A machine of mass $m$ has a rotating device, which generates a harmonic forcing excitation $f(t)$ in

the vertical direction. The machine is mounted on the factory floor using a vibration isolator of

stiffness $k$ and damping constant $b.$ The harmonic component of the force that is transmitted to

the floor, due to the forcing excitation, is $f_s(t).$ A simplified model of the system is shown in the

following figure. The corresponding force transmissibility magnitude $|T_f|$ from $f$ to $f_s$ is given by

$|T_f|=\sqrt[\phantom{2}]{\frac{1+4{}^2{r}^2}{(1-r^2{)}^2+4{}^2{r}^2}},$ where $r=\frac{}{{}_n},$ is the damping ratio, ${}_n$ is the undamped

natural frequency of the system, and $$ is the excitation frequency $($of $f(t).$

Suppose that $m=100kg$ and $k=1\mathrm{.}0{10}^6\frac{N}{m}.$ Also, the frequency of the excitation force $f(t)$

in the operating range of the machine is known to be $200ra\frac{d}{s}$ or higher. Determine the damp$-$

ing constant $b$ of the vibration isolator so that the force transmissibility magnitude is not more

than $0\mathrm{.}5.$

Using MATLAB, plot the resulting transmissibility function and verify that the design require$-$

ments are met.

Note: $2\mathrm{.}0=6dB$;$\sqrt[\phantom{2}]{2}=3dB$;$\frac{1}{\sqrt[\phantom{2}]{2}}=-3dB$;$0\mathrm{.}5=-6dB.$