Understanding Coherence in Vibration Analysis

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Coherence — also called the coherence function — is a signal-processing tool used in vibration analysis to judge the quality and validity of a two-channel measurement. It is a number between 0 and 1, computed frequency by frequency, that tells you how much of the output signal at each frequency is genuinely and linearly caused by the input signal. In effect, coherence is the analyst’s confidence meter: it answers the question “can I trust this measurement, or is noise contaminating it?” before any conclusions are drawn from the data.

1. Definition: What is Coherence?

Coherence quantifies the linear cause-and-effect relationship between two simultaneously measured signals at every frequency in the spectrum. The scale is intuitive:

• A coherence of 1.0 at a given frequency means a perfect linear relationship between the two signals — 100% of the output at that frequency is caused by the input.
• A coherence of 0.5 means only 50% of the output’s energy at that frequency is linearly related to the input. The remaining half comes from other factors: noise, non-linearities, or other unmeasured inputs.
• A coherence of 0.0 means there is no linear relationship at all between the two signals at that frequency.

Mathematically, coherence is derived from the cross-power spectral density of the two channels together with the auto-spectrum of each, normalised so the result always falls between 0 and 1. Crucially, it is an averaged quantity: a meaningful coherence value requires several averages of the measurement, which is why it can only be produced by a multi-channel analyser capable of acquiring two signals at once.

2. Validating Frequency Response Function (FRF) Measurements

The most common and critical use of coherence is to validate a Frequency Response Function (FRF). When performing an impact test — also known as a bump test — to measure how a structure responds across frequency, the coherence plot is essential for deciding whether the captured data is worth keeping.

• Good measurement: for a valid FRF the coherence should sit very close to 1.0 at the frequencies of the resonant peaks. High coherence — say above 0.95 — gives the analyst confidence that the measured response was truly caused by the hammer impact and not by background vibration or measurement noise.
• Poor measurement: if coherence drops sharply at a resonant peak, the measurement is suspect. The cause might be a poor hammer strike, a noisy environment, or a genuinely non-linear structural response. The right move is to reject that impact and try again.

One subtlety must not be mistaken for a fault: coherence naturally falls at anti-resonances — the valleys between peaks in the FRF — because the structure barely moves there and the response is dominated by noise. Low coherence in those valleys is normal and expected. This is exactly why coherence is read alongside FRF data in modal analysis, where confirming the true natural frequencies of a machine or structure depends on clean, trustworthy peaks.

3. Source Identification

Coherence can also reveal whether vibration from one machine is driving the vibration of another. Suppose a pump and a motor share a common base, and you suspect the motor is shaking the pump:

• Procedure: place one accelerometer on the motor (the input) and a second on the pump (the output), measure both simultaneously, and compute the coherence between them.
• Interpretation: if coherence is high at the motor’s running speed, that is strong evidence the vibration is being transmitted from the motor to the pump through their shared structure. If coherence is low at that frequency, the pump’s vibration is most likely caused by its own problems — its own unbalance or cavitation, for instance — rather than by the motor.

Used this way, coherence helps map vibration transmission paths and stops an analyst from chasing the wrong machine — a frequent and expensive mistake when two coupled units vibrate at similar speeds.

4. Factors That Reduce Coherence

Several distinct mechanisms can pull the coherence value below 1.0, and recognising which one is at work is part of the diagnosis:

• Measurement noise: extraneous noise contaminating either the input or output channel — the most common culprit, and one that better sensor mounting or more averages can often reduce.
• Non-linear systems: coherence only measures the linear relationship. If the system behaves non-linearly — because of looseness, a crack, or fluid-structure interaction — coherence will be low even when a genuine causal relationship exists.
• Time delays: a significant delay between input and output signals reduces coherence unless the analyser is set up to account for it.
• Other unmeasured inputs: if the output is driven by more than one source and you measure only one of them as the input, the unmeasured energy shows up as lost coherence.

5. Coherence as a Quality-Control Tool

In practice, coherence functions less like a diagnosis and more like a gatekeeper that protects every diagnosis built on two-channel data. It is closely related to the transfer function and FRF it accompanies — the FRF tells you how a structure responds, while coherence tells you how much to believe that answer at each frequency. Routine field balancing and single-channel spectrum work with a portable analyser such as the Balanset-1A do not need a coherence plot, but the moment an investigation moves into impact testing, resonance hunting, or source tracing on a multi-channel system, coherence becomes the parameter that separates a reliable result from a misleading one. In summary, the coherence function is a vital quality-control tool for advanced vibration measurements: it provides confidence in the validity of FRF data and helps identify the paths along which vibration travels through a machine.

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