Reaction forces and friction forces in the dynamics of systems with geometric singularities

Abstract

The properties of the holonomic mechanical systems motion with parameters are discussed. For some (critical) parameter values, the configuration space of a mechanical system is a manifold with singularities. For other parameter values, the configuration space is a smooth manifold. It is assumed that the sliding friction force according to the Amonton-Coulomb model can act upon one of the material points of the mechanical system. When the parameters of a mechanical system differ from critical values, then the classical Lagrange equations could be applied to describe its dynamics. The point of interest is the motion on smooth manifolds near points which transform into singular points as the parameters of the mechanical system tend to critical values. The behavior of reaction forces and Lagrange multipliers for such “pre-singular” points is considered. Two types of configuration spaces with singularities are studied: the union of two intersecting curves in the plane and the union of two tangent curves in the plane. For the first time, various variants of the Lagrange multipliers behavior are shown using the example of a given type of perturbation of configuration spaces with singularities. In general, it is proven that for a singularity of the intersection type, the Lagrange multipliers become unlimited near the singular point (on a manifold with singularities), regardless of the influence of the friction force. For a tangency singularity type, there are different variants with taking into account the friction force. For one type of perturbation of the configuration space, the resulting Lagrange multipliers are limited. For other type of perturbation of the configuration space, the resulting Lagrange multipliers are unlimited. The general property of friction force for the considered mechanical systems is derived. If the friction force is taken into account, then there are two solutions for reaction forces when moving near a singular point in one direction, but there are no solutions when moving in the other direction.

Keywords:

constraint reaction, friction force, hinge mechanism, singular point, holonomic constraint, Lagrange multipliers, manifolds with singularities