Nonlinear Dynamics

, Volume 63, Issue 3, pp 395–416 | Cite as

Analysis of the cutting tool on a lathe

Original Paper


We use a systematic approach combining a path-following scheme, the method of multiple scales, the method of harmonic balance, Floquet theory, and numerical simulations to investigate the local and global dynamics and stability of cutting tool on a lathe due to the regenerative mechanism. First, we use the method of multiple scales to determine the normal form of the Hopf bifurcation at all of the stability boundaries and calculate the limit cycles generated by the bifurcation. Then, we use a combination of a path-following scheme and the method of harmonic balance to continue the branch of generated limit cycles. Thus, we calculate small- and large-amplitude limit cycles and ascertain their stability using Floquet theory. We validate these results using numerical simulations. Then, we search for isolated branches of large-amplitude solutions coexisting with the linearly stable trivial solution. We use all of the results to generate bifurcation diagrams consisting of multiple large-amplitude stable and unstable branches of limit cycles coexisting with the trivial response, indicating three regions of operation, as in the experimental observations. Then, we investigate bifurcation control using cubic-velocity feedback and show that the unconditionally stable region can be expanded at the expense of the conditionally stable region.


Machine-tool chatter Hopf bifurcation Bifurcation control Normal form Method of multiple scales Harmonic balance Floquet theory 


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  1. 1.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley-Interscience, New York (1979) MATHGoogle Scholar
  2. 2.
    Hanna, N.H., Tobias, S.A.: A theory of nonlinear regenerative chatter. ASME J. Eng. Ind. 96, 247–255 (1974) CrossRefGoogle Scholar
  3. 3.
    Shi, H.M., Tobias, S.A.: Theory of finite amplitude machine tool instability. Int. J. Mach. Tool Des. Res. 24, 45–69 (1984) CrossRefGoogle Scholar
  4. 4.
    Nayfeh, A.H., Chin, C.M., Pratt, J.R.: Perturbation methods in nonlinear dynamics: applications to machining dynamics. J. Manuf. Sci. Eng. 119, 485–493 (1997) CrossRefGoogle Scholar
  5. 5.
    Nayfeh, A.H.: Perturbation Methods. Wiley-Interscience, New York (1973) MATHGoogle Scholar
  6. 6.
    Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley-Interscience, New York (1981) MATHGoogle Scholar
  7. 7.
    Pratt, J.R., Nayfeh, A.H.: Design and modeling for chatter control. Nonlinear Dyn. 19, 49–69 (1999) MATHCrossRefGoogle Scholar
  8. 8.
    Pratt, J.R., Nayfeh, A.H.: Chatter control and stability analysis of a cantilever boring bar under regenerative cutting conditions. Philos. Trans. R. Soc., Math. Phys. Eng. Sci. 359, 759–792 (2001) MATHCrossRefGoogle Scholar
  9. 9.
    Moon, F.C., Kalmár-Nagy, T.: Nonlinear models for complex dynamics in cutting materials. Philos. Trans. R. Soc. Lond. 359, 695–711 (2001) MATHCrossRefGoogle Scholar
  10. 10.
    Kalmár-Nagy, T., Stépán, G., Moon, F.C.: Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dyn. 26, 121–142 (2001) MATHCrossRefGoogle Scholar
  11. 11.
    Gilsinn, D.: Estimating critical Hopf bifurcation parameters for a second-order delay differential equation with application to machine tool. Nonlinear Dyn. 30, 103–154 (2002) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nayfeh, A.H.: Order reduction of retarded nonlinear systems: The method of multiple scales versus center-manifold reduction. Nonlinear Dyn. 51, 483–500 (2008) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Stépán, G.: Modeling nonlinear regenerative effects in metal cutting. Proc. R. Soc. Lond. A 359, 739–757 (2001) MATHGoogle Scholar
  14. 14.
    Wahi, P., Chatterjee, A.: Regenerative tool chatter near a codimension 2 Hopf point using multiple scales. Nonlinear Dyn. 40, 323–338 (2005) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Chandiramani, N.K., Pothala, T.: Dynamics of 2-dof regenerative chatter during turning. J. Sound Vib. 290, 448–464 (2006) CrossRefGoogle Scholar
  16. 16.
    Faassen, R.P.H., van de Wouw, N., Nijmeijer, H., Oosterling, J.A.J.: An improved tool path model including periodic delay for chatter prediction in milling. J. Comput. Nonlinear Dyn. 2, 167–179 (2007) CrossRefGoogle Scholar
  17. 17.
    Long, X.-H., Balachandran, B., Mann, B.P.: Dynamics of milling processes with variable time delay. Nonlinear Dyn. 47, 49–63 (2007) MATHCrossRefGoogle Scholar
  18. 18.
    Long, X.-H., Balachandran, B.: Stability analysis for milling process. Nonlinear Dyn. 49, 349–359 (2007) MATHCrossRefGoogle Scholar
  19. 19.
    Insperger, T., Stépán, G., Turi, J.: State-dependent delay in regenerative turning processes. Nonlinear Dyn. 47(1–3), 275–283 (2007) MATHGoogle Scholar
  20. 20.
    Insperger, T., Barton, D.A.W., Stépán, G.: Criticality of Hopf bifurcation in state-dependent delay model of turning processes. Int. J. Non-Linear Mech. 43, 140–149 (2008) CrossRefGoogle Scholar
  21. 21.
    Taylor, F.W.: On the art of cutting metals. ASME 28, 31–350 (1907) Google Scholar
  22. 22.
    Kalmár-Nagy, T., Pratt, J.R., Davies, M.A., Kennedy, M.D.: Experimental and analytical investigation of the subcritical instability in metal cutting. In: Balachandran, B., Kurdila, A., Pratt, J.R., Murphy, K. (eds.) Proceedings of DETC’99, 17th ASME Biennial Conference on Mechanical Vibration and Noise, Las Vegas, NV, September 12–15, pp. 1–9. ASME, New York (1999) Google Scholar
  23. 23.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley-Interscience, New York (1995) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Engineering Science and MechanicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Northrup GrummanArlingtonUSA

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