1. Consider the following rotational system.
T(t)
Gear 1
$N_1 = 4$
$b_1 = 1 N-m-s/rad$
$J_1 = 2 kg-m^2$
$N_2 = 12$
Gear 2
$b_2 = 2 N-m-s/rad$
Gear 3
$N_3 = 4$
$J_2 = 1 kg-m^2$
$b_3 = 32 N-m-s/rad$
$N_4 = 16$
Gear 4
$\theta_3(t)$
$K = 64 N-m/rad$
$J_3 = 16 kg-m^2$
a) (25 points) Find the transfer function $\theta_3(s) / T(s)$ in symbolic form without using the shortcut method (i.e., drawing the free-body diagrams, using D'Alembert's principle, etc.).
b) (10 points) Substitute the numerical values in the transfer function found in (a).
c) (15 points) Verify your answer in (b) by using the shortcut method for reflecting impedances and torque.
Solution Hint: Either write equations for each gear system ($\theta_1, \theta_2$, and $\theta_3$), and solve three equations along with using gear teeth ratio to transfer T or $\theta$
or find equivalent Inertia and impedances at $\theta_3$ system and solve for $\theta_3$.
$J\ddot{\theta} + b\dot{\theta} + K\theta = T(t)eq$
