Regarding Vibration Probes

Position (displacement), velocity, and acceleration are all related to one another by the calculus functions of time-integration and time-differentiation. Integrating acceleration gets you to velocity; integrating velocity gets you to position. Differentiating position gets you to velocity; differentiating velocity gets you to acceleration. However, there are practical problems limiting these techniques.

An accelerometer-style probe inherently measures acceleration. If you take the time-integral of this signal, you get a velocity signal. This is commonly done with piezoelectric sensors and build-in analog integrator circuits. The Bently-Nevada "Velomitor" probe works this way if memory serves correctly.

If you tried to integrate the velocity signal (i.e. double-integrate the acceleration signal), the errors and zero-drift that you'd encounter would be severe. This is especially true since for displacement measurement on rotating equipment we're interested in thousandths of an inch clearance. Any zero-bias in the velocity signal would inevitably "wind up" with integration over time and lead to gross displacement errors.

This is why practical displacement measurement always uses a proximity-type probe where the signal directly represents gap between the probe tip and the machine shaft -- no math required.

In theory, you could time-differentiate this position signal to get velocity, and do it again to get acceleration, but the problem here is you end up amplifying any high-frequency noise present on the original signal. Remember the rule of differentiating a sine function: d/dt sin kt = k cos kt. Any multiple of frequency (k) ends up becoming an amplifying factor in the differentiated function. This makes noise a severe problem when trying to time-differentiate a real-world signal.

This is why we generally see acceleration-style probes *and* proximity-style probes both used for vibration measurement: it's too impractical to derive all three variables from a single probe.