A case of forming chaotic attractors in cutting dynamic system

Aleksandra Gubanova; Vilor Zakovorotnyy; Khristoforova   Veronika

doi:doi:10.12737/11588

A case of forming chaotic attractors in cutting dynamic system

Published в Vestnik of Don State Technical University · Volume 15, Issue 2, 2015 · Pages 11–22 · Rubrics: MACHINE BUILDING AND MACHINE SCIENCE

DOI 10.12737/11588

Received: 11.06.2015 Accepted: 30.06.2015 Published: 30.06.2015 Language of publication: RUS

Authors

Aleksandra Gubanova · Vilor Zakovorotnyy · Khristoforova Veronika

ABSTRACT

The conditions under which the chaotic dynamics is formed during the materials processing by cutting are analyzed. In the previous studies, in case of loss of sta-bility of the cutting pro-cess, limit cycles or invariant tori are generated in the neigh-borhood of the equilibrium system. Contrastingly to these studies, the case when the tool properties are such that a nonlin-ear positive feed-back is formed by flexural deformations is considered. A mathematical system model is provided for this case. On the basis of the numerical simulation, using the MATLAB application program package, the dynamic model parameters effect is explored under the conditions of the chaotic dynamics formation. The research results show that with increasing the parameters characterizing the formation of a positive feedback, the system undergoes a series of period-doubling bifurcations in the system of strange attractors. They are located in the vicinity of the equilibrium points and have a limited area. It is shown that the tool chaotic oscillations lead to the chaotic work surface forming, therefore, in the application sector, it is necessary to choose the parameters under which the chaotic dynamics is not formed. Although the considered examples relate to the cutting process, the obtained results are of general validity for the dynamic systems interacting with various environments, for example, with a tribological environment.

KEYWORDS

material cutting process dynamic system invari-ant varieties chaotic attractors bifurcations.

Information Text References

Authors:

Type:

Article

Pages:

from 11 to 22

Status:

Published

Language:

Russian

Keywords:

material cutting process , dynamic system, invari-ant varieties, chaotic attractors, bifurcations.

Text

Введение. Динамическая система процесса резания, базовая модель которой характеризует взаимодействие пространственной динамической модели подсистемы инструмента с динамической связью, формируемой процессом обработки, может служить примером, иллюстрирующим различные эффекты нелинейной динамики. В известных иссле-дованиях в области динамики процессов обработки на металлорежущих станках главное внимание уделялось рассмотрению устойчивости точки равновесия системы, а также автоколебаниям системы [1–18]. Во всех этих работах не принимался во внимание случай, когда за счет существенных изгибных деформационных смещений инструмента, вызывающих уменьшение переднего угла режущего инструмента, силы резания не уменьшаются, как в отмеченных выше работах, а возрастают. Тем самым формируется положительная обратная связь, способствующая самовозбужде-нию системы резания. Ниже будет показано, что в этом случае, как правило, в окрестностях равновесия образуются странные (хаотические) аттракторы. Приводимый в статье материал дополняет известные примеры образования хао-тических аттракторов [19–24].

References

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