Nonlinear Regenerative Machine Tool Vibrations

Abstract

Abstract The existence and the nature of the Hopf bifurcation is presented in the delay-differential equation model of the so-called regenerative machine tool vibration. The relevant nonlinearity is considered at the cutting force dependence on the chip thickness. The delayed terms show a special algebraic structure in the nonlinear part of the equation of motion. This results in a surprisingly simple and useful analytical formula in the end of the lengthy calculation based on center manifold reduction in the corresponding infinite dimensional phase space. The result gives a simple way to estimate the domain of attraction of the stable stationary cutting as well as an estimation of that technological parameter domain, where the cutting is globally stable.