Abstract: Grazing bifurcation is a special bifurcation behavior ofnon-smooth dynamical systems. In the present paper, grazing bifurcations ofa cantilever system with one-sided impact is investigated. Firstly, thelocal discontinuity mapping and Poincar\'e mapping of the vibrating systemare developed. Secondly, numerical simulation with the impacting clearanceas the bifurcation parameter are carried out to obtain two types ofgrazing bifurcations of the system, which are different from those of rigidimpact systems. Finally, it is further shown that the local discontinuitymapping can be also used to investigate grazing bifurcation of non-smoothdynamical systems with high order nonlinear terms.