Effect of spindle group wavering on attracting sets of deformation tool displacements under turning

Introduction. In a large body of research, problems of the cutting process dynamics are considered in the uniformity of the static stability in the moving coordinate system defined by the trajectories of the machine executive elements, and also of the attracting sets properties generated in the vicinity of the balance point. In contrast to these investigations, the given paper discusses the problem of forming attracting sets that are generated in the dynamic cutting system in which the movement trajectories of the machine executive elements are perturbed periodic vibration displacements of the spindle group. These perturbations are largely determined by the spindle group wavering. The latter causes periodic oscillation displacements of the tool relating to the work-in-process, and, combined with the dynamic system properties, leads to different dynamic effects. First, due to the parametric self-excitation, the instability of the steady-state stationary trajectories of the periodic deformation displacements may occur. Secondly, depending on the frequency and amplitude of the spindle group spatial beats, the formation of attracting sets of the limit cycle type and invariant tori, as well as their degeneration may happen. Finally, with the variations of frequency, the oscillation flip bifurcations, and, consequently, the formation of chaotic attractors occur. Moreover, their generation is extremely sensitive both to the parameters interacted through the cutting process of subsystems on the side of the tool and the work-in-process, and to the parameters of the dynamic link generated by the cutting process. Materials and Methods. The properties of the attracting sets and the laws of their formation are studied on the digital simulation basis. For this, a mathematical model of the perturbed system is given, as well as the results of studying the formation of attracting sets in a dynamic cutting system using the example of the turning process. Research Results. The examples of beats effect on the system attracting sets are provided including the system bifurcations depending on the parameters perturbations.