On generalized Coulomb–Amontons’ law in the context of rigid body dynamics

Abstract

A generalization of Coulomb–Amontons’ law of dry friction recently proposed by V. V. Kozlov is considered in the context of rigid body dynamics. Universal requirements for dry friction tensor formulated by V. V. Kozlov are complemented by a condition taking into account the contact nature of dry friction, and applied to several models. For the famous Painlevé problem, a generalized Coulomb–Amontons’ force without singularities, yet such that the dissipation takes place only at the point of contact, is found. By the example of the motion of a rigid ball on a plane with a single point of contact, it is shown that these principles are consistent with the well-known equations, studied by G.-G. Coriolis. Further, a ball simultaneously touching two perpendicular planes at two points of contact is considered. The corresponding equations of motion are derived and analyzed. An exact particular solution that describes a technique used in practice in billiards is obtained. It is shown that unlike the single contact case, the Lagrange multipliers can depend on friction coefficients. With this in mind, a generalization of the conditions on the tensor of dry friction for the case of an arbitrary number of contacts is proposed.