Stability problems of the stationary trajectories of the tool elastic deformational displacement under the longitudinal endmilling are considered. In the moving coordinates which displacement is determined by the actuators motion, the stationary steady state is not an equilibrium point (as when turning), but some closed tra-jectory. The synergetic concept of the analysis of trajectories executed in two stages is used. At the first stage, the established stationary trajectories are calculated according to the offered technique. At the second stage, the stability of these trajectories is analyzed. A case, when the equation parameters in variations under the fixed trajectory can be considered constant within the system impulsive reaction, is considered. Features of the stationary trajectories formation are studied; conditions under which they converge to some steady trajectories are obtained. Besides, some general properties of the loss of balance are received. An example of analysis is cited, and recommendations for ensuring stability of a trajectory of the established form-building motions are given.
endmilling dynamics, stationary trajectories, stability, synergetics, variable parameters.
Одной из центральных проблем науки во второй половине ХХ века стало формирование синергетической парадигмы эволюции и самоорганизации [1–4]. В известных работах [5, 6] предложено использовать ее для управления сложными нелинейными объектами, в том числе для анализа и синтеза динамической системы резания [7–9]. Система резания рассматривается как взаимодействие подсистем инструмента и обрабатываемой детали через связь, формируемую процессом обработки [10–16]. Данная связь является нелинейной с периодически изменяющимися параметрами и обладает свойством эволюционной изменчивости [7–9]. В литературе описаны проблемы устойчивости и многообразий, формируемых в окрестностях равновесия. Рассматриваются автоколебания [17, 18], инвариантные торы [19–21] и хаотические аттракторы [19–23]. Показано, что упругие деформационные смещения не могут быть скалярными, если необходимо раскрыть их основные динамические свойства. Они, как минимум, должны анализироваться в плоскости [24–27]. На динамические свойства системы оказывает влияние и тип процесса резания [7, 8, 26–48].
Из рассматриваемых в данном контексте процессов наиболее сложным является фрезерование [26–47]. Это обусловлено его нестационарностью, периодическими изменениями параметров длины и толщины слоя, срезаемого каждым режущим лезвием фрезы. Поэтому в подвижной системе координат, перемещение которой определяется траекториями исполнительных элементов, стационарным установившимся состоянием является не точка равновесия (как при точении), а некоторая замкнутая траектория. В связи с этим уравнение динамики в общем случае имеет периодически изменяющиеся коэффициенты. Кроме того, учитывается влияние запаздывающих аргументов [28–39]. В указанных работах для изучения устойчивости используется теория Флоке для — периодических процессов. Изучается процесс фрезерования на станках, имеющих до пяти координат управления при обработке деталей, матрицы жесткости которых существенно изменяются вдоль траектории движения инструмента [40–47]. В настоящей статье результаты отмеченных выше работ рассматриваются в отношении полных нелинейных математических моделей с учетом периодического изменения параметров. При этом ставится задача исследования устойчивости не точки в подвижной системе координат, а стационарной траектории формообразующих движений. Здесь можно рассматривать два случая. Первый относится к обработке с малыми частотами вращения шпинделя, когда в системе в пределах импульсной реакции параметры можно считать замороженными. Это явление рассматривается в рамках данной работы. Второй случай относится к обработке с большими частотами вращения инструмента, когда параметры нельзя считать замороженными. Соответствующий материал будет рассмотрен в следующей статье.
1. Prigozhin, I., Stengers, I.; Arshinov, V.I., Klimontovich, Y.L., Sachkov, Y.V., eds. Poryadok iz khaosa. [Order out of chaos.] 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, G. 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, G. 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, А.А. Sinergeticheskaya teoriya upravleniya. [Synergetic control theory.] Moscow: Energoatomizdat, 1994, 344 p. (in Russian).
6. Kolesnikov, А.А. ed. Sinergetika i problemy teorii upravleniya. [Synergetics and control theory problems.] Mos-cow: 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 Interacting with the Medium. International Journal of Mechanical Engineering and Automation, 2014, vol. 1, no. 5, pp. 271-285.
10. Tlusty, I. Avtokolebaniya v metallorezhushchikh stankakh. [Self-oscillations in machine tools.] Moscow: Mash-giz, 1956, 395 p. (in Russian).
11. Tlusty, I. Selbsterregte Schwingungen an Werkzeugmaschinen. Berlin: Veb Verlag Technik Berlin, 1962, 320 р.
12. Tobias, S.-A. Machine Tool Vibrations. London: Blackie, 1965, 350 р.
13. Kudinov, V.А. Dinamika stankov. [Machine dynamics.] Moscow: Mashinostroenie, 1967, 359 p. (in Russian).
14. Elyasberg, М.Е. Avtokolebaniya metallorezhushchikh stankov: teoriya i praktika. [Self-oscillations of machine tools: theory and practice.] St. Petersburg: OKBS, 1993, 182 p. (in Russian).
15. Veits, V.L., Vasilkov, D.V. Zadachi dinamiki, modelirovaniya i obespecheniya kachestva pri mekhanicheskoy obrabotke malozhestkikh zagotovok. [Problems of dynamics, simulation and quality assurance under machining of nonrigid workpieces.] STIN, 1999, no. 6, pp. 9-13 (in Russian).
16. 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).
17. Sokolovskiy, А.P. Vibratsii pri rabote na metallorezhushchikh stankakh. Issledovanie kolebaniy pri rezanii metal-lov. [Vibration at work on machine tools. Study on vibrations in metal cutting] Moscow: Mashgiz, 1958, pp. 15-18 (in Russian).
18. Murashkin, L.S., Murashkin, S.L. Prikladnaya nelineynaya mekhanika stankov. [Applied nonlinear machine engi-neering.] Leningrad: Mashinostroenie, 1977, 192 p. (in Russian).
19. 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.
20. 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).
21. 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).
22. Stepan, G.; Moon F.C., ed. Delay-differential equation models for machine tool chatter. New York: John Wiley, 1998, pp. 165-192.
23. 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.
24. Tobias, S.-A., Fishwick, W. Theory of regenerative machine tool chatter. The Engineer, 1958, vol. 205, pp. 199-203.
25. Merritt, H.-E. Theory of self-excited machine tool chatter. ASME. Journal of Engineering for Industry, 1965, vol. 205, no. 11, pp. 447-454.
26. 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.
27. Altintas, Y., Budak, E. Analytical prediction of stability lobes in milling. CIRP Annals, 1995, vol. 44, no. 1, pp. 357-362.
28. 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.
29. Minis, I., Yanushevsky, T. A new theoretical approach for the prediction of machine tool chatter in milling. ASME. Journal of Engineering for Industry, 1993, vol. 115, no. 2, pp. 1-8.
30. Insperger, T., Stepan, G. Stability of the milling process. Periodical Polytechnic-Mechanical Engineering, 2000, vol. 44, no. 1, pp. 47-57.
31. 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.
32. 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.
33. 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. , pp. 459-466.
34. Insperger, T. Stability of up-milling and down-milling. Part 1: Alternative analytical methods. International Jour-nal of Machine Tools and Manufacture, 2003, vol. 43, no. 1, pp. 25-34.
35. Kline, W.-A., Devor, R.-E, Shareef, I.-A. The prediction of surface ac¬curacy in end milling. ASME. Journal of Engineering for Industry, 1982, vol. 104, no. 5, pp. 272-278.
36. Elbestawi, M.-A., Sagherian, R. Dynamic modeling for the prediction of surface errors in milling of thin-walled sections. Theoretical and Computational Fluid Dynamics, 1991, vol. 25, no. 2, pp. 215-228.
37. Campomanes, M.-L., Altintas, Y. An improved time domain simulation for dynamic milling at small radial im-mersions. ASME. Journal of Manufacturing Science and Engineering, 2003, vol. 125, no. 3, pp. 416-425.
38. Paris, H., Peigne, G., Mayer, R. Surface shape prediction in high-speed milling. International Journal of Machine Tools and Manufacture, 2004, vol. 44, no. 15, pp. 1567-1576.
39. Altintas, Y., Lee, P. A general mechanics and dynamics model for helical end mills. CIRP Annals, 1996, vol. 45, no. 1, pp. 59-64.
40. Ozturk, E., Budak, E. Modeling of 5-axis milling processes. Machining Science and Technology, 2007, vol. 11, no. 3, pp. 287-311.
41. Budak, E., Ozturk, E., Tunc, L.-T. Modeling and simulation of 5-axis milling processes. CIRP Annals. Manufac-turing Technology, 2009, vol. 58, no. 1, pp. 347-350.
42. 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.
43. Weinert, K., et al. Modeling regenerative workpiece vibrations in five-axis milling. Production Engineering. Re-search and Development, 2008, no. 2, pp. 255-260.
44. Biermann, D., Kersting, P., Surmann, T. A general approach to simulating workpiece vibrations during five-axis milling of turbine blades. CIRP Annals. Manufacturing Technology, 201, vol. 59, no. 1, pp. 125-128.
45. Voronov, S.A., Kiselev, I.A., Arshinov, S.V. Metodika primeneniya chislennogo modelirovaniya dinamiki mnog-okoordinatnogo frezerovaniya slozhnoprofil´nykh detaley pri proektirovanii tekhnologicheskogo protsessa. [Application methods of numerical simulation of multi-axis milling dynamics of figurine-shaped detection under process designing.] Vestnik MGTU im. N. E. Baumana, Machine Building, 2012, spec. iss. no. 6, pp. 50-69 (in Russian).
46. Voronov, S.A., Nepochatov, A.V., Kiselev, I.A. Kriterii otsenki ustoychivosti protsessa frezerovaniya nezhestkikh detaley. [Estimation criteria of process stability of non-stiff workpieces milling.] Proceedings of Higher Educational Institu-tions. Machine Building, 2011, no. 1 (610) pp. 50-62 (in Russian).
47. Voronov, S., Kiselev, I. Dynamics of flexible detail milling. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 2011, vol. 225, no. 3, pp. 1177-1186.
48. 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).
49. Pontryagin, L.S. Izbrannye trudy. T. 2 [Selecta. Vol. 2.] Moscow: Nauka, 1988, 576 p. (in Russian).
50. Tikhonov, А.N., Vasilyev, A.B., Volosov, V.M. Differentsial´nye uravneniya, soderzhashchie malyy parametr. [Differential equations with a small parameter.] Trudy mezhdunarodnogo simpoziuma po nelineynym kolebaniyam. [Proc. Int. Symposium on nonlinear oscillations.] Kiev: Izd-vo AN USSR, 1963, pp. 56-61 (in Russian).
51. Zakovorotny, V.L., Pham Dinh Tung, Nguen Xuan Chiem. Modelirovanie deformatsionnykh smeshcheniy instru-menta otnositel´no zagotovki pri tochenii. [Modeling of tool deformation offsetting to workpiece in turning.] Vestnik of DSTU, 2010, vol. 7 (50), pp. 1005-1015 (in Russian).
52. Danzhelo, R. Lineynye sistemy s peremennymi parametrami. [Linear systems with variable parameters.] Moscow: Mashinostroenie, 1974, 287 p. (in Russian).
53. Berezkin, Е.N. Lektsii po teoreticheskoy mekhanike. [Lectures on theoretical mechanics.] Moscow: Izd-vo MGU, 1968, 279 p. (in Russian).
54. 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).
