Reasons for stable milling zones

The stability lobe diagram for milling shown in Figure 1 illustrates characteristic features of all stability lobe diagrams for milling. The diagram shows stable (chatter-free) and unstable (chattering) cutting conditions for a given radial DOC when slotting. The combinations of axial DOC and spindle speed that will cause chatter are shaded red, while those that will not cause chatter are shown in white. 

Figure 1. The red-shaded areas in a stability lobe diagram show where the spindle speeds will cause chatter, while those that will not cause chatter are shown in white.

Notable stable zones occur around 8,000, 6,800, 5,700 and 4,850 rpm. There are additional, smaller peaks, which diminish as the spindle speed decreases. Finally, there is a large stable zone below about 800 rpm. 

Why do these stable zones appear the way they do? The answer is connected to the physical mechanism that causes chatter. The cutting tool is stiff, but it is not infinitely stiff. When an individual tooth contacts the workpiece, the tool may deflect and vibrate because of the cutting forces (Figure 2). The vibrating tool causes the teeth to leave behind a wavy surface. 

All images courtesy of S. Smith

Figure 2. Regeneration of waviness occurs when an individual cutting tool tooth contacts the workpiece and the tool deflects and vibrates because of the cutting forces.

The next tooth to pass over the wavy surface encounters a variable chip thickness, in part because of the waviness left on the surface and in part because of the current tool vibration. The variation in the chip thickness causes a variation in the cutting force, which, in turn, causes a vibration that generates a wavy surface. This “regeneration of waviness” is the primary mechanism responsible for chatter when milling. 

Depending on the conditions, the vibration may increase, which is chatter, or decrease. The variation in the chip thickness depends on the alignment between the wave previously left on the surface and the current tool motion. Some alignments cause a large variation in chip thickness (Figures 3a and 3b). However, when the wave left on the surface is exactly aligned with the tool motion, the chip thickness appears as if there was no vibration (Figure 3c). Although the tool is moving, the thickness of the hatched area is constant.

Figures 3a, 3b and 3c. Alignments of surface waviness and tool vibration.

This last kind of favorable alignment stops the mechanism that causes chatter, and it happens whenever there is exactly an integer number of vibration cycles between the passage of subsequent teeth. The largest stable zone happens when there is exactly one vibration cycle between subsequent teeth. This happens when the tooth passing frequency equals the natural frequency, the frequency at which the tool would like to vibrate. 

The next most stable speed occurs when the tooth passing frequency is equal to half of the natural frequency, meaning there are exactly two vibration cycles between the passage of subsequent teeth. That occurs at a spindle speed that is half of the best spindle speed. The next stable zone is at a third of the largest stable peak, followed by a quarter of the largest stable peak, meaning three and four waves between teeth, respectively, and so on. 

In the stability lobe diagram shown in Figure 1, the best speed would have been at about 34,000 rpm, which is beyond the capability of the measured machine and is off the chart. The first stable zone within the spindle-speed range of the machine is at the right edge of the chart, part of a zone that would have a peak at 8,500 rpm (34,000/4). That is part of the fourth zone from the best, but it still offers a substantially large and stable axial DOC. The next stable peak is at 6,800 rpm (34,000/5), followed by 5,667 rpm (34,000/6), and so on. The peaks all occur at integer fractions of the best speed. 

The stable zone below about 800 rpm is different than the other stable zones. It occurs because as the number of waves between subsequent teeth becomes large, the wavelength becomes short. Eventually, the wavelength becomes so short that the tool begins to look dull in comparison to the surface waviness, and the tool cannot copy the wave anymore. The regeneration of waviness mechanism is lost. This is the “process damping” region, and a large axial DOC is possible if the spindle drive has sufficient power. CTE

About the Author: Dr. Scott Smith is a professor at the William States Lee College of Engineering, University of North Carolina at Charlotte, specializing in machine tool structural dynamics. Contact him at kssmith@uncc.edu.