Bifurcations of stationary manifolds formed in the neighborhood of equilibrium in the dynamic cutting system

Abstract

The work objective is to study the formation of orbitally asymptotically stable limit cycles and two-dimensional invariant tori including bifurcations near the attracting sets. The investigators use primarily methods based on the mathematic simulation of the dynamic systems. Some problems of the nonlinear dynamics of the material cutting are considered. A mathematical model of the dynamic system considering the dynamic link formed by the cutting process is offered. Here, the following key features of the dynamic coupling are taken into account: dependence of the cutting forces on the area of a cut-off layer, delay of forces towards the elastic deformation shifts of the tool in relation to the workpiece, restrictions imposed on the tool movements when the back of the instrument is approaching the treated part of the workpiece, forces - cutting velocity relation. The dynamic subsystem of the tool is presented by a linear dynamic system in the plane orthogonal to a cutting surface. Following the research, some guidelines for designing systems with the required stationary manifold in the state space are provided. Importantly, in the neighborhood of equilibrium, various criteria of set causing regular or irregular features of the formed in-cut surface can develop depending on the models interacting under processing.