Rapid stability analysis of serrated end mills using graphical-frequency domain methods

Serrated tools result in an irregular distribution of chip thickness and varying time delays along their flutes that disturb the regenerative chatter mechanism and maximize productivity in rough machining operations. The chatter stability simulation with such tools is complex due to the delay differential equations with multiple and distributed delays. Although the time domain numerical integration methods and the semi-discretization method are accurate, they are computationally expensive and hence inhibit the optimal design of serrated cutters. Despite the advantages that serrated tools offer, there has been little attention paid to rapidly predict stability lobes diagram for these tools to help instruct their optimal designs. To address this need, new graphicalfrequency methods are presented to solve for stability of serrated tools with arbitrary geometry. The graphical method averages the number of teeth in cut to make the system time-invariant and results in a closed-form analytical solution. It reduces the simulation time by up to -99% compared to the semi-discretization method. Generalized solutions to the classical zero-order approximation and the multi-frequency methods are also presented by using the Nyquist stability criterion. The use of the zero-order approximation method and the multi-frequency method reduces simulation times by up to -97% and -79% respectively compared to the semidiscretization method. These are significant improvements over reports in the previous literature. Stability predictions are validated experimentally and benchmarked with high-fidelity time domain methods. The presented methods are general and can be applied to any end mill with non-uniform pitch and helix angles as well.