4 Surface Roughness

EXAMPLE 14-4 Surface Roughness

  Consider the surface roughness experiment originally de-

  and the sum of squares for A is found using Equation 14-15:

  scribed in Example 14-2. This is a 2 3 factorial design in the

  1Contrast A 2 1272

  factors feed rate (A), depth of cut (B), and tool angle (C ), with

  SS A

  k n 45.5625 2 replicates. Table 14-16 presents the observed surface n 2 182

  roughness data.

  It is easy to verify that the other effects are

  The effect of A, for example, is

  A AB 1.375

  Examining the magnitude of the effects clearly shows that

  feed rate (factor A) is dominant, followed by depth of cut (B)

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  CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

  Table 14-16 Surface Roughness Data for Example 14-4

  Treatment

  Design Factors

  Surface

  Combinations

  A B C Roughness Totals

  and the AB interaction, although the interaction effect is rela-

  freedom, giving the total 3 in the column headed “DF.’’ The

  tively small. The analysis of variance, summarized in Table

  column headed “Seq S’ (an abbreviation for sequential sum

  14-17, confirms our interpretation of the effect estimates.

  of squares) reports how much the model sum of squares in-

  Minitab will analyze 2 k factorial designs. The output

  creases when each group of terms is added to a model that

  from the Minitab DOE (Design of Experiments) module for

  contains the terms listed above the groups. The first number in

  this experiment is shown in Table 14-18. The upper portion of

  the “Seq S’ column presents the model sum of squares for

  the table displays the effect estimates and regression coeffi-

  fitting a model having only the three main effects. The row la-

  cients for each factorial effect. However, the t-statistic com-

  beled “2-Way Interactions’’ refers to AB, AC, and BC, and the

  puted from Equation 14-18 is reported for each effect instead

  sequential sum of squares reported here is the increase in the

  of the F-statistic used in Table 14-17. To illustrate, for the

  model sum of squares if the interaction terms are added to a

  main effect of feed Minitab reports t

  4.32 (with eight

  model containing only the main effects. Similarly, the sequen-

  degrees of freedom), and t 2 2

  18.66, which is

  tial sum of squares for the three-way interaction is the increase

  approximately equal to the F-ratio for feed reported in Table

  in the model sum of squares that results from adding the term

  14-17 (F 18.69). This F-ratio has one numerator and eight

  ABC to a model containing all other effects.

  denominator degrees of freedom.

  The column headed “Adj S’ (an abbreviation for ad-

  The lower panel of the Minitab output in Table 14-18 is

  justed sum of squares) reports how much the model sum of

  an analysis of variance summary focusing on the types of

  squares increases when each group of terms is added to a

  terms in the model. A regression model approach is used in the

  model that contains all the other terms. Now since any 2 k de-

  presentation. You might find it helpful to review Section 12-2.2,

  sign with an equal number of replicates in each cell is an

  particularly the material on the partial F-test. The row entitled

  orthogonal design, the adjusted sum of squares will equal the se-

  “main effects’’ under source refers to the three main

  quential sum of squares. Therefore, the F-tests for each row in

  effects feed, depth, and angle, each having a single degree of

  the Minitab analysis of variance table are testing the significance

  Table 14-17 Analysis of Variance for the Surface Finish Experiment

  Source of

  Degrees of

  Variation

  Sum of Squares

  Freedom

  Mean Square

  f 0 P-Value

  A 45.5625

  B 10.5625

  C 3.0625

  AB 7.5625

  AC 0.0625

  BC 1.5625

  ABC

  Error

  Total

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  14-5 2 k FACTORIAL DESIGNS

  Table 14-18 Minitab Analysis for Example 14-4 Estimated Effects and Coefficients for Roughness Term

  Effect

  Coef

  StDev Coef

  Analysis of Variance for Roughness Source

  Main Effects

  2-Way Interactions

  3-Way Interactions

  Residual Error

  Pure Error

  of each group of terms (main effects, two-factor interactions,

  that feed rate and depth of cut have large main effects, and

  and three-factor interactions) as if they were the last terms to

  there may be some mild interaction between these two factors.

  be included in the model. Clearly, only the main effect terms are

  Therefore, the Minitab output is in agreement with the results

  significant. The t-tests on the individual factor effects indicate

  given previously.