Nonlinear Dynamics

, Volume 47, Issue 1–3, pp 275–283 | Cite as

State-dependent delay in regenerative turning processes

  • Tamás Insperger
  • Gábor Stépán
  • Janos Turi
Original Article


Stability of a two degrees of freedom model of the turning process is considered. An accurate modeling of the surface regeneration shows that the regenerative delay, determined by the combination of the workpiece rotation and the tool vibrations, is in fact state-dependent. For that reason, the mathematical model considered in this paper is a delay-differential equation with state-dependent time delay. In order to study linearized stability of stationary cutting processes, an associated linear system, corresponding to the state-dependent delay equation, is derived. Stability analysis of this linear system is performed analytically.

A comparison between the state-dependent delay model and the previously used constant or time-periodic delay models shows that the incorporation of the state-dependent delay into the model slightly affects the linear stability properties of the system in certain parameter domains.


Machine tool chatter Regenerative effect State-dependent delay Linearization Stability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Tlusty, J., Polacek, A., Danek, C., Spacek, J.: Selbsterregte schwingungen an Werkzeugmaschinen. VEB Verlag Technik, Berlin (1962)Google Scholar
  2. 2.
    Tobias, S.A.: Machine Tool Vibration. Blackie, London (1965)Google Scholar
  3. 3.
    Balachandran, B., Zhao, M.X.: A mechanics based model for study of dynamics of milling operations. Meccanica 35(2), 89–109 (2000)CrossRefMATHGoogle Scholar
  4. 4.
    Peigne, G., Paris, H., Brissaud, D., Gouskov, A.: Impact of the cutting dynamics of small radial immersion milling operations on machined surface roughness. Int. J. Mach. Tools Manuf. 44(11), 1133–1142 (2004)CrossRefGoogle Scholar
  5. 5.
    Stépán, G.: Retarded dynamical systems. Longman, Harlow (1989)Google Scholar
  6. 6.
    Altintas, Y., Budak, E.: Analytical prediction of stability lobes in milling. Ann. CIRP 44(1), 357–362 (1995)CrossRefGoogle Scholar
  7. 7.
    Insperger, T., Mann, B.P., Stépán, G., Bayly, P.V.: Stability of up-milling and down-milling, Part 1: Alternative analytical methods. Int. J. Mach. Tools Manuf. 43(1), 25–34 (2003)CrossRefGoogle Scholar
  8. 8.
    Bayly, P.V., Halley, J.E., Mann, B.P., Davies, M.A.: Stability of interrupted cutting by temporal finite element analysis. J. Manuf. Sci. Eng. 125(2), 220–225 (2003)CrossRefGoogle Scholar
  9. 9.
    Faassen, R.P.H., van de Wouw, N., Oosterling, J.A.J., Nijmeijer, H.: Prediction of regenerative chatter by modeling and analysis of high-speed milling. Int. J. Mach. Tools Manuf. 43(14), 1437–1446 (2003)CrossRefGoogle Scholar
  10. 10.
    Szalai, R., Stépán, G.: Stability boundaries of high-speed milling corresponding to period doubling are essentially closed curves. Proceedings of ASME International Mechanical Engineering Conference and Exposition, Washington D.C., USA, paper no. IMECE2003-42122 (2003)Google Scholar
  11. 11.
    Corpus, W.T., Endres, W.J.: Added stability lobes in machining processes that exhibit periodic time variation – Part 1: An analytical solution. J. Manuf. Sci. Eng. 126(3), 467–474 (2004)Google Scholar
  12. 12.
    Merdol, S.D., Altintas, Y.: Multi frequency solution of chatter stability for low immersion milling. J. Manuf. Sci. Eng. 126(3), 459–466 (2004)CrossRefGoogle Scholar
  13. 13.
    Gradišek, J., Kalveram, M., Insperger, T., Weinert, K., Stépán, G., Govekar, E., Grabec, I.: On stability prediction for milling. Int. J. Mach. Tools Manuf. 45(7–8), 769–781 (2005)CrossRefGoogle Scholar
  14. 14.
    Long, X.-H., Balachandran, B.: Milling model with variable time delay. Proceedings of the 2004 ASME International Mechanical Engineering Congress and Exposition, Anaheim, CA, paper no. IMECE2004-59207 (2004)Google Scholar
  15. 15.
    Long, X.-H., Balachandran, B., Mann, B.P.: Dynamics of milling processes with variable time delay. Nonlinear Dyn., in this issue (2006)Google Scholar
  16. 16.
    Faassen, R., van de Wouw, N., Oosterling, H., Nijmeijer, H.: Updated tool path modelling with periodic delay for chatter prediction in milling. Fifth EUROMECH Nonlinear Dynamics Conference, ENOC 2005, Eindhoven, The Netherlands, pp. 1080–1089 (2005)Google Scholar
  17. 17.
    Sexton, J.S., Milne, R.D., Stone, B.J.: A stability analysis of single point machining with varying spindle speed. Appl. Math. Modelling 1, 310–318 (1977)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Jayaram, S., Kapoor, S.G., DeVor, R.E.: Analytical stability analysis of variable spindle speed machining. J. Manuf. Sci. Eng. 122(3), 391–397 (2000)CrossRefGoogle Scholar
  19. 19.
    Insperger, T., Stépán, G.: Stability analysis of turning with periodic spindle speed modulation via semi-discretization. J. Vib. Control 10(12), 1835–1855 (2004)CrossRefMATHGoogle Scholar
  20. 20.
    Insperger, T., Stépán, G.: Semi-discretization method for delayed systems, Int. J. Numer. Methods Eng. 55(5), 503–518 (2002)Google Scholar
  21. 21.
    Insperger, T., Stépán, G.: Updated semi-discretization method for periodic delay-differential equations with discrete delay, Int. J. Numer. Methods Eng. 61(1), 117–141 (2004)Google Scholar
  22. 22.
    Győri, I., Hartung, F.: On the exponential stability of a state-dependent delay equation. Acta Sci. Math. 66, 71–84 (2000)Google Scholar
  23. 23.
    Krisztin, T., Arino, O.: The 2-dimensional attractor of a differential equation with state-dependent delay. J. Dynam. Differ. Equ. 13, 453–522 (2001)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Hartung, F., Turi, J.: Linearized stability in functional-differential equations with state-dependent delays. Proceedings of the conference Dynamical Systems and Differential Equations, added volume of Discrete and Continuous Dynamical Systems, pp. 416–425 (2000)Google Scholar
  25. 25.
    Luzyanina, T., Engelborghs, K., Roose, D.: Numerical bifurcation analysis of differential equations with state-dependent delay. Int. J. Bifurcation Chaos 11(3), 737–753 (2001)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Hartung, F.: Linearized stability in periodic functional differential equations with state-dependent delays J. Comput. Appl. Math. 174, 201–211 (2005)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Richard, T., Germay, C., Detournay, E.: Self-excited stick-slip oscillations of drill bits. Comptes rendus Mecanique 332(8), 619-626 (2004)Google Scholar
  28. 28.
    Germay, C., van de Wouw, N., Sepulchure, R., Nijmeijer, H.: Axial stick-slip limit cycling in drill-string dynamics with delay, Fifth EUROMECH Nonlinear Dynamics Conference, ENOC 2005, Eindhoven, The Netherlands pp. 1136-1143 (2005)Google Scholar
  29. 29.
    Insperger, T., Stépán, G., Hartung, F., Turi, J.: State-dependent regenerative delay in milling processes. in Proceedings of ASME International Design Engineering Technical Conferences, Long Beach CA, (2005), paper no. DETC2005-85282 (2005)Google Scholar
  30. 30.
    Hartung, F., Turi, J.: On differentiability of solutions with respect to parameters in state-dependent delay equations. J. Differ. Equ. 135(2), 192–237 (1997)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Applied MechanicsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Programs in Mathematical SciencesUniversity of Texas at DallasRichardsonUSA

Personalised recommendations