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Research Papers

Derived Nodes Approach for Improving Accuracy of Machining Stability Prediction

[+] Author and Article Information
Le Cao, Tao Huang

State Key Laboratory of Digital Manufacturing
Equipment and Technology,
Huazhong University of Science and Technology,
Wuhan 430074, China

Xiao-Ming Zhang

Professor
State Key Laboratory of Digital Manufacturing
Equipment and Technology,
Huazhong University of Science and Technology,
Wuhan 430074, China
e-mails: zhangxm.duyi@gmail.com;
cheungxm@hust.edu.cn

Han Ding

Professor
State Key Laboratory of Digital Manufacturing
Equipment and Technology,
Huazhong University of Science and Technology,
Wuhan 430074, China

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 11, 2017; final manuscript received December 15, 2017; published online February 9, 2018. Assoc. Editor: Julian Rimoli.

J. Vib. Acoust 140(3), 031017 (Feb 09, 2018) (8 pages) Paper No: VIB-17-1451; doi: 10.1115/1.4038947 History: Received October 11, 2017; Revised December 15, 2017

Machining process dynamics can be described by state-space delayed differential equations (DDEs). To numerically predict the process stability, diverse piecewise polynomial interpolation is often utilized to discretize the continuous DDEs into a set of linear discrete equations. The accuracy of discrete approximation of the DDEs generally depends on how to deal with the piecewise polynomials. However, the improvement of the stability prediction accuracy cannot be always guaranteed by higher-order polynomials due to the Runge phenomenon. In this study, the piecewise polynomials with derivative-continuous at joint nodes are taken into consideration. We develop a recursive estimation of derived nodes for interpolation approximation of the state variables, so as to improve the discretization accuracy of the DDEs. Two different temporal discretization methods, i.e., second-order full-discretization and state-space temporal finite methods, are taken as demonstrations to illustrate the effectiveness of applying the proposed approach for accuracy improvement. Numerical simulations prove that the proposed approach brings a great improvement on the accuracy of the stability lobes, as well as the rate of convergence, compared to the previous recorded ones with the same order of interpolation polynomials.

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References

Ismail, F. , and Soliman, E. , 1997, “ A New Method for the Identification of Stability Lobes in Machining,” Int. J. Mach. Tools Manuf., 37(6), pp. 763–774. [CrossRef]
Solis, E. , Peres, C. , Jimenez, J. , Alique, J. , and Monje, J. , 2004, “ A New Analytical–Experimental Method for the Identification of Stability Lobes in High-Speed Milling,” Int. J. Mach. Tools Manuf., 44(15), pp. 1591–1597. [CrossRef]
Quintana, G. , Ciurana, J. , Ferrer, I. , and Rodríguez, C. , 2009, “ Sound Mapping for Identification of Stability Lobe Diagrams in Milling Processes,” Int. J. Mach. Tools Manuf., 49(3), pp. 203–211. [CrossRef]
Smith, S. , and Tlusty, J. , 1993, “ Efficient Simulation Programs for Chatter in Milling,” CIRP Ann.-Manuf. Technol., 42(1), pp. 463–466. [CrossRef]
Minis, I. , and Yanushevsky, B. , 1993, “ A New Theoretical Approach for the Prediction of Machine Tool Chatter in Milling,” ASME J. Eng. Ind., 115(1), pp. 1–8.
Altintaş, Y. , and Budak, E. , 1995, “ Analytical Prediction of Stability Lobes in Milling,” CIRP Ann.-Manuf. Technol., 44(1), pp. 357–362. [CrossRef]
Merdol, S. , and Altintas, Y. , 2004, “ Multi Frequency Solution of Chatter Stability for Low Immersion Milling,” ASME J. Manuf. Sci. Eng., 126(3), pp. 459–466. [CrossRef]
Hajdu, D. , Insperger, T. , Bachrathy, D. , and Stepan, G. , 2017, “ Prediction of Robust Stability Boundaries for Milling Operations With Extended Multi-Frequency Solution and Structured Singular Values,” J. Manuf. Processes, 30, pp. 281–289. [CrossRef]
Dutterer, B. , and Burns, T. , 2002, “ Stability Prediction for Low Radial Immersion Milling,” ASME J. Manuf. Sci. Eng., 124(9), pp. 217–225.
Bayly, P. , Halley, J. , Mann, B. , and Davies, M. , 2003, “ Stability of Interrupted Cutting by Temporal Finite Element Analysis,” ASME J. Manuf. Sci. Eng., 125(2), pp. 220–225. [CrossRef]
Khasawneh, F. , Patel, B. , and Mann, B. , 2009, “ A State-Space Temporal Finite Element Approach for Stability Investigations of Delay Equations,” ASME Paper No. SMASIS2009-1263.
Khasawneh, F. , Bobrenkov, O. , Mann, B. , and Butcher, E. , 2012, “ Investigation of Period-Doubling Islands in Milling With Simultaneously Engaged Helical Flutes,” ASME J. Vib. Acoust., 134(2), p. 021008. [CrossRef]
Insperger, T. , and Stépán, G. , 2002, “ Semi-Discretization Method for Delayed Systems,” Int. J. Numer. Methods Eng., 55(5), pp. 503–518. [CrossRef]
Insperger, T. , and Stépán, G. , 2004, “ Updated Semi-Discretization Method for Periodic Delay-Differential Equations With Discrete Delay,” Int. J. Numer. Methods Eng., 61(1), pp. 117–141. [CrossRef]
Insperger, T. , Stépán, G. , and Turi, J. , 2008, “ On the Higher-Order Semi-Discretizations for Periodic Delayed Systems,” J. Sound Vib., 313(1), pp. 334–341. [CrossRef]
Lehotzky, D. , and Insperger, T. , 2012, “ Stability of Turning Processes Subjected to Digital PD Control,” Period. Polytech. Eng. Mech. Eng., 56(1), pp. 33–42. [CrossRef]
AsI, F. , and Ulsoy, A. , 2003, “ Analysis of a System of Linear Delay Differential Equations,” ASME J. Dyn. Syst. Meas. Control, 125(2), pp. 215–223. [CrossRef]
Sun, Y. , Nelson, P. , and Ulsoy, A. , 2007, “ Delay Differential Equations Via the Matrix Lambert W Function and Bifurcation Analysis: Application to Machine Tool Chatter,” Math. Biosci. Eng., 4(2), p. 355. [CrossRef] [PubMed]
Li, M. , Zhang, G. , and Huang, Y. , 2013, “ Complete Discretization Scheme for Milling Stability Prediction,” Nonlinear Dyn., 71(1–2), pp. 187–199. [CrossRef]
Niu, J. , Ding, Y. , Zhu, L. , and Ding, H. , 2014, “ Runge–Kutta Methods for a Semi-Analytical Prediction of Milling Stability,” Nonlinear Dyn., 76(1), pp. 289–304. [CrossRef]
Butcher, E. , Bobrenkov, O. , Bueler, E. , and Nindujarla, P. , 2009, “ Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels,” ASME J. Comput. Nonlinear Dyn., 4(3), p. 031003. [CrossRef]
Engelborghs, K. , Luzyanina, T. , Hout, K. , and Roose, D. , 2001, “ Collocation Methods for the Computation of Periodic Solutions of Delay Differential Equations,” SIAM J. Sci. Comput., 22(5), pp. 1593–1609. [CrossRef]
Ding, Y. , Zhang, X. , and Ding, H. , 2015, “ A Legendre Polynomials Based Method for Stability Analysis of Milling Processes,” ASME J. Vib. Acoust., 137(2), p. 024504. [CrossRef]
Ding, Y. , Niu, J. , Zhu, L. , and Ding, H. , 2016, “ Numerical Integration Method for Stability Analysis of Milling With Variable Spindle Speeds,” ASME J. Vib. Acoust., 138(1), p. 011010. [CrossRef]
Ding, Y. , Zhu, L. , Zhang, X. , and Ding, H. , 2010, “ A Full-Discretization Method for Prediction of Milling Stability,” Int. J. Mach. Tools Manuf., 50(5), pp. 502–509. [CrossRef]
Huang, T. , Zhang, X. , Zhang, X. , and Ding, H. , 2013, “ An Efficient Linear Approximation of Acceleration Method for Milling Stability Prediction,” Int. J. Mach. Tools Manuf., 74, pp. 56–64. [CrossRef]
Khasawneh, F. , and Mann, B. , 2011, “ A Spectral Element Approach for the Stability of Delay Systems,” Int. J. Numer. Methods Eng., 87(6), pp. 566–592. [CrossRef]
Lehotzky, D. , Insperger, T. , Khasawneh, F. , and Stepan, G. , 2017, “ Spectral Element Method for Stability Analysis of Milling Processes With Discontinuous Time-Periodicity,” Int. J. Adv. Manuf. Technol., 89(9), pp. 2503–2514. [CrossRef]
Ding, Y. , Zhu, L. , Zhang, X. , and Ding, H. , 2010, “ Second-Order Full-Discretization Method for Milling Stability Prediction,” Int. J. Mach. Tools Manuf., 50(10), pp. 926–932. [CrossRef]
Quo, Q. , Sun, Y. , and Jiang, Y. , 2012, “ On the Accurate Calculation of Milling Stability Limits Using Third-Order Full-Discretization Method,” Int. J. Mach. Tools Manuf., 62, pp. 61–66. [CrossRef]
Ozoegwu, C. , Omenyi, S. , and Ofochebe, S. , 2015, “ Hyper-Third Order Full-Discretization Methods in Milling Stability Prediction,” Int. J. Mach. Tools Manuf., 92, pp. 1–9. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

The relationship between ||μ|−|μ0|| and the number of intervals m with different combination of spindle speed Ω and depth of cut w. The error curves of the previous second-order TFEA and FDM are marked with hollow triangle and hollow circle, while the proposed second-order TFEA and FDM are marked with solid triangle and solid circle, respectively.

Grahic Jump Location
Fig. 2

The sketch map of interpolation polynomials

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