$(2)$ The two pendulums are connected by a spring k$.$ The links are massless, but they are rigid $($not flexible cables$).$ The spring is unstretched when both links are vertical. The rotational damping constants at the joints are $\backslash (\backslash$mathrm$\{$C$\}\_\{1\}\backslash )$ and $\backslash (\backslash$mathrm$\{$C$\}\_\{2\}\backslash ),$ respectively. Derive the equations of motion and put the equations in matrix form with $\backslash (\backslash$mathrm$\{$M$\},\backslash$mathrm$\{$C$\},\backslash$mathrm$\{$K$\}\backslash )$ matrices and a right$-$hand side vector $($even when the right$-$hand vector is a zero vector for free vibration$).$