and phase, so they can reveal resonant frequencies, anti-resonances, damping, mode shapes, boundary-condition effects and test quality.
A frequency response function (FRF) describes how much a system responds to a known excitation at each frequency, including phase. In structural dynamics and vibration testing, the excitation is often force and the response may be displacement, velocity, acceleration or strain.
For a measured force input F(\omega) and response Y(\omega), the basic complex ratio is:
The value of H(\omega) is complex. Its magnitude states response per unit input. Its phase states the lead or lag between response and input. An FRF is therefore not just a curve height; it is a frequency-dependent complex measurement with units, locations, directions and boundary conditions.
Engineering Role
FRFs are the core data product of many modal tests, shaker tests, impact-hammer tests and structural dynamic validation campaigns. They help engineers identify natural frequencies, damping, anti-resonances, mode-shape information, force paths, boundary-condition errors and finite-element correlation issues.
Common FRF types include:
| FRF type | Ratio | Typical unit | Use |
|---|---|---|---|
| Receptance | X/F | \text{m}/\text{N} | displacement response and flexibility |
| Mobility | V/F | (\text{m}/\text{s})/\text{N} | velocity response and damping interpretation |
| Accelerance | A/F | (\text{m}/\text{s}^2)/\text{N} | accelerometer-based modal testing |
| Strain FRF | \epsilon/F | strain per newton | fatigue-sensitive response locations |
The same structure can have different FRFs depending on input point, response point, direction, boundary condition, load path, sensor type and operating condition. A point FRF uses the same input and response location. A transfer FRF uses different locations.
FRF Matrix and Reciprocity
A modal test with many input and response coordinates produces an FRF matrix, not a single curve. One entry can be written as:
where F_k is the force applied at coordinate k and Y_j is the response measured at coordinate j. A column of the matrix shows how many response points react to one input. A row shows how one response point reacts to different inputs.
For a linear, passive, reciprocal mechanical structure measured with compatible directions and units, a transfer FRF should satisfy approximately:
Reciprocity is a useful quality check, but it is not automatic. It can fail or become hard to interpret when the setup has nonreciprocal components, rotating machinery gyroscopic effects, active control, follower forces, nonlinear joints, inconsistent coordinate signs, different sensor loading, shaker-structure interaction or different boundary conditions between measurements.
Estimating an FRF from Test Data
In measured data, FRFs are often estimated from spectra rather than from a single complex division. For response y(t) and input force f(t), a common estimator is:
where G_{yf} is the cross-spectrum between response and force, and G_{ff} is the force autospectrum. This estimator is often preferred when output noise is significant.
A second common estimator is:
where G_{yy} is the response autospectrum. This estimator is more sensitive to input-noise assumptions. In practice, the estimator choice should match the expected noise source, test method and decision being made. A report that gives FRF curves without estimator, window, averaging and coherence information is incomplete.
The magnitude-squared coherence is often reported with the FRF:
High coherence supports the repeatability of the linear input-output relation at that frequency, but it does not prove that the test setup, coordinate convention or boundary condition is correct.
Worked Example: Convert Accelerance to Mobility and Receptance
An impact-hammer test on a bracket gives a force spectrum magnitude of:
At 42\ \text{Hz}, an accelerometer measures acceleration response:
with phase:
The accelerance magnitude is:
Angular frequency is:
The equivalent mobility magnitude is:
The equivalent receptance magnitude is:
For phase, velocity lags acceleration by 90^{\circ}:
Displacement differs from acceleration by 180^{\circ}:
which is usually wrapped into the [-180^{\circ},180^{\circ}] interval and reported as:
Engineering comment: the three FRFs describe the same physical measurement in different response variables, but their magnitudes and phases are not interchangeable. A modal test report must state whether it is reporting receptance, mobility, accelerance or another response/input ratio.
Validation and Common Mistakes
A defensible FRF states input location, response location, directions, excitation type, force calibration, response sensor calibration, boundary conditions, operating point, frequency range, frequency resolution, windowing, averaging, estimator, units, phase convention, coherence or quality metric, repeatability and uncertainty.
Useful review checks include repeated impacts, drive-level sensitivity, reciprocity, sensor mounting inspection, force-window quality, overload checks, anti-aliasing and sampling review, boundary-condition confirmation and comparison with expected resonances or finite-element trends. If an FRF will support modal assurance criterion, effective modal mass, damping extraction or aeroelastic clearance, the test evidence should match that decision consequence.
Common mistakes include:
- reporting magnitude without phase;
- mixing receptance, mobility and accelerance units;
- comparing FRFs measured with different boundary conditions or excitation amplitudes;
- accepting data with double hammer hits, force dropout, clipping or sensor overload;
- ignoring sensor mass loading, cable motion, poor mounting or shaker-structure interaction;
- treating a nonlinear or amplitude-dependent response as a single linear FRF;
- using high coherence as the only quality check without reviewing reciprocity, repeatability and physical plausibility.
Distinction from Related Terms
Frequency response function is not generic frequency response. Frequency response is the broader relationship between sinusoidal input frequency and output magnitude and phase. An FRF is a specific complex response/input function, often measured between defined test points with physical units.
Frequency response function is not a Bode plot. A Bode plot is one way to display magnitude and phase versus frequency. The FRF is the underlying complex data.
Frequency response function is not modal analysis. Modal analysis uses FRFs, curve fitting and mode-shape information to estimate modal parameters. The FRF is the measured or computed response function used by the analysis.
Frequency response function is not a transfer function in every context. A transfer function is a mathematical system model, usually written in the Laplace domain. A measured FRF may approximate a transfer function along the frequency axis, but it also includes test setup, sensors, boundary conditions and noise.
Frequency response function is not a mode shape. Mode shapes are spatial deformation patterns. FRFs can be measured at many points and fitted to estimate those shapes.