This article is from the Acoustics FAQ, by Andrew Silverman with numerous contributions by others.
Vibration is monitored with an accelerometer. This is a device that is securely attached by some means to the surface under investigation. The accelerometer produces a tiny electrical charge output, proportional to the surface acceleration, which is then amplified by a charge amplifier and recorded or observed with a meter. The frequencies of interest are generally lower than sound, and range from below 1 Hz to about 1 kHz.
It is sometimes more useful to know the velocity or displacement rather than the acceleration. In the case of velocity, it is necessary to integrate the acceleration signal. A second integration will provide a displacement output. If the vibration is sinusoidal at a known frequency, f, then an integration is easily calculated by dividing the original by 2 x pi x f (noting that there is a phase change)
Example: A machine is vibrating sinusoidally at 79.6 Hz with an rms acceleration of 10 m/s^2. Its rms velocity is therefore 10/(2 x pi x 79.6) = 20 mm/s Its rms displacement is 10/(4 x pi^2 x 79.6^2) = 0.04 mm
 
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