Delay differential equations via the matrix lambert W function and bifurcation analysis: Application to machine tool chatter

Mathematical Biosciences and Engineering

Volumn 4, Issue 2, 2007, Pages 355-368

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

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Bifurcation analysis; Delay differential equation; Lambert W function; Regenerative tool chatter

Indexed keywords

EID: 34547256518     PISSN: 15471063     EISSN: 15510018     Source Type: Journal    
DOI: 10.3934/mbe.2007.4.355     Document Type: Article

References (22)

1. S. A. Tobias, MACHINE-TOOL VIBRATION. Blackie, London, UK, 1965.
2. J. Tlusty, MANUFACTURING PROCESS AND EQUIPMENT. Prentice Hall, New Jersey, 2000.
3. H. E. Merrit, THEORY OF SELF EXCITED-TOOL CHATTER: RESEARCH 1. ASME J Eng Ind 81 (1965) 447-54.
4. H. Optiz and F. Bernardi, INVESTIGATION AND CALCULATION OF THE CHATTER BEHAVIOR OF LATHES AND MILLING MACHINES. Ann CIRP 18 (1970) 335-43.
5. I. E. Minis, E. B. Magrab, and I. O. Pandelidis, IMPROVED METHODS FOR THE PREDICTION OF CHATTER IN TURNING-PART 3: A GENERALIZED LINEAR THEORY. ASME J Eng Ind 112 (1990) 28-35.
6. S. G. Chen, A. G. Ulsoy, and Y. Koren, COMPUTATIONAL STABILITY ANALYSIS OF CHATTER IN TURNING. ASME J Manuf Sci Eng 119 (1997) 457-60.
7. G. Stepan, RETARDED DYNAMICAL SYSTEMS. Longman, Harlow, 1989.
8. G. Stepan, DELAY DIFFERENTIAL EQUATION MODELS FOR MACHINE TOOL CHATTER, in Dynamics and Chaos in Manufacturing Process F. C. Moon (ed). Wiley, New York, 1997, pp. 165-191.
9. Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics. Academic Press, Cambridge, 1993.
10. A. Nayfeh, C. Chin, and J. Pratt, APPLICATION OF PERTURBATION METHODS TO TOOL CHATTER DYNAMICS, in Dynamics and Chaos in Manufacturing Processes F. C. Moon (ed), Wiley, New York, 1997, pp. 193-213.
11. T. Kalmar-Nagy, G. Stepan, and F. C. Moon, SUBCRITICAL HOPF BIFURCATION IN THE DELAY EQUATION MODEL FOR MACHINE TOOL VIBRATIONS. Nonlinear Dynamics 26 (2001) 121-42.
12. M. S. Fofana, DELAY DYNAMICAL SYSTEMS AND APPLICATIONS TO NONLINEAR MACHINE-TOOL CHATTER. Chaos Solitons and Fractals, 17 (2003) 731-47.
13. R. Sipahi and N. Olgac, DEGENERATE CASES IN USING THE DIRECT METHOD. ASME J Dyn Syst Meas Cont 125 (2003) 194-201.
14. N. Olgac and R. Sipahi, A UNIQUE METHODOLOGY FOR CHATTER STABILITY MAPPING IN SIMULTANEOUS MACHINING. ASME J Manuf Sci Eng 127 (2005) 791-800.
15. F. M. Asl and A. G. Ulsoy, ANALYSIS OF A SYSTEM OF LINEAR DELAY DIFFERENTIAL EQUATIONS. ASME J Dyn Syst Meas Cont 125 (2003) 215-23.
16. S. Yi and A. G. Ulsoy, 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.
17. J. K. Hale, THEORY OF FUNCTIONAL DIFFERENTIAL EQUATIONS. Springer-Verlag, New York, 1977.
18. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, ON LAMBERT'S W FUNCTION. Advances in Computational Mathematics 5 (1996) 329-359.
19. H. Shinozaki and T. Mori, ROBUST STABILITY ANALYSIS OF LINEAR TIME-DELAY SYSTEMS BY LAMBERT W FUNCTION: SOME EXTREME POINT RESULTS. Automatica 42 (2006) 1791-99.
20. M. Malek-zavarei and M. Jamshidi, TIME-DELAYED SYSTEMS. North Holland, New York, 1987.
21. J. Forde and P. W. Nelson, APPLICATION OF STURM SEQUENCES TO BIFURCATION ANALYSIS OF DELAY DIFFERENTIAL EQUATION MODELS. J Math Anal Appl 300 (2004) 273-84.
22. A. Prestel and C. N. Delzell, POSITIVE POLYNOMIALS: FROM HILBERT'S 17TH PROBLEM TO REAL ALGEBRA. Springer-Verlag, Berlin, 2001.