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Using modified Nyquist contours, the dominant poles of the closed loop delay-differential equation for machining systems such as milling are identified. Contours with constant damping ratio of the dominant poles are constructed using this method. These contours are similar in shape to the stability lobes, but move upwards and to the right as the instability parameter increases. Additionally, it is possible to study the movement of the dominant poles to the right-hand side of the complex plane as the system becomes unstable by increasing the depth of cut at a constant spindle speed. The movement of the dominant pole is shown to be towards the right (unstable) and upward (higher vibration frequency) of the complex plane. In some cases, there would be a jump of vibration frequency due to the change of the lobe number. It is also shown that the damping ratio of the structure strongly affects both the vibration frequency and the damping ratio of the dominant poles in the closed loop system. Finally, in two milling experiments with two different spindle speeds and continuously increasing depth of cuts, vibration frequencies are measured and compared to the theoretical predictions. The measurements agree with the theoretical predictions, particularly in the unstable cutting conditions

Frequency Domain Study of Vibrations above and under Stability Lobes in Machining Systems