Criticality of Hopf bifurcation in state-dependent delay model of turning processes

T. Insperger, David A W Barton, G. Stépán

Research output: Contribution to journalArticle

65 Citations (Scopus)

Abstract

In this paper the non-linear dynamics of a state-dependent delay model of the turning process is analyzed. The size of the regenerative delay is determined not only by the rotation of the workpiece, but also by the vibrations of the tool. A numerical continuation technique is developed that can be used to follow the periodic orbits of a system with implicitly defined state-dependent delays. The numerical analysis of the model reveals that the criticality of the Hopf bifurcation depends on the feed rate. This is in contrast to simpler constant delay models where the criticality does not change. For small feed rates, subcritical Hopf bifurcations are found, similar to the constant delay models. In this case, periodic orbits coexist with the stable stationary cutting state and so there is the potential for large amplitude chatter and bistability. For large feed rates, the Hopf bifurcation becomes supercritical for a range of spindle speeds. In this case, stable periodic orbits instead coexist with the unstable stationary cutting state, removing the possibility of large amplitude chatter. Thus, the state-dependent delay in the model has a kind of stabilizing effect, since the supercritical case is more favorable from a practical viewpoint than the subcritical one.

Original languageEnglish
Pages (from-to)140-149
Number of pages10
JournalInternational Journal of Non-Linear Mechanics
Volume43
Issue number2
DOIs
Publication statusPublished - Mar 2008

Fingerprint

State-dependent Delay
Hopf bifurcation
Criticality
Hopf Bifurcation
Periodic Orbits
Chatter
Orbits
orbits
Numerical Continuation
Model
Bistability
spindles
Nonlinear Dynamics
Numerical analysis
Numerical Analysis
Vibration
Unstable
numerical analysis
vibration
Range of data

Keywords

  • Hopf bifurcation
  • Machine tool chatter
  • State-dependent delay

ASJC Scopus subject areas

  • Mechanical Engineering
  • Statistical and Nonlinear Physics

Cite this

Criticality of Hopf bifurcation in state-dependent delay model of turning processes. / Insperger, T.; Barton, David A W; Stépán, G.

In: International Journal of Non-Linear Mechanics, Vol. 43, No. 2, 03.2008, p. 140-149.

Research output: Contribution to journalArticle

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