The Painlevé paradox and geometric singular perturbation theory

Painlevé 悖论和几何奇异微扰理论

基本信息

批准号：
1939397
负责人：
金额：
--
依托单位：
University of Bristol
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=studentship-1939397
关键词：
Painlev paradox geometric singular perturbation

项目摘要

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.

当一块粉笔被拖到黑板上时，这是一个常见的问题，而且通常是不愉快的，粉笔可以射击并有时会发出高音尖叫。这种行为与Painlevé悖论有关（Painlevé1905）。从物理上讲，接触点处的摩擦扭矩足以克服刚性表面的电阻，这意味着粉笔应该进入黑板。由于这不可能发生，粉笔会跳跃。最近发现悖论可能发生在机器人操纵器中，在机器人操纵器中，它影响可控性，以及一些出色的实验证据（Zhao等，2008），已证明对这个旧问题具有强烈的现代兴趣。该项目旨在处理与ParaclevéParadox有关的一些出色问题。对于在粗糙的表面上滑动的细长杆，刚体方程中的不确定性或不一致表示建模的故障。假定的刚度必须放松。 Hogan＆Kristiansen（2016）已经表明，当存在某种依从性时，就会出现像物理上这种行为（例如，从董事会瞬间跳跃的瞬时跳跃）。这种合规性（或正则化）非常小，结果方程式导致了一个缓慢的系统，其中有大量现有的数学理论。但是，要捕获刚体的分段平滑（PWS）极限，我们需要几何奇异的扰动理论，在该理论中有很多进步。最近开发的“爆炸方法”（Krupa＆Szmolyan 2001）可以识别与调节相关的量表，这是在动态系统理论中对经典还原方法进行的框架。杜邦（Dupont＆Yamajako）（1997）认为，这个项目将旨在解决该项目的一个重大问题是两个粗糙表面之间的杆。目的是建立在Hogan＆Kristiansen（2016）中的框架之上，在该框架上，刚性身体动力学的基本建模假设放松，PWS系统被平滑的模型通过调节代替。然后，爆炸将用于对问题的分析。