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Nonlinear Delay Equations With Fluctuating Delay: Application to Regenerative Chatter

[+] Author Affiliations
Ali Demir, N. Sri Namachchivaya

University of Illinois, Urbana, IL

W. F. Langford

University of Guelph, Guelph, ON, Canada

Paper No. IMECE2002-32389, pp. 311-321; 11 pages
doi:10.1115/IMECE2002-32389
From:
  • ASME 2002 International Mechanical Engineering Congress and Exposition
  • Design Engineering
  • New Orleans, Louisiana, USA, November 17–22, 2002
  • Conference Sponsors: Design Engineering Division
  • ISBN: 0-7918-3628-2 | eISBN: 0-7918-1691-5, 0-7918-1692-3, 0-7918-1693-1
  • Copyright © 2002 by ASME

abstract

The mathematical models representing machine tool chatter dynamics have been cast as differential equations with delay. The suppression of regenerative chatter by spindle speed variation is attracting increasing attention. In this paper, we study nonlinear delay differential equations with periodic delays which models the machine tool chatter with continuously modulated spindle speed. 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. We make use of Lyapunov-Schmidt Reduction method to determine the periodic solutions and analyze the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions close to the new stability boundary.

Copyright © 2002 by ASME

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