The Nonsmooth Dynamics of Combined Slip and Spin Motion Under Dry Friction

• Published in: Journal of nonlinear science Vol. 32; no. 4; p. 58
• Main Authors: Antali, Mate, Varkonyi, Peter L.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2022; Springer Nature B.V
• Subjects: Analysis; Approximation; Classical Mechanics; Contact force; Coulomb friction; Discontinuity; Dry friction; Dynamical systems; Economic Theory/Quantitative Economics/Mathematical Methods; Equations of motion; Fields (mathematics); Flat surfaces; Force distribution; Friction; Kinematics; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Rigid structures; Slip; Spherical harmonics; Spin dynamics; Theoretical; Nonsmooth dynamics; 70E55; 34A26; 34A36; 70G60; Slow–fast systems; Slip–stick transition; 70E18; Dry friction
• ISSN: 0938-8974; 1432-1467; 1432-1467
• DOI: 10.1007/s00332-022-09812-x

We investigate the motion of rigid bodies subject to combined slipping and spinning over a rough flat surface in the presence of dry friction. Integration of Coulomb friction forces over the contact area gives rise to a dynamical system with an isolated discontinuity of codimension 3. Recent results about such vector fields are applied to the motion of flat bodies under the assumption of known, time-independent distributions of normal contact forces and to general bodies where kinematic constraints enforce a state-dependent normal contact force distribution with a discontinuity at the sticking state. In both cases, the equations of motion are transformed into a smooth slow–fast dynamical system. The fixed points of the fast flow indicate the possible directions of combined slip–spin motion immediately before a body stops. We also introduce an approximation of the frictional forces and moments by the leading-order term of a spherical harmonic expansion, which allows for an explicit formulation of the equations of motion. The approximate model captures important empirically observed features of the motion. It is proven analytically and illustrated by examples that the number of fixed points in the approximate model is 2, 4, or 6.