Problem $1.$ If you have a spring$-$mass$-$damper system that is harmonically excited, its equation of motion will be in the form:

$m{x}^{}+c{x}^{}+kx=F_{0}cos(t)$

We found out that the force transmissibility $($force transmitted into the support of the system relative to the input force$)$ can be written as:

$\frac{F_T}{F_0}=\sqrt[\phantom{2}]{\frac{1+(2r{)}^2}{(1-r^2{)}^2+(2r{)}^2}}$

Use the MATLAB file titled 'AmplitudeBodePlot$\_$HarmonicExcitation.mlx$\text{'}$ as a starting point to plot the force transmissibility $(\frac{F_T}{F_0})$ vs normalized frequency $(r)$ :

a$.$ Plot force transmissibility $(\frac{F_T}{F_0})$ vs normalized frequency $(r)$ in linear scale.

b$.$ force transmissibility $(\frac{F_T}{F_0})$ in $dB$ vs normalized frequency $(r)($use log scale for $r).$

For both problems above:

Draw the response for $=0\mathrm{.}01,0\mathrm{.}05,0\mathrm{.}1,0\mathrm{.}3,0\mathrm{.}8$ and for a range of $r$ from $0\mathrm{.}01$ to $10$ with a step size of $0\mathrm{.}01.\{$These values are already used in the code. You don't have to change them

Label your axes properly.

Title your figures.

Draw grids on each figure.

Add a figure legend to mark each $$ value.

List your observations related to your force transmissibility $(\frac{F_T}{F_0})$ vs normalized frequency $(r).$

What happens at low $r$ and why?

What happens at large $r$ and why?

What happens at $r$~~$1$ and why?

What happens as $$ increases and why?

Suppose that you want to design a system that isolates a sensitive device from harmonic forces. You would like your system to have as low force transmissibility as when exposed to a harmonic force with a particular frequency $.$ How do you choose ${}_n$ and $$ for your system based on your observations in part $(2)?$