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.
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
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:
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
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)
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).
