The high speed milling represents the most important process to produce parts in different fields such as aeronautics, automotive and mould. It allows obtaining parts with complex form and with the best surface quality. So, it remains essential to understand the impact of both the cutting force model and its parameters on the tool tip response. Consequently, the mastering of chatter vibration in milling and surface properties of the work piece is essential. In this paper, the simulation of machining is applied to determine the cutting forces distribution. A spindle system modeling is presented using a new approach: Both rigid and flexible modes of the spindle's shaft are taken into account. The shaft is discretized with the Timoshenko beam finite elements with different circular sections. Nonlinear electromagnetic loads exerted by AMBs are computed in terms of the nominal air gap between bearings and spindle, the control current and the displacement of each node. A parametric study is performed to determine the influence of some parameters, such as the feed rate, the tangential cutting coefficient, the spindle speed and the axial depth of the cut on the cutting forces and the chatter vibrations in milling.

Xiong proved that if f: I → I is any map of the unit interval I, then the depth of the centre of f is at most 2, and Ye proved that for any map f: T → T of a finite tree T, the depth of the centre of f is at most 3. It is natural to ask whether the result can be dendrites. In this note, we show that there is dendrite D such that for any countable ordinal number λ there is a map f: D →D such that the depth of centre of f is λ. As a corollary, we show that for any countable ordinal number λ there is a map (respectively a homeomorphism) f of a 2-dimensional ball B2 (respectively a 3-dimensional ball B3) such that the depth of centre of f is λ.