Q$.4$ A DC servomotor operates under different gear ratios and load inertias. A model of the motor and gearbox system is given by:

$\frac{(s)}{V_i(s)}=G(s)=\frac{1}{s(\frac{R}{m}{K}_m{K}_g{?}_{K_m}s+K_m{K}_g)}$

where $(s)$ is the angle in radians of the output shaft, $V_i(s)$ is the voltage applied to the armature, $R_m=2\mathrm{.}6$ is the armature resistance, $K_m=0\mathrm{.}00767N\frac{m}{A}$ is the motor torque constant, $K_g$ is the gear ratio, $J_{eq}$ is the equivalent inertia seen at the output of the gearbox, which is given by:

$J_{eq}=K_g^2{J}_m+J_l$

where $J_m=3\mathrm{.}87{10}^{-7}Kg{m}^2$ is the armature inertia and $J_l$ is the load inertia. The nominal value of the gear ratio is $K_g=14$ and the nominal value of the load inertia is $J_l=0\mathrm{.}00003Kg{m}^2.$ The extreme value of the gear ratio is $K_g=70$ and the extreme value of the load inertia is $J_l=0\mathrm{.}001Kg{m}^2.$

a$)$ Calculate the nominal transfer function $G(s).$

b$)$ Write the upper and lower bounds equations of the system magnitude and phase then calculate their values at $w=2ra\frac{d}{s}.$

c$)$ Find the multiplicative uncertainty that describe the set of possible plants.

$\frac{2}{3}$