I want to compute the motion of a gyroscope system like in this figure to show the different motions for different initial condition and then study precession and nutation motions. The problem is that I can't find the equations of motion and compute the trajectory in the inertial sytem (located at the support point). One way could be use Euler angles but I'm not sure about how can I do it
if you neglect the centrifugal and Coriolis torques this means that your system is rotate slowly , you get the equations of motion.
$$\ddot{\varphi}= -{\frac {\sin \left( \psi \right) T_{{\vartheta }}}{J_{{\vartheta }} \cos \left( \vartheta \right) }}+{\frac {\cos \left( \psi \right) { \it uu}_{{1}}}{\cos \left( \vartheta \right) J_{{\varphi }}}} $$
$$\ddot{\vartheta}={\frac {\cos \left( \psi \right) T_{{\vartheta }}}{J_{{\vartheta }}}}+ {\frac {\sin \left( \psi \right) {\it uu}_{{1}}}{J_{{\varphi }}}} $$
$$\ddot{\psi}={\frac {T_{{\psi}}}{J_{{\psi}}}}+{\frac {\sin \left( \vartheta \right) \sin \left( \psi \right) T_{{\vartheta }}}{J_{{\vartheta }} \cos \left( \vartheta \right) }}-{\frac {\cos \left( \psi \right) \sin \left( \vartheta \right) J_{{\vartheta }}{\it uu}_{{1}}J_{{\psi} }-J_{{\vartheta }}J_{{\varphi }}\cos \left( \vartheta \right) {\it uu }_{{3}}}{J_{{\vartheta }}J_{{\varphi }}J_{{\psi}}\cos \left( \vartheta \right) }} $$
with
$$uu_1=-\cos \left( \vartheta \right) g\cos \left( \varphi \right) \left( aM-bm \right) $$ $$uu_3=-g \left( \cos \left( \psi \right) \sin \left( \varphi \right) \sin \left( \vartheta \right) +\sin \left( \psi \right) \cos \left( \vartheta \right) \right) \left( aM-bm \right) $$
$J_\varphi\,,J_\vartheta\,,J_\psi$ are the inertia about the axes and $T_\vartheta\,,T_\psi$ are the torques given in body fixed frame.
to do the animation you need the rotation matrix between body fixed frame and inertial frame
$$R= \left[ \begin {array}{ccc} 1&0&0\\ 0&\cos \left( \varphi \right) &-\sin \left( \varphi \right) \\ 0 &\sin \left( \varphi \right) &\cos \left( \varphi \right) \end {array} \right] \, \left[ \begin {array}{ccc} \cos \left( \vartheta \right) &0&\sin \left( \vartheta \right) \\ 0&1&0 \\ -\sin \left( \vartheta \right) &0&\cos \left( \vartheta \right) \end {array} \right] \,\left[ \begin {array}{ccc} \cos \left( \psi \right) &-\sin \left( \psi \right) &0\\ \sin \left( \psi \right) &\cos \left( \psi \right) &0\\ 0&0&1\end {array} \right] $$
notice that you get singularity if $\vartheta=\pi/2$
