Chatter stability analysis using the matrix Lambert function and bifurcation analysis

Proceedings of the International Conference on Manufacturing Science and Engineering

Volumn 2006, Issue , 2006, Pages

  Yi, Sun ^a   Nelson, Patrick W ^a   Ulsoy, A Galip ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL METHODS; DIFFERENTIAL EQUATIONS; FUNCTION EVALUATION; GRAPHIC METHODS; SYSTEM STABILITY;

BIFURCATION METHOD; DELAY DIFFERENTIAL EQUATIONS; GEOMETRIC ANALYSIS; LAMBERT FUNCTION;

MACHINE TOOLS;

EID: 33751281777     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1115/msec2006-21130     Document Type: Conference Paper

References (20)

1. Tobias, S. A., 1965, Machine-Tool Vibration, Blackie, London, UK.
2. Tlusty, J., 2000, Manufacturing Process and Equipment, Prentice Hall, New Jersey.
3. Merrit, H. E., 1965, "Theory of self excited-tool chatter: Research 1," J. Eng., Ind., 81, pp. 447-454.
4. Optiz, H. and Bernardi, F., 1970, "Investigation and calculation of the chatter behavior of lathes and milling machines," Annals of the CIRP, 18, pp. 335-343.
5. Minis, I. E., Magrab, E. B., and Pandelidis, I. O., 1990, "Improved methods for the prediction of chatter in turning-Part 3: A generalized linear theory," J. Eng., Ind., 112, pp. 28-35.
6. Chen, S. G., Ulsoy, A. G., and Koren, Y., 1997, "Computational stability analysis of chatter in turning," J. Manuf. Sci. Eng., 119(4), pp. 457-460.
7. Stépán, G., 1989, Retarded Dynamical Systems, Longman, Harlow.
8. Stépán, G., 1997, "Delay differential equation models for machine tool chatter," in Dynamics and Chaos in Manufacturing Process F. C. Moon (ed). Wiley, New York, pp. 165-191.
9. Nayfeh, A., Chin, C., and Pratt, J., 1997, "Application of perturbation methods to tool chatter dynamics," in Dynamics and Chaos in Manufacturing Processes, F. C. Moon (ed), Wiley, New York, pp. 193-213.
10. Kalmar-Nagy, T., Stépan, G., and Moon, F. C., "Subcritical hopf bifurcation in the delay equation model for machine tool vibrations," Nonlinear Dynamics, 26, 2001, pp. 121-142.
11. Fofana, M. S., 2003, "Delay dynamical systems and applications to nonlinear machine-tool chatter," Chaos Solitons & Fractals, 17, pp. 731-747.
12. Sipahi, R. and Olgac, N., 2003, "Degenerate cases in using the Direct Method', J. Dyn. Syst., Meas., Control, 125, pp. 194-201
13. Olgac, N. and Sipahi, R., 2005, "A unique methodology for chatter stability mapping in simultaneous machining," J. Manuf. Sci. Eng., 127, pp. 791-800.
14. Asl, F. M. and Ulsoy, A. G., 2003, "Analysis of a system of linear delay differential equations," J. Dyn. Syst., Meas., and Control, 125 (2), pp. 215-223.
15. Yi, S. and Ulsoy, A. G., 2006, 'Solution of a system of linear delay differential equation using the matrix Lambert function', in Proceedings of the American Control Conference, Minneapolis, MN, 2006, pp. 2433-2438.
16. Corless, R. M., Gonnet, G. H., Hare, D. E. G., 1996, Jeffrey, D. J., and Knuth, D. E., "On Lambert's W function," Advances in Computational Mathematics, 5(4), pp. 329-359.
17. Shinozaki, H., 2003, Robust Stability Analysis of Linear Ttime-delay Systems by Lambert W Function, Master's thesis, Dpt. Electro. Info. Sci., Kyoto Institute of Technology, Kyoto, Japan.
18. Malek-zavarei, M. and Jamshidi, M., 1987, Time-delayed Systems, North Holland, New York.
19. Forde, J. and Nelson, P., 2004, "Application of sturm sequences to bifurcation analysis of delay differential equation models," J. Math. Anal. Appl. 300, pp. 273-284.
20. Prestel, A. and Delzell, C. N., 2001, Positive polynomials: from Hilbert's 17th problem to real algebra, Springer-Verlag, Berlin.