Estimation of the Bistable Zone for Machining Operations for the Case of a Distributed Cutting-Force Model

Tamás G. Molnár, Tamás Insperger, S. John Hogan, Gábor Stépán

Research output: Contribution to journalArticle

9 Citations (Scopus)


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 are shown numerically 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 number051008
JournalJournal of Computational and Nonlinear Dynamics
Issue number5
Publication statusPublished - Sep 2016


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

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Mechanical Engineering
  • Applied Mathematics

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