In my work with gyroscopes, I have correctly derived the Lagrangian of a 3d top or gyroscope as
$L=\frac{1}{2}I_{xy}(\dot{\phi}sin^2(\theta)+\dot{\theta}^2) + \frac{1}{2}I_{zz}(\dot{\phi}cos^2(\theta)+2\dot{\phi}\dot{\psi}cos(\theta)+\dot{\psi}^2)-mgZ_gcos(\theta) $.
Solutions to the typical Euler-Lagrange Equation are relatively trivial, however when considering friction about the z-axis, and so because the spin is $\dot{\psi}$, the friction is some function $F_\mu (t) = k\dot{\psi}$.
To my understanding, this allows me to include a dissipation function $D(\dot{\psi}) = \frac{1}{2}k_\mu \dot{\psi}^2$ subject to $\frac{d}{dt}\frac{\partial L}{\partial \dot{q_i}} +\frac{\partial D}{\partial \dot{q_i}} = \frac{\partial L}{\partial q_i}$. I would love to see a proper solution to this system, and for someone to explain if the dissipation function is erroneous. Thanks.
