How to Find Eigenvalues and Eigenvectors of Masses on Springs in a Circle?

[technial]

Hello, I am a having a problem finding the solution of the follow:

3 equal masses are connected by springs in a circle, and are confined to move in the circle. Find the eigenvalues and eigenvectors of the system.

I'm really at a bit of a loss here, I don't know whether you begin by assuming SHM occurs and get answers from there or something completely different. The question is from a general paper so you can use anything to get an answer! Any help would be greatly appreciated.
Thanks everyone.

 

[tiny-tim]

… I don't know whether you begin by assuming SHM occurs and get answers from there or something completely different …

Hello technial!

Call the spring lengths θ φ and (2π - θ - φ).

Do F = ma on each mass, and you should end up with an equation in θ'' φ" θ and φ which I suspect (i haven't worked it out) will be separable into two x'' = -kx equations with x (I'm guessing) = θ ± φ.

So no need to make assumptions … the SHM should come out of the elementary F = ma equations.

What do you get? ​