Geometry
$\lambda = r/b$, $b = r/\lambda$.
Kinematics
\[ \alpha = \dot\alpha\, t \]
\[ s(t) = r + b - r\cos\alpha - b\sqrt{\,1 - \lambda^2\sin^2\alpha\,} \]
\[ s'(t) = \dot\alpha \left( r\sin\alpha + \frac{b\lambda^2\sin 2\alpha}{2\sqrt{1-\lambda^2\sin^2\alpha}} \right) \]
\[ s''(t) = \dot\alpha^2 \left( r\cos\alpha + \frac{b\lambda^2}{2}\left[ \frac{2\cos 2\alpha}{\sqrt{1-\lambda^2\sin^2\alpha}} + \frac{\lambda^2\sin^2 2\alpha}{2\bigl(1-\lambda^2\sin^2\alpha\bigr)^{3/2}} \right] \right) \]
Reference times
\[ z = \frac{-1 + \sqrt{1+8\lambda^2}}{4\lambda}, \qquad \alpha_1 = \arccos z, \quad \alpha_2 = 2\pi - \alpha_1 \]
\[ t_\pi = \frac{\pi}{\dot\alpha}, \quad t_1 = \frac{\alpha_1}{\dot\alpha}, \quad t_2 = \frac{\alpha_2}{\dot\alpha}, \quad T = \frac{2\pi}{\dot\alpha} \]
Force
\[ F = m\, s'' + 2 m\,\frac{h}{a}\, f\, \mathrm{sign}(s')\, |s''| \]