$3.(30$ points$)$

For the $2-$mass system shown below, the degrees of freedom are displacements $\backslash ($ q$\_\{1\}\backslash )$ and $\backslash ($ q$\_\{2\}\backslash ).$ Assume k$1=12$ N$/$m$,$ k$2=8$N$/$m$,$ c$1=\backslash (3\backslash$mathrm$\{$Ns$\}/\backslash$mathrm$\{$m$\},\backslash$mathrm$\{$c$2\}=7\backslash$mathrm$\{$Ns$\}/\backslash$mathrm$\{$m$\}\backslash ).$

$($a$)$ Derive the expressions for the mass matrix, M $,$ the stiffness matrix, K $,$ and the damping matrix, C $.$

$($b$)$ Write and solve $($by hand calculations $-$ NOT MATLAB$)$ the eigen value problem and obtain the natural frequencies and mode shapes of the system.

$($c$)$ Give an example of the initial conditions that will be needed to get an initial response of the system.