Stability prediction of thin-walled workpiece made of Al7075 in milling based on shifted Chebyshev polynomials

  • Zhenghu Yan
  • Zhibing LiuEmail author
  • Xibin Wang
  • Biao Liu
  • Zhiwen Luo
  • Dongqian Wang


With the rapid development of aerospace technology, Al7075 has been widely used for structural components. High-speed milling is one of the most effective ways to improve machining efficiency of Al7075. During the milling process, regenerative chatter which restricts the milling quality and productivity often occurs. With the aim of avoiding regenerative chatter, stability lobe diagram (SLD) is widely used to obtain chatter-free parameters. This work presents a stability prediction method by using shifted Chebyshev polynomials. The milling dynamics with consideration of the regenerative effect is described by time periodic delay-differential equations (DDEs). The transition matrix of the milling system is constructed with the help of Chebyshev–Gauss–Lobatto (CGL) points. In order to demonstrate the accuracy of the proposed method, the rate of convergence of the proposed method is compared with that of the classical benchmark methods. On the other hand, in the process of thin-walled workpiece milling, the dynamic behavior of the workpiece depends on the tool position. To study the influence of the tool position dependent dynamics on the chatter stability of the thin-walled workpiece, a three-dimensional SLD is obtained. The verification experiments are conducted to verify the reliability of the proposed method. The results show that the experimental results are consistent with the predicted results.


Milling stability Chebyshev polynomials Thin-walled workpiece Position dependent dynamics 


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  1. 1.
    Subramanian M (2014) A study on process characteristics of CNC end milling process for Al7075-T6 aluminium alloy. Dissertation, Anna UniversityGoogle Scholar
  2. 2.
    Quintana G, Ciurana J (2011) Chatter in machining process: a review. Int J Mach Tools Manuf 51(5):363–376. doi: 10.1016/j.ijmachtools.2011.01.001 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.
    Merdol SD, Altintas Y (2004) Multi-frequency solution of chatter stability for low immersion milling. J Manuf Sci Eng 126(3):459–466. doi: 10.1115/1.1765139 CrossRefGoogle Scholar
  5. 5.
    Shorr MJ, Liang SY (1996) Chatter stability analysis for end milling via convolution modeling. Int J Adv Manuf Technol 11(5):311–318. doi: 10.1007/BF01845689 CrossRefGoogle Scholar
  6. 6.
    Bayly PV, Halley JE, Mann BP, Davies MA (2003) Stability of interrupted cutting by temporal finite element analysis. J Manuf Sci Eng 125(2):220–225. doi: 10.1115/1.1556860 CrossRefGoogle Scholar
  7. 7.
    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 Nonlinear Dyn 4(3):031003. doi: 10.1115/1.3124088 CrossRefGoogle 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(1):117–141. doi: 10.1002/nme.1061 MathSciNetCrossRefzbMATHGoogle 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.
    Long XH, Balachandran B (2007) Stability analysis for milling process. Nonlinear Dyn 49(3):349–359. doi: 10.1007/s11071-006-9127-8 CrossRefzbMATHGoogle Scholar
  11. 11.
    Long XH, Balachandran B, Mann BP (2007) Dynamics of milling processes with variable time delays. Nonlinear Dyn 49(1–3):49–63. doi: 10.1007/s11071-006-9058-4 zbMATHGoogle Scholar
  12. 12.
    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
  13. 13.
    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
  14. 14.
    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
  15. 15.
    Ding Y, Zhu LM, Zhang XJ, Ding H (2011) Milling stability analysis using the spectral method. Sci China Technol Sci 54(12):3130–3136. doi: 10.1007/s11431-011-4611-x CrossRefzbMATHGoogle Scholar
  16. 16.
    Ding Y, Zhu LM, Zhang XJ, Ding H (2011) Numerical integration method for prediction of milling stability. J Manuf Sci Eng 133(3):031005. doi: 10.1115/1.4004136 CrossRefGoogle Scholar
  17. 17.
    Ding Y, Zhu LM, Zhang XJ, Ding H (2013) Stability analysis of milling via the differential quadrature method. J Manuf Sci Eng 135(4):044502. doi: 10.1115/1.4024539 CrossRefGoogle Scholar
  18. 18.
    Ding Y, Zhang XJ, Ding H (2015) A Legendre polynomials based method for stability analysis of milling process. J Vib Acoust 137(2):024504. doi: 10.1115/1.4029460 CrossRefGoogle Scholar
  19. 19.
    Li MZ, Zhang GJ, Huang Y (2013) Complete discretization scheme for milling stability prediction. Nonlinear Dyn 71(1–2):187–199. doi: 10.1007/s11071-012-0651-4 MathSciNetCrossRefGoogle Scholar
  20. 20.
    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. doi: 10.1007/s11071-013-1127-x MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ozoegwu CG (2014) Least squares approximated stability boundaries of milling process. Int J Mach Tools Manuf 79:24–30. doi: 10.1016/j.ijmachtools.2014.02.001 CrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    Li HZ, Li PX, Chen Q (2003) A novel chatter stability criterion for the modeling and simulation of the dynamic milling process in the time domain. Int J Adv Manuf Technol 22:619–625. doi: 10.1007/s00170-003-1562-9 CrossRefGoogle Scholar
  24. 24.
    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 68:1967–1976. doi: 10.1007/s00170-013-4813-4 CrossRefGoogle Scholar
  25. 25.
    Tangjitsitcharoen S, Pongsathornwiwat N (2013) Development of chatter detection in milling processes. Int J Adv Manuf Technol 65:919–927. doi: 10.1007/s00170-012-4228-7 CrossRefGoogle Scholar
  26. 26.
    Wan M, Ma YC, Zhang WH, Yang Y (2015) Study on the construction mechanism of stability lobes in milling process with multiple modes. Int J Adv Manuf Technol 79:589–603. doi: 10.1007/s00170-015-6829-4 CrossRefGoogle Scholar
  27. 27.
    Yang YQ, Liu Q, Zhang B (2014) Three-dimensional chatter stability prediction of milling based on the linear and exponential cutting force model. Int J Adv Manuf Technol 72:1175–1185. doi: 10.1007/s00170-014-5703-0 CrossRefGoogle Scholar
  28. 28.
    Wang MH, Gao L, Zheng YH (2014) Prediction of regenerative chatter in the high-speed vertical milling of thin-walled workpiece made of titanium alloy. Int J Adv Manuf Technol 72:707–716. doi: 10.1007/s00170-014-5641-x CrossRefGoogle Scholar
  29. 29.
    Zhang XJ, Xiong CH, Ding Y, Zhang XM (2010) Stability analysis in milling of thin-walled workpieces with emphasis on the structural effect. P I Mech Eng B-J Eng 224(4):589–608. doi: 10.1243/09544054JEM1696 Google Scholar
  30. 30.
    Li X, Zhao W, Li L, He N, Chi SW (2015) Modeling and application of process damping in milling of thin-walled workpiece made of titanium alloy. Shock Vib 228(11):1359–1371. doi: 10.1177/0954405414522216 Google Scholar
  31. 31.
    Tang AJ, Liu ZQ (2009) Three-dimensional stability lobe and maximum material removal rate in end milling of thin-walled plate. Int J Adv Manuf Technol 43(1–2):33–39. doi: 10.1007/s00170-008-1695-y CrossRefGoogle Scholar
  32. 32.
    Song QH, Ai X, Tang WX (2011) Prediction of simultaneous dynamic stability limit of time–variable parameters system in thin-walled workpiece high-speed milling processes. Int J Adv Manuf Technol 55:883–889. doi: 10.1007/s00170-010-3139-8 CrossRefGoogle Scholar
  33. 33.
    Thevenot V, Arnaud L, Dessein G, Cazenave-Larroche G (2006) Integration of dynamic behaviour variations in the stability lobes method: 3D lobes construction and application to thin-walled structure milling. Int J Adv Manuf Technol 27:638–644. doi: 10.1007/s00170-004-2241-1 CrossRefGoogle Scholar
  34. 34.
    Zhang XJ, Xiong CH, Ding Y, Huang XD, Ding H (2014) A synthetical stability method for cutting parameter optimization to assure surface location accuracy in flexible part milling. Int J Adv Manuf Technol 75:1131–1147. doi: 10.1007/s00170-014-6151-6 CrossRefGoogle Scholar
  35. 35.
    Yang X, Wang ZQ (2015) A Chebyshev-Gauss spectral collocation method for ordinary differential equations. J Comput Math 33(1):59–85. doi: 10.4208/jcm.1405-m4368 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    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. IMECE 2002–34116:351–363. doi: 10.1115/IMECE2002-39116 Google Scholar
  37. 37.
    Shen J, Tang T (2006) Spectral and high-order methods with application. Science Press, BeijingzbMATHGoogle Scholar
  38. 38.
    Wang MH, Lei G, Zheng YH (2014) An examination of the fundamental mechanics of cutting force coefficients. Int J Mach Tools Manuf 78:1–7. doi: 10.1016/j.ijmach CrossRefGoogle Scholar
  39. 39.
    Tan JY, Liu GJ, Li GH (2009) Experimental study on adhesive wear of milling insert with complex groove. Int J Adv Manuf Technol 44:631–637. doi: 10.1007/s00170-008-1856-z CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2016

Authors and Affiliations

  • Zhenghu Yan
    • 1
  • Zhibing Liu
    • 1
    Email author
  • Xibin Wang
    • 1
  • Biao Liu
    • 1
  • Zhiwen Luo
    • 1
  • Dongqian Wang
    • 1
  1. 1.Key Laboratory of Fundamental Science for Advanced MachiningSchool of Mechanical Engineering, Beijing Institute of TechnologyBeijingChina

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