# Special Cases Of The Wide Ranging Factorial Design Biology Essay

Published:

This essay has been submitted by a student. This is not an example of the work written by our professional essay writers.

In previous chapter, the detailed overview the experimental procedure, experimental table, investigational result is has discuss. In this chapter the design of experiment predictions and residuals performance parameter regression analysis for Ra, communication matrix plot for Ra, surface plot for Ra, contour plot for Ramost important effect plot for Ra, cube plot of the communication for Ra, scatter plot for Ra, pareto chart for Ra, comparison of predicted and experimental surface roughness experimental modus operandi, experimental table, experimental result are discuss

## 6.1 DESIGN OF EXPERIMENT

Factorial design has widely used in experiments concerning several factors where it is necessary to study the joint effect of these factor on a response. There are special cases of the wide-ranging factorial design that are very important in the thesis work. The first of these special cases that k factors each at only two levels. These levels may be quantitative, such as two principles of feed, speed and depth of cut; they may be qualitative, such as one machine, the high and low levels of a reason, or perhaps the presence and absence of factor. Box-Behnken designs are rotatable designs that also fit a full quadratic model but use just three levels of each cause. It can be second-hand to optimize the factor.

## 6.1.1Initial Design Information

Table 6.1: Initial Design Information

Factors

3

Block

1

Central Point Per Block

3

Replicates

1

Observations

15

Responses

1

## 6.1.2 Factor Properties

Table 6.2: Factor Properties

## Factor

## Name

## Low Level

## High Level

A

Speed(rpm)

49

186

B

Feed(mm/rev)

7

16

C

Depth of cut(mm)

0.5

1

## 6.2 PREDICTIONS AND RESIDUALS

The prediction and residuals board shows, for each observation, its weight, the value of the qualitative illuminating variable, if there is only one, the observed value of the dependent variable, the model's prediction, the residuals, the confidence intervals together with the en suite prediction and Cook's D if the corresponding options had been activate in the dialog box. Two types of confidence interval are display: a confidence interval approximately the mean (analogous to the case where the prediction would be made for an infinite number of observations with a set of given values for the instructive variables) and an interval around the out-of-the-way prediction (corresponding to the case of an inaccessible prediction for the values given for the explanatory variables). The second intermission is always greater than the first, the random values being larger. If the legalization data have been selected, they are displayed at the end of the table.

Table 6.4: Predictions and residuals

## S.N.

## Actual Value

## (Ra)

## Fitted Value

## (Ra)

## Residual

## Standardized

## Residual

## Studentized

## Residual

## External Studentized

## Residual

## Leverage

## Cook's Distance

1

3.7

3.475

0.225

0.2268

0.4537

0.4144

0.75

0.0617

2

2.4

2.35

0.05

0.0504

0.1008

0.0903

0.75

0.003

3

2.6

2.65

-0.05

-0.0504

-0.1008

-0.0903

0.75

0.003

4

4.4

4.625

-0.225

-0.2268

-0.4537

-0.4144

0.75

0.0617

5

2.6

3.1375

-0.5375

-0.5419

-1.0838

-1.1083

0.75

0.3524

6

2.5

2.8625

-0.3625

-0.3655

-0.7309

-0.6918

0.75

0.1603

7

1.9

1.5375

0.3625

0.3655

0.7309

0.6918

0.75

0.1603

8

3.2

2.6625

0.5375

0.5419

1.0838

1.1083

0.75

0.3524

9

3.8

3.4847

0.3125

0.3151

0.6301

0.5874

0.75

0.1191

10

4.1

3.5125

0.5875

0.5923

1.1846

1.2493

0.75

0.421

11

1.3

1.8875

-0.5875

-0.5923

-1.1846

-1.2493

0.75

0.421

12

3

3.3125

-0.3125

-0.3151

-0.6301

-0.5874

0.75

0.1191

13

4.8

4.0333

0.7667

0.7729

0.9467

0.9346

0.3333

0.0448

14

2.6

4.0333

-1.4333

-1.4451

-1.7698

-2.5901

0.3333

0.1566

15

4.7

4.0333

0.6667

0.6721

0.8232

0.7919

0.3333

0.0339

## 6.3 PERFORMANCE PARAMETER

By untrustworthy input parameter according to Box-Behnken design we get surface roughness (Ra) by using mild steel as work piece and high speed steel as tool. base on calculations for poles apart input parameters, the output parameters are calculated and the result is analyzed according to two level full factorial designs. ANOVA and Response Surface Methodology is perform by the use of DOE++ software intended for Surface Roughness.

## 6.3.1Response Surface Roughness (Ra)

ANOVA & Regression information are represented below in Table 6.5 and Table 6.6

Table 6.5: ANOVA for Ra

## Source of variation

## Degree of freedom

## Sum of squares

## Mean squares

## F Ratio

## P Value

Model

9

10.2902

1.1434

1.1621

0.4579

Linear effects

3

3.0325

1.0108

1.0274

0.4551

Interaction effects

3

3.3825

1.1275

1.146

0.4158

Quadratic effects

3

3.8752

1.2917

1.3129

0.3677

Residual

5

4.9192

0.9838

## -

## -

Lack of fit

3

1.8325

0.6108

0.3958

0.7726

Pure error

2

3.0867

1.5433

## -

## -

Total

14

15.2093

## -

## -

## -

S = 0.9919 Press = 36.265

R-sq = 67.66 R-sq (pred) = 0

R-sq = 9.44

Table 6.6: Regression information for Ra (surface Roughness)

## Term

## Coefficient

## Standard Error

## Low CI

## High CI

## T Value

## P Value

Intercept

4.0333

0.5727

2.8794

5.1873

7.0431

0.0009

A:Speed

0.2125

0.3507

-0.4941

0.9191

0.606

0.571

B:Feed

0.3625

0.3507

-0.3441

1.0691

1.0337

0.3487

C:D.O.C

-0.45

0.3507

-1.1566

0.2566

-1.2832

0.2557

AB

0.775

0.4959

-0.2243

1.7743

1.5627

0.1789

AC

0.35

0.4959

-0.6493

1.3493

0.7057

0.5119

BC

0.35

0.4959

-0.6493

1.3493

0.7057

0.5119

AA

-0.6292

0.5162

-1.6693

0.411

-1.2189

0.2773

BB

-0.1292

0.5162

-1.1693

0.911

-0.2502

0.8124

CC

-0.8542

0.5162

-1.8943

0.186

-1.6547

0.1589

## 6.4 REGRESSION ANALYSIS FOR Ra (SURFACE ROUGHNESS)

ANOVA based chronological sum of squares test was done to select the most inappropriate model to be fitted. The model significance can be tested by comparing the corresponding p value to the threshold p value. The threshold p value depends on the selected connotation level which was set here. The highest order polynomial for which the supplementary terms were significant and the model was not aliased was chosen. Based on the test outcome, the two factor interaction representation was chosen for fitting. The regression statistics for the full two aspect interaction model are shown in Table. The predicted R2 value and the in the swing of things R2 value were found to be in slam agreement. Adjusted R2 is a compute of the amount of distinction about the mean which is explain by the model. When there is a large difference in the values of predicted R2 the familiar R2, it indicate that some non-significant terms have been included in the model and the model would pick up on excluding such terms. To check if the fitting would improve on plummeting some terms, a condensed two factor interaction model was fitted. Backward step-wise fitting was used for model fitting with term-dropping p value superior than. It was found that the dissimilarity between the predicted R2 value and the adjusted R2 value decreased on dropping some of the terms. Hence the condensed two factor interaction model was chosen.

The ANOVA tables for the summary two factor interaction model is shown in Table Values of probability indicate representation terms are significant. The regression model for Ra cannot be twisted because there is no significant factor in the regression term. The final regression equation for surface roughness in terms of the genuine parameter values is:

## 5.0937+0.0011*Speed-113.8177*Feed+8.0775*Depth of cut +1.1024 *AB +0.0378 *AC +

## 73.6842*BC-0.0005*AA-357.8024*BB-13.6667*CC

## 6.5 INTERACTION MATRIX PLOT FOR Ra (SURFACE ROUGHNESS)

Interaction plot show the deviation of surface roughness due to interaction between cutting speed and feed rate (v*f), feed rate and distance downward of cut (f*d) and spindle speed and depth of cut (v*d). Interaction occur when one factor does not produce the matching effect on the response at different levels of an additional factor. Therefore, if the lines of two factors are parallel, there is no interaction. A figure 6.1 highlight the main factor plots for Ra. Surface roughness Ra appears to be diminishing function of spindle speed, feed, and depth of cut.

Figure 6.1 Interaction Matrix Plot for Ra (surface Roughness)

From the above plot it is understandable that when feed rate is increased then there is decrease in irregularity on higher level and increase on lower level.

## 6.6 SURFACE PLOT FOR Ra (SURFACE ROUGHNESS)

In this graph, each value of spindle speed (rpm) and feed (mm/min) generate a surface roughness value. This three-dimensional diagram shows the response surface from the side and it is called a response surface plot. Figures 6.2 plots the predict value of surface smoothness (Ra) in terms of the feed rate and spindle speed by the regression model. It has shown that greater than ever the feed rate and spindle speed increasing of surface roughness.

Figure 6.2 3D Surface Roughness as a Function of cutting speed and feed

From the above discussion it is clear that surface roughness increases with increase the value of spindle speed and feed. The highest value of Ra 4.6541(at speed=170.776 rpm, feed=16mm/min).

## 6.7 CONTOUR PLOT FOR Ra (SURFACE ROUGHNESS)

Figure 6.3 shows the interaction upshot of spindle speed and feed rate on surface roughness. It is comprehensible that the surface roughness depend on spindle speed and feed. The inclination gets reversed for feed rate where surface roughness increases with increases in spindle speed.

Figure 6.3 Contour graph of interaction effect of spindle speed and feed rate

Figure 6.3 shows the probable response surface process parameters of feed and spindle speed. It can be seen from the figure 6.3, the surface roughness tends to increase, when increase in spindle speed and feed. Hence, highest surface roughness is obtained at high feed (16 mm/min) and spindle speed (185.9998 rpm).

## 6.8 MAIN EFFECT PLOT FOR Ra (SURFACE ROUGHNESS)

The plots show the distinction of individual and interaction responses with the three parameter i.e. spindle speed, feed rate, and depth of cut. In the plots, x-axis show the value of both parameter at three levels and y-axis the response values.

Figure 6.4 Main effect plot for Ra (surface roughness)

Figure 6.4 shows the main effect contrive for surface roughness the effect of spindle speed, feed rate, and depth of cut. The results show that augment spindle speed, feed rate, depth of cut, there is permanent decrease in surface roughness.

## 6.9 CUBE PLOT OF THE INTERACTION FOR Ra

Figure 6.5 represent the cube plot which depicts the three-factor interaction in the midst of depth of cut (A), spindle speed (B), and feed rate (C). According to the plot, the exterior roughness is drastically minimized when the depth of cut is set to the low level (1.4708) feed rate and spindle speed are high (2.2458 mm/min and 3.5208 rpm respectively).

Figure 6.5 Cube plot of the interaction ABC

The feed was the most foremost cutting condition on surface roughness, followed by spindle speed and depth of cut. Interaction effect between spindle speed and feed will also give a high effect on surface roughness standards.

## 6.10 SCATTER PLOT FOR Ra (SURFACE ROUGHNESS)

AÂ scatter plot is a type mathematicalÂ usingÂ Cartesian coordinatesÂ to put on show values for twoÂ Â for a set of data. The data is display as a collection of points, every one having the value of one variable determining the arrangement on the horizontal axis and the value of the other variable influential the position on the vertical axis. This brand ofÂ plotÂ is also called aÂ scatter chart,Â scatter illustrationÂ andÂ scatter graph. If a parameter exist that is systematically incremented and/or decremented by the other, it is called theÂ be in charge of parameterÂ orÂ independent variableÂ and is as a matter of course plotted along the horizontal axis

Figure 6.6 Scatter plot for Ra (surface roughness)

Figure 6.6 shows the interaction effect of, spindle speed on surface roughness. It is clear that the outside roughness variable with increase speed.

## 6.11 PARETO CHART FOR Ra (SURFACE ROUGHNESS)

The length of every one bar in the Pareto chart is proportional to the critical value of its associated regression coefficient or homogeneous effect. The numerical estimate of the effects point toward that the effect of feed has positive direction. The positive direction means that the surface finish deteriorate with increasing the cutting feed. Figure 6.7 shows the end product of spindle speed. The Positive direction revenue that greater than ever the spindle speed improves the exterior finish. The depth of cut also has negative value, which indicates that increasing the depth of cut decrease the surface terminate.

Figure 6.7 Pareto chart for Ra (surface roughness)

## 6.12 COMPARISON OF PREDICTED AND EXPERIMENTAL SURFACE ROUGHNESS

Figure 6.8 shows facade bumpiness values obtained by experimentation and values predicted. It is obvious that the predicted values by are very close to the investigational readings.

Figure 6.8 Plot of predicted and experimental surface roughness

It is found that Response Surface Methodology predicts standards which are very close to experimental values. Response Surface Methodology consumes lesser time with high degree of accuracy. Hence, it can be accomplished that Response Surface Methodology is very effective for predicting surface finish.

## Chapter 7

## CONCLUSION

In this chapter, the conclusion of statically psychotherapy of surface roughness for milling machine is approved out using response surface methodology was discussed. The experiments were conducted on milling machine using high speed tool and mild steel work piece. Surface roughness is premeditated for various combinations of speed, feed, and depth of cut applied on experimental work to improve the surface irregularity. Surface roughness plays an important role in many areas and is a factor of great importance in the assessment of machining accuracy. Surface finish has been one of the most significant considerations in determining the machinability of materials. Surface roughness and dimensional accuracy have been imperative factors to predict machining performances of any machining operation. The conclusions of the present work are:

Higher depth of cut gives poor surface finish with respect to lower depth of cut because chip thickness increases. As a result, cutting temperature also rises.

Surface finish obtained at higher cutting speed is better than lower cutting speed.

Cutting speed, feed rate and depth of cut have a major impact on surface roughness. Smoother surfaces are fashioned when machined with a higher cutting speed, smaller feed rate and depth of cut. The result obtained confirms the established facts.

It can be observed from the figure6.3 that lower speed as feed rate increase the surface roughness decreases. On the other supply at higher speed as feed rate increases the surface roughness also increases due to the increases in chip thickness so that cutting force and vibration also increases.

Further, prediction of the response unpredictables is made using regression analysis. It is found that Response Surface Methodology predicts values which are very put up the shutters to experimental values. Response Surface Methodology consumes lesser time with high degree of exactness. Hence, it can be concluded that Response Surface Methodology is very effective for predicting surface finish.

The scope of future work in the present work is :

The discrepancy of Speed, Feed and Depth of cut on work piece can also be considered to study the effect on surface roughness of work piece.

In this present work, the mild steel used as work material, the similar work can be performed on other materials.

The Response Surface Methodology can also be used for prediction of surface finish obtains in other machine tools such as Lathe, Grinding machine etc.

The present work can be extended by Taguchi method