Abstract
On the basis of the classical Runge-Kutta method and the complete discretization method, a Runge-Kutta-based complete discretization method (RKCDM) is proposed in the paper to predict the chatter stability of milling process, in which the regenerative effect is taken into consideration. Firstly, the dynamics model of milling process is simplified as a 2-DOF vibration system in the two orthogonal directions, which can be expressed as coefficient-varying periodic differential equations with a single time delay. Then, all parts of the delay differential equation (DDE), including delay term, time-domain term, parameter matrices, and most of all the differential terms are discretized using the classical fourth-order Runge-Kutta iteration method to replace the direct integration scheme used in the classical semi-discretization method (C-SDM) and the classical complete discretization scheme with the Euler method (C-CDSEM), which can simplify the complexity of the discretization iteration formula greatly. Lastly, the Floquet theory is adopted to predict the stability of milling process by judging the eigenvalues of the state transition matrix corresponding to certain cutting conditions. Comparing RKCDM with C-SDM and C-CDSEM, the numerical simulation results show that RKCDM has the highest convergence rate, computation accuracy, and computation efficiency. As dichotomy search rather than sequential search is used in the algorithm, the calculation time for obtaining the stability lobe diagrams (SLDs) is greatly reduced. As a result, it is practical to determine the optimal chatter-free cutting conditions for milling operation in shop floor applications.
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References
Sridhar R, Hohn RE, Long GW (1968) A stability algorithm for the general milling process: contribution to machine tool chatter research. J Eng Ind 90:330–334
Tobias SA (1965) Machine tool vibration. Blackie, Glazgow
Altintas Y, Budak E (1995) Analytical prediction of stability lobes in milling. CIRP Ann 44(1):357–362
Ismail F, Soliman E (1997) A new method for the identification of stability lobes in machining. Int J Mach Tools Manuf 37(6):763–774
Altintas Y, Engin S, Budak E (1999) Analytical stability prediction and design of variable pitch cutters. J Manuf Sci E-T ASME 121(2):173–178
Altintas Y, Budak E (1995) Analytical prediction of stability lobes in milling. CIRP ANN-Manuf Techn 44(7):357–362
Li ZQ, Liu Q, Ming XZ, Wang X, Dong YF (2014) Cutting force prediction and analytical solution of regenerative chatter stability for helical milling operation. Int J Adv Manuf Technol 73(1-4):433–442
Budak E, Altintas Y (1998) Analytical prediction of chatter stability in milling—part I: general formulation. J Dyn Syst-T ASME 120(1):22–30
Merdol SD, Altintas Y (2004) Multi frequency solution of chatter stability for low immersion milling. J Manuf Sci E-T ASME 126(3):459–466
Bayly PV, Mann BP, Schmitz TL, Peters DA, Stepan G, Insperger T (2002) Effects of radial immersion and cutting direction on chatter instability in end milling. American Society of Mechanical Engineer, Manufacturing Engineering Division, MED, pp. 351–363
Bayly PV, Mann BP, Schmitz TL, Peters DA, Stepan G, Insperger T (2002) Effects of radial immersion and cutting direction on chatter instability in end-milling. Am Soc Mech Eng 13:351–363
Insperger T, Stepan G (2002) Semi-discretization method for delayed systems. Int J Nume Meth Eng 55(5):503–518
Insperger T, Stepan G (2004) Updated semi-discretization method for periodic delay-differential equations with discrete delay. Int J Nume Meth Eng 61(1):117–141
Ding Y, Zhu LM, Zhang XJ, Ding H (2010) A full-discretization method for prediction of milling stability. Int J Mach Tools Manuf 50(5):502–509
Ding Y, Zhu LM, Zhang XJ, Ding H (2011) Numerical integration method for prediction of milling stability. J Manuf Sci Eng 133(3):255–267
Li ZQ, Liu Q (2008) Solution and analysis of chatter stability for end milling in the time-domain. Chin J Aeronaut 21(2):169–178
Li M, Zhang G, Huang Y (2013) Complete discretization scheme for milling stability prediction. Nonlinear Dynam 71(1-2):187–199
Niu JB, Ding Y, Zhu LM, Ding H (2013) Runge–Kutta methods for a semi-analytical prediction of milling stability. Nonlinear Dynam 76(1):289–304
Tamas I (2010) Full-discretization and semi-discretization for milling stability prediction: some comments. Int J Mach Tools Manuf 50(7):658–662
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Li, Z., Yang, Z., Peng, Y. et al. Prediction of chatter stability for milling process using Runge-Kutta-based complete discretization method. Int J Adv Manuf Technol 86, 943–952 (2016). https://doi.org/10.1007/s00170-015-8207-7
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DOI: https://doi.org/10.1007/s00170-015-8207-7