Painlevé paradox during oblique impact with friction

• Published in: European journal of mechanics, A, Solids Vol. 30; no. 4; pp. 457 - 467
• Main Authors: Shen, Yunian, Stronge, W.J.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Masson SAS 01.07.2011; Elsevier
• Subjects: Acceleration; Contact; Contact angle; Dynamic tests; Dynamics; Engineering Sciences; Exact sciences and technology; Frictional impact; Fundamental areas of phenomenology (including applications); Impact dynamics; Jam; Mathematical models; Mechanical contact (friction...); Mechanics; Painlevé paradox; Paradoxes; Physics; Rigid-body dynamics; Sliding; Solid dynamics (ballistics, collision, multibody system, stabilization...); Solid mechanics; Structural and continuum mechanics; Impact dynamics; Sliding; Painlevé paradox; Frictional impact; Jam; Rigid bodies; Point contact; Deformability; Oblique incidence; Roughness; Solid dynamic; Mechanical contact; Contact surface; Relative motion; Multiple solution; Nonsmooth analysis; Friction; Sliding contact; Dynamic stability; Hybrid model; sliding; frictional impact; impact dynamics; jam
• ISSN: 0997-7538; 1873-7285
• DOI: 10.1016/j.euromechsol.2011.03.001

In analyses using non-smooth dynamics, oblique impact of rough bodies in an unsymmetrical configuration can result in self-locking or “jam” at the sliding contact if the coefficient of friction is sufficiently large; this has been termed, Painlevé’s paradox. In the range of configurations and coefficients of friction where Painlevé’s paradox occurs, analyses based on rigid body dynamics give results indicating that either there are multiple solutions or the solution is nonexistent. This conundrum has been resolved by considering that the contact has small normal and tangential compliance which is representative of deformability in a local region around the contact point. An analysis using a hybrid model which includes local compliance of the contact region has calculated the time-dependent changes in relative motion of colliding bodies for a range of incident angles of obliquity, tan −1[− V 1(0)/ V 3(0)] where V 1(0)and V 3(0) are the incident tangential and normal relative velocities at the contact point, respectively. The paradox is shown to result from a negative relative acceleration of the contact points during an initial period of sliding – a negative acceleration that is inconsistent with the assumption of rigid-body contact.