Non-linear Oscillations of Milling

The principal features of two mathematical models that can be used to study non-linear oscillations of a workpiece – tool system during a milling operation are presented and explained in this article. These models are non-linear, non-homogeneous, delay-differential systems with time-periodic coefficients. In the treatment presented here, the sources of non-linearities are the multiple regenerative effect and the loss-of-contact effect. The time-delay effect is taken into account, and the dependence of this delay effect on the feed rate is modelled. A variable time delay is introduced to capture the influence of the feed-rate in one of the models. Two formulations that can be used to carry out stability analysis of periodic solutions are presented. The models presented and the stability-analysis formulations are relevant for predicting and understanding chatter in milling.