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Coupled Oscillations

  1. Mar 19, 2009 #1
    I have a burning question,

    I was trying to find the solutions for a double mass coupled oscillation. So I found out the eigenvectors and then I arrived at this step

    [tex] \left( \begin{array}{c} \ddot{x_1} \\ \ddot{x_2} \end{array} \right)=\lambda \left( \begin{array}{c} x_1 \\ \ x_2 \end{array} \right) [/tex]
    (the second matrix is without the accents, I think the latex code will take a while to refresh)

    ok so my question is, why is one of the solutions displayed as

    [tex] x_{1}+x_{2}=A_{1}\cos{(\omega t+\phi)} [/tex]

    when from the first equation, it is evident that

    [tex] \ddot{x_1}=\lambda{x_1} [/tex]

    so

    [tex] x_1=A_1\cos{(\omega t+\phi)} [/tex]

    I simply don't understand why the above is not acceptable. Also, I am having trouble in relating the addition of the equations (equation 2) to the solution for the eigenvectors. By the way, I also know the solution for the eigenvectors.
     
    Last edited: Mar 19, 2009
  2. jcsd
  3. Mar 19, 2009 #2
    Isn't X_1 + X_2 a mode coordinate?

    The 2 mode coordinates being the sum and difference of X_1 and X_2. They are ways of looking at the motion of the system as a whole, not X_1 and X_2 individually.
     
  4. Mar 19, 2009 #3
    thank you for your reply,

    isn't the matrix constructed from x_1 and x_2 individually?

    I also do not understand what is a mode coordinate, could you explain this to me if this is important?
     
  5. Mar 19, 2009 #4

    tiny-tim

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    Hi Oerg! :wink:

    It is acceptable, but it's not as simple nor as conceptually deep as using normal modes such as x1 ± x2

    From http://en.wikipedia.org/wiki/Coupled_oscillation#Coupled_oscillations
    See also http://en.wikipedia.org/wiki/Normal_mode :smile:
     
  6. Mar 19, 2009 #5
    thanks for your reply too

    there was another solution that is

    [tex] x_{1}+x_{2}=A_{2}\cos{(\omega t+\phi _{1})} [/tex]

    and then with the first equation in the original post, x1 and x2 is then given as

    [tex] x_1=\frac{1}{2}(A_{1}\cos{(\omega _{0}t-\phi)} +A_{2}\cos{(\sqrt{3}\omega _{0}t-\phi _{1})}) [/tex]

    by the way,

    [tex] \lambda =1[/tex]

    and

    [tex] \lambda =3[/tex]

    are the eigenvalues for the problem. So there seems to be a discrepancy for the equations for x_1 and x_2.Where have I gone wrong :confused:
     
  7. Mar 20, 2009 #6
    help im drowning arghhhhhhh
     
  8. Mar 20, 2009 #7

    tiny-tim

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    i don't follow :confused:

    what discrepancy are you referring to?
     
  9. Mar 20, 2009 #8
    why is the last equation from my last post different from the correct solution to the de?

    Also, how do I obtain the correct solutions from the eigenvector
     
  10. Mar 21, 2009 #9
    I think i understand a little now, the eigenvalues that I found was when all the masses displayed the same frequency of oscillation.

    But how do I prove that the equations for the positions of the masses are a superposition of normal modes with the eigenvectors?
     
  11. Mar 21, 2009 #10

    tiny-tim

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    Hi Oerg! :smile:

    Stop using all these technical words

    x1 + x2 = A cosBt, x1 - x2 = C cosDt,

    obviously x1 = (AcosBt + CcosDt)/2 … that's year-1 arithmetic! :wink:

    We solve it that way round because you only have to look at the formula for x1 on its own to see that it's much more difficult to solve than x1 + x2 :smile:

    but there's no magic of "superposition" or "normal modes" to understand
     
  12. Mar 21, 2009 #11
    thanks for your reply

    hmm, i know about the equations "x1 + x2 = A cosBt, x1 - x2 = C cosDt" for a two mass system, it is just the addition and subtraction and then the acceleration and the position are common terms.

    But what about a system with a higher number of masses and springs? How do i know
    x?+x?+x?+..=Acoswt+phi

    So I was trying to see how by finding out the eigenvectors for a two mass system for simplicity, that x1+x2=Acoswt+phi. Im still at a loss though.
     
  13. Mar 21, 2009 #12

    tiny-tim

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    i'm confused :confused: … i think you're answering your own question …

    every matrix has eigenvectors, and each eigenvector is a combination of "basis" vectors, and by definition of eigenvector that combination is going to satisfy the shm equation ∑'' = -w2∑, so ∑ = Acoswt+phi :smile:
     
  14. Mar 21, 2009 #13
    so we have a Ax=b and b can be expressed as a linear combination of the eigenvalues multiplied by the respective eigenvectors? This is because eigenvectors are orthogonal. so in this spirit we have the solutions for a 3 mass system as

    [tex]\left( \begin{array}{cc} \ddot{x_1} \\ \ddot{x_2} \\ \ddot{x_3} \end{array} \right)=\lambda _{1}v_{1}x+\lambda _{2}v_{2}x+\lambda _{3}v_3x [/tex]

    where

    [tex] x=\left( \begin{array}{cc} x_1 \\ x_2 \\ x_3 \end{array}\right)[/tex]

    is this correct?
     
  15. Mar 22, 2009 #14
    ahh i think i understand now, I read up on a chapter on diagonalization and now I understand how it can be applied to solve the system of differential equations. Thanks for your help tiny tim.
     
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