Attracting manifolds forming features in neighborhood of cutdown endmilling trajectories

Abstract

The work objective is to analyze the attracting manifolds generated under the endmilling. The formation of the periodic stationary trajectories of the tool deformation displacements in relation to the workpiece and their sustainability is previously considered. In this case, the movements form the attracting manifold defined by the trajectories of the periodic changes in the thickness and width of the cut-off layer by each cutter tooth taking into account the deformation displacements. As opposed to the previously considered cases, the paper focuses on the attracting manifolds generated at the buckling failure of the stationary trajectory. It is shown that depending on the system parameters and the process conditions in the dynamic milling system, the attracting manifolds of limit cycles, invariant tori, and strange (chaotic) attractors can be formed. In this context, two cases are analyzed. The first relates to the system parameters (primarily the cutting speed) which allow neglecting the coefficient variations in the differential equations within the impulsive reaction of the system. In the second case, the system parameters vary within the impulsive reaction of the system, and an additional source of the parametric self-excitation is formed in it. Considerable attention is paid to the analysis of the attracting manifold bifurcations in a parameter space: an overview and examples are provided. The attracting sets are analyzed from the perspective of their impact on the quality parameters of the parts production.