Regenerative Chatter Stability and Hopf Bifurcation Analysis in Milling System

Abstract:

The shifted Chebyshev polynomials and Floquet theory are both adopted for the prediction regenerative chatter stability and Hopf bifurcation in milling. The influences of the system parameter on the stability of the milling system have been analyzed. The stability lobe diagrams are obtained. The result shows that the shifted Chebyshev polynomials method is more accurate than the semi-discretion scheme for spindle speed lower than 3500 round per minutes. The stability in milling can well be predicted by the cutting depth and feed rate lobes diagrams. Only Hopf bifurcations are detected by the Eigen-value analysis. The stable solution transform from the stable equilibrium point to the quasi-periodic oscillation after Hopf bifurcation.

• Periodical: Advanced Materials Research (Volume 739)
• Pages: 400-407
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.739.400
• Online since: August 2013
• Authors: D. Zhao, Q. Zhang
• Keywords: Chebyshev Polynomials, Hopf Bifurcation, Milling, Regenerative Chatter

References

[1] S. A. Tobias, Machine Tool Vibration, Blackie and Sons Ltd, (1965).

[2] Y. Altintas, E. Budak, Analytical Prediction of Stability Lobes in Milling, Annals of the CIRP 44(1) (1995) 357-362.

DOI: 10.1016/s0007-8506(07)62342-7

[3] Y. Altintas, Analytical Prediction of Three Dimensional Chatter Stabolity in Milling, Japan Society of Mechanical Engineers International 44 (2001) 717-723.

[4] Y. Altintas, G. Stepan, D. Merdol, Z. Dombovari, Chatter Stability of Milling in Frequency and Discrete Time Domain, CIRP Journal of Manufacturing Science and Technology 1 (2008) 35-44.

DOI: 10.1016/j.cirpj.2008.06.003

[5] W. X. Tang, Q. H. Song, S. Q. Yu, S. S. Sun, B. B. Li, B. Du, X. Ai, Prediction of Chatter Stability in High-speed Finishing End Milling Considering Multi-mode Dynamics, Journal of Material Processing Technology, 209 (2009) 2585-2591.

DOI: 10.1016/j.jmatprotec.2008.06.003

[6] G. Totis, RCPM-A New Method for Robust Chatter Prediction in Milling, International Journal of Machine tools & Manufacture 49 (2009) 273-284.

DOI: 10.1016/j.ijmachtools.2008.10.008

[7] R. P. H. Faassen, N. Van de Wouw, J. A. J. Oosterling, H. Nijmaijer, Prediction of Regenerative Chatter by Modelling and Analysis of High-speed Milling, International Journal of Machine tools & Manufacture 43 (2003) 1437-1446.

DOI: 10.1016/s0890-6955(03)00171-8

[8] G. Quintana , J. Ciurana, D. Teixidor, A New Experimential Methodology for Identification of Stability Lobes Diagram in Milling Operations, International Journal of Machine tools & Manufacture 48 (2008) 1637-1645.

DOI: 10.1016/j.ijmachtools.2008.07.006

[9] J. Gradisek, M. Kalveram, T. Insperger, K. Weinert, G. Stépán, E. Govekar, I. Grabec, On Stability Prediction for Milling, International Journal of Machine tools & Manufacture 45 (2005) 769-781.

DOI: 10.1016/j.ijmachtools.2004.11.015

[10] T. Insperger, G. Stépán, P. V. Bayly, B. P. Mann, Multiple Chatter Frequencies in Milling Processes, Journal of Sound and Vibration, 262 (2003) 333-345.

DOI: 10.1016/s0022-460x(02)01131-8

[11] E. A. Butcher, H. T. Ma, E. Bueler, V. Averina, Z. Szabo, Stability of Linear Time-Periodic Delay-Fifferential Equations via Chebyshev Polynomials, International Journal for Numerical Methods in Engineering, 59 (2004) 895-922.

DOI: 10.1002/nme.894

[12] S. C. Sinha, D. H. Wu, An Efficient Computational Scheme for the Analysis of Periodic Systems, Journal of Sound and Vibration, 151 (1991) 91-117.

DOI: 10.1016/0022-460x(91)90654-3

[13] E. A. Butcher, S. C. Sinha, A Hybrid Formulation for the Analysis Time Preiodic Linear systems via Chebyshev Polynomials, Journal of Sound and Vibration 195(3) (1996) 518-527.

DOI: 10.1006/jsvi.1996.0441

[14] S. C. Sinha, E. A. Butcher, Symbolic computation of fundmental solution Matrices for Linear Time-Periodic Dynamical Systems, Journal of Sound and Vibration, 26(1) (1997) 61-8.

DOI: 10.1006/jsvi.1997.1079