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Figure S1-2 The two-factor interaction between cure time and cure temperature.
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thickness. The cure temperature effect can be evaluated by comparing the average of the eight runs in the top of the cube (where temperature 100 F) with the average of the eight runs in the bottom (where temperature 70 F), or 14.125 13.425 0.7. Thus, the effect of increasing the cure temperature is to increase the average pull-off force by 0.7 pounds. Thus, if the engineer s objective is to design a connector with high pull-off force, there are apparently several alternatives, such as increasing the wall thickness and using the standard curing conditions of 1 hour and 70 F or using the original 3 32-inch wall thickness but specifying a longer cure time and higher cure temperature. There is an interesting relationship between cure time and cure temperature that can be seen by examination of the graph in Fig. S1-2. This graph was constructed by calculating the average pull-off force at the four different combinations of time and temperature, plotting these averages versus time and then connecting the points representing the two temperature levels with straight lines. The slope of each of these straight lines represents the effect of cure time on pull-off force. Notice that the slopes of these two lines do not appear to be the same, indicating that the cure time effect is different at the two values of cure temperature. This is an example of an interaction between two factors. The interpretation of this interaction is very straightforward; if the standard cure time (1 hour) is used, cure temperature has little effect, but if the longer cure time (24 hours) is used, increasing the cure temperature has a large effect on average pull-off force. Interactions occur often in physical and chemical systems, and factorial experiments are the only way to investigate their effects. In fact, if interactions are present and the factorial experimental strategy is not used, incorrect or misleading results may be obtained. We can easily extend the factorial strategy to more factors. Suppose that the engineer wants to consider a fourth factor, type of adhesive. There are two types: the standard adhesive and a new competitor. Figure S1-3 illustrates how all four factors, wall thickness, cure time, cure temperature, and type of adhesive, could be investigated in a factorial design. Since all four factors are still at two levels, the experimental design can still be represented geometrically as a cube (actually, it s a hypercube). Notice that as in any factorial design, all possible combinations of the four factors are tested. The experiment requires 16 trials.
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Figure S1-3 A four-factorial experiment for the connector wall thickness problem.
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Generally, if there are k factors and they each have two levels, a factorial experimental design will require 2k runs. For example, with k 4, the 24 design in Fig. S1-3 requires 16 tests. Clearly, as the number of factors increases, the number of trials required in a factorial experiment increases rapidly; for instance, eight factors each at two levels would require 256 trials. This quickly becomes unfeasible from the viewpoint of time and other resources. Fortunately, when there are four to ve or more factors, it is usually unnecessary to test all possible combinations of factor levels. A fractional factorial experiment is a variation of the basic factorial arrangement in which only a subset of the factor combinations are actually tested. Figure S1-4 shows a fractional factorial experimental design for the four-factor version of the connector experiment. The circled test combinations in this gure are the only test combinations that need to be run. This experimental design requires only 8 runs instead of the original 16; consequently it would be called a one-half fraction. This is an excellent experimental design in which to study all four factors. It will provide good information about the individual effects of the four factors and some information about how these factors interact. Factorial and fractional factorial experiments are used extensively by engineers and scientists in industrial research and development, where new technology, products, and processes are designed and developed and where existing products and processes are improved. Since so much engineering work involves testing and experimentation, it is essential that all engineers understand the basic principles of planning ef cient and effective experiments. We discuss these principles in 13. 14 concentrates on the factorial and fractional factorials that we have introduced here.
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