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Milling instability, or chatter, is one factor that limits material removal rates, because the stable depth of cut is restricted by the system dynamics. Using stability maps, however, stable combinations of spindle speed and axial depth of cut can be selected. Larger stable axial depths of cut are generally obtained at higher spindle speeds. A second primary limitation to high material removal rates is tool wear. Because diffusive tool wear is temperature-driven and higher cutting speeds lead to increased cutting temperatures, hard-to-machine materials may cause the higher spindle speeds — which provide access to increased depths of cut — to be inaccessible due to unacceptable wear rates. Together, chatter and tool wear combine to increase machining costs and are, therefore, the subject of widespread modeling and experimental efforts.

One beneficial phenomenon that occurs at low spindle speeds and increases the allowable depth of cut is process damping. Process damping has been identified as energy dissipation due to interference between the cutting tool clearance face and machined surface during relative vibrations between the two. Because process damping enables increased material removal rates at low cutting speeds, it is an important consideration when modeling machining operations for hard-to-machine materials.

The purpose of machining models is to select optimal operating parameters at the process planning stage. For today’s job shops, process planning begins with a solid model of the part to be produced. A CAM software package is then used to generate the CNC tool paths that reveal the desired solid model geometry from the stock model geometry (prismatic bar stock or additive preform, for example). For milling operations, the tool geometry, spindle speed and feed per tooth must be specified in the CAM software. Additionally, the axial depth of cut (stepdown) and radial stepover must also be selected for 2.5D tool paths. Because material is removed layer-by-layer in 2.5D tool paths, the axial depth of cut is fixed and the radial depth of cut is defined by the selected stepover. For this reason, it is beneficial to determine the limiting radial depth of cut as a function of spindle speed for a fixed axial depth. However, traditional stability analyses assume a fixed radial depth and identify the maximum chatter-free axial depth for the selected range of spindle speeds.

In this article the effects of process damping are included in a stability map that describes the limiting radial depth of cut as a function of spindle speed. A milling stability model that includes process damping is applied to generate spindle speed versus axial depth stability limits for multiple radial depths. These limits are then combined to identify the corresponding spindle speed versus radial depth stability map for the selected axial depth. The spindle speed versus radial depth of cut stability map is produced using the following sequence of steps.

1. Specify the system dynamics, tool geometry and force model, including both the cutting force and process damping coefficients.
2. Select the desired spindle speed range and axial depth of cut.
3. Generate the spindle speed versus axial depth stability map for the selected dynamic system using the smallest desired radial depth of cut.
4. Determine the spindle speeds at which the limiting axial depth is equal to the desired axial depth from step 2. Store these {spindle speed, radial depth} pairs.
5. Increment the radial depth of cut to a larger value and repeat steps 2-4. Continue until the radial depth is increased to the tool diameter.
6. Collect all {spindle speed, radial depth} pairs from steps 2-5. The result is the limiting radial depth of cut as a function of spindle speed. Because the axial depth of cut stability analysis includes process damping, the final radial depth stability limit also incorporates process damping effects.

The procedure steps are demonstrated through an example and the corresponding figures. For the example, the selected spindle speed range is zero to 10,000 rpm and the desired axial depth of cut (stepdown) is 3 mm. The spindle speed versus axial depth of cut stability map for an up (conventional) milling radial depth of cut equal to 25% of the tool diameter (that is, a 25% radial immersion) is displayed in Figure 1. This represents the result from step 3. The spindle speeds where the limiting axial depth is equal to the desired axial depth (3 mm) for the final radial depth stability lobe diagram are identified in Figure 2 (step 4).

For comparison purposes, the step 4 result for a 50% radial immersion is shown in Figure 3. It is observed that number of speeds is reduced with the increased radial depth because the axial depth stability limit is lowered. The final radial depth of cut stability map (step 6) is displayed in Figure 4. This represents the collection of limiting axial depths of cut identified over the full range of radial immersions. The corresponding diagram for a desired axial depth of 5 mm is provided in Figure 5. As expected, the radial depth stability limit is lowered with the increased axial depth.