Question $3(20$ points$)$: Calculate the natural frequency and damping ratio for the system in the

Figure below. The system's key parameters are m$=10$kg$,$c$=100$k$($g$)/($s$),$k$\_(1)=4000($N$)/($m$),$k$\_(2)=200$

$($N$)/($m$),$ and k$\_(3)=1000($N$)/($m$).$ Assume that no friction acts on the rollers.

$(1)$ Derive the equation of motion of this system.

$(2)$ Is the system overdamped, critically damped, or underdamped?

$(3)$ Use Matlab to simulate the displacement and velocity responses of this system, when the

initial displacement x$\_(0)$ is $0\mathrm{.}1$ m and the initial velocity v$\_(0)$ is $0\mathrm{.}5($m$)/($s$)$