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Predictive models are required to select optimal milling parameters, including axial and radial depths of cut, spindle speed and feed per tooth, at the process planning stage. The intent is to select parameters that enable first part correct performance to avoid costly and time-consuming trial and error. This first-part-correct performance requires that chatter, or self-excited vibration, does not occur and acceptable geometric accuracy and surface finish are achieved. Predictive models are available for milling dynamics that rely on the tool-toolholder-spindle-machine vibration response and a force model that related the cutting force to the chip area.

Another class of predictive models relates the milling parameters to the cutting and noncutting times given the part geometry. These models use the peak or average cutting force, which depends on the part geometry, CNC tool path and milling parameters, to modify the cutting and noncutting times by updating the instantaneous feed rate along the tool path to maintain a constant average force. The outcome is optimized cutting and noncutting times for maximum productivity.

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To date, however, these two predictive capabilities have remained separate. Machining dynamics models do not typically include the time-dependent cutting conditions imposed by CNC tool paths. They tend to focus on, for example, a fixed radial depth to determine stable combinations of spindle speed and axial depth in the graphical form of a stability map. Feedrate scheduling solutions do consider the variable cutting conditions for arbitrary three-axis and five-axis tool paths, but do not include the effects of relative vibration between the cutting tool and workpiece on the milling stability. This relative vibration occurs because the tool and workpiece are not rigid and a complete solution requires more than geometry.

To demonstrate the combination of these two predictive models, the part geometry displayed in Figure 1 was selected. It provides a continuously variable radial depth of cut with a fixed axial depth. The combination of varying radial depth with fixed axial depth mimics traditional three-axis, 2.5D CNC machining tool paths, where the material is removed with X-Y planar tool paths that implement the user-selected stepover (radial depth) and advance the stepdown in the Z direction (axial depth) between each planar tool path. The Figure 1 geometry was machined multiple times using a different axial depth to transition from stable (low axial depth) to unstable, or chatter (high axial depth), cutting conditions. The workpiece material was 6061-T6 aluminum in all cases.

For the part geometry shown in Figure 1, the variation in radial depth of cut with cutting time is displayed in the top panel of Figure 2. The constant radial depth of 3.18 mm is observed at the beginning and end of the toolpath. The radial depth increases from 3.18 mm (25% radial immersion) to 12.7 mm (slotting) at the center of the cut. The variation in angle of engagement is shown in the bottom panel of Figure 2. The angle is 60 degrees for the 25% radial immersion portion of the tool path and increases to 180 degrees for the slotting condition in the middle of the tool path.

Machining tests were performed where a 6061-T6 aluminum workpiece was bolted to a cutting force dynamometer so the in-process cutting force could be measured. The workpiece geometry was the ramp profile shown in Figure 1 with the variation in radial depth displayed in Figure 2. The axial depths were 7 mm and 12 mm. The spindle speed was 7,000 rpm and the feed per tooth was 0.05 mm. Flood coolant was applied to evacuate chips.

Results for the 7-mm axial depth are displayed in Figure 3. The black line is the predicted time-dependent cutting force, the red line is the measured time-dependent cutting force, and the blue line is the peak cutting force (assuming a rigid tool). Figure 3 shows that the force profile mimics the variation in radial depth of cut in Figure 2. The 7-mm axial depth provides stable cutting conditions.

A magnified view of the beginning of the Figure 3 cut is shown in Figure 4. Figure 4 shows that the cutting force constantly varies as the teeth enter and exit the cut and the radial depth of cut increases to the constant 3.18 mm value.

Figure 5 displays results for the second ramp case. The axial depth is now 12 mm and the cutting conditions are unstable near the middle of the tool path. It is observed that the predicted and measured cutting force grows dramatically as the radial depth approaches 12.7 mm (slotting) and chatter occurs. However, the peak force predicted by feed rate scheduling with the rigid tool assumption does not show the chatter condition.

Figure 6 displays a magnified view of the transition from stable to chatter conditions. The force increases substantially due to the self-excited vibration and poor surface finish is obtained. As noted, the peak force predicted by feed rate scheduling with the rigid tool assumption does not predict chatter.