I have heard that an O'neill cylinder rotating about the central axis of the cylinder can be unstable. O'neill, himself, initially solved this problem by having two adjoining cylinders rotating in opposite directions.
Let's take a cylinder with R1 (the radius of the outer shell from the axis of rotation) to be 1050m, R2 (the radius of the inner shell to be 1000m and the length of the cylinder to be 3000m. Approximate the shell to be constant density (or evenly distributed mass). It will be in a constant tangential velocity orbit of the Earth.
I tried to learn the math of moment of inertia, but it baffled me. It seems to me that the axis with the higher moment of inertia would be the cylinder axis and so that the rotating cylinder would be stable. Could somebody do the math for me? Does the stability vary with the length to diameter ratio of the cylinder?