fully solve this question:

Problem $1$ System A in Fig. P$1($a$)$ was designed for a mechanical positioning system where $x(t)$ is a displacement

input; and $(m,k,$ and $b)$ are the $($mass$,$ spring stiffness, and damping coefficient$).$ During manufacturing, the spring

and damper were accidentally interchanged $($System B$).$ An engineer suggests modifying the design to System C

making the spring and damper components interchangeable to avoid potential errors. He further suggests that the

effects of the system zeros on the transient response can be analyzed using an initial response test with $f(t)=0.$

a$)$ Derive the transfer functions $Y\frac{s}{x}(s)$ for both Systems A and B $,$ and the Laplace transformed output $Y(s)$ for

System C which is initially at rest but displaced by $y(t=0)=1.$ Show that all three systems have the same

characteristic equation of the form, $s^2+2{}_{n}s+{}_n^2=0.$ Express the damping ratio $$ and natural frequency ${}_n$ in

terms of $m,k,$ and $b.$

b$)$ The damping coefficient is designed such that $b=b_{\text{c}\text{r}\text{i}\text{t}}=2\sqrt[\phantom{2}]{mk}$ and $\frac{b}{m}=8$ You are asked to sketch and compare

the time responses of Systems A and B to a unit$-$step change in $x(t)$ and that of System C $($with $f=0)$ instantly

relieved from its initial unit displacement.

c$)$ Present Systems A and B by the block diagram in Fig. P$1($b$).$ For each system, express the variables and transfer

functions in terms of the system parameters $(m,b,$ and $k).$

Fig. P$1$