Gearboxes vibrate often / Pixabay: gears-4191907_1920 edited

Access to Accelerometers

What is Vibration?

Basics of Vibration 2

​Relation between Acceleration, Velocity and Displacement

So far we have set all amplitudes to "one unit" i.e. we have arbitrarily chosen the scale to plot the curves in a way that they appeared uniform with the same amplitude. This to make it easier to show the phase shift phenomena between a, v and d.

However the amplitudes of acceleration, velocity and displacement are always in a determined relation to each other which is given by the frequency.

In the following we want to explore this law  in more details.

In a harmonic vibration we can choose one amplitude (for example the acceleration) and the frequency. With this set the other amplitudes (velocity and displacement) will be in a fixed relation as follows:

The notation of the acceleration was:

 

With constant acceleration and increasing frequency...

the velocity decreases proportionally with the inverse frequency:

 

the displacement decreases with the

inverse frequency squared:

Relation between acceleration, velocity and displacement vs frequency.

The respective amplitudes become then:

Or using the frequency notation: 

The linear graph is not very legible.

That's why we normally use logarithmic scales for the frequency and the amplitude.

With         V = ω ⁻¹ ·A

the velocity amplitude V decreases -1 decade per decade

and with   D = ω ⁻² ·A

the displacement amplitude D decreases -2 decades per decade

Logarithmic relation between acceleration, velocity and displacement vs frequency.
 
 

Dimensions of Acceleration, Velocity and Displacement

In the chapter about linear acceleration we have seen the dimensions of the vibration parameters

which are:

displacement :    meters (m) or milli-meters (mm)

velocity :             meters per second (m/s) or milli-meters per second (mm/s)

acceleration :      meters per second per second (m/s²)

These are also the correct dimensions to use for the vibration terms in the SI-system

(SI = International System of Units)   

However in wide parts of the industry particularly in aeronautics we use also an English system with the following units:

displacement :    inch (in) or mils (in/1000)

velocity :             inch/second (ips)

acceleration :      g ( = acceleration of gravity)

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1g  =  9.81 m/s²

An additional particularity is that the displacement is normally measured in "peak to peak" (pk-pk) values

while the velocity and acceleration are mostly given in "peak" (pk).

Sometimes you see RMS values and very rarely "average"

A particularity is that the displacement is normally measured in "peak to peak" (pk-pk) values while the velocity and acceleration are mostly given in "peak" (pk).