Formulating an Eigenvector Equation

[2.718281828459]

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:$(K-M\omega^{2})\alpha=0.$ 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 $K\alpha=M\omega^{2}\alpha$ which has a matrix on each side. Naturally I thought to multiply by the inverse of M on both sides to get $M^{-1}K\alpha=\omega^{2}\alpha.$ However, this didn't give the correct result. Why not? How should it be done?

 

[maajdl]

You must have K or M wrong, or both.
Check again, or submit the original problem and your approach here.

 

[AlephZero]

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.

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