motion equations of a disc rotating freely around its center (3d)

bluekuma

1. The problem statement, all variables and given/known data

The system is made of a disc the center of which is pinned to the origin (so the disc cannot translate), and some weights that can be stuck on the disc to make it tilt (weights do not translate on the disc) (see images attached).
There is no friction whatsoever. The only force is gravitational force, with direction opposite to the z-axis'.

Let's start with the disc at rest with its axis parallel to axis z. Now, if you put a weight on it, the disc starts oscillating just as if it was a pendulum. Then, at time t=t₀, you put another weight on it.

If ω⃗ is the rotational speed vector and θ⃗ is the rotation vector of the disc (meaning the direction of the disc's axis is always the z-versor rotated by θ radians around θ⃗ ) what's the expression of f⃗ (t,ω,θ) in:

dω⃗ /dt = f(t,ω,θ),
dθ⃗ /dt = ω⃗

Given the initial values ω⃗ (t₀) =ω⃗₀≠0 and θ⃗ (t₀=θ⃗₀≠0, dω⃗ (t₀)/dt≠0,
that would give me a way to simulate the system's motion through a standard Runge-Kutta integration method.

MIGHT HELP TO KNOW:
- I'm pretty sure there is a way to divide the two vectorial equations in six (three systems of two) linear equations
- z-component of momentum M⃗ (where dω⃗ /dt = M⃗ /I ) is always 0 (zero) as M is the result of a cross product between a vector r (x, y, x) and the gravitational force (0, 0, -mg), therefore z-component of ω⃗ and θ⃗ are also 0.


2. Relevant equations

d[itex]\vec{ω}[/itex]/dt = [itex]\vec{M}[/itex]/I
where [itex]\vec{M}[/itex] is the momentum and I the moment of inertia.


3. The attempt at a solution

ehr...i'm actively looking for a system of coordinates in which the vectorial equations can be separated in three linear equations, that would solve my problem. Obviously I had no success so far.

bluekuma

I tried with Lagrangian and Euler-Lagrange equation but I really don't know where to start writing kinetic and potential energies in cartesian or spherical coordinates.

I tried with a coordinate system that moves together with the disc (longitudinal and vertical axis, as it is usually done when studying motion of airplanes). There I'd have that the weights are still but force of gravity keeps changing its direction, that only adds to the confusion.

Any idea?