Nonlinear Dynamics

, Volume 30, Issue 2, pp 103–154 | Cite as

Estimating Critical Hopf Bifurcation Parameters for a Second-Order Delay Differential Equation with Application to Machine Tool Chatter

  • David E. Gilsinn


Nonlinear time delay differential equations are well known to havearisen in models in physiology, biology and population dynamics. Theyhave also arisen in models of metal cutting processes. Machine toolchatter, from a process called regenerative chatter, has been identifiedas self-sustained oscillations for nonlinear delay differentialequations. The actual chatter occurs when the machine tool shifts from astable fixed point to a limit cycle and has been identified as arealized Hopf bifurcation. This paper demonstrates first that a class ofnonlinear delay differential equations used to model regenerativechatter satisfies the Hopf conditions. It then gives a precisecharacterization of the critical eigenvalues on the stability boundaryand continues with a complete development of the Hopf parameter, theperiod of the bifurcating solution and associated Floquet exponents.Several cases are simulated in order to show the Hopf bifurcationoccurring at the stability boundary. A discussion of a method ofintegrating delay differential equations is also given.

center manifolds delay differential equations exponential polynomials Hopf bifurcation limit cycle machine tool chatter normal form semigroup of operators subcritical bifurcation 


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  1. 1.
    van der Heiden, U., 'Delays in physiological systems', Journal of Mathematical Biology 8, 1979, 345-364.Google Scholar
  2. 2.
    MacDonald, N. Biological Delay Systems: Linear Stability Theory, Cambridge University Press, Cambridge, 1989.Google Scholar
  3. 3.
    Kuang, Y., Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA, 1993.Google Scholar
  4. 4.
    Tlusty, J., 'Machine dynamics', in Handbook of High-speed Machine Technology, Robert I. King (ed.), Chapman and Hall, New York, 1985, pp. 48-153.Google Scholar
  5. 5.
    Davies, M., Balachandran, B., 'Impact dynamics in milling of thin-walled structures', in Proceedings of the Nonlinear Dynamics and Controls Symposium, International Mechanical Engineering Congress and Expedition, Atlanta, GA, November 12-22, A. K. Bajaj, N. S. Namachchivaya, and M. Franchek (eds.), DE-Vol. 91, ASME, New York, 1996, pp. 67-72.Google Scholar
  6. 6.
    Davies, M. A., Dutter, B., Pratt, J., Schaut, A. J., and Bryan, J., 'On the dynamics of high-speed milling with long slender endmills', Annals of the CIRP 47, 1998, 293-296.Google Scholar
  7. 7.
    Zhao, M. X., Balachandran, B., Davies, M. A., and Pratt, J. R., 'Dynamics and stability of partial immersion milling operation', in Proceedings of DETC '99, 1999 ASME Design Engineering Technical Conferences, Las Vegas, NV, September 12-15, B. Balachandran, A. Kurdila, J. Pratt, and K. Murphy (eds.), ASME, New York, 1999, Paper No. DETC99/VIB-8058.Google Scholar
  8. 8.
    Balachandran, B., 'Nonlinear dynamics of milling processes', Philosophical Transactions of the Royal Society of London, Series A 359, 2001, 793-819.Google Scholar
  9. 9.
    Balachandran, B. and Zhao, M. X., 'A mechanics based model for study of dynamics of milling operations', Meccanica 35, 2000, 89-109.Google Scholar
  10. 10.
    Zhao, M. X. and Balachandran, B., 'Dynamics and stability of milling process', International Journal of Solids and Structures 38, 2001, 2233-2248.Google Scholar
  11. 11.
    Hassard, B. D., Kazarinoff, N. D., and Wan, Y. H., Theory and Applications of Hopf Bifurcations, Cambridge University Press, Cambridge, 1981.Google Scholar
  12. 12.
    Kalmár-Nagy, T., Pratt, J. R., Davies, M. A., and Kennedy, M.D., 'Experimental and analytical investigation of the subcritical instability in metal cutting', in Proceedings of DETC'99, 17th ASME Biennial Conference on Mechanical Vibration and Noise, Las Vegas, NV, September 12-15, B. Balachandran, A. Kurdila, J. Pratt, and K. Murphy (eds.), ASME, New York, 1999, pp. 1-9.Google Scholar
  13. 13.
    Tlusty, J. and Ismail, F., 'Basic non-linearity in machining chatter', Annals of the CIRP 30, 1981, 299-304.Google Scholar
  14. 14.
    Jemielniak, K. and Widota, A., 'Numerical simulation of non-linear chatter vibration in turning', International Journal of Machine Tools and Manufacture 29, 1989, 239-247.Google Scholar
  15. 15.
    Hanna, N. H. and Tobias, S. A., 'A theory of nonlinear regenerative chatter', ASME Journal of Engineering for Industry 96, 1974, 247-255.Google Scholar
  16. 16.
    Nayfeh, A. H., Chin, C. M., and Pratt, J., 'Perturbation methods in nonlinear dynamics-Applications to machining dynamics', Journal of Manufacturing Science and Engineering 119, 1997, 485-493.Google Scholar
  17. 17.
    Pratt, J. R., Davies, M. A., Evans, C. J., and Kennedy, M. D., 'Dynamic interrogation of a basic cutting process', Annals of the CIRP 48(1), 1999, 39-42.Google Scholar
  18. 18.
    Stépán, G. and Kalmár-Nagy, T., 'Nonlinear regenerative machine tool vibrations', in Proceedings of DETC'97, 1997 ASME Design Engineering Technical Conferences, Sacramento, CA, September 14-17, ASME, New York, 1997, pp. 1-11.Google Scholar
  19. 19.
    Marsh, E. R., Yantek, D. S., Davies, M. A., and Gilsinn, D. E., 'Simulation and measurement of chatter in diamond turning', Journal of Manufacturing Science and Engineering 120, 1998, 230-235.Google Scholar
  20. 20.
    Yantek, D. S., Marsh, E. R., Davies, M. A., and Gilsinn, D. E., 'Simulation and measurement of chatter in diamond turning', ASME, Manufacturing Science and Technology MED-Vol. 6-2 1997, 389-394.Google Scholar
  21. 21.
    Chow, C. and Mallet-Paret, J., 'Integral averaging and bifurcation', Journal of Differential Equations 26, 1977, 112-159.Google Scholar
  22. 22.
    Claeyssen, J. R., 'The integral-averaging bifurcation method and the general one-delay equation', Journal of Mathematical Analysis and Applications 78, 1980, 429-439.Google Scholar
  23. 23.
    Ioos, G. and Joseph, D. D., Elementary Stability and Bifurcation Theory, Springer-Verlag, New York, 1980.Google Scholar
  24. 24.
    Hale, J. K. and Lunel, S. M. V., Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.Google Scholar
  25. 25.
    Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamic Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York, 1990.Google Scholar
  26. 26.
    Kalmár-Nagy, T., Stépán, G., and Moon, F. C., 'Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations', Nonlinear Dynamics 26, 2001, 121-142.Google Scholar
  27. 27.
    Kalmár-Nagy, T., Moon, F. C., and Stépán, G., 'Regenerative machine tool vibrations', Dynamics of Continuous, Discrete and Impulsive Systems, to appear.Google Scholar
  28. 28.
    Campbell, S. A., Bélair, J., Ohira, T., and Milton, J., 'Limit cycles, tori and complex dynamics in a second-order differential equation with delayed negative feedback', Journal of Dynamics and Differential Equations 7, 1995, 213-236.Google Scholar
  29. 29.
    Liao, X., Wong, K.-W., and Wu, Z., 'Hopf bifurcation and stability of periodic solution for van der Pol equation with distributed delay', Nonlinear Dynamics 21, 2001, 23-44.Google Scholar
  30. 30.
    Kazarinoff, N. D., Wan, Y.-H., and van den Driessche, P., 'Hopf bifurcation and stability of periodic solutions of differential-difference and integro-differential equations', Journal of the Institute of Mathematics and its Applications 21, 1978, 461-477.Google Scholar
  31. 31.
    Avellar, C. E. and Hale, J. K., 'On the zeros of exponential polynomials', Journal of Mathematical Analysis and Applications 73, 1980, 434-452.Google Scholar
  32. 32.
    Bellman, R. and Cooke, K. L., Differential-Difference Equations, Academic Press, New York, 1963.Google Scholar
  33. 33.
    Pinney, E., Ordinary Difference-Differential Equations, University of California Press, Berkeley, CA, 1958.Google Scholar
  34. 34.
    Stepan, G., Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific & Technical, Essex, 1989.Google Scholar
  35. 35.
    Altintas, Y. and Budak, E., 'Analytic prediction of stability lobes in milling', Annals of the CIRP 44, 1995, 357-362.Google Scholar
  36. 36.
    Apostol, T. M., Mathematical Analysis, Addison-Wesley, Menlo Park, CA, 1974.Google Scholar
  37. 37.
    Pennisi, L.L., Elements of Complex Variables, Holt, Rinehart and Winston, New York, 1963.Google Scholar
  38. 38.
    Driver, R. D., Ordinary and Delay Differential Equations, Springer-Verlag, New York, 1977.Google Scholar
  39. 39.
    Hille, E. and Phillips, R. S., Functional Analysis and Semi-Groups, American Mathematical Society, Providence, RI, 1957.Google Scholar
  40. 40.
    Krasovskii, N. N., Stability of Motion, Stanford University Press, Stanford, CA, 1963.Google Scholar
  41. 41.
    Yosida, K., Functional Analysis, Springer-Verlag, Berlin, 1965.Google Scholar
  42. 42.
    Hale, J., Functional Differential Equations, Springer-Verlag, New York, 1971.Google Scholar
  43. 43.
    Hale, J. K., 'Linear functional-differential equations with constant coefficients', Contributions to Differential Equations II, 1963, 291-317.Google Scholar
  44. 44.
    Halanay, A., Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, 1966.Google Scholar
  45. 45.
    Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.Google Scholar
  46. 46.
    Carr, J., Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981.Google Scholar
  47. 47.
    Nayfeh, A. H., Method of Normal Forms, Wiley, New York, 1993.Google Scholar
  48. 48.
    Hassard, B. and Wan, Y. H., 'Bifurcation formulae derived from center manifold theory', Journal of Mathematical Analysis and Applications 63, 1978, 297-312.Google Scholar
  49. 49.
    Taylor, J. R., On the Art of Cutting Metals, ASME, New York, 1906.Google Scholar
  50. 50.
    Tobias, S. A. and Fishwick, W., 'The chatter of lathe tools under orthogonal cutting conditions', Transactions of the ASME 80, 1958, 1079-1088.Google Scholar
  51. 51.
    El'sgol'ts, L. E. and Norkin, S. B., Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York, 1973.Google Scholar
  52. 52.
    Atkinson, K. E., An Introduction to Numerical Analysis, Wiley, New York, 1978.Google Scholar
  53. 53.
    Oberle, H. J. and Pesch, H. A., 'Numerical treatment of delay differential equations by Hermite interpolation', Numerische Mathematik 37, 1981, 235-255.Google Scholar
  54. 54.
    Shampine, L. F. and Watts, H. A., 'The art of writing a Runge-Kutta code, II', Applied Mathematics and Computation 5, 1979, 93-121.Google Scholar
  55. 55.
    Forsythe, G. E., Malcolm, M. A., and Moler, C. B., Computer Methods for Mathematical Computations, Prentice-Hall, Englewood Cliffs, NJ, 1977.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • David E. Gilsinn
    • 1
  1. 1.Mathematical and Computational Sciences DivisionNational Institute of Standards and TechnologyGaithersburgU.S.A

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