Understanding the Frequency Response Function (FRF)
ದಿ Frequency Response Function (FRF) describes how a structure, component or system responds to an applied excitation force as a function of frequency. Put plainly, it tells you how much a system will vibrate at each frequency when you “hit” it with a known force. The FRF is a cornerstone of structural dynamics, modal analysis and resonance detection — and it is the single most direct way to find a machine’s natural frequencies before they cause trouble.
Mathematically the FRF is a transfer function that relates a measured output response (most often acceleration) to a measured input force:
FRF = Output Response / Input Force
Both the output and the input are functions of frequency, and the FRF itself is a complex function — it carries both amplitude and phase information at every frequency line. That phase content is what makes the FRF far more informative than an ordinary operating spectrum, which records the response but not the force that caused it.
1. Definition: What the FRF Really Measures
An ordinary vibration spectrum tells you how hard a machine is shaking, but not why. The FRF answers a different and more fundamental question: what is the structure’s inherent tendency to amplify motion at each frequency, independent of how hard it happens to be driven? Because it normalises the response by the known input force, the FRF is a property of the structure itself — its mass, stiffness and damping — not of whatever forces are present on a given day. Depending on the response unit used, the same measurement is named differently: accelerance (acceleration/force), mobility (velocity/force) or receptance (displacement/force), but all are forms of the FRF.
2. How is an FRF Measured?
The classic field method is the bump test, also called an impact test:
- An accelerometer is mounted on the structure at the point where the response is to be measured.
- The structure is struck at a chosen point with an instrumented hammer — a hammer with a force sensor (load cell) built into its tip that measures the input force of each impact.
- A multi-channel vibration analyzer records the input signal from the hammer and the output signal from the accelerometer simultaneously.
- The analyzer performs an FFT on both signals and computes the ratio of output to input at each frequency line. That ratio is the FRF.
The process is repeated over several impacts and the results are averaged, which suppresses random noise and yields a clean, reliable measurement. The coherence function is computed alongside the FRF as a quality check: a coherence close to 1.0 across the band of interest confirms that the measured response really was caused by the measured input and not by extraneous noise, a poorly seated sensor, or a double hammer strike.
3. Interpreting an FRF Plot
An FRF is normally displayed as a pair of plots that must be read together:
- Magnitude plot: shows the amplitude of the FRF against frequency. It contains distinct peaks, and the frequency of each peak is a natural (resonant) frequency of the structure. The height and sharpness of each peak indicate how much amplification occurs there and how much damping is present — a tall, narrow peak means light damping and strong amplification, a low, broad peak means heavy damping.
- Phase plot: shows the phase shift between response and input force against frequency. As the frequency sweeps through a resonance, the phase makes a characteristic 180° shift, passing through 90° exactly at the natural frequency. This phase behaviour is the definitive confirmation that a peak truly is a resonance and not, say, a measurement artefact.
Reading both plots in tandem is the safeguard: a genuine mode shows both a magnitude peak and the matching phase rollover, whereas spurious peaks generally do not.
4. Applications in Vibration Diagnostics
The FRF is an indispensable tool for diagnosing and curing resonance problems in machinery and supporting structures:
- Identifying natural frequencies: its primary use — pinpointing the natural frequencies of a machine, its baseplate, connected piping, or the surrounding support structure.
- Confirming resonance: if a machine vibrates heavily at a particular frequency in service, an FRF measurement reveals whether that operating frequency coincides with a structural natural frequency. When a peak in the running spectrum lines up with a peak in the FRF, resonance is confirmed as the root cause of the high vibration — a far more decisive answer than spectrum data alone can give.
- Modal analysis: by taking FRF measurements at many points across a structure, a full model of its vibration modes — its mode shapes, or operating deflection shapes at resonance — can be built. This model shows not just the frequency of each mode but the shape in which the structure deforms.
- Structural modification (“what-if” analysis): once a resonance is confirmed, the modal model can simulate the effect of candidate fixes — adding a stiffener or a tuning mass, for example — before any metal is cut, so the chosen remedy is known to work in advance.
5. Why the FRF Matters in Rotating Machinery
Resonance is one of the most common reasons a rotor that has been correctly balanced still vibrates too much. If a machine’s running speed happens to coincide with a structural natural frequency, even a tiny residual unbalance is hugely amplified, and no amount of further balancing will bring the vibration down. This is why an FRF or bump test belongs in the balancing engineer’s toolkit: when a rotor refuses to balance, the FRF reveals whether the real culprit is a resonant support rather than the rotor itself. In the field this often unfolds with a single instrument — a portable two-channel analyser such as the ಬ್ಯಾಲೆನ್ಸೆಟ್-1ಎ can capture the 1× amplitude and phase that characterise the running condition, while a bump test on the stationary structure identifies any nearby natural frequency that the operating speed might be exciting. Confirming the separation between running speed and the structure’s resonances, with help from a natural frequency calculator, often explains a stubborn vibration that balancing alone could never resolve.
