## 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