How is vibration measured ?

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