Closed-form estimations of the bistable region in metal cutting via the method of averaging

Tamas G. Molnar, Tamas Insperger, Gabor Stepan

Research output: Contribution to journalArticle


Machine tool vibrations in turning processes are analyzed by taking into account the nonlinearity of the cutting force characteristics. Unstable limit cycles are computed for the governing nonlinear delay-differential equation in order to determine the bistable technological parameter region where stable stationary cutting and large-amplitude machine tool vibrations coexist. Simple closed-form formulas are derived for the amplitude of limit cycles and for the size of the bistable region considering a general cutting force characteristics. The analytical results are determined by the method of averaging, which can be used to treat the nonlinearities without their third-order approximation. The results are confirmed by numerical continuation and using Melnikov's integral.

Original languageEnglish
Pages (from-to)49-56
Number of pages8
JournalInternational Journal of Non-Linear Mechanics
Publication statusPublished - Jun 2019



  • Averaging
  • Bistability
  • Machine tool chatter
  • Nonlinear dynamics

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics

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