The rocking can: a reduced equation of motion and a matched asymptotic solution

TL;DR

This paper models the rocking can problem using a reduced equation of motion and matched asymptotic solutions, revealing complex dynamics including angle of turn and contact locus trajectories, with implications for slip conditions.

Contribution

It introduces a Frobenius-based reduced equation of motion and a matched asymptotic solution for the rocking can problem, capturing complex behaviors and contact dynamics.

Findings

Excellent agreement between analytical and numerical solutions.

Identification of the angle of turn phenomenon.

Derived lower bound for coefficient of friction to prevent slip.

Abstract

The rocking can problem consists of a empty drinks can standing upright on a horizontal plane which, when tipped back to a single contact point and released, rocks down towards the flat and level state. At the bottom of the motion, the contact point moves quickly around the rim of the can. The can then rises up again, having rotated through some finite angle of turn $\Delta\psi$. We recast the problem as a second order ODE and find a Frobenius solution. We then use this Frobenius solution to derive a reduced equation of motion. The rocking can exhibits two distinct phenomena: behaviour very similar to an inverted pendulum, and dynamics with the angle of turn. This distinction allows us to use matched asymptotic expansions to derive a uniformly valid solution that is in excellent agreement with numerical calculations of the reduced equation of motion. The solution of the inner problem was used to investigate of the angle of turn phenomenon. We also examine the motion of the contact locus $\underline{x}_l$ and see a range of different trajectories, from circular to petaloid motion and even cusp-like behaviour. Finally, we obtain an approximate lower bound for the required coefficient of friction to avoid slip.