Period Doubling Bifurcation and Center Manifold Reduction in a Time-periodic and Time-delayed Model of Machining

Abstract

A closed-form calculation is presented for the analysis of the period-doubling bifurcation in the time-periodic delay-differential equation model of interrupted machining processes such as milling where the nonlinearity is essentially nonsymmetric. We prove the subcritical sense of this period-doubling bifurcation and approximate the emerging period-two oscillations by the Lyapunov—Perron method for computing the center manifold and by calculating the Poincaré—Lyapunov constant of the bifurcation analytically at certain characteristic parameter values. The existence of the unstable period-two oscillations around the stable stationary cutting is confirmed using a numerical continuation algorithm developed for time-periodic delay-differential equations.