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Delay Induced Canards in High Speed Machining

We consider here a model from [1] 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 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 [2]. We use numerical simulations and numerical continuation to support and verify our analysis.