An Analytical and Geometrical Study of the Dzhanibekov and Tennis Racket Phenomena

We consider the torque-free rotation of a rigid body with three distinct moment of inertia values and angular velocity components. As can be seen in the Dzhanibekov and tennis racket phenomena, rotations about the largest and smallest principal moments of inertia create stable rotations. However, when rotating about the principal intermediate moment of inertia, an unstable rotation is produced that leads to the basis of this phenomena. In this publication, the above phenomena are examined and explained analytically and applied to a satellite system to observe the change in trajectory as the solar panels and reflectors are deployed. To begin the derivation, the Euler torque-free equations for a rotating rigid body are formulated using the moving frame method. The derived equations are then non-dimensionalized and a complete analytical solution, including an expression for the non-dimensional period, is presented. Second, we further look at the limiting axisymmetric cases and examine the effect as the intermediate moment of inertia is varied. Lastly, the analytical expressions are compared with the numerical simulation to validate the results. The complete solution is then summarized and shown to clearly prove that the conservation of angular momentum is indeed preserved in the phenomena.