# Basic Kinematic Designs

A list of basic kinematic couple designs and their effect on constraint, load carrying capacity, repeatability and cost, is in order.

We will take the first designs from the most famous author, in this field, Lord Kelvin. The Kelvin clamp consists of a ball or spherical surface in a conical cup, a ball in a vee block and a ball against a flat surface.

The Kelvin clamp or coupling has several unique and very powerful features. At the same time that we enumerate these exceptional properties we must recognize that this design is not symmetrical.

The most obvious problem caused by this lack of symmetry, is the ill effects on positioning caused by temperature changes.

The three constraint couple of the Kelvin clamp will fix the point center of its respective sphere, so that all expansion or contraction in the "X" and "Y" directions, will occur away from or towards this point.

## Ball And Flat

We have already covered the concept of the ball against a flat surface in a previous paragraph.

## The Conical Cup

The next couple to be considered is the conical cup and a sphere. This couple constrains the three degrees of linear freedom "X", "Y" and "Z". A commercial quality conical seat is very inexpensive to produce.

Because a cone has an infinitely changing internal radius, very high quality conical cups are nearly impossible to produce. For this reason a cone is literally the poorest choice when good repeatability is the important consideration. Some kinematic practitioners contend that the conical cup is not even a true kinematic element, because it forms an over constrained line of contact.

A 360-degree contact with the spheres gives the conical seat excellent load carrying capacity.

In order to minimize the poor geometric qualities of a cone, its surface can be undercut vertically, every 120 degrees, to leave 3 equilateral pads. This three-point contact will greatly improve the repeatability of this worst of couples, but at considerable monetary expense and a great reduction in its load carrying capacity.

## Other Designs

While we are on the subject of the conical seats, we will explore some of the other three constraint devices that can take the place of the cone.

## Tri-Hedral Prism

To alleviate the quality problems with the cone and sphere couple, Lord Kelvin postulated the use of an exceptionally high quality internal trihedral prism, which he found at that time, was impossible to produce. Economical production of this three faceted internal prism had eluded us until recently, when the urgent need of a customer spawned a solution to this age-old problem. By precision lapping and polishing the flat angled surfaces of three truncated cylindrical posts and then epoxy gluing the cylindrical sections into three deep counter bores in the kinematic platform, the problem has been solved. This device is very rigid, has excellent repeatability, and good load carrying capacity; but it is somewhat expensive to produce and install.

## Three Cylinders

High quality cylinders are comparatively easy and inexpensive to produce. By inclining three equally spaced cylinders to form a conical configuration, an excellent kinematic couple is generated. The three-cylinder contact with the sphere gives good load carrying capacity with excellent repeatability.

Rigidly mounting the three cylinders in the conical configuration is a relatively difficult and expensive machining and mounting process.

## Three Spheres

The three ball kinematic couple is the ultimate design for a three constraint highly repeatable configuration. The three balls form a nest that surrounds the load ball. The load carrying capacity of this device is determined by the yield strength of the materials used, the stiffness, the diameters of the spheres and the contact angle or slope between the load ball and those of the couple. diameter of the balls used for the seat, the smaller will be the contact area, so the better the repeatability of the couple. This will obviously only work down to the point, where permanent plastic deformation of the balls occurs and then this rule will no longer apply.

As a general statement, within certain limits, the smaller the diameter of the balls used for the seat, the smaller will be the contact area, so the better the repeatability of the couple. This will obviously only work down to the point, where permanent plastic deformation of the balls occurs and then this rule will no longer apply.

Ploys to increase the load carrying capacity of this excellent location device include, using high stiffness materials with inherent high yield strength, such as ceramics or cemented tungsten carbide balls and or using balls of large spherical diameter. The use of very large diameter balls is severely limited by the huge size of such a couple.

## The Rose Bud

One of the recent developments addresses the huge size of large diameter three ball couples by using three small segments of large diameter high alloy steel or cemented tungsten carbide spheres, glued into deep trenches placed at 120-degree increments in the kinematic platform. Users have named this couple the Rose Bud, because of its general appearance. This device, being a form of the three ball kinematic coupling, gives it excellent repeatability and good load carrying capacity; but it is a good deal more expensive to produce and install than simple modified balls.

## Mounting The Balls

There are several good methods of mounting balls in general and for the three ball kinematic coupling in particular. The simplest method is to glue complete balls into countersinks or spherical cavities. As an additional safety net, a ball with a drilled hole in the bottom can be glued over a steel pin in the center of the counter sink or spherical cavity. If even better fail-safe performance is required, the hole in the ball can be threaded and the ball can be glued over a threaded stud in the center of the counter sink or spherical cavity. A popular, inexpensive, method of mounting balls is to glue a truncated ball into a shallow counter bore. The rim of the counter-bore acts to prevent the ball from being sheared off by a slight bump. A further improvement is to glue a truncated ball with a hole in it over a pin. The most popular of all ball mounting method is to use a truncated and threaded ball held in position by a threaded fastener from behind. This mounting technique has an additional advantage because the fastener can be loosened, and the ball rotated to a new position to compensate for wear or damage. This truncated and threaded ball can also be mounted by tightening it onto a setscrew in the platform as if it were a nut. The non-marring Lotus wrench can be used for this purpose. Another popular approach is to drill and counter bore a hole through a truncated ball on the platform with a socket head cap screw from the front.

## Vee Block

The third element in the Kelvin clamp is a vee block and a sphere. This device constrains two degrees of freedom. A monolithic vee block suffers from a similar problem to the cone. The quality of the vee block surfaces is limited to a good ground surface, as there is no practical method of lapping the flat vee faces. The quality of these surfaces can be improved by breaking up the lay of the surface texture by grinding the flat prismatic faces with the side face of a grinding wheel, instead of the periphery.

## Prismatic Vee Block

It was realized that in generating the truncated flat lapped surface on the cylindrical posts for the trihedral prism we had also solved the problem of producing really high quality vee blocks. As with the trihedral, two of these devices are glued into two deep counter bores, 180 degrees across from each other. These components are produced in both hard Martensetic stainless steel and Tungsten Carbide. This vee block scenario gives excellent repeatability and high load carrying capacity; but it is more expensive to produce and install than a monolithical vee block, especially if Tungsten Carbide is used.

## Included Angle

A question that must be answered in designing the vee block is, what should the included angle of the vee block surfaces be? This discussion also applies to the cone and its derivations. This brings us back to the slippery slope concept. Both theoretical and experimental methods have both determined the most stable and most repeatable tangent angle of contact with the sphere to be 120 degrees. For the faces of the vee block to be tangent at this angle, the included angle of the vee block will be 60 degrees. Like everything in engineering, there is a buyoff here also. When the angle of the vee block gets any steeper than 60 degrees, the vector forces go way up and the load carrying capacity goes way down. Around this angle, the sphere actually begins, developing a tendency to stick or wedge in the vee when subjected to high loads.

Going in the other direction, what effect does flattening the included angle of the vee block have? As the angle flattens the load carrying capacity goes up rapidly. The repeatability falls off very slowly until we reach about 100 degrees and then it degrades very rapidly until at 120 degrees it is ineffective as a location device. What it comes down to is that the conventional 90 degrees included angle, which has good load carrying capacity and still gives excellent repeatability is a good general purpose angle and is used in most catalog vee blocks.

We must keep in mind that the very broad spectrum of kinematic applications don't all have the same goals.

For many of these applications, I may be using the term kinematic loosely, as deterministic location might be the more proper term.

In many manufacturing processes, the resistance to an upsetting force may be a paramount requirement. An upsetting force is one that is at roughly 90 degrees to the kinematic axis. The resistance of a kinematic coupling to this upsetting force is directly related to the steepness of the included angles of the individual couples. The dilemma here is to keep the system kinematic or at least semi-kinematic while resisting heavy side loading forces. From our experience, I would speculate that an included angle steeper than 50 degrees would not be functional.

## Cylindrical Vee Block

An excellent substitute for the prismatic vee block, is to use two high quality ground and lapped cylinders mounted in parallel to form a cradle for locating the sphere.

From a technical standpoint, the choice of material for the cylinders is influenced by the need for high stiffness, good wear resistance, corrosion resistance and low susceptibility to fretting. The other important parameter is on the cost side of manufacturing, with the associated ready availability of the raw material. This pattern is typical of all the kinematic elements. The choices come down to a high chrome, high carbon Martensetic stainless steel that is hard and corrosive resistant but still inexpensive or Cemented Tungsten Carbide, which is ideal; but it is several times more expensive. When the lack of magnetic permeability or the good electrical insulating properties of ceramic are needed, it can perform adequately.

Two parallel cylinders will give superior repeatability to prismatic vee blocks, but their load carrying capacity will be lower with the value determined by their outside diameter, yield strength, Young's Modulus of elasticity, and the contact angle.

## Mounting The Cylinders

There are several good methods for mounting precision cylinders for kinematic applications.

Two complete cylinders can be glued into a precision- machined trench, giving then very rigid location.

A truncated cylinder can be glued into a shallow recess on the top surface of the kinematic platform. A truncated cylinder with two small holes in the ends, can be glued over two pins on the surface of the platform.

This is a good time to acknowledge that more sophisticated modifications of the cylinders that make them simpler and more reliable in use require more sophisticated machining and therefore cost considerably more money.

The next improvement in the mounting method is to drill and tap a hole at each end of a truncated cylinder and clamp it down with two screws from behind. This is by far the most popular mounting method for cylinders.

A similar approach is to drill and counter bore a hole at each end of the truncated cylinder and to clamp it down with socket head cap screws from the top.

Cylinders have been successfully press fit or shrink fit into kinematic platforms.

## Quarter Round

The Quarter Round was developed to reduce the footprint of large two cylinder couples, while retaining the high load carrying capacity of these large diameter cylinders. By producing these component parts that have a cross section that are only one quarter of a complete cylinder, we dramatically reduce the physical volume of the platform that would be required for a complete large diameter cylinder. The quarter round is usually mounted by gluing it into a precision-machined trench in the kinematic platform. This component is not the small improvement that it might seem, but actually opens up entire new vistas of repeatability, load carrying capacity and an enormous increase in the resonate frequency.

## Two Ball Couple

Among the devices that will constrain two degrees of mechanical freedom, is the two-ball kinematics couple. By cradling a cylinder between two balls, a very positive, extremely self-balancing couple is created. This couple has good load carrying capacity as determined by the diameter of the elements, the yield strength and the Young's Modulus of the material as well as the contact angle. A combination of three cylinders and six balls will produce an excellent self-centering coupling that will constrain all six degrees of freedom. It may be less obvious, but the two ball kinematic couple will work very well against prismatic surfaces. The flat surfaces will give a higher load carrying capacity in the same size envelope. This more complex arrangement of prismatic surfaces may be used for optical or electrical reasons.

## Crossed Cylinders

The points of contact between two cylindrical surfaces are concentrated in one very small area. Crossed cylinders are a true kinematic couple. The cylinders are usually oriented at 90 degrees to each other, but alternative angles may be chosen in order to spread the preload. Cylinders have a higher load carrying capability than spheres of the same diameter. Very high quality cylinders can be economically produced from kinematicaly compatible materials. The same high chrome, high carbon Martensetic stainless that has already been mentioned is the first choice. Cemented Tungsten Carbide cylinders are readily available in many different diameters so the advantages of this material can be economically exploited.

## Cylindrical Vees

By canting or standing up two cylinders at 90 degrees to each other, an excellent vee cradle for locating a third cylinder is formed. This design is very forgiving and will capture severely mis-aligned kinematic platforms. By using three cylindrical vees, each formed by two inclined cylinders and three cylinders that intersect these vees at 90 degrees, we achieve a perfectly kinematic coupling, with good center-ability and high load carrying capacity that is not prohibitively expensive. A good ploy for mounting cylinders that stick out at 90 degrees is to glue them into a close fitting counter bore with ceramic filled epoxy glue.

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