The stationary shape of a rolling, circularly-closed elastic ribbon has been studied at relatively high speeds [1]. However, at lower speeds, the ribbon undergoes a periodic movement that breaks this stationary shape.
This instability appears to be triggered by curvature imperfections on the ribbon, which induces a parametric forcing on the rolling dynamics.
The phenomenon has been captured using numerical simulations of 2D rods with frictional contact [2], and analytical and experimental comparisons are also provided.
[1] P.S. Raux, P. M. Reis, J. W. M. Bush, and C. Clanet. “Rolling Ribbons.” Physical Review Letters 105, no. 4 (2010): 044301. https://doi.org/10.1103/PhysRevLett.105.044301.
[2] E. Hohnadel, T. Métivet, and F. Bertails-Descoubes. "Mixed Super-Circles." Computer Graphics Forum, to appear in 2026 (Proc. of Eurographics'26). Source code freely available at https://gitlab.inria.fr/elan-public-code/circonflex.
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