Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Dynamics Beyond Dynamic Jam; Unfolding the Painleve Paradox Singularity
KTH, School of Engineering Sciences (SCI), Mechanics.ORCID iD: 0000-0002-9934-2945
Univ Technol & Econ, Dept Mech Mat & Struct, Fac Architecture & Engn, H-1111 Budapest, Hungary..
Univ Bristol, Dept Engn Math, Bristol BS8 1UB, Avon, England..
2018 (English)In: SIAM Journal on Applied Dynamical Systems, ISSN 1536-0040, E-ISSN 1536-0040, Vol. 17, no 2, p. 1267-1309Article in journal (Refereed) Published
Abstract [en]

This paper analyzes in detail the dynamics in a neighborhood of a Genot-Brogliato point, colloquially termed the G-spot, which physically represents so-called dynamic jam in rigid body mechanics with unilateral contact and Coulomb friction. Such singular points arise in planar rigid body problems with slipping point contacts at the intersection between the conditions for onset of lift-off and for the Painleve paradox. The G-spot can be approached in finite time by an open set of initial conditions in a general class of problems. The key question addressed is what happens next. In principle, trajectories could, at least instantaneously, lift off, continue in slip, or undergo a so-called impact without collision. Such impacts are nonlocal in momentum space and depend on properties evaluated away from the G-spot. The answer is obtained via an analysis that involves a consistent contact regularization with a stiffness proportional to 1/epsilon(2) for some epsilon. Taking a singular limit as epsilon -> 0, one finds an inner and an outer asymptotic zone in the neighborhood of the G-spot. Matched asymptotic analysis then enables continuation from the G-spot in the limit epsilon -> 0 and also reveals the sensitivity of trajectories to epsilon. The solution involves large-time asymptotics of certain generalized hypergeometric functions, which leads to conditions for the existence of a distinguished smoothest trajectory that remains uniformly bounded in t and epsilon. Such a solution corresponds to a canard that connects stable slipping motion to unstable slipping motion through the G-spot. Perturbations to the distinguished trajectory are then studied asymptotically. Two distinct cases are distinguished according to whether the contact force becomes infinite or remains finite as the G-spot is approached. In the former case it is argued that there can be no such canards and so an impact without collision must occur. In the latter case, the canard trajectory acts as a dividing surface between trajectories that momentarily lift off and those that do not before taking the impact. The orientation of the initial condition set leading to each eventuality is shown to change each time a certain positive parameter beta passes through an integer. Finally, the results are illustrated in a particular physical example, namely the frictional impact oscillator first studied by Leine, Brogliato, and Nijmeijer.

Place, publisher, year, edition, pages
SIAM PUBLICATIONS , 2018. Vol. 17, no 2, p. 1267-1309
Keywords [en]
friction, impact, Painleve paradox, singularity, asymptotics, contact mechanics
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-232286DOI: 10.1137/17M1141242ISI: 000436995800006Scopus ID: 2-s2.0-85049252739OAI: oai:DiVA.org:kth-232286DiVA, id: diva2:1233755
Note

QC 20180719

Available from: 2018-07-19 Created: 2018-07-19 Last updated: 2018-07-19Bibliographically approved

Open Access in DiVA

No full text in DiVA

Other links

Publisher's full textScopus

Search in DiVA

By author/editor
Nordmark, Arne
By organisation
Mechanics
In the same journal
SIAM Journal on Applied Dynamical Systems
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 17 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf