Robust Chatter Stability Prediction of the Milling Process considering Uncertain Machining Positions

Table 1

Stable or unstable conditions for different distributions of the vertices in the complex plane.

Case no.	Milling condition	Distributions of the vertices in the complex plane
1	Stable	All vertices are located in the same quadrant, and the origin is outside the polygon.
2	Stable	Vertices are located in two adjacent quadrants, such as I-II, II-III, III-IV, or IV-I.
3	Stable	Vertices are located in diagonal quadrants such as I–III or II–IV, and the system is stable only when -αm^_ < π or -αm^+ > π. m is the mth quadrant, and and αm^_ are the minimum and maximum quadrant angles.
4	Stable	Vertices are located in three quadrants as I-II-III, II-III-IV, III-IV-I, and IV-I-II where only the following four conditions correspond to the stable system: -α1^_ < π in I-II-III; -α2^_ < π in II-III-IV; α3^_-α1^+ < π in III-IV-I; α4^_-α2^+ < π in IV-I-II.
5	Unstable	Vertices are located in four quadrants, and the origin is inside the polygon.