Phase transitions in rolling of irregular cylinders and spheres

TL;DR

This paper investigates phase transitions in the rolling behavior of irregular cylinders and spheres on inclined planes, revealing complex dynamics, hysteresis, and scaling laws that depend on shape irregularity and inertia.

Contribution

It introduces a theoretical framework for phase transitions in rolling objects with irregular shapes, supported by experiments on cylinders and spheres that confirm the predictions.

Findings

Identification of phase transitions as a function of shape and inertia

Experimental validation of lag time scaling in cylinders

Observation of orbit behaviors and period-doubling in spheres

Abstract

When placed on an inclined plane, a perfect 2D disk or 3D sphere simply rolls down in a straight line under gravity. But how is the rolling affected if these shapes are irregular or random? Treating the terminal rolling speed as an order parameter, we show that phase transitions arise as a function of the dimension of the state space and inertia. We calculate the scaling exponents and the macroscopic lag time associated with the presence of first and second order transitions, and describe the regimes of co-existence of stable states and the accompanying hysteresis. Experiments with rolling cylinders corroborate our theoretical results on the scaling of the lag time. Experiments with spheres reveal closed orbits and their period-doubling in the overdamped and inertial limits respectively, providing visible manifestations of the hairy ball theorem and the doubly-connected nature of SO(3), the space of 3-dimensional rotations. Going beyond simple curiosity, our study might be relevant in a number of natural and artificial systems that involve the rolling of irregular objects, in systems ranging from nanoscale cellular transport to robotics.