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

Research output: Contribution to journalJournal articleResearchpeer-review

### 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 language English 20160773 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473 2202 18 1364-5021 https://doi.org/10.1098/rspa.2016.0773 Published - 2017

### Keywords

• Impact without collision
• Compliance
• Regularization

### Cite this

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}",
year = "2017",
doi = "10.1098/rspa.2016.0773",
language = "English",
volume = "473",
journal = "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",
issn = "1364-5021",
publisher = "The/Royal Society",
number = "2202",

}

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 - 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

M1 - 20160773

ER -