Problem #$2.(25$ points$).$ In a mechanical structure, a $3-$degree$-$of$-$freedom system representing a simplified

model of a vibrating structure is governed by the following mass and stiffness matrices:

$M=\left[\begin{array}{ccc}4& 0& 0\\ 0& 2& 0\\ 0& 0& 1\end{array}\right],K=\left[\begin{array}{ccc}12& -3& 0\\ -3& 8& -1\\ 0& -1& 4\end{array}\right]$

The natural frequencies of vibration are determined by solving the generalized eigenvalue problem in Eq$.(1)$:

$Kx=Mx$

Where:

$K$:$=$ is the stiffness matrix,

$M$:$=$ is the mass matrix,

$$:$=$ represents the eigenvalues related to the square of the system's natural frequencies, and

$x$:$=$ are the corresponding eigenvectors.

$(10$ points$).$ Solve the generalized eigenvalue problem to find the eigenvalues $.$ These eigenvalues

represent the square of the natural frequencies of the system.

$(5$ points$).$ Find the eigenvectors corresponding to the computed eigenvalues.

$(5$ points$).$ From the eigenvalues, determine the natural frequencies of the system in rad$/$s$.$ Discuss the

physical meaning of the eigenvectors in the context of structural vibration.

$(5$ points$).$ Use a numerical method or tool $($such as MATLAB$)$ to compute the eigenvalues and

eigenvectors.