Treatment of impact with friction in planar multibody mechanical systems

Frictional impact analysis of multibody mechanical systems has traditionally relied on the use of Newton's hypothesis for the definition of the coefficient of restitution. This approach has in some cases shown energy gains inherent in the use of Newton's hypothesis. This paper presents a general formulation, consistent with energy conservation principles, for the analysis of impact problems with friction in any planar multibody mechanical system. Poisson's hypothesis is instead utilized for the definition of the coefficient of restitution. A canonical form of Cartesian momentum/impulse-balance equations are assembled and solved for the changes in the momenta using an extension of Routh's graphical method for the normal and tangential impulses. Impulse process diagrams are numerically generated, and the Cartesian velocity or momenta jumps are calculated by balancing the accumulated system momenta during the contact period. This formulation recognizes the correct mode of impact, i.e., sliding, sticking, and reverse sliding. Impact problems are classified into seven cases, based on these three modes and the conditions during the compression and restitution phases of impact. Expressions are derived for the normal and tangential impulses corresponding to each impact case. The developed formulation is shown to be an effective tool in analyzing some frictional impact problems including frictional impact in a two-body system, an open-loop system, and a closed-loop system.