Diagonalize a coupled damped driven oscillator

  • #1

Homework Statement


I am trying to follow a paper, https://arxiv.org/pdf/1410.0710v1.pdf, I want to get the results obtained in equations 5 and 6 but can't quite work out how eq 3 has been diagonalized.

Homework Equations


eq 3

The Attempt at a Solution


As the system is driven i thought I'd try a solution first and then try to rearrange into an eigenvalue problem. I tried the solution ##x=Ae^{i\omega t}## where ##\omega## is the driving frequency and A some constant, I tried this solution as the system will oscillate at the driving frequency?
 

Answers and Replies

  • #2
Dr Transport
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I think that eq 3 (in the [ ] at the beginning) should be [itex]\frac{d^2}{dt^2} + \gamma \frac{d}{dt} + \Omega^2 _0 [/itex]
 
  • #3
I didn't even notice that mistake and have been reading it as you have corrected. Would you suggest trying a solution and rearranging is the best method?
 
  • #4
Dr Transport
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on further review, maybe they don't treat the damping initially, [itex] \gamma [/itex] doesn't appear in the diagonalized matrix.....
 
  • #5
Yes you're right, if i don't consider damping I get the correct expressions for the eigenvalues, why would they not include the damping term? Also I'm struggling to understand the transformation matrix U, Is this a rotating frame transformation?
 
  • #6
Dr Transport
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[itex] U [/itex] allows for a normal mode analysis..., thus decoupling the equations of motion.
 

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