High-order full-discretization methods for milling stability prediction by interpolating the delay term of time-delayed differential equations

  • Kai Zhou
  • Pingfa Feng
  • Chao Xu
  • Jianfu Zhang
  • Zhijun Wu


The full-discretization method (FDM) has been proven effective for prediction on the regenerative chatter in many papers. However, the previous studies towards FDM just focused on high-order Lagrange interpolation for state term of time-delayed differential equations (DDEs), which formulates the dynamics model in milling process. It is well known that the discretization error caused by the delay term of DDEs would transmit to the state term inevitably; higher-order Lagrange interpolation for delay term is thus vital. In this paper, second-order, third-order, and fourth-order full-discretization methods using Lagrange interpolation for the delay term of DDEs (DFDMs) were firstly proposed. Then, influence on the accuracy, computational efficiency, and convergence rate of the proposed DFDMs was discussed in detail as the change of interpolation order. It was found that rise in accuracy and convergence rate of the proposed DFDM nearly stopped when the interpolation order for delay term was up to fourth order. Next, some researches on 2-degree-of-freedom (2-DOF) of dynamic system was studied and the results show that the proposed method using fourth-order Lagrange interpolation for the delay term of DDEs (4th DFDM) was effective. Finally, this paper verified the 4th DFDM by experiment and analyzed the prediction error of 4th DFDM, which may be caused by the modeling process of cutting force. The proposed DFDMs are developed to find a better method, which can update the existing FDM and make regenerative chatter’s prediction more efficient and precise.


Milling Machine tool dynamics Chatter prediction Full-discretization method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Koenigsberger F, Tlusty J (1970) Machine tool structures. Pergamon Press, LondonGoogle Scholar
  2. 2.
    Sridhar R, Hohn RE, Long GW (1968) A stability algorithm for the general milling process. ASME J Eng Ind 90(2):330–334. doi: 10.1115/1.3604637 CrossRefGoogle Scholar
  3. 3.
    Altintas Y, Budak E (1995) Analytical prediction of stability lobes in milling. CIRP Ann Manuf Technol 44(1):357–362. doi: 10.1016/S0007-8506(07)62342-7 CrossRefGoogle Scholar
  4. 4.
    Tang WX, Song QH, Yu SQ, Sun SS, Li BB, Du B, Ai X (2009) Prediction of chatter stability in high-speed finishing end milling considering multi-mode dynamics. J Mater Process Technol 209(5):2585–2591. doi: 10.1016/j.jmatprotec.2008.06.003 CrossRefGoogle Scholar
  5. 5.
    Gradišek J, Kalveram M, Insperger T, Weinert K, Stépán G, Govekar E, Grabec I (2005) On stability prediction for milling. Int J Mach Tools Manuf 45(7–8):769–781. doi: 10.1016/j.ijmachtools.2004.11.015 CrossRefGoogle Scholar
  6. 6.
    Budak E, Altintas Y (1998) Analytical prediction of chatter stability in milling—part I: general formulation. J Dyn Syst Meas Control 120(1):22–30. doi: 10.1115/1.2801317 CrossRefGoogle Scholar
  7. 7.
    Insperger T, Stépán G (2002) Semi-discretization method for delayed systems. Int J Numer Methods Eng 55:503–518. doi: 10.1002/nme.505 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Insperger T, Stépán G (2004) Updated semi-discretization method for periodic delay-differential equations with discrete delay. Int J Numer Methods Eng 61:117–141. doi: 10.1002/nme.1061 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Insperger T, Stépán G, Turi J (2008) On the higher-order semi-discretizations for periodic delayed systems. J Sound Vib 313(1–2):334–341. doi: 10.1016/j.jsv.2007.11.040 CrossRefGoogle Scholar
  10. 10.
    Ahmadi K, Ismail F (2012) Modeling chatter in peripheral milling using the semi discretization method. CIRP J Manuf Sci Technol 5(2):77–86. doi: 10.1016/j.cirpj.2012.03.001 CrossRefGoogle Scholar
  11. 11.
    Ding Y, Zhu LM, Zhang XJ, Ding H (2011) Numerical integration method for prediction of milling stability. J Manuf Sci Eng 133. doi: 10.1115/1.4004136
  12. 12.
    Liang XG, Yao ZQ, Luo L, Hu J (2013) An improved numerical integration method for predicting milling stability with varying time delay. Int J Adv Manuf Technol 68(9–12):1967–1976. doi: 10.1007/s00170-013-4813-4 CrossRefGoogle Scholar
  13. 13.
    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. doi: 10.1016/j.ijmachtools.2010.01.003 CrossRefGoogle Scholar
  14. 14.
    Ding Y, Zhu LM, Zhang XJ, Ding H (2010) Second-order full-discretization method for milling stability prediction. Int J Mach Tools Manuf 50(10):926–932. doi: 10.1016/j.ijmachtools.2010.05.005 CrossRefGoogle Scholar
  15. 15.
    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. doi: 10.1016/j.ijmachtools.2012.05.001 CrossRefGoogle Scholar
  16. 16.
    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. doi: 10.1016/j.ijmachtools.2015.02.007 CrossRefGoogle Scholar
  17. 17.
    Liu YL, Fischer A, Eberhard P, Wu BH (2015) A high-order full-discretization method using Hermite interpolation for periodic time-delayed differential equations. Acta Mech Sinica 31(3):406–415. doi: 10.1007/s10409-015-0397-6 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Tang XW, Peng FY, Yan R, Gong YH, Li YT, Jiang LL (2016) Accurate and efficient prediction of milling stability with updated full-discretization method. Int J Adv Manuf Technol 88(9–12):2357–2368. doi: 10.1007/s00170-016-8923-7 Google Scholar
  19. 19.
    Li ZQ, Yang ZK, Peng YR, Zhu F, Ming XZ (2015) Prediction of chatter stability for milling process using Runge-Kutta-based complete discretization method. Int J Adv Manuf Technol 86(1–4):943–952. doi: 10.1007/s00170-015-8207-7 Google Scholar
  20. 20.
    Xie QZ (2015) Milling stability prediction using an improved complete discretization method. Int J Adv Manuf Technol 83(5–8):815–821. doi: 10.1007/s00170-015-7626-9 Google Scholar
  21. 21.
    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. Paper presented at the ASME 2002 International Mechanical Engineering Congress and Exposition, New OrleansGoogle Scholar
  22. 22.
    Kuang J, Cong Y (2005) Stability of numerical methods for delay differential equations. Science Press, BeijingGoogle Scholar
  23. 23.
    Insperger T (2010) Full-discretization and semi-discretization for milling stability prediction: some comments. Int J Mach Tools Manuf 50(7):658–662. doi: 10.1016/j.ijmachtools.2010.03.010 MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhao MX, Balachandran B (2001) Dynamics and stability of milling process. Int J Solids Struct 38:2233–2248. doi: 10.1016/S0020-7683(00)00164-5 CrossRefMATHGoogle Scholar
  25. 25.
    Long XH, Balachandran B (2010) Stability of up-milling and down-milling operations with variable spindle speed. J Vib Control 16(7–8):1151–1168. doi: 10.1177/1077546309341131 MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Yue C, Liu XL, Liang SY (2016) A model for predicting chatter stability considering contact characteristic between milling cutter and workpiece. Int J Adv Manuf Technol 88(5–8):2345–2354. doi: 10.1007/s00170-016-8953-1 Google Scholar
  27. 27.
    Balachandran B (2001) Nonlinear dynamics of milling process. Philos Trans R Soc Lond A 359(1781):793–819. doi: 10.1098/rsta.2000.0755 CrossRefMATHGoogle Scholar
  28. 28.
    Balachandran B, Gilsinn D (2005) Non-linear oscillations of milling. Math Comput Model Dyn Syst 11(3):273–290. doi: 10.1080/13873950500076479 CrossRefMATHGoogle Scholar
  29. 29.
    Long XH, Balachandran B (2007) Stability analysis for milling process. Nonlinear Dyn 49(3):349–359. doi: 10.1007/s11071-006-9127-8 CrossRefMATHGoogle Scholar
  30. 30.
    Moradi H, Vossoughi G, Movahhedy MR (2014) Bifurcation analysis of nonlinear milling process with tool wear and process damping: sub-harmonic resonance under regenerative chatter. Int J Mech Sci 85:1–19. doi: 10.1016/j.ijmecsci.2014.04.011 CrossRefGoogle Scholar
  31. 31.
    Tyler CT, Troutman JR, Schmitz TL (2016) A coupled dynamics, multiple degree of freedom process damping model, part 2: milling. Precis Eng 46:73–80. doi: 10.1016/j.precisioneng.2016.03.018 CrossRefGoogle Scholar
  32. 32.
    Wang JJ, Uhlmann E, Oberschmidt D, Sung CF, Perfilov I (2016) Critical depth of cut and asymptotic spindle speed for chatter in micro milling with process damping. CIRP Ann Manuf Technol 65(1):113–116. doi: 10.1016/j.cirp.2016.04.088 CrossRefGoogle Scholar
  33. 33.
    Tyler CT, Troutman J, Schmitz TL (2015) Radial depth of cut stability lobe diagrams with process damping effects. Precis Eng 40:318–324. doi: 10.1016/j.precisioneng.2014.11.004 CrossRefGoogle Scholar
  34. 34.
    Rahnama R, Sajjadi M, Park SS (2009) Chatter suppression in micro end milling with process damping. J Mater Process Technol 209(17):5766–5776. doi: 10.1016/j.jmatprotec.2009.06.009 CrossRefGoogle Scholar
  35. 35.
    Li ZY, Sun YW, Guo DM (2016) Chatter prediction utilizing stability lobes with process damping in finish milling of titanium alloy thin-walled workpiece. Int J Adv Manuf Technol 89(9–12):2663–2674. doi: 10.1007/s00170-016-9834-3 Google Scholar
  36. 36.
    Budak E, Tunç LT, Alan S, Özgüven HN (2012) Prediction of workpiece dynamics and its effects on chatter stability in milling. CIRP Ann Manuf Technol 61(1):339–342. doi: 10.1016/j.cirp.2012.03.144 CrossRefGoogle Scholar
  37. 37.
    Liu YL, Wu BH, Ma JJ, Zhang DH (2016) Chatter identification of the milling process considering dynamics of the thin-walled workpiece. Int J Adv Manuf Technol 89(5). doi: 10.1007/s00170-016-9190-3

Copyright information

© Springer-Verlag London Ltd. 2017

Authors and Affiliations

  • Kai Zhou
    • 1
  • Pingfa Feng
    • 1
    • 2
  • Chao Xu
    • 1
  • Jianfu Zhang
    • 2
  • Zhijun Wu
    • 2
  1. 1.Division of Advanced Manufacturing, Graduate School at ShenzhenTsinghua UniversityShenzhenChina
  2. 2.Department Mechanical EngineeringTsinghua UniversityBeijingChina

Personalised recommendations