Question

Hi, can you help me provide the step-by-step working solution for part (b) of the above question?

12 A nickel-titanium alloy is used to make components for jet turbine aircraft engines. Cracking is a potentially serious problem in the final part, as it can lead to non-recoverable failure. A test is run at the parts producer to determine the effects of four factors on cracks. The four factors are pouring temperature (A), titanium content (B), heat treatment method (C), and the amount of grain refiner used (D). Two replicated of a 24 design are run, and the length of crack (in um) induced in a sample coupon subjected to a standard test is measured. The data are shown below: Replicate Replicate A B C D | + + + + + + - + + + + + + + 7.037 14.707 11.635 17.273 10.403 4.368 9.360 13.440 8.561 16.867 13.876 19.824 11.846 6.125 11.190 15.653 6.376 15.219 12.089 17.815 10.151 4.098 9.253 12.923 8.951 17.052 13.658 19.639 12.337 5.904 10.935 15.053 + - + + + + + + - - - + + + + + + + + + + + (a) Estimate the main and interaction) factor effects. Which factor effects appear to be large? (Hint: calculate the percent contribution of each factor). (b) Based on the most significant factors, suggest a regression model for this experiment and perform the curve-fitting (you can use the Regression option of the Data Analysis toolkit in EXCEL). Also provide the normality plot for this regression.

Answer 1

A titanium-nickel alloy is a superalloy. It is mostly used for making jet turbine aircraft engines. Jet turbines work from high cyclic dynamic, thermal loads are acting on the jet engine body. Therefore,  cracking is occurs in their parts and it is a very sober problem So, that we make an experimental test for four factors on cracks.

The four fours are pouring temperature (A), titanium (B), heat treatment (C), amount of grain refiner used (D). The best optimum technique can be a 24 Experiment design technique, which is developed by Dr. Taguchi.

The data are shown below in the design table and total of the replica I and II and the effect of factors

A	B	C	D	Replicate I	Replicate II	Total
-1	-1	-1	-1	7.037	6.376	13.413
1	-1	-1	-1	14.7.7	15.219	15.219
-1	1	-1	-1	11.635	12.089	23.724
1	1	-1	-1	17.273	17.815	35.088
-1	-1	1	-1	10.403	10.151	20.554
1	-1	1	-1	4.368	4.098	8.466
-1	1	1	-1	9.36	9.253	18.613
1	1	1	-1	13.44	12.923	26.363
-1	-1	-1	1	8.561	8.951	17.512
1	-1	-1	1	16.867	17.052	33.919
-1	1	-1	1	13.876	13.658	27.534
1	1	-1	1	19.824	19.639	39.463
-1	-1	1	1	11.846	12.337	24.183
1	-1	1	1	6.125	5.904	12.029
-1	1	1	1	11.19	10.935	22.125
1	1	1	1	15.653	15.053	30.706

A B AB CAC BC ABC D AD BD ABD CD ACD BCD ABCD TOTAL -1 -1 1 1 1 1 -1 -1 1 1 -1 1 -1 - 1 1 13.413 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 29.926 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 23.724 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 35.088 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20.554 8.466 -1 1 1 1 1 1 -1 - 1 1 -1 1 -1 1 -1 1 18.613 1 1 1 1 1 1 1 1 1 1 1 1 26.363 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 17.512 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 33.919 - 11 1 - 11 -1 1 1 -1 1 -1 -1 1 1 1 27.534 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 39.463 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 24.183 1 .1 - 1 1 1 1 1 1 1 1 1 1 1 1 1 12.029 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 22.125 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 30.706

Effectr = Contrast/2k – 1)n = Contrast/8 * 2)

EffectA = 48.302/16 = 3.018875SSa = (48.302*48.302) 32 = 72.908850126

Effect AB = 30.946/16 = 1.934126 SSAB = 30.946 * 30.946)/32 = 29.926716125

EffectABC = 50.2/16 = 3.1375SSAB = (50.2 * 50.2)/32 = 78.75125

Effect ABCD = 0.226/16 = 0.014125SS ABCD = (0.226 -0.226)/32 = 0.001596126

Interaction Plot for Replicate I & II Data Means

The Factors effect plot

Main Effects Plot for Replicate I & II Data Means А в 22.5-1 20.0 Mean 25.0 22.5 20.0

Fitted Line plot for Factor A

Fitted Line Plot A = -0.6622 +0.02872 Replicate I & II 1.0- . . 1.03631 6.0% R-59 R-Sq(adj) 0.0% -1.0 - 10 15 20 25 Replicate I & II 30 35 40

Fitted Line plot for Factor B

Fitted Line plot for Factor C

Fitted Line Plot B = - 1.544 + 0.06696 Replicate I & II R-59 R-Sq(adj) 0.876503 32.8% 28.0% -1.0 - 10 .... 15 . . 20 25 Replicate I & I 30 35 40

Fitted Line Plot C = 0.8443 -0.03662 Replicate I & II .. 1.0 . . R-59 R-Sq(adj) 1.01529 9.8% 3.4% с -1.04 10 15 20 25 Replicate I & II 30 35

Fitted Line plot for Factor D

Fitted Line Plot D = -0.9074 + 0.03935 Replicate I & II 1.04 . . R-54 R-Sa(adj) 1.00671 11.3% 5.0% D 0.0- -1.0 . 10 .. 15 .. . . 20 25 Replicate I & II 30 35

Minitab Output

Fitted Line: D versus Replicate I & II

Regression Analysis: Replicate I & II versus A, B, C, D

The regression equation is
Replicate I & II = 23.1 + 2.10 A + 4.90 B - 2.68 C + 2.88 D

Predictor Coef SE Coef T P
Constant 23.057 1.632 14.13 0.000
A 2.100 1.632 1.29 0.225
B 4.895 1.632 3.00 0.012
C -2.677 1.632 -1.64 0.129
D 2.877 1.632 1.76 0.106

S = 6.52716 R-Sq = 59.9% R-Sq(adj) = 45.4%

Analysis of Variance

Source DF SS MS F P
Regression 4 701.02 175.26 4.11 0.028
Residual Error 11 468.64 42.60
Total 15 1169.66

Source DF Seq SS
A 1 70.54
B 1 383.39
C 1 114.67
D 1 132.43