Abstract
We study the dynamics of a slender rigid rod slipping with unilateral constraint on a rough surface, able to move and rotate in 3D, an extension of the classical (planar) Painlevé problem. We demonstrate that the potential nonexistence and nonuniqueness of forward-time solutions, due to the conflict between the rigid-body assumptions and the the resulting dynamics, seen in the 2D problem, persist in this 3D problem. We are the first to identify crucial aspects of the geometry of the problem and the importance of the azimuthal angular velocity to the dynamics. We show that the planar problem is a singular subset of the full 3D problem, and that crucial results that apply to the 2D problem lose significance in the 3D extension. Like Champneys and Várkonyi, we also study a particular type of orbit, not present in the planar problem, that reaches inconsistency from slipping, and we give a geometric justification for their results. We prove that, unlike in the planar problem, these orbits are typical.We then proceed to resolve the “paradox”, generalising results obtained by Hogan and Kristiansen through regularising the 2D problem with compliance to the 3D problem. This compliance introduces a small parameter and we use geometric singular perturbation theory (GSPT) to study the resulting singularly perturbed problem. With the incorporation of this compliance, we recover impact without collision (IWC). We follow certain significant orbits from the rigid-body problem and find that they also undergo IWC. To facilitate studying this system, instead of viewing spatial Coulomb friction as a piecewise-smooth (PWS) system, we take it to be the limit of a smooth system, using blowup in the analysis.
The de-facto approach to PWS dynamical systems, with codimension–1 discontinuity sets, relies on the Filippov framework. But this framework does not generalise to systems with higher codimension discontinuities, such as those with spatial Coulomb friction. We study a general system with an isolated codimension–2 discontinuity set, regularising the nonsmooth system and viewing it as the limit of a smooth one, using blowup to study the dynamics. We present a framework for studying these problems, giving the local dynamics, and generalising Filippov sliding, crossing, and sliding vector fields. Whilst motivated by Coulomb friction, the approach is sufficiently general as to apply to any nonsmooth dynamical system with finite-time approach to a codimension-2 discontinuity. We also present a particular class of codimension-2 discontinuity problem, for which we give a more complete classification. This work, with a mathematically rigorous foundation, expands upon the formal work of Antali and Stépán.
Generally, we present a framework for the study of mechanical systems with Painlevé paradoxes, nonsmoothness, or other causes of nonexistence or nonuniqueness of solutions. This approach is to regularise the problem (through smoothing or an incorporation of compliance or other physics) and to use the framework of smooth dynamical systems to study the resulting problem (particularly GSPT and geometric blowup).
| Date of Award | 21 Jun 2022 |
|---|---|
| Original language | English |
| Awarding Institution |
|
| Supervisor | John Hogan (Supervisor) & K Uldall Kristiansen (Supervisor) |