Technology / Technical guide
Vibration Displacement, Velocity & Acceleration
Displacement, velocity and acceleration describe different aspects of the same motion. Their units differ, their peaks occur at different phases, and their amplitudes depend on frequency in different ways. A vibration value is therefore incomplete without the quantity, frequency content and amplitude convention that it represents.
The same motion can produce very different numerical values.
A single sinusoid gives a clear relationship; a broadband signal cannot be converted using one arbitrary frequency.
01 / Motion model
A sinusoid needs an amplitude and a time scale.
Consider x(t) = A cos(ωt), where x is displacement in metres, A its peak amplitude and ω angular frequency in rad/s.
Consider x(t) = A cos(ωt), where x is displacement in metres, A its peak amplitude and ω angular frequency in rad/s. Frequency in hertz is f = ω/(2π). The cosine reference places maximum positive displacement at t = 0.
OpenStax presents the displacement, velocity and acceleration relationships for simple harmonic motion. Here the sinusoid is a declared mathematical example. A real vibrating structure can contain multiple frequencies, changing amplitudes and offsets that do not follow this single expression.
02 / Derivatives
Velocity and acceleration carry different frequency weights.
Differentiating gives v(t) = −Aω sin(ωt) and a(t) = −Aω² cos(ωt).
Differentiating gives v(t) = −Aω sin(ωt) and a(t) = −Aω² cos(ωt). Peak speed is Aω, while peak acceleration magnitude is Aω². Their units are m/s and m/s² respectively, rather than alternative unit labels for displacement.
Acceleration is opposite in sign to displacement in this example. Velocity is zero at maximum displacement and largest in magnitude at zero displacement. The phase relationships mean the three quantities do not reach their positive peaks together, even though they describe the same sinusoidal motion.
03 / Toy values
Units and frequency determine the numerical comparison.
Take hypothetical A = 0.001 m and f = 5 Hz, giving ω ≈ 31.416 rad/s.
Take hypothetical A = 0.001 m and f = 5 Hz, giving ω ≈ 31.416 rad/s. The peak displacement is 1 mm, peak speed about 0.0314 m/s and peak acceleration about 0.987 m/s². These are calculated values from an invented sinusoid.
Doubling the frequency while retaining A doubles peak speed and quadruples peak acceleration. No increased displacement is required for that change. This explains why two displacement traces with similar amplitudes can imply substantially different acceleration magnitudes.
04 / RMS meaning
The peak-to-RMS relationship has waveform conditions.
For each zero-offset sinusoid over complete cycles, RMS equals its peak magnitude divided by √2.
For each zero-offset sinusoid over complete cycles, RMS equals its peak magnitude divided by √2. The toy velocity RMS is therefore about 0.0222 m/s, and acceleration RMS about 0.698 m/s². The calculation uses the stated pure waveform and averaging interval.
A finite interval that does not span complete cycles can give a different value. An offset also changes RMS, while waveform distortion changes the relationship between peak and RMS. Quoting √2 as a universal vibration conversion would omit those conditions.
05 / Broadband motion
Different components need their own frequency weighting.
For a sum of sinusoids, differentiation applies to each component.
For a sum of sinusoids, differentiation applies to each component. Higher-frequency displacement components receive larger velocity and acceleration weights. Multiplying a whole broadband displacement RMS by one selected angular frequency generally does not reproduce the corresponding velocity RMS.
Under suitable stationary and differentiable-process assumptions, derivative power spectra carry ω² weighting for velocity and ω⁴ for acceleration relative to displacement. Integrating each spectrum over the represented bandwidth produces its corresponding mean-square value. The bandwidth is therefore part of a broadband comparison.
06 / Scope
Integration leaves information about initial conditions unresolved.
Integrating acceleration determines velocity only up to an initial velocity; integrating again also requires an initial displacement.
Integrating acceleration determines velocity only up to an initial velocity; integrating again also requires an initial displacement. Measurement offsets can create growing integrated terms. Those mathematical ambiguities differ from proving that the structure has physically moved by the integrated amount.
No MARSTON vibration limit, sensor accuracy or vibration-monitoring function is specified. This page gives no live-data integration or sensor adjustment procedure. It explains quantity relationships and their waveform conditions without labelling any amplitude as acceptable or abnormal.
Related technical guides: Numerical Derivatives: Signal Slope & Noise Amplification · Modal Superposition: Coordinates, Coupling & Truncation · Vibration Envelopes: Modulation, Sidebands & Interpretation.
General background: OpenStax College Physics: harmonic motion kinematics. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Displacement | Metres or a stated length unit | Reference position and peak/RMS convention |
| Velocity | m/s; sinusoidal peak Aω | Defined frequency and differentiable motion |
| Acceleration | m/s²; sinusoidal peak Aω² | Same sinusoidal model and angular frequency |
| RMS | Square-root mean-square quantity | Waveform, offset and averaging interval stated |
| Broadband conversion | Component-dependent frequency weighting | One arbitrary frequency cannot represent the whole signal |
Engineering questions.
Is velocity simply displacement in another unit?
No. It is the rate of change and depends on frequency.
What changes when frequency doubles at fixed displacement?
For a sinusoid, peak speed doubles and peak acceleration quadruples.
Do the three positive peaks occur together?
No. Their phase relationships differ.
Is RMS always peak divided by √2?
Only under the stated zero-offset sinusoidal and averaging conditions.
Can one frequency convert broadband RMS?
Generally not; components carry different derivative weights.
Are actual turbine vibration limits provided?
No. The values come from an illustrative sinusoid.
Technical enquiry
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