# Solutions to Non-Ideal Gyroscopic Equations

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.

• "Solutions to the typical Euler-Lagrange Equation are relatively trivial" I don't think you have analytical solution for the gyroscope differential equations, even without friction. – Eli Jul 9 at 14:38
• Obviously considering that they form a system of 3 second order non-linear partial differential equations, I doubt that analytical solutions exist, and they certaintly havent been found. I use the word "trivial" in this context mostly because the Lagrangian yields two conserved momenta when friction is not considered, and substitution allows $\ddot{\theta}$ to be expressed as a function of $\theta$, which allows numerical solutions to come quite easily. In the case of friction, my main issue is that i'm entirely unsure if the dissipation function is correct or being used correctly. – BooleanDesigns Jul 9 at 18:00