Steep Trouble

Author E. Rivin
April 01,1999 - 11:00am

Related Glossary Terms

  • axial force

    axial force

    When drilling, a force that is directed axially—along the direction of machining. The magnitude of an axial force rises with the drill’s diameter and the chisel edge’s width. Axial force is also known as thrust. When turning and boring, the term “feed force” is commonly used instead of “axial force.” See cutting force.

  • calibration


    Checking measuring instruments and devices against a master set to ensure that, over time, they have remained dimensionally stable and nominally accurate.

  • centers


    Cone-shaped pins that support a workpiece by one or two ends during machining. The centers fit into holes drilled in the workpiece ends. Centers that turn with the workpiece are called “live” centers; those that do not are called “dead” centers.

  • chatter


    Condition of vibration involving the machine, workpiece and cutting tool. Once this condition arises, it is often self-sustaining until the problem is corrected. Chatter can be identified when lines or grooves appear at regular intervals in the workpiece. These lines or grooves are caused by the teeth of the cutter as they vibrate in and out of the workpiece and their spacing depends on the frequency of vibration.

  • clearance


    Space provided behind a tool’s land or relief to prevent rubbing and subsequent premature deterioration of the tool. See land; relief.

  • collet


    Flexible-sided device that secures a tool or workpiece. Similar in function to a chuck, but can accommodate only a narrow size range. Typically provides greater gripping force and precision than a chuck. See chuck.

  • depth of cut

    depth of cut

    Distance between the bottom of the cut and the uncut surface of the workpiece, measured in a direction at right angles to the machined surface of the workpiece.

  • endmill


    Milling cutter held by its shank that cuts on its periphery and, if so configured, on its free end. Takes a variety of shapes (single- and double-end, roughing, ballnose and cup-end) and sizes (stub, medium, long and extra-long). Also comes with differing numbers of flutes.

  • endmilling


    Operation in which the cutter is mounted on the machine’s spindle rather than on an arbor. Commonly associated with facing operations on a milling machine.

  • facemilling


    Form of milling that produces a flat surface generally at right angles to the rotating axis of a cutter having teeth or inserts both on its periphery and on its end face.

  • feed


    Rate of change of position of the tool as a whole, relative to the workpiece while cutting.

  • flat ( screw flat)

    flat ( screw flat)

    Flat surface machined into the shank of a cutting tool for enhanced holding of the tool.

  • gang cutting ( milling)

    gang cutting ( milling)

    Machining with several cutters mounted on a single arbor, generally for simultaneous cutting.

  • grinding


    Machining operation in which material is removed from the workpiece by a powered abrasive wheel, stone, belt, paste, sheet, compound, slurry, etc. Takes various forms: surface grinding (creates flat and/or squared surfaces); cylindrical grinding (for external cylindrical and tapered shapes, fillets, undercuts, etc.); centerless grinding; chamfering; thread and form grinding; tool and cutter grinding; offhand grinding; lapping and polishing (grinding with extremely fine grits to create ultrasmooth surfaces); honing; and disc grinding.

  • inches per minute ( ipm)

    inches per minute ( ipm)

    Value that refers to how far the workpiece or cutter advances linearly in 1 minute, defined as: ipm = ipt 5 number of effective teeth 5 rpm. Also known as the table feed or machine feed.

  • micron


    Measure of length that is equal to one-millionth of a meter.

  • milling


    Machining operation in which metal or other material is removed by applying power to a rotating cutter. In vertical milling, the cutting tool is mounted vertically on the spindle. In horizontal milling, the cutting tool is mounted horizontally, either directly on the spindle or on an arbor. Horizontal milling is further broken down into conventional milling, where the cutter rotates opposite the direction of feed, or “up” into the workpiece; and climb milling, where the cutter rotates in the direction of feed, or “down” into the workpiece. Milling operations include plane or surface milling, endmilling, facemilling, angle milling, form milling and profiling.

  • milling machine ( mill)

    milling machine ( mill)

    Runs endmills and arbor-mounted milling cutters. Features include a head with a spindle that drives the cutters; a column, knee and table that provide motion in the three Cartesian axes; and a base that supports the components and houses the cutting-fluid pump and reservoir. The work is mounted on the table and fed into the rotating cutter or endmill to accomplish the milling steps; vertical milling machines also feed endmills into the work by means of a spindle-mounted quill. Models range from small manual machines to big bed-type and duplex mills. All take one of three basic forms: vertical, horizontal or convertible horizontal/vertical. Vertical machines may be knee-type (the table is mounted on a knee that can be elevated) or bed-type (the table is securely supported and only moves horizontally). In general, horizontal machines are bigger and more powerful, while vertical machines are lighter but more versatile and easier to set up and operate.

  • precision machining ( precision measurement)

    precision machining ( precision measurement)

    Machining and measuring to exacting standards. Four basic considerations are: dimensions, or geometrical characteristics such as lengths, angles and diameters of which the sizes are numerically specified; limits, or the maximum and minimum sizes permissible for a specified dimension; tolerances, or the total permissible variations in size; and allowances, or the prescribed differences in dimensions between mating parts.

  • shank


    Main body of a tool; the portion of a drill or similar end-held tool that fits into a collet, chuck or similar mounting device.

  • steep-taper toolholder

    steep-taper toolholder

    Common method of securing the cutting tool body to the spindle in a machine tool. Comes in various styles, including CAT V-flange, British Taper (BT) and ISO.

  • stiffness


    1. Ability of a material or part to resist elastic deflection. 2. The rate of stress with respect to strain; the greater the stress required to produce a given strain, the stiffer the material is said to be. See dynamic stiffness; static stiffness.

  • tolerance


    Minimum and maximum amount a workpiece dimension is allowed to vary from a set standard and still be acceptable.

  • toolholder


    Secures a cutting tool during a machining operation. Basic types include block, cartridge, chuck, collet, fixed, modular, quick-change and rotating.

The vast majority of machine tools making chips in U.S. shops today employ a 7/24-taper (steep) interface between the spindle and the toolholder. This steep-taper interface, which can be found on machining centers and other machine tools, represents a familiar, widely used, and firmly established technology.

The steep taper has been standardized by ANSI, ISO, DIN, and other standardization bodies. And there are many thousands of machine tool spindles with steep-taper holes, as well as many millions of 7/24-taper toolholders.

Until recently, the steep-taper interface satisfied the basic requirements of most machining operations. But manufacturing trends and competitive pressures are giving way to machining practices that call for a much more accurate and reliable interface.

With the increasing use of hard-to-machine structural materials, tighter tolerances, and high-performance cutting tools and inserts, shops are starting to rely on high-accuracy, high-power, high-speed machine tools to remain productive.

While these machine tools were modified to be stiffer, the toolholder/ spindle interface was not. As a result, the toolholder and the interface have become the weakest links in the machining system.

Steep Taper’s Good and Bad Points
Despite the drawbacks of a steep-taper interface when used in a high-performance machining system, there is much merit in the design. Steep-taper toolholders are not self-locking. Because they do not become jammed into the spindle with use, a kick-out device is not required to force the toolholder out of the spindle. This allows tools to be held in place and released with a simple drawbar. Once the toolholder is tight in the tapered hole of the spindle, the tool is secured. The solid taper allows the designer to locate the mechanism that holds the tool deep within the body of the holder. This reduces tool overhang from the spindle face.

Standard steep-taper toolholders also are relatively simple and inexpensive to make, because the taper angle is the only dimension that has to be machined with a high degree of precision.

There are shortcomings to the steep-taper interface, however. One of the most serious is that it relies on the tapered surface to play two important roles simultaneously. The surface must precisely locate the toolholder relative to the spindle, and it must provide enough contact between the spindle and the toolholder to clamp the toolholder rigidly in place.

Radial-location accuracy is inadequate, because the interface’s design tolerances are not tight enough. The tolerances specify a “minus” deviation of the hole angle and a “plus” deviation of the toolholder angle. These deviations create clearance at the back of the connection, between the male taper and the spindle hole. ISO’s standards for an AT4 toolholder (the IS1947 standards specify quality levels from AT1 to AT9, with AT1 being the most accurate) specify 13 ang. sec. tolerance on each angle. Such a deviation can produce a radial clearance as high as 13µm at the back end of the connection. At 3" from the spindle’s face, a 13µm deviation will cause the tool’s axis to shift a maximum of 20mm from the axis of the spindle, and it could also throw the tool off balance.

Face contact between the spindle and the toolholder cannot be relied on to supply rigidity or to accurately locate the toolholder, because the taper’s steepness makes it almost impossible to manufacture interchangeable toolholders that have both face and taper contact with the spindle. To provide simultaneous taper and face contact, spindles and toolholders have to be machined to fractional-micron tolerances. If a toolholder is machined to fit a specific spindle, simultaneous contact can be achieved. Some toolholder manufacturers provide such precision fitting for critical machining operations. However, precision fitting is impractical for the existing inventory of toolholders and spindles, since spindle gage diameters can vary by tens of microns.

Instead of trying to achieve simultaneous contact, the industry has adopted a steep-taper interface design that guarantees a large clearance between the face of the spindle and the toolholder flange.

With no face contact between the toolholder and spindle to restrain the components, the radial clearance at the back of the taper allows the toolholder to move when it is under heavy cutting forces. These microscopic shifts accelerate wear on the front part of the spindle hole—a problem known as bell mouthing. In extreme cases, the oscillations can further accelerate wear through fretting corrosion, which is the gradual oxidation and disintegration of the contacting surfaces.

The clearance between the spindle face and the toolholder flange also results in unpredictable variations in the axial positioning of the tool. A deviation from 25µm to 50µm can occur between the axial positioning of one tool and another or even between the original location of a tool and its location after use. This indeterminacy has been found to be the result of friction variations between different toolholders and spindles, inconsistent axial preload forces, disparities in spindle gage diameters (a dimension that has not been standardized), and small differences in spindle-hole taper angles. The axial uncertainty also makes it difficult to measure parts with a touch probe mounted on a machining center’s spindle. In such cases, a special procedure is required for axial calibration of the probe.

The stiffness of the steep-taper connection is sensitive to minute differences in the taper angle and to axial preload. A No. 50-taper toolholder is typically held in place with 2000 to 5500 lbs. of axial force. Researchers have found that it is possible to increase stiffness by 20% to 50% with a 400% to 800% increase in preload force. However, when toolholders are frequently connected and disconnected, the repeated application of high axial force to the steep-taper connection rapidly wears the spindle hole, and the radial expansion of the spindle caused by this extreme force can damage the front spindle bearing.

High-Performance Requirements
The drawbacks of the steep-taper design make it difficult for the interface to meet the requirements of today’s high-accuracy, high-power, high-speed machine tools. To develop an interface that will meet these needs, a designer must keep in mind that the requirements are not necessarily the same from job to job. Although some machining centers can combine high precision with high horsepower and high spindle speeds, few applications require all three capabilities. Because each type of machining places its own demands on the toolholder/spindle interface, these jobs require different interface characteristics.

To machine to tight tolerances, the interface must be able to position the tool accurately in both the radial and axial directions. Drilling and many types of endmilling do not require this degree of precision, and a standard steep-taper interface will provide adequate accuracy for these applications. But because of runout and face clearance, the interface cannot achieve the extreme precision needed for tight-tolerance work.

High-power machining, which is frequently performed at lower spindle speeds, requires an interface that provides stiffness and damping. These qualities help prevent chatter while promoting accurate tool positioning.

High-speed operations require an interface that maintains its characteristics at elevated spindle speeds. At high speeds, centrifugal force causes the front end of a steep-taper spindle to expand. A No. 30-taper spindle, for example, will expand as much as 5µm at 30,000 rpm. However, the toolholder inside the spindle does not expand as much, so the connection between the toolholder and spindle becomes slightly looser. The loose connection increases the effective length of the cantilever tool, reduces its stiffness, and changes the axial position of the toolholder. Expansion of the spindle may also lead to toolholder imbalance.

For a steep-taper interface to maintain reliable contact between the toolholder and spindle at high speeds, the initial interference must be quite high. A No. 40 taper will require an interference of as much as 20µm. A high degree of interference along the length of the connection can also eliminate the clearance at the back end of the taper caused by the taper tolerances. However, for a standard solid-steel, AT4-grade taper, this interference would have to be at least 13µm to eliminate the clearance. Such magnitudes of interference would require extremely high drawbar forces and make it nearly impossible for the operator to change the tool. High interference would also cause the spindle to bulge enough to damage the spindle bearing.

A more practical way to hold the spindle and toolholder together at high speeds is to generate axial flange contact between them. If the friction forces between the spindle face and toolholder flange are higher than the centrifugal forces, the contact will prevent spindle expansion.

High-speed machining also requires precisely balanced toolholders. But the keys and keyslots of a standard steep-taper interface (as well as the runout caused by the clearance at the back of the connection) make it difficult to achieve the requisite balance. The keys are needed to transmit torque and, in some operations, to correctly orient the toolholder. But if the connection itself can generate enough friction to transmit torque, the keys can be eliminated.

The Alternatives
When a technological system has problems, it’s usually best to make minor modifications and draw on the system’s internal resources to remedy the problems.

In solving the toolholder/spindle-interface problem, however, the industry has introduced drastic changes that make existing spindles and the huge inventory of steep-taper toolholders currently used by U.S. shops obsolete. To adopt these innovations on just one machine tool, a shop would have to purchase duplicates of many of the toolholders already in its inventory.

Among the solutions that have reached the market are designs that use flat spiral gears on both the spindle and toolholder to create a stiff connection, and a number of schemes that provide simultaneous face and taper contact. Some of these systems have required a nonstandard spindle, nonstandard toolholders, or both. Generally, the components of these systems also have cost significantly more than standard steep-taper toolholders and spindles.

One of the most heavily promoted designs is the HSK interface. It is also the interface with the fastest growing number of users. The basic HSK design uses a 1/10 taper that is shorter than the standard steep taper. The hollow HSK toolholders have thin walls for resiliency. Tolerances on the toolholder’s and spindle’s critical dimensions must be within 2µm to 6µm. Due to the interface’s tight tolerances, shallow taper angle, and thin-wall design, the application of axial force results in simultaneous contact between the taper and face surfaces. This axial force is quite high. In some HSK designs, the axial force also causes the taper to shrink and the spindle to bulge, further ensuring contact between the surfaces.

The extended contact between the toolholder and the spindle, together with increased axial force, results in precise axial positioning and a stiffer connection than what a standard steep-taper interface provides. HSK’s developers also claim that the interface is not affected by high speeds. The interface’s patented drawbar uses a wedge-like connection to attach to the taper. This design applies a higher axial load as the spindle speed increases.

While the HSK interface does provide some performance advantages, it also has shortcomings. It is not compatible with existing spindles and toolholders, and the HSK system’s exacting tolerances and intricate design make it 150% to 200% costlier than standard steep-taper toolholders.

In addition to complex toolholders, the HSK interface employs a more involved spindle design. Because HSK’s shallow taper self-locks in the spindle, the drawbar must incorporate a mechanism to kick out the toolholder. This additional hardware increases the drawbar’s complexity and expense.

The HSK interface also introduces other problems that hinder the ability of the spindle/toolholder combination to perform accurately and reliably. The hollow design of the HSK toolholder forces manufacturers to locate tool-clamping mechanisms, such as collets, completely in front of the spindle. This effectively increases the overhang of the toolholder, thus reducing overall end-of-tool stiffness.

Additionally, some researchers have found the chatter stability of HSK toolholders to be low. And the interface’s large continuous contact area makes the accuracy of the connection vulnerable to small amounts of dirt on the contacting surfaces, especially on the taper. The expansion of the HSK spindle when a toolholder is inserted can damage the spindle bearings. And the transitional area between the thin-wall tapered segment and the massive flange that contacts the spindle face is prone to extreme stress when the tapered shaft deforms during clamping.

One study of HSK interfaces found that the thin walls of the toolholder’s back section curved inward when the drawbar applied force to the toolholder. Centrifugal forces cause a similar reaction at high spindle speeds. As a result of these two actions, there may no contact at all between the toolholder and spindle. In other cases, contact may be reduced to only two narrow circular bands. Evidence of this limited contact can be seen on the toolholder’s taper after a few insertions.

Such minimal contact decreases the toolholder’s bending stiffness, especially when the tool is subjected to radial (cutting) forces large enough to cause the toolholder flange to separate from the spindle face. Flange/face contact can be maintained by increasing interference by 15µm to 20µm. But this much interference absorbs most of the drawbar force, leaving little to produce adequate pressure where the flange meets the spindle face. Such high interference can also increase the spindle’s bulge.

New Solution
We, the authors of this article, set out to address the toolholder/spindle-interface problem. Our mission was to develop a system that performed better at high speeds and was stiffer and more accurate than conventional interfaces. The proposed system would also be fully compatible with existing toolholders and spindles and would not be prohibitively expensive.

We had two goals: Develop a toolholder that would contact the spindle at both the taper and the face, and modify existing toolholders to make them stiffer and decrease runout. We considered the second goal to be especially important, because it could upgrade the performance characteristics of millions of existing toolholders with minimal changes in the system.

After extensively surveying the available systems, we developed and tested seven designs. They employ a “virtual taper” concept, in which discrete points or lines define a tapered surface and contact the tapered spindle hole. Elastic design elements at the point of contact allow axial drawbar force to bring the toolholder into face contact with the spindle without deforming the whole toolholder shank. Such an approach achieves the desired interference without absorbing too much drawbar force or making the toolholder vulnerable to contamination.

We rejected most of these designs because they introduced elements that required expensive precision machining. One that we kept is the WSU-1 version (Figure 1). The accuracy required to machine this version is not greater than what is needed to manufacture standard steep-taper toolholders.

Figure 1: The WSU-1 interface features a cage holding steel balls that is attached to the tapered shaft of the toolholder. These balls are deformed as the toolholder is pulled into the spindle.

WSU-1’s shank is designed with the same 7/24 taper as a standard steep-taper toolholder. The shank’s diameter is smaller than that of a standard toolholder, but WSU-1 and steep-taper toolholders of the same class have the same flange diameter.

A metal or plastic cage containing a number of precision balls is attached to the shank. The cage has lips that protect the inside area from dirt. When the toolholder is out of the spindle, the gage diameter defined by the balls is 5µm to 10µm larger than the gage diameter of the spindle. As the drawbar applies axial force, the balls are deformed in a uniform manner and the toolholder moves into the spindle. It stops when the toolholder flange and spindle face meet.

The WSU-1 design generates high axial forces that, in turn, produce high friction forces between the flange and the spindle face. As a result, keys are needed only as a safety measure to prevent sliding when dynamic overloads occur during milling. Applying a coating to the toolholder face that enhances friction might make it possible to eliminate the keys entirely.

The WSU-1 design solves the problems associated with steep-taper interfaces while avoiding the shortcomings of other interfaces. The WSU-1 does not require a new spindle design, and the shank is no more difficult to manufacture than a standard shank. There is little difference in cost between WSU-1 toolholders and standard holders. Tool-clamping components can be located deep inside the holder, and the holder can be clamped securely without bulging the spindle. Either a conventional knob or an HSK-like drawbar can be used to clamp the toolholder. To reduce tool-changing time, the taper can be shortened.

The WSU-1 system benefits from the use of uniformly sized precision balls as elastic elements. Balls made of various materials are available at very reasonable prices. (The retail price of 100 precision steel balls in the United States is about $3.50.) The sphericities and diameters of the balls are accurate to within fractions of a micron. For the toolholder shown in Figure 1, balls with a sphericity within 0.25µm (medium accuracy) were used. The high dimensional accuracy of the ball-taper connection is due to Hertzian deformation between the balls and the two tapered surfaces, and the solid-body deformation of the balls.

If steel balls are used, the toolholder’s allowable axial direction is 35µm. Once the flange and spindle face meet, the contact is tightened by drawbar force. Since relatively low force is required to deform the balls, it takes less axial force to achieve face contact than is required with the HSK design. This makes more drawbar force available for face clamping.

For proper performance with WSU-1 toolholders, the spindle nose must have a gage-diameter tolerance within 10µm to 15µm, and the drawbar must provide sufficient axial force (4500 to 6500 lbs. for a No. 50 taper). Less drawbar force is needed if the toolholder is used to hold a touch probe or to make light but accurate cuts. We conducted tests to determine if contact with the balls under this much axial force would dent the surfaces of the spindle or toolholder, but we did not find any evidence that the design would cause damage.

When we measured the runout of the toolholder, we found it to be less than that of any standard toolholder. The reduced runout is due to the elimination of the clearance at the back of the taper. We also found that the flange/spindle-face contact enhanced stiffness to the level we were expecting, based on the work of others who have researched such interfaces.

We also tested the performance of the WSU-1 design using a horizontal milling machine equipped with a manually operated drawbar. Strain gages measured the axial force during these tests. Both facemilling and slot milling were performed at parameters that created the maximum allowable load on the cutters.

The facemilling was performed with a 150mm-dia., 8-insert cutter at 139 rpm, a feed of 230mm/min., and a 6.35mm depth of cut (DOC). The workpiece material was gray cast iron. The flatness of the machined surface was measured on a CMM. We found that we could achieve significantly better flatness with the WSU-1 toolholder than we could with conventional systems.

Slot-milling tests were performed on cast iron using a 25.4mm-dia., 5-flute endmill. The cuts were made at 204 rpm, a feed of 0.4mm/tooth, and a DOC of 12.7mm. The tool was clamped in a collet that was integral with the toolholder. In this test, we found the toolholder produced better flatness and surface finish than conventional systems.

A Simpler Interface
When axial indexing of the toolholder is not required, the interface design can be simplified. In these cases, the main interface problems are reduced stiffness, runout, and fretting. The clearance at the back of the taper connection causes these problems.

The WSU-2 toolholder that we developed specifically to address these shortcomings has a standard taper. At the back end of the taper, a coaxial groove was machined to receive one or more rows of precision balls. The balls, which are held in place with rubber or plastic filling, protrude enough to exceed the maximum possible clearance that can exist between the toolholder and the spindle (Figure 2).


Figure 2: The WSU-2 interface features a standard steep-taper toolholder modified with a groove to receive rows of balls that deform to bridge the clearance at the back of the taper when the toolholder is inserted into the spindle.

When a WSU-2 toolholder is inserted in a spindle, the balls deform to bridge the clearance and precisely locate the toolholder in the hole. The balls also provide additional stiffness at the end of the tool by preventing the toolholder from pivoting about the contact area at the front of the connection. The micromotions of the shank that can cause increased wear also are reduced.

WSU-2 No. 50-taper toolholders were tested with a load applied 40mm in front of the spindle face. We found that the stiffness of the WSU-2 was comparable to (and in some cases exceeded) the stiffness of interfaces that offered both taper and face contact. We also found that runout with the balls installed was 10% to 50% less than when the toolholder was inserted in a spindle without the balls. In facemilling tests, the WSU-2 interface produced cuts with a measured flatness that was better than a conventional steep-taper toolholder could produce. The flatness was not as good as that produced by the WSU-1 interface, however.

We also tested a No. 40-taper WSU-2 toolholder under typical shop-floor conditions at a compressor manufacturing plant. The toolholder, which was manufactured to an AT3 accuracy grade, was fitted with two rows of 5mm-dia. titanium balls. Using a shrink-fit connection, a 0.437"-dia., 1.6"-long endmill was loaded into the toolholder, and the assembly was then used to machine 0.437"-wide slots in cast iron at a speed of 2500 rpm and a feed of 20 ipm, which are typical parameters for this type of application. The machining left 1.500"-high, 0.125"-thick ribs in the as-cast part.


The performance of the WSU-2 toolholder was compared to the performance of a standard collet holder. To measure vibration levels, we ran both toolholders in a free-running spindle at speeds ranging from 1000 to 6000 rpm. The vibration level of the spindle carrying the WSU-2 shrink-fit tool was identical to the vibration level of the spindle without any tool. The reference tool generated at least twice the vibration.

When we measured rib thickness after machining, we found deviations from 0.0055" to 0.0084" in the ribs machined with the reference tool. The WSU-2 toolholder produced ribs that deviated from 0.0004" to 0.0033", well within the tolerance range.

We are planning to run more tests to compare the performance of these modified toolholders. Existing toolholders can be modified so that they conform to the WSU-2 design. Although the modification is very simple and inexpensive, it increases the tool’s effective stiffness significantly—especially at low drawbar forces—and reduces runout.

As our work has shown, it is possible to develop a steep-taper interface that is capable of handling heavy-duty, high-speed, high-accuracy work without making existing spindles and toolholders obsolete. The WSU-1 design can be adopted by ensuring that the spindle-hole gage diameter is within specified tolerances and by enhancing the drawbar force for applications that require particularly high stiffness. The WSU-2 interface can be adopted without any spindle changes. With only a minor retrofit of the toolholder, this interface can solve most of the problems associated with conventional steep-taper toolholders.

About the Authors
E. Rivin is a professor at Wayne State University, Detroit. J. Agapiou is a senior project engineer at General Motors Corp.’s North American Operations Manufacturing Center, Warren, MI. C. Xie is an engineer at Lyndex Corp., Northbrook, IL. K.R. Cagle teaches tool-and-cutter grinding courses at Jackson Community College, Jackson, MI.


E. Rivin is a professor at Wayne State University, Detroit.