CHAPTER 6 MACHINABILITY MODELS WITH THREE INDEPENDENT VARIABLES

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1 CHAPTER 6 MACHINABILITY MODELS WITH THREE INDEPENDENT VARIABLES 6.1 Introduction It has been found from the literature review that not much research has taken place in the area of machining of carbon silicon carbide composite and comprehensive mathematical models representing the machinability parameters are yet to be developed. Almost all researchers have dealt with the effect of each of the independent variables on one or more outputs changing one independent variable while fixing the other independent variables at constant levels. Hence development of mathematical models for various machinability parameters is of great practical significance. Estimation of machinability responses like electrode wear rate (EWR), overcut (OC), and material removal rate (MRR) while machining carbon silicon carbide are of considerable interest to production planners. In many cases the required response function values may act as a constraint on the machine setting for composite machining. It is established that the machining parameters influencing machinability are spindle speed, feed rate, drill size etc. Although in case of carbon silicon carbide composite, no research has been done on machining these composites and studying the influence on response functions MRR, surface finish, hole circularity and overcut. Apart from this, models representing a wider perspective of machining of carbon-silicon carbide are also not available. An attempt has been made here to establish empirical models in terms of the machining parameters for 101

2 material removal rate, overcut and hole circularity for machining carbon silicon carbide composite. Independent variables considered are spindle speed, feed rate and drill size. The range of values of variables is selected based on machine capability. The study of various stated responses by response surface methodology generally follows the useful sequence of most scientific research. This investigation will cover the following steps: 1. Postulation of mathematical models 2. Design of experiments 3. Choice of actual cutting conditions 4. Conducting experiments 5. Estimation of parameters 6. Check on the adequacy of the postulated model 7. Estimation of confidence intervals 8. Identification of optimum region 9. Conducting experiments in the optimum region of variables 10. Developing new models 11. Check adequacy of fitted models. 6.2 Postulation of Machinability Models A functional relationship between machinability parameters and the independent variables under investigation could be represented by R = C 0 + C 1 X 1 + C 2 X 2 +C 3 X 3 (6.1) Where X 1, X 2, X 3 represent spindle speed, feed rate and drill size respectively and R is the response function like material removal rate (MRR), hole circularity and 102

3 overcut (OC). A linear mathematical model that might describe this relationship is η = β 0 x 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 (6.2) where, η = true value of the response functions, x 0 =1 (a dummy variable),x 1, x 2, x 3 are the coded values of the process variables and β 0, β 1, β 2, β 3 are the parameters to be estimated. The equation 6.2 however can be expressed as y-e = b 0 x 0 + b 1 x 1 + b 2 x 2 + b 3 x 3 (6.3) where y is the measured response function, e is the experimental error and b 0, b 1, b 2, b 3 are the estimates of parameters β 0, β 1, β 2, β 3. Equation 6.3 is a polynomial of the first degree. The coefficients of this linear equation can be estimated by the method of least squares. When lack of fit of the first order response function models is indicated, it is possible to extend the equation 6.2 as follows: 2 η = β 0 x 0 β 1 x 1 + β 2 x 2 + β 3 x 3 + β 11 x 1 + β 22 x β 33 x β 12 x 1 x 2 + β 23 x 2 x 3 + β 31 x 3 x 1 (6.4) The equation 6.4 may be given by 2 yˆ = b 0 x 0 + b 1 x 1 + b 2 x 2 + b 3 x 3 + b 11 x 1 + b 22 x b 33 x b 12 x 1 x 2 + b 23 x 2 x 3 + b 31 x 3 x 1 (6.5) Where yˆ is the estimated value of the various responses while the b values are estimates of the β s. 6.3 Design of Experiment To develop first order models for the responses MRR, OC and hole circularity a design consisting of twelve experiments is used. Eight of the twelve experiments represent a 2 3 factorial design and may be represented by the numbered vertices of a cube, 103

4 while the remaining four experiments indicate an added center point to the cube, repeated four times to estimated error. This represents a central composite rotatable design. This design provides three levels for each independent variable coded -1 as low level, 0 as the centre level and +1 as the high level. The design matrix X i for such an arrangement can be written as shown in the Table 6.1. The design matrix for such a design is shown in Table Estimation of Parameters The parameters of the approximating polynomials in equation 6.2 and 6.4 can be estimated by the method of least squares, using the formula, B = (X T X) -1 X T Y (6.6) Where, B = matrix of parameter estimates X= matrix of individual variables or design matrix X T = transpose of matrix X and Y = matrix of measured responses The least square objective function S is given by S= (y i - yˆ) 2 (6.7) Where, y i = the measured or experimental value of the response for the i th observation i = the predicted or calculated value of response for i th observation N = the total number of experimental points 104

5 Table 6.1 Design Matrix Exp.No. X 1 X 2 X Table 6.2 First order Design for K=3 Standard Order X=Design matrix of X-variables X 0 X 1 X 2 X 3 Measured response Y Y 2 Y 3 Y 4 Y 5 105

6 Y 6 Y 7 Y 8 Y 9 Y 10 Y 11 Y 12 It may be noted that as S represents the sum of squares of error e, the best value of the model parameters is obtained when the objective function S is minimized. The predictive first order model is yˆ = b 0 + b 1 x 1 + b 2 x b k x k (6.8) Where b 0, b 1, b k are the parameter estimates based on the least square method First Order Design The parameters in equation 6.8 may be obtained by using equation 6.6. In the case of response surface designs considered in section 6.3, these equations can be directly computed using the following equations, b 0 = 0 y/ N (6.9) bi = x i y/ n e (6.10) Where N= the total number of experimental points n c = the corner points and 106

7 i= 1, 2, k. 6.5 Checking the Adequacy of the Postulated Models The usual method of testing the adequacy of a model is to compare F-ratio of lack of fit to pure error and compare it with the standard value. Making an analysis of variance table, which includes sum of squares, degree of freedom and mean square, can check the adequacy most conveniently. In order to perform the analysis of variance the sum of squares of the y s usually partitioned into contributions from the first order terms, the second order terms, the lack of fit and experimental error(or pure error). Lack of fit measures the deviations of the responses from the fitted surface. Experimental error is obtained from the replicated points at the centre. The sum of squares of lack of fit and sum of squares of pure error constitute the residual sum of squares. The detailed formulae for the variance analysis are given in Table 6.3 for the first order models. In these formulae g is the grand total of observed responses, N the total number of experimental points, n 0 the centre points, n c the corner points, K the number of factors, y ou is the observed responses at centre points with mean y 0, C ii is the inverse of the sum of squares of the i th column of X matrix and (Oy), (Iy), (2y) etc. are the sum of products of each column in the X matrix with the column of y values. 107

8 Table 6.3 Formulae for Analysis of Variance for First Order Model Source Sum of Squares Degree of Freedom First Order Terms b 2 i /C ii K Lack of Fit By Subtraction n c -K Experimental Error (y ou - y 0 ) 2 n o -K Total Y u 2 -g 2 / N N Test for Significance of Individual Parameters Estimated error variance based on residual sum of squares is given by S 2 = s / df (6.11) Where s is the estimated standard error, S is the residual sum of squares and df is the degree of freedom. The variance of estimated parameters is V (b) = (X T X) -1 s 2 (6.12) Or V (b) = C ii S 2 (6.13) Where C ii is the diagonal element of (X T X) -1 matrix. Thus the standard error, SE, of the estimated parameters SE (b i ) = υ(b i) (6.14) The 95% confidence interval for b s are given by b + ( t df.0.05/2. SE(b)) (6.15) Hence, b c = t df.0.0/2. SE (b) A parameter b is said to be significant if b is greater than or equal to b c. 108

9 The following tests are also considered to assess the validity of a predicted model. (1) The least squares objective function i.e., the residual sum of squares S, should be as small as possible and ideally zero. (2) The multiple correlation coefficients, R 2, should be near to 1 and ideally it should be 1 to predict the fit. (3) Further treatment combinations of the factors may be chosen randomly and tested to get the response, which may be compared with the predicted values from the model. 6.7 Experimental Conditions for Tests RSM is utilized for obtaining predictive models for MRR, OC and hole circularity of machined carbon silicon carbide composite. The general procedure adopted is in accordance with that shown in Table Design of Experiment for First Order Models First order model demands at least eight experiments i.e. 2 3 factorial design. Independent variables considered are (1) spindle speed (2) feed rate (3) drill size. Measured responses are material removal rate (mm 3 /min), overcut (mm), and hole circularity (mm). Experimental points considered are corners of the cube and four replicated central points for the estimation of the pure error. The ranges of values of the independent variables are selected based on the capability of the machine used. The 109

10 physical units, different levels used in the experiments and the coding of the independent variables are shown in Table 6.4 Table 6.4 Variables and their levels Variable Levels Unit Code Parameters Low (-1) Centre (0) High (+1) Spindle speed rpm X Feed rate mm/min X Drill size mm X Randomisation In an experiment, the order of the application of different combinations should be a random one. Randomisation avoids possible bias due to the influence of systematic disturbances. Randomising is done in the experiment by using a table of random numbers. Run number thus obtained are given in the second column of Table 6.5 Table 6.5 Experimental Conditions with Run numbers Standard Order Run No. X 1 X 2 X

11 Experimental Conditions According to the capability of the commercial machine available for machining this material the range and the number of levels of the parameters selected are given as in Table 6.4 Experiments were performed randomly according to the factorial design approach using four centre points on the C/SiC plate. The diameter of the hole produced is measured using an Axiovision systems Axiotech microscope with a CCD camera having a least count of 10 µm at 5 X magnification. A number of readings at different angular positions of the hole are taken and diametric overcut (D - d) is calculated by average value where D is hole circularity and d is diameter of the tool. The MRR is calculated by taking the difference between initial weight and final weight of the work piece using Libror AEG 220 balance, Shimadzu make with accuracy of g. The results obtained from conducting the above experiment are tabulated in Table

12 6.7.3 Estimation of Parameters It is in the interest of process planners to develop models and explore the relationships of the variables. Regression methods are used to analyse the data. The main effects first order model to be fitted is, Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + ε (6.16) Where X 1, X 2, X3 represents spindle speed, feed rate and drill size respectively. Hence the fitted linear regression model for hole circularity is: Y^ = X X X 3 (6.17) In a similar manner using regression analysis the fitted regression model for diametric overcut is Y^ = X X X 3 (6.18) The fitted linear regression model for material removal rate is Y^ = X X X 3 (6.19) Checking the Adequacy of Models The ANOVA analysis for the parameters is as shown below in Table 6.6, 6.7 and 6.8. Table 6.6 ANOVA analysis for hole circularity Source of Sum of Degrees of Mean variation squares freedom square A B C AB AC F 0 112

13 BC ABC Error Total Table 6.7 ANOVA analysis for diametric overcut Source of Sum of Degrees of Mean variation squares freedom square A B C AB AC BC ABC Error Total F 0 Table 6.8 ANOVA analysis for material removal rate Source of Sum of Degrees of Mean F 0 variation squares freedom square A B C AB AC BC ABC Error Total

14 The response surfaces generated using MATLAB software for MRR is shown in Figure 6.1, 6.2 and 6.3. Fig.6.1 Response surface as function of feed rate & speed Fig.6.2 Response surface as function of drill size & speed Figure 6.3 Response surface as a function of drill size & feed rate. The response surfaces generated using MATLAB software for hole circularity are shown in Figure 6.4, 6.5 and

15 Figure 6.4 Response surface as function of feed rate & speed Figure 6.5 Response surface as function of drill size and speed Figure 6.6 3D Response surface as function of drill size & feed rate The response surfaces generated for diametric overcut using MATLAB software are shown in Figure 6.7, 6.8 and

16 Figure 6.7 Response surface as function of feed rate & speed. Fig.6.8 Response surface as function of Drill size and speed Fig. 6.9 Response surface as function of drill size and feed rate 116

17 6.8 Conclusion The first order models developed for material removal rate and hole circularity are adequate and that of overcut is inadequate. The order of significance of parameters of the circularity model is drill size, feed and spindle speed and the order of significance for the material removal rate is spindle speed, feed rate and drill size. The response surface generated using MATLAB software, show curvature for the experiments which have been conducted. Hence the first order regression models are not adequately representing the process, hence a non-linear model would be able to represent this process better, for which further experiments would have to be planned. This experiment establishes that C- SiC can be drilled using HSS drills and a relatively clean cut is observed and further investigation is required to analyse the effect of machining on the matrix and reinforcement used. 117