1. Introduction

A detailed Study has been done on theoretical aspects of Design of Experiments (DOE)

which gives the following understanding.

1.1 Designing Experiments

Using the factors and levels determined in the previous step, the experiments now can be

designed and the method carrying them out established. To design the experiment,

implement the following:

-Select the appropriate orthogonal array.

In the present study, the interaction between the welding parameters is considered.

Therefore, degrees of freedom owing to the three level welding process parameters were

evaluated. Tht degrees of freedom for the orthogonal array should be greater than or at

least equal to those for the process parameters. In this study, L 18 orthogonal arrays were

used.

1.2 Running Experiment

All the experiments of laser welding were carried out (during joining process only) in

random order of the developed matrices by the software to avoid any systematic error

during the experiments. After the joining process the responses, mentioned earlier in this

work, were tested and measured in sequential order following the standard procedures

when available for each response. An average of at least three (in most cases) recorded

measurements in calculated and considered for more analysis.

1.3 Analyzing Results

Before analysis, the raw experimental data might have to be combined into an overall

evaluation criterion. This is particularly true when there are multiple criteria of evaluation.

Analysis is performed to determine the following:

The optimum design.

Influence of individual factors.

Performance at the optimum condition.

Relative influence of individual factors.

The steps in this analyzing stage are following in this sequence:

1.3 Developing the mathematical model

Design expert software develops and exhibits the possible modules which can fit input data

and suggest the model that best fits the experiment data.

Estimating of the coefficients of the model independent factors

Regression analysis is carried out by software to estimate the coefficients for all factorsin

each experiment.

1.5 The Signal-to-noise (S/N) ratio analysis

A signal to noise ratio in the ANOVA Table is presents as an Adequate Precision.

Equations 3.15 and 3.16 are applied to the model to compares the range of the predicted

values at the design points to the average prediction error. Ratios greater than 4 indicate

adequate model discrimination.

Adequate Precision max(Y) min(Y)

4V(Y)

…(3.15)

2n

f 1

1 PV(Y) V(Y)

n n

…(316)

P = number of model parameters, 2 = residual MS from ANOVA table, n = number of

experiments.

1.6 ANOVA Outputs

The analyses of variances (ANOVA) were applied to test adequacy of the developed

models. Each term in developed models was examined by the following statistical

significance tools using Eq. 3.15-3.20 [140]:

VF value: Test for comparing model variance with residual (error) variance. When the

variances are close to each other, the ratio will be close to one and it is less likely that any

of the factors have a significant effort on the response. Model VF =Value and associated

probability value (Prob.> VF ) to confirm model significance. VF value is calculated by

term mean square divided by residual mean square.

Prob.> VF : Probability of seeing the observed VF value if the null hypothesis is true (there

is no factor effect). If the Prob.> VF of the model and/or of each term in the model does not

exceed the level of significance (for chosen a = 0.05 in this work) then the model can be

considered adequate within the confidence interval (1-a).

Precision of a parameter estimate is based on the number of independent samples of

information which can be determined by degree of freedom f(d ).

Degree of Freedom f(d ) : the degree of freedom equals to the number of experiments minus

the number of experiments minus the number of additional parameters estimated for that

calculation.

The same tables show also the other adequacy measures 2R , adjusted 2R and adequacy

precision 2R for each response. In this study, all adequacy measures were close to 1, which

indicates adequate models.

The Adequate Precision compares the range of the predicated value at the design points to

the average predicted error. The adequate precision ratio above 4 indicates adequate model

discrimination. In this study, the values of adequate precision are significantly greater than

4.

2 r

M r

SSR 1

SS SS

…(3.17)

2 2n 1Adj. R (1 R )

n p

…(3.18)

2

r M

PRESSPredicted R 1

SS SS

…(3.19)

1

n2

f i ,f 1

PRESS (Y Y )

…(3.20)

1.7 Model reduction

Model reduction consists of eliminating those terms that are not desired or which are

statistically insignificant. In this case it was done automatically by the software used. For

each response regression the starting model can be edited by specifying fewer candidate

terms than the full model would contain. In the three automatic regression variations, those

terms which are forced into the model regardless of their entry/exit a value could be

controlled. There are three basic types of automatic model regression: Step-Wise: A term

is added, eliminated or exchanged at each step. Step-wise regression is a combination of

forward and backward regressions. Backward elimination: A term is eliminated at each

step. The backward method may be the most robust choice since all model terms will be

given a chance of inclusion in the model. Conversely, the forward selection procedure

starts with a minimal core model, thus some terms never get included. Forward selection:

A term is added at each step.

1.8 Development of final model form

The program automatically defaults to the “Suggested” polynomial model which best fits

the criteria discussed in the Fit Summary section. The responses could be predicted at any

midpoints using the adequate model. Also, essential plots, such as Contour, 3D surface,

and perturbation plots of the desirability function at each optimum can be used to explore

the function in the factor space. Also, any individual response

1.9 Running Confirmation Experiments

The final step is to predict and verify the improvement of the response using the optimal

level of the welding process parameters. In addition, to verify the satisfactoriness of the

developed models, at least three confirmation experiments were carried out using new test

conditions at optimal parameters conditions, obtained using the Design Expert software.

2. Proposed Parameters

2.1 Independent parameters along with their levels

Three types of parameters have been considered for the experiment. Machining

parameters, Material property parameter and Geometrical parameter. Machining

Parameters comprises of Cutting Speed (m/min), Feed rate (mm/rev) and Depth of cut

(mm). BHN (Brinell Hardness Number) has been considered as Material Property

parameter and Cutting point angle has been considered as Geometrical parameters.

All parameters along with their level have been shown below. Magnitude of parameter

quantity are proposed and subjected to change as per situation or requirement.

Sl

No. Parameter (Unit) Symbol Level 1 Level 2 Level 3

1 Cutting Speed

(m/min) A 100 150 200

2 Feed rate

(mm/rev) B 0.5 0.7 1.0

3 Depth of Cut

(mm) C 0.2 0.25 0.3

4 Material (BHN) D Aliminium MS Iron

5 Cutting point

angle (Degree) E 85 90 95

2.2 Response variables

Two response variables have been considered for study. These are i) Avarage Surface

Roughness (ASR) measured in m and ii) Material Removal Rate (MRR) measured in

mm3/min

3. Proposed Orthogonal Array

Here in this study least array which will be considered is L27 and largest array which may

be considered is L64.

L27 Orthogonal array for ASR

Exp

No.

Parameters ASR

A B C D E Mean SD Log of

SD S/N

1 100 0.5 0.20 Al 85

2 100 0.5 0.25 MS 90

3 100 0.5 0.30 Fe 95

4 100 0.7 0.20 Al 85

5 100 0.7 0.25 MS 90

6 100 0.7 0.30 Fe 95

7 100 1.0 0.20 Al 85

8 100 1.0 0.25 MS 90

9 100 1.0 0.30 Fe 95

10 150 0.5 0.20 Al 85

11 150 0.5 0.25 MS 90

12 150 0.5 0.30 Fe 95

13 150 0.7 0.20 Al 85

14 150 0.7 0.25 MS 90

15 150 0.7 0.30 Fe 95

16 150 1.0 0.20 Al 85

17 150 1.0 0.25 MS 90

18 150 1.0 0.30 Fe 95

19 200 0.5 0.20 Al 85

20 200 0.5 0.25 MS 90

21 200 0.5 0.30 Fe 95

22 200 0.7 0.20 Al 85

23 200 0.7 0.25 MS 90

24 200 0.7 0.30 Fe 95

25 200 1.0 0.20 Al 85

26 200 1.0 0.25 MS 90

27 200 1.0 0.30 Fe 95

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