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Analytic Bounds for Instability Regions in Periodic Systems With Delay via Meissner's Equation

Publication ,  Journal Article
Butcher Eric, A; Mann Brian, P
Published in: JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS
January 2012

A method for obtaining analytic bounds for period doubling and cyclic fold instability regions in linear time-periodic systems with piecewise constant coefficients and time delay is suggested. The method is based on the use of transition matrices for Meissner's equation corresponding to the desired type of instability. Analytic expressions for the disconnected regions of fold and flip instability for two- and three-segment coefficients including both complex and real eigenvalues in Meissner's equation are obtained. The proposed method when applied to the example of two-segment interrupted turning with complex eigenvalues in each segment yields the same results as those obtained recently for the boundaries of the flip regions (Szalai and Stepan, 2006, ``Lobes and Lenses in the Stability Chart of Interrupted Turning,{''} J Comput. Nonlinear Dyn., 1, pp. 205-211). Next, the period-doubling instability regions for a particular delay differential equation related to the damped Meissner's equation and the fold instabilities for a model of delayed position feedback control are analytically obtained. Finally, we extend the method to a single degree-of-freedom milling model with a three-piecewise-constant-segment approximation to the true specific cutting force in which lower bounds for and horizontal locations of the regions of flip instability are obtained. The analytic results are verified through numerical stability charts obtained using the temporal finite element method. Conditions for the existence of islands of instability are also obtained. {[}DOI: 10.1115/1.4004468]

Duke Scholars

Published In

JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS

DOI

ISSN

1555-1423

Publication Date

January 2012

Volume

7

Issue

1

Related Subject Headings

  • Acoustics
  • 4903 Numerical and computational mathematics
  • 4017 Mechanical engineering
  • 0913 Mechanical Engineering
  • 0103 Numerical and Computational Mathematics
 

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Butcher Eric, A., & Mann Brian, P. (2012). Analytic Bounds for Instability Regions in Periodic Systems With Delay via Meissner's Equation. JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS, 7(1). https://doi.org/10.1115/1.4004468
Butcher Eric, A., and P. Mann Brian. “Analytic Bounds for Instability Regions in Periodic Systems With Delay via Meissner's Equation.” JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS 7, no. 1 (January 2012). https://doi.org/10.1115/1.4004468.
Butcher Eric A, Mann Brian P. Analytic Bounds for Instability Regions in Periodic Systems With Delay via Meissner's Equation. JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. 2012 Jan;7(1).
Butcher Eric, A., and P. Mann Brian. “Analytic Bounds for Instability Regions in Periodic Systems With Delay via Meissner's Equation.” JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS, vol. 7, no. 1, Jan. 2012. Manual, doi:10.1115/1.4004468.
Butcher Eric A, Mann Brian P. Analytic Bounds for Instability Regions in Periodic Systems With Delay via Meissner's Equation. JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. 2012 Jan;7(1).

Published In

JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS

DOI

ISSN

1555-1423

Publication Date

January 2012

Volume

7

Issue

1

Related Subject Headings

  • Acoustics
  • 4903 Numerical and computational mathematics
  • 4017 Mechanical engineering
  • 0913 Mechanical Engineering
  • 0103 Numerical and Computational Mathematics