Stability and bifurcation analysis of a nonlinear DDE model for drilling

Journal of Nonlinear Science

Volumn 14, Issue 1, 2004, Pages 27-57

  Stone, E ^a   Campbell, S A ^b  

^a UTAH STATE UNIVERSITY   (United States)

^b UNIVERSITY OF WATERLOO   (Canada)

Author keywords

centre manifold reduction; chatter; delay differential equations; drilling

Indexed keywords

EID: 84867969858     PISSN: 09388974     EISSN: 14321467     Source Type: Journal    
DOI: 10.1007/s00332-003-0553-1     Document Type: Article

References (48)

1. Altintas, Y., and E. Budak, Analytical prediction of stability lobes in milling. Annals of the CIRP 44(1) (1995) 357-362.
2. Bayly, P., Halley, J. E., Davies, M. A., and J. R. Pratt, Stability analysis of interrupted cutting with finite time in cut. ASME Publication MED-Vol. 11, Proceedings of the ASME Manufacturing Engineering Division (2000) 989-996.
3. Bayly, P. V., S. A. Metzler, A. J. Schaut, and K. A. Young, Theory of torsional chatter in twist drills: Model, stability analysis and comparison to test. ASME J. Management Sci. & Eng. 123 (2001) 552-561. (Pubitemid 33715853)
4. Bayly, P. V., K. A. Young, and J. E. Halley, Analysis of tool oscillation and hole roundness error in a quasi-static model of reaming. ASME J. Management Sci. & Eng. 123 (2001) 387-396. (Pubitemid 33715832)
5. Bayly, P. V., M. T. Lamar, and S. G. Calvert, Low frequency regenerative vibration and the formation of lobed holes in drilling. ASME J. Management Sci.&Eng. 124 (2001) 275-285. (Pubitemid 35202616)
6. Belair, J., and S. A. Campbell, Stability and bifurcations of equilibria in a multiple-delayed differential equation. SIAM J. Appl. Math. 54 no. 5 (1994) 1402-1424.
7. Bhatt, S. and C. Hsu, Stability criteria for second order dynamical systems with time lag. J. Appl. Mech. 33 (1966) 113-118.
8. Campbell, S. A., Stability and bifurcation in the harmonic oscillator with multiple delayed feedback loops. Dyn. Cont. Disc. Impul. Sys. 5 (1999) 225-235.
9. Campbell, S. A., and J. Belair, Analytical and symbolically assisted investigation of Hopf bifurcations in delay-differential equations. Can. Appl. Math. Quart. 3 no. 2 (1995) 137-154.
10. Campbell, S. A., J. Belair, T. Ohira, and J. Milton, Limit cycles, tori, and complex dynamics in second-order differential equations with delayed negative feedback. J. Dyn. Diff. Eqs. 7 no. 1 (1995) 213-236.
11. Campbell, S. A., J. B'elair, T. Ohira, and J. Milton, Complex dynamics and multistability in damped harmonic oscillator with delayed negative feedback. Chaos 5 no. 4 (1995) 640-645. (Pubitemid 126012921)
12. Cooke, K., and Z. Grossman, Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86 (1982) 592-627.
13. Ema, S., and E. Marui, Suppression of chatter vibration in drilling. J. Management Sci. & Eng. 120 (1998) 200-202.
14. B. Ermentrout,XPPAUT5.52-The Differential Equations Tool. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA (2002).
15. Faria, T., and L. Magalh̃aes, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. JDE 122 (1995) 181-200.
16. Faria, T., and L. Magalh̃aes, Normal forms for retarded functional differential equations applications to Bogdanov-Takens singularity. JDE 122 (1995) 201-224.
17. Golubitsky, M., andW. F. Langford, Classification and unfoldings of degenerate Hopf bifurcation. JDE 41 (1981) 525-546.
18. Guckenheimer, J., and P. Holmes, Nonlinear Oscillation, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York (1983).
19. Hale, J. K., Flows on center manifolds for scalar functional differential equations. Proc. Roy. Soc. Edinburgh 101A (1985) 193-201.
20. Hale, J. K., and S. M. Verduyn Lunel, Introduction to Functional Differential Equations. Springer-Verlag, New York (1993).
21. Hsu, C., and S. Bhatt, Stability charts for second-order dynamical systems with time lag. J. Appl. Mech. 33 (1966) 119-124.
22. Kalm'ar, T., G. St'ep'an, and F. Moon, Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlin. Dynam. 26 (2001) 121-142. (Pubitemid 34058435)
23. Koenigsberger, F., and J. Tlusty, Machine Tool Structures,Vol. I. Pergamon Press, N.Y. (1970) p. 122.
24. Longtin, A., and J. G. Milton, Complex oscillations in the human pupil light reflex with mixed and delayed feedback. Math. Biosci. 90 (1988) 183-199.
25. Merchant, M. E., Mechanics of the cutting process. J. Appl. Phys. 16 267 (1945).
26. Milton, J. G., and A. Longtin, Evaluation of pupil constriction and dilation from cycling measurements. Vision Res. 30 (1990) 515-525. (Pubitemid 20089833)
27. Minis, I., and R. Yanushevsky, A new theoretical approach for the prediction of chatter in milling. J. Eng. Ind. 115 (1993) 1-8. (Pubitemid 23631743)
28. Moon, F. C., and M. A. Johnson, Nonlinear dynamics and chaos in manufacturing processes. Dynamics and Chaos in Manufacturing Processes, ed. F. C. Moon,Wiley, New York (1998) 3-32.
29. Nayfeh, A. H., C.-M. Chin, and J. Pratt, Perturbation methods in nonlinear dynamics-Applications of machining dynamics. J. Man. Sci. Eng. 1194A (1997) 485-493. (Pubitemid 127825755)
30. Oxley, P. L. B., The Mechanics of Machining. Ellis Horwood Ltd., Chincester (1989).
31. Pratt, J., and A. H. Nayfeh, Design and modeling for chatter control. Nonlin. Dynam. 19 (1999) 49-69.
32. Rincon, D. M., and A. J. Ulsoy, Effects of drill vibrations on cutting forces and torques. Annals of the CIRP 43 (1994) 59-62.
33. Rincon, D. M., and A. J. Ulsoy, Complex geometry, rotary inertia and gyroscopic moment effects on drill vibrations. J. Sound & Vib. 188 (1995) 701-715.
34. Shridar, R., R. E. Hohn, and G. W. Long, A general formulation of the milling process equation. Trans. ASME 79 (1957) 139-148.
35. Shridar, R., R. E. Hohn, and G.W. Long, A stability algorithm for the general milling process. J. Eng. Ind. 90 (1968b) 330-334.
36. Stepan, G., Delay-differential equation models for machine tool chatter. Dynamics and Chaos in Manufacturing Processes, ed. F. C. Moon, Wiley, New York (1998) 165-191.
37. Stepan, G., Retarded Dynamical Systems: Stability and Characteristic Functions. Pitman Research Notes in Mathematics,Vol. 210. Longman Scientific and Technical, London (1989).
38. Stephenson, D. A., and J. S. Agapiou, Calculation of main cutting edge forces and torque for drills with arbitrary geometries. Int. J. Machine Tools Man. 32 4 (1992) 521-538. (Pubitemid 23573194)
39. Stephenson, D. A., and S. M.Wu, Computer models for the mechanics of three-dimensional cutting processes-Part II: Results for oblique end turning and drilling. J. Eng. Ind. 110 (1988) 38-43. (Pubitemid 18566676)
40. Stone, E., and A. Askari, Nonlinear models of chatter in drilling processes. Dynam. Sys. 17 no. 1 (2002) 65-85.
41. Takens, F., Unfoldings of certain singularities of vector fields: Generalized Hopf bifurcations. JDE 14 (1973) 476-493.
42. Tarng, Y. S., and T. C. Li, Detection and suppression of drilling chatter. J. Dyn. Sys. Meas. & Control 116 (1994) 729-734.
43. Tarng, Y. S., and T. C. Li, Adaptive recognition of drilling chatter. J. Materials Proc. Tech. 48 (1995) 247-253.
44. Tekinalp, O., and A. G. Ulsoy, Modeling and finite element analysis of drill bit vibration. ASME J. Eng. Ind. 111 (1989) 148-155.
45. Tlusty, J., The dynamics of high-speed milling. J. Eng. Ind. 108 (1986) 59-67.
46. Tobias, S. A., Machine Tool Vibration. Wiley, New York (1965).
47. Waterloo Maple Software, Maple 7. University of Waterloo, Waterloo, Canada (2001).
48. Wischert, W., A. Wunderlin, A. Pelster, M. Olivier, and J. Groslambert, Delay-induced instabilities in nonlinear feedback systems. Phys. Rev. E 49 no. 1 (1994) 203-219.