THE PROBLEM OF CONSTRAINED IMPACT

The general laws governing collisions between two rigid bodies when th eir displacements are subject to certain restrictions are discussed an d 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 th e body being fixed. It is shown that in the presence of dry friction t he impact may be of the cut-off type, that is, the contact stresses do not disappear. Conditions are obtained for cut-off impact in terms of the geometry of the fixed points. Another peculiarity of the collisio ns of bodies with fixed points is the change in the physical meaning o f the coefficient of restitution: it depends on the configuration of t he system. The second type is represented by problems of impact when t here is a unilateral constraint-one of the bodies is supported on a ma ssive 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 no n-unique, and which resemble the well-known Painleve paradoxes. The fo llowing conclusion is reached: for an adequate description of the phen omenon of constrained impact, allowance must be made for the complianc e of the colliding bodies not only directly in the impact pair, but al so at points of contact with other bodies. In the general case, the us e of wave theory to describe constrained impact creates immense mathem atical difficulties and one must first work with simplified deformatio n models, which lead to systems of ordinary differential equations. Ex amples are considered, namely the impact of a physical pendulum on a w all and the Coriolis problem of colliding billiard balls. (C) 1997 Els evier Science Ltd. All rights reserved.