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Accurate and efficient stability prediction for milling operations using the Legendre-Chebyshev-based method

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Abstract

Stability prediction with both high computational accuracy and speed is still a challenging issue and has been attracting significant attention from the academia and industry. This study presents a Legendre-Chebyshev-based stability analysis method (LCM) for milling operations. According to the cutting state, it divides the system period of milling model into the free and the forced vibration time periods. By introducing appropriate transformation, the latter time interval is further discretized nonuniformly into the Chebyshev-Gauss-Lobatto points, which has explicit expression. Then, the state term over the discrete time points is approximated with the Legendre expansion, and its corresponding derivative is acquired via a novel and efficient algorithm. Thereafter, Floquet matrix within the system period of milling model can be determined for predicting the system stability via the known Floquet theory. Finally, we validate the effectiveness of the LCM by employing the single and two degrees of freedom (DOF) milling operations and making detailed comparisons with the recent representative algorithms, which indicates that the presented Legendre-Chebyshev-based method has both high prediction accuracy and speed.

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References

  1. 1.

    Cui XB, Zhao B, Jiao F, Zheng JX (2016) Chip formation and its effects on cutting force, tool temperature, tool stress, and cutting edge wear in high- and ultra-high-speed milling. Int J Adv Manuf Technol 83:55–65

  2. 2.

    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

  3. 3.

    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

  4. 4.

    Quintana G, Stepan CJ (2011) Chatter in machining processes: a review. Int J Mach Tools Manuf 51(5):363–376

  5. 5.

    Tao JF, Qin CJ, Xiao DY, Shi HT, Ling X, Li BC, Liu CL (2019) Timely chatter identification for robotic drilling using a local maximum synchrosqueezing-based method. J Intell Manuf. https://doi.org/10.1007/s10845-019-01509-5

  6. 6.

    Altintas Y (2012) Manufacturing automation: metal cutting, mechanics, machine tool vibrations, and CNC design. Cambridge University Press, New York

  7. 7.

    Tao JF, Qin CJ, Xiao DY, Shi HT, Liu CL (2019) A pre-generated matrix-based method for real-time robotic drilling chatter monitoring. Chin J Aeronaut 32(12):2755-2764

  8. 8.

    Dong XF, Qiu ZZ (2019) Stability analysis in milling process based on updated numerical integration method. Mech Syst Signal Process. https://doi.org/10.1016/j.ymssp.2019.106435

  9. 9.

    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

  10. 10.

    Hajdu D, Insperger T, Stepan G (2017) Robust stability analysis of machining operations. Int J Adv Manuf Technol 88(1–4):45–54

  11. 11.

    Tao JF, Qin CJ, Liu CL (2019) A synchroextracting-based method for early chatter identification of robotic drilling process. Int J Adv Manuf Technol 100:273–285

  12. 12.

    Tao JF, Zeng HW, Qin CJ, Liu CL (2019) Chatter detection in robotic drilling operations combining multi-synchrosqueezing transform and energy entropy. Int J Adv Manuf Technol 105(7–8):2879–2890

  13. 13.

    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

  14. 14.

    Smith S, Tlusty J (1993) Efficient simulation programs for chatter in milling. CI RP Ann Manuf Techn 42:463–466

  15. 15.

    Li ZQ, Liu Q (2008) Solution and analysis of chatter stability for end milling in the time-domain. Chin J Aeronaut 21:169–178

  16. 16.

    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

  17. 17.

    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

  18. 18.

    Roukema JC, Altintas Y (2006) Time domain simulation of torsional-axial vibrations in drilling. Int J Mach Tools Manuf 46(15):2073–2085

  19. 19.

    Altintas Y, Budak E (1995) Analytical prediction of stability lobes in milling. CIRP Ann 44(1):357–362

  20. 20.

    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

  21. 21.

    Merdol SD, Altintas Y (2004) Multi frequency solution of chatter stability for low immersion milling. J Manuf Sci Eng 126(3):459–466

  22. 22.

    Wu Y, You Y, Jiang J (2019) A stability prediction method research for milling processes based on implicit multistep schemes. Int J Adv Manuf Technol 105(7–8):3271–3288

  23. 23.

    Butcher EA, Ma HT, Bueler E, Averina V, Szabo Z (2004) Stability of linear time-periodic delay-differential equations via Chebyshev polynomials. Int J Numer Methods Eng 59(7):895–922

  24. 24.

    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

  25. 25.

    Insperger T, Stepan G (2002) Semi-discretization method for delayed systems. Int J Numer Methods Biomed Eng 55(5):503–518

  26. 26.

    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

  27. 27.

    Insperger T, Stepan G, Turi J (2008) On the higher-order semidiscretizations for periodic delayed systems. J Sound Vib 313(1):334–341

  28. 28.

    Dong XF, Zhang W, Deng S (2016) The reconstruction of a semi-discretization method for milling stability prediction based on Shannon standard orthogonal basis. Int J Adv Manuf Technol 85:1501–1511

  29. 29.

    Dong XF, Zhang WM (2019) Chatter suppression analysis in milling process with variable spindle speed based on the reconstructed semi-discretization method. Int J Adv Manuf Technol 105:2021–2037. https://doi.org/10.1007/s00170-019-04363-0

  30. 30.

    Jiang SL, Sun YW, Yuan XL, Liu WR (2017) A second-order semi-discretization method for the efficient and accurate stability prediction of milling process. Int J Adv Manuf Technol 92(1–4):583–595

  31. 31.

    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

  32. 32.

    Sun Y, Xiong Z (2017) High-order full-discretization method using Lagrange interpolation for stability analysis of turning processes with stiffness variation. J Sound Vib 386(1):50–64

  33. 33.

    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

  34. 34.

    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

  35. 35.

    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

  36. 36.

    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

  37. 37.

    Qin CJ, Tao JF, Liu CL (2018) A predictor-corrector-based holistic-discretization method for accurate and efficient milling stability analysis. Int J Adv Manuf Technol 96:2043–2054

  38. 38.

    Qin CJ, Tao JF, Liu CL (2019) A novel stability prediction method for milling operations using the holistic-interpolation scheme. Proc IME Part C: J Mechanical Engineering Science 233(13):4463–4475

  39. 39.

    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

  40. 40.

    Qin CJ, Tao JF, Shi HT, Xiao DY, Li BC, Liu CL (2019) A novel Chebyshev-wavelet-based approach for accurate and fast prediction of milling stability. Precis Eng 62:244-255

  41. 41.

    Ding Y, Zhu LM, Zhang XJ, Ding H (2011) Numerical integration method for prediction of milling stability. J Manuf Sci Eng 133(3):031005

  42. 42.

    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

  43. 43.

    Li MZ, Zhang GJ, Huang Y (2013) Complete discretization scheme for milling stability prediction. Nonlinear Dyn 71:187–199

  44. 44.

    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):943–952

  45. 45.

    Ding Y, Zhu LM, Ding H (2014) A wavelet-based approach for stability analysis of periodic delay-differential systems with discrete delay. Nonlinear Dyn 79(2):1049–1059

  46. 46.

    Lu YA, Ding Y, Peng ZK, Chen ZZC, Zhu LM (2017) A spline-based method for stability analysis of milling processes. Int J Adv Manuf Technol 89:2571–2586

  47. 47.

    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

  48. 48.

    Zhang XJ, Xiong CH, Ding Y, Ding H (2017) Prediction of chatter stability in high speed milling using the numerical differentiation method. Int J Adv Manuf Technol 89:2535–2544

  49. 49.

    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

  50. 50.

    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

  51. 51.

    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

  52. 52.

    Zhang W, Ma H (2008) The Chebyshev-Legendre collocation method for a class of optimal control problems. Int J Comput Math 85(2):225–240

  53. 53.

    Canuto C, Hussaini MY, Quarteroni A, Zang TA (2012) Spectral methods in fluid dynamics. Springer Science & Business Media

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Acknowledgments

This work was partially supported by the National Key R&D Program of China (Grant No. 2018YFB1702503), the Science and Technology Planning Project of Guangdong Province (Grant No. 2017B090914002), and the China Postdoctoral Science Foundation (Grant No. 2019M661496).

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Correspondence to Jianfeng Tao.

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Qin, C., Tao, J., Xiao, D. et al. Accurate and efficient stability prediction for milling operations using the Legendre-Chebyshev-based method. Int J Adv Manuf Technol (2020). https://doi.org/10.1007/s00170-020-05040-3

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Keywords

  • Stability
  • Milling processes
  • Legendre-Chebyshev-based method
  • Delay differential equations
  • Floquet theory