Hand Deriving the Maglev Dynamics

I derived the initial force-balance equation using Newton’s second law and then linearized it around the equilibrium point using a multivariable Taylor expansion. Applying the Laplace transform exposed the right-half-plane pole, mathematically proving why the system is naturally unstable without active correction. Setting up the closed-loop stability condition using the Routh-Hurwitz criterion felt familiar from my EE and calc coursework, but keeping track of the coefficients during the partial derivatives tripped me up for a second. I ultimately got it sorted and now have the exact mathematical thresholds required for my proportional and derivative gains.

Session 1 Timelapse: https://lapse.hackclub.com/timelapse/9LWAljPJ38Sr
The math images are on the GitHub repo!