Problem $1(40$ Pts$)$

For the mechanical system below, assume that $M_1=2kg,M_2=1kg,K_1=5\frac{N}{m},K_2=10\frac{N}{m},$ and $B_1=B_2=10N-\frac{s}{m}.u$ is an externally$-$applied input force. The desired outputs are: the position of mass $M_1,$ the acceleration of mass $M_1$ and the viscous damping force due to $B_2.$

$($a$)$ Using the equations provided above, determine state$-$space equations for this system. Note that the state variables are: $x_1,v_1,x_2,$ and $v_2*(20PTS)$

$($b$)$ Determine the output equations for this system. $(10$ PTs$)$

$($c$)$ Determine state$-$space matrices $($A$,$ B$,$ C$,$D$)$ for this system. $(10$ PTs$)$