What is the maximum length of a cylindrical space station for stability versus small asymmetries in the distribution of mass?

The early US satellite Explorer 1 was a long cylinder that was intended to rotate around its geometric axis of symmetry, i.e. the long axis running down the center of its length. Due to a small asymmetry in the distribution of mass, the satellite began to precess, and then started foolishly tumbling end-over-end.

There is a maximum for the ratio of length to diameter for the rotational stability of cylinders rotating around their geometric axis of symmetry. What is that maximum ratio?

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  • Long cylinders, initially rotating around their physical axis of symmetry (a line running down the length through the middle) are rotationally unstable versus small deviations in the internal distribution of mass. They start to precess, and then they tumble end-over-end, just as the Explorer 1 satellite did.

    Short, squat cylinders are, on the other hand, rotationally stable around their physical axis of symmetry.

    Where's the break-even point in frustum length between cylinders that rotate stably around their physical axis of symmetry, and those that are unstable? Rotating objects tend to seek the rotational axis having the greatest moment of inertia, which is usually the shortest axis.

    The moments of inertia for a hollow cylinder of inner radius Rᵢ , outer radius Rₒ , and length h are

    Rotation around the physical axis of symmetry (desired)

    I∥ = (M/2)(Rᵢ²+Rₒ²)

    Rotation perpendicular to the physical axis of symmetry (to be avoided)

    I⟂ = (M/12) [ 3(Rₒ²+Rᵢ²) + h² ]

    Set them equal to each other.

    (M/2)(Rᵢ²+Rₒ²) = (M/12) [ 3(Rₒ²+Rᵢ²) + h² ]

    6Rᵢ² + 6Rₒ² = 3Rₒ² + 3Rᵢ² + h²

    3Rᵢ² + 3Rₒ² = h²

    Assuming that the thickness of the frustum wall (Rₒ−Rᵢ) is negligible compared with the diameter of the cylinder,

    6R² ≈ h²

    h = √6 R = 2.449489743 R

    When the length, h, of the cylinder is less than √6 R, the cylinder is rotationally stable versus small asymmetries in mass distribution around the frustum. But when h is greater than √6 R, the cylinder is rotationally unstable.

    Wasn't that simple?

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