Delay equations with fluctuating delay related to the regenerative chatter

Ali Demir, Alemdar Hasanov, N Sri Namachchivaya

Research output: Contribution to journalArticlepeer-review

Abstract

The mathematical models representing machine tool chatter dynamics have been cast as differential equations with delay. In this paper, non-linear delay differential equations with periodic delays which model the machine tool chatter with continuously modulated spindle speed are studied. The explicit time-dependent delay terms, due to spindle speed modulation, are replaced by state-dependent delay terms by augmenting the original equations. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The reduced bifurcation equation is obtained by making use of Lyapunov-Schmidt Reduction method. By using the reduced bifurcation equations, the periodic solutions are determined to analyze the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions near the new stability boundary.

Original languageEnglish (US)
Pages (from-to)464-474
Number of pages11
JournalInternational Journal of Non-Linear Mechanics
Volume41
Issue number3
DOIs
StatePublished - Apr 1 2006

Keywords

  • Bifurcation equation
  • Chatter suppression
  • Functional differential equation
  • Lyapunov-Schmidt reduction
  • Periodic solutions
  • Spindle speed variation
  • Stability boundaries

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics

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