- #1

happyparticle

- 400

- 20

- Homework Statement
- Find the eigenvalues of a 3x3 matrix

- Relevant Equations
- ##\begin{pmatrix} - \omega^2 +\frac{k}{m} &-\frac{k}{m} &0\\ -\frac{k}{M}& - \omega^2 + 2 \frac{k}{M} &-\frac{k}{M} \\

0&- \frac{k}{m} & -\omega^2 + \frac{k}{m}

\end{pmatrix} \begin{pmatrix}A_1\\A_2\\A_3 \end{pmatrix} =0##

Hi,

I have a 3 mass system. ##M \neq m##

I found the forces and I get the following matrix.

I have to find ##\omega_1 , \omega_2, \omega_3## I know I have to find the values of ##\omega## where det(A) = 0, but with a 3x3 matrix it is a nightmare. I can't find the values.

I'm wondering if there's another way to have the eigenvalues.

##\begin{pmatrix} - \omega^2 +\frac{k}{m} &-\frac{k}{m} &0\\ -\frac{k}{M}& - \omega^2 + 2 \frac{k}{M} &-\frac{k}{M} \\

0&- \frac{k}{m} & -\omega^2 + \frac{k}{m}

\end{pmatrix} \begin{pmatrix}A_1\\A_2\\A_3 \end{pmatrix} =0

##

I have a 3 mass system. ##M \neq m##

I found the forces and I get the following matrix.

I have to find ##\omega_1 , \omega_2, \omega_3## I know I have to find the values of ##\omega## where det(A) = 0, but with a 3x3 matrix it is a nightmare. I can't find the values.

I'm wondering if there's another way to have the eigenvalues.

##\begin{pmatrix} - \omega^2 +\frac{k}{m} &-\frac{k}{m} &0\\ -\frac{k}{M}& - \omega^2 + 2 \frac{k}{M} &-\frac{k}{M} \\

0&- \frac{k}{m} & -\omega^2 + \frac{k}{m}

\end{pmatrix} \begin{pmatrix}A_1\\A_2\\A_3 \end{pmatrix} =0

##