Problem $5.$ Consider the mass$-$spring$-$damper system

shown in the figure $($model of a "skyhook" damper$),$

with mass $m,$ damping coefficient $b,$ stiffness $k,$

position $x,$ and displacement $r(t)$ at the end of the

spring as input. The spring is unstretched for $x=r.$

a$)$ Derive the EoM for $x$ using free body diagram and

fundamental law of dynamics.

b$)$ For ICs $x(0)=0$ and $x^{}(0)=0,$ derive the

impulse response of the system by considering

$r(t)=r_0(t)$ and solving the EoM analytically.

Calculate the solution for damping ratios

$1,$ and $>1.$

c$)$ Plot the impulse response using Matlab for $m=2$

$kg,b=20N\frac{s}{m},k=300\frac{N}{m},r_0=0\mathrm{.}1m,$ and

tin$[0,2]s.$ Submit the figure and your codes.