Stability prediction in radial immersion for milling with symmetric structure

• 2D directional coefficient matrix is reduced to a scalar coefficient.
• Analytical expression for the stability boundary in radial immersion is presented.
• Flowchart for establishing stability chart in radial immersion is established.
• Unstable islands and secondary lobes can exist in radial stability diagram.
• Symmetric milling is more stable than up/down milling for the same MRR.

In this study, an analytical approach is presented to find stability limits in terms of radial immersion for a given axial depth of cut, and vice versa. Under the assumption of axis-symmetric structure and using the zero order force model, the direction coefficient matrix is decoupled to reduce the 2D milling system to a 1D stability problem. The effect of the radial immersion and radial cutting coefficient on the system stability are explicitly represented through the eigenvalue function of the directional coefficient matrix. The resulting characteristic equation allows the limiting radial immersion be solved for a given axial immersion. A procedure is presented in obtaining the radial stability diagram, in which additional unstable island and secondary lobes are shown to exist besides the traditional lobes. Stability diagrams in both axial and radial immersion are presented to demonstrate the physical insights offered by the presented method. The model is validated by comparing with results from the existing analytical and numerical models.