Chatter instability analysis of spinning micro-end mill with process damping effect via semi-discretization approach

In this paper, the stability of delay differential equations (DDEs), describing self-excited vibrations in a micro-milling process, is investigated based on semi-discretization (SD) method. Due to the stubby geometry of micro-tools, the shear deformation and rotary inertia effects are considered for modeling the structure. The extended Hamilton’s principle is used to derive a detailed dynamical model of the spinning micro-tool with the support of misalignment in which the gyroscopic effects cause coupling of equations. Considering the actual geometry of the micro-end mill, exact dynamic stiffness (DS) formulations are developed to investigate the tool’s free vibration characteristics. The extracted mode shapes obtained from DS method are utilized as base functions in a Galerkin approach. Having considered regenerative cutting force, imposing the Galerkin method reduces the governing PDEs of the system to a set of DDEs. The resulting equations are discretized by means of SD procedure. Finally, numerical Floquet theory is utilized to investigate the stability of the system. Also, the effects of process damping on the stability diagrams are explored. The results show the efficiency of the proposed model and delineate the considerable influence of process damping on the stability borders of the system especially at low spindle speed.