Kevin N. Otto

Modular product architecture

Abstract This paper presents an approach to architecting a product family that shares inter-changeable modules. Rather than a fixed product platform upon which derivative products are created through substitution of add-on modules, the approach here permits the platform itself to be one of several possible options. We first develop function structures for each product. After comparing function structures for common and unique functions, rules are applied to determine possible modules. This process defines possible architectures. Each architecture is represented using a matrix of functions versus products, with shared/unique function levels indicated. This provides a systematic approach to generating architectures.

Trade-off strategies in engineering design

A formal method to allow designers to explicitly make trade-off decisions is presented. The methodology can be used when an engineer wishes to rate the design by the weakest aspect, or by cooperatively considering the overall performance, or a combination of thesestrategies. The design problem is formulated with preference rankings, similar to a utility theory or fuzzy sets approach. This approach separates the designtrade-off strategy from the performance expressions. The details of the mathematical formulation are presented and discussed, along with two design examples: one from the preliminary design domain, and one from the parameter design domain.

Engineering design calculations with fuzzy parameters

Uncertainty in engineering analysis usually pertains to stochastic uncertainty, i.e. ,v ariance in product or process parameters characterized by probability (uncertainty in truth). Methods for calculating under stochastic uncertainty are well documented. It has been proposed by the authors that other forms of uncertainty exist in engineering design. Imprecision, or the concept of uncertainty in choice, is one such form. This paper considers real-time techniques for calculating with imprecise parameters. These techniques utilize interval mathematics and the notion of -cuts from the fuzzy calculus. The extremes or anomalies of the techniques are also investigated, particularly the evaluation of singular or multi-valued functions. It will be shown that realistic engineering functions can be used in imprecision calculations, with reasonable computational performance.

Imprecision in Engineering Design

Methods for incorporating imprecision in engineering design decision-making are briefly reviewed and compared. A tutorial is presented on the Method of Imprecision (MoI), a formal method, based on the mathematics of fuzzy sets, for representing and manipulating imprecision in engineering design. The results of a design cost estimation example, utilizing a new informal cost specification, are presented. The MoI can provide formal information upon which to base decisions during preliminary engineering design and can facilitate set-based concurrent design.

Tuning parameters in engineering design

In the design and manufacture of mechanical devices, there are parameters whose values are determined by the manufacturing process in response to errors introduced in the device’s manufacture or operating environment. Such parameters are termed tuning parameters, and are distinct from design parameters which the designer selects values for as a part of the design process. This paper introduces tuning parameters into the design methods of: optimization, Taguchi’s method, and the method of imprecision [10]. The details of the mathematical formulation, along with a design example, are presented and discussed. Including tuning parameters in the design process can result in designs that are more tolerant of variational noise.

Constructing membership functions using interpolation and measurement theory

Imprecision is well suited for representing uncertainty in choice during an engineering design process, and in particular for computer-aided engineering design and analysis tools. To propagate imprecise understanding through engineering tools, however, first the membership functions must be constructed based on the understandings of the design engineer. For this purpose, measurement theory offers an axiomatically based, easy to use method. For any given variable, the best and worst values are determined, and the remaining values are assigned a degree of membership by comparison with the best and the worst. On real variables, however, this would require an infinite number of questions. Instead, a continuity assumption can be made, and the remaining membership values determined through interpolation. Traditional interpolation schemes, however, fail to satisfy the restrictions of a membership function. The [0,1] boundedness condition and the fuzzy-convex property in particular present difficulty. A simple and efficient constrained interpolation scheme is developed for fitting a membership function to a finite number of known membership values. Thus, a simple method enabling design engineers to construct membership functions is presented.

Approximating a-cuts with the vertex method

If f:Rn → R is continuous and monotonic in eachvariable, and if μi is a fuzzy number on the ith coordinate, then the membership on R induced by ƒ and by the membership onfRn given by μ(x) = min(μ1(x1), …, μn(xn)) can be evaluated by determining the membership at the endpoints of the level cuts of each μi. Here more general conditions are given for both the function ƒ and the manner in which the fuzzy numbers {μi} are combined so that this simple method for computing induced membership may be used. In particular, a geometric condition is given so that the α-cuts computed when the fuzzy numbers are combined using min is an upper bound for the actual induced membership.

Measurement methods for product evaluation

Among the many tasks designers must perform, evaluation of product options based on performance criteria is fundamental. Yet I have found that the methods commonly used remain controversial and uncertain among those who apply them. In this paper, I apply mathematical measurement theory to analyze and clarify common design methods. The methods can be analyzed to determine the level of information required and the quality of the answer provided. Most simple, a method using an ordinal scale only arranges options based on a performance objective. More complex, an interval scale also indicates the difference in performance provided. To construct an interval scale, the designer must provide two basic a priori items of information. First, a base-pointdesign is required from which the remaining designs are relatively measured. Second, the deviation of each remaining design is compared from the base point design using a metricdatum design. Given these two datums, any other design can be evaluated numerically. I show that concept selection charts operate with interval scales. After an interval scale, the next more complex scale is a ratio scale, where the objective has a well-defined zero value. I show that QFD methods operate with ratio scales. Of all measurement scales, the most complex are extensively measurable scales. Extensively measurable scales have a well defined base value, metric value and a concatenation operation for adding values. I show that standard optimization methods operate with extensively measurable scales. Finally, it is also possible to make evaluations with non-numeric scales. These may be more convenient, but are no more general.

Design Parameter Selection in the Presence of Noise

Themethod of imprecision is a design method whereby a multi-objective design problem is resolved by maximizing the overall degree ofdesigner preference: values are iteratively selected based on combining the degree of preference placed on them. Consider, however, design problems that exhibit multiple uncertainty forms (noise). In addition to degrees of preference(imprecision) there areprobabilistic uncertainties caused by, for example, measuring and fabrication limitations. There are also parameters that can take on any valuepossible within a specified range, such as a manufacturing or tuning adjustment. Finally, there may be parameters which mustnecessarily satisfly all values within the range over which they vary, such as a horsepower requirement over a motors different speeds. This paper defines a “best” set of design parameters for design problems with such multiple uncertainty forms and requirements.

Swept envelopes of cutting tools in integrated machine and workpiece error budgeting

This paper presents closed form equations for calculating discrete points on the surface of a machined part given the shape and the motion function of the cutting tool. This permits rapid calculation of the effect of machine tool geometric errors on the shape of the machined part. A valve grinding machine serves as an example application of the equations. The proper profile of the grinding wheel is derived based on the desired profile of the valve. The model is also used to estimate acceptable magnitudes of errors in the machine based on desired part tolerances.