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.
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January 2012
Research Papers
Analytic Bounds for Instability Regions in Periodic Systems With Delay via Meissner’s Equation
Eric A. Butcher,
Eric A. Butcher
Department of Mechanical and Aerospace Engineering,
New Mexico State University
, Las Cruces, NM 88003
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Brian P. Mann
Brian P. Mann
Department of Mechanical Engineering and Materials Science,
Duke University
, Durham, NC 27708
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Eric A. Butcher
Department of Mechanical and Aerospace Engineering,
New Mexico State University
, Las Cruces, NM 88003
Brian P. Mann
Department of Mechanical Engineering and Materials Science,
Duke University
, Durham, NC 27708J. Comput. Nonlinear Dynam. Jan 2012, 7(1): 011004 (10 pages)
Published Online: August 9, 2011
Article history
Received:
August 27, 2010
Revised:
June 10, 2011
Online:
August 9, 2011
Published:
August 9, 2011
Citation
Butcher, E. A., and Mann, B. P. (August 9, 2011). "Analytic Bounds for Instability Regions in Periodic Systems With Delay via Meissner’s Equation." ASME. J. Comput. Nonlinear Dynam. January 2012; 7(1): 011004. https://doi.org/10.1115/1.4004468
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