Phase-Space and Dissipation Studies of the Dancing Wing Nut

Abstract

In this paper, we address the nonlinear dynamics of Dzhanibekov’s dancing wing nut and the related intermediate axes theorem governing the motion of a tossed tennis racket. Drawing first on the comprehensive investigations by Murakami, Rios and Impelluso [J. Appl. Mech. 83 (2016), https://doi./org/10.1115/1.4034318] we extend the exploration of the wing nut’s spherical phase portrait by incorporating basins of attraction and their heteroclinic connections. We then turn our attention to the significant damping studies of Español, Thachuk and de la Torre [Eur. J. Mech. A Solids 103, 105184 (2024)] who derive from first principles a generalized stochastic version of the Euler equations which can incorporate both oriental realignment (which they call oriental diffusion) and viscoelastic effects. Applied to the wing nut dancing, they showed that the former is the more significant effect and as the authors say “Had Dzhanibekov observed his wing nut for a sufficiently long period, he would have found it spinning around the major axis. In other words, precession relaxation kills the Dzhanibekov effect in the long run.”