Displacement, Velocity and Acceleration: What Each Vibration Measurement Tells You

From Mechanical Maintenance to Vibration Analysis - Part 3

In Part 2 - From a Measurement to a Maintenance Decision, we followed the complete vibration-analysis workflow: understand the machine, collect repeatable data, validate the measurement, analyse the evidence, recommend an action and verify the result.

Now we address a question every new analyst meets:

Should I measure displacement, velocity or acceleration?

All three describe the same vibrating motion, but they emphasize different parts of the frequency range. Selecting the right quantity can make a machine condition easier to see. Selecting the wrong one can reduce or hide useful evidence.

Begin with one simple motion

Imagine a point on a bearing housing moving back and forth around its normal position:

• Displacement tells us how far the point moves.
• Velocity tells us how fast it moves.
• Acceleration tells us how quickly its velocity changes.

For a simple sinusoidal vibration at frequency f:

Velocity peak = 2 π f × displacement peak

Acceleration peak = 2 π f × velocity peak

Therefore, acceleration peak = (2 π f)² × displacement peak

This relationship explains the main selection principle:

• Displacement emphasizes lower-frequency motion.
• Velocity provides a useful balance across a broad middle-frequency range.
• Acceleration emphasizes higher-frequency activity.

This is not a rule that one quantity is always better than another. The right choice depends on the machine, bearing type, sensor, expected fault, speed and frequency range of interest.

1. Displacement: how far the vibration moves

Displacement is the change in position of the vibrating surface or shaft. Common units include:

• micrometres, or µm;
• millimetres, or mm; and
• mils, where 1 mil equals 0.001 inch.

Displacement is often displayed as peak-to-peak because that value represents the total travel from one extreme of the waveform to the other. Always confirm the convention in the instrument or software; µm peak, µm peak-to-peak and µm RMS are not interchangeable.

Where displacement is especially useful

• Low-frequency vibration and slow movement
• Shaft-relative vibration on machines with fluid-film bearings
• Large turbo-machinery where non-contact proximity probes observe shaft motion relative to the bearing housing
• Clearance-related questions where actual movement matters

A proximity probe measures the changing gap between the probe tip and the shaft. This is a different physical measurement from an accelerometer mounted on the bearing housing. One represents shaft-relative motion; the other normally represents absolute casing vibration. They should not be treated as if they were the same measurement.

Important limitation

As frequency increases, the displacement produced by high-frequency impacts can become extremely small. A developing rolling-element bearing defect may therefore be difficult to recognize in a displacement plot even though it is clear in acceleration or an appropriate demodulated measurement.

2. Velocity: how fast the vibration moves

Velocity is the rate of change of displacement. Common units are:

• millimetres per second, or mm/s; and
• inches per second, or in/s.

Overall casing velocity is frequently displayed as RMS. RMS is useful because it represents the effective energy of a varying signal, but you must still confirm the instrument configuration and frequency band before comparing values.

Where velocity is especially useful

• General condition monitoring of many rotating machines
• Broad assessment of casing vibration over a middle-frequency range
• Common mechanical conditions such as unbalance, misalignment and looseness when their frequencies fall inside the configured measurement band
• Trending overall machine condition under comparable operating states

Velocity does not give every frequency equal importance in every real measurement. Sensor response, integration, filtering and the selected frequency band still matter. However, compared with displacement and acceleration, velocity often provides a practical balance for general machinery monitoring.

Important limitation

An overall velocity value can tell you that vibration energy changed, but it cannot identify the cause by itself. Two machines can have the same overall velocity and completely different spectra. The analyst must review frequency, direction, location, phase, waveform, process condition and machine history.

3. Acceleration: how quickly velocity changes

Acceleration is the rate of change of velocity. Common units include:

• metres per second squared, or m/s²; and
• g, where 1 g is approximately 9.81 m/s².

Industrial accelerometers commonly use piezoelectric sensing elements. They are versatile, robust and capable of measuring a wide frequency range when the sensor, mounting and acquisition settings are suitable.

Where acceleration is especially useful

• Higher-frequency vibration
• Rolling-element bearing impacts
• Gear-mesh activity
• Blade- or vane-related frequencies
• Impulsive events and resonance excited by impacts

Acceleration is also commonly the original signal collected by a portable data collector. The instrument or software may mathematically integrate that signal to display velocity and, in some applications, displacement.

Important limitation

High-frequency acceleration can be sensitive to mounting quality. A hand-held probe, magnet, adhesive pad and stud do not provide identical frequency response. A loose or inconsistent mounting method can distort the very high-frequency information the analyst wants to examine.

A practical comparison table

Quantity	What it describes	Common display units	Often useful for
Displacement	How far the vibration moves	µm peak-to-peak, mils peak-to-peak	Low-frequency motion and shaft-relative vibration
Velocity	How fast the vibration moves	mm/s RMS, in/s peak or RMS	General casing-vibration condition monitoring
Acceleration	How quickly velocity changes	g peak, g RMS, m/s2	High-frequency, bearing, gear and impact-related activity

The entries are starting points, not universal rules. A specific monitoring program must follow the machine design, sensor specifications, applicable standards, original-equipment-manufacturer guidance, site procedures and engineering judgement.

Worked example: the same 1X vibration in three quantities

Consider a motor running at 1,500 rpm:

Frequency = 1,500 / 60 = 25 Hz

Suppose the 1X displacement is 100 µm peak-to-peak. For a simple sinusoid:

1. Displacement peak is half of peak-to-peak: 50 µm, or 0.000050 m.
2. Velocity peak = 2 π × 25 × 0.000050 = 0.00785 m/s, or 7.85 mm/s peak.
3. Velocity RMS = 7.85 / √2 = approximately 5.55 mm/s RMS.
4. Acceleration peak = 2 π × 25 × 0.00785 = approximately 1.23 m/s², or 0.126 g peak.

The motion has not changed. Only the quantity and amplitude convention used to describe it have changed.

Never compare 100 µm peak-to-peak directly with 5.55 mm/s RMS or 0.126 g peak as if they were competing severity numbers. They are different descriptions of the same sinusoidal motion.

Why frequency changes what you see

For the same displacement amplitude, increasing frequency increases velocity in direct proportion to frequency and acceleration in proportion to frequency squared.

This explains why:

• a slow shaft movement may look large in displacement but modest in acceleration;
• a high-frequency bearing impact may look small in displacement but strong in acceleration; and
• velocity often serves as a useful middle ground for general rotating-machine casing vibration.

It also explains why a single overall value cannot cover every failure mode equally well. A monitoring program may need overall velocity, acceleration spectra, bearing-condition measurements and shaft-relative displacement, depending on the asset.

Peak, peak-to-peak and RMS: do not ignore the convention

For a pure sine wave:

• Peak-to-peak = 2 × peak
• RMS = peak / √2

These simple conversions apply exactly to a pure sinusoid. Real machine vibration is usually a combination of frequencies, impacts and noise. Do not convert a broadband overall RMS value to peak or peak-to-peak using the sine-wave factors and assume the result represents the real waveform.

Before comparing readings, confirm all of the following:

• same physical quantity;
• same unit;
• same amplitude convention;
• same frequency band and filtering;
• same sensor and mounting method;
• same measurement point and direction; and
• comparable speed and load.

The sensor and the displayed quantity are not always the same

An accelerometer measures acceleration, but software can integrate its signal once to display velocity and twice to display displacement. This does not mean every converted result is equally reliable.

• Integration can magnify low-frequency noise, drift and sensor-settling effects.
• High-pass filters may remove low-frequency content.
• Sensor mounting and frequency response limit the usable high-frequency range.
• The selected acquisition range and resolution determine what enters the calculation.

Always check the actual sensor, its mounting, the acquisition setup and the displayed engineering units. Software cannot reconstruct information that the sensor and acquisition system did not capture correctly.

A practical selection guide

1. Define the machine and bearing type. A rigid-bearing motor and a fluid-film-bearing turbine do not require identical measurements.
2. Identify the failure mode of interest. Are you looking for slow shaft movement, general mechanical vibration, or high-frequency impacts?
3. Estimate the relevant frequency range. Use running speed, bearing geometry, gear teeth, blade or vane count and known excitation frequencies.
4. Select a suitable sensor and mounting. Confirm its usable frequency and amplitude range.
5. Select the quantity and convention. Record whether the result is displacement, velocity or acceleration and whether it is RMS, peak or peak-to-peak.
6. Trend like with like. Keep the point, direction, operating condition, band and setup consistent.
7. Use more than one view when needed. Combine overall trends with spectra, waveforms and other diagnostic techniques.

Three mistakes new analysts should avoid

Mistake 1: Asking which quantity is best

There is no universal best quantity. Ask which one is most sensitive and meaningful for the machine condition and frequency range being investigated.

Mistake 2: Comparing values without reading the units

A value of 5 can mean 5 mm/s RMS, 5 µm peak-to-peak or 5 g peak. The number alone is meaningless.

Mistake 3: Treating an overall value as a diagnosis

Overall vibration is useful for screening and trending. Diagnosis requires the frequency content, location, direction, time behaviour, operating condition and machine context.

Final takeaway

Displacement, velocity and acceleration are not three unrelated measurements. They are three connected ways of describing vibration.

Displacement answers how far. Velocity answers how fast. Acceleration answers how quickly the velocity changes.

The analyst's job is not to select a familiar unit automatically. It is to match the measurement quantity, sensor, frequency range and amplitude convention to the machine and the suspected condition.

Coming in Part 4

Part 4: Understanding the Time Waveform and FFT Spectrum. We will explain what each plot shows, how frequency relates to machine speed, what 1X and harmonics mean, and why a spectrum pattern should be treated as evidence rather than proof.

Discussion question: Which quantity do you use most often in your plant - displacement, velocity or acceleration - and what type of machine are you monitoring?

References and further learning

• Emerson, Vibration Analysis for Machinery Health Diagnosis, especially the comparison of typical sensors, units and applications.
• SKF, Introduction Guide to Vibration Monitoring, sections on displacement, velocity and acceleration.
• NI, What Is Vibration Monitoring?, sensor and measurement overview.
• Emerson Process Management, Basic Vibration Analysis - Course 2031, measurement-parameter fundamentals.
• Vibration Analysis Guide, beginner sections on amplitude, frequency and measurement quantities.

Educational note: The worked example assumes a pure sinusoidal signal and is intended to demonstrate the mathematical relationship between quantities. It is not a universal alarm limit. Apply approved site criteria, applicable standards, manufacturer guidance and qualified engineering judgement to real machinery.