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.
endmilling operation, attracting manifolds, bifurcations, control, quality parameters of parts production.
В последние два десятилетия при изучении свойств эволюции и самоорганизации широко используется синергетическая парадигма [1–4]. Ее применение для управления сложными нелинейными объектами нашло свое отражение в работах [5–8]. При создании систем управления динамической системой резания также используются основы синергетической теории управления [7–14]. В этом случае управление, в том числе на основе построения программы ЧПУ, включает определение желаемых траекторий формообразующих движений и соответствующих им траекторий движения исполнительных элементов станка. При решении этой задачи принципиально важно знать свойства тех притягивающих множеств, которые самостоятельно образуются в окрестности формообразующих движений инструмента относительно обрабатываемой детали. Необходимо также уметь управлять этими множествами, влияющими на качество поверхности, формируемой при резании. Наконец, раскрытие свойств притягивающих множеств, формирующих сигнал виброакустической эмиссии, открывает новые пути построения информационных моделей для диагностирования процесса [15].
Изучение притягивающих множеств связано с рассмотрением динамической системы резания, изучению которой посвящено множество известных исследований [16–32]. Фрезерование является наиболее сложным процессом резания [33–46]. Это обусловлено его нестационарностью. Параметры длины и толщины срезаемого слоя каждым режущим лезвием периодически изменяются. Поэтому в подвижной системе координат, движение которой определяется траекториями исполнительных элементов, стационарным, установившимся состоянием является не точка равновесия (как при точении), а замкнутая траектория. В связи с этим уравнение динамики имеет периодически изменяющиеся коэффициенты.
Рассмотрим процесс фрезерования на станках, имеющих до пяти координат управления при обработке деталей, параметры жесткости которых существенно изменяются вдоль траектории движения инструмента [44–46]. В настоящей статье, в отличие от известных исследований, учитываются несколько источников самовозбуждения и связи, обусловленные взаимодействиями передней и задней поверхностей инструмента с деталью. Здесь можно рассматривать два случая. Первый — обработка с малыми частотами вращения шпинделя, когда в системе в пределах импульсной реакции параметры можно считать замороженными. Второй — обработка с большими частотами вращения, когда параметры нельзя считать замороженными.
1. Prigozhin, I., Stengers, I. Poryadok iz khaosa. [Order out of chaos.] Arshinov, V.I., Klimontovich, Y.L., Sachkov, Y.V., eds. Moscow: Progress, 1986, 193 p. (in Russian).
2. Prigozhin, I., Stengers, I. Poryadok iz khaosa. Novyy dialog cheloveka s prirodoy. [Order out of chaos. A new dia-logue of man with nature.] Moscow: Editorial URSS, 2003, 312 p. (in Russian).
3. Haken, H. Sinergetika. Ierarkhiya neustoychivostey v samoorganizuyushchikhsya sistemakh i ustroystvakh. [Synergetics. Hierarchy of instabilities in self-organizing systems and devices.] Moscow: Mir, 1985, 424 p. (in Russian).
4. Haken, H. Tayny prirody. Sinergetika: uchenie o vzaimodeystvii. [Secrets of nature. Synergetics: theory of inter-action.] Moscow; Izhevsk: Institute of Computer Science, 2003, 320 p. (in Russian).
5. Kolesnikov, A.A. Sinergeticheskaya teoriya upravleniya. [Synergetic theory of management.] Moscow: Ener-goatomizdat, 1994, 344 p. (in Russian).
6. Kolesnikov, A.A., ed. Sinergetika i problemy teorii upravleniya. [Synergetics and problems of control theory.] Moscow: Fizmatlit, 2004, 504 p. (in Russian).
7. Zakovorotny, V.L., Flek, M.F. Dinamika protsessa rezaniya. Sinergeticheskiy podkhod. [Cutting process dynam-ics. Synergetic approach.] Rostov-on-Don: Terra, 2006, 880 p. (in Russian).
8. Zakovorotny, V.L., et al. Sinergeticheskiy sistemnyy sintez upravlyaemoy dinamiki metallorezhushchikh stankov s uchetom evolyutsii svyazey. [Synergetic system synthesis of controlled dynamics of machine tools with coupling evolution.] Rostov-on-Don: DSTU Publ. Centre, 2008, 324 p. (in Russian).
9. Zakovorotny, V. L., Lukyanov, A. D. The Problems of Control of the Evolution of the Dynamic System Interact-ing with the Medium. International Journal of Mechanical Engineering and Automation, 2014, vol. 1, no. 5, pp. 271-285.
10. Zakovorotny, V.L., Panov, E.Y., Potapenko, P.N. Svoystva formoobrazuyushchikh dvizheniy pri sverlenii glubokikh otverstiy malogo diametra. [Properties of forming movements when drilling deep pinholes.] Vestnik of DSTU, 2001, vol. 1, no. 2, pp. 81-93 (in Russian).
11. Zakovorotny, V.L., Lapshin, V.P., Turkin, I.A. Upravlenie protsessom sverleniya glubokikh otverstiy spiral´nymi sverlami na osnove sinergeticheskogo podkhoda. [Process control drilling deep holes twist drills based on the synergetic ap-proach.] University News. North-Caucasian region. Technical Sciences Series, 2014, no. 3 (178), pp. 33-41(in Russian).
12. Zakovorotny, V.L., Pham Dinh Tung. Osobennosti preobrazovaniya traektoriy ispolnitel´nykh elementov stanka v traektorii formoobrazuyushchikh dvizheniy instrumenta otnositel´no zagotovki. [Features of transformation of machine actuator trajectories into the trajectories of forming tool movements relative to the workpiece.] University News. North-Caucasian region. Technical Sciences Series, 2011, no. 4, pp. 69-75(in Russian).
13. Zakovorotny, V.L., Flek, M.F., Pham Dinh Tung. Sinergeticheskaya kontseptsiya pri postroenii sistem upravleniya tochnost´yu izgotovleniya detaley slozhnoy geometricheskoy formy. [Synergetic concept in construction of accuracy control systems for manufacturing parts of complex geometric forms.] Vestnik of DSTU, 2011, vol. 11, no. 10 (61), pp. 1785-1797 (in Russian).
14. Zakovorotny, V.L., Sankar, T., Borodachev, E.V. Sistema optimal´nogo upravleniya protsessom glubokogo sverleniya otverstiy malogo diametra. [The optimal control system of deep drilling of pinholes.] STIN, 1994, no. 12, pp. 22 (in Russian).
15. Zakovorotny, V.L., Ladnik, I.V. Postroenie informatsionnoy modeli dinamicheskoy sistemy metallorezhushchego stanka dlya diagnostiki protsessa obrabotki. [Building of data model of the machine tool dynamic system for treatment process diagnostics.] Journal of Machinery Manufacture and reliability, 1991, no. 4, pp. 75-81(in Russian).
16. Tlusty, I. Avtokolebaniya v metallorezhushchikh stankakh. [Self-oscillations in machine tools.] Moscow: Mash-giz, 1956, 395 p. (in Russian).
17. Tlusty, I., et al. Selbsterregte Schwingungen an Werkzeugmaschinen. Berlin: Veb Verlag Technik, 1962, 320 р.
18. Tobias, S. A. Machine Tool Vibrations. London: Blackie, 1965, 350 р.
19. Kudinov, V.А. Dinamika stankov. [Machine dynamics.] Moscow: Mashinostroenie, 1967, 359 p. (in Russian).
20. Elyasberg, M.E. Avtokolebaniya metallorezhushchikh stankov: Teoriya i praktika. [Self-oscillations of machine tools: Theory and practice.] St. Petersburg: OKBS, 1993, 182 p. (in Russian).
21. Veits, V.L., Vasilkov, D.V. Zadachi dinamiki, modelirovaniya i obespecheniya kachestva pri mekhanicheskoy obrabotke malozhestkikh zagotovok. [Problems of dynamics, simulation and quality assurance for machining low-rigid work-pieces.] STIN, 1999, no. 6, pp. 9-13 (in Russian).
22. Sokolovskiy, A.P. Vibratsii pri rabote na metallorezhushchikh stankakh. Issledovanie kolebaniy pri rezanii metal-lov. [Vibrations at work on machine tools. Researching vibrations under metal cutting.] Moscow: Mashgiz, 1958, pp. 15-18 (in Russian).
23. Murashkin, L.S., Murashkin, S.L. Prikladnaya nelineynaya mekhanika stankov. [Applied nonlinear mechanics of machines.] Leningrad: Mashinostroenie, 1977, 192 p. (in Russian).
24. Zakovorotny, V. L. Bifurcations in the dynamic system of the mechanic processing in metal-cutting tools. Journal of Transactions on Applied and Theoretical Mechanics, 2015, vol. 10, pp. 102-116.
25. Zakovorotny, V. L., Pham D.-T., Bykador, V.S. Samoorganizatsiya i bifurkatsii dinamicheskoy sistemy obrabotki metallov rezaniem. [Self-organization and bifurcations of dynamical metal cutting system.] Izvestia VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, no. 3, pp. 26-40 (in Russian).
26. Zakovorotny, V. L., Pham D.-T., Bykador, V.S. Vliyanie izgibnykh deformatsiy instrumenta na samoorganizatsiyu i bifurkatsii dinamicheskoy sistemy rezaniya metallov. [Influence of a flexural deformation of a tool on self-organization and bifurcations of dynamical metal cutting system.] Izvestia VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, no. 3, pp. 40-53 (in Russian).
27. Stepan, G.; F.-C. Moon, ed. Delay-differential equation models for machine tool chatter. New York: John Wiley, 1998, pp. 165-192.
28. Stepan, G., Insperge, T., Szalai, R. Delay, Parametric excitation, and the nonlinear dynamics of cutting processes. International Journal of Bifurcation and Chaos, 2005, vol. 15, no. 9, pp. 2783-2798.
29. Tobias, S.-A., Fishwick, A. Theory of regenerative machine tool chatter. The Engineer, 1958, vol. 205, pp. 199-203.
30. Merritt, H.-E. Theory of self-excited machine tool chatter. ASME Journal of Engineering for Industry, 1965, vol. 205, no. 11, pp. 447-454.
31. Sridhar, R., Hohn, R.-E., Long, G.-W. A stability algorithm for the general milling process: Contribution to machine tool chatter research-7. ASME Journal of Engineering for Industry, 1968, vol. 90, no. 2, pp. 330-334.
32. Altintas, Y., Budak, E. Analytical prediction of stability lobes in milling. Annals of the CIRP, 1995, vol. 44, no. 1, pp. 357-362.
33. Tlusty, J., Ismail, F. Special aspects of chatter in milling. ASME Journal of Vibration, Stress, and Reliability in Design, 1983, vol. 105, no. 1, pp. 24-32.
34. Minis, I., Yanushevsk y, T. A new theoretical approach for the prediction of machine tool chatter in milling. Trans. ASME Journal of Engineering for Industry, 1993, vol. 115, no. 2, pp. 1-8.
35. Insperger, T., Stepan, G. Stability of the milling process. Periodical Polytechnic-Mechanical Engineering, 2000, vol. 44, no. 1, pp. 47-57.
36. Budak, E., Altintas, Y. Analytical prediction of chatter stability in milling. Part I: General formulation. ASME Journal of dynamic systems measurement and control, 1998, vol. 120, no. 6 (1), pp. 22- 30.
37. Budak, E., Altintas, Y. Analytical prediction of chatter stability conditions for multi-degree of systems in milling. Part II: Applications. ASME Journal of dynamic systems measurement and control, 1998, vol. 120, no. 6 (1), pp. 31-36.
38. Merdol, D., Altintas, Y. Multi-frequency solution of chatter stability for low immersion milling. ASME Journal of Manufacturing Science and Engineering, 2004, vol. 126, no. 3, pp. 459-466.
39. Insperger, T., et al. Stability of up-milling and down-milling. Part 1: Alternative analytical methods. International Journal of machine tools and manufacture, 2003, vol. 43, no. 1, pp. 25-34.
40. Kline, W.-A., Devor, R.-E., Shareef, I.-A. The prediction of surface accuracy in end milling. ASME Journal of engineering for industry, 1982, vol. 104, no. 5, pp. 272-278.
41. Elbestawi, M.-A., Sagherian, R. Dynamic modeling for the prediction of surface errors in milling of thin-walled sections. Theoretical computational fluid dynamics, 1991, vol. 25, no. 2, pp. 215-228.
42. Campomanes, M.-L., Altintas, Y. An improved time domain simulation for dynamic milling at small radial im-mersions. Trans. ASME Journal of manufacturing science and engineering, 2003, vol. 125, no. 3, pp. 416-425.
43. Paris, H., Peigne, G., Mayer, R. Surface shape prediction in high-speed milling. International Journal of machine tools and manufacture, Paris, 2004, vol. 44, no. 15, pp. 1567-1576.
44. Altintas, Y., Lee, P. A general mechanics and dynamics model for helical end mills. Annals of the CIRP, 1996, vol. 45, no. 1, pp. 59-64.
45. Ozturk, E., Budak, E. Modeling of 5-axis milling processes. Machining Science and Technology, 2007, vol. 11, no. 3, pp. 287-311.
46. Budak, E., Ozturk, E., Tunc, L.-T. Modeling and simulation of 5-axis milling processes. Annals of CIRP. Manu-facturing Technology. 2009, vol. 58, no. 1, pp. 347-350.
47. Bravo, U., et al. Stability limits of milling considering the flexibility of the workpiece and the machine. Interna-tional Journal of machine tools and manufacture, 2005, vol. 45, pp. 1669-1680.
48. Weinert, K., et al. Modeling regenerative workpiece vibrations in five-axis milling. Production Engineering. Re-search and Development, 2008, no. 2, pp. 255-260.
49. Biermann, D., Kersting, P., Surmann, T. A general approach to simulating workpiece vibrations during five-axis milling of turbine blades. CIRP Annals. Manufacturing Technology. 2010, vol. 59, no. 1, pp. 125-128.
50. Voronov, S.A., Nepochatov, A.V., Kiselev, I.A. Kriterii otsenki ustoychivosti protsessa frezerovaniya nezhestkikh detaley. [Stability criteria evaluation for milling process of non-rigid parts.] Proceedings of Higher Educational Institutions. Маchine Building, 2011, no. 1 (610), pp. 50-62 (in Russian).
51. Voronov, S., Kiselev, I. Dynamics of flexible detail milling. Proceedings of the Institution of Mechanical Engi-neers. Part K: Journal of Multi-body Dynamics, 2011, vol. 225, no. 3, pp. 1177-1186.
52. Zakovorotny, V.L., Pham Dinh Tung, Nguen Xuan Chiem. Matematicheskoe modelirovanie i parametricheskaya identifikatsiya dinamicheskikh svoystv podsistemy instrumenta i zagotovki. [Mathematical modeling and parametric identifi-cation of dynamic properties of the subsystem of the cutting tool and workpiece in the turning.] University News. North-Caucasian region. Technical Sciences Series, 2011, no. 2, pp. 38-46 (in Russian).
53. Zakovorotny, V.L., et al. Modelirovanie dinamicheskoy svyazi, formiruemoy protsessom tocheniya, v zadachakh dinamiki protsessa rezaniya (skorostnaya svyaz´). [Dynamic coupling modeling formed by turning in cutting dynamics problems (velocity coupling).] Vestnik of DSTU, 2011, vol. 1, no. 2 (53), pp. 137-146 (in Russian).
54. Zakovorotny, V.L., et al. Modelirovanie dinamicheskoy svyazi, formiruemoy protsessom tocheniya (pozitsion-naya svyaz´). [Dynamic coupling modeling formed by turning in cutting dynamics problems (positional coupling).] Vestnik of DSTU, 2011, vol. 11, no. 3 (54), pp. 301-311 (in Russian).
55. Zakovorotny, V.L., Pham Dinh Tung, Nguen Xuan Chiem. Modelirovanie deformatsionnykh smeshcheniy in-strumenta otnositel´no zagotovki pri tochenii. [Modeling of tool deformation offsetting.] Vestnik of DSTU, 2010, vol. 10, no. 7 (50), pp. 1005-1015 (in Russian).
56. Danzhelo, R. Lineynye sistemy s peremennymi parametrami. [Linear systems with variable parameters.] Moscow: Mashinostroenie, 1974, 287 p. (in Russian).
57. Pontryagin, L.S. Izbrannye trudy. Т. 2. [Selecta. Vol. 2.] Moscow: Nauka, 1988, 576 p. (in Russian).
58. Tikhanov, А.N., Vasilyev, A.B., Volosov, V.M. Differentsial´nye uravneniya, soderzhashchie malyy parame-ter. [Differential equations containing a small parameter.] Tr. mezhdunar. simpoziuma po nelineynym kolebaniyam. [Proc. Int. Symposium on nonlinear vibrations.] Kiev: Izd-vo AN USSR, 1963, pp. 56-61 (in Russian).
59. Zakovorotny, V.L., Pham D.T. Parametricheskoe samovozbuzhdenie dinamicheskoy sistemy rezaniya. [Parametric self-excitation of cutting dynamic system.] Vestnik of DSTU, 2013, vol. 13, no. 5/6 (74), pp. 97-103 (in Russian).
60. Feigenbaum, M.-J. Quantitative universality for a class of nonlinear transformations. Journal of Statistical Phys-ics, 1978, vol. 19, pp. 25- 52.
