A satellite antenna positioning system uses a dc motor driven by a rack$-$and$-$pinion mechanism with a return spring as illustrated in Figure $1.$ The rack and pinion converts radial motion to linear motion and torque to force according to the following equations:

$x=r$

$F=\frac{T}{r}$

where, $x$ is the linear displacement, $$ is the motor angular position, $$ the angular speed, $T$ is the shaft torque and F is the driving force applied to the rack.

The dc motor and antenna parameters are.

$M=0\mathrm{.}45Kg$: total mass,

$K=200\frac{N}{m}$ : spring constant,

$Jm=210-5Kg*m2$ : total inertia of the motor and pinion,

$Bm=210-3$ N$.$s$/$m: viscous damping coefficient

$r=2cm$ : pinion radius.

$K_t=0\mathrm{.}1N\frac{m}{A}$

$K_b=0\mathrm{.}1\frac{V}{ra\frac{d}{s}}$

$R=2$

$L=50mH$

Write the equations of motion of the motor and antenna mechanism. The wind force $F_w$ is

considered as a disturbace.