Vibration Units: Displacement, Velocity, Acceleration and How to Convert Them

Vibration is expressed with three quantities: displacement (µm, mil), velocity (mm/s, in/s) and acceleration (m/s², g). For a sine wave at frequency f they are related simply: v = 2πf·d, a = 2πf·v. Amplitude is given as RMS, peak or peak-to-peak; for a sine wave peak = 1.414·RMS and peak-to-peak = 2.828·RMS. A wideband overall level cannot be converted this way.

Three quantities and their units

The form and period of the vibration are the same whether you measure displacement, velocity or acceleration. The amplitude and phase differ. Modern instruments usually measure acceleration with an accelerometer and obtain velocity and displacement by integration.

Quantity	SI units	Imperial units	Typical use
Displacement (d)	µm, mm	mil (0.001 inch)	Low frequencies, clearances, shaft motion measured with proximity probes
Velocity (v)	mm/s	in/s	Overall machine vibration, ISO 10816/20816 limits
Acceleration (a)	m/s²	g, in/s²	High frequencies: rolling element bearings, gears

The unit g is outside the SI system but is still widely used. Brüel & Kjær notes that a factor of almost 10 (9.81) relates the two units, so mental conversion within about 2% is easy.

RMS, peak and peak-to-peak

The same vibration can be described in four ways:

• Peak-to-peak (pk-pk): from minimum to maximum. Useful where a clearance or stress in a part matters, for example how far a shaft moves inside its bearing.
• Peak (0-peak): the maximum excursion from zero. Shows short shocks but says nothing about the time history of the signal.
• Rectified average: of little practical interest.
• RMS: takes the whole time history into account and is directly related to the energy of the vibration, and so to its destructive capability. That is why Brüel & Kjær calls RMS the most relevant measure of amplitude.

For a pure sine wave the factors are:

From	To RMS	To peak	To peak-to-peak
RMS	1	× 1.414 (√2)	× 2.828 (2√2)
Peak	× 0.707	1	× 2
Peak-to-peak	× 0.354	× 0.5	1

For real signals with impacts, beats and several frequencies these factors are wrong: peak and RMS have to be measured, not converted into each other.

Conversion formulas for a sine wave

At a single frequency f (Hz) the quantities are related through 2πf:

1. Velocity from displacement: v = 2πf · d
2. Acceleration from velocity: a = 2πf · v
3. Acceleration from displacement: a = (2πf)² · d
4. Back again: v = a / (2πf), d = v / (2πf)

The formulas hold for any consistent amplitude measure: RMS in gives RMS out. Watch the units: with d in metres you get v in m/s.

Unit conversion constants:

Conversion	Multiply by
g → m/s²	9.80665
in/s → mm/s	25.4
mil → µm	25.4
rpm (CPM) → Hz	÷ 60

A mil is one thousandth of an inch and a unit of displacement; it is unrelated to the millimetre.

Worked examples

Example 1. Velocity to displacement and acceleration. A 1500 rpm motor, running speed 25 Hz, shows 4.5 mm/s RMS at that frequency on a bearing.

1. 2πf = 2 × 3.1416 × 25 = 157.1 rad/s.
2. Displacement: d = 4.5 / 157.1 = 0.0286 mm = 28.6 µm RMS. Peak-to-peak: 28.6 × 2.828 = 81 µm, or 81 / 25.4 = 3.19 mil.
3. Acceleration: a = 0.0045 m/s × 157.1 = 0.707 m/s² RMS = 0.072 g RMS. Peak: 1.0 m/s² ≈ 0.10 g.
4. In inches: 4.5 / 25.4 = 0.177 in/s RMS, 0.251 in/s peak.

Example 2. The same velocity at 250 Hz. At 4.5 mm/s RMS and 250 Hz the displacement is 10 times smaller (2.9 µm RMS) and the acceleration 10 times larger (7.07 m/s² RMS, 0.72 g). The same velocity at a high frequency barely shows in displacement and shows clearly in acceleration.

4.5 mm/s RMS at	Displacement, peak-to-peak	Acceleration, RMS
25 Hz	81 µm (3.19 mil)	0.707 m/s² (0.072 g)
250 Hz	8.1 µm (0.32 mil)	7.07 m/s² (0.72 g)

Example 3. Peak-to-peak displacement to RMS velocity. A proximity probe on a shaft reads 50 µm peak-to-peak at 50 Hz. RMS displacement: 50 / 2.828 = 17.7 µm. Velocity: 0.0177 mm × 2π × 50 = 5.55 mm/s RMS. But this is shaft velocity relative to the bearing housing and must not be compared with a housing limit (see below).

When conversion does not work

• Wideband overall level. A value measured, for example, over 10–1000 Hz is made of components at different frequencies. Each would need its own 2πf, so a single formula cannot convert it. Integrate the spectrum line by line or measure the required quantity directly.
• Relative versus absolute. A proximity probe measures shaft motion relative to the housing, while accelerometers and velocity sensors measure absolute housing motion. Converting housing velocity to displacement will not give you the shaft probe reading.
• Non-sinusoidal signal. The 1.414 and 2.828 factors do not apply to an impulsive signal.

Which quantity to use

Task	Quantity	Why
Overall machine condition	RMS velocity, mm/s, 10–1000 Hz	Velocity relates to energy, low and high frequencies are weighted roughly equally; ISO 20816-1 uses it as the primary indicator
Rolling element bearings, gears, high frequencies	Acceleration, m/s² or g	Acceleration gives weight to high frequencies
Low-speed machines, clearances, shafts in fluid film bearings	Displacement, µm or mil	Appreciable displacements occur only at low frequencies; proximity probes always output displacement

Brüel & Kjær puts it this way: displacement gives the low-frequency components most weight, acceleration the high-frequency ones. Most of the vibration energy of machines lies between 10 and 1000 Hz, but interesting components often reach 10 kHz, so velocity or acceleration is normally used for frequency analysis: their spectrum is flatter and makes better use of the instrument's dynamic range.

According to Emerson, shaft vibration on fluid film bearings is measured with eddy current probes and expressed as peak-to-peak displacement, while axial shaft position is usually given as 0-peak. An electrodynamic (seismic) velocity sensor is usable from about 4 to 1000 Hz: below 4 Hz the mass moves with the sensor body. Piezoelectric velocity sensors usually include a high-pass filter that removes the first 2–3 Hz, because integration inflates amplitudes at the lowest frequencies.

Zone limits for motors and pumps are collected in Electric motor vibration limits per ISO 10816-3, and what each sensor measures is covered in How to choose a vibration sensor.

Common mistakes

1. Amplitude type not stated. "3 mm/s" as RMS or as peak differs by a factor of 1.414. Record both the type and the frequency band.
2. Converting an overall level with one frequency. The result means nothing.
3. Mixing up mil and mm. 3 mil is 76 µm, not 3 mm.
4. Using g as m/s² without conversion. The difference is almost 10 times.
5. Comparing shaft peak-to-peak displacement with a housing RMS velocity limit. These are different quantities measured at different points.

What to measure with

According to the manufacturer's datasheet, the Bently Nevada vb7 portable analyzer displays acceleration in g or m/s², velocity in in/s or mm/s and displacement in mil, mm or µm, with amplitude as 0-peak, peak-to-peak or RMS, so no manual conversion is needed. For shaft vibration and position on fluid film bearings there is the 3300 XL proximity system. The difference between a simple vibration meter and an analyzer is explained in Vibration meter or vibration analyzer. KEG TRK supplies Bently Nevada equipment in Kazakhstan. Price on request.

Sources

• Brüel & Kjær. Measuring Vibration (BR 0094).
• Emerson. Measurement Types: Online Machinery Monitoring.
• Sensemore. Vibration Unit Converter.
• Specto Technology. Machine Vibration Measurements.
• Bently Nevada. vb7 Portable Data Collector, Analyzer and Balancer, datasheet 116M5255 Rev. F.