4-disk Squared Gyroscope problem

[RicardoMP]

Homework Statement

A rigid, square-shaped, structure with negligible mass contains 4 disks in rotation as you can see in the figure. Each disk has mass m, moment of inertia I about its rotation axis and angular velocity ws. Also, the plane of the structure coincides with the horizontal plane and it can move freely about O. The system is precessing uniformly with angular velocity \Omega about a vertical axis that goes through O. Considering the inertial frame with origin at point O, z axisperpendicular to the strutucture plane and the x axis, at a certaint moment, is along the square diagonal in the OC direction. The objective is to write the angular momentum in that instant.

The Attempt at a Solution

The first thing that came to my mind was to consider the inertial frame and a rotating frame sharing the z axis with the first frame. I was inclined to apply the rotating coordinates transformation, but quickly arrived to the conclusion that that wasn't the correct way.
Then I tried to check for all the contributions to the system's angular momentum which I suppose are the 4 spin angular momentums of the 4 disks, plus the angular momentum along the z axis. But how do I relate them with the total angular momentum? And is there any contribution I'm missing? The structure's weight produces torque, which means there is another contribution to the angular momentum, right?
I'm really sorry for the long post, but I would be really thankful if someone could give me a couple of clues for the right path to take in solving this problem.
Thank you in advance.

 

[haruspex]

how do I relate them with the total angular momentum?

Umm... add them vectorially?

The structure's weight produces torque, which means there is another contribution to the angular momentum, right?

But it doesn't rotate in that direction (out of the plane)... why not, do you think?