I wished to understand a particular case of Euler's equation applied in the following cylindrical body:
where $I_{1,2,3}$ are the moments of inertia. By symmetry, $I_1=I_2=I_T$.
Here I consider that no external forces are applied to the cylinder. Stating the equations by their components, we have that:
$$I_1\dot{\omega}_1−(I_2−I_3)\omega_2\omega_3=0$$ $$I_2\dot{\omega}_2−(I_3−I_1)\omega_3\omega_1=0$$ $$I_3\dot{\omega}_3−(I_1−I_2)\omega_1\omega_2=0$$
where $\omega_i$ is the rotation speed on axis $\hat{b}_i$.
My question is the following: Considering that initially the cylinder spins with speed $\omega_3$, and it also moves around axis $\hat{b}_1$ and/or $\hat{b}_2$, that is, $\omega_1$ and/or $\omega_2$ are different than zero. According to the third equation, $\omega_3$ should remain constant, because $I_1−I_2=0$. But I imagine that to conserve its momentum, as the cylinder is rotated to a "laid" position around axis $\hat{b}_1$ or $\hat{b}_2$, it would keep spinning around $\hat{b}_3$, but with a different speed because $I_3\neq I_T$.
Am I interpreting this equation wrong?
