Delay induced canards in a model of high speed machining

Abstract

To cite this Article Campbell, Sue Ann, Stone, Emily and Erneux, Thomas(2009) 'Delay induced canards in a model of high speed machining ',Dynamical Systems,24:3,[373][374][375][376][377][378][379][380][381][382][383][384][385][386][387][388][389][390][391][392] To link to this Article: DOI: 10.1080/14689360902852547 URL: http://dx.doi.org/10.1080/14689360902852547Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden.The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Vol. 24, No. 3, September 2009, 373-392 Delay induced canards in a model of high speed machining (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. Dynamical Systems

Mixed-Mode Oscillations with Multiple Time Scales

“…In principle, this should be expected at least for the case of a finite number of fixed delays, for which the DDE does not feature a continuous spectrum [96]. Indeed a positive answer was recently obtained by Campbell, Stone, and Erneux [32] for a two-dimensional DDE model of high-speed machining. In their system a small delay induces perturbation from a degenerate Hopf bifurcation, which results in a canard explosion as discussed in section 2.2; see also [34] for details of the underlying theory for slow-fast DDEs with small delay.…”

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“…A similar system has been investigated in a mechanical context [6], where the behavior of a machine is described by a delay differential equation with a slow variable. The authors focus on the behavior of the system in the limit of small delays.…”

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