How Flat Is Flat?

Two common methods can measure surface plate flatness, but not without problems

by Philip Stein

Imagine you are stranded on a desert island. After appropriate work on your part, your basic needs such as shelter and food have been satisfied, and you decide to begin building an industrial society (OK, I know that's a bit far-fetched, but work with me here). Since this is a measurement column, I'm most interested in how to build a measurement system for an industrial society. Where do you start?

Let's start by establishing a basis for measurement of physical dimensions. That's a good place to start because many of the subsequent tasks will be based on dimensions. To measure dimensions, start with a straight line and mark it off in equal units. Straight lines are pretty easy to generate: Just stretch a string (or a vine, if you wish to continue with the tale), and if you pull hard enough, it will be straight enough for most purposes. If you stretch it vertically, gravity won't interfere and you won't even have to pull very hard--just enough to get out the kinks.

Next, you'll want to deal with two and three dimensions. First, you need a flat surface to serve as a reference plane so you can compare two objects' dimensions; then you need a right angle so you have a third dimension perpendicular to your flat. This is where this article is really headed--generating and measuring flats and right angles.

Making a flat surface

Generating a flat surface is a bit tricky, certainly harder than stretching a string. In fact, the approach is quite elegant. If you take two stones, put some grit between them and rub randomly for a long time, the grinding action will make them fit together quite smoothly. If you want a flat surface, it's easier to start with stones that are sort of flat already. That will speed up the process.

Grinding two stones (call them A and B) randomly will produce mating parts. They could both be flat, or one could be convex and the other concave. Since these are your only standards, you don't know whether they're flat or curved. The tricky part is to add a third stone, C. Alternately grinding A on B, B on C and C on A will produce three mating parts, and the only way for them all to fit one another is if they're all flat.

Once you have a flat stone, you can take three roughly square stones and, laying them on the flat surface, grind them against each other (A-B-C) until they all mate. Again, the only way they will all fit each other is if they're all perpendicular to the flat stone.

Flats and squares are still made this same way, by grinding stones (usually granite). They're not built using the principle shown above, however. Machines (and for the most precise results, people) grind the stones so they're flat (or square) and then measure the flatness with special tools.

Generally, the flat stones are called surface plates, and they can be bought in sizes from 6 x 6 inches up to 8 x 16 feet or even larger. Any area on a flat that appears to be standing above the overall surface can be smoothed with grit, and the flatness measurement can then be repeated.

Measuring flatness

Measuring flatness of surface plates and repairing worn ones are a thriving business. The surface plate is a tool universally used in machining and manufacturing, and any shop will have one or more. Frequent use can make them wear unevenly, and because they are used as reference standards, they require periodic maintenance and calibration.

There are two common methods for measuring flatness. In the first method, a very sensitive mechanical or optical level is used to determine the tilt of the surface at each point in a grid. One grid point is defined as being at zero elevation, and the surface is then measured as going uphill or downhill from that point to the next, then to the next and so on. Knowing the angle of uphill or downhill tilt and combining that with the grid spacing distance, you can calculate the height difference at each grid point.

The optical instruments for measuring these small angles are known as autocollimators. Metrology lasers outfitted with angle mirrors can also be used for this purpose. The mechanical instruments usually contain precision pendulums.

In the second method, a calibrated straightedge is supported above the surface. A sensitive mechanical or electronic indicator is used to measure the distance between the straightedge and the surface at each grid point. The straightedge is moved to various locations over the table until every grid point is covered. This method is best for small plates because for larger ones, a very long straightedge is needed. These straightedges are usually made of granite so they don't droop in the middle. Long ones would be too heavy to handle. The optical method can handle the largest plates.

A third method uses a small straightedge just long enough to bridge the gap from one grid point to the next and to measure the deviation from the tilt expected from the starting grid point. This technique measures the repeatability of the surface curvature, but will indicate a constant value on a spherical surface--a uniform curvature but not one that's flat. Often, this instrument is mistakenly used as a flatness measure, which it is not. If you know most of a plate is flat and you want to find local areas for refinishing, this repeatability gage is very useful.

Both flatness methods are advantageous because they are relatively easy to carry out, and the calculations necessary to reduce the resulting data are pretty simple and can be done with a pencil and paper. The original publication of these methods, and many subsequent ones, includes a test for closure. Since several of the measured lines cross each other, a practitioner can compare the answers obtained separately from each line at the intersection. Usually, if the answers are within a tolerance, the measurement is accepted. Otherwise, some or all of the procedure must be repeated.

Errors accumulate

Both of these methods were invented more than 40 years ago. At that time, the concept of measurement uncertainty was deemed less important than it is now, and the closure test was usually the only way to assure the quality of the measurement result. Today, however, uncertainty is considered of prime importance. Laboratories that calibrate surface plates under accreditation need to be able to calculate the measurement uncertainty of their processes, and this has uncovered a primary issue in flatness.

It's easiest to explain using the autocollimator. Each grid point elevation is calculated from the elevation of the previous point plus the height of the slope (up or down) given by the measured angle. Any uncertainty in the angle measurement results in an uncertainty of the calculated height.

A point located 10 grid points away from the origin (the arbitrary zero elevation) has been arrived at through an imperfect process repeated 10 times over. The uncertainties of individual measurements are not likely to be correlated among themselves, and so a 10-step process is uncertain by the error of one step multiplied by the square root of 10.

It's best to assign the origin to the center of the plate. That way, the number of steps to the furthest measured point is smallest. Taking bigger steps also reduces uncertainty, but gives you less detail about the table profile.

The laser and pendulum methods accumulate errors in exactly the same way. The straightedge method usually gets to any point with fewer steps, depending on the procedure used. However, the uncertainty still grows at least as fast as the square root of the number of steps.

The ideal approach to reducing the errors of surface plate calibration is to use a procedure that measures each point from a variety of approaches. Rather than pick an arbitrary zero point, a statistical procedure called a two-dimensional least-squares fit will find the plane that minimizes the overall measurement variation.

Such a technique, however, is beyond pencil and paper methods and requires a computer. While portable computers and even palm computers are commonplace these days, carrying and using one would be a big step for the typical surface plate shop because the math is hard to visualize.

Further development of error analyses of these measurement methods may soon yield an approach that is more accurate, simple to use and easy to understand.

PHILIP STEIN is a metrology and quality consultant in private practice in Pennington, NJ. He holds a master's degree in measurement science from the George Washington University in Washington, DC, and is an ASQ Fellow.

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