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Formulating an Eigenvector Equation

  1. Mar 16, 2014 #1
    Hello. I am working on a project with a double pendulum and I am currently looking for the normal mode frequencies. I don't think that's too important to answer my question, but in the derivation I hit a point that look like this:[itex](K-M\omega^{2})\alpha=0.[/itex] Here, K and M are 2x2 square matrices. I want to solve for the eigenvalue here, but this doesn't follow the form that I normally have with eigenvector equations. If I rearrange this, I get [itex]K\alpha=M\omega^{2}\alpha[/itex] which has a matrix on each side. Naturally I thought to multiply by the inverse of M on both sides to get [itex]M^{-1}K\alpha=\omega^{2}\alpha.[/itex] However, this didn't give the correct result. Why not? How should it be done?
     
  2. jcsd
  3. Mar 16, 2014 #2

    maajdl

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    Gold Member

    You must have K or M wrong, or both.
    Check again, or submit the original problem and your approach here.
     
  4. Mar 16, 2014 #3

    AlephZero

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    Finding the eigenvalues of ##M^{-1}K## works. You must have made a mistake somewhere.

    But you don't need to do that. Just find the determinant of ##K-\omega^2M##, and find the values of ##\omega## that make it zero.

    $$\left|\begin{matrix} k_{11}-\omega^2m_{11} & k_{12}-\omega^2m_{12} \\
    k_{21}-\omega^2m_{21} & k_{22}-\omega^2m_{22} \end{matrix} \right| = (k_{11}-\omega^2m_{11})(k_{22}-\omega^2m_{22}) - (k_{12}-\omega^2m_{12})(k_{21}-\omega^2m_{21}) $$
     
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