On the bistable zone of milling processes

Zoltan Dombovari, G. Stépán

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

A modal-based model of milling machine tools subjected to time-periodic nonlinear cutting forces is introduced. The model describes the phenomenon of bistability for certain cutting parameters. In engineering, these parameter domains are referred to as unsafe zones, where steady-state milling may switch to chatter for certain perturbations. In mathematical terms, these are the parameter domains where the periodic solution of the corresponding nonlinear, time-periodic delay differential equation is linearly stable, but its domain of attraction is limited due to the existence of an unstable quasiperiodic solution emerging from a secondary Hopf bifurcation. A semi-numerical method is presented to identify the borders of these bistable zones by tracking the motion of the milling tool edges as they might leave the surface of the workpiece during the cutting operation. This requires the tracking of unstable quasiperiodic solutions and the checking of their grazing to a time-periodic switching surface in the infinitedimensional phase space. As the parameters of the linear structural behaviour of the tool/machine tool system can be obtained by means of standard modal testing, the developed numerical algorithm provides efficient support for the design of milling processes with quick estimates of those parameter domains where chatter can still appear in spite of setting the parameters into linearly stable domains.

Original languageEnglish
Article number20140409
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume373
Issue number2051
DOIs
Publication statusPublished - Sep 28 2015

Fingerprint

Milling (machining)
Machine tools
Chatter
machine tools
Milling machines
Quasi-periodic Solutions
Hopf bifurcation
Machine Tool
Numerical methods
Differential equations
Linearly
Unstable
Switches
milling machines
Cutting Force
Bistability
Domain of Attraction
Testing
grazing
Delay Differential Equations

Keywords

  • Bistable
  • Chatter
  • Delay
  • Milling
  • Time-periodic
  • Torus

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)
  • Engineering(all)

Cite this

On the bistable zone of milling processes. / Dombovari, Zoltan; Stépán, G.

In: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 373, No. 2051, 20140409, 28.09.2015.

Research output: Contribution to journalArticle

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