A three-dimensional model for the dynamics of micro-endmills including bending, torsional and axial vibrations

Attainable geometric accuracy and surface finish in a micromilling operation depends on predicting and controlling the vibrations of micro-endmills. The specific multi-scale geometry of micro-endmills results in complexities in dynamic behavior, including three-dimensional vibrations, which cannot be accurately captured using one-dimensional (1D) beam models. This paper presents an analytically based three-dimensional (3D) model for micro-endmill dynamics, including actual cross-section and fluted (pretwisted) geometry. The 3D model includes not only bending, but also coupled axial/torsional vibrations. The numerical efficiency is enhanced by modeling the circular cross-sectioned shank and taper sections using 1D beam equations without compromising in model accuracy, while modeling the complex cross-sectioned and pretwisted fluted section using 3D linear elasticity equations. The boundary-value problem for both 1D and 3D models are derived using a variational approach, and the numerical solution for each section is obtained using the spectral-Tchebychev (ST) technique. Subsequently, component mode synthesis is used for joining the individual sections to obtain the dynamic model for the entire tool. The 3D model is validated through modal experimentation, by comparing natural frequencies and mode-shapes, for two-fluted and four-fluted micro-endmills with different geometries. The natural frequencies from the model was seen to be within 2% to those from the experiments for up to 90 kHz frequency. Comparison to numerically intensive, solid-element finite-elements models indicated that the 3D and FE models agree with less than 1% difference in natural frequencies. The 3D-ST model is then used to analyze the effect of geometric parameters on the dynamics of micro-endmills.