Bifurcations of attracting sets of deformation displacement of cutting tool depending on the spindle group beats

V. L. Zakovorotnyj ; V. E. Gvindzhilija

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 25 (2017) no. 6, pp. 38-56.

Résumé

Subject of the study. Bifurcations of the attracting sets of the deformation displacement of the tool in the dynamic system of the turning machine depending on the beats periodic trajectory of the spindle group are considered in the article. The dynamic system is represented by the two interact mechanical subsystems through the dynamic link formed by the cutting process. Through the link is represented by the forces model in the coordinate condition, the trajectories of the executive elements and the trajectories of the spindle beats group. The model of this link is nonlinear. Therefore the system characteristics (stability of the trajectories and the forming attracting sets of the deformation displacement) depend on the parameters of the beats. In this system not only the conditions for the parametrical self-excitation are created but the different attracting sets of the deformation displacement (limit cycles, invariant tori, chaotic attractors) depending on the parameters of the beats are formed. Substantiation of address to studying of this question. Disturbances by the spindle group depending on the accuracy and condition of the machine cause formation of the various attracting sets of the deformation displacements of the tool during work-process of the detail. These deformation displacements change the shape-generating trajectories of the tool movement relative to the part which influence the geometric topology of the forming surface. The research intends to discover the influence of the dynamic of the cutting process in the unity of the parameters of the beats and the attracting sets with the geometric topology of the work surface of the detail determined its consistency with the specified technological requirements. Method of research. Characteristics of the attracting sets of the deformation displacement and the laws of its formation are studied on the basis of the digital simulation. Mathematical model of the perturbed system was proposed for this purpose. Novelty. In contrast to previously done researches the transformation of the attracting sets of the deformation displacement depending on the parameters of the beats are analyzed in the article. Mathematical model of the system is performed as perturbed, the forces are the sum of the forces acted on the front surface of the tool and forces formed in the contact area of its back side plane. Results. For instance, the influence of the spindle group beats with the attracting sets of the deformation displacement of the system are presented. Bifurcations of the system depending on the parameters of the disturbance are also represented. Conclusions. The complicated transformation of the attracting sets takes place while disturbance parameters changes. Therefore it’s necessary to consider the effect of perturbations on attainable quality of the detail during controlling of the cutting process and in a process of CNC program creation.

Détail

Keywords: Bifurcation, attracting sets, dynamic cutting system, beats of the spindle.