top of page
Decoration-concentric waves as symboll for vibration. Image source:

Piezo Accelerometer Tutorial

What is Vibration?

Basics of Vibration

 Here is a simplified version of this page 


A body is said to vibrate when the body as a whole or parts of it are oscillating about a rest position. (For the moment we are looking at oscillations that are cyclic or periodic.)


The number of times a complete motion cycle takes place during one second is called frequency.

The dimension of the frequency is called Hertz

1 Hertz equals to 1/sec (or 1 cycle per second)


If the oscillating motion consists of a single component occurring at a single frequency, like for example a tuning fork, the motion is called harmonic or sinusoidal.


The vibration of a real machine is normally much more complex and consists of many components with different frequencies occurring simultaneously.


Trigger_Button to open additional info_edited_edited.png

Three different Ways to describe Sinusoidal Vibration

In the following we will explore the sinusoidal vibration in more details. There are three different notations to describe and quantify the vibration: Displacement, velocity and acceleration


The correct notation of a sinusoidal displacement d (motion) is:


d = D sin (ωt𝛗)


D = amplitude

ω = angular frequency ω = 2π∙f

f = frequency   f = 1 / T

φ = phase

We can normally consider that φ = 0

so d becomes:


d = D  sin (ωt)

The displacement vs time is a sine or sinus

The notation "sin (ωt) " is arbitrary, sometimes "cos (ωt) " is used in place.



The velocity is another mean to describe vibration

The velocity is of course also a sine function. It is leading* the displacement with a phase shift of π/2.

v = V ∙ sin (ωt + π / 2)


this is identical to


v = V ∙ cos (ωt)


V = velocity amplitude

ω = angular frequency ω = 2π∙f

= frequency = 1/T

*) We could also say the displacement is lagging the velocity


Note: The scale of the velocity amplitude has been chosen in a way that it appears eaqual to the displacement amplitude



A third mean finally to describe vibration is the acceleration

The acceleration is sinusoidal with a phase shift of half a cycle or π with respect to the displacement.


a = A ∙ sin (ωt + 𝛑)


this is identical to

a = − A ∙ sin (ωt)*


* The negative sign is indicating that for the harmonic motion the acceleration is always in opposition to the displacement.

The acceleration points exactly in the opposite direction of the displacement

Note: The scale of the acceleration amplitude has been chosen in a way that it appears equal to the displacement amplitude

Relation between A / V / D

​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:

a = − A ∙ sin (ωt)

With constant acceleration and increasing frequency ....

the velocity decreases proportionally with the inverse frequency:

v = A/ωcos (ωt)

the displacement decreases with the

inverse frequency squared:

d = A/ω² ∙ sin (ωt)

Relation between acceleration, velocity and displacement vs frequency.


The respective amplitudes become then:      V=1ω·A           D=1ω²·A

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)

Link to Wikipedia

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).

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).
Dimensions of A, V, D

 This is to contunue on the yellow, more advanced path 

 This is the continuation of the green, easy path 

bottom of page