Assignment $[15$ marks$]$

Figure $1$ shows a $3-$DOF mass$-$spring system. It is found that the eigenvalues are $6\mathrm{.}34,33\mathrm{.}33,$ and $73\mathrm{.}67,$ whereas the eigenvectors for the respective eigenvalues are $\{[1],[1\mathrm{.}1571],[1]{\}}_{{}_1}=6\mathrm{.}34,\{[-1],[0],[1]{\}}_{{}_2}=33\mathrm{.}33,$ and $\{[1],[-1\mathrm{.}7285],[1]{\}}_{{}_3}=73\mathrm{.}67.$

a$)$ Based on the information given, what is the relationship between the dynamic characteristics of the system $($i$.$e$.,$ natural frequencies, mode shapes$)$ and the eigenvalues and eigenvectors? Describe the dynamic characteristics of the system. $(3$ marks$)$

b$)$ Briefly describe the procedures needed to obtain the eigenvalues and eigenvectors. $(2$ marks$)$