Vibration Unit Converter

The measurement

Amplitude

Frequency

The frequency of the component being converted, not the span of the spectrum.

Amplitude convention

RMSPeakPeak to peak

The same motion, every other way

Enter an amplitude and a frequency.

	RMS	Peak	Peak to peak
Acceleration (g)	—	—	—
Acceleration (m/s²)	—	—	—
Velocity (mm/s)	—	—	—
Velocity (in/s)	—	—	—
Displacement (µm)	—	—	—
Displacement (mil)	—	—	—

Which one to quote Velocity is normally reported RMS, displacement peak to peak, and acceleration either RMS for a trended overall or peak for a time waveform. The relationships here hold for a single sine wave: peak is RMS × √2, and peak to peak is twice peak.

This converts one frequency component. An overall level covering 10 to 1000 Hz cannot be converted this way, because the components inside it sit at different frequencies and each would be divided by a different 2πf. Integrate the spectrum instead, or measure the quantity you want to report.

Give one amplitude and the frequency it belongs to, and the calculator returns the same motion as acceleration, velocity and displacement, in six units and all three amplitude conventions. Acceleration, velocity and displacement are not three separate measurements of a machine. They are three descriptions of one motion, and at a single frequency each is one step of calculus away from the next.

Three quantities, one motion

A point vibrating at a single frequency traces a sine wave. Displacement is how far it moves, velocity is how fast it is moving, and acceleration is how quickly that speed is changing. Differentiating a sine multiplies its amplitude by the angular frequency, so with f in hertz:

a = v × 2πf
v = a / 2πf
d = v / 2πf

That factor of 2πf is the entire conversion. Everything else is a change of unit.

The consequence is worth stating plainly, because it decides which sensor you fit. Dividing by 2πf twice makes displacement collapse as frequency rises, which is why a gear mesh at several kilohertz measures as a fraction of a micrometre while it destroys the gearbox. Multiplying by 2πf twice makes acceleration collapse as frequency falls, which is why a slow shaft with a serious unbalance barely registers in g. Neither quantity is wrong. Each one simply hides faults at the end of the range where its own factor works against it.

The unit conversions

Two constants do all the remaining work.

Conversion	Factor
g to m/s²	× 9.80665 (standard gravity)
in/s to mm/s	× 25.4
mil to µm	× 25.4
CPM to Hz	÷ 60

The inch based units share the same 25.4 because a mil is a thousandth of an inch and a micrometre is a thousandth of a millimetre, so the ratio survives the change of prefix. Note that the mil is a unit of displacement and has nothing to do with the millimetre; confusing the two is a factor of 25.4 error in a clearance figure.

RMS, peak and peak to peak

For a pure sine wave:

peak = RMS × √2
peak to peak = 2 × peak = RMS × 2√2

These follow from the shape of a sine and from nothing else. A real vibration signal is not a sine, and the ratio between its peak and its RMS is a measurement in its own right rather than a constant you can apply. When a single frequency component is being converted, as it is here, the sine relationships are exact enough to use.

Which convention to quote is a matter of trade practice rather than physics. Velocity is normally reported RMS, displacement peak to peak, and acceleration either RMS for a trended overall or peak for a time waveform. The important discipline is to say which one you used. A report that gives 4.5 mm/s without saying whether that is RMS or peak has left a factor of 1.41 for the reader to guess at, and readers guess wrong.

What this conversion cannot do

It converts one frequency component. It cannot convert an overall level.

An overall reading covering 10 to 1000 Hz contains components spread across that whole band, and each of them would have to be divided by a different 2πf. Picking one frequency to represent the lot produces a number that looks like an answer and is not one. If you need the overall in another quantity, integrate the spectrum, or set the instrument to measure the quantity you intend to report in the first place.

The same caution applies to converting a velocity spectrum’s overall into displacement in order to compare it with a proximity probe reading. The probe measures shaft motion relative to the housing; a seismic sensor on the housing measures something else. The arithmetic will complete, and the comparison will still be meaningless.

A worked example

A velocity reading of 4.5 mm/s RMS at 25 Hz.

The angular frequency is 2π × 25 = 157.08 rad/s, and 4.5 mm/s is 0.0045 m/s.

Acceleration: 0.0045 × 157.08 = 0.7069 m/s² RMS, which is 0.7069 / 9.80665 = 0.07208 g RMS
Displacement: 0.0045 / 157.08 = 2.865 × 10⁻⁵ m, which is 28.65 µm RMS
Displacement peak to peak: 28.65 × 2√2 = 81.03 µm, or 81.03 / 25.4 = 3.190 mil
Velocity in inch units: 4.5 / 25.4 = 0.1772 in/s RMS
Acceleration peak: 0.07208 × √2 = 0.1019 g

Now hold the velocity at 4.5 mm/s RMS and move the frequency to 250 Hz. Acceleration becomes 0.7208 g RMS and displacement becomes 2.865 µm RMS: ten times larger in one and ten times smaller in the other, from a reading that has not changed. That is the whole argument for reporting machine condition in velocity, and the whole reason a displacement alarm is useless above a few hundred hertz.

Frequently asked questions

How do you convert mm/s to g?

Through the frequency of the component. Convert the velocity to metres per second, multiply by 2πf to get acceleration in m/s², then divide by 9.80665 to get g. At 25 Hz, 4.5 mm/s RMS works out as 0.0045 × 157.08 = 0.7069 m/s² RMS, which is 0.0721 g RMS. The same 4.5 mm/s at 250 Hz is ten times that in g, because the factor 2πf has grown by ten.

Why does converting vibration units need a frequency?

Because velocity is the rate of change of displacement and acceleration is the rate of change of velocity, and differentiating a sine wave multiplies its amplitude by the angular frequency 2πf. Without a frequency there is no factor to multiply or divide by. A reading of 4.5 mm/s is a different displacement and a different acceleration at every frequency it could have come from, so a converter that does not ask for one is guessing.

What is the difference between RMS, peak and peak to peak?

They are three ways of measuring the size of the same waveform. RMS is the root mean square, which relates to the energy in the signal. Peak is the largest excursion from zero, and peak to peak is the full swing from the lowest point to the highest. For a single sine wave, peak is RMS times the square root of two, and peak to peak is twice peak. For anything that is not a sine those relationships no longer hold.

Can an overall vibration level be converted from velocity to acceleration?

No. An overall level is the sum of every component in a band, typically 10 to 1000 Hz, and each of those components would have to be divided by its own 2πf. There is no single frequency to use, so any answer produced by picking one is wrong by whatever margin the spectrum happens to allow. Integrate the spectrum instead, or set the instrument to measure the quantity that has to be reported.

Should vibration be reported as displacement, velocity or acceleration?

Velocity, usually RMS, is the normal reporting quantity for machine condition because it stays reasonably flat across the frequency range where most faults live. Displacement, quoted peak to peak, suits low frequency work such as shaft motion and clearance. Acceleration suits high frequency work such as bearings and gears, reported RMS for a trended overall and peak for a time waveform.

The study material behind this tool

The calculator gives you the number. These course books explain what the number means and how the measurement that produced it should be taken.

Other tools for this subject

All calculators