(Solved): The gearbox shown has three gears on three shafts. The only significant inertia is J4 as shown. ...

The gearbox shown has three gears on three shafts. The only significant inertia is $$J_4$$ as shown. The gears have $$N_2$$ and $$N_3$$ teeth, as indicated. The torsional spring constants for the relevant sections of the shafts are given, as are the tooth numbers for the three gears. (Note that the torsional stiffnesses of the shafts at the top and bottom and the gears on them are identical.) Recall that the angular positions of the gears will be related by $${?}_2=\left(N_3/N_2\right){?}_3$$. What is the natural frequency of the system? (Hint: In terms of finding equivalent springs, it may be helpful to understand that the rotational system for this problem is analogous to the linear mass with massless compound pulley shown below it.)