# The Painlevé paradox and geometric singular perturbation theory

Lead Research Organisation:
University of Bristol

Department Name: Engineering Mathematics

### Abstract

When a piece of chalk is dragged across a blackboard, it is a matter of common, and usually unpleasant, experience that the chalk can judder and sometimes emit a high-pitched squeal. Such behaviour is related to the Painlevé paradox (Painlevé 1905). Physically, the frictional torque at the point of contact is high enough to overcome the resistance of the rigid surface, implying that the chalk should enter the blackboard. Since this cannot happen, the chalk jumps.The recent discovery that the paradox can occur in robotic manipulators, where it effects controllability, together with some excellent experimental evidence (Zhao et al. 2008), have provoked strong modern interest in this old problem.This project aims to deal with some outstanding issues relating to the Painlevé paradox. For a slender rod slipping on a rough surface, indeterminacy or inconsistency in the rigid body equations represent failures in modelling. The assumed rigidity must be relaxed. It has been shown by Hogan & Kristiansen (2016) that behaviour like that seen physically (e.g. instantaneous jumping of the chalk away from the board) arises when there is some compliance at the point of contact. This compliance (or regularization) is extremely small, and the resulting equations lead to a slow-fast system for which there is a wealth of existing mathematical theory. However, to capture the piecewise-smooth (PWS) limit of the rigid body, we need geometric singular perturbation theory, in which there have been many advances. The recently developed "blowup method" (Krupa & Szmolyan 2001) enables the identification of scales associated with the regularization, in a framework amendable to classical reduction methods in dynamical system theory. One outstanding problem that this project will aim to resolve was posed by Dupont & Yamajako (1997) of a rod between two rough surfaces. The aim is to build upon the framework in Hogan & Kristiansen (2016), where the underlying modelling assumptions of rigid body dynamics are relaxed and the PWS system is replaced by a smooth one through regularization. Then blowup will be used in the analysis of the problem.

## People |
## ORCID iD |

Stephen John Hogan (Primary Supervisor) | |

Noah Cheesman (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509619/1 | 01/10/2016 | 30/09/2021 | |||

1939397 | Studentship | EP/N509619/1 | 18/09/2017 | 30/09/2021 | Noah Cheesman |

Description | The Painlevé paradox is classically a planar (2D) problem whereby the rigid body formulation of the problem results in equations that can lack existence or uniqueness of forward solutions in time. Recent developments in the field have shown that this problem can be "resolved" through the incorporation of compliance (relaxing the rigid body assumptions). Nevertheless, very little work has been done to understand the full problem (a rod allowed to move and rotate in 3D). The work done so far in this award goes some way to extending some of the recent developments in the understanding of the 2D problem with compliance to 3D. This is done with the use of mathematical theory and techniques such as slow-fast theory, geometric singular perturbation theory & the method of geometric blowup. |

Exploitation Route | It is expected that there may be some unanswered questions in the specific problem approached by this work. Furthermore, this work promotes a framework for dealing with similar problems in mechanics where the rigid body equations may lack existence or uniqueness of forward solutions in time. In addition, the work gives a potentially useful way of dealing with mechanical problem that involve spatial Coulomb friction (a useful model for objects moving along surfaces in 3D). |

Sectors | Manufacturing, including Industrial Biotechology |