Problem $7.$ Consider the setup in Problem $6.$

a$)$ For the EoM:

$m{x}^{}+b{x}^{}+kx=F(t)$

construct the transfer function $($TF$)$ by taking $F$ as

input and $x$ as output.

b$)$ For the reaction force equation:

$F_r=b{x}^{}+kx$

construct the TF by taking $x$ as input and $F_r$ as output.

c$)$ Determine the overall TF with $F$ as input as $F_r$ as

output. $($Hint: take the product of the two transfer

functions obtained above.$)$ Express the poles, zeros,

and DC gain of the TF using ${}_n$ and $.$

d$)$ Determine the corresponding frequency response

function $($FRF$)$ and express it as a function of $$ and $.$

Calculate the magnitude $M$ and the phase $$ of the

FRF$.$ Compare it to the result from Problem $6.$

e$)$ Plot the Bode diagrams in Matlab for $=0\mathrm{.}3$ and

${}_n=2\frac{}{3}.($Hint: use the tf and bode commands.

Observe that the maximum magnitude and minimum

phase angle match those on the resonance curves.$)$

Submit the figure and your codes.