Q$2(50$p$)$

a$)$ Determine the time constant $()$ and damping coefficient $()$ for the overall system given

below.

b$)$ A unit step change is introduced $(x_0(t)=1)$ determine the response $x_2(t)$ using below transfer

function?

$\frac{x2(s)}{x0(s)}=\frac{10}{{}^2{s}^2+2s+1}=\frac{10}{28s^2+11s+1}$

c$)$ Determine the name of response of the transfer function to a step change $($Underdamped$,$

critically damped, Overdamped$)$

d$)$ The qualitative nature of the solution $($a$),x2(t)$ can be related to the location of the roots of

the denominator of $2(s)$ in the complex plane. These roots are the roots of the characteristic

equation and the roots of the denominator of the transformed forcing function.

Draw the complex plane. Please label roots with their coordinates.

e$)$ What is the final value of $x2$ if a unit step change is given to input? $(xO(s)=\frac{1}{s}($apply final

value theorem$).$