National Physical Laboratory

Why is a partial arc a bad datum?

Man Question Mark Leaning

A partial feature is one that constitutes a fraction of a complete feature:

  • An actual feature might be of this form
  • Access difficulties makes it possible to only measure part of the feature

Examples:

  • Arc of a circle
  • Patch of a sphere
  • Frustum of a cone (a solid figure formed from a cone by removing a slice at the top parallel to the base)

Such features are more difficult to measure than full features. Because of the incomplete nature of the surface, errors can occur when trying to predict the centre and the radius of the best-fit circle from co-ordinate data from a CMM.

Such a feature is a bad choice for a datum.

Try this!

Measure a circular item of 25 mm radius, by contacting the surface at 20 points around the circumference. Note the centre co-ordinates and radius. Now measure again but this time contact at 20 points on a 45° sector of the ring. Depending on the form deviations in the surface the results could be quite different.

Measuring partial arcs on CMM

If a partial feature is to be measured relative to some other datum then it is often better to fit a circle equal to the specified radius and the look at the deviations of form from this circle.

Consider a partial (circular) arc, measured at ten points uniformly spaced over part of a nominally circular feature.

Partial arc

Suppose a least-squares circle is fitted to these points to obtain the radius and centre co-ordinates of the circle of which the arc is part. These circle parameters will have uncertainties associated with them as a consequence of the CMM measurement uncertainties. These uncertainties can be considerably greater when determined from such partial arc data as opposed to the use of measurements giving sensible coverage of a complete circle.

A partial arc will subtend a certain angle at the centre of a circle.  Suppose the length of an arc is halved, thus halving the subtended angle. Ten uniformly spaced measurements are taken as before, but over this shorter arc. The resulting uncertainty of the computed radius is increased by a factor of approximately four, with a comparable statement concerning the uncertainty of the centre co-ordinates. This result applies for circular arcs that subtend any angle up to approximately 80°.

The significance of the result can be seen by applying it to an arc subtending say 80°, and then subtending, say 5°. The uncertainty in the radius determined for the latter case is greater than that for the former by a factor of over 250.

Design consideration - partial arcs

To determine the radius and centre of a 5°arc to within a small uncertainty might require an accuracy of co-ordinate measurement that is not readily available.

A design specification in terms of such quantities can be regarded as unreasonable. A specification that required the form deviation (departure from circularity) to meet a certain tolerance would be much more reasonable.

Design consideration partial arcs

Determining whether the measurements of the arc indicate acceptance in this sense is simpler and forms a better approach. The arc would not be a good choice as datum.

Least squares or minimum zone

When evaluating form on CMMs the least-squares method has traditionally been used.

Today the reference feature should be determined by the minimum zone method:

The minimum zone is defined for a circle as positioned to just enclose the measured profile such that their radial departure is a minimum.

If a reference feature is determined by least squares it can be assumed that the peak to valley is larger than the minimum zone requirement.

Least square should only be used where minimum zone is not available (on older CMMs) or when the method has been agreed in advance. If no evaluation rule is indicated for form testing then the minimum zone method should be used especially in arbitration cases.

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