The work objective is to consider the stability problem of steady-state paths of the elastic deformational tool displacement under the longitudinal end milling. The authors analyze the case of high speed cutting in contrast to the previously discussed stability problems that analyze the case of slow movements for which the system parameters can be considered frozen in the equations in variations relative to the stationary path. In this case, the stability analysis must consider the linearized system in variations with periodically varying coefficients. With speeding-up the tool rotation in many cases there is a parametric self-excitation of oscillations. Therefore, the main attention is paid to studying the parametric excitation conditions of a dynamic endmilling system. It is shown that the parametric excitation condition is affected by the technological cutting modes, both the tool rotation frequency and the tool geometry which determines the matrix angular coeffi-cients of the cutting forces orientation. Examples of stability areas depending on changes in the system settings are given.
endmilling process, stationary trajectories, periodi-cally-varying parameters, stability, parametric excitation.
При изучении динамики процесса резания, в частности, фрезерования, рассматривают упругие подсистемы со стороны режущего инструмента и обрабатываемой детали, которые взаимодействуют между собой через динамическую связь, формируемую процессом обработки [1–9]. В свою очередь, динамическая связь характери-зует модель сил резания, представленную в координатах состояния системы и технологических режимах [10, 11].
1. 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).
2. Zakovorotny, V.L., Lukyanov, A.D., Nguen Xuan Chiem, Pham Dinh Tung. Sinergeticheskiy sistemnyy sintez up-ravlyaemoy 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).
3. Zakovorotny, V. L., Lukyanov, A.D. The Problems of Control of the Evolution of the Dynamic System Interacting with the Medium. Int. J. of Mechanical Engineering and Automation, 2014, vol. 1, no. 5, pp. 271 - 285.
4. Tlusty, I. Avtokolebaniya v metallorezhushchikh stankakh. [Self-oscillations in machine tools.] transl. from Czech. Moscow: Mashgiz, 1956, 395 p. (in Russian).
5. Tlusty, I., Polacek, M., Danek, О., Spacek, L. Selbsterregte Schwingungen an Werkzeugmaschinen. Veb Verlag Technik, Berlin, 1962, 320 р.
6. Tobias, S. A. Machine Tool Vibrations. Blackie, London, 1965, 350 р.
7. Kudinov, V.А. Dinamika stankov. [Machine dynamics.] Moscow: Mashinostroenie, 1967, 359 p. (in Russian).
8. 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).
9. Weiz, 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 non-rigid workpieces.] STIN, 1999, no. 6, pp. 9-13 (in Russian).
10. 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).
11. 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).
12. Sokolovskiy, А.P. Vibratsii pri rabote na metallorezhushchikh stankakh. Issledovanie kolebaniy pri rezanii metal-lov. [Vibrations at work on machine tools. Study on vibrations under metal cutting]. Moscow: Mashgiz, 1958, 158 p. (in Rus-sian).
13. Murashkin, L.S., Murashkin, S.L. Prikladnaya nelineynaya mekhanika stankov. [Applied nonlinear machine me-chanics.] Leningrad: Mashinostroenie, 1977, 192 p. (in Russian).
14. 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 (in Russian).
15. Zakovorotny, V.L., Lukyanov A.D., Bykador V.S. Dynamic self-organization in cutting process. Proceedings of the 6-th International Conference on Mechanics and Materials in Design. Delgada/Azores, 26-30 July, 2015, pp. 119 - 134.
16. 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).
17. 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).
18. Stepan, G. Delay-differential equation models for machine tool chatter. Moon, F.C., ed. John Wiley, NY, 1998, pp. 165-192.
19. Stepan, G., Insperge, T., and Szalai, R. Delay, Parametric excitation, and the nonlinear dynamics of cutting pro-cesses. Int. J. of Bifurcation and Chaos, 2005, vol. 15, no. 9, pp. 2783 - 2798.
20. 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.
21. Altintas, Y., Budak, E. Analytical prediction of stability lobes in milling. Annals of the CIRP, 1995, vol. 44, no., 1, pp. 357 - 362.
22. 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.
23. Minis, I., Yanushevsky, 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. 24. Insperger, T., Stepan, G. Stability of the milling process. Periodical Polytechnic-Mechanical Engineering, 2000, vol. 44, no. 1, pp. 47 - 57. 25. Budak, E., Altintas, Y. Analytical prediction of chatter stability in milling. Part I: General formulation. ASME J. Dyn. Syst., Meas., Control. 1998, vol. 120, no. 6(1), pp. 22 - 30. 26. Budak, E., Altintas, Y. Analytical prediction of chatter stability conditions for multi-degree of systems in milling. Part II: Applications. ASME J. Dyn. Syst., Meas., Control. 1998, vol. 120, no. 6 (1), pp. 31 - 36. 27. Merdol, D., Altintas, Y. Multi-frequency solution of chatter stability for low immersion milling. ASME J. Manuf. Sci. Eng., 2004, vol. 126, no. 3, pp. 459 - 466.
24. Insperger, T., Mann, B., Stepan, G., Bayly, P.V. Stability of up-milling and down-milling. Part 1: Alternative ana-lytical methods. Int. J. of Machine Tools and Manufacture, 2003, vol. 43, no. 1, pp. 25 - 34. 29. Kline, W.A., Devor, R. E., Shareef I. A. The prediction of surface accuracy in end milling. ASME J. Eng. Ind., 1982, vol. 104, no. 5, pp. 272 - 278. 30. Elbestawi, M. A., Sagherian, R. Dynamic modeling for the prediction of surface errors in milling of thin-walled sections. Theor. Comput. Fluid Dyn., 1991, vol. 25, no. 2, pp. 215 - 228. 31. Campomanes, M.L., Altintas, Y. An improved time domain simulation for dynamic milling at small radial immer-sions. Trans. ASME. J. of Manuf. Sci. and Eng., 2003, vol. 125, no. 3, pp. 416 - 425. 32. Paris, H., Peigne, G., Mayer, R. Surface shape prediction in high-speed milling. Int. J. of Machine Tools and Man-ufacture, 2004, vol. 44, no. 15, pp. 1567 - 1576. 33. 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. 34. Ozturk, E., Budak, E. Modeling of 5-axis milling processes. Machining Science and Technology, 2007, vol. 11, no. 3, pp. 287 - 311. 35. Budak, E., Ozturk, E., Tunc, L.T. Modeling and simulation of 5-axis milling processes. Annals of CIRP. Manufac-turing Technology, 2009, vol. 58, no. 1, pp. 347 - 350.
25. Bravo, U., Altuzarra, O., Lopez de Lacalle, L.N., Sanchez, J.A., Campa, F.J. Stability limits of milling considering the flexibility of the workpiece and the machine. Int. J. of Machine Tools and Manufacture, 2005, vol. 45, pp. 1669 - 1680.
26. Weinert, K., Kersting, P., Surmann, T., Biermann D. Modeling regenerative workpiece vibrations in five-axis mill-ing. Prod. Eng. Res. Devel., 2008, no. 2, pp. 255 - 260. 38. 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. 39. Voronov, S.А, Kiselev, I.A., Arshinov, S.V. Metodika primeneniya chislennogo modelirovaniya dinamiki mnog-okoordinatnogo frezerovaniya slozhnoprofil´nykh detaley pri proektirovanii tekhnologicheskogo protsessa. [Application meth-ods of numerical modeling of the dynamics of multi-axis milling of complex profile parts in technological process design.] Vestnik MSTU. Series Machine Building. 2012, spec. iss. no. 6, pp. 50-69 (in Russian).
27. Voronov, S.А, Nepochatov, A.V., Kiselev, I.A. Kriterii otsenki ustoychivosti protsessa frezerovaniya nezhestkikh detaley. [Stability criteria evaluation process of milling of non-rigid parts.] Proceedings of Higher Educational Institutions. Маchine Building, 2011, no. 1 (610), pp. 50-62 (in Russian).
28. Zakovorotny, V.L., Gubanova, A.A., Lukyanov, A.D. Sinergeticheskiy podkhod pri izuchenii ustoychivosti formoobrazuyushchikh traektoriy poputnogo frezerovaniya bokovymi granyami kontsevykh frez (sluchay maloy skorosti re-zaniya). [Synergetic approach to studying stability of form-building trajectories of climb milling by side edges of endmills (low cutting speed case).] Vestnik of DSTU, 2016, no. 1 (85), pp. 50-62 (in Russian).
29. Lyapunov, А М. Obshchaya zadacha ob ustoychivosti dvizheniya. [A general problem on motion stability.] Mos-cow: Gostekhizdat, 1950, 167 p. (in Russian).
30. D’Angelo, G. Lineynye sistemy s peremennymi parametrami. [Linear systems with variable parameters.] Moscow: Mashinostroenie, 1974, 287 p. (in Russian).
31. 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).
