Abstract
We consider here a model from Stone and Askari [Nonlinear models of chatter in drilling process, Dyn. Syst. 17 (2002), pp. 65-85] for regenerative chatter in a drilling process. The model is a nonlinear delay differential equation where the delay arises from the fact that the cutting tool passes over the metal surface repeatedly. For any fixed value of the delay, a large enough increase in the width of the chip being cut results in a Hopf bifurcation from the steady state, which is the origin of the chatter vibration. We show that for zero delay the Hopf bifurcation is degenerate and that for a small delay this leads to a canard explosion. That is, as the chip width is increased beyond the Hopf bifurcation value, there is a rapid transition from a small amplitude limit cycle to a large relaxation cycle. Our analysis relies on perturbation techniques and a small delay approximation of the DDE model due to Chicone [Inertial and slow manifolds for delay differential equations, J. Diff. Eqs 190 (2003), pp. 364-406]. We use numerical simulations and numerical continuation to support and verify our analysis.
Original language | English |
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Pages (from-to) | 373-392 |
Number of pages | 20 |
Journal | Dynamical Systems |
Volume | 24 |
Issue number | 3 |
DOIs | |
State | Published - Sep 2009 |