"Darwin123" escreveu na mensagem
...

However, your spin inducing
apparatus may be producing a strange torque for reasons that have
nothing to do with Eulers equations.
T1(w1,w2,w3), T2(w1,w2,w3), T3(w1,w2,w3) ?

The main issue here is "Conservation of Energy".
We are to assume that energy is conserved, that's for sure.

How do we express the conservation of energy?

The truth is that one cannot draw energy conclusions, since
we are working with velocities (w1, w2, w3) and we have
no means to know the angular displacements (theta, phi, psi).
One cannot integrate and, even if we do, the facts are that
the gyro produces tens of turns and in no way an integral
can ever express that number of turns.

My point is that Power is Conserved, instead.
Power conservation follows from Energy conservation
since time is not relativistic.
My point is that: T1 w1 = Tz wz = T2 w2

That is, T3 and w3 is out of the Energy Conservation
problem.

I'm to assume that T3, w3 are an isolated system,
which must hold true since axis 1 and 2 are independent.
If axis 1 and 2 are independent, why should axis 3 be also
independent?

Therefore, we are to assume all axes (1, 2, 3) to be
independent, one from each other.
That is, in the process of precession, nutation and
variable external torques (that we can calculate), the
gyro main angular speed w3 remains constant (except
for bearings friction). Hence, T3 is only due to the main
bearings friction, we can neglect or else compensate
to keep angular velocity w3 constant.

Do you agree with the above picture?

Do you believe it's possible that a torque T3 can arise in
the process, in order to spin up/down the gyro main
angular velocity w3?