This paper extends analytical modeling of the milling process to include process damping effects. Two cutting mechanisms (shearing and plowing mechanisms) and two process damping effects (directional and magnitude effects) are included. The directional effect is related to vibration energy dissipation due to directional variation of cutter∕workpiece relative motion. The magnitude effect is associated with change in force magnitude due to variation of rake angle and clearance angle. Process damping is summarized as containing these separate components: direction-shearing, direction-plowing, magnitude-shearing, and magnitude-plowing. The total force model including the process damping effect is obtained through convolution integration of the local forces. The analytical nature of this model makes it possible to determine two unknown dynamic cutting factors from measured vibration signal during milling. The effects of cutting conditions (cutting speed, feed, axial and radial depths of cut) on process damping are systematically examined. It is shown that total process damping increases with increasing feed, axial and radial depths of cut, but decreases with increasing cutting velocity. Predictions based on the analytical model are verified by experiment. Results show that plowing mechanism contributes more to the total damping effect than the shearing mechanism, and magnitude-plowing effect has by far the greatest influence on total damping.

This paper investigated the scribing process characteristics of the hard and brittle materials including single crystal silicon, STV glass, and sapphire substrate. Under various cutting angles, major process characteristics are examined including the groove geometry, specific cutting energy, and critical depth of cut at the onset of ductile-to-brittle cutting transition. As the cutting depth increases, groove geometry clearly reveals the ductile-to-brittle transition from the plastic deformation to a brittle fracture state. The material size effect in the ductile region as well as the transition in scribing behavior is well reflected by change in the specific cutting energy. Further, it is shown that the change of specific cutting energy as a function of the cutting depth can serve as a criterion for estimating the critical depth of cut. Such estimated critical depth of cut is confirmed by measurement from a 3D confocal microscope. The critical depths of cut for these hard materials are found to be between $0.1μm$ and $0.5μm$ depending on the materials and cutting angles.

The frequency response function (FRF) method has been well used to determine the worst spindle speeds and their critical limiting chip width for turning operation by finding the maximum negative real part of the FRF. In this study, a modified FRF concept is adapted for a 2 DOF milling system of planar isotropic dynamics to determine the worst spindle speeds and the critical limiting axial depth of cut in explicit, analytic formulas. Analogous to the formulation of worst spindle speeds, similar expression for the best spindle speeds is also obtained. The modified FRF is obtained by multiplying the original FRF of the structure with a complex scaling factor, corresponding to a scaling and a rotation of its original Nyquist plot. The scaling factor is determined analytically from the system characteristic equation with the radial cutting constant and radial immersion angle as the major system parameters. Through the presented method, it is also shown that the worst spindle speeds for a milling operation can be found without the prior knowledge of modal dynamics and stability lobe diagram. The proposed analytical expressions are confirmed by the existing stability models and experimentally verified.