Vibration
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.
Heinrich Hertz © by Wikipedia, the free encyclopedia
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
Displacement
The correct notation of a sinusoidal displacement d (motion) is:
d = D ∙ sin (ωt−𝛗)
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 notation "sin ωt " is arbitrary, sometimes "cos ωt " is used in place.
Velocity
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
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
Acceleration
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.
Note: The scale of the acceleration amplitude has been chosen in a way that it appears equal to the displacement amplitude