Joint Mechanical Engineering-Mathematics Seminar

Gabor Stepan
Department of Applied Mechanics, Budapest University

Dynamics of high-speed milling

Abstract: As an introduction, basic stability results are summarised for delay differential equations with constant and time-periodic coefficients in second order systems modeling mechanical oscillators with low damping and delayed feed-back. Then the lecture briefly addresses the classical theory of regenerative vibrations in cutting processes, including experiments for thread cutting, stability charts, stable and unstable periodic motions as well as quasi-periodic and chaotic ones, and their sensibility on stochastic peturbations.

High-speed milling is often modeled as a kind of highly interrupted machining, when the ratio of time spent cutting to not cutting can be considered as a small parameter. In these cases, the classical regenerative vibration model breaks down to a simplified discrete mathematical model. The linear analysis of this discrete model leads to the recognition of the doubling of the so-called instability lobes in the stability charts of the machining parameters. This kind of lobe-doubling is related to the appearance of period doubling vibrations originated in a flip bifurcation. This is a new phenomenon occurring primarily in low-immersion high-speed milling along with the Neimark-Sacker bifurcations related to the classical self- excited vibrations or Hopf bifurcations. The nonlinear vibrations in case of period doubling is investigated and compared to the well- known subcritical nature of the Hopf bifurcations in turning processes. The identification of the global attractor in case of unstable cutting leads to contradiction between experiments and theory. This contradiction draws the attention to the limitations of the small parameter approach related to the highly interrupted cutting condition.

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