 Initial acceleration of three stacked cylinders

Three identical cylinders initially form an equilateral triangle on a horizontal floor and are released from rest. If each has moment of inertia $\beta m{R}^2$, the initial top downward acceleration is $g/(7+6\beta )$ for no-slip rough ground and smooth cylinder contacts, or $g/(7+8\beta )$ for smooth ground and no-slip rough cylinder contacts. In both cases the lower cylinders separate with outward acceleration $u=\sqrt{3}{a}_{\mathrm{top}}$. The corresponding lower-cylinder angular acceleration magnitudes are $u/R$ and $2a_{\mathrm{top}}/R$, with opposite rotation senses between the cases. Sufficient static friction is required; a finite coefficient allowing sliding changes the problem.