Hopf Bifurcation in Machining

The classical theorem of Hopf Bifurcation for the study of periodic solutions of ordinary differential equations (ODEs) at an equilibrium point has become the standard mathematical tool for reducing higher dimensional dynamical systems to lower dimensional systems, while at the same time preserving the emanating dynamics. The attraction for this tool is due to the fact that the classical Hopf theorem imposes generic conditions on the systems for the bifurcation to occur, and also, it provides explicit algorithms for the characterization of the possible dynamics. In this paper, we employ Hopf’s theorem and the centre manifold theorem to reduce the infinite-dimensional character of delay differential equations, describing the instability machining induced by regenerative chatter in orthogonal machining process, to a four-dimensional ODEs. We will conduct numerical simulations of the cutting force as a function of the shear angle, tool geometry and the cutting conditions in the future when specific values for the model parameters have been determined experimentally.