Displacement, Velocity and Acceleration
A vibration signal plotted as displacement vs. frequency can be converted into a plot of velocity vs. frequency by a process of differentiation, as we have defined earlier. Differentiation involves a multiplication by frequency, and this means the vibration velocity at any frequency is proportional to the displacement times the frequency. For a given displacement, if the frequency is doubled, the velocity will also double, and if the frequency is increased tenfold, the velocity is also increased by a factor of ten.
In order to obtain acceleration from velocity, another differentiation is required, and this results in another multiplication by frequency. The result is that for a given displacement, the acceleration is proportional to the frequency squared. This means that the acceleration curve slopes upward twice as steeply as the velocity curve.
To illustrate these relationships, consider how easy it is to move your hand back and forth over a distance of one foot at one cycle per second, or 1 Hz.. It might be possible to attain the same hand displacement at 5 or 6 Hz. But consider how fast your hand would be moving if it had the same 1 foot displacement at 100 Hz, or 1000 Hz!
Now consider the great force that would be required to move your hand a foot at these higher frequencies. Force equals mass times acceleration according to Newton, so the force required goes up as the square of the frequency. This is the reason we never see high acceleration levels combined with high displacement values. The very large forces that would be required are simply not found in practice.
From these considerations, it can be seen that the same vibration data plotted in displacement, velocity, and acceleration will have very different appearances. The displacement curve will greatly emphasize the lowest frequencies, and the acceleration curve will greatly emphasize the highest frequencies at the expense of the lowest ones.
The relationship between levels of displacement, velocity, and acceleration versus frequency in standard English units of mils peak-to-peak, inches per second peak, and G RMS are expressed by the following equations:
The three curves shown above display the same information, but the emphasis is changed. Note that the displacement curve is difficult to read at higher frequencies, and acceleration has enhanced higher frequency levels. The velocity curve is the most uniform in level over frequency. This is typical of most rotating machinery, but in some cases the displacement or acceleration curves will be the most uniform. It is a good idea to select the units so the flattest curve is attained -- this provides the most visual information to the observer. Velocity is the most commonly used vibration parameter for machine diagnostic work.
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