View Full Version : I still don't get the surface plate three point mount thing...

I've read the Third Hand thread...but I still don't get the geometric significance of a three point mount over say a four point mount, or a levelled pair of scraped knife edges or something like that. And now, leafing through Foundations of Mechanical Accuracy, I see that Moore mounts their plates by the four corners...but at one end there is a cross member that is supported in the middle by one point making it overall a three point support. So why support the table at four points but mount it at three? I just don't get it, stuff eludes me (stuff in general, not just this!)

[This message has been edited by abn (edited 02-27-2004).]

yorgatron

02-27-2004, 07:21 AM

why is it easier to build a 3 legged stool that keeps all its feet on the floor than a 4 legged one?

What yorgatron said: 3 points determine a plane, so with 3 points, it will always sit solidly.

As far as the location of the 3 points: they're positioned to result in minimum sag between them. Doesn't the Moore book talk about the "Airy points"?

Paul Alciatore

02-29-2004, 04:40 AM

First, we are talking about idealized points. In reality they can only be approximated. That having been said:

Any three dimensional object has six degrees of freedom: it can move in the three directions commonly referred to as X, Y, and Z and it can rotate about the three mutually perpendicular axis that are in those same directions. So it has three lateral and three rotational ways to move for a total of six. This is the common coordinate system used to describe positions in mathematics.

By fixing a single point of that object it can no longer move in any of the three lateral directions – if it did, then the point would not be “fixed”. If you fix a second point, it looses two of the rotational degrees of freedom but can still rotate about an axis that goes through the two points. Remember we are talking idealized points here, not a real mounting pad which has a real area and therefore many, many actual points. Finally, by assigning a third point, it is completely fixed in space. No motion of any type can take place without one or more of those points becoming not fixed. That’s the mathematical theory behind three point suspension. But that’s all theoretical and in the real world, things can become more complicated.

Real objects are not completely rigid. They will flex under the influence of gravity and any other forces that are acting on them. Three points may not be enough to adequately support the weight or to contain the forces generated by a machine in operation. Three points may not provide a safe base to avoid tipping over unless they extend a significant distance outside of the object itself. (Think of a camera on a tripod. The camera may be only about six inches but the ends of the tripod legs are on a 2 or 3 foot circle. This intrudes into operator space.)

So a true, three point suspension is not always practical. But when we go to a fourth point, we are taking a risk of adding an additional force (torque) on the object if the four points on the object and on the mount are not perfectly aligned. The first three fix the object in space and the fourth, if not perfectly aligned, will torque it; perhaps bending it out of shape. This is why we have to shim the fourth leg of a lathe to get the bed completely flat and parallel. No table (mount) in the world is going to be flat within tenths.

The suspension you mention with two points on one end and a beam with two points on the object and only the center held up is a way around this situation. There are many others. Some objects need to be supported very uniformly over their entire area but also with no torques to distort their shape. Telescope mirrors are one example of this. A 100 or 200 inch mirror will bend and flex in a most unacceptable manner if suspended by only three points no matter which three you choose. This is complicated by the fact that it is tilted in use and may be anywhere from horizontal to near vertical. At every angle the shape can change in a different manner. But it is equally bad to add more “fixed” suspension points because the forces they apply will change as it is moved. One solution that is used is to start with three primary points and at each of these a three armed support extends out to three more points in triangular fashion. That makes a total of nine points. For really big mirrors, each of those nine points can have either a two ended arm for a total of eighteen points or another triangular plate with three points each for a total of 27. All the distances are carefully calculated to provide the best balance of forces. A vehicle suspension is another example of four or more points of suspension. All the wheels should remain on the road for best operation and the various elements in the suspension system allow this condition to be met.

Most objects we deal with are not this well mounted. In many cases the distortion does not matter. In others, the mount itself is flexible enough to allow the object to not distort to a significant degree. Each case must be considered on it’s own.

One practical consideraton here is that any object that is carefully finished to a particular shape (lathe bed, surface plate, telescope mirror, etc.) is likely to return to the closest to that shape when it is supported in exactly the same manner as it was when that finishing was being done. Any other suspension will likely produce some distortion from that "ideal" shape. So it may do no good to provide a better suspension than was used to make the object. It may actually make things worse.

Paul A.

Herb Helbig

02-29-2004, 11:47 AM

Very nice summary, Paul A.! A couple of key search terms are "kinematic design" and "overconstraint".

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Right on Paul! Couldn't have said it better myself.