The problem of constrained impact

Abstract

The general laws governing collisions between two rigid bodies when their displacements are subject to certain restrictions are discussed, and the legitimacy of using various mathematical models to describe such collisions is considered. Two types of constraint are discussed. The first—bilateral constraints—are conditional on one or two points of the body being fixed. It is shown that in the thepresence of dry friction the impact may be of the cut-off type, that is, the contact stresses do not disappear. Conditions are obtained for cut-off cut-off in terms of the geometry of the fixed points. Another peculiarity of the collisions of bodies with fixed points is the change in the physical meaning of the coefficient of restitution: it depends on the configuration of the system. The second type is represented by problems of impact when there is a unilateral constraint-one of the bodies is supported on a massive base; it is shown that dry friction at the point of support may lead to situations in which a solution is either non-existent or is non-unique, and which resemble the well-known Painlevé paradoxes. The following conclusion is reached: for an adequate description of the phenomenon of constrained impact, allowance must be made for the compliance of the colliding bodies not only directly in the impact pair, but also at points of contact with other bodies In the general case, the use of wave theory to describe constrained impact creates immense mathematical difficulties and one must first work with simplified deformation models, which lead to systems of ordinary differential equations. Examples are considered, namely the impact of a physical pendulum on a wall and the Coriolis problem of colliding billiard balls.