Lee,
I've got a couple of questions. What is the tolerance for feature size and what is the geometric tolerance for position? Is the position tolerance constant or variable (does it have a MMC modifier attached to the tolerance)?
I don't suppose that you are questioning about the upper and lower control limits for feature size.
Process control examination for the position tolerance (I think) is best accomplished by monitoring the X,Y,Z coordinates of the feature location separately. It provides two important benefits, It helps to distinguish whether the individual coordinates are accurate (mean centered on the basic location) or not and it helps to discover how precise each axial deviation is and how much each is contributing to process variation.
You commented about the position tolerance being non-linear scattered about 360 degrees. The position tolerance describes the zone that the axis, median plane, center, etc. of the feature must reside within. The zones can be described in a number of ways (spherical, circular, cylindrical, rectangular, square, cubic, etc.) all determined by the symbols used in (or absent from) the feature control frame and from the way the tolerance "leader lines" are depicted on the specification.
The individual coordinates can be normally distributed when examined separately but when they are combined to determine the displacement from the basic location the resultant radial separation (or doubled "diametrical deviation") is typically a skewed distribution that is 'nearer to' and 'truncated by' zero and tailed toward the USL. Monitoring the derived position deviation for process control is not good for a couple of reasons. The position deviation does not distinguish between process parameters "X, Y, & Z" that are accurate but not precise where mean-shifts are no help and ones that are precise but not accurate where mean-shifts may improve. The other reason is that the deviation is not typically a normal distribution.
The control limits for the coordinates are established by the process variability so there is no relation to the drawing specification for position. Process control can be done effectively
by monitoring the X Y & Z of geometric deviations.
To predict the process capability for the true position deviation one must first establish that the process is "in-control" and that can be done with the individual coordinates. If the geometric tolerance is constant one can predict capability by applying the appropriate distribution function that best fits the skewed true position deviations and transforming the data. If the geometric tolerance is variable you can use a method that compares the (USL plus the Mean variable tolerance "bonus tolerance" minus the Mean geometric deviation) to (three times the square root of the combined variances for size and geometric deviation). It assumes normality for both but it demonstrates prediction error margins comparable to predicting the capability of a constant tolerance with the Weibull Method.
I would not recommend using: the residual tolerance method described in "Simple Process Capability" Quality Magazine by me, the percent of tolerance method described in "Calculating MMC Cpk" by Marty Ambrose, or the adjusted true position method "Calculation of Cpk under conditions of Variable tolerances" Quality Engineering by Glen Gruner because each one of those methods uses an individual pair of variables for size and geometric deviation to produce a surrogate variable the can be compared to a constant limit. In so doing the underlying variation from the independent sources can either be amplified or moderated in the surrogate.
I will be publishing another paper soon.
confused: )
