Estimation of the bistable zone for machining operations for the case of a distributed cutting-force model

Tamás G. Molnár, Tamas Insperger, John Hogan, Gabor Stepan

Research output: Contribution to journalArticle (Academic Journal)peer-review

20 Citations (Scopus)
484 Downloads (Pure)


Regenerative machine tool chatter is investigated for a single-degree-of-freedom model of turning processes. The cutting force is modeled as the resultant of a force system distributed along the rake face of the tool, whose magnitude is a nonlinear function of the chip thickness. Thus, the process is described by a nonlinear delay-differential equation, where a short distributed delay is superimposed on the regenerative point delay. The corresponding stability lobe diagrams are computed, and it is shown that a subcritical Hopf bifurcation occurs along the stability boundaries for realistic cutting-force distributions. Therefore, a bistable region exists near the stability boundaries, where large-amplitude vibrations (chatter) may arise for large perturbations. Analytical formulas are obtained to estimate the size of the bistable region based on center manifold reduction and normal form calculations for the governing distributed-delay equation. The locally and globally stable parameter regions are computed numerically as well using the continuation algorithm implemented in DDE-BIFTOOL. The results can be considered as an extension of the bifurcation analysis of machining operations with point delay.
Original languageEnglish
Article number 051008-1
Number of pages10
JournalJournal of Computational and Nonlinear Dynamics
Issue number5
Publication statusPublished - 3 Feb 2016

Structured keywords

  • Engineering Mathematics Research Group


  • metal cutting
  • turning
  • delay-differential equation
  • distributed delay
  • Hopf bifurcation
  • center manifold reduction
  • bistable zones


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