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 [Hz]

1 Hz 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 ways to describe and quantify the vibration:

Displacement, velocity and acceleration

### Displacement

The curve we obtain over the time line is called a sine or sinus whereby

D = amplitude

= largest excursion positive or negative

T = period

= time to complete one complete cycle

f = frequency f = 1/T

= number of cycles per second

### Velocity

The velocity is another mean to describe vibration.

The velocity is also a sine function but it is leading the displacement by a certain phase shift.

The phase shift is 1/4 of a period.

We say also the phase angle is = 90°

(the complete period is referred to 360°)

V = velocity amplitude

Frequency f = 1/T

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

### Acceleration

A third mean finally to describe vibration is the acceleration

The acceleration is sinusoidal like the displacement however it points exactly in the opposite direction.

We can also say the acceleration is leading the displacement half a cycle or the phase shift is 180°

A = acceleration amplitude

Frequency f = 1/T

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 chosen the scale to plot the curves in a way that they appeared uniform with the same amplitude. This made it easier to show the basic characteristics of sinusoidal vibration and particularly the phase shift between a, v and d.

However the amplitudes of acceleration, velocity and displacement are always in a determined relation to each other. This relation 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.

With constant acceleration and increasing frequency

the velocity decreases proportionally with the inverse frequency:

The displacement decreases with the

inverse frequency squared:

For low frequencies the values for v and d become very large and for high frequencies very small.

For this reason we normally use logarithmic scales for the frequency and the amplitude.

A logarithmic scale is non-linear. The values increase always a distinct factor from division to division. For example a factor 10 like on the sample on the right.

This allows showing the values over several decades.

The velocity amplitude V decreases -1 decade per decade of the frequency

and the displacement amplitude D decreases -2 decades per decade of the 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)

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).