$($A$)$

$($C$)$

$($B$)$

$($D$)$

Determine all of the displacements, $x(t),$ for each system above, and mark them on the diagrams.

For the system in Fig. A$,$

a$)$ Derive the equation of motion for the system.

b$)$ Considering $M=1kg.\frac{s^2}{m},D=6N.\frac{s}{m},$ and $K=5\frac{N}{m},$ derive the transfer function between $f(t)$ and $x(t).$ Note: Transfer function, $T(s)=C\frac{s}{R}(s)$ : Output$/$Input; Laplace of $e^{-at}.u(t)=\frac{1}{s+a}$

c$)$ Find the natural response of the displacement, $x(t),$ in time domain.

d$)$ Is this a stable system? Why?

$($a$)$ For Fig $($B$),$ considering the same values for the system components as in $2($b$)$ and a perfect elasticity for the spring, derive the transfer function for the system.

b$)$ Without solving for $x(t),$ describe the motion of the mass when the force, $f(t)$ is eliminated, and briefly explain why.