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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