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

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SUMMARY

The discussion focuses on finding the eigenvalues and eigenvectors of a system consisting of three equal masses connected by springs arranged in a circle. Participants suggest starting with Newton's second law (F = ma) to derive the equations of motion for the masses, specifically using the angles θ and φ to represent the spring lengths. The consensus is that simple harmonic motion (SHM) will emerge from these equations without needing to make initial assumptions about SHM. This approach leads to separable equations that can be analyzed to find the desired eigenvalues and eigenvectors.

PREREQUISITES
  • Understanding of Newton's second law (F = ma)
  • Familiarity with simple harmonic motion (SHM)
  • Basic knowledge of eigenvalues and eigenvectors
  • Concept of systems of differential equations
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technial
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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.
 
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technial said:
… I don't know whether you begin by assuming SHM occurs and get answers from there or something completely different …

Hello technial! :smile:

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 :wink: … the SHM should come out of the elementary F = ma equations.

What do you get? :smile:​
 

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