Vibration frequencies in stable and unstable milling

• Cauchy's argument principle combined with an algorithm to divide the complex plane can find the dominant poles of closed loop machining systems.
• 3D plot of the closed loop damping ratio plots shows the peaks of stability.
• 3D plot of vibration frequency shows discontinuities and their relationship to the normalized tooth passing frequencies and peaks of stability.
• The method's results match the experiment's results, particularly in the chattering zone.

Vibration frequencies in machining may be employed for calculation of natural frequencies of the dominant modes in chatter and selection of chatter-free spindle speeds with large material removal rates. In this approach, it is important to investigate the relationship between the vibration frequencies, the natural frequencies, spindle speeds and depth of cuts for both stable and unstable cutting conditions. In this paper, the dominant poles of the closed loop time delay differential equation of a milling operation are calculated by successive sectioning of the complex plane and using Cauchy's argument principle. Vibration frequency and damping ratio of the closed loop machining system for each cutting condition is calculated based on the position of the dominant pole on the complex plane which provides 3D plots of the vibration frequency and closed loop damping ratio over any range of depth of cuts and spindle speeds. Finally, the findings of the analytical approach are compared to a machining experiment and a time domain simulation and differences and similarities in their predictions are discussed.

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