After the modeling process, we get

$q=R^2{y}^{}$

$m{x}^{}-2k(y-x)-b(y^{}-x^{})=0$

$($a$)$ The input to the whole system is the fluid volume $q,$ and the output is $x.$

Find the transfer function between the input and the output.

$($b$)$ Assume that $m=1,b=4,k=1\mathrm{.}5,R^2=1$ and $q$ is unit step, draw schematically

the output $x(t).($You may use initial and$/$or final value theorems and the system

damping ratio.$)$