Acceleration, velocity and displacement
Vibration can be described by how far a surface moves (displacement), how fast it moves (velocity) or how quickly its speed changes (acceleration). For a single sinusoidal vibration at frequency f, the three are linked by the angular frequency ω = 2πf:
v = a / (2πf) d = v / (2πf)
The same velocity therefore means a large displacement at low frequency and a large acceleration at high frequency. This is why displacement is used for slow shafts and structures, velocity for general machine condition and acceleration for bearings and gears.
Peak, RMS and peak-to-peak
Each quantity can be stated three ways:
- Peak: the largest value in one cycle
- RMS: the root-mean-square value, peak / √2 for a sinusoid
- Peak-to-peak: the total swing, 2 × peak
By convention, acceleration is usually given as peak (g), velocity as RMS (mm/s or in/s) and displacement as peak-to-peak (µm or mils). Always check which one a data sheet or a limit uses: confusing RMS and peak-to-peak displacement is an error of 2.8 times.
Worked example
A pump bearing shows 4.5 mm/s RMS at 50 Hz (3000 rpm):
- Velocity peak 6.36 mm/s, peak-to-peak 12.73 mm/s
- Acceleration peak 0.204 g (2.00 m/s²)
- Displacement peak-to-peak 40.5 µm
- A 100 mV/g accelerometer gives 20.4 mV peak, 14.4 mV RMS
For a 15–300 kW machine on a rigid foundation, 4.5 mm/s is the boundary between zones C and D in ISO 10816-3.
Severity zones
ISO 10816-3 evaluates machines above 15 kW by the broadband RMS velocity measured on the bearing housings, 10 to 1000 Hz:
- Zone A: newly commissioned machines
- Zone B: acceptable for unrestricted long-term operation
- Zone C: not suitable for continuous long-term operation; plan maintenance
- Zone D: severe enough to cause damage
The boundaries depend on machine size and on whether the foundation is rigid or flexible. ISO 20816-3:2022 has replaced ISO 10816-3; contracts and site standards often still refer to the older edition, so use the one that applies to your acceptance test.
Choosing an accelerometer
The converter shows the acceleration the sensor will see. Pick a measuring range with headroom for shocks and higher harmonics, and a frequency range that covers the machine’s running speed and its bearing frequencies. General-purpose sensors are 100 mV/g; use 500 mV/g or more for slow machines with small accelerations. For loops into a PLC, 4–20 mA vibration transmitters output RMS velocity directly; scale them with the 4–20 mA converter.
Common mistakes
Applying single-frequency conversions to overall values. A broadband reading contains many frequencies; dividing an overall acceleration by 2πf at running speed does not give the overall velocity.
Mixing peak and RMS. Some portable meters show “peak” values calculated as √2 × RMS (“derived peak”), which differs from the true peak of a real signal.
Ignoring the frequency range. Two instruments with different filters can show different overall values for the same machine.
Frequently asked questions
Which units are supported?
Acceleration in g and m/s², velocity in mm/s and in/s, displacement in µm and mils (thousandths of an inch).
Is 1 g exactly 9.81 m/s²?
The conversion uses standard gravity, 9.80665 m/s².
Does the zone check work for any machine?
Only for the machine groups and conditions of ISO 10816-3. Reciprocating machines, wind turbines and others have their own parts of the standard.