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

Tamas G. Molnar, T. Insperger, G. Stépán

Research output: Contribution to journalArticle

Abstract

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
JournalInternational Journal of Non-Linear Mechanics
DOIs
Publication statusPublished - Jan 1 2019

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Cutting Force
Metal cutting
Machine Tool
Limit Cycle
Averaging
Closed-form
Vibration
Metals
Nonlinearity
Numerical Continuation
Machine tools
Delay Differential Equations
Nonlinear Differential Equations
Unstable
Differential equations
Approximation

Keywords

  • Averaging
  • Bistability
  • Machine tool chatter
  • Nonlinear dynamics

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics

Cite this

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title = "Closed-form estimations of the bistable region in metal cutting via the method of averaging",
abstract = "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.",
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T1 - Closed-form estimations of the bistable region in metal cutting via the method of averaging

AU - Molnar, Tamas G.

AU - Insperger, T.

AU - Stépán, G.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Averaging

KW - Bistability

KW - Machine tool chatter

KW - Nonlinear dynamics

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