A load inertia $I_2$ is driven through gears by a motor with inertia $I_1($Figure P$4\mathrm{.}14).$ The shaft inertias are $l_3$ and $4$ ; the gear inertias are $I_5$ and $5.$ The gear ratio is $5$:$1($the motor shaft has

the greater speed$).$ The motor torque is $T_1$ and the viscous damping coefficient is $c=1\mathrm{.}2lb$ ft $-$sec$/$rad$.$ Neglect elasticity in the system, and use the following inertia values $($in $se{c}^2-ft-l\frac{b}{r}ad)$:

$I_1=0\mathrm{.}01,I_2=0\mathrm{.}5$

$I_3=0\mathrm{.}001,I_4=0\mathrm{.}005$

$I_5=0\mathrm{.}02,I_6=0\mathrm{.}03$

Figure P$4\mathrm{.}14$

$($a$)$ Derive the model for the motor shaft speed ${}_1$ with $T_1$ as input.

$($b$)$ Derive the model for the load shaft speed ${}_2$ with $T_1$ as input.