A Theoretical and Numerical Study of the Dzhanibekov and Tennis Racket Phenomena1

Author:

Murakami Hidenori1,Rios Oscar1,Impelluso Thomas Joseph2

Affiliation:

1. Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, San Diego, CA 92093-0411 e-mail:

2. Department of Mechanical and Marine Engineering, Bergen University College, Bergen 5020, Norway e-mail:

Abstract

This paper presents a complete explanation of the Dzhanibekov and the tennis racket phenomena. These phenomena are described by Euler's equation for an unconstrained rigid body that has three distinct moment of inertia values. In the two phenomena, the rotations of a body about the principal axes that correspond to the largest and the smallest moments of inertia are stable. However, the rotation about the axis corresponding to the intermediate principal moment of inertia becomes unstable, leading to the unexpected rotations that are the basis of the phenomena. If this unexpected rotation is not explained from a complete perspective which accounts for the relevant physical and mathematical aspects, one might misconstrue the phenomena as a violation of the conservation of angular momenta. To address this, the phenomenon is investigated using more precise mathematical and graphical tools than those employed previously. The torque-free Euler equations are integrated using the fourth-order Runge–Kutta method. Then, a recovery equation is applied to obtain the rotation matrix for the body. By combining the geometrical solutions with numerical simulations, the unexpected rotations observed in the Dzhanibekov and the tennis racket experiments are shown to preserve the conservation of angular momentum.

Publisher

ASME International

Subject

Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics

Reference23 articles.

1. Du mouvement de rotation des corps solides autour d'un axe variable,1758

2. Du mouvement de rotation des corps solides autour d'un axe variable,1964

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