ITLApplied  Computational Mathematics Division
ACMD Seminar Series
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Estimating Critical Hopf Bifurcation Parameters for a Second Order Delay Differential Equation with Application to Machine Tool Chatter

David E. Gilsinn

Thursday, September 26, 2002 15:00-16:00,
Room 145, NIST North (820)
Thursday, September 26, 2002 13:00-14:00,
Room 4511

Abstract: Nonlinear time delay differential equations are well known to have arisen in models in physiology, biology and population dynamics. They have also arisen in models of metal cutting processes. Machine tool chatter, from a process called regenerative chatter, has been identified as self sustained oscillations for nonlinear delay differential equations. The actual chatter occurs when the machine tool shifts from a stable fixed point to a limit cycle and has been identified as a realized Hopf bifurcation. This paper demonstrates first that a class of nonlinear delay differential equations used to model regenerative chatter satisfies the Hopf conditions. It then gives a precise characterization of the critical eigenvalues on the stability boundary and continues with a complete development of the Hopf parameter, the period of the bifurcating solution and associated Floquet exponents. I will present several simulation cases are in order to show the Hopf bifurcation occurring at the stability boundary. I will also discuss a method of integrating delay differential equations.
Contact: A. J. Kearsley

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