Problem $1$: The problem shown below was also discussed in class. Please refer to class notes.

For the rotational system with gears shown, determine the transfer function

$G(s)={}_2\frac{s}{T}(s)$

The angular dsiplacement ${}_l(t)$ is generated by the applied Torque $T(t)$ of the motor with moment inertia $J_1.$

Indeed the gear $1$ will have a load torque of $T_1(t).$ There will indeed be a Torque $T_2(t)$ on gear $2$ that is driving

the shaft $2.$ The angular displacement of shaft $2$ is ${}_2(t).$ It is clear from the description written below that $N_1$

is the driving gear and $N_2$ is the driven gear.

Gear Ratio is $N$

$N=\frac{{}_1}{{}_2}=\frac{{}_1}{{}_2}=\frac{D_2}{D_1}$

$\frac{T_1{}_1}{{}_1}=T_2{}_2=\frac{{}_2}{{}_2}=\frac{T_2}{T_1}=\frac{N_1}{N_1}$

Gear $is$ Ratio

nsidered rigid for $k_{t_1}=k_{t_2}=0$

$$aged

The moment of inertia of the two gears

are considered negligible. The motor

moment of inertia is $J_{\text{I}}$

and that of the impeller is $J_2.$ The

torsional damping coefficient of

shaft $1$ is $b_{t1}$ and for shaft $2$ is $b_{t2}.$