Understanding the influence of several input factors on the relationship between bivariate responses with one directional and one linear component is an important part of some industrial applications and has not been considered often in the literature. A model based on the C-linear form suggested by Mardia and Sutton (1978) for factorial designs with a cylindrical response is presented. A strategy adapted from backwards stepwise regression is also given for obtaining a parsimonious model to highlight numerically and graphically the main and interaction effects. The proposed method is used to analyze data from an experiment involving the balancing of automotive flywheels.

Keywords: Circular-linear Data, Directional Data, Factorial Designs.

*by* **Christine M. Anderson-Cook, Virginia Polytechnic
Institute and State University, Blacksburg, VA 24061-0439
**

*INTRODUCTION*

Cylindrical data occurs in a number of industrial applications, including those that involve rotating or circular parts. In the automotive industry, parts are produced which rotate during operation of the vehicle (such as wheels, brake rotors, engine flywheels, etc.). One of the final stages of production is to precisely balance these parts to minimize the vibration transmitted through the system by these rotating components. Hence, to determine how a correction to the part should be made, a cylindrical response is obtained. A cylindrical response consists of one directional component that measures the angle or direction of the imbalance, and a linear component quantifies the magnitude of the observed imbalance (namely, how much weight should be removed at a particular location to balance the part). Since the data can be thought of as a location on a unit circle and a linear measure, this data is often called cylindrical or "circular-linear."

Frequently, a number of input factors can be adjusted during production, which may influence the components and possibly the interrelationship between the components. In this paper we address the case where the factors influence the interrelationship between the components. To summarize the relationship between components a function relating the two components is used. For example (as with simple linear regression), a first-order Taylor series approximation is often assumed to adequately describe the major features of the data. The equivalent "simple" approximation for a circular component is the first-order Fourier series approximation, which models the linear response as related to the circular component by a single cosine curve. Across the range of the circle, this curve has a single maximum and a single minimum for the linear component.

The next section of the paper introduces the example from the automotive industry, and it reviews the basic circular-linear model for the relationship between the factors. For more details on the background of the basic model and its development, see Anderson-Cook (1999). Formal tests are given for the factorial case with several factors to determine if any of the factors are influential. Adaptations of linear regression methods are presented to develop an overall model. Finally, a parsimonious model, which identifies factors significantly influencing the cylindrical relationship, is obtained by a variation of backwards stepwise regression. These factors can then be evaluated using subject matter knowledge to assess whether the variables remaining in the reduced model are important.

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