Mathematical model of rigid body motion for the research of the “Dzhanibekov's effect”

An alternative set of angles for determining the position of a rigid body, instead of Euler angles, is proposed. These angles were used to model the motion of a rigid body (rotor) rotating around a horizontal axis. For the proposed angles, equations similar to Euler's kinematic equations were derived. Most machines with rotating shafts are designed horizontally, and the proposed angles are convenient for studying these machines using Euler's dynamic equations. Equations were derived that allow one to obtain Euler angles from known proposed angles and to obtain proposed angles from known Euler angles. For a freely rotating rigid body, one can observe the Dzhanibekov's effect, which can be classified as the periodic motion of an asymmetrical top and a type of tennis racket effect. The horizontal movement of the rotor, which is accompanied by the Dzhanibekov effect, well illustrates the advantages of the proposed angles. Based on Euler's equations and using the proposed angles, two types of systems of differential equations were obtained. These systems do not degenerate, are tested on a well-known classical example and are applicable to any motion of a rigid body. Systems of differential equations were solved numerically using the Matlab software package using the ode45 subroutine (the simplest solver). The resulting solutions are stable and repeatable. The numerical results obtained are compared with the results obtained using the well-known modified predictor-corrector method. The preference of using the proposed coordinate system for numerical solutions of a system of differential equations compared to Eulerian angles is shown. For the free movement of an asymmetrical top, an analytical solution is known, obtained in the form of elliptic functions. This solution is extended to the Dzhanibekov's effect and the results obtained are compared with the results of the numerical simulation of the Dzhanibekov's effect. The formulas were obtained to determine the time before the first 180-degree turn and the time between the next 180-degree turns. The formula for determining the time to the first turn was adjusted for the case of non-zero initial conditions for the three angular velocities, taking into account their sign. The known formula for the period of an asymmetrical top movement process during Dzhanibekov's effect was confirmed.