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.
material cutting process , dynamic system, invari-ant varieties, chaotic attractors, bifurcations.
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].
1. Drozdov, N. A. K voprosu o vibracijah stanka pri tokarnoj obrabotke. [On the issue of vibration machine for turn-ing.] Stanki i instrument, 1937, no. 22, pp. 12–17 (in Russian).
2. Kashirin, A. I. Issledovanie vibracij pri rezanii metallov. [Study of vibrations in metal cutting.] Moscow: AN SSSR, 1944, 282 p. (in Russian).
3. Sokolovskiy, A. P. Vibracii pri rabote na metallorezhushhih stankah. [Vibrations at work on machine tools.] Issle-dovanie kolebanij pri rezanii metallov: sb. tr. [Study on fluctuations in metal cutting: coll. of papers.] Moscow: Mashgiz, 1958, pp. 15–18 (in Russian).
4. Murashkin, L. S., Murashkin, S. L. Prikladnaja nelinejnaja mehanika stankov. [Applied nonlinear mechanics of machines.] Leningrad : Mashinostroenie, 1977, 192 p. (in Russian).
5. Albrecht, P. Dinamika processa rezanija metalla.[Dynamics of the Metal Cutting Process.] Konstruirovanie i tehnologija mashinostroenija: Trudy amerikanskogo obshhestva inzhenerov-mehanikov ASME. [Engineering Design and Technology: Proc. of ASME.] Moscow: Izd-vo Mir, 1965, vol.87, ser. B, no. 4, pp. 40–54. (in Russian).
6. Zharkov, I. G. Vibracii pri obrabotke lezvijnym instrumentom. [Vibrations at edge tool cutting.] Leningrad: Mashi-nostroenie, 1987, 184 p. (in Russian).
7. Tlustyy, I. Avtokolebanija v metallorezhushhih stankah. [Self-oscillations in machine tools.] Moscow : Mashgiz, 1956, 395 p. (in Russian).
8. Kudinov, V. A. Dinamika stankov. [Dynamics of machines.] Moscow: Mashinostroenie, 1967, 359 p. (in Russian).
9. Elyasberg, M. E. Avtokolebanija metallorezhushhih stankov: Teorija i praktika. [Self-oscillations of machine tools: theory and practice.] SPb.: OKBS, 1993, 182 p. (in Russian).
10. Weiz, V. L., Vasilkov, D. V. Zadachi dinamiki, modelirovanija i obespechenija kachestva pri mehanicheskoj obrabotke malozhestkih zagotovok. [Problems of dynamics, modeling and quality assurance at mechanical treatment of slender workpieces.] STIN, 1999, no. 6, pp. 9–13 (in Russian).
11. Zakovorotny, V. L., Flek, M. B. Dinamika processa rezanija. Sinergeticheskij podhod . [Dynamics of cutting pro-cess. Synergetic approach.] Rostov-on-Don: DSTU Publ. Centre, 2006, 876 p. (in Russian).
12. Zakovorotny, V. L., Pham Dinh Tung, Nguyen Xuan Chiem. Matematicheskoe modelirovanie i parametriche-skaja identifikacija dinamicheskih svojstv podsistemy instrumenta i zagotovki. [Mathematical modeling and parametric identi-fication of dynamic properties of the tool – workpiece subsystem.] Izvestiya vuzov. Severo-Kavkazskiy region. Technical Sci-ences, 2011, no. 2, pp. 38–46 (in Russian).
13. Zakovorotny, V. L., Bordachev, E. V., Alekseychik, M. I. Dinamicheskij monitoring sostojanija processa rezanija. [Dynamic monitoring of the cutting process.] STIN, 1998, no. 12, pp. 6–12 (in Russian).
14. Zakovorotny, V. L., Pham Dinh Tung, Nguyen Xuan Chiem. Modelirovanie deformacionnyh smeshhenij instru-menta ot-nositel´no zagotovki pri tochenii. [Modeling of tool deformation offsetting to workpiece in turning.] Vestnik of DSTU, 2010, vol. 10, no. 7, pp. 1005–1015 (in Russian).
15. Altintas, Y., Budak, E. Analytical prediction of stability lobes in milling. Ann. CIRP, 1995, vol. 44, iss. 1, pp. 357–362.
16. Balachandran, B. Non-linear dynamics of milling process. Philos. Trans. Roy. Soc. 359, 2001, pp. 793–820.
17. Davies, M. A., Pratt, J. R. The stability of low immersion milling, Ann. CIRP 49, 2000, pp. 37–40.
18. Gouskov, A. M., Voronov, S. A., Paris, H., Batzer, S. A. Nonlinear dynamics of a machining system with two in-terdependent delays. Commun. Nonlin. Sci. Numer. Simul., 2002, vol. 7, iss. 4, pp. 207–221.
19. Anishchenko, V. S. Slozhnye kolebanija v prostyh sistemah. [Complex oscillations in simple systems.] Moscow: Nauka, 1990, 312 p. (in Russian).
20. Anishchenko, V. S. Attraktory dinamicheskih sistem. [Attractors of dynamical systems.] Izvestija vuzov. Priklad-naja nelinejnaja dinamika, 1997, vol. 5, no. 1, pp. 109–127 (in Russian).
21. Neymark, Y. I., Landa, P. S. Stohasticheskie i haoticheskie kolebanija. [Stochastic and chaotic oscillations.] Mos-cow: Nauka, 1987, 424 p. (in Russian).
22. Li, T., Yorke, J. A. Period Three Implies Chaos. Amer. Math. Monthly, 1975, vol. 82, no. 3, pp. 982–985.
23. Lorens, E. N. Deterministic Nonperiodic Flow. J. Atmos. Sci., 1963, vol. 20, no. 5, pp. 130–141.
24. Dorfman, J. R. An Introduction to Chaos in Nonequilibrum Statistical Mechanics. Cambridge University Press, 1999, 288 p.
25. Feigenbaum, M. J. The transition to a periodic behavior in turbulent systems. Commun. Math. Phys., 1980, vol. 77, no. 1, pp. 65–86.
26. Bobrov, V. F. Osnovy teorii rezanija metallov. [Fundamentals of the metal cutting theory.] Moscow: Mashi-nostroenie, 1975, 344 p. (in Russian).
27. Zakovorotny, V. L., Pham Dinh Tung, Nguyen Xuan Chiem, Ryzhkin, M.N. Modelirovanie dinamicheskoj svjazi, formiruemoj processom tochenija, v zadachah dinamiki (skorostnaja svjaz´). [Dynamic coupling modeling formed by turning in cutting dynamics problems (velocity coupling).] Vestnik of DSTU, 2011, vol.11, no. 2, pp. 137–147 (in Russian).
28. Zakovorotny, V. L., Pham Dinh Tung, Nguyen Xuan Chiem, Ryzhkin, M. N. Modelirovanie dinamicheskoy svyazi, formiruemoy protsessom tocheniya, v zadachakh dinamiki (pozitsionnaya svyaz´). [Dynamic coupling modeling formed by turning in cutting dynamics problems (position coupling).] Vestnik of DSTU, 2011, vol.11, no. 3, pp. 30–38 (in Russian).
29. Kabaldin, Y. G. Samoorganizacijaja i nelinejnaja dinamiki v processah trenija i iznashivanija instrumenta pri re-zanii.[ Self-organization and nonlinear dynamics in processes of friction and tool wear under cutting.] Komsomolsk-na-Amure: Izd-vo KnAGTU, 2003, 175 p. (in Russian).
