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

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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
Issue number2202
Number of pages18
Publication statusPublished - 2017


  • Painlevééé paradox
  • Impact without collision
  • Compliance
  • Regularization

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