P$3.(30$ points$)$ Consider the system shown in the figure below $($let $p_{\text{l}\text{o}\text{a}\text{d}}=0).$ Assume $D_p=1\mathrm{.}5m^3\frac{s}{r}ad,R_{\text{l}\text{e}\text{a}\text{k}}=1Pa*\frac{s}{m^3},R=1Pa*\frac{s}{m^3},b_p=300N*m*\frac{s}{r}ad,J=0\mathrm{.}2N*m.\frac{s^2}{r}ad,C_{\text{a}\text{c}\text{c}}=1{10}^{-4}\frac{m^3}{P}a,$ and $R_{eq}=(\frac{1}{R_{\text{l}\text{e}\text{a}\text{k}}}+\frac{1}{R}{)}^{-1}=0\mathrm{.}5Pa*\frac{s}{m^3}$

The transfer function of this second$-$order system $($for $p_{\text{l}\text{o}\text{a}\text{d}}=0)$ is:

$\frac{q(s)}{T_m(s)}=\frac{\frac{D_p}{R}}{J{C}_{\text{a}\text{c}\text{c}}{s}^2+(b_p{C}_{\text{a}\text{c}\text{c}}+\frac{J}{\text{R}\text{e}\text{q}})s+(D_p^2+\frac{b_p}{R_{eq}})}$

a$)$ Find both analytical $($formula using parameters in equation above$)$ and numerical values for the static gain, natural frequency, and damping ratio of this system.

b$)$ Use the bode command in MATLAB to draw the Bode plots for this system using the given numerical values.