Stability and bifurcation analysis of a nonlinear DDE model for drilling

We study a model for chatter in twist drills derived by Stone and Askari [Dynam. Sys., 17, 1 (2002), 65-85], in which a linear vibration mode interacts with nonlinear cutting forces. This results in a delay differential equation describing an oscillator that is nonlinear in damping and with cross-terms in the damping and the delay. Linear stability analysis of the model with significant nonlinear terms is performed, and an analysis of the nonlinear stability of the primary Hopf bifurcation is observed. The latter is done via the construction, using symbolic algebra, of a two-dimensional centre manifold in the infinite dimensional space employing an algorithm developed by Campbell and B́elair [SIAM J. Appl. Math., 54, 5 (1994), 1402-1424; and Can. Appl. Math. Quart., 3, 2 (1995), 137-154]. Our analysis shows that the stability of the Hopf bifurcation depends on the type of vibration in question and on the cutting speed. These results are confirmed numerically and further bifurcations in the high-speed limit are also explored numerically, with tantalizing results that could be the basis of much future work.