Analytic bounds for instability regions in periodic systems with delay via Meissner's equation

Eric Butcher, Brian P. Mann

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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-constantsegment 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.

Original languageEnglish (US)
Article number011004
JournalJournal of Computational and Nonlinear Dynamics
Volume7
Issue number1
DOIs
StatePublished - 2012
Externally publishedYes

Fingerprint

Time varying systems
Periodic Systems
Flip
Fold
Period Doubling
Chart
Eigenvalue
Cutting Force
Position Control
Transition Matrix
Numerical Stability
Coefficient
Delay Differential Equations
Convergence of numerical methods
Damped
Position control
Feedback Control
Lens
Linear Time
Time Delay

ASJC Scopus subject areas

  • Mechanical Engineering
  • Applied Mathematics
  • Control and Systems Engineering

Cite this

Analytic bounds for instability regions in periodic systems with delay via Meissner's equation. / Butcher, Eric; Mann, Brian P.

In: Journal of Computational and Nonlinear Dynamics, Vol. 7, No. 1, 011004, 2012.

Research output: Contribution to journalArticle

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