We consider gauge repeatability and reproducibility (R&R) analyses for two-dimensional data when the engineering tolerance associated with measurements is a circle. We develop summaries for repeatability, reproducibility, and R&R by employing the diameters of circles that provide 99% capture rates. We derive an inequality between the results of a one-dimensional gauge R&R method and this two-dimensional method. We also show that the additivity-of-variance in the one-dimensional gauge R&R case becomes a sub-additivity-of-variance in the two-dimensional case under certain conditions. We use measurements of unbalance of rotating devices, where the tolerance must be a circle, to motivate the ideas and to illustrate the technique. Our results apply to corresponding coordinate-measuring-machine data when the engineering tolerance is a circle. The method can be extended to other problems, including square and spherical tolerances. We recommend that this method be used in conjunction with graphical, other univariate, and multivariate analyses.Key Words: Measurement Systems Analysis, Tolerances, Two-Dimensional Data.

*By* **JOSEPH G. VOELKEL The John D. Hromi Center for
Quality and Applied Statistics, RIT, Rochester, NY 14623-5604**

**Introduction**

IN this paper, we use measurements of so-called *unbalance*
to illustrate our ideas. A common example
of unbalance occurs when a new automobile tire
is put on a rim. The amount of unbalance is detected
by a spin balancer and then corrected by attaching
weights to selected sections of the rim. This process
is repeated if needed until the so-called *residual unbalance*
is determined to be small. Unbalance in rotating
devices such as tires or pumps creates a force
that can lead to wobble, vibration, accelerated wear,
and loss of functionality. The examples we use in
this paper are based on work in balancing rotors in
centrifugal pumps.

We illustrate these ideas schematically in Figure 1. Assume that a disc, represented by the large circular area, is initially perfectly balanced. The larger solid circle represents placing a weight of 3.0 oz. a distance of 0.5 in. from the center of this disc. This creates an unbalance of 1.5 oz.-in. An equivalent amount of unbalance would be created if a weight of 0.75 oz. were placed at a distance of 2.0 in. from the center, as indicated by the smaller solid circle. Either of these unbalances can be denoted by placing a symbol of unit weight, in this case 1.0 oz., at 1.5 in. from the center, as indicated by the other two symbols.

Graphs using this unit-weight symbolism are used
in the remainder of this paper to represent unbalance
measurements. Each of these unbalance measurements
is actually a vector. This vector can be
measured in Cartesian, that is (*x, y*), coordinates, or
in polar, that is (*r*, _{}), coordinates. We can convert
from Cartesian to polar coordinates by r = _{}
and _{} = arctan (*y/x*). Figure 1 is based on polar coordinates,
in which the axis label provides the values
of *r* and the angle _{} is measured counterclockwise
from the horizontal line on the right side. In this
example, the unbalance in polar coordinates would
be (1.50, 40°). Although the unbalance reading itself
is two-dimensional, the word "unbalance" usually
refers only to the "*r*" portion of the reading, as
the mass-length unit assignments above suggest.

Unbalance can be reduced with a variety of methods. These include the removal of mass by drilling or grinding; the addition of mass, such as solder; and occasionally the re-centering of mass. For our purposes, we assume that such balancing methods have already been employed, that the amount of residual unbalance is to be measured, and that a suitable measuring device is available. For more details, see Schenck-Trebel (1990) or Paddock et al. (1999).

The objective of this article is to discuss how the variation of these two-dimensional measurements can be translated into useful summary measures of variation. We illustrate the approach using gauge R&R terminology, which we brie.y review. Extensions to more complex cases, such as those involving additional components of variance, follow along the same lines of reasoning.

This work was motivated by measurements of unbalance, for which the tolerance is always a circle. (Note that we always use "tolerance" in the engineering sense in this paper. In particular, we do not use the term to refer to statistical tolerance regions.) However, the results can be applied to most twodimensional measurements of points, including twodimensional coordinate-measurement data, so long as the tolerance is a circle.

We now illustrate an important distinction between
unbalance data and coordinate-measurement
data. In Figure 1, both the × and + symbols represent
unbalance scenarios of *r* = 1.5 units, but at
different angles _{}. For purposes of reducing unbalance,
it is important to know both *r* and _{}. But
when a pump is rotating—that is, when the part is
being used—both scenarios are equivalent. If this
had been coordinate-measurement data, these two
scenarios would typically not be equivalent when the
part was in use.

A related paper (Heaphy and Gruska (1982)) that considers geometric dimensioning and tolerancing illustrates how to estimate the fraction of parts out of tolerance when the two-dimensional set of points has a bivariate normal distribution and the tolerance region is a circle. For two-dimensional measurements when the measurement variation is circular normal, a special case of our problem, see Wang and Lam (1997). For measurement issues related to coordinate-measuring machines, see Dowling, Grif fin, Tsui, and Zhou (1997).

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