On the regularization of impact without collision: the Painlevé paradox and compliance

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Abstract

We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevéé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.
Original languageEnglish
Article number20160773
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume473
Issue number2202
Number of pages18
ISSN1364-5021
DOIs
Publication statusPublished - 2017

Keywords

  • Painlevééé paradox
  • Impact without collision
  • Compliance
  • Regularization

Cite this

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title = "On the regularization of impact without collision: the Painlevé paradox and compliance",
abstract = "We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevéé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.",
keywords = "Painlevééé paradox, Impact without collision, Compliance, Regularization",
author = "Hogan, {S. J.} and Kristiansen, {K. Uldall}",
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language = "English",
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journal = "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",
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On the regularization of impact without collision: the Painlevé paradox and compliance. / Hogan, S. J.; Kristiansen, K. Uldall.

In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 473, No. 2202, 20160773, 2017.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - On the regularization of impact without collision: the Painlevé paradox and compliance

AU - Hogan, S. J.

AU - Kristiansen, K. Uldall

PY - 2017

Y1 - 2017

N2 - We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevéé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.

AB - We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevéé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.

KW - Painlevééé paradox

KW - Impact without collision

KW - Compliance

KW - Regularization

U2 - 10.1098/rspa.2016.0773

DO - 10.1098/rspa.2016.0773

M3 - Journal article

VL - 473

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 1364-5021

IS - 2202

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