# [Physics] How to calculate the damped frequencies of a linear system

linear systemsvibrations

I am dealing with a vibration problem. The system is free to oscillate and its mass, stiffness and damping matrices are
$$M = \begin{bmatrix} 60 & 23.5 & 0\\ 23.5 & 15.996 & 0\\ 0 & 0 & 3.507 \end{bmatrix}$$
$$K= \begin{bmatrix} 600000 & 117500 & 0\\ 117500 & 117010.4 & -2000\\ 0 & -2000 & 2000 \end{bmatrix}$$
$$C= \begin{bmatrix} 600 & 117.5 & 0\\ 117.5 & 319.01 & -200\\ 0 & -200 & 200 \end{bmatrix}$$
The three matrices are not simultaneously diagonalisable, so the classical modal analysis gives superficial insight of the problem. The natural frequencies of the associate undamped system are
$$\omega_n= \begin{Bmatrix} 143.078\\ 82.2742\\ 23.6099 \end{Bmatrix}$$
Through FFT I can see the frequency spectrum but I would like to have a more analytical approach. I know how to decouple equations of motion by complex modal analysis but I still don't understand ho to get the damped frequencies.

You can write the differential equation as a system of first order differential equations,

$$\frac{d}{dt} \textbf y = A\ \textbf y, \tag{1}$$

where,

$$\textbf y = \begin{bmatrix} \vec{x} \\ \dot{\vec{x}} \end{bmatrix}, \tag{2}$$

and

$$A = \begin{bmatrix} 0 & I\\ -M^{-1}K & -M^{-1}C \end{bmatrix}. \tag{3}$$

The damped frequencies of this system can be calculated from the eigenvalues of $A$.