An analytical dynamics approach for mechanical systems involving a single frictional contact using b-geometry

This study focuses on the development of a new formulation, which describes the dynamics of a class of mechanical systems involving a single contact with friction. The whole process is performed within the general framework of analytical dynamics. At the same time, the efforts are assisted and enhanced by employing some fundamental tools of differential geometry. This provides a strong theoretical foundation, allowing for a systematic and consistent application of Newton's law of motion to systems possessing configuration manifolds with boundary. The manifold boundary is first defined by using the unilateral constraint representing the contact event. Then, it is shown that the corresponding contact phase takes place inside a thin layer near this boundary, where the dominant dynamics is described by a suitable set of three ordinary differential equations. The explicit appearance of time in the formulation introduces a short but finite time scale, which permits the consideration of contact of any combination of mechanical bodies, including rigid and discretized deformable bodies. Finally, the study concludes with presentation of a selected set of examples. In all cases, emphasis is put on illustrating the basic steps of the analysis as well as on capturing and investigating phenomena arising during central or eccentric collision of bodies.