Abstract
Regenerative chatter is the most important factor affecting the stability of the milling process. It is core for suppressing chatter and improving production efficiency to accurately and efficiently identify the stable region of milling chatter. Therefore, according to the theory of predictor-corrector, three predictor–corrector methods (PCM) are, respectively, proposed for the milling stability region by applying the fourth-order Adams-Bashforth-Moulton formula, Simpson formula, and Hamming formula. Firstly, the regenerative chatter milling process is described as a second-order time-delay differential equation (DDE) with periodic coefficients. Thus, the forced vibration time can uniformly be discretized as a time node set. Secondly, the fourth-order Adams-Bashforth formula is used to predict the displacement at every time node, whereas the fourth-order Adams-Moulton formula can be employed to correct this predicted value. In addition, the fourth-order Simpson formula and Hamming formula can also correct the predicted value. Thus, a higher precision discrete prediction-correction expansion is constructed for the transformation of DDE into the state transition express. The Floquet theory can be depended on to present the judgment criterion of milling stability. Moreover, finally, under the same milling process parameters, comparisons of both the stability lobe curve and the local discrete error curve show that the PCM has a faster convergence rate than the 1st-SDM (first-order semi-discretization method) and 2nd-FDM (second-order full-discretization method). This shows that the PCM can obtain better computational accuracy under the same discrete number, whereas the PCM is significantly higher computational efficiency over 1st-SDM and 2nd-FDM. Meanwhile, considering the actual machining environment, helix angle effect and multiple modes effect of the tool are analyzed; experimental verification considering multiple modes with helix angle further indicates the applicability of the PCM.
Similar content being viewed by others
Availability of data and material
All included in the paper.
Code availability
Not applicable.
References
Altintas Y, Weck M (2004) Chatter stability of metal cutting and grinding. CIRP Ann Manuf Technol 53(2):619–642
Altintas Y (2012) Manufacturing automation: metal cutting mechanics, machine tool vibrations, and CNC design. Cambridge University Press, Cambridge
Faassen RPH, Van de WN, Oosterling JAJ, Nijmeijer H (2003) Prediction of regenerative chatter by modelling and analysis of high-speed milling. Int J Mach Tools Manuf 43(14):1437–1446
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
Balachandran B (2001) Nonlinear dynamics of milling processes. Philos Trans R Soc A 359(1781):793–819
Wiercigroch M, Budak E (2001) Sources of nonlinearities, chatter generation and suppression in metal cutting. Philos Trans R Soc A 359(1781):663–693
Li H, Shin YC (2006) A comprehensive dynamic end milling simulation model. Trans ASME J Manuf Sci Eng 128(1):86–95
Minis I, Yanushevsky R (1993) A new theoretical approach for the prediction of machine tool chatter in milling. Trans ASME J Manuf Sci Eng 115(1):1–8
Campomanes ML, Altintas Y (2003) An improved time domain simulation for dynamic milling at small radial immersions. Trans ASME J Manuf Sci Eng 125(3):416–422
Davies MA, Pratt JR, Dutterer BS (2000) The stability of low radial immersion milling. CIRP Ann Manuf Technol 49(1):37–40
Davies MA, Pratt JR, Dutterer BS (2002) Stability prediction for low radial immersion milling. Trans ASME J Manuf Sci Eng 124(2):217–225
Altintas Y, Budak E (1995) Analytical prediction of stability lobes in milling. CIRP Ann Manuf Technol 44(1):357–362
Merdol S, Altintas Y (2004) Multi-frequency solution of chatter stability for low immersion milling. Trans ASME J Manuf Sci Eng 126(3):459–466
Altintas Y, Lee P (1998) Mechanics and dynamics of ball end milling. Trans ASME J Manuf Sci Eng 120(4):684–692
Altintas Y, Engin S, Budak E (1999) Analytical stability prediction and design of variable pitch cutters. Trans ASME J Manuf Sci Eng 121(2):173–178
Altintas Y (2001) Analytical prediction of three dimensional chatter stability in milling. JSME Int J Ser C 44(3):717–723
Turner S, Merdol D, Altintas Y, Ridgway K (2007) Modelling of the stability of variable helix end mills. Int J Mach Tools Manuf 47(9):1410–1416
Ko JH, Altintas Y (2007) Dynamics and stability of plunge milling operations. Trans ASME J Manuf Sci Eng 129(1):32–40
Bayly PV, Halley JE, Mann BP, Davies MA (2003) Stability of interrupted cutting by temporal finite element analysis. Trans ASME J Manuf Sci Eng 125:220–225
Insperger T, Stepan G (2002) Semi-discretization method for delayed systems. Int J Numer Methods 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 Eng 61:117–141
Insperger T, Stepan G, Turi J (2008) On the higher-order semi-discretizations for periodic delayed systems. J Sound Vib 313(1):334–341
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
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
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
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(1):61–66
Liu YL, Zhang DH, Wu BH (2012) An efficient full-discretization method for prediction of milling stability. Int J Mach Tools Manuf 63:44–48
Sun YX, Xiong ZH (2017) High-order full-discretization method using Lagrange interpolation for stability analysis of turning processes with stiffness variation. J Sound Vib 386:50–64
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
Tang XW, Peng FY, Yan R, Gong YH, Li YT, Jiang LL (2017) Accurate and efficient prediction of milling stability with updated full-discretization method. Int J Adv Manuf Technol 88(9–12):2357–2368
Tang XW, Peng FY, Yan R, Gong YH, Li X (2016) An effective time domain model for milling stability prediction simultaneously considering multiple modes and cross-frequency response function effect. Int J Adv Manuf Technol 86(1):1037–1054
Zhou K, Zhang JF, Xu C, Feng PF, Wu ZJ (2018) Effects of helix angle and multi-mode on the milling stability prediction using full-discretization method. Precis Eng 54:39–50
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(9):2535–2544
Li MZ, Zhang GJ, Huang Y (2013) Complete discretization scheme for milling stability prediction. Nonlinear Dyn 71(1–2):187–199
Ding Y, Zhu LM, Zhang XJ, Ding H (2011) Numerical integration method for prediction of milling stability. Trans ASME J Manuf Sci Eng 133(3):031005
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, De Lacallec LL, Rodrígueza CA, Campa FJ (2017) Determination of the stability lobes in milling operations based on homotopy and simulated annealing techniques. Mechatronics 24(3):177–185
Zhang Z, Li HG, Meng G (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, Liu CL (2019) A novel stability prediction method for milling operations using the holistic-interpolation scheme. Proc IME C J Mech Eng Sci 233(13):4463–4475
Dai YB, Li HK, Xing X, Hao B (2018) Prediction of chatter stability for milling process using precise integration method. Precis Eng 52:152–157
Dong XF, Qiu ZZ (2020) Stability analysis in milling process based on updated numerical integration method. Mech Syst Signal Process 137:106435
Lou WD, Qin GH, Zuo DW (2021) Investigation on Cotes-formula-based prediction method and its experimental verification of milling stability. J Manuf Process 64:1077–1088
Wu Y, You YP, Jiang JJ (2020) New predictor-corrector methods based on piecewise polynomial interpolation for milling stability prediction. Mach Sci Technol 24(5):688–718
Stoer J, Bulirsch R (2013) Introduction to numerical analysis. Springer Science & Business Media, London
Mann BP, Young KA, Schmitz TL, Dilley DN (2005) Simultaneous stability and surface location error predictions in milling. Trans ASME J Manuf Sci Eng 127(3):446–453
Bayly PV, Mann BP, Schmitz TL, Peters DA, Stepan G (2002) Insperger T (2002) Effects of radial immersion and cutting direction on chatter instability in end-milling. ASME IMECE 3641:351–363
Zatarain M, Munoa J, Peigne G, Insperger T (2006) Analysis of the influence of mill helix angle on chatter stability. CIRP Ann 55(1):365–368
Mann BP, Edes BT, Easley SJ, Young KA, Ma K (2008) Chatter vibration and surface location error prediction for helical end mills. Int J Mach Tools Manuf 48(3–4):350–361
Funding
This work is supported by the National Nature Science Foundation of China (Grant No. 51765047), the Major Discipline Academic and Technical Leader Training Plan Project of Jiangxi Province (Grant No. 20172BCB22013), and the Key Research and Development Plan Project of Jiangxi Provincial Science and Technology Department (Grant No. 20203BBE53049).
Author information
Authors and Affiliations
Contributions
Guohua Qin: conceptualization, methodology, investigation, writing—review and editing. Weida Lou: conceptualization, methodology, formal analysis, writing—original draft. Huamin Wang: writing, review; data curation. Zhuxi Wu: language modification, data analysis.
Corresponding author
Ethics declarations
Ethics approval
Our research does not involve humans and animals, so there is no need to provide ethical approval.
Consent to participate and publish
All authors agreed to participate and publish.
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Qin, G., Lou, W., Wang, H. et al. High efficiency and precision approach to milling stability prediction based on predictor-corrector linear multi-step method. Int J Adv Manuf Technol 122, 1933–1955 (2022). https://doi.org/10.1007/s00170-022-09952-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00170-022-09952-0