Abstract
Chatter vibration in milling has been one crucial factor hindering the realization of high-performance machining. The corresponding stability analysis is of great significance for obtaining chatter-free machining parameters. Based on the predictor-corrector scheme, this paper develops an accurate and efficient holistic-discretization method for the stability analysis of milling processes. According to the system state equation, the period of the time-periodic coefficient matrix is divided into two time intervals. The forced vibration time period is then equidistantly discretized. Working as a holistic unit, the time-periodic parameter matrix, the state term, and the delay term are approximated over two different subintervals by the second-order Lagrange interpolations. Finally, the Floquet transition matrix can be constructed by taking advantage of the predictor-corrector scheme, and the milling stability can be semi-analytically determined by utilizing the Floquet theory. The computational accuracy of the proposed method is analyzed theoretically and illustrated by making comparisons with the first-order semi-discretization method (1st SDM), the second-order, and the third-order updated full-discretization methods (2nd UFDM and 3rd UFDM). The stability lobes for two benchmark milling models and the computational efficiency of these methods are presented to further verify the effectiveness of the proposed method. Theoretical analysis and numerical results validate that the proposed predictor-corrector-based holistic-discretization method achieves both high computational accuracy and efficiency for milling stability analysis. In conclusion, the proposed semi-analytical algorithm has a high potential for industrial applications.
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Munoa J, Beudaert X, Dombovari Z, Altintas Y, Budak E, Brecher C, Stepan G (2016) Chatter suppression techniques in metal cutting. CIRP Ann Manuf Techn 65(2):785–808
Quintana G, Stepan CJ (2011) Chatter in machining processes: a review. Int J Mach Tools Manuf 51(5):363–376
Altintas Y (2012) Manufacturing automation: metal cutting, mechanics, machine tool vibrations, and CNC design. Cambridge University Press, New York
Kilic ZM, Altintas Y (2016) Generalized mechanics and dynamics of metal cutting operations for unified simulations. Int J Mach Tools Manuf 104:1–13
Kilic ZM, Altintas Y (2016) Generalized mechanics and dynamics of metal cutting operations for unified simulations. Int J Mach Tools Manuf 104:14–25
Lauro CH, Brandão LC, Baldo D, Reis RA, Davim JP (2014) Monitoring and processing signal applied in machining processes—a review. Measurement 58:73–86
Siddhpura M, Paurobally R (2012) A review of chatter vibration research in turning. Int J Mach Tools Manuf 61:27–47
Wiercigroch M, Budak E (2001) Sources of nonlinearities, chatter generation and suppression in metal cutting. Philos Trans R Soc A 359(1781):663–693
Altintas Y, Weck M (2004) Chatter stability of metal cutting and grinding. CIRP Ann 53(2):619–642
Altintas Y, Stepan G, Merdol D, Dombovari Z (2008) Chatter stability of milling in frequency and discrete time domain. CIRP J Manuf Sci Technol 1(1):35–44
Ding H, Ding Y, Zhu LM (2012) On time-domain methods for milling stability analysis. Chinese Sci Bull 57(33):4336–4345
Smith S, Tlusty J (1993) Efficient simulation programs for chatter in milling. CI RP Ann Manuf Techn 42:463–466
Davies MA, Pratt JR, Dutterer B, Burns TJ (2000) Stability of low radial immersion milling. CIRP Ann Manuf Techn 49:37–40
Davies MA, Pratt JR, Dutterer B, Burns TJ (2002) Stability prediction for low radial immersion milling. J Manuf Sci E-T ASME 124:217–225
Campomanes ML, Altintas Y (2003) An improved time domain simulation for dynamic milling at small radial immersions. J Manuf Sci E-T ASME 125:416–422
Urbikain G, Olvera D, López de Lacalle LN (2017) Stability contour maps with barrel cutters considering the tool orientation. Int J Adv Manuf Technol 89(9–12):2491–2501
Altintas Y, Budak E (1995) Analytical prediction of stability lobes in milling. CIRP Ann 44(1):357–362
Budak E, Altintas Y (1998) Analytical prediction of chatter stability in milling—part II: application of the general formulation to common milling systems. ASME J Dyn Syst Meas Control 120(1):31–36
Merdol SD, Altintas Y (2004) Multi frequency solution of chatter stability for low immersion milling. J Manuf Sci Eng 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. In: Proceedings of the international mechanical engineers conference and exposition. New Orleans, Paper No.IMECE2002-39116
Butcher EA, Ma H, Bueler E, Averina V (2004) Stability of time-periodic delay-differential equations via Chebyshev polynomials. Int J Numer Methods Eng 59(7):895–992
Butcher EA, Bobrenkov OA, Bueler E, Nindujarla P (2009) Analysis of milling stability by the Chebyshev collocation method: algorithm and optimal stable immersion levels. J Comput Nonlin Dyn 4:031003
Insperger T, Stepan G (2002) Semi-discretization method for delayed systems. Int J Numer Methods Biomed Eng 55(5):503–518
Insperger T, Stepan G (2004) Updated semi-discretization method for periodic delay-differential equations with discrete delay. Int J Numer Methods Biomed Eng 61(1):117–141
Insperger T, Stepan G, Turi J (2008) On the higher-order semidiscretizations for periodic delayed systems. J Sound Vib 313(1):334–341
Long XH, Balachandran B, Mann BP (2007) Dynamics of milling processes with variable time delays. Nonlinear Dyn 47:49–63
Seguy S, Insperger T, Arnaud L, Dessein G, Peign G (2010) On the stability of high-speed milling with spindle speed variation. Int J Adv Manuf Technol 48:883–895
Wan M, Zhang WH, Dang JW, Yang Y (2010) A unified stability prediction method for milling process with multiple delays. Int J Mach Tools Manuf 50:29–41
Ahmadi K, Ismail F (2012) Stability lobes in milling including process damping and utilizing multi-frequency and semi-discretization methods. Int J Mach Tools Manuf 54:46–54
Insperger T, Stepan G (2011) Semi-discretization for time-delay systems: stability and engineering applications. Springer, New York
Ding Y, Zhu LM, Zhang XJ, Ding H (2010a) 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 (2010b) Second-order full-discretization method for milling stability prediction. Int J Mach Tools Manuf 50:926–932
Quo Q, Sun YW, Jiang Y (2012) On the accurate calculation of milling stability limits using third-order full-discretization method. Int J Mach Tools Manuf 62:61–66
Ozoegwu CG, Omenyi SN (2016) Third-order least squares modelling of milling state term for improved computation of stability boundaries. Prod Manuf Res 4(1):46–64
Ozoegwu CG, Omenyi SN, Ofochebe SM (2015) Hyper-third order full-discretization methods in milling stability prediction. Int J Mach Tools Manuf 92:1–9
Ozoegwu CG (2014) Least squares approximated stability boundaries of milling process. Int J Mach Tools Manuf 79:24–30
Ding Y, Zhu LM, Zhang XJ, Ding H (2011) Numerical integration method for prediction of milling stability. J Manuf Sci Eng 133(3):031005
Ding Y, Niu JB, Zhu LM, Ding H (2016) Numerical integration method for stability analysis of milling with variable spindle speeds. ASME. J Vib Acoust 138(1):011010
Li MZ, Zhang GJ, Huang Y (2013) Complete discretization scheme for milling stability prediction. Nonlinear Dyn 71:187–199
Xie QZ (2016) Milling stability prediction using an improved complete discretization method. Int J Adv Manuf Technol 83(5–8):815–821
Li ZQ, Yang ZK, Peng YR, Zhu F, Ming XZ (2016) Prediction of chatter stability for milling process using Runge-Kutta-based complete discretization method. Int J Adv Manuf Technol 86(1–4):943–952
Ding Y, Zhu LM, Zhang XJ, Ding H (2013) Stability analysis of milling via the differential quadrature method. J Manuf Sci Eng 135:044502
Niu JB, Ding Y, Zhu LM, Ding H (2014) Runge–Kutta methods for a semi-analytical prediction of milling stability. Nonlinear Dyn 76(1):289–304
Olvera D, Elías-Zúñiga A, Martínez-Alfaro H, López de Lacalle LN, Rodríguez CA, Campa FJ (2014) Determination of the stability lobes in milling operations based on homotopy and simulated annealing techniques. Mechatronics 24:177–185
Compeán FI, Olvera D, Campa FJ, López de Lacalle LN, Elías-Zúñiga A, Rodríguez CA (2012) Characterization and stability analysis of a multi variable milling tool by the enhanced multistage homotopy perturbation method. Int J Mach Tools Manuf 57:27–33
Zhang Z, Li HG, Meng G, Liu C (2015) A novel approach for the prediction of the milling stability based on the Simpson method. Int J Mach Tools Manuf 99:43–47
Qin CJ, Tao JF, Li L, Liu CL (2017) An Adams-Moulton-based method for stability prediction of milling processes. Int J Adv Manuf Technol 89(9–12):3049–3058
Tao JF, Qin CJ, Liu CL (2017) Milling stability prediction with multiple delays via the extended Adams-Moulton-based method. Math Probl Eng 2017:1–15. https://doi.org/10.1155/2017/7898369
Qin CJ, Tao JF, Liu CL (2017) Stability analysis for milling operations using an Adams-Simpson-based method. Int J Adv Manuf Technol 92(1–4):969–979
Zhang XJ, Xiong CH, Ding Y, Ding H (2016) Prediction of chatter stability in high speed milling using the numerical differentiation method. Int J Adv Manuf Technol 89:2535–2544. https://doi.org/10.1007/s00170-016-8708-z
Lu YA, Ding Y, Peng ZK, Chen ZZC, Zhu LM (2016) A spline-based method for stability analysis of milling processes. Int J Adv Manuf Technol 89:2571–2586. https://doi.org/10.1007/s00170-016-9757-z
Tang X, Peng F, Yan R, Gong Y, Li Y, Jiang L (2017) Accurate and efficient prediction of milling stability with updated full-discretization method. Int J Adv Manuf Technol 88(9–12):2357–2368
Yan Z, Wang X, Liu Z, Wang D, Jiao L, Ji Y (2017) Third-order updated full-discretization method for milling stability prediction. Int J Adv Manuf Technol 92(5–8):2299–2309
Funding
This work was partially supported by the project of Shanghai Science and Technology Commission (Grant No. 17511109203), the National Key Research and Development Program of China (Grant No. 2017YFB1302601), and the Innovation Fund of National Business Aircraft Manufacturing Engineering Technology Research Center (Grant No. SAMC14-JS-15-046).
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Qin, C., Tao, J. & Liu, C. A predictor-corrector-based holistic-discretization method for accurate and efficient milling stability analysis. Int J Adv Manuf Technol 96, 2043–2054 (2018). https://doi.org/10.1007/s00170-018-1727-1
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DOI: https://doi.org/10.1007/s00170-018-1727-1