Modes

Abstract

You have seen how spotting a pair of state variables can enable you to analyse a single-degree-of-freedom system that involves just a single frequency.

An example of such a system is a simple pendulum, swinging from side to side. But that pendulum has another degree of freedom where it swings towards us or away, or any combination of the two, swinging diagonally or in circles or ellipses. It has two frequencies, although in this case they are the same.

In general there will be numerous modes. Consider the system of a plate hanging on two springs that you met in the previous chapter.

As you well know, the mass has six degrees of freedom – three of position and three more of rotation. So there will be six ways in which it can vibrate or swing.

These are:

• 1. Up and down bouncing on the springs;

• 2. Rotary twisting on the springs about an axis ‘into the page’;

• 3. Sideways swinging;

• 4. Rotation about a vertical axis;

• 5. A combination of swinging towards us and away, where the plate tilts in the same direction as the swing;

• 6. Another combination where the tilt opposes the swing.

These can all happen at the same time!

Each of the first four can be analysed individually, using the techniques you have used for single degree of freedom systems.

But the last two require something more special.