Bifurcation of stationary manifolds formed in the neighborhood of the equilibrium in a dynamic system of cutting

The problems related to nonlinear dynamics of material processing by cutting are reviewed in this study. A mathematical model of a dynamic system that considers the dynamic link, formed by the cutting process, is proposed. The following key features of the dynamic links are examined: the dependence of the cutting forces on the area of the shear layer, lag of forces with respect to the elastic deformation displacement of the tool relative to the workpiece, the restrictions imposed on the movement of the tool toward the rear end of the instrument with the treated part of the workpiece, the dependence of the forces on the cutting speed, and the change of force components at varying angles of the tool with respect to the direction of movement of the tool relative to the workpiece.The dynamic subsystem of the tool is presented by a linear dynamic system in the plane normal to the cutting surface. The focus of this study is on the analysis of attracting sets formed near the equilibrium point (orbitally asymptotically stable limit cycles, two-dimensional invariant tori, and chaotic attractors). It is shown that by considering the bending deformation of the tool, there is a possibility of branching of equilibrium points during changes of control parameters. Data on the bifurcations of the parametric space and the space of control parameters are shown. The general laws of buckling equilibrium of the system are reviewed.