# Gyroscope Equations of Motions

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 $$\vartheta=\pi/2$$

• Oh sorry, I forgot to draw a small mass on the left side of the bar. With that you have a torque there, and the system precess. Also I want to describe the nutation.At the end, all this is just to make an animation of this motion, but I need to find the equation of motion Mar 24, 2020 at 13:40