On the Dynamics of a Nutation Ring Damper

The paper derives, in a body-fixed frame chosen so e1 points to the slug, the torque-free angular-momentum transport law ḣ = −Ωs × h with Ωs = Ω + β̇ e3 and shows |h| is conserved (the angular-momentum sphere) . It then rewrites this as ḣ = h × [(Ir + Is)^{-1}(h + Ir β̇ e3)] by inverting h = Ir Ω + Is Ωs . With a linear tangential friction model F̄r,s = −Cd(R − d) β̇ ēβ, the paper obtains β̈ = −Cd(R − d)^2 β̇/Izs for the slug and Ω̇z = +Cd(R − d)^2 β̇/Izr for the ring via action–reaction torques , and then the final ODE for β̇, β̈ = −β̇ Cd (R − d)^2 (Izs + Izr)/(Izs Izr) + hx hy/((Iyr + Iys) Izs) − hx hy/((Ixr + Ixs) Izs) . The candidate solution reproduces exactly these steps, formulas, and equilibria (six points with β̇ = 0 and h on principal axes), and uses the same modeling assumptions (diagonal inertias with Ixs = 0, Iys = Izs, and the same body frame) . Hence both are correct and essentially the same argument.