Most NC programming software and NC verification packages make their computations based purely on geometry. This means the tool is modeled as a cylinder and the workpiece as a prism, and as long as the tool moves past the prism in the right places, the part will be made correctly.

The mathematics that underlie the computations consist of subtracting the space removed by the tool motion from the space occupied by the workpiece. However, the physics of metalcutting are considerably different from these geometric assumptions. To start, tools and workpieces are generally stiff but not infinitely stiff. The tool and workpiece deflect in response to the cutting force. As a result, the tool removes less material than planned.

Surface location error for a 6-flute endmill when milling at three radials DOCs.

Machinists and many NC programmers are well aware of this, and they often include "spring passes" in their programs—finish passes where the tool moves over the same path it just cut. Sometimes, multiple spring passes are employed for one area, each generating a surface closer to the specified dimension.

Most people would assume the error in surface location is proportional to the metal-removal rate, but that is not true. Even if tool deflection is simply proportional to cutting force (the static deflection case), generation of the surface is more complicated. The tool is not a cylinder but rather a collection of teeth bound and rotated together.

An endmill with six straight teeth is shown in Figure 1 in three different radial immersions. The desired surface, marked "S," is not generated continuously but in a series of instants when a tooth is in the position marked "A." A tooth in position A imprints the tool deflection on surface S. The tool deflection in response to the cutting force at that moment matters but the deflection of the tool at other times does not because the other deflections are imprinted on material that is later removed.

In the first case (left), the radial DOC is small enough that only one tooth is in contact with the workpiece when a tooth is in position A. The chip thickness at A is close to zero, so the force on the tool is close to zero, and, therefore, the surface is almost perfectly located.

If the radial DOC increases, the picture at the instants in time when the desired surface is generated does not change until the second tooth becomes involved in the cut (center). Now, there are two teeth cutting simultaneously, and the force on the second tooth (Ft2) causes deflection of the tool. The error in surface location undergoes a sudden step change when the second tooth begins to cut. If the radial DOC increases still further, a third tooth becomes involved in the cut, and the surface location error changes in another discrete step.

For slow-speed milling, applying a tool with six straight teeth, where the radial DOC is gradually increasing, the surface location error does not gradually increase. It arises in two discrete steps.

In reality, the situation is usually more complicated. The teeth are generally not straight but aligned in a helix. The contact point between tool and desired surface rises along the tool axis as the tool rotates. If tool deflection changes with time, the surface location error changes from the bottom to the top of a wall created by the endmill. In addition, if the tooth passing frequency is high enough, the tool behaves dynamically—it vibrates. In that case, timing is what counts: the key is the location of the tool in its cycle of motion when a tooth is in position to generate the surface of interest. While the surface location error is not intuitive, it is predictable. CTE

About the Author: Dr. Scott Smith is a professor and chair of the Department of Mechanical Engineering at the William States Lee College of Engineering, University of North Carolina at Charlotte, specializing in machine tool structural dynamics. Contact him via e-mail at [email protected]. A more detailed account of this topic is in his book "Machining Dynamics—Frequency Response to Improved Productivity," co-authored by Tony Schmitz.