(Solved):   3. The mass matrix and the stiffness matrix of a three DOF system are \( \mathbf{M}=\left[ ...

 

3. The mass matrix and the stiffness matrix of a three DOF system are \( \mathbf{M}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1\end{array}\right] \) and \( \mathbf{K}=\left[\begin{array}{ccc}3 & -2 & 0 \\ -2 & 3 & -1 \\ 0 & -1 & 3\end{array}\right] \) respectively. (1) Determine the natural angular frequencies (5 pts). (2) Determine the mode shapes (3 pts). (3) Determine the modal matrix ( \( 1 \mathrm{pt}) \). (4) Draw the diagram illustrating the mode shape vectors (1 pt).