A non-standard analysis approach to systems involving friction

Many physical systems involve discontinuous forces. The conventional numerical discretization approach does not always yield a consistent prediction of the behavior of these systems. Here, as an example of a system involving discontinuous forces, a simple frictional system subjected to continuous as well as discontinuous excitations is considered. A set of orthogonal tracking functions is introduced, and the discontinuous excitation is expanded in a series of these tracking functions. Using non-standard analysis, the discontinuous functions are rigorously represented by continuous hyperreal approximations. The transformation of discontinuous forces to continuous ones allows consistent prediction of the system's behavior, thus providing a reliable technique for simulation of systems involving discontinuous forces.