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

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