Component tolerances for assembled products are often set with the help of the error transmission formula. However, this approach requires knowledge of the partial derivatives of the functional relationship between the component dimensions and the assembly quality characteristic. In many practical situations, those quantities are not easily obtained. In this article we will demonstrate how a combined use of computer-aided design (CAD) and design of experiments (DOE) can be used to obtain partial derivatives of the functional relationship without knowing an explicit mathematical expression for it. With knowledge of the partial derivatives, we can use the error transmission formula to establish functional tolerances. The intent of the present article is to demonstrate, with some examples, an idea and a set of techniques that can be used to set functional tolerances for mechanical components and assemblies.

Keywords: Computer-Aided Design, Design of Experiments, Estimation.

*by* **Søren Bisgaard, Institute for Technology
Management, University of St. Gallen 9000, Switzerland**;**
Spencer Graves, Productive Systems Engineering, San José,
CA 95126**; and** Garrick Shin, Hewlett-Packard Company,
Agilent, Rohnert Park, CA 94928**

*INTRODUCTION*

Mass production and assembly lines, a vital basis for our modern economy and high standard of living, are founded fundamentally on the subtle concept of producing components within narrow tolerance limits. Tolerances allow for the achievement of acceptable variation in complex assemblies despite variability in the individual components. In fact, Hounshell (1984) shows that it was largely the inability to manufacture components consistently within narrow limits that held back the widespread use of interchangeable parts and, hence, mass production from its modern beginning in the late eighteenth century to the early twentieth century.

Unfortunately, the science of setting realistic tolerances for complex products is today still in an embryonic state. Many manufacturers use rules of thumb that often result in tolerances that are too tight and needlessly expensive or too wide, resulting in unsatisfactory assembly performance.

A more formal approach is based on theoretical calculations of the assembly’s variance. From the contributions coming from the individual components’ variances and knowledge of the functional relationship between input and output, we can use the first order error transmission formula to estimate the variance of the output (see, e.g., Montgomery (1996)). This approach has been used at least since Shewhart (1931, p. 256) and provides the theoretical foundation for much research in statistical tolerancing, including ours and many of the recent computer-based methods. See Bjørke (1989) for computer-based tolerancing and Srinivasan and Voelcker (1993) for a recent state-of-the-art assessment.

An important new term is *functional tolerancing.*
In this approach, the output functionality of a particular
system is bracketed with tolerances for how much variability
can be allowed without appreciably affecting the ability of
the product to function and meet customers’ needs. A
transfer function relationship between design or input variables
and output quality is used to translate the tolerances on
functionality into specifications for the input variables.
These concepts are discussed in Srinivasan, Wood, and McAdams
(1995); Taylor and Henderson (1994); and Zhang, et al. (1997).
We will therefore use the term "functional tolerancing"
to describe tolerancing based on a model that relates characteristics
of components to the functionality of the assembly in meeting
customers’ needs.

One problem common to the approaches described above is that they require a precise mathematical model relating component parameter values to the assembly quality characteristic of interest. They are therefore of limited practical use when such relationships are not known; this is frequently the case because these relationships are either impossible to specify or too cumbersome to establish. Many practical engineers ultimately resort to physical tests, cumbersome and expensive as they may be, to verify the usefulness of a particular set of tolerances.

A new experimental approach for functional tolerancing, useful when no theoretical transfer function is available, has recently been demonstrated by Bisgaard (1997). With this approach, several prototype components are made to exacting specifications either a little below or above the nominal dimensions according to a designed experiment. These perturbed physical prototype components are then assembled either in all possible ways or according to some other experimental design. The quality characteristic of interest is then measured. From the experiments we then get empirical estimates of the partial derivatives; those can in turn be used with the error transmission formula (Equation (1) below) to set realistic tolerances.

One disadvantage of this, and for that matter any experimental approach, is that it requires the manufacture and assembly of costly physical prototypes. When the assembly quality characteristic is a geometric dimension and the problem is one of setting geometric tolerances, we will show that it might not be necessary to make costly physical prototypes. Using instead computer-aided design (CAD) software, we might perturb the component drawings using two-level factorial experiments, assemble the parts on the screen to a final assembly, and measure, again on the screen, the geometric influence of these perturbations. The evaluation of perturbed assemblies is facilitated by modifying a master file containing the nominal design drawing. In the CAD drawing, the geometric deviations of the assembly characteristic from the nominal will be obtained from the CAD drawings and used to estimate the partial derivatives of the design function with respect to each of the design variables. This will give us the information needed to use the error transmission formula to set tolerances.

In this article, we will outline the idea of setting tolerances using CAD and DOE. First we will introduce the theoretical foundation for our approach. This is followed by an application to the design of a simple square bracket. For this particular design it is easy to establish the functional relation. We can therefore compare the experimental results with those obtained from theoretical calculations. After this initial example, we apply this approach to a more complex design problem where the theoretical function is harder to establish. In the final section of this article, we speculate about future applications of our approach, its possible disadvantages, and what would be needed in terms of software to make it useful for the general design engineering public.

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