Skip to main content
SpringerLink
Log in

An updated full-discretization milling stability prediction method based on the higher-order Hermite-Newton interpolation polynomial

  • ORIGINAL ARTICLE
  • Published:
The International Journal of Advanced Manufacturing Technology Aims and scope Submit manuscript

Abstract

Chatter is undesirable self-excited vibrations, which always lead to adverse effects during milling process. Selecting a reasonable combination of cutting parameters is an effective way to avoid chatter. Based on the mathematical model of milling process and the Floquet theory, the stable cutting area can be determined. The stability lobe diagrams (SLD) could be obtained by different interpolation methods. To study the effect of higher order interpolation methods on the accuracy and efficiency of milling stability prediction, the state item, the time-delayed item, and the periodic-coefficient item of the state-space equation are approximated by different higher order interpolation methods, respectively. The calculations show that when the state item is approximated by the third-order Hermite interpolation polynomial, third-order Newton interpolation of the time-delayed item can improve the accuracy of SLD, while higher order interpolation of periodic-coefficient item has negative effect on improving effectiveness and efficiency compared to high-order interpolation of the state item and the time-delayed item. In order to obtain the SLD of milling process more accurately, an updated full-discretization milling stability prediction method which based on the third-order Hermite-Newton interpolation polynomial approximation is proposed in this paper. By dividing the tooth passing period equally into a finite set of time intervals, the third-order Hermite interpolation polynomial and the third-order Newton interpolation polynomial are utilized in each time interval to estimate the state item and the time-delayed item, respectively. The comparison of convergence rate of the critical eigenvalues and the SLD of the proposed method between the existing methods is carried out. The results indicate that the proposed method show a faster convergence rate than that of other methods, and its SLD is more close to the ideal ones with small number of time intervals.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kayhan M, Budak E (2009) An experimental investigation of chatter effects on tool life. Proc Inst Mech Eng PartB: J Eng Manuf 223(11):1455–1463. https://doi.org/10.1243/09544054 JE M1506

    Article  Google Scholar 

  2. Olgac N, Hosek M (1998) A new perspective and analysis for regenerative machine tool chatter. Int J Mach Tools Manuf 38(7):783–798. https://doi.org/10.1016/S0890-6955(97)00062-X

    Article  Google Scholar 

  3. Altintas Y (2000) Manufacturing automation: metal cutting mechanics, machine tool vibrations, and CNC design. Cambridge University Press, New York

    Google Scholar 

  4. Quintana G, Ciurana J (2011) Chatter in machining processes: a review. Int J Mach Tools Manuf 51(5):363–376. https://doi.org/10.1016/j.ijmachtools.2011.01.001

    Article  Google Scholar 

  5. Yan ZH, Wang XB, Liu ZB, Wang DQ, Ji YJ, Jiao L (2017) Orthogonal polynomial approximation method for stability prediction in milling. Int J Adv Manuf Technol 91(9-12):4313–4330. https://doi.org/10.1007/s00170-017-0067-x

    Article  Google Scholar 

  6. Altintas Y, Budak E (1995) Analytical prediction of stability lobes in milling. Annals of the CIRP 44(1):357–362. https://doi.org/10.1016/S0007-8506(07)62342-7

    Article  Google Scholar 

  7. Li ZY, Sun YW, Guo DM (2017) 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. https://doi.org/10.1007/s00170-016 -983 4-3

    Article  Google Scholar 

  8. Merdol SD, Altintas Y (2004) Multi frequency solution of chatter stability for low immersion milling. J Manuf Sci Eng -Trans ASME 126(3):459–466. https://doi.org/10.1115/1.1765139

    Article  Google Scholar 

  9. Gradišek J, Govekar E, Grabec I, Kalveram M, Weinert K, Insperger T, Stépán G (2005) On stability prediction for low radial immersion milling. Mach Sci Technol 9(1):117–130. https://doi.org/10.1081/MST-200051378

    Article  Google Scholar 

  10. Zatarain M, Bediaga I, Muñoa J, Insperger T (2010) Analysis of directional factors in milling: importance of multi-frequency calculation and of the inclusion of the effect of the helix angle. Int J Adv Manuf Technol 47(5–8):535–542. https://doi.org/10.1007/s00170-009-2230-5

    Article  Google Scholar 

  11. Bayly PV, Halley JE, Mann BP, Davies MA (2003) Stability of interrupted cutting by temporal finite element analysis. Transactions of the ASME 125(2):220–225. https://doi.org/10.1115/1.155 6860

    Google Scholar 

  12. Insperger T, Stépán G (2004) Updated semi-discretization method for periodic delay-differential equations with discrete delay. Int J Numberical Methods Eng 61(1):117–141. https://doi.org/10.1002/nme.1061

    Article  MathSciNet  MATH  Google Scholar 

  13. 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. https://doi.org/10.1016/j.jsv.2007.11.040

    Article  Google Scholar 

  14. Insperger T (2010) Full-discretization and semi-discretization for milling stability prediction: some comments. Int J Mach Tools Manuf 50(7):658–662. https://doi.org/10.1016/j.ijmachtools.2010. 03.010

    Article  MathSciNet  Google Scholar 

  15. 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(1):29–41. https://doi.org/10.1016/ j.ijmachtools.2009.09.009

    Article  Google Scholar 

  16. 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. https://doi.org/10.1007/s11071-013-1127-x

    Article  MathSciNet  MATH  Google Scholar 

  17. 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. https://doi.org/10.1007/s00170-015-82 07-7

    Google Scholar 

  18. Li M, Zhang G, Huang Y (2013) Complete discretization scheme for milling stability prediction. Nonlinear Dynam 71(1–2):187–199. https://doi.org/10.1007/s11071-012-0651-4

    Article  MathSciNet  Google Scholar 

  19. Xie QZ (2015) Milling stability prediction using an improved complete discretization method. Int J Adv Manuf Technol 83(5–8):815–821. https://doi.org/10.1007/s00170-015-7626-9

    Google Scholar 

  20. 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. https://doi.org/10.1016/j.ijma chtools.2010.01.003

    Article  Google Scholar 

  21. 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. https://doi.org/10.1016/j.ijma chtools.2010.05.005

    Article  Google Scholar 

  22. Guo 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. https://doi.org/10.1016/j.ijma chtools.2012.05.001

    Article  Google Scholar 

  23. Ozoegwu CG (2014) Least squares approximated stability boundaries of milling process. Int J Mach Tools Manuf 79:24–30. https://doi.org/10.1016/j.ijmachtools.2014.02.001

    Article  Google Scholar 

  24. 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. https://doi.org/10.1016/j.ijmachtools. 2015.02.007

    Article  Google Scholar 

  25. 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. https://doi.org/10.1007/s00170-013-4813-4

    Article  Google Scholar 

  26. Li ZQ, Wang ZK, Shi XF (2017) Fast prediction of chatter stability lobe diagram for milling process using frequency response function or modal parameters. Int J Adv Manuf Technol 89(9-12):2603–2612. https://doi.org/10.1007/s00170-016-9959-4

    Article  Google Scholar 

  27. Liu YL, Zhang DH, BH W (2012) An efficient full-discretization method for prediction of milling stability. Int J Mach Tools Manuf 63:44–48. https://doi.org/10.1016/j.ijma chtools.2012.07.008

    Article  Google Scholar 

  28. Ozoegwu CG (2016) High order vector numerical integration schemes applied in state space milling stability analysis. Appl Math Comput 273:1025–1040. https://doi.org/10.1016/j.amc.2015.10.069

  29. Lakshmikantham V, Trigiante D (1988) Theory of difference equations, numerical methods and applications. Academic Press, London

    MATH  Google Scholar 

  30. Kolmanovskii VB, Nosov VR (1986) Stability of functional differential equations. Academic Press, London

    Google Scholar 

Download references

Acknowledgements

This work is jointly supported by the National Natural Science Foundation of China (grant nos. 51375055 and 51575055), the high-end CNC machine tools and basic manufacturing equipment Science and Technology Major Project of China (grant no. 2015ZX04001002).

Author information

Authors and Affiliations

  1. Key Laboratory of Fundamental Science for Advanced Machining, Beijing Institute of Technology, No. 5 South Zhongguancun Street, Haidian District, Beijing, 100081, People’s Republic of China

    Yongjian Ji, Xibin Wang, Zhibing Liu & Zhenghu Yan

  2. Key Laboratory of Modern Measurement and Control Technology, Beijing Information Science and Technology University, No.12 East Qinghexiaoying Road, Beijing, 100192, People’s Republic of China

    Hongjun Wang

Authors
  1. Yongjian Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xibin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhibing Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hongjun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Zhenghu Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Yongjian Ji or Zhibing Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ji, Y., Wang, X., Liu, Z. et al. An updated full-discretization milling stability prediction method based on the higher-order Hermite-Newton interpolation polynomial. Int J Adv Manuf Technol 95, 2227–2242 (2018). https://doi.org/10.1007/s00170-017-1409-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00170-017-1409-4

Keywords

Search

Navigation