model

Physical model from simplified circuit Kirchoff + Newton

Derivation

Terms

$$ \begin{align} J_m = \textnormal{motor inertia vs torque coefficient} \newline R_a = \textnormal{armature resistance} \newline K_b = \textnormal{back-emf vs angular velocity coefficient} \newline K_t = \textnormal{torque vs current coefficient} \newline D_m = \textnormal{viscous damping term} \newline L_a = \textnormal{armature inductance} \end{align} $$

Differential equation

Solution according to Sympy

Notes: Bounds are incomplete!

First derivative of solution according to Sympy

Notes: First derivative appears identical to the ode solution w(t), think this is clear wrong, would expect perhaps to see a term for de_a(t)/dt. Perhaps that does not appear as eliminated via doing IBP Su'v = uv - Sv'u