Imperfections, impacts, and the singularity of Euler's disk.

Area of Science:

• Physics of complex systems
• Friction and wear dynamics
• Rotational dynamics

Background:

• The final phase of a spinning rigid disk's motion on a flat surface involves a finite-time singularity due to energy dissipation.
• The primary energy absorption mechanism during this final phase has been debated for two decades, with air drag and friction as leading candidates.

Purpose of the Study:

• To investigate the role of impacts, caused by geometric imperfections in the disk or surface, as an energy absorption mechanism.
• To analyze the dynamics of polygonal disks with unilateral point contacts to understand impact effects.

Main Methods:

• Analysis of the dynamics of polygonal disks with unilateral point contacts.
• Determination of energy absorption rates assuming a regular motion pattern (precession-free rolling).
• Numerical investigation of irregular motion emerging from instability in the impact model.

Main Results:

• Impacts can become the dominant energy absorption mechanism for specific parameter ranges (small radii of gyration or small restitution coefficients).
• The asymptotic stability of the regular motion pattern depends on the parameters of the impact model.
• Instability can lead to irregular motion, which was investigated numerically.

Conclusions:

• Impacts caused by geometric imperfections can dominate energy dissipation in the final phase of a spinning disk's motion under certain conditions.
• However, for typical parameters of a homogeneous disk on a hard surface, impact effects are not dominant compared to other mechanisms.