## 3.3.7 A Mixed-Effects Model for the Analysis of Circular Measurements

Jack C.M. Wang

Statistical Engineering Division, ITL

C.T. Lam

Department of Industrial and Operations Engineering, University of Michigan

A circular feature in a mechanical object is one of the most basic geometric primitives. Its specification can be described easily by a center and a radius. A circular feature has several functional advantages: it has uniform strength in any direction, and its symmetry offers simplicity in assembly. However, due to imperfections introduced in manufacturing, machined parts will not be truly circular. For example, uncertainty in the positioning of the tool will cause variability in the center location; tool wear and vibration can affect the radius and the circularity of the produced features or machined parts. To estimate the geometric parameters, discrete sets of measurements are taken from machined parts. A computer controlled Coordinate Measuring Machine is commonly used for this task.

We present a statistical model for circular measurements. The proposed model captures the variation in center location of different machined parts. The radii of machined parts are assumed to be different and will be estimated from measurements. The major difference between the proposed model and the other statistical models for circular measurements studied before in the literature is that the latter models assume that the center of the true circle is fixed but unknown and hence the models do not include the between-part variation of a circular feature manufacturing process.

Under the assumption that the angular differences between measurements are known, the model is simplified to a linear model. Maximum likelihood estimates are derived for both the within and the between-part variations, as well as for the geometric characteristics. The geometric parameter estimates are compared statistically with the nominal values. A two-sided confidence interval for the between-part variability and a tolerance region which captures the population of the center of machined parts are also provided. A simple sampling scheme is obtained which minimizes the variance of the center estimate and takes into consideration the sampling cost of adding an extra machined part to the sample relative to that of taking extra measurements from machined parts. Based on this sampling scheme, statistical process control procedures can be developed to monitor the performance of the manufacturing process over time. An example on the automobile transmission gear carrier is given to illustrate the use of the results derived.

This work will appear in the May issue of Technometrics.

Date created: 7/20/2001
Last updated: 7/20/2001