Abstract
The attractive sets formed in the vicinity of equilibrium (asymptotically stable orbital limiting cycles, two-dimensional invariant tori, and chaotic attractors) are analyzed. The bifurcations of the system in parameter space and the space of control parameters are considered. General laws of stability loss of the equilibrium in a cutting system are outlined.
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References
Nicolis, G. and Prigogine, I., Exploring Complexity: An Introduction, New York: W.H. Freeman, 1989.
Haken, H., Information and Self-Organization. A Macroscopic Approach to Complex Systems, Berlin: Springer-Verlag, 1988.
Sinergetika i problemy teorii upravleniya (Synergetic and Problems of Management Theory), Kolesnikov, A.A., Ed., Moscow: Fizmatlit, 2004.
Tlustý, J., Samobuzené Kmity v Obrábecích Strojích, Praha: Cesk. Akad. Ved., 1954.
Tlustý, J., Polacek, A., Danek, C., and Spacek, J., Selbsterregte Schwingungen an Werkzeugmaschinen, Berlin: VEB Verlag Technik, 1962.
Tlusty, J., Manufacturing Processes and Equipment, Upper Saddle River, NJ: Prentice Hall, 2000.
Tobias, S.A., Machine Tool Vibrations, London: Blackie, 1965.
Kudinov, V.A., Dinamika stankov (Dynamics of Machines), Moscow: Mashinostroenie, 1967.
El’yasberg, M.E., Avtokolebaniya metallorezhushchikh stankov: teoriya i praktika (Auto-Oscillations of Machine Tools: Theory and Practice), St. Petersburg: Osoboe Konstr. Byuro Stankostr., 1993.
Veits, V.L. and Vastil’kov, D.V., Dynamics, modeling, and quality maintenance at the machine treatment of small hard billets, Stanki Instrum., 1999, no. 6, pp. 9–13.
Zakovorotnyj, V.L. and Flek, M.B., Dinamika protsessa rezaniya. Sinergeticheskii podkhod (Dynamics of Cutting Process: Synergetic Approach), Rostov-on-Don: Terra, 2006.
Zakovorotnyj, V.L., Fam Din Tung, and Nguen Suan Tiem, Mathematical modeling and parametric identification of dynamic properties of the tool and billet subsystem, Izv. Vyssh. Uchebn. Zaved., Sev.-Kavk. Reg., Tekh. Nauki, 2011, no. 2, pp. 38–46.
Stepan, G., Delay-differential equation models for machine tool chatter, in Nonlinear Dynamics of Material Processing and Manufacturing, Moon, F.C., Ed., New York: Wiley, 1998, pp. 165–192.
Stepan, G., Insperge, T., and Szalai, R., Delay, parametric excitation, and the nonlinear dynamics of cutting processes, Int. J. Bifurcation Chaos Appl. Sci. Eng., 2005, vol. 15, no. 9, pp. 2783–2798.
Zakovorotnyj, V.L., Luk’yanov, A.D., Nguen Dong An, and Fam Din Tung, Sinergeticheskii sistemnyi sintez upravlyaemoi dinamiki metallorezhushchikh stankov s uchetom evolyutsii svyazei (Synergetic System Synthesis of Controlled Dynamics of Metal Cutting Machines, Taking into Account the Evolution of Relations), Rostov-on-Don: Donsk. Gos. Tekh. Univ., 2008.
Zharkov, I.G., Vibratsii pri obrabotke lezviinym instrumentom (Vibrations Occurred during Edge Tool Implementation), Leningrad: Mashinostroenie, 1987.
Kashirin, A.I., Issledovanie vibratsii pri rezanii metallov (Analysis of Vibration at the Metal Cutting), Moscow: Akad. Nauk SSSR, 1944.
Sokolovskii, A.P., Vibratsii pri rabote na metallorezhushchikh stankakh. Issledovanie kolebanii pri rezanii metallov (Vibration at Working on Machines: Analysis of Vibrations in Metal Cutting), Moscow: Mashgiz, 1958, pp. 15–18.
Corpus, W.T. and Endres, W.J., Added stability lobes in machining processes that exhibit periodic time variation. Part 1: An analytical solution, J. Manuf. Sci. Eng., 2004, no. 126, pp. 467–474.
Gouskov, A.M., Voronov, S.A., Paris, H., and Batzer, S.A., Nonlinear dynamics of a machining system with two interdependent delays, Commun. Nonlinear Sci. Numer. Simul., 2002, no. 7, pp. 207–221.
Zakovorotnyj, V.L. and Fam Thy Hyong, Parametric self-excitation of dynamic cutting system, Vestn. Donsk. Gos. Tekh. Univ., 2013, vols. 74–75, nos. 5–6, pp. 97–104.
Mathieu, E., Memoire sur le mouvement vibratoire d'une membrane de forme elliptique, J. Math. Pure Appl., 1968, no. 13, pp. 137–203.
Floquet, M.G., Equations differentielles lineaires a coefficients peridiques, Ann. Sci. Ec. Norm. Super., 1883, no. 12, pp. 47–89.
Litak, G., Chaotic vibrations in a regenerative cutting process, Chaos, Solitons Fractals, 2002, no. 13, pp. 1531–1535.
Grabec, I., Chaotic dynamics of the cutting process, J. Mach. Tools Manuf., 1988, no. 28, pp. 19–50.
Litak, G., Chaotic vibrations in a regenerative cutting process, Chaos, Solitons Fractals, 2002, no. 13, pp. 1531–1535.
Wiercigroch, M., Chaotic vibrations of a simple model of the machine tool-cutting system, ASME J. Vib. Acoust., 1997, no. 119, pp. 468–542.
Wiercigroch, M. and Cheng, A.H.-D., Chaotic and stochastic dynamics of orthogonal metal cutting, Chaos, Solitons Fractals, 1997, no. 8, pp. 715–740.
Kabaldin, Yu.G., Samoorganizatsiya i nelineinaya dinamika v protsessakh treniya i iznashivaniya instrumenta pri rezanii (Self-Organization and Nonlinear Dynamics in the Friction and Wear Processes of Cutting Instrument), Komsomolsk-on-Amur: Komsomolsk-na-Amur. Gos. Tekh. Univ., 2003.
Feigenbaum, M.J., Quantitative universality for a class of nonlinear transformations, J. Stat. Phys., 1978, vol. 19, pp. 25–52.
Murashkin, L.S. and Murashkin, S.L., Prikladnaya nelineinaya mekhanika stankov (Applied Nonlinear Mechanics of Machines), Leningrad: Mashinostroenie, 1977.
Warminski, J., Litak, G., Cartmell, M. P., Khanin, R., and Wiercigroch, M., Approximate analytical solutions for primary chatter in the non-linear metal cutting model, J. Sound Vib., 2003, vol. 259, no. 4, pp. 917–933.
van der Pol, F. and Strutt, M.J.O., On the stability of the solutions of Mathieu's equation, Philos. Mag. J. Sci., 1928, no. 5, pp. 18–38.
Zakovorotnyj, V.L., Fam Din Tung, and Nguen Suan Tiem, Mathematical modeling and parametric identification of dynamic properties of the tool and the billet subsystem at turning, Izv. Vyssh. Uchebn. Zaved., Sev.- Kavk. Reg., Tekh. Nauki, 2011, no. 2, pp. 38–46.
Zakovorotnyj, V.L., Fam Din Tung, and Nguen Suan Tiem, Simulation of deformation shifts of the tool regards to the billet during turning, Vestn. Donsk. Gos. Tekh. Univ., 2010, vol. 10, no. 7, pp. 1005–1015.
Lyapunov, A.M., Obshchaya zadacha ob ustoichivosti dvizheniya. Sobranie sochinenii (General Problem of Motion Stability: Collection of Research Works), Moscow: Akad. Nauk SSSR, 1956, vol. 2.
Merkin, D.R., Vvedenie v teoriyu ustoichivosti dvizheniya (Introduction to the Theory of Motion Stability), Moscow: Nauka, 1987.
Likhadanov, V.M., The effect of forces on the stability of motion, Prikl. Matem. Mekh., 1974, vol. 38, pp. 246–253.
Likhadanov, V.M., Stabilization of potential systems, Prikl. Matem. Mekh., 1975, vol. 39, pp. 53–58.
Zakovorotnyj, V.L., Bordachev, E.V., and Alekseichik, M.I., Dynamic monitoring of the status of the cutting process, Stanki Instrum., 1998, no. 12, pp. 6–12.
Zakovorotnyj, V.L. and Ladnik, I.V., Compilation of information model of the dynamic system of the metalcutting machine for the diagnosis of treatment process, Probl. Mashinostr. Nadezhnosti Mash., 1991, no. 4, pp. 75–79.
Zakovorotnyj, V.L. and Bordachev, E.V., Information support of the dynamic diagnostics system of the cutting tool wear by the example of turning, Probl. Mashinostr. Nadezhnosti Mash., 1995, no. 3, pp. 95–101.
Ostaf’ev, V.A., Antonyuk, V.S., and Tymchik, G.S., Diagnosis of the metal treatment process, Tekhnika, 1991, pp. 54–55.
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Original Russian Text © V.L. Zakovorotnyi, V.S. Bykador, 2015, published in STIN, 2015, No. 12, pp. 18–24.
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Zakovorotnyi, V.L., Bykador, V.S. Cutting-system dynamics. Russ. Engin. Res. 36, 591–598 (2016). https://doi.org/10.3103/S1068798X16070182
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DOI: https://doi.org/10.3103/S1068798X16070182