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An asymmetrical body: example of analytical solution for the rotation matrix in elementary functions and Dzhanibekov effect
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We solved the Poisson equations, obtaining their exact solution in elementary functions for the rotation matrix of a free asymmetrical body with angular velocity vector lying on separatrices. This allows us to discuss the temporal evolution of Dzhanibekov's nut directly in the Laboratory system, where it is observed. The rotation matrix depends on two parameters with clear physical interpretation as a frequency and a damping factor of the solution. Qualitative analysis of the solution shows that it properly describes a single-jump Dzhanibekov effect.
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An alternative interpretation of the Grioli gyroscope suspension points
An asymmetric heavy gyroscope can regularly precess only when its suspension point lies on one of two lines marking where the intermediate and largest moments of inertia exchange order, with exactly two motions.
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