# Characterizing the Geometry of Monocrystalline Diamond Contouring Tools

Highly accurate contoured surfaces can be produced using diamond cutting tools, which are themselves extremely precise devices. The tool tip geometry can now be generated with an accuracy of a few micro-inches, thanks to the latest developments in the fabrication of diamond cutting tools.

Knowing the cutting tool geometry at the micro-inch level is essential to make components with contour accuracies to a similar degree of accuracy. This article discusses a method of characterizing the geometry of diamond cutting tools to accurately fabricate a predefined workpiece by calculating of a tool path.

## Types of Diamond Tools

Like other cutting tools, a value needs to be stated for the tool radius of diamond tools. The characteristics of the diamond tools could be determined by considering the values such as tool radius, sweep angle, clearance angle, and top rake angle. However, these data are not sufficient to define the diamond tools properly.

One technique is to form a cylindrical surface on the tool nose, at an angle to the intended cutting plane, to generate the clearance angle on a contouring tool. Another technique involves generating a conical surface with an axis at right angle to the cutting plane to form the clearance angle on a contouring tool. Both methods have their own advantages and drawbacks.

The radius of a tool with a cylindrical clearance face remains constant when the tool’s top face is relapped, which is a key advantage of cylindrical clearance tools. They are considered to be more rugged when compared to conical clearance tools. The drawback of a cylindrical clearance tool is that when the cut moves away from the tool axis, the clearance angle declines and approaches 0°.

Conversely, the clearance angle does not change for conical clearance tools when the cut moves away from the tool axis. However, the radius of the conical clearance tools changes whenever a conical clearance tool is relapped. Hence, it is necessary to re-qualify the conical clearance tools.

Most of the diamond tools have an elliptical form instead of a round form. Figure 1 shows a zero degree top rake, cylindrical clearance tool, which has a clearance angle of 10°, where the top rake face of the tool is positioned in the cutting plane with an angle of intersection of 10° with the clearance cylinder.

Figure 1. Even a tool with a zero degree top rake is an ellipse which differs significantly from the circle which is commonly used to approximate it.

This causes an error in the tool radius of 0.000011 at a point 40° off the tool axis. All cylindrical rake tools are ellipses, except for the case in which the top rake is -1 times the clearance angle. The tool also forms an ellipse when the cutting edge of the tool is projected into the cutting plane.

Conical clearance tools have a top rake of 0° and are therefore circular. Nevertheless, a conical clearance tool with a top rake equivalent to -2 times the clearance angle is also circular. However, the projection of the cutting edge onto the cutting plane is an ellipse.

A diamond tool analyzer is used for measuring the tool radius by rotating the tool under a microscope, and adjusting the positions of both the microscope and the tool in relation to the center of rotation. The tool edge is centered first, followed by the center of the circle that best fits the edge. The deviation of these two positions provides the radius of the best-fit circle or the tool radius.

## Characterization of Ellipse Formed

After measuring the tool radius, the tool is tipped to allow its top rake surface to be in the plane of rotation, thus ensuring that the tool edge is in focus across the full range of rotation. The angle of rotation of the tool must be noted, along with the tool radius.

Figure 2. A negative top rake tool with a conical clearance angle. The tool is shown as a cross section taken along its axis of symmetry.

Figure 2 presents a negative rake, conical clearance tool. The length of line J, which is one axis of the ellipse, is:

From this the other axis, K, can be expressed as follows:

The formula for the ellipse is as follows:

Equation 4 is solved to get the expression for p:

The eccentricity of the ellipse (e) is expressed as follows:

where a = major axis of the ellipse and b = minor axis of the ellipse.

When K is substituted for a, and J for b, the fraction a/b consists of only functions of the angles α and β. The evaluation of Equation 7 reveals that K is major axis of the ellipse, not J, when its value goes beyond 1. If this is the case, before solving Equation 6 for ‘e’, Equation 7 must be inverted so that its value is below 1.

Values for X and Y must be obtained to solve Equation 5. These values are obtained from the best-fit circle which was identified using the diamond tool analyzer. If, for example, this circle was fit to the tool using ±35° of sweep, it is tangent to the tool at its axis and intersects the ellipse 35° either side of the center line. The coordinates of the intersection are expressed below:

where σ is equivalent to half of the total sweep angle of the diamond tool. When these values are substituted into Equation 5, p can be determined. After solving for p, the values for K, L, and J can be determined using the following expressions:

If Equation 7 exceeds 1, as is the case for negative rake tools with a top rake angle less than two times the clearance angle, then the ellipse’s minor axis will lie along the tool axis. For this case, the formula for the ellipse is as follows:

Here,

and

The ellipse’s major axis always lies along the tool axis for a diamond tool with a cylindrical clearance angle as illustrated in Figure 3. The calculations are the same as mentioned before, except for e. Once e is determined, Equation 5 is solved for p, and then the values for K and J are determined.

Figure 3. A tool with a negative top rake and a cylindrical clearance angle

where R = Radius of the clearance cylinder.

By this, the ellipse created by the tool nose is characterized. This is followed by projecting this ellipse into the cutting plane of the machine tool. However, it is necessary to foreshorten the K dimension by the cosine of the top rake angle, β, in order to project the nose of the tool into this plane. The projection of the tool shape into the cutting plane is the last point to be considered.

## Conclusion

For tools with a top rake other than zero, the point on the tool which cuts the workpiece is positioned outside of the cutting plane by an amount. This amount is a function of the tool tip dimensions J and K, the top rake angle (β), the slope of the work at its center of rotation, the instantaneous slope of the work, and the tool orientation angle. Although the errors due to this are diminutive, it is necessary to build corrections into the tool path to compensate for the consequences of this geometry.

Precitech began operations in 1992, but continues the rich history of ultra-precision machine tool building dating back to 1962, when Pneumo Precision was founded. In October of 1997, the Pneumo ultra-precision machine tool division of Taylor Hobson (formerly Rank Taylor Hobson / Rank Pneumo) was merged with Precitech. The Precitech name was retained for this corporate entity and all offices and manufacturing facilities are now located at 44 Blackbrook Road in Keene, New Hampshire.

Our facility staffs approximately 100 talented individuals in a recently designed 60,000 Sq. Ft. building.

Precitech is a member of AMT (The Association of Manufacturing Technology) and has corporate affiliations with several professional societies and academic institutions such as Germany’s Research Community for Ultra Precision Technology at the Fraunhofer Institute, ASPE the American Society for Precision Engineering, and EUSPEN the European Society for Precision Engineering and Nanotechnology.

This information has been sourced, reviewed and adapted from materials provided by Precitech.

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