Estimating Critical Hopf Bifurcation Parameters for a Second Order
Delay Differential Equation with Application to Machine Tool
Chatter
David E. Gilsinn
NIST/MCSD/OCGG
Thursday, September 26, 2002 15:00-16:00,
Room 145, NIST North (820)
Gaithersburg
Thursday, September 26, 2002 13:00-14:00,
Room 4511
Boulder
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. KearsleyNote: Visitors from outside NIST must contact
Robin Bickel; (301) 975-3668;
at least 24 hours in advance.
|
