Vibration

Abstract

This chapter is all about vibration.

• When there is a force or couple that accelerates something towards some central position, things vibrate.

• If there is no ‘damping’ to soak up energy, they will vibrate forever.

• If the force or couple is proportional to the displacement, then the movement will be a sinusoidal (sine or cosine) function of time.

• If there is some damping, with a force or couple proportional to velocity, the sine-wave will ‘decay’ exponentially.

As an engineer, you will probably wish to add some damping to limit any unwanted vibration or to change the ‘resonant’ frequency away from that of a disturbance.

As a musician, you would wish to shape the vibration to something that sounds pleasant!

So to deal with a vibrating system you should:

• 1. Look for the ‘variables’ that describe what is happening.

• 2. Find some equations for their rates-of-change in terms of all such variables and any inputs.

• 3. Either eliminate all the variables but one, to get a differential equation, or else find the characteristic equation of a first order matrix equation to find the eigenvalues.

• 4. Solve this equation to analyse what will happen, in terms of the frequencies involved.

Now there are two ways to go about step 3. You can mess about with simultaneous equations and algebra, or you can use the power of matrices to help you.

• Write the ‘state equations’ in matrix form.

• Look for ‘eigenvalues’ and ‘eigenvectors’, where the rate-of-change of an eigenvector is just the eigenvalue times the vector itself.

This gives an exponential solution that is easy to recognise.