MTRL 460-2014-TutRep3-Team9

MTRL 460Monitoring and Optimization of

Materials Processing

Tutorial 3: Introduction to Designed Experiments

Group 9:Muhammad Arshad Hassni

Vishal SharmaBennet LimIgor Vranjes

Muhammad Harith Mohd FauziDaniel Shim

Date: December 8, 2014

INTRODUCTION

Design of Experiments (DOE) is a class of statistically based techniques to organize experimentation to obtain the maximum amount of information at the minimum cost and time expenditure. Among the many different DOE techniques, two-level factorial experiments (2FD) are amongst the most effective in engineering applications. They can be easily used to (1) identify the effects and interactions of relatively few (k=2- 4) variables or to (2) screen many (k>5) variables in order to identify the few (typically 4 or less) significant ones.

2FD are however NOT the best choice to empirically model the process in terms of the few important variables; for the modeling we will be using Response Surface Methods (RSM). These two procedures (i.e. 2FD, RSM) will be aided by use of commercial software Design Expert (DX8). The best outcome of well-organized experiments is an empirical (or semi-empirical) model of the process that allows a prediction of the response Y as a function of the independent variables (factors) Xi at a specified confidence level.

Some of the features of efficient experiments include:

The experiments are carefully planned, with a significant portion of time and resources (~25%) spent at the planning stage. The problem should be well understood. The right questions should be asked, as even “an approximate answer to the right question is worth a great deal more than a precise answer to the wrong question”.

The objectives of the experimentation are clearly defined and communicated to everybody involved; the task should focus on the roots of real problems, with possible numerical targets to be reached as a result of the experimentation

The process that is experimented on is stable, well defined and understood by experimenters. In industrial environment focused on process (or product) optimization, SPC should be introduced, run and utilised to stabilise the process before attempting experimental program on process optimization

TUTORIAL 3A(i): BLIND BALANCE

Task (i): Two-level Factorial Design with 2 Variables

OBJECTIVE To determine the effects of the support and body rotations on body “blind balance” abilities.

EXPERIMENTAL PLAN

Variables

X1 Surface type X1 (low level {-}): floor X1 (high level {+}: foam

X2 Rotations X2 (low level {-}): 0 rotations 0R

X2 (high level {+}): 1 rotation 1R

1 Response: Y in-balance time (seconds)

DATA COLLECTION

Exp # X1 [surface]

X2 [rotations]

X1*X2 Y[sec]A Y[sec]B Y[sec]C Y[sec] Average

1 floor {-} 0 rot {-} {+} 51.63 45.38 50.92 49.312 foam {+} 0 rot {-} {-} 33.85 41.33 29.18 34.793 floor {-} 1 rot {+} {-} 37.08 42.76 16.42 32.094 foam {+} 1 rot {+} {+} 73.20 48.75 83.27 68.41

Average 46.15

DATA ANALYSIS

Square Plot

Total “effect” L

49.31

34.79

32.09

68.41

The total “effect” L of any given variable on the experiment outcome is calculated as the difference between all responses when the variable is at high and low levels.

The effect of X1 surface, L1:

LX1 = 34.79 + 68.41 – 49.31 – 32.09 = 21.80 s47.23% of average: weak positive effect of surface

The effect of X2 rotation, L2:

LX2 = 32.09 + 68.41 – 49.31 – 34.79 = 16.40 s35.53% of average: weak positive effect of rotations

Effect of interaction

The effect of the interaction is calculated by taking the difference between (1) the effects when all variables are the same level (i.e. both at high or both at low level), and (2) the effects when all variables are at mixed levels (i.e. one at high level and another at low level).

The effect of interaction of X1 surface and X2 rotation, L12:

LX1X2 = 49.31 + 68.41 – 32.09 – 34.79 = 50.84 s110% of average: strong interaction effect

CONCLUSIONSBoth variables have weak positive effects. The interaction between the two variables is strong on the other hand.

TUTORIAL 3A(ii): BLIND COORDINATION

Task (ii): Two-level Factorial Design with 3 Variables

OBJECTIVE To determine the effects of the hands and body rotations, on your body “blind coordination” abilities.

EXPERIMENTAL PLAN2FD3 with the following variables and response:

Variables

X1 hand-to-target distance X1 low level {-}: close X1 high level {+}: far

X2 hand X2 low level {-}: right X2 high level {+}: leftX3 rotations X3 low level {-}: 0 rot 0R X3 high level {+}: 1 rot 1R

“Response” Y: shots at target (%)

DATA COLLECTION

Exp #

MainInteractions Group

DataFactorsDistance Hand Rotation

X1 X2 X3 X1X2 X1X3 X2X3 X1X2X3 Y(%)1 -1 -1 -1 1 1 1 -1 202 1 -1 -1 -1 -1 1 1 903 -1 1 -1 -1 1 -1 1 904 1 1 -1 1 -1 -1 -1 105 -1 -1 1 1 -1 -1 1 106 1 -1 1 -1 1 -1 -1 707 -1 1 1 -1 -1 1 -1 508 1 1 1 1 1 1 1 10

Effects 10 -30 -70 -250 30 -10 50 Ave=44

DATA ANALYSIS

Cube plot

The results of such 2FD3, with 3-variables at 2 levels (23=8 experiments) factorial design of experiments can be conveniently plotted on a “cube plot”, with the response values (i.e. average % of successful hits) at the cube corners.

Total “effect” L

The effect of X1 distance, L1:

L1 = -20 + 90 – 90 + 10 – 10 + 70 – 50 + 10 = 1022.7% of average: weak positive effect of distance

The effect of X2 hand, L2:

L2 = -20 - 90 + 90 + 10 – 10 - 70 + 50 + 10 = -3068.2% of average: weak negative effect of hand

The effect of X3 rotation, L3:

L3 = -20 - 90 – 90 - 10 + 10 + 70 + 50 + 10 = -70159% of average: strong positive effect of rotation

Effect of interaction

X3

X2

X1 9020

9010

10 70

50 10

The effect of interaction of X1 distance and X2 hand, L12:

L12 = 20 - 90 – 90 + 10 + 10 - 70 – 50 + 10 = -250568% of average: strong negative effect of distance and hand: Strong Interaction Effect

The effect of interaction of X1 distance and X3 rotation, L13:

L13 = 20 - 90 + 90 - 10 – 10 + 70 – 50 + 10 = 3068.2% of average: weak positive effect of distance and rotation: Weak Interaction Effect

The effect of interaction of X2 hand and X3 rotation, L23:

L23 = 20 + 90 – 90 - 10 – 10 - 70 + 50 + 10 = -1022.7% of average: weak negative effect of hand and rotation: Weak Interaction Effect

The effect of interaction of X1 distance and X2 hand and X3 rotation, L123:

L123 = -20 + 90 + 90 - 10 + 10 - 70 – 50 + 10 = 50114% of average: strong positive effect of distance, hand and rotation:

Strong Interaction Effect

CONCLUSION

From the results above, we can see that L12 and L123 have strong interaction effect whilst L13 and L23 have weak interaction effect to our experiments.

TUTORIAL 3B: FRACTIONAL FACTORIALS FOR VARIABLES SCREENING

Task #9: Titanium Di-Boride for Composite Aluminum Machining Applications

OBJECTIVE To design the screening experiments to identify the four significant variables out of the pool of the tentative seven variables by using DX program.

Tut3B includes the following 4 steps:

1. The Scenario Advanced Titanium Di-boride (TiB2) ceramic is one of the primary candidates for applications in high-wear environments, at low and intermediate temperatures and in contact with corrosive liquid or solid aluminum. The primary advantages of TiB2 are extremely high hardness, and stability against solid and liquid aluminum. It is presently being considered as an alternative (to diamond) cutting tool for the highly abrasive composites based on aluminum, such as SiC-Al and Al2O3-Al. For the metal cutting applications, the following three properties of TiB2 are required:

R1 Maximum hardness (Target R1T = 27GPa)R2 Maximum fracture toughness (Target R2T = 7MPa√mR3 Maximum Weibull Modulus (Target R3T = 13)

The following seven process variables X1 to X7 were tentatively proposed as those controlling the required properties R1, R2 and R3:

X1 Carbon additive content (2 to 6 wt%)X2 Heating rate (10 to 30 C/min)X3 Powder milling time (5 to 10 hr)X4 Oxygen impurity content in TiB2 (1 to 5 wt%)X5 Sintering temperature (1,600 to 1,900 C)X6 Sintering time (1 to 3 hr)X7 TiB2 powder grain diameter (0.5 to 2 μm)

Using the DX program, design the screening experiments (27-3 fractional factorials) to identify the 4 significant variables out of the pool of the tentative 7 variables.

2. Screening Experiment using DX8

The tentative 7 variables and their range of variation:

The table of responses (R1, R2, R3):

3. Run the Design Experiment Using MTRL460LABSIM

The data below shows one of the simulations conducted using the LAB program.

Lab 9 Simulation completed on: 16-11-14 at 00:03:30:

Variables Values S.D.X1: Carbon Content % 3.00 wt% 0.1 %X2: Heating Rate 25.00 °C/min 0.1 %X3: Milling Time 7.00 hr 0.1 %X4: Oxygen Content in TiB2 4.00 wt% 0.1 %X5: Sintering Temperature 1700.00 °C 0.1 %X6: Sintering Time 2.50 hr 0.1 %X7: TiB2 Grain Diameter 0.75 um 0.1 %

Err. of Measurement 0.2 %

Hardness [GPa] Toughness [MPa•√m] Modulus

Average 23.938 5.901 11.754S.D. 0.395 0.023 0.057

Values Values Values1 23.086 5.857 11.6572 23.471 5.859 11.6633 23.503 5.864 11.6724 23.523 5.884 11.6885 23.670 5.888 11.6986 23.740 5.893 11.7247 23.774 5.895 11.7258 23.794 5.896 11.7559 23.817 5.903 11.758

10 23.853 5.904 11.75811 24.009 5.905 11.77312 24.025 5.907 11.77413 24.069 5.909 11.77814 24.114 5.909 11.77915 24.195 5.911 11.78116 24.251 5.912 11.78517 24.268 5.922 11.78818 24.375 5.923 11.81919 24.453 5.926 11.84820 24.768 5.947 11.860

The average value of hardness, fracture toughness, and Weibull modulus obtained from each simulation using LAB program is then transferred into DX8 Data Entry Table as shown below.

DX8 Data Entry Table:

Design summary:

4. Analysis and the 4 significant variables

Normal probability plots:

Using DX8, we generated a normal probability plot for each of the three response variables. Outliers on these plots can be identified, showing us the significant variables in this process.

Figure 4: Maximum hardness half-normal plot.

Design-Expert® SoftwareMaximum Hardness

Shapiro-Wilk testW-value = 0.825p-value = 0.001A: Carbon Additive ContentB: Heating RateC: Powder Milling TimeD: Oxygen Impurity Content in TiB2E: Sintering TemperatureF: Sintering timeG: TiB2 power grain diameter

Positive Effects Negative Effects

0.00 0.60 1.21 1.81 2.42 3.02 3.63 4.23 4.84 5.44 6.05

0

10

20

30

50

70

80

90

95

99

Half-Normal Plot

|Standardized Effect|

Ha

lf-N

orm

al

% P

rob

ab

ilit

y

A-Carbon Additive Content

D-Oxygen Impurity Content in TiB2

E-Sintering Temperature

G-TiB2 power grain diameter

AD

AGDEDG

Figure 5: Maximum fracture toughness half-normal plot.

Figure 6: Maximum Weibull modulus half-normal plot.

From figures 4-6, we can see that the outliers on the half-normal plots are carbon additive content (A), sintering temperature (E), oxygen impurity content in TiB2 (D), and TiB2 powder grain diameter (G). Therefore, these 4 variables are significant and they are the main effects of the process. The other outliers on the half-normal plot are various combinations of 2 of these main effects. This means that the interaction effects between these 4 variables are also significant.

Design-Expert® SoftwareMaximum Fracture Toughness



0.00 0.30 0.61 0.91 1.21 1.52 1.82 2.12 2.43 2.73 3.03

0

10

20

30

50

70

80

90

95

99

Half-Normal Plot


Ha

lf-N

orm

al

% P

rob

ab

ilit

y





ADAE

AG

DEDF

Design-Expert® SoftwareMaximum Weibull Modulus



0.00 1.52 3.04 4.56 6.08

0

10

20

30

50

70

80

90

95

99

Half-Normal Plot


Ha

lf-N

orm

al

% P

rob

ab

ilit

y





ADAE

AGDE

DF

DG

Figure 4 shows that the positive effects on maximum hardness are oxygen impurity content in TiB2 (D), sintering temperature (E), the interaction between oxygen impurity content in TiB2 and TiB2 powder grain diameter (DG), and the interaction between carbon additive content and oxygen impurity content in TiB2 (AD). The negative effects on maximum hardness are carbon additive content (A), TiB2 powder grain diameter (G), the interaction between oxygen impurity content in TiB2 and sintering temperature (DE), and the interaction between carbon additive content and TiB2 powder grain diameter (AG).

Figure 5 shows that the positive effects on maximum fracture toughness are carbon additive content (A) and interactions AE, AD, DE, and DF. The negative effects on maximum fracture toughness are sintering temperature (E), oxygen impurity content in TiB2 (D), TiB2 powder grain diameter (G), and interaction AG.

Figure 6 shows that the positive effects on maximum Weibull modulus are the interactions AE, AD, DE, DF, and DG. The negative effects on maximum Weibull modulus are all 4 of the main effects and the interaction AG.

Figure 7: Cube plot for A, D, E and their interactions affecting maximum hardness.

Design-Expert® SoftwareFactor Coding: ActualMaximum Hardness (GPa)X1 = A: Carbon Additive ContentX2 = D: Oxygen Impurity Content in TiB2X3 = E: Sintering Temperature

Actual FactorsB: Heating Rate = 20C: Powder Milling Time = 8F: Sintering time = 2G: TiB2 power grain diameter = 1.125

CubeMaximum Hardness (GPa)

A: Carbon Additive Content (wt%)D:

Ox

yg

en

Im

pu

rity

Co

nte

nt

in T

iB2

(w

t%)

E: Sintering Temperature (oC)

A-: 3 A+: 5D-: 2

D+: 4

E-: 1700

E+: 1800

23.3469

28.5839

22.3922

24.9181

16.2389

21.4759

20.0308

22.5567

Figure 8: Cube plot for A, D, E and their interactions affecting maximum fracture toughness.

Figure 9: Cube plot for A, D, E and their interactions affecting Weibull modulus.Figures 7-9 show the cube plots for A, D, E significant variables and their interactions and their effects on the three response variables. All three cube plots confirm the half-normal plots findings on which main effects and interactions cause positive effects and which main effects and interactions cause negative effects. However, our process has 4 significant variables and cube plots can only show the effects of 3 significant variables. Therefore, several more cube plots would have to be made with different combinations of 3 significant variables. The results would show the same as the half-normal plots.

Design-Expert® SoftwareFactor Coding: ActualMaximum Fracture Toughness (MPa*sqrt(m))X1 = A: Carbon Additive ContentX2 = D: Oxygen Impurity Content in TiB2X3 = E: Sintering Temperature


CubeMaximum Fracture Toughness (MPa*sqrt(m))


Ox

yg

en

Im

pu

rity

Co

nte

nt

in T

iB2

(w

t%)


A-: 3 A+: 5D-: 2

D+: 4

E-: 1700

E+: 1800

9.13019

7.51006

5.14231

4.84394

8.72181

7.41894

5.31819

5.33706

Design-Expert® SoftwareFactor Coding: ActualMaximum Weibull ModulusX1 = A: Carbon Additive ContentX2 = D: Oxygen Impurity Content in TiB2X3 = E: Sintering Temperature


CubeMaximum Weibull Modulus


Ox

yg

en

Im

pu

rity

Co

nte

nt

in T

iB2

(w

t%)


A-: 3 A+: 5D-: 2

D+: 4

E-: 1700

E+: 1800

13.4728

10.4799

9.60641

8.48684

12.0567

9.68084

9.57859

9.07616

TUTORIAL 3C: RESPONSE SURFACE METHOD FOR PROCESS MODELLING AND OPTIMIZATION

OBJECTIVE

Using Response Surface Methodology (RSM) principles, Central Composite Design CCD of DX8 software, and the LAB simulation program, empirically model, plot and examine the response surfaces for R1, R2 and R3 as a function of the four significant variables.

Examine and discuss the statistical significance tests for the models Optimize the process using the model: identify the optimum conditions of the

process, that would result in a combination of the responses R1, R2 and R3 closest to the target values R1T, R2T and R3T; verify the optimum processing and/or use conditions through running LAB simulation program.

In tutorial 3B, the 4 significant variables that affect the 3 responses were determined to be carbon additive, oxygen impurity, sintering temperature and titanium diboride (TiB2) powder grain diameter. Using these variables and the DX9 program, we are able to plan a Central Composite Design (CCD) to model the process. The CCD will run 30 experiments providing different sets of data for the 4 variables.

The new series of CCD experiments are run again using the LAB simulation program with the previously determined values for the 4 variables. Each simulation will occur at various values for the 4 variables while the 3 minor variables will be kept at constant values at the midpoint of their respective ranges. The LABSIM program will generate averaged values for the 3 responses (hardness, fracture toughness and Weibull modulus) for each individual experiment.

The following is the design summary of the CCD experiments

The CCD table for the 30 experiments is shown below.

Now that we have our data we can obtain empirical models of the process with the CCD module in DX9. We will be producing 3 models, one for each response that we want to maximize.

Response 1: Hardness

The following is the model for maximizing hardness.

The following is the ANOVA table generated in the CCD module.

With this ANOVA table we can assess the quality of the model. As indicated, the F-ratio has a value of 229.93 and the correlation coefficient r2 has a value of 0.9954. The standard requirement of a qualified model is a F-ratio larger than 10 and correlation coefficient larger than approximately 0.9.

Since this model qualifies, we do not need to perform further re-evaluation such as using transformations or different ranges of insignificant variables.

We can now proceed to observe some of the plots of this model.

Response 2: Fracture Toughness

The following is the model for maximizing fracture toughness.

The generated ANOVA table for model 2 is shown below.

With this ANOVA table we can assess the quality of the model. As indicated, the F-ratio has a value of 1173.81 and the correlation coefficient r2 has a value of 0.9991. The values are well above the requirements and compared to model 1, as the quality of the model can be considered to be much higher. Therefore, no further action needs to be taken.

The following are graphical representations of the model.

Response 3: Weibull Modulus

The following is the generated model for our third response: Weibull modulus.

Correspondingly, the ANOVA table of the model is shown below.

As indicated in the table, the F-ratio has a value of 674.46 and the correlation coefficient r2 has a value of 0.9984. We can confirm the quality of the model as these two values exceed the standard requirement.

Finally we represent our model graphically for the third response.

Optimization of CCD

We set the 4 significant variables in the range and the goal is to maximize the 3 responses. This is done in the “Criteria” tab under optimization in DX9.

We order the solutions in terms of desirability, which means that the underlined set of data is providing the best data that maximizes our target responses. The ideal solution is shown below:

The red dots indicate the 4 significant variables whereas the blue dots indicate the our maximized responses. The desirability is 86.7%, which means that we are 86.7% certain that the values for the 4 variables gives us the best end result. Since there are other non-identified errors in statistics, it would be rather difficult to approach a perfect desirability of 100%. Therefore, we can still be satisfied with these results as it is relatively close to a perfect 100% desirability. An alternative representation is shown below as a bar graph

Consequently, we can obtain contour and cube plots of the results from the ideal solution. Below is the contour plot of the 4 significant variables, showing the ranges of different desirability.

We can also express our results as a cube plot, again in terms of the 4 variables.

A summary of our optimization results is shown below, 3 confirmation runs were run to ensure consistent results.

Finally, we use LABSIM again at the predicted optimum level of the 4 variables, to verify the model predictions.

Hardness [GPa] Toughness [MPa•m^(1/2)] Modulus Variables Values S.D.

Average 28.190 9.724 17.302 X1: Carbon Content % 3.49 wt% 0.1 %S.D. 0.177 0.033 0.069 X2: Heating Rate 25.00 °C/min 0.1 %

X3: Milling Time 7.00 hr 0.1 %Values Values Values X4: Oxygen Content in TiB2 2.00 wt% 0.1 %

1 27.803 9.653 17.172 X5: Sintering Temperature 1725.82 °C 0.1 %2 27.968 9.673 17.184 X6: Sintering Time 2.50 hr 0.1 %3 28.015 9.680 17.216 X7: TiB2 Grain Diameter 0.75 um 0.1 %4 28.037 9.685 17.236 Err. of Measurement5 28.074 9.700 17.255

0.2 %

Our model has presented that the average for hardness; toughness and modulus are given as below:

1) Hardness= 28.2818 GPa

From the data obtained in the model, our mode averagel is about 0.32% closed to the lab simulation result for hardness value. According to the graph above, the lab data is within the range of specification and its average is reliable to verify our model.

2) Fracture toughness = 9.6739MPa*m0.5

From the data obtained in the model, our model average is about 0.52% closed to the lab simulation result for fracture toughness value. According to the graph above, the lab data is within the range of specification and its average is reliable to verify our model

3) Weibull Modulus = 17. 1969 MPa

From the data obtained in the model, our model average is about 0.61% closed to the lab simulation result for Weibull Modulus value. According to the graph above, the lab data is within the range of specification and its average is reliable to verify our model

The lab data average is approximately equal to the model data average as the changes are just less than 1% differences. The changes are maybe due to the systematic and random error occurring during measurement. For the experiment, you can optimize the data value by having a well-calibrated high quality measuring tool.

Documents

MTRL 460-2014-TutRep3-Team9