## 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:

a = − A ∙ sin (ωt)

With constant acceleration and increasing frequency ....

the velocity decreases proportionally with the inverse frequency:

v = A/ω ∙ cos (ωt)

the displacement decreases with the

inverse frequency squared:

d = A/ω² ∙ sin (ωt)

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

## 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).